Evaluation of a hybrid FEM/DEM approach for determination of rock

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Evaluation of a hybrid FEM/DEM approach for

determination of rock mass strength using a

combination of discontinuity mapping and

fracture mechanics modelling, with particular

emphasis on modelling of jointed pillars

Davide Elmo BEng (Hons) FGS

Submitted by Davide Elmo to the University of Exeter as a thesis for the degree of Doctor of

Philosophy in Geomechanics in the School of Geography, Archaeology and Earth Resources,

March 2006.

This thesis is available for Library use on the understanding that it is copyright material and that

no quotation from the thesis may be published without prior acknowledgement.

I certify that all material in this thesis, which is not my own work has been identified and that

no material is included for which, a degree has previously been conferred upon me.

...................................... Davide Elmo

Abstract

This thesis presents a new numerical modelling approach for naturally fractured rock masses,

with particular emphasis on the modelling of jointed mine pillars. Numerical methods and

computing techniques have now become integrated components in studies for rock mechanics

and rock engineering and the use of numerical modelling provides an opportunity to increase

fundamental understanding of the factors governing rock mechanics problems. A major drive

of research and development in rock mechanics is currently the representation of rock fractures

in numerical models, both as individual entities and as a collective system. In this context, the

advancing of computer power allows nowadays for realistic simulations of naturally fractured

rock masses.

This thesis discusses the evaluation of a new recent hybrid continuum/discontinuum approach,

linking different numerical techniques such as FEM and DEM, allowing for large scale analysis

of locally large displacements along fracture planes and fracturing of the initially continuum

meshed problem. The hybrid approach is coupled with a discrete fracture network model that

maximises the quality of representation of existing jointing geometry and fully accounts for the

style of naturally fractured rock masses.

Novel aspects and contributions of the current research were:

i. Reviewing the process of developing a stochastic DFN model using the code FracMan

(Golder, 2005) from initial field data; this included the introduction of an updated

workflow and the use of a simplified method to derive synthetic fracture radius

distributions from mapped fracture traces.

ii. Reviewing the principal numerical parameters for the hybrid FEM/DEM code ELFEN

(Rockfield, 2005), in order to verify how the proposed approach depends on modelling

parameters. An explicit combination of numerical parameters was proposed

specifically for the jointed pillar models considered in the current research.

iii. The synthesised FracMan model represented the source of 2D trace sections which were

subsequently imported in the hybrid FEM/DEM code ELFEN. An important novelty of

the proposed research is that the numerical method presented in this thesis does not in

general lead to the creation of complete blocks created by continuous joint sets within

strict geometry and topology, an unrealistic weakness of some current DEM models.

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Following a detailed discussion on modelling constraints and requirements for a careful

calibration process of the proposed ELFEN pillar model, it is believed that the current research

provided novel contributions to improved understanding of modelled brittle behaviour of rock,

in particular the modelling fractured pillars, in relation to the following:

iv. Capturing the continuum to discontinuum transition typical of brittle failure as a result

of local degradation of the intact rock material deriving from the insertion of new

fractures.

v. Modelling pre-fractured models of 2D pillars, with fracture geometries derived from

discrete fracture network systems generated in FracMan and based on actual field

mapping data, to capture the key mechanical role of pre-existing fractures and their

effects on the modelled pillar strength and deformation.

vi. Demonstrating that, for the modelled 2D jointed pillars, the influence of the natural

fractures diminishes with increasing pillar width. The simulated results were also

encouraging when compared with previous empirical mass strength and width-to-height

ratio models.

vii. Demonstrating that, for a given width-to-height ratio, the modelled pillar loading

capacity was ultimately related to the fracture intensity of the existing fracture network.

A qualitative formulation was proposed combining intact rock behaviour, joint surface

conditions (i.e. joint properties), fracture intensity and shape effects in one single

expression characterising rock mass strength.

viii. By extending the analysis to the consideration of a larger number of different pre-

fractured (slender) pillar models, it was concluded that current pillar strength equations

developed from empirical studies could not predict, for slender pillars, significantly

different strength estimates deriving from the variability associated with fracture

intensity and jointing conditions.

ix. Suggesting that the mapped fracture intensity parameter could be potentially used

as a readily measurable indicator of the structural character of the rock mass.

Additionally, the results were interpreted as partly confirming the coupled GSI-RocLab

approach as a reliable measure of rock mass strength.

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x. Discussing how the ELFEN code could also be used in analyses concerning more

typical shear mechanisms, with asperity breakages and shearing in the simulated models

due to growth and coalescence of tensile fractures.

Whereas at this stage of the research the loading of the pillar models was simulated as if they

were subjected to uniaxial laboratory loading conditions, it is anticipated that the analysis could

be extended to consider pillars created by staged excavation within pre-stressed 2D and 3D

solids containing the same fracture systems.

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List of contents

Abstract................................................................................................................................... ii

List of accompanying material.............................................................................................. xiv

Author’s declaration .............................................................................................................. xv

List of principal symbols and abbreviations........................................................................ xvii

Acknowledgements ................................................................................................................ xix

Chapter 1 - Introduction

1.1 Framework ......................................................................................................................... 1

1.2 Research methodology....................................................................................................... 2

1.3 Thesis structure .................................................................................................................. 3

Chapter 2 - A review of principal numerical modelling techniques

2.1 Introduction........................................................................................................................ 6

2.2 Rock mechanics modelling - The general context ............................................................. 6

2.3 Numerical methods for rock mechanics problems............................................................. 8

2.4 Modelling discontinuities in numerical analysis................................................................ 11

2.5 Numerical modelling of fracture processes ....................................................................... 14

2.6 Numerical modelling in 2D and 3D space - A comparative overview .............................. 16

2.7 Constitutive laws in numerical modelling ......................................................................... 17

2.7.1 Modelling brittle failure of rocks................................................................................. 21

2.8 Summary and research framework .................................................................................... 23

Chapter 3 - The use of a DFN model combined with a hybrid FEM/DEM method as a new

numerical modelling approach for naturally fractured rock masses

3.1 Introduction........................................................................................................................ 25

3.2 An introduction to specific FracMan terminology............................................................. 26

3.3 The development of a work-flow for the generation of a DFN FracMan model ............... 26

3.3.1 Choice of a specific DFN generation model................................................................ 29

3.3.2 Fracture orientation...................................................................................................... 29

3.3.3 Fracture termination..................................................................................................... 29

3.3.4 Fracture shape and size ................................................................................................ 30

3.3.5 Fracture intensity ......................................................................................................... 35

3.4 A case example - The development of a DFN model for Middleton mine........................ 39

3.4.1 Mapping considerations............................................................................................... 39

3.4.2 Validation of the Enhanced Baecher DFN model for the Middleton mine case.......... 40

3.4.3 Discontinuity mapping and orientation distributions for the Middleton mine case..... 41

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3.4.4 Fracture radius distribution for the Middleton mine case ............................................ 43

3.4.5 Fracture intensity parameters for the Middleton mine case......................................... 45

3.4.6 FracMan DFN model for Middleton mine................................................................... 47

3.4.7 Comparison between mapped and simulated traces on pillars .................................... 49

3.4.8 Transfer of fracture geometry data from FracMan to ELFEN..................................... 53

3.5 Summary ............................................................................................................................ 54

Chapter 4 - The use of a hybrid FEM/DEM method to model continuum to discontinuum

transition

4.1 Introduction........................................................................................................................ 56

4.2 The use of a hybrid approach to model continuous/discontinuous transformations .......... 57

4.3 A rock fracture mechanics approach to model rock brittle failure..................................... 58

4.4 ELFEN Material constitutive models ................................................................................ 65

4.5 ELFEN modelling of rock brittle failure............................................................................ 68

4.5.1 Modelling tensile fracturing in ELFEN....................................................................... 68

4.5.2 Modelling the continuum/discontinuum transition of rock brittle failure in ELFEN .. 74

4.6 ELFEN - Principal numerical parameters for modelling applications............................... 75

4.6.1 ELFEN loading data structure ..................................................................................... 76

4.6.2 ELFEN discrete element data structure ....................................................................... 76

4.6.2.1 The Mohr-Coulomb contact friction criterion ....................................................... 78

4.6.2.2 Contact damping for contacting surfaces .............................................................. 79

4.6.2.3 General limitations of the node-to-facet contact and penalty method................... 80

4.6.2.4 The ELFEN data structure for contact properties.................................................. 81

4.6.3 The use of displacement damping in ELFEN.............................................................. 82

4.6.4 The ELFEN critical time step concept......................................................................... 82

4.7 Summary and discussion.................................................................................................... 83

Chapter 5 - Initial ELFEN modelling of fractured pillars and dependence of modelling

results to numerical parameters

5.1 Introduction........................................................................................................................ 85

5.2 Development of a strategy for a sensitivity analysis in ELFEN ........................................ 86

5.3 Combined effects of displacement damping and loading rates on ELFEN analysis of pillar

behaviour ................................................................................................................................. 88

5.3.1 Model Set-up ............................................................................................................... 89

5.3.2 Analysis of results and discussion ............................................................................... 93

5.4 Effects of the ELFEN contact penalty method on the analysis of jointed pillar behaviour

................................................................................................................................................. 101

5.4.1 Possible correlation between joint stiffness parameters and penalty coefficients ....... 102

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5.4.2 Analysis of the sensitivity of the model to varying the applied normal penalty.......... 105

5.4.3 Analysis of results........................................................................................................ 108

5.4.4 Conclusions on the effects of varying the applied normal penalty for the jointed pillar

models................................................................................................................................... 117

5.5 Sensitivity analysis for material parameters used in the modelling of jointed pillars........ 120

5.5.1 Model set-up ................................................................................................................ 120


5.6 Summary ............................................................................................................................ 124

Chapter 6 - Numerical modelling of the progressive shear behaviour of rock joints with

tooth-shaped asperities using a combined FEM/DEM method

6.1 Introduction........................................................................................................................ 127

6.2 ELFEN constitutive criterion to model shear strength of discontinuities .......................... 127

6.3 Effects of discontinuity geometry on shear strength and deformation............................... 129

6.4 ELFEN modelling of the progressive shear behaviour of rock joints with tooth-shaped

asperities .................................................................................................................................. 131

6.4.1 Model set-up ................................................................................................................ 131


6.5 Summary and discussion.................................................................................................... 140

Chapter 7 - A new modelling approach for determination of rock mass strength of jointed

pillars and implications for rock mass strength characterisation

7.1 Introduction........................................................................................................................ 143

7.2 Engineering aspects of the design of hard-rock pillars ...................................................... 144

7.2.1 Pillar stress analysis..................................................................................................... 144

7.2.2 Pillar strength............................................................................................................... 145

7.2.3 Numerical analysis of the failure of rock pillars.......................................................... 147

7.3 2D modelling of pillars with one single intersecting fracture............................................ 150

7.3.1 Model set up ................................................................................................................ 150


7.4 A 2D geomechanical model of naturally jointed pillars in ELFEN................................... 157

7.4.1 Model set-up ................................................................................................................ 157

7.4.2 Modelling results ......................................................................................................... 162

7.4.3 Comparison with empirical pillar formulae................................................................. 168

7.4.4 Analysis of the relationship between rock mass strength and fracture intensity for the

modelled pillars .................................................................................................................... 172

7.4.5 Implications of the proposed hybrid approach in terms of characterisation of rock mass

strength ................................................................................................................................. 179

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7.5 Summary and conclusions ................................................................................................. 181

Chapter 8 - Evaluation of the applicability of the hybrid FEM/DEM code ELFEN for 3D

analysis of jointed pillars

8.1 Introduction........................................................................................................................ 185

8.2 Hardware considerations for 3D analysis .......................................................................... 185

8.3 ELFEN fracturing algorithm for 3D analysis..................................................................... 186

8.4 3D modelling of UCS rock specimen with and without the use of mobilised material

parameters with plastic strain................................................................................................... 189

8.5 3D analysis of jointed pillars ............................................................................................. 194

8.6 Summary and conclusions ................................................................................................. 197

Chapter 9 - Conclusions and recommendations for further work

9.1 Introduction........................................................................................................................ 199

9.2 The use of a hybrid FEM/DEM approach to model the behaviour of a fractured rock mass,

with a particular emphasis on jointed rock pillars ................................................................... 199

9.3 Final conclusions on FracMan modelling.......................................................................... 202

9.4 Final conclusions on the use of a hybrid FEM/DEM code with fracture insertion capability

to model rock mass behaviour ................................................................................................. 203

9.4.1 ELFEN – The use of an appropriate displacement damping coefficient and suitable

loading rate ........................................................................................................................... 205

9.4.2 ELFEN - Use of specific contact properties for the discretised elements ................... 207

9.4.3 ELFEN - Selection of the minimum element mesh size.............................................. 208

9.4.4 ELFEN - Influence of surface roughness on the mechanical behaviour of a rock joint

.............................................................................................................................................. 209

9.4.5 ELFEN - Hardware specifications and computational limitations .............................. 209

9.5 Modelling of naturally fractured pillars using the hybrid code ELFEN ............................ 210

9.6 Recommendations for further work ................................................................................... 212

Appendix I

A - Discrete Fracture Network models in FracMan................................................................. 215

B - Field mapping at Middleton mine (Derbyshire, UK)......................................................... 221

C - The ELFEN explicit solution scheme ................................................................................ 231

D - A brief review of the principal rock mass failure criteria .................................................. 245

List of references .................................................................................................................... 251

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List of Figures

Figure 2.1: Qualitative classification of intact and fractured rock masses............................... 11

Figure 2.2: Suitability of different numerical methods for an excavation in a rock mass in

relation to the existing fractures system geometry .................................................. 12

Figure 2.3: Representation of a fractured rock mass in (a) FDM, (b) FEM, (c) BEM and (d)

DEM models............................................................................................................ 13

Figure 2.4: Representation of typical constitutive laws; (a) Linear elastic, (b) non-linear elastic,

(c) elasto-plastic and (d) rigid-plastic. ..................................................................... 18

Figure 2.5: Different plastic behaviours: elasto-perfectly-plastic (a), positive-hardening (b),

negative-hardening/normal-softening (c), critical and subcritical-softnening (1d and

2d respectively)........................................................................................................ 20

Figure 3.1: Graphical illustration of FracMan fracture quantification parameters .................. 26

Figure 3.2: Proposed work-flow for a fully stochastic analysis in FracMan ........................... 28

Figure 3.3: Type of termination styles recognised in FracMan ............................................... 30

Figure 3.4: Circle of equivalent area for a polygonal fracture ................................................. 31

Figure 3.5: Schematic representation of truncation effects for fracture traces in a circular

window .................................................................................................................... 32

Figure 3.6: Example of the FNGraph plot of the first and third moment of a negative

exponential trace length distribution with mean 1................................................... 35

Figure 3.7: Example of the general anisotropy of the parameter ... .................................... 36 21P

Figure 3.8: Schematic representation of the estimation process for a FracMan analysis . 38 32P

Figure 3.9: Typical view of Middleton Mine, Derbyshire (UK).............................................. 39

Figure 3.10: (a) Data for Scanline F1 on Panel 1-Level 1 at Middleton mine and (b) test for

negative exponential fracture spacing ..................................................................... 41

Figure 3.11: Stereoplot (lower hemisphere) of the fractures mapped at Middleton during the

2003 and 2004 site visits ......................................................................................... 42

Figure 3.12: Histograms of trace lengths measured for each fracture set in two parallel windows

that give the maximum number of data points ........................................................ 44

Figure 3.13: Box Region and Pillar region settings for the Middleton DFN model in FracMan

................................................................................................................................. 47

Figure 3.14: FracMan 3D model of a Middleton mine pillar................................................... 49

Figure 3.15: Comparison between mapped and simulated traces in FracMan for Set 1a ........ 50

Figure 3.16: Comparison between mapped and simulated traces for Set 1b. .......................... 51

Figure 3.17: Comparison between mapped and simulated traces for Sets 2a and 2b .............. 51

Figure 3.18: Comparison between mapped and simulated traces for Sets 3a and 3b .............. 52

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Figure 4.1: Generalised quasi-brittle response obtained from conventional triaxial test. ........ 59

Figure 4.2: Equivalence between laboratory experiments and in-situ mining conditions ....... 60

Figure 4.3: Evolution of micro-cracking and formation of macroscopic failure planes for a

hypothetic cylindrical rock specimen under uniaxial compression. ........................ 61

Figure 4.4: Fundamentals modes of fracture ........................................................................... 63

Figure 4.5: The compressive fracture model, the isotropic Mohr-Coulomb yield surface with

softening anisotropic tensile planes. ........................................................................ 66

Figure 4.6: a) compressive loading with confining stress, b) relationship between axial and

volumetric strain and c) compressive failure with associated lateral extensional

inelastic strain causing fracture and dilation............................................................ 66

Figure 4.7: Wing-crack initiation process for the 2D case....................................................... 69

Figure 4.8: Geometric representation of the UCS model used in the analysis, with indication of

principal dimensions and loading directions ........................................................... 70

Figure 4.9: Intact UCS model; fracture evolution at different stages of the simulation. ......... 71

Figure 4.10: Intact UCS model; fracture evolution at different stages of the simulation, showing

details of fracture initiation and wing cracks development at the tips of previously

generated fractures................................................................................................... 72

Figure 4.11: Examples of initiation of wing cracks for models with a pre-existing fracture... 73

Figure 4.12: UCS model. (a) Axial strain-stress response; (b) Simulated volumetric strain-axial

stress response; (c) axial strain-simulated volumetric strain response. ................... 75

Figure 4.13: The penalty contacting couple in ELFEN as an equivalent spring system.......... 78

Figure 5.1: Flowchart showing the methodology implemented to develop the sensitivity analysis

for various ELFEN numerical parameters. .............................................................. 87

Figure 5.2: ELFEN model for displacement damping analysis ............................................... 90

Figure 5.3: Different loading functions used in the analysis ................................................... 91

Figure 5.4: Typical stress strain behaviour for a slender rock pillar........................................ 93

Figure 5.5: Simulated axial strain-axial stress curves for the ELFEN models corresponding to 1,

4 and 8 seconds loading time respectively .............................................................. 94

Figures 5.6: Observed behaviour in terms of estimated maximum axial stress for different

loading rates (a) 0.0, (b) 0.01, (c) 0.03 and (d) 0.1 displacement damping............. 96

Figure 5.7: Effects of displacement damping and ELFEN loading rates for pillar models

corresponding to loading times of 1, 4 and 8 seconds respectively......................... 96

Figures 5.8: Fracturing development at peak stress or near peak stress for the models with 0 and

0.01, 0.03 displacement damping and for loading functions of 1, 4 and 8 seconds. 97

Figure 5.9: Kinetic energy variation with measured axial strain of the pillar. Models with (a)

0.0, (b) 0.01, (c) 0.03 and (d) 0.1displacement damping respectively..................... 98

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Figure 5.10: Effects of displacement damping and different loading conditions on the fracture

pattern and geometry for the initial jointed pillar model. ........................................ 99

Figure 5.11: (a) Diagrammatic representation of a block containing a single discontinuity and

(b) the node-to-facet penalty couple in ELFEN as an equivalent spring system..... 101

Figure 5.12: Geometry and mesh definition for the model testing the equivalence between

normal penalty and normal joint stiffness. .............................................................. 102

Figure 5.13: Initial elastic stress-strain response for the models with 1GPa/m and 5GPa/m

penalty respectively. .............................................................................................. 104

Figure 5.14: ELFEN penalty method model; loading functions and selected displacement

damping factors for each stage of the simulation. ................................................... 106

Figure 5.15: H model axial stress-strain curves. (a) tn 10PP = and (b) ........................ 108 tn PP =

Figure 5.16: L model axial stress-strain curves. (a) and (b) . ..................... 108 tn PP 5= tn 2PP =

Figure 5.17: H model (contact damping 0.5), fracturing evolution at different stages of the

simulation. Cases with ............................................................................ 109 tn 10PP =

Figure 5.18: H model (contact damping 0.5), fracturing evolution at different stages of the

simulation. Cases with . .............................................................................. 110 tn PP =

Figure 5.19: L model (contact damping 0.3), fracturing evolution at different stages of the

simulation. Cases with .............................................................................. 111 tn 2PP =

Figure 5.20: L model (contact damping 0.3), fracturing evolution at different stages of the

simulation. Cases with .............................................................................. 112 tn PP 5=

Figure 5.21: Variation of pillar strength and deformation modulus with normal penalty for the H

and L models. .......................................................................................................... 114

Figure 5.22: Stress-strain curve for the L2 model with intact and pre-fractured geometries... 118

Figure 5.23: Example of the locking-up effects observed in the simulations, following fracturing

of the intact rock material and blocks ejection ........................................................ 119

Figure 5.24: (a, b, c) Stress-strain curves for the QF model and (d) variation of the estimated

pillar strength with fracture cohesion for a given value of friction fc fφ ............. 121

Figure 5.25: Sensitivity analysis for material parameters. Fracture evolution at peak stress for

the QF model ........................................................................................................... 123

Figure 6.1: Schematic representation of the Mohr-Coulomb criterion for shear strength of

discontinuities.......................................................................................................... 128

Figure 6.2: Basic shear stress-displacement curves of joint asperity....................................... 130

Figure 6.3: Example of simple joint specimen used in laboratory testing to study shear

mechanisms and strength degradation ..................................................................... 131

Figure 6.4: Shear model; geometrical assembly and dimensions for the (a) 15 and (b) 30 degrees

asperity models respectively.................................................................................... 133

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Figure 6.5: Shear model; (a) loading directions and (b) loading functions for the applied vertical

and horizontal stresses respectively......................................................................... 133

Figure 6.6: Mesh set-up for the (a) 15 and (b) 30 degrees asperity model respectively .......... 134

Figure 6.7: Shear stress-displacement curves for joint models with 15 degrees asperities and

varying normal stress............................................................................................... 135

Figure 6.8: Shear stress-displacement curves for joint models with 30 degrees asperities and

varying normal stress............................................................................................... 135

Figures 6.9: Comparison between simulated and experimental maximum shear stresses at failure

for the (a) 15 degrees and (b) 30 degrees model respectively ................................. 136

Figure 6.10: Shear stress-displacement curves for joint with (a) 15 and (b) 30 degrees asperities

respectively ............................................................................................................. 136

Figure 6.11: Typical failure modes observed from the simulated 15 degrees joint model at

different values of normal stress.............................................................................. 138

Figure 6.12: Typical failure modes observed from the simulated 30 degrees joint model at

different values of normal stress.............................................................................. 139

Figure 6.13: Comparison between simulated and published shear stress-shear strain curves . 140

Figure 7.1: Effect of confining stress on compressive strength of intact and fractured rocks . 146

Figure 7.2: Pillar width-to-height ratio models for different empirical pillar strength formulae

................................................................................................................................. 146

Figure 7.3: Effect of rock quality on pillar strength................................................................. 147

Figure 7.4: Failure modes of naturally jointed pillars.............................................................. 148

Figure 7.5: Model definition and loading directions for the pillar with a single intersecting

fracture..................................................................................................................... 150

Figure 7.6: Loading function for the 2D modelling of pillars with a single discontinuity ...... 152

Figure 7.7: Modelled variation of maximum estimated pillar strength with dip angle α of the

joint (a) and theoretical curve (b). ........................................................................... 154

Figure 7.8: Progressive fracture evolution for Case A ( ) ......................................... 155 o30=cφ

Figure 7.9: Progressive fracture evolution for Case B ( )............................................ 156 o5=cφ

Figure 7.10: Pillar region in FracMan with indication of the sampling planes used to define the

2D fracture traces models for ELFEN..................................................................... 158

Figure 7.11: Typical pillar dimensions and fracture geometries for the simulated 2.8m x 7m

pillar......................................................................................................................... 159

Figure 7.12: Typical pillar dimensions and fracture geometries for the simulated 7m x 7m pillar

................................................................................................................................. 160

Figure 7.13: Typical pillar dimensions and fracture geometries for the simulated 14m x 7m

pillar......................................................................................................................... 161

Figure 7.14: Stress-strain curves for the jointed pillar models ................................................ 163

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Figure 7.15: Fracturing evolution at peak stress for the 2.8m x 7m pillar models .................. 164

Figure 7.16: Fracturing evolution at peak stress for the 7m x 7m pillar models ..................... 164

Figure 7.17: Fracturing evolution at peak stress for the 14m x 7m pillar models ................... 165

Figure 7.18: Axial stress contour plots at peak stress level for the 2.8m x 7m pillar models.. 166

Figure 7.19: Axial stress contour plots at peak stress level for the 7m x 7m pillar models..... 167

Figure 7.20: Axial stress contour plots at peak stress level for the 14m x 7m pillar models... 168

Figure 7.21: Potential implications of the numerical results in terms of rock mass strength

characterisation as a function of fracture intensity and failure mechanisms ........... 170

Figure 7.22: Comparison between the results of several simulations described in the text and

empirical results for a variety of pillar types. .......................................................... 171

Figure 7.23: Comparison between simulated results and width-to-height ratio models for an

equivalent rock type................................................................................................. 172

Figure 7.24: Correlation between estimated pillar strength and fracture intensity ............ 172 21P

Figure 7.25: Diagrammatic explanation for the proposed general expression for rock mass

strength characterisation. ......................................................................................... 174

Figure 7.26: Different geometrical definitions for the models with ratio of 0.4. ...... 175 2120 P:P

Figure 7.27: Different geometrical definition for the models with varying ratio. ...... 176 2120 P:P

Figure 7.28: (a) Variation of the simulated pillar strength with fracture intensity for the different

2.8mx 7m pillar models; (b) and (c) Correlation between fracture intensity and

simulated pillar strength, based on the proposed grouping into critical and non critical

jointing conditions. .................................................................................................. 177

Figure 7.29: Results for the pillar models intersect by a single through-going fracture (case with

fφ = 30 degrees and cφ = 5 degrees) and multi-fractured pillar models. ............... 178

Figure 7.30: Combined GSI-RocLab approach for the determination of rock mass strength, with

= 12 and GSI in the range of 70-80 (a) and 40-50 (b) respectively. Comparison

between the ELFEN modelled response and the RocLab-GSI approach for the models

with fracture intensity of 1.8 (c) and 2.6 (d) respectively. ................................ 180

im

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Figure 8.1: Wing-crack initiation process for the 3D case....................................................... 186

Figure 8.2: ELFEN results for the model attempting to generate wing crack extension at the tips

of an existing fracture in 3D space .......................................................................... 188

Figure 8.3: Fracturing evolution for an intact 3D model equivalent to the 2D UCS model

described in Section 4.5.1........................................................................................ 189

Figure 8.4: Geometrical definition for the ELFEN UCS model in a 3D space........................ 191

Figure 8.5: (a) Simulated axial stress strain-curves for the 3D UCS models and (b) comparison

with the results obtained for the equivalent 2D models........................................... 192

Figure 8.6: Fracture propagation steps for the 3D UCS models described in Figure 8.5(a). ... 193

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Figure 8.7: Typical 3D pillar model with a single intersecting fracture. ................................. 194

Figure 8.8: Stress-strain curves for some of the 3D models. The stages A, B, C, D and E refer to

the fracture propagation steps showed in Figure 8.9. .............................................. 195

Figure 8.9: Fracture propagation steps for the 3D models described in Figure 8.8. ................ 196

List of Tables

Table 2.1: Summary of numerical methods for rock mechanics problems.............................. 9

Table 3.1: Expressions for determining Dµ and Dσ from fµ and fσ respectively. .......... 33

Table 3.2: Expressions for determining the ratio ( ) ( )24 / DEDE . ............................................. 34

Table 3.3: Mean dip and dip-direction and coefficient K for the fracture sets identified for the

pillars mapped at Middleton mine. .......................................................................... 42

Table 3.4: Fracture radius distributions for Middleton fracture sets, based on the analytical

solution proposed by Zhang and Einstein (2000) and the simplified approximation

method discussed in Section 3.3.4. .......................................................................... 45

Table 3.5: Results for the estimated (volumetric intensity) corresponding to each fracture

set mapped at Middleton mine (Level 1) ................................................................. 46

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Table 3.6: Middleton mine DFN model; original and corrected orientations for the pillar faces in

accordance with the condition of parallelism between Box and Pillar Region ....... 47

Table 3.7: FracMan input parameters for the Middleton DFN model. .................................... 48

Table 3.8: Areal intensity ( ) values for the simulated and mapped windows for fracture Sets

1b, 2a and 2b combined, 3a and 3b combined......................................................... 52

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Table 4.1: Material parameters that have to be specified in ELFEN when using the Mohr-

Coulomb with Rankine tensile cut-off material model............................................ 67

Table 4.2: Rock material properties for the model illustrated in Figure 4.8 ............................ 70

Table 4.3: Platen material properties and discrete contact parameters for the model illustrated in

Figure 4.8................................................................................................................. 71

Table 4.4: Material parameters which are required as part of the contact-surface-properties data

structure to define the contact surface properties of the discrete fracture planes .... 82

Table 5.1: ELFEN displacement damping model; material properties for the intact rock material

and rock fractures .................................................................................................... 92

Table 5.2: ELFEN displacement damping model; platen material properties and discrete contact

parameters for the models illustrated in Figure 5.2 ................................................. 92

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Table 5.3: Normal penalty test model; material properties for the intact rock material and rock

fractures ................................................................................................................... 103

Table 5.4: ELFEN penalty method model; material properties for the intact rock material and

rock fractures. Rock fracture normal and shear stiffnesses were varied throughout the

analysis using the combinations listed in Table 5.5................................................. 105

Table 5.5: ELFEN penalty method model; combinations of and values for rock fracture

used in the analysis for models H and L respectively.............................................. 107

nP tP

Table 5.6: List of some empirical equations suggested for estimating the modulus of

deformation of a fractured rock mass; also indicated are the corresponding estimates

for a jointed pillar model equivalent to that used in the current ELFEN analysis... 116

Table 5.7: Summary of results of the ELFEN analysis investigating the effects of varying

normal penalty, : ratio and contact damping.................................................. 117 nP tP

Table 5.8: Different combinations of and fc fφ used in the analysis..................................... 121

Table 5.9: Proposed ELFEN loading/displacement damping two-stages set-up ..................... 125

Table 5.10: Proposed ELFEN normal penalty , contact damping and : ratio

combination ............................................................................................................. 126

nP nP tP

Table 6.1: ELFEN shear model; material properties used in the analysis are based on the

material parameters for the rock specimen tested by Huang et al. (2002)............... 132

Table 6.2: Shear model; properties of the rock joint contact and fracture elements................ 134

Table 7.1: Jointed pillar model; material and contacts properties used in the analysis ........... 152

Table 7.2: Different fracture friction angles and rock/platen friction values used in the analysis

................................................................................................................................. 153

Table 7.3: Comparison of empirical with modelled average pillar strengths .......................... 170

Table 8.1: ELFEN 3D UCS model; material parameters and rock-platens contact properties.190

Table 8.2: ELFEN 3D UCS model; mobilised parameters with effective plastic strain for the

intact material .......................................................................................................... 190

List of accompanying material

This thesis consists of 1 volume and 1 appended DVD. The single volume includes the full

thesis with associated appendix and list of references, whilst the appended DVD contains a

digital copy of the thesis and the movie files showing the fracture pattern development for the

different ELFEN models described in the text.

- xiv -

Author’s Declaration

The author has not been registered for another award of a University. The material contained in

this thesis is the result of original investigation. All authors and works quoted are fully

acknowledged.

……………………………………… Candidate

Davide Elmo

……………………………………… Supervisor

Dr J.S. Coggan (Camborne School of Mines)

……………………………………… Supervisor

Prof. R.J. Pine (Camborne School of Mines)

Supervisors

Dr. J.S. Coggan

Senior Lecturer in Rock Engineering - Camborne School of Mines

School of Geography Archaeology and Earth Resources

University of Exeter in Cornwall

Tremough Campus

Cornwall TR10 9EZ

Prof. R.J. Pine

Professor of Geotechnical Engineering - Head of Camborne School of Mines



Tremough Campus

Cornwall TR10 9EZ

- xv -

Examiners

Mr. Dean Millar

Lecturer in Mining Engineering - Camborne School of Mines



Tremough Campus Penryn

Cornwall

TR10 9EZ

Dr. John Harrison

Senior Lecturer in Rock Mechanics

Department of Earth Science and Engineering

Imperial College of Science, Technology and Medicine

London

SW7 2BP

- xvi -

List of principal symbols and abbreviations

UCS: Uniaxial compressive strength.

GSI: Geological Strength Index.

RMR: Rock mass rating index.

Q-index: Q index from the Barton et al. (1974) rock mass classification.

nJ , , , and SRF: Parameters used in the Q-index classification system. rJ aJ wJ

bm : Hoek-Brown criterion constant.

s : Hoek-Brown criterion constant.

a : Hoek-Brown criterion constant.

im : Hoek-Brown criterion constant.

D: Hoek-Brown criterion degree of disturbance.

JRC: Joint roughness coefficient.

JCS: Joint wall compressive strength.

1σ : Major principal stress.

2σ : Intermediate stress.

3σ : Minor principal stress.

nσ : Normal stress acting on the failure plane.

τ : Shear stress acting on a failure plane.

c and/or : rock internal cohesion. cc

φ and//or cφ : rock internal friction.

β : Angle of the failure plane relative to the minor principal stress.

ciσ : Intact rock strength, equivalent to the uniaxial compressive strength.

cσ : Uniaxial compressive strength of the rock mass.

tσ : Tensile strength.

cmσ : Global rock mass strength.

E : Intact material Young’s Modulus.

mE : Rock mass Young’s Modulus.

nF and/or : Normal contact force. nf

sF and/or : Shear contact force. sf

nk : Joint normal stiffness.

sk : Joint shear stiffness.

- xvii -

fc : Cohesion along the fracture plane.

fφ : Basic friction angle for the fracture plane.

rfφ : Residual angle of friction for the fracture plane.

ψ : Dilation angle for the fracture plane.

ns : Contribution of failure of intact material, associated with roughness on the fracture plane.

iK : Stress intensity factor.

,IK IIK and : Fracture toughness Mode I, Mode II and Mode III respectively. IIIK

G : Crack extension force.

fG : Fracture Energy.

ν : Poisson’s ratio.

µ : Shear modulus.

ρ : Material density.

21P : FracMan notation for length of fracture traces per unit area of sampling plane.

32P : FracMan notation for area of fractures per unit volume of rock mass.

nP : ELFEN normal penalty coefficient.

tP : ELFEN tangential penalty coefficient.

W and/or : Pillar width. pW

H : Pillar height.

HW : : Pillar width-to-height ratio.

- xviii -

Acknowledgements

I am especially grateful to my supervisors, Prof. Robert Pine and Dr. John Coggan, for their

suggestions and support during the course of this research.

I am indebted to my parents for their love and encouragement, and sacrifices they have made for

the betterment of my life and career. I also want to thank all the members of my family and my

friends for their continuous support.

Special thanks to Amy and her family (John, Linda, Lauren and Aidan), and to Simon and Val,

which have all contributed to make me feel like home here in Cornwall.

Professor Roger Owen and his team at University of Wales at Swansea were responsible for

code developments in ELFEN. Support and material relating to ELFEN and FracMan, provided

by Rockfield Software and Golder Associates respectively, was greatly appreciated. OMYA

kindly provided access and logistical support at Middleton mine. I am also grateful to Dr. Zara

Flynn for helping me with the mapping at Middleton mine and for her work developing the

FracMan model for Middleton mine.

The research described in this thesis has been conducted as part of a project funded through a

grant from the United Kingdom Engineering and Physical Sciences Research Council

(GR/S04970/01).

I would like to thank the Camborne School of Mines Trust and the Institute of Materials,

Minerals and Mining for the award of travel scholarships that allowed me to carry out my field

visits and to attend the 40th U.S. Rock Mechanics Symposium in Alaska in June 2005.

- xix -

“The idea is to try to give all the information to

help others to judge the value of your

contribution; not just the information that leads to

judgment in one particular direction or another.”

Richard P. Feynman

- xx -

Introduction

1

Introduction

1.1 Framework

The design and analysis of underground structures poses many difficult problems; underground

excavations are normally created in fractured rock masses, but conventional numerical

modelling approaches are limited by their inability to incorporate the effects of pre-existing

discontinuities and propagation of new fractures adjacent to the periphery of the excavation as a

result of stress redistribution. State of the art numerical techniques for rock mechanics include

continuum methods (Finite difference method, FDM, Finite element method, FEM and

Boundary element method, BEM), discontinuum methods (Discrete elements method, DEM and

Discrete fracture network method, DFN) and hybrid models (Hybrid FEM/BEM, Hybrid

FEM/DEM and other hybrid models).

This thesis introduces a new approach to the modelling of rock mass behaviour, with

incorporation of both intact rock material behaviour and the effects of a pre-existing fracture

network. The aim is to provide an improved understanding of the mechanical behaviour of

fractured rock masses. The proposed approach is based on the utilisation of the hybrid

FEM/DEM software ELFEN (Rockfield, 2005) coupled with the DFN code FracMan (Golder,

2005). The latter was used for the interpretation and synthesis of a three-dimensional (3D)

fracture system geometry derived from in-situ qualitative and quantitative data. An important

novelty of the proposed research is that the numerical method presented in this thesis does not

in general lead to the creation of complete blocks created by continuous joint sets within strict

geometry and topology, an unrealistic weakness of some current DEM models.

The research, undertaken at the Camborne School of Mines (University of Exeter), has been

financially supported by the Engineering and Physical Sciences Research Council (EPSRC,

Grant No. GR/S04970/01) and has involved collaboration between academic and industrial

- 1 -

Introduction

partners, including the University of Swansea, Rockfield Software, Golder Associates, OMYA

and RioTinto.

1.2 Research methodology

Remarkable effort has been given to develop constitutive models for rock fractures; still further

research is necessary in order for currently available models to be able to predict fracture

behaviour with a reasonable level of confidence. The proposed approach utilizes the code

ELFEN implemented with the code FracMan to simulate the behaviour of a pre-fractured rock

mass.

The code ELFEN, which has recently found increasing use in rock mechanics, can

accommodate a variety of constitutive behaviours, including a Rankine tensile strength criterion

and, for tension/compression stress states, a capped Mohr-Coulomb criterion in which the

softening response is coupled to the tensile model. It can also simulate the transition of a rock

mass from a continuum to a discontinuum by generation of new fractures as a result of

exceeding a defined failure criterion.

The program FracMan has been used on numerous projects for civil and mining applications,

where a good understanding of the effects of fracture networks is required.

The numerical analysis undertaken as part of this research initially concentrated on the analysis

of the behaviour of a simple intact rock block under uniaxial loading conditions. The effects of

individual fractures and multi-fracture geometries were then investigated. Subsequently, the

code FracMan was used to generate a representative 3D fracture network based on field

mapping carried out at an operational underground mine located in the United Kingdom. The

synthesised DFN model represents the input data of 2D trace sections and 3D fracture planes for

the ELFEN geomechanical modelling as part of the study on the mechanical behaviour of

jointed rock masses.

The current research primarily focused on the modelling of progressive pillar failure. In this

context mine pillars represent a typical example of the importance of estimating correct rock

mass strength values; techniques for estimating pillar strength include the use of empirical

formulae based on survey data from actual mining conditions, whilst the conventional approach

to modelling pillar behaviour is to adopt some constitutive criteria to represent the rock material

behaviour, such as a cohesion weakening or cohesion weakening-friction strengthening model.

- 2 -

Introduction

Modelled pillar strengths have been compared with existing empirical mass strength approaches

for determination of pillar strength. Validity of the model rationale has also been investigated

through the strength reduction associated with the effects of variation in angle of a pre-existing

single discontinuity. The results of the ELFEN modelling showed how the modelled pillar

loading capacity was ultimately related to the fracture intensity of the existing fracture network.

The mechanical response of jointed pillars with varying fracture intensity was also found to be

in good agreement with the approach based on the Hoek-Brown failure criterion and Geological

Strength Index (GSI). In this context, it was concluded that the combined use of the FracMan

and ELFEN codes could be used to provide a stronger link between mapped fracture systems

and rock mass strength than is possible with current rock mass classifications. Overall, the

analysis demonstrated that the combined FracMan/ELFEN approach has wider application for

the design of most excavations in fractured rock masses.

From the initial modelling of progressive pillar failure, it is expected that the analysis could be

easily extended to consider other design applications, such as tunnel roof behaviour and

simulation of discontinuity-controlled caving mechanics. Suggestions for further work have

been discussed, that includes providing a stronger link between mapped fracture systems and

rock mass strength than is possible with current rock mass classifications.

It is expected that the proposed research could provide the civil and mining community with a

novel computational decision support system, validated through an extensive experimental

programme, which could assist in the design of underground excavations.

1.3 Thesis structure

This thesis consists of 9 Chapters, one Appendix and one enclosed DVD. The present Chapter

serves as an introduction to the entire dissertation, while Chapter 9 provides a summary of this

research, together with recommendations for future research. Two papers, which were written

during the work of this thesis and are related to the material presented in Chapter 3 and 7, are

included in the enclosed DVD, together with a digital copy of this thesis and movie files

showing the fracture pattern development for the different ELFEN models described in the text.

A brief outline of each remaining chapters is as follows:

Chapter 2 reviews some of the fundamental aspects of numerical modelling in rock

mechanics, with a particular attention to numerical modelling of fractured rock masses.

State of the art numerical methods for rock mechanics problems are introduced, with a

discussion on the differences between continuum and discontinuum methods and on the

- 3 -

Introduction

use of numerical methods to investigate brittle failure of rock. Classical constitutive

models, based on the theory of elasticity and plasticity, are also reviewed.

Chapter 3 introduces the use of the Discrete Fracture Network (DFN) code FracMan to

generate a 3D stochastical model of fracture networks based on qualitative and

quantitative data collected at a specific site location. Using an appropriate interface, the

synthesised FracMan model represents the source of 2D trace sections and 3D fracture

planes which can be subsequently imported in the hybrid FEM/DEM code ELFEN as

part of the study on the mechanical behaviour of jointed rock masses. A particular

emphasis is given on the use of the Zhang and Einstein (2000) method to derive the

fracture radius distribution from the mapped fracture traces. A new approach is

described which uses an approximate solution of the published mathematical

formulations by means of a simple software tool.

Chapter 4 reviews the computational characteristics of the hybrid code ELFEN

(Rockfield, 2005) and its applications for the modelling of the continuum/discontinuum

transformation typical of rock brittle failure, showing how the ELFEN fracture

generation/insertion capability can be used to simulate the process of fracture growth,

extension and coalescence. Numerical examples are given of the applicability of the

code ELFEN in terms of modelling the brittle response of laboratory-scale rock

specimen, showing in particular the potential of the ELFEN fracture generator

algorithm for Mode I (tensile) fracturing, which can effectively reproduce fracture

propagation in a plane parallel to the maximum principal compressive stress. The

ELFEN data structure used to define the necessary parameters for the numerical

analysis is also introduced, with indication of the most important modelling options that

may be combined to create models for differing classes of problems.

Chapter 5 presents numerical examples using the ELFEN code. This Chapter describes

the methodology that was adopted to investigate the effects of various input parameters

on model behaviour using a series of numerical tests on a 2D model of a pre-fractured

rock pillar. The results of the sensitivity analysis were used to provide a modelling

strategy for future ELFEN applications in the field of rock-engineering.

Chapter 6 presents the results of the numerical simulation of simple shear tests using the

code ELFEN, with the objective to study the capability of the proposed hybrid approach

to simulate the progressive shear behaviour of rock joints with tooth-shaped asperities.

The numerical analysis was performed to evaluate also the potential limitations of the

ELFEN rotating crack model when simulating shear mechanism.

Chapter 7 illustrates the applicability of the proposed FEM/DEM method to the analysis

of the mechanical behaviour of pre-fractured mine pillars. The analysis was carried out

mainly in 2D plane strain conditions, and initially it included numerical modelling of

the failure mechanism of a pillar intersected by a single plane of weakness. The

- 4 -

Introduction

computational analysis was then extended to investigate the effects of multiple pre-

existing discontinuities on the overall pillar strength, by studying 2D pillar models

containing different fracture geometries, which were derived from discrete fracture

networks systems generated in FracMan. Chapter 7 also discusses the relationship

between the modelled pillar loading capacity and fracture intensity, examining how the

mapped fracture intensity parameter could be potentially used as a readily

measurable indicator of the structural character of the rock mass.

21P

Chapter 8 briefly reviews simple 3D models consisting of pillars intersected by, or

containing a single fracture to further investigate the capability of the ELFEN 3D

solution algorithm.

- 5 -

A review of techniques for numerical modelling of rock masses

- 6 -

2

A review of principal numerical modelling

techniques

2.1 Introduction

This Chapter reviews the main constitutive models for stress analysis of rock mechanics

problems. It is not intended as an exhaustive review or history of all available different

approaches and applications of numerical modelling. The objective is to present the basic

concepts of the principal numerical models used to simulate the mechanical behaviour of rock

masses, with a particular attention on numerical modelling of fractured rock masses. In the last

decades, constitutive modelling of fractured rock masses has been a subject of interest, with

numerous models being developed in an attempt to simulate their mechanical response. The

review does not include a detailed presentation of mathematical formulations for each specific

numerical method; the aim is to present the assumptions made in the different methods,

describing their applicability and their main advantages and limitations.

2.2 Rock mechanics modelling - The general context

Numerical modelling has played and will continue to play an important role in the design of

underground openings (Hoek et al., 1990). The creation of any underground excavation in rock

will ultimately result in a modification of the stress state of the surrounding rock, and an

analysis of the stresses and associated displacements is required in order to understand the

behaviour of such excavations. As a result of the creation of an underground opening, complex

processes are started, involving the deformation and fracturing of intact pieces of rock, the

displacement of large individual blocks and the possibility of rock joint surfaces opening,

closing and moving relatively to each other.


- 7 -

The scope of the model is not to represent such processes in their entirety, rather the objective of

the analyst is to determine which process need to be considered explicitly and which can be

represented in an average way (Hoek et al., 1990). Quoting Starfield and Cundall (1988), “we

build models because the real world is too complex for our understanding; it does not help if we

build models that are also too complex”. The key components of any satisfactory modelling

approach are data and understanding; in their discussion about methodology for rock mechanics

modelling, Starfield and Cundall (1988) added the following comments:

A model should try and simplify reality rather than trying to create a perfect imitation.

The construction of the model should be driven by the questions that the model is

supposed to answer rather than by the details of the system that is being modelled.

Simple models may be more appropriate to analyse different aspects of the problem or

address the same questions from a different perspectives.

Jing (2003) emphasised how the model does not have to be complete and perfect; it has to be

adequate for the purpose, enhancing the understanding of the processes involved, particularly

the changes that result from the perturbations introduced by the excavation. The same author

provided a comprehensive review of state of the art numerical modelling techniques for rock

mechanics and rock engineering, including a detailed literature source.

These introductory remarks emphasize how, provided they are used correctly, numerical models

can prove to be very useful in designing underground excavations. Any complex model

conceptualisation should try and reflect the inherent Discontinuous, Inhomogeneous,

Anisotropic, and Not-Elastic nature of the rock mass (DIANE concept, after Harrison and

Hudson, 2000), including all the features that are deemed necessary for the purpose. In relation

to the intrinsic discontinuous nature of rock masses, the numerical models should also

incorporate a characterization of pre-existing fractures, as well as fractures induced by changes

in the original state of stress. The physical processes and the modelling techniques chosen will

eventually influence the extent to which these features can be incorporated in the model.

Parameter representability associated with sample size, representative elemental volume and

homogenisation/upscaling represent fundamental problems associated with modelling. For this

reason, any modelling and subsequent rock engineering design will have to include some form

of subjective judgements (Jing, 2003).

The introduction of computer-based numerical modelling has overcome the limitations

presented by closed-form solutions in calculating stresses, displacements and failure of rock

masses surrounding complex underground excavations. However, when restricted to very


- 8 -

simple geometries and material behaviour, thus limiting their practical value in engineering

design, closed-form solutions still have a great value for conceptual understanding of the rock

mass behaviour and for the testing and calibration of numerical models. Analyses in the field of

rock mechanics are influenced by an intrinsic limitation in terms of available data, which also

explains why empirical approaches such as classification systems have been developed and are

still required for design purposes.

2.3 Numerical methods for rock mechanics problems

Numerical modelling is essential for studying the fundamental processes occurring in rocks and

for rock engineering design. Two main approaches are used for the numerical modelling of

fractured rock masses, based on the concept that the deformation of a rock mass subjected to

applied external loads can be considered as being either continuous or discontinuous. The main

differences between the continuum and discontinuum analysis techniques lie in the

conceptualisation and modelling of the fractured rock mass and the subsequent deformation that

can take place in it.

When considering a given rock mechanics problem, some regions of the rock mass could be

treated as continuous, whilst discontinuum analysis would explicitly apply to other elements

like discontinuities. A continuum model would reflect mainly material deformation of the

system, whilst a discontinuum model would reflect the component movement of the system.

The concepts of continuum and discontinuum are, however, not absolute but relative and

problem specific, depending on the problem scale. Based on these observations, the most

common types of numerical models that have found application in the solving of rock

mechanics problems can be grouped as follows:

Continuum methods: Boundary Element Method (BEM), Finite Element Method (FEM)

and Finite Difference Method (FDM).

Discontinuum methods: Discrete Elements Method (DEM), Discontinuous Deformation

Analysis (DDA) and Discrete Fracture Network Method (DFN).

Hybrid models: Hybrid BEM/DEM, Hybrid FEM/BEM, Hybrid FEM/DEM and other

hybrid models.

Table 2.1 summaries applicability, advantages and disadvantages of the most common available

computational methods.


- 9 -

Continuum Analysis

BEM

Concept Discretisation of the boundary of the problem. Definition and solution of a problem entirely in terms of surface values of traction and displacements

Applicability Dynamic and static problems. Underground and surface excavations in rock. Modelling of tabular ore bodies.

Advantages Useful where linear elastic behaviour can be assumed for a rock mass, or where continuous planes of weakness separate elastic domains. Rapid assessment of designs and stress concentrations. Capability of 3D modelling.

Disadvantages Normally elastic analysis only, tough non-linear and time dependent options are available.

FEM

Concept Definition of a problem domain surrounding an excavation and division of the domain into an assembly of discrete interacting elements with an assigned constant state of strain. Strains are expressed in terms of nodal displacements.

Applicability Applications in non-linear mechanics and geotechnical and rock engineering. Analysis of underground and surface excavations in rock and soil.

Advantages Flexibility in handling material inhomogeneity and anisotropy, complex boundary conditions and dynamic problems. Moderate efficiency in dealing with complex constitutive models and fractures. Capability of 3D modelling. Able to simulate both saturated and unsaturated flow/water pressures. Complicated models can now be PC-based and requiring reasonable run-time periods. Can incorporate coupled dynamic/groundwater analysis. Suitable for soil, rock or mixed soil-rock analysis. Time dependent deformation readily simulated.

Disadvantages A considerate amount of time may be required to prepare input data. Computationally expensive. User must be aware of model/software limitations including effects of mesh size, boundaries and symmetry restrictions. Simple fractured structures can be modelled using interfaces, but generally not suitable for highly jointed media.

FDM

Concept The conventional FDM is a simple and efficient method for solving partial differential equations (PDEs) in problem regions with simple boundaries

Applicability The FEM and FDM are similar modelling techniques, which have found wide application in the field of rock engineering.

Advantages Conceptually simple, the method can handle material non-linearity. Useful for solving fluid flow equations and for coupled THM problems of large scale.

Disadvantages Regular grid systems used in conventional FDM models limit the method applicability in terms of fractures integration, description of complex boundary conditions and material inhomogeneity.

Table 2.1: Summary of numerical methods for rock mechanics problems (based on Staub et al., 2002;

Jing, 2003 and Coggan and Stead, 2005).


- 10 -

Discontinuum Analysis

DEM

Concept Formulation and solution of equations of motion of rigid and/or deformable blocks using implicit and explicit formulations. The rock mass is considered as an assemblage of rigid or deformable discrete blocks.

Applicability The DEM has been widely used in rock engineering problems.

Advantages Suitable for modelling media undergoing large displacements and fracturing, especially for loosely jointed media. Able to model complex behaviour, including both block deformation and relative movement of blocks. 3D modelling is possible.

Disadvantages Uncertainty about the fracture system geometry (Spacing, persistence). Limited data on joint stiffness available.

DDA

Concept Originally derived to determine a best fit to a deformed configuration of a block system from measured displacements and deformations.

Applicability Tunnelling, caverns and fracturing processes of rock material. Determine the mechanisms that cause displacements and deformations, rock falls, fracture propagation.

Advantages Especially appropriate when the mechanisms involved are a combination of different modes. Disadvantages Joint opening assumption / Rigid blocks.

DFN

Concept Developed from a need to represent more realistic fracture system geometries in 3D. Applicability Rock engineering. Understanding and quantifying the geological and physical uncertainties. Advantages Valuable tool for generic studies for quantitatively evaluating the impact of fracture system

variations on the model output. Large-scale DFN calculations are easier to run because the number of degrees of freedom of DFN models is much less compared with FEM.

Disadvantages Dependent on the quality of the field data collected and on the interpretation of the in situ fracture system geometry. Mixing of flow and transmissivity change due to stress and displacement discontinuities at intersections not yet properly solved. Lack of matrix flow and stress influence.

Hybrid Models (Incorporating intact rock fracture capability)

Concept Combine the above methods in order to eliminate undesirable characteristics while retaining as many advantages as possible.

Applicability Mechanical behaviour of underground excavations and slope stability analysis. Advantages Harmonization of the geometry of the required problem resolution with the numerical

techniques available. Effective representation of the effects of the far-field to the near-field rocks. Able to allow for extension of existing fractures and creation of new fractures through intact rock. 3D modelling is possible. Can incorporate dynamic effects.

Disadvantages Care should be taken to ensure continuity conditions in the interface between regions of different models, especially when different material assumptions are made. Limited use and validation. Little data available for contact properties and fracture mechanics properties. Use of parallel processing recommended for complex geometries.

Table 2.1 (Continued): Summary of numerical methods for rock mechanics problems. (Based on Staub

et al., 2002; Jing, 2003 and Coggan and Stead, 2005).


- 11 -

2.4 Modelling discontinuities in numerical analysis

As recognised by Cai and Horii (1993), constitutive models of fractured rock masses have long

been subject of interest, with numerous models attempting to simulate their mechanical

response.

As illustrated in Figure 2.1, on a qualitative basis, the rock mass can be classified into three

groups: (i) intact rock mass, (ii) fractured rock mass and (iii) highly fractured or weathered rock

mass. The mechanical behaviour of (i) can be investigated according to models based on a

continuum approach, whilst discontinuous model may be used for analyzing type (ii) and type

(iii) rock masses, although there are some limitations with the number of details that a

discontinuous model can effectively handle. Assuming that a highly fractured rock mass

behaves like a continuous body in a global sense, type (iii) rock masses could also be treated as

a continuum, with equivalent material properties reflecting the effects of pre-existing fractures;

the properties of the continuum should reflect the overall response of both the intact rock

material and the fractures (Curran and Ofoegbu, 1993).

Figure 2.1: Qualitative classification of intact and fractured rock masses.

The continuum approach may circumvent some of the difficulties associated with the discrete

method, in terms of complexity of the model and impracticality of modelling every fracture in a

deterministic way. However, an inherent limitation of the equivalent continuum approach is

that the stress acting on a specific fracture is usually not the same as that deduced from the

overall stress, since it depends on the stiffness of the fracture itself and on the stiffness of the

fracture’s surrounding matrix (Cai and Horii, 1993). In addition, blocks relative displacements

and interlocking, with associated internal moments produced by block rotations, cannot be

adequately accounted for in a continuum model. The continuum approach trades material

complexity for geometrical simplicity, requiring proper homogenization techniques to identify


- 12 -

the material parameters associated with specified constitutive equations for the equivalent

continuum; the homogenization process is usually very complex and valid only over a certain

representative elementary volume or REV (Jing, 1998).

Numerical modelling of fractured rock masses using models of continua have been described,

among the others, by Zienkiewicz and Pande (1977), Gerrard (1982), Fossum (1985), Cundall

(1987) in which the approach is referred to as ubiquitous joint model, Cai and Horii (1992),

Pande (1993), Sitharam et al. (2001), Sitharam et al. (2005) and Yufin et al. (2005).

Problem scale and fracture system geometry affect the choice between continuum and discrete

methods. Figure 2.2 illustrates the application of different numerical methods for different

fracture system geometries; a continuum approach is suitable to characterise a rock mass with

either no fractures or many fractures, the latter case being based on the equivalent continuum

concept. The continuum approach is also possible is the case in which few fractures are present,

but no fracture opening or block detachment is allowed. Moderately fractured rock masses,

where displacements of individual blocks are possible, are better described by a discrete

approach.

Figure 2.2: Suitability of different numerical methods for an excavation in a rock mass in relation to the

existing fractures system geometry: (a) continuum method, (b) either continuum with fracture elements or

discrete method, (c) discrete method and (d) continuum method with equivalent properties (based on

Figure 4 from Jing, 2003, page 290).


- 13 -

Figure 2.3 illustrates the way pre-existing fracture networks can be included within current

available numerical methods, showing the associated specific discretisation concepts.

Figure 2.3: Representation of a fractured rock mass in (a) FDM), (b) FEM, (c) BEM and (d) DEM

models (based on Figure 3 from Jing, 2003, page 289).

As discussed by Curran and Ofoegbu (1993), irrespective of the approach chosen to incorporate

fractures in a numerical analysis, variability and uncertainties associated with the constitutive

relations (stresses versus displacements) defined for the characterisation of the fracture surfaces

affect the quality of the overall results. The problem of scaling laboratory data to obtain a

description of the in-situ mechanical behaviour of fracture surfaces could also affect the quality

of the analysis.

Representation of fractures within a BEM model is achieved considering the fracture surfaces as

opposite boundaries; the relationships between shear and normal stress, shear and normal

displacement for the fracture planes are represented by boundaries conditions.

FEM models allow an explicit consideration of several fractures and/or modelling of their

constitutive behaviour. Fractures can be explicitly modelled within a FEM model using specific

fracture elements and fracture surfaces constitutive relations; the intact rock material in between

fractures would be represented by solid elements. This approach is the so-called Goodman joint

element (Goodman et al., 1968), which has been widely implemented in FEM codes and applied

to many practical rock engineering problems. Ghaboussi et al. (1973) introduced a FEM joint

element based on the theory of plasticity, using relative displacements between two opposite


- 14 -

surfaces of fractures as the independent system unknowns. As reported in Jing (2003), the

Ghaboussi joint element formulation is more robust in terms of numerical ill-conditioning as

compared with the one proposed by Goodman et al. (1968), since the latter is based on

continuum assumptions, hence it does not permit large-scale opening, sliding, and complete

detachment of elements. Zienkiewicz et al. (1970) proposed a six-node fracture element with

two additional nodes in the middle section of the element, and a small thickness, this allowing

the representation of curved joint elements.

Compared with continuum modelling, discontinuum modelling allows for a better consideration

of the role of the discontinuities within the rock mass. In a DEM model, a pre-fractured rock

mass is represented as an assemblage of discrete blocks and rock fractures are modelled

numerically as interfaces between blocks. This means that the rock fracture is considered

equivalent to a boundary condition, rather than representing a special element, like the case of

FEM model approach. Complex constitutive relations can be used in the DEM to define

contacts forces and displacements at the interface of adjacent stressed blocks and specific

fracturing criterion can also be implemented. For a DEM model, the simplest constitutive

relation for the contact interface is linear stiffness subject to constraints of no tension and

limiting shear stress according to a Mohr-Coulomb friction law (Curran and Ofoegbu, 1993).

Beside the problem of obtaining reliable data for the location, orientation and intensity of the

fracture network, some difficulties could be associated with the definition of material behaviour

at the contact points and the damping of the system.

2.5 Numerical modelling of fracture processes

Numerical simulation of the fracture process in rock material requires robust numerical methods

allowing the efficient resolution of multiple interacting cracks and rigorous fracture models that

can reflect the material fabric characteristics (Liu, 2003). As discussed by Owen et al. (2004b),

for problems in which material failure takes place due to progressive damaging resulting in the

formation of either single or multiple fractures, the current position of computational modelling

is not so established. This class of problems include the study of the development of a single

fracture, where, although eventual failure is manifested by such a discrete event, the conditions

leading to the onset of crack propagation are controlled by damage based micro-cracking

mechanisms. In addition to that are problems concerning with the multi-fracturing of quasi-

brittle materials such as ceramics, rock and concrete.

Different numerical approaches have been applied to the analysis of the rock fracture process,

among which are:


- 15 -

i. BEM models and in particular the displacement discontinuity method have found

application into the modelling of rock fractures. A coupled DDM/splitting model was

proposed by Tan et al. (1996) to simulate side crack propagation and indentation, whilst

Blair and Cook (1998) used a coupled DDM/statistical analysis model to analyse

compressive fracture in rock. De Bremaecker and Ferris (2004) described a 2D

numerical approach based on the DDM method to model shear fractures.

ii. FEM models for simulation of rock fractures include SICRAP (Saouma and Kleinoski,

1984), CRACKER (Swenson and Ingraffea, 1998), FRAN2D (Wawryznek and

Ingraffea, 1989), DIANA and NUMA (Alehossein and Hood, 1996) and R-T2D (Liu,

2003).

iii. The FDM code FLAC (Itasca, 2005) include some plastic model groups that allow rock

fracture to be simulated.

iv. The DEM code PFC2D has been used to simulate the fracturing of intact rock and the

fracturing of simple pre-fractured rock masses (Kulatilake et al., 2001; Diederichs,

2002). Applications of the DDA model to block fracturing have been presented in

Amadei et al. (1994), whilst Ke and Goodman (1994) introduced new capabilities of the

DDA analysis technique for simulating fracture propagation in intact rocks.

The variety of numerical methods and fracture models developed in an attempt to simulate the

fracture process of rock material reflects the complexity of the fracture process itself; not all

numerical models can visually simulate the progressive fracture process, including crack

initiation, propagation, interaction and coalescence (Liu, 2003). As discussed by Stead et al.

(2004) and Coggan and Stead (2005), although continuum and discontinuum models can

provide useful analysis for interpretation of failure around underground openings, neither

approach can capture the interaction of existing discontinuities and the creation of new fractures

through fracturing of the intact rock material.

In this context, the hybrid FEM/DEM code ELFEN (Rockfield, 2005) used as part of the current

research allows the simulation of fracture process initiation, fracture growth and extension in

both intact and pre-fractured rock material. Examples of the applicability of the code ELFEN

are illustrated in Klerck (2000) and Klerck et al. (2004), where the model is specifically applied

to triaxial and plane strain tests, punch tests and borehole breakouts; Owen et al. (2004a), with

examples of simulations of dragline operations; Stead et al. (2004), Eberhardt et al. (2004),

Coggan and Stead (2005), where the method is applied to rock slopes; Coggan et al. (2003),

Elmo et al. (2005) and Pine et al. (2006), where the effectiveness of the proposed numerical

model is assessed by application to problems related to the limit load analysis of a series of

mining pillars.


- 16 -

The success of modelling fracture processes in heterogeneous rock materials ultimately depends

of the understanding of the fracture mechanism, the soundness of a universal fracture criterion

and the effectiveness of the numerical techniques used (Liu, 2003). It is expected that the rapid

development of computing power, including also topological data structure update, will enable

to overcome some of the limitations of the currently available models, though the challenge of

representing the inherent rock material heterogeneity would still require the numerical models to

incorporate some form of simplification.

2.6 Numerical modelling in 2D and 3D space - A comparative overview

In the preceding paragraphs the discussion has focused on different numerical models. No

specific mention was given to the geometric representation of a certain rock engineering

problem in terms of distinction between two-dimensional (2D) and three-dimensional (3D)

analysis.

It is known that 2D models are limited to approximations whether the consideration of a plane

strain or plane stress section is adequate to represent the given geometry; on the other hand 3D

models allow the consideration of far more general situations but may require several orders of

magnitude more computer resources for the same problem resolution. Most of all numerical

methods presented in the preceding paragraphs are capable to carry out 3D simulations, but due

to the CPU time and memory required this is usually limited to simple problem domains.

2D numerical modelling which incorporates plane strain elastic solutions has been used

extensively for design of tunnels and mine openings (Hoek and Brown, 1980), and the

techniques can be used to calculate the stress distribution around a tunnel of infinite length;

however, a 2D numerical solution cannot be employed to calculate representative stress

distributions around a tunnel face-end or where complex 3D geometries characterise the

problem domain.

An example of FEM 2D and 3D elasto-plastic analysis for coal pillar design and its application

to highwall mining can be found in Duncan Fama et al. (1995). Meyer (2000) presented a direct

comparison between 2D and 3D numerical simulations of the mechanical behaviour of coal

roadways using the proprietary codes FLAC2D and FLAC3D (Itasca, 2005). Dhawana et al

(2002) discussed the differences between 2D and 3D FEM models for underground excavations

in an inhomogeneous rock mass.


- 17 -

This research involves the use of the code FracMan (Golder, 2005), which allows the generation

of 3D stochastic fracture network models, coupled with the hybrid FEM/DEM code ELFEN

(Rockfield, 2005) to investigate the mechanical behaviour of fractured rock masses. The

transferring of 3D assemblages of fractures from FracMan into ELFEN is relatively

straightforward in terms of the 3D geometrical definition of the problem; however, limitations

have to be considered in terms of meshing and subsequent modelling of block

displacement/rotation. The possibility to reproduce the fracturing of the intact material and the

interaction between newly generated and existing fractures further increases the complexity of

the whole model, since it requires the topological update of the mesh when a fracture is inserted

in the domain.

A 2D simulation of a pre-fracture rock mass will inherently assume a continuity of the

discontinuity surfaces in the plane of the 2D section, which may lead to an underestimate of the

effective maximum load at failure that can be taken by the fractured rock mass in question;

however, 3D pillars may be created due to less confinement. A 2D representation of a specific

geometry domain allows a more time-efficient computation, whilst at the same time yielding

important insights into the mechanical behaviour of fractured rock masses. For these reasons

this research has focused primarily on 2D geomechanical modelling of pre-fractured rock

masses; a direct comparison between 2D and 3D models has been limited to the consideration of

models consisting of intact rock material or simple pre-fractured geometries.

As discussed by Owen et al. (2004b), due to the extremely intensive computations involved in

hybrid continuous/discrete simulations of realistic applications, parallelisation becomes an

obvious option for significantly increasing existing computational capabilities. Significant

advances in the development of parallel computer hardware, particularly the emergence of

commodity PC clusters, make such a parallel computing option feasible and attractive.

Although some progress has been made in the development of appropriate algorithms, the same

authors argued that, however, considerable further work is required to provide general robust

and efficient solution procedures that account for the complexities encountered in industrial

problems.

2.7 Constitutive laws in numerical modelling

The stress-strain behaviour of the rock mass defines its strength and deformability; it is

reasonable to assume that the concepts of rock mass strength and deformability are not unique

but depend on the instances of the problem under consideration: the behaviour of intact rock

material, the occurrence of discontinuities and/or assemblages of blocks are all factors


- 18 -

contributing to their definition. Adequate constitutive models, relating stress and strain, have to

be incorporated in the study of rock mass behaviour by numerical models to represent the

combined mechanical/physical properties of the intact rock material and discontinuity planes.

The classical constitutive models of rocks fractures and fractured rock masses are based on the

theory of elasticity and plasticity, with special considerations for fracture effects. Jing (2003)

presented an introduction to the most recent developments in the area, supported by literature

sources. Constitutive models for fractured rock masses are typically developed from

corresponding constitutive models for intact rock and rock fractures (Pande, 1993).

Constitutive models for rock mechanics can be subdivided into two groups, based upon the

reversible or non-reversible nature of the material behaviour. Materials are defined elastic if at

the end of a loading-unloading cycle they reach their initial state in terms of stress-strain;

materials are considered to be plastic if at the end of a loading-unloading cycle a permanent

deformation has occurred. A brief introduction to elastic and plastic models is here given, with

different constitutive laws being illustrated in Figure 2.4. The ELFEN code, which constitutes

the core of the research studies presented in this thesis, allows the use of a large range of

material modelling options, which can also be combined to create material models for differing

classes of problem. Materials models available in the ELFEN code will be described in more

detail in Chapter 4.

Figure 2.4: Representation of typical constitutive laws; (a) Linear elastic, (b) non-linear elastic, (c)

elasto-plastic and (d) rigid-plastic (based on Cividini, 1993).


- 19 -

Citing also work by other authors, Desai (1993) discussed how linear elastic and non-linear

elastic constitutive models have found application in the computation of elastic, homogeneous

and isotropic rock masses. However, these models are limited by not being able to simulate the

behaviour of rock masses that exhibit a non-linear and non-elastic response. Referring to Figure

2.4, for a uniaxial state of stress (i.e. in a 2D space), plastic deformation occurs when a specific

stress eσ (elastic limit) is reached. For multi-axial stress state, the scalar quantity eσ is

replaced by a function (yield surface), whose shape is dependent upon the specific definition

considered.

Besides simple models for linear elastic, isotropic and homogeneous materials, more complex

constitutive models of anisotropic elasticity can be developed in closed-form by allowing for

alternative elastic symmetry conditions for intact rocks (such as transversely isotropic elasticity)

or equivalent continuum elastic rocks intersected by orthogonal sets of fractures (Jing, 2003).

Desai (1993) has also reported how various investigators have proposed non-linear and

plasticity based models to characterise the behaviour of fractured rock masses. Plasticity based

models, such as Von Mises, Mohr-Coulomb and Drucker-Praeger, have been used to simulate

the behaviour of plastic and/or inelastic deformation in rocks. Since the 1970s plasticity and

elasto-plasticity models have been developed and widely applied to fractured rocks, based

mainly on the classical theory of plasticity, with typical models using Mohr-Coulomb and

Hoek-Brown failure criteria (Hoek, 1983; Hoek and Brown, 1980, 1997). The failure criteria of

rocks are important components of constitutive relations and are usually used as yield surfaces

or/and plastic potential functions in a plasticity model.

Expressions for the elasto-plastic material models consist of three main parts: (i) a yield

condition, represented by a surface in the stress state defining the stress at which plastic

deformation may occur, (ii) a hardening law, which describes the possible changes in shape,

size and position of the yield surface and (iii) a flow rule, which governs the increment of the

plastic strains. Elasto-perfectly plastic material behaviour is described in Figure 2.5, according

to which the elastic limit does not depend on the stress and/or plastic strain history. Positive-

hardening and normal-softening are the two main features of plastic behaviour of rocks, with the

latter typically observed under uniaxial compression test conditions. Positive-hardening and

normal-softening behaviours are also illustrated in Figure 2.5.


- 20 -

Figure 2.5: Different plastic behaviours: elasto-perfectly-plastic (a), positive-hardening (b), negative-

hardening/normal-softening (c), critical and subcritical-softnening (1d and 2d respectively) (based on

Cividini, 1993).

According to Cividini (1993), an important difference exists between some of the schemes

illustrated in Figure 2.5. Perfectly plastic and hardening models intend to represent the

behaviour of a continuous and homogeneous material, i.e. can be considered as actual

constitutive laws; conversely, the softening behaviour can be considered as the outcome of

some substantial modifications that take place in the sample during testing. The same author

has affirmed that the softening behaviour is typically observed in hard, stiff rocks, in which a

series of fractures develops and propagates when the load approaches its peak value. A

practical implication of the softening behaviour is that the relevant data from laboratory test are

markedly influenced by the size of the specimen and by the characteristics of the testing

equipment (Hudson et al., 1972, reported in Cividini, 1993). Depending on the material under

examination, the softening behaviour is associated with a loss of continuity of the sample due to

the formation of fractures or shear bands. This assumption of shear failure is reflected in many

traditional continuum failure models (e.g. Mohr-Coulomb and Hoek-Brown); in these models

cohesion and friction components are assigned and assumed to act simultaneously to define the

maximum deviatoric stress limit or yield strength at any given confinement (Diederichs, 2002).

According to the same author, because of their inherent internal structure, the compressive

damage in competent hard rocks at low or moderate confinement occurs primarily as extensile


- 21 -

cracking. The tensile nature of the damage process leads to a constitutive separation of cohesive

and frictional strength components, which explains the limitations for yield prediction for hard

rock materials using conventional continuum failure models. Under low confinement, this

Mode I process of growth and extension of stress-induced fractures extension results in spalling

and slabbing, which are a typical mechanism associated with brittle failure; a discussion on the

modelling of brittle failure is presented in the following section.

2.7.1 Modelling brittle failure of rocks

Brittle failure typically results from the accumulation and growth of stress-induced fractures

growing parallel to the excavation boundary. In a modelling perspective, brittle failure

corresponds to a transition from a continuum to a discontinuum state, which is extremely

difficult to capture in current numerical models despite the advances in discontinuum

modelling.

An alternative approach is to capture in continuum models the fundamentals of brittle failure.

Using conventional continuum modelling approaches, Wagner (1987), Pelli et al. (1991), Castro

et al. (1996), Martin (1997) and Grimstad and Bhasin (1997) have investigated the subject

considering traditional failure criteria and assuming that the mobilization of the cohesion and

frictional strength components is instantaneous. As stated by Martin et al. (2001), this approach

overlooks a fundamental observation of brittle failure, that the formation of tensile cracks is the

first step in the failure process.

Following work by Martin and Chandler (1994), the attention of investigators has turned to an

approach based on the concept that in brittle failure peak cohesion and friction are not mobilised

together and most of the cohesion is lost before peak friction is mobilised. At present it is a

common approach to use continuum analyses to simulate brittle failure of rocks and adopting

constitutive criteria such as elastic-brittle-plastic or strain-softening. Current applications of

continuum analyses to modelling of brittle-failure around underground excavations can be

found in Fang and Harrison (2002), Hajiabdolmajid et al. (2000, 2002), Martin and Maybee

(2000) and Meyer et al. (2001).

Hajiabdolmajid et al. (2002) adopted a “cohesion weakening and friction strengthening”

(CWFS) criterion to predict the extent and depth of brittle failure of rocks, whilst Fang and

Harrison (2002) implemented a local material degradation approach to model brittle fracture of

mine pillars using the code FLAC (Itasca, 2005). Similarly, Martin et al. (2001) used a model

with cohesion weakening and friction hardening as a function of plastic strain implemented in


- 22 -

the code FLAC (Itasca, 2005) to numerically model the shape and extent of brittle failure for the

AECL’s Mine-by test tunnel constructed in Lac du Bonnet granite. The utilisation of a

continuum approach to modelling a process that is ultimately discontinuum has intrinsic

limitations and cannot capture all the subtleties of brittle failure.

Stacey and De Jongh (1977) found that the conventional Mohr and Griffith criteria for brittle

fracture were not able to capture fracturing associated with tunnel boring in hard rocks. The

former authors stated that fracture of the rock occurs in indirect tension when the tensile strain

exceeds a limiting value, which is dependent on the properties of the rock, and thus introducing

a tensile strain criterion. Stacey (1981) changed the terminology from tensile strain to extension

strain. According to the extension strain criterion, fracturing in brittle rocks will initiate when

the total extension strain in the rock exceeds a critical value cε , with fracture forming in planes

normal to the direction of the extension strain, which corresponds to the direction of the

minimum principal stress. The criterion does not require the presence of a tensile stress field;

even though all the three principal stresses are compressive, the triaxial state of stress can result

in a tensile strain. It is noted how the extension criterion also takes into consideration the

effects of the intermediate stress. The occurrence of extension strains would be inhibited by

compressive confining stress; hence at high stresses fracturing will typically occur in proximity

of the boundary of the excavation. Stacey (1981) successfully implemented the extension strain

criterion to recreate slabbing conditions of mine haulage sidewalls.

Diederichs (1999 and 2002) using a DEM approach demonstrated how the brittle response

resulting in compression spalling can be the result of purely Mode I (tensile) crack initiation and

growth; the consequences of this tensile damage process is that, under low confinement,

extensile crack growth and Mode I crack opening removes the possibility for frictional

resistance between the crack surfaces and, as a result, cohesive strength is lost before frictional

strength can be mobilised.

This research reviews the process of rock brittle failure using a coupled FEM/DEM approach,

with the intention to capture the transition from a continuum to a discontinuum state occurring

as a result of crack initiation, accumulation and interaction; this is achieved using the hybrid

FEM/DEM code ELFEN (Rockfield, 2005), which have been recently used by many authors to

investigate brittle failure (Coggan et al., 2003; Klerck, 2000; Klerck et al., 2004 and Owen et

al., 2004b). Although crack growth, accumulation and coalescence is permitted in ELFEN

under Mode I (tensile) failure only, the code allows to investigate the interaction between newly

generated and pre-existing fractures, capturing also the subsequent displacement and/or rotation

of independent blocks in a typical DEM fashion.


- 23 -

2.8 Summary and research framework

This Chapter has reviewed different numerical methods used for modelling the behaviour of

fractured rock masses; two key analysis techniques, continuum and discontinuum, were

introduced. Limit equilibrium analysis and analytical methods were briefly mentioned as

alternative method used to predict rock mass behaviour. Table 2.1 has summarised

applicability, advantages and disadvantages of the most common available computational

methods. As discussed by Staub et al. (2002), the suitability of a numerical model is ultimately

related to the reciprocal size of the engineering works compared to the geometry of the fractured

rock mass; highly fractured rock masses will be better represented by a discontinuum approach

with special emphasis on how contacts are handled, whereas homogeneous media or media

sparsely jointed can be modelled as continuous.

The review has show how nowadays numerical methods and computing techniques have

become integrated components in studies for rock mechanics and rock engineering. At the same

time, the discussion has focused on the key point of representing in numerical models the

inherent discontinuous nature of a rock mass; this clearly requires the current numerical

modelling techniques to further advance and improve in order to be able to represent the

behaviour of individual fractures and the interaction between intersecting fractures. A major

drive of research and development in rock mechanics is the representation of rock fractures in

numerical models, both as individual entities and as a collective system. At present the use of

DEM models, especially for 3D analysis, is limited by the large computational requirements for

realistic simulations. Ultimately, as argued by Cundall (2001), with substantial advances in

computer power, this type of approach could be suitable for simulations of large-scale fractured

rock masses. As reported by Pariseau (1993), Hart (2003) and more recently by Owen et al.

(2004b), the introduction of a hybrid continuum/discontinuum approach, linking different

numerical techniques such as FEM and DEM, could allow for large scale analysis and locally

large displacements along fracture planes, overcoming both the extremely intensive memory

capacity and calculations times required.

The main objective of this research is the introduction of a new approach to the modelling of

rock mass behaviour, with incorporation of both rock material behaviour and the effects of a

pre-existing fracture network; this is achieved via utilisation of the hybrid continuum/discrete

numerical method code ELFEN (Rockfield, 2005) in conjunction with the FracMan code for the

interpretation of the in-situ fracture system geometry. This modelling approach has the

advantage of not considering only complete blocks, like those created within the code UDEC,

but rather to allow the investigation of the mechanical response of a rock mass including a


- 24 -

network of pre-existing intersecting and isolated fractures. Barton et al. (1994) discussed how

the constraints of a pre-defined geometry of discrete blocks and continuous discontinuity in

UDEC do not allow a realistic modelling of the physics of rock deformation and failure. It is

intended that the proposed discontinuum approach can better reflect the key role of rock

fractures in the rock mass response to loading and unloading.

The use of the Discrete Fracture Network code FracMan to represent 3D fracture geometry systems

3

The use of a DFN model combined with a hybrid

FEM/DEM method as a new numerical modelling

approach for naturally fractured rock masses

3.1 Introduction

One of the primary aspects of the current research was the development of a new approach for

modelling fractured rock masses. The objective is to maximise the quality of representation of

the geometry of existing rock jointing and to use this within a geomechanical model that fully

accounts for the style of jointing. The platform for data capture and synthesis was the Discrete

Fracture Network (DFN) code FracMan (Golder, 2005; Dershowitz et al., 1998). The code

FracMan was used to generate a 3D stochastical model of fracture networks based on qualitative

and quantitative data collected at a specific site location. Using an appropriate interface, the

synthesised FracMan model represented the source of 2D trace sections which were

subsequently imported in the hybrid FEM/DEM code ELFEN (Rockfield, 2005) as part of the

study on the mechanical behaviour of jointed rock masses. The latter analysis is described in

the subsequent chapters.

The current Chapter describes the fundamental aspects associated with the synthesis of a DFN

model. A complete review of the various options in FracMan exceeds the scope of this thesis;

the objective is here to present a newly developed work-flow for a FracMan analysis, which

includes a novel approximate solution for the method proposed by Zhang and Einstein (2000)

for the determination of the fracture radius distribution.

- 25 -


3.2 An introduction to specific FracMan terminology

The specific FracMan terminology used to quantify fracture characteristics is graphically

illustrated in Figure 3.1. Measures included in the shaded boxes are the most used in FracMan

for typical Discrete Fracture Network (DFN) characterisation.

Dimension of feature

0 1 2 3

0 P00 Length0 Number of fractures

Point

Measure

1

P10 Length-1 Number of fractures per unit length of scanline (Frequency or linear intensity)

P11 Length0 Length of fractures intersects per unit length of scanline

Linear Measure

2

P20 Length-2 Number of traces per unit area of sampling plane (Areal density)

P21 Length-1

Length of fracture traces per unit area of sampling plane (Areal intensity)

P22 Length0 Area of fractures per unit area of sampling plane

Areal Measure

Dim

ensi

on o

f sam

plin

g re

gion

3

P30 Length-3 Number of fracture per unit volume of rock mass (Volumetric density)

P32 Length-1 Area of fractures per unit volume of rock mass (Volumetric intensity)

P33 Length-2 Volume of fractures per unit volume of rock mass

Volumetric Measure

Figure 3.1: Graphical illustration of FracMan fracture quantification parameters (Based on Slide No. 20

from http://fracman.golder.com/Gallery/guidtour.asp).

3.3 The development of a work-flow for the generation of a DFN FracMan model

Basic input data for generation of a stochastic DFN model in FracMan include:

i. Definition of a generation model

ii. Fracture orientation distribution

iii. Fracture termination

iv. Fracture size

v. Fracture intensity

- 26 -


The code FracMan has recently found an increase use for various applications, both in the field

of civil and mining engineering; hence the definition of the points listed above may seem a

seemingly straightforward process. However, the process of specifying the correct input

parameters for a FracMan analysis requires a certain degree of user intervention, which involves

the use of techniques not directly implemented in a general FracMan work-flow. This is

particularly true in the case of the determination of the fracture radius distribution (fracture

size).

The flowchart in Figure 3.2 summarises the specific FracMan work-flow developed as part of

the current research and the following sections provide a description of the fundamental

parameters considered. It is noted how the proposed approach specifically refers to a stochastic

analysis in FracMan: fractures are generated using a stochastic selection of their orientation,

diameter, and spatial location such that the prescribed statistical distributions of the fracture sets

are adhered to. Conditional DFN model generation techniques, which exactly reproduce the

mapped fracture traces, are also implemented in the code FracMan; however, these were not

used as part of the current research.

The stochastic nature of the process is such that there are infinitely many possible realisations of

the 3D fracture system based on the mapped data. Indeed, the mapping process is itself random

by the nature of how fractures are presented in available windows. With the exception of very

explicit modelling of an individual fracture or simplified fracture sets, which cannot be wholly

representative, the stochastic approach provides the best option for creating realistic geometric

models of fracturing.

- 27 -


Figure 3.2: Proposed work-flow for a fully stochastic analysis in FracMan.

- 28 -


3.3.1 Choice of a specific DFN generation model

A comprehensive review of the principal DFN models was outside the scope of the current

research. Literature sources for conceptual DFN models include, amongst the others,

Dershowitz and Einstein (1988), Staub et al. (2002) and Jing (2003). DFN models available in

the current release version of FracMan are described in Appendix I. In general terms, the main

difference between the various conceptual DFN models is a function of the way fracture

characteristics are considered. Most of the models involve the same consideration for specific

fracture characteristics, such as shape (generally polygons), size and termination at

intersections. The differences between the models rely on the specific distribution laws utilised

to simulate fracture orientation and fracture location.

The choice of a specific DFN model is typically based on assumptions made from field data and

geological observations. In a general case the Enhanced Baecher model provides an appropriate

solution for different applications; this model corresponds to a Poisson process leading to an

exponential distribution function for the fracture spacing along a sampling line. Scanline

surveys were incorporated into the fieldwork methodology in order to substantiate the use of the

Enhanced Baecher model as typical generation model.

3.3.2 Fracture orientation

DFN models are generated separately for each fracture set and then combined to obtain the

overall representation of the fracture network. In the specific context of this research, the

software DIPS (Rocscience, 2005) is used to process the discontinuity data in order to identify

the different fracture sets. Typically, mean value and standard deviation for both the dip and dip

direction for each set is determined. These parameters can be used in FracMan in association

with the bivariate normal option or alternatively fracture orientation can be modelled specifying

the mean dip and dip direction in addition to the relative Fisher dispersion coefficient.

3.3.3 Fracture termination

Fracture termination is taken into account considering different types of termination styles

(Figure 3.3) and estimated according to the following equation:

ends ObservedT of Number

nTerminatio of Percentage s= [3.1]

- 29 -


where refers to the T-type of termination style shown in Figure 3.3, whilst the observed-ends

represent the number of fracture traces end-points within the mapped window.

sT

T X

Figure 3.3: Type of termination styles recognised in FracMan.

The value calculated using Equation [3.1] is then entered in FracMan as a percentage. Other

useful parameters can be derived using Equations [3.2] and [3.3] and used for comparison

purposes between mapped fracture traces and simulated DFN models.

( )Area Sampling

X T intensity onIntersecti ss += [3.2]

( )ss

s

X TT

y Probabilit nTerminatio+

= [3.3]

3.3.4 Fracture shape and size

Fracture shape and size can be assessed by analytical methods applied to data sampled on

exposed rock faces (Mauldon, 1998; Zhang and Einstein, 1998; Zhang and Einstein, 2000).

The process is based on basic assumptions:

i. Fractures are considered to be planar (Priest and Hudson, 1976; Baecher et al., 1977;

Warburton, 1980 and Kulatilake, 1993).

ii. Fractures can be represented by thin circular discs. Robertson (1970), based on field

observations, implied the possibility that discontinuities are equidimensional. On the

contrary, Bridges (1975) and Einstein et al. (1979) indicated that discontinuities are not

equidimensional. For mathematical convenience, Baecher et al. (1977), Warburton,

(1980) and Kulatilake (1993) assumed fractures to be equivalent to circular discs. On

these bases, fractures are here assumed to be polygons with n sides and their size is to

- 30 -


be represented by the radius of a circle of equivalent area (Figure 3.4). For a polygon

with sides and , the equivalent radius is defined as: n 3n >

πfrac

e

AR = [3.4]

Figure 3.4: Circle of equivalent area for a polygonal fracture.

Within the code FracMan, fracture shape can also be more specifically described

indicating a particular aspect ratio, which is the ratio of long to short axis in the

direction of elongation, and a preferential direction of elongation. It is, however,

extremely difficult to obtain field data on aspect ratio.

iii. Fracture centres are randomly and independently distributed in space. This is a widely

accepted assumption (Zhang and Einstein, 2000), leading to an exponential distribution

form for the fracture spacing along a scanline. It is noted that this assumption is in

agreement with several field studies (Priest and Hudson, 1976; Call et al., 1976;

Baecher et al., 1977 and Einstein et al., 1979).

iv. Fracture size distribution is independent of spatial location. Assumed by Zhang and

Einstein (2000) for a question of simplicity, this assumption is also considered to be

implicit in most rock mechanics analyses (Baecher et al., 1977).

It is accepted that measured trace lengths are typically biased, therefore there is the need to

correct sampling bias in order to derive true trace lengths, which are then employed to estimate

fracture size distribution. The analytical method proposed by Zhang and Einstein (2000) has

been implemented in the current study of field data. The method is based on the analysis of data

mapped within circular windows, which have the advantage over scanline and rectangular

windows of automatically eliminating orientation bias.

- 31 -


The distribution form for the length of the mapped traces is initially estimated using the module

GeoFractal (Golder, 2005), which is part of the code FracMan. This distribution is denoted with

the notation ( )lg . Either or Kolmogorov-Smirnov goodness-of-fit tests can be used to

validate the estimated distribution form. Mean (

2χ

gµ ) and standard deviation ( gσ ) of ( )lg are

then used in addition to truncation effects (Figure 3.5) to calculate , which is the true trace

length distribution, with associated mean and standard deviation

( )lf

fµ and fσ respectively.

Note that fµ is calculated using the method firstly developed by Mauldon (1998) and later

revised by Zhang and Einstein (1998), and according to:

( )( )ct

ctf NNN

NNNc

+−−+

=2πµ [3.5]

fσ is calculated from:

( gff COV )µσ = [3.6]

In Equation [3.5], represents the total number of fractures intersecting the window, is

the number of fracture completely transecting the window and is the number of fracture

entirely enclosed in the window; the term is the radius of the sampling window. The term

in Equation [3.6] represents the coefficient of variation for , corresponding to the

ratio of the standard deviation

N tN

cN

c

( gCOV ) ( )lg

gσ to the mean gµ . Note that ( )lf and are assumed to

have the same distribution form.

( )lg

Figure 3.5: Schematic representation of truncation effects for fracture traces in a circular window.

- 32 -


For area sampling of rock discontinuities, the stereological relationship between the true trace

length distribution and the discontinuity diameter distribution is expressed as

(Warburton, 1980):

( )lf ( )Dg

( ) ( )∫∞

−

∂=

td lD

DDglf22

1µ

[3.7]

The procedure to derive from the calculated ( )Dg ( )lf is illustrated in the paper by Zhang and

Einstein (2000), pages 823 to 825. Note that the distribution forms considered for ( )Dg are

Lognormal, Negative Exponential and Gamma. The discontinuity diameter distribution ( )Dg

may or may not have the same distribution form as ( )lf .

It is noted how the method proposed by Zhang and Einstein (2000) involves the solution of

complex integrals. A diverse approach was implemented as part of the current research that

simplified the required numerical integrations by using a simple and easy-to-use numerical

code, FNGraph. The FNGraph code (Minza, 2002) allows the integration of complex

mathematical functions over a specified range. However, the proposed approach considered

only Lognormal and Negative Exponential distribution forms. The application of the proposed

simplified method consists then in:

Step 1: Assume a distribution form for the fracture radius distribution and compute

the mean and standard deviation,

( )Dg

Dµ and Dσ respectively, by using the relationships

proposed by Zhang and Einstein (2000) and listed in Table 3.1.

Distribution form of

( )Dg Dµ Dσ

Lognormal ( )22

3

3

128

ff

f

σµπ

µ

+

( )( )2226

624222

9

1281536

ff

ffff

σµπ

µµσµπ

+

−+

Negative

Exponential fµ32

22⎟⎠⎞

⎜⎝⎛

fµπ

Table 3.1: Expressions for determining Dµ and Dσ from fµ and fσ respectively (after Zhang

and Einstein, 2000).

- 33 -


Step 2: For each assumed distribution form calculate the ratio of the forth and second

moment of the underlying discontinuity diameter distribution , ( )Dg ( )4DE and

( )2DE respectively, by using the following relationships (Zhang and Einstein, 2000).

Distribution form of ( )Dg ( )( )2

4

DEDE

Lognormal ( )

8

522

D

DD

µσµ +

Negative Exponential 212 Dµ

Table 3.2: Expressions for determining the ratio ( ) ( )24 / DEDE (after Zhang and Einstein,

2000).

Step 3: Calculate the first and third moment of the trace length distribution ( )lf , ( )lE

and ( )3lE respectively. This step requires solving the following integrals:

( ) ( )∫∞

=0

2

41 dDDgDlE

Dµπ

[3.8]

( ) ( )∫∞

=0

43

163 dDDgDlE

Dµπ

[3.9]

For a Lognormal distribution, ( )Dg has the form:

( ) ( ) 22 2/ln

21

DDD

D

eD

Dg σµ

πσ−−= [3.10]

For a Negative Exponential distribution, ( )Dg has the form:

( ) DeDg λλ −= where [3.11] 1−= Dµλ

- 34 -


Using the code FNGraph (Minza, 2002), Equations [3.8] and [3.9] are entered in their

analytical form with the specific form for ( )Dg as expressed in Equations [3.10] and

[3.11] and then integrated over a relative large range, which in this way yields and

approximate solution to the limit to infinite of the original integral functions (Figure 3.6).

The ratio ( ) ( )lElE /3 is then computed. As discussed in Zhang and Einstein (2000), the

best distribution form for ( )Dg is the form for which the calculated ratios in Steps 2 and

3 respectively are the closest to each other. Whereas it was not considered as part of the

current research, Equations [3.8] and [3.9] could also be solved for a Gamma distribution

form and using the proposed approximation method if the explicit form of the Gamma

distribution form is known.

Figure 3.6: Example of the FNGraph plot of the first and third moment of a negative exponential

trace length distribution with mean 1. For this particular case, the ratio ( )3lE to ( )lE was equal to

8.8 over the range [0, 30].

3.3.5 Fracture intensity

The generation of a DFN model requires the estimation of a parameter known as fracture

intensity. In the context of the specific FracMan terminology (see also Figure 3.1), the

volumetric fracture intensity is defined as the ratio of total fracture area to unit volume, and

according to:

VAP i∑

=32 [3.12]

- 35 -


The parameter (m2/m3) can be assessed on the basis of a linear correlation with (m/m2),

which is the total trace length of fractures per unit area, or with (m-1), which is the total

number of fractures along a scanline or borehole (i.e. fracture frequency). The linear

correlations are defined as (Dershowitz and Herda, 1992):

32P 21P

10P

212132 PCP = [3.13]

101032 PCP = [3.14]

The constants of proportionality and depend on the orientation and radius size

distribution of the fractures and on the orientation of the sampling panel ( ) or

scanline/borehole ( ). Conversely, the parameter is independent of the orientation and

radius size distribution of fractures (Figure 3.7). As reported by Zhang and Einstein (2000),

does not depend on the size of the sampled region, as long as it is representative of the fracture

network.

21C 10C

21P

10P 32P

32P

Figure 3.7: Example of the general anisotropy of the parameter . The synthetic realisation refers to

fracture set 1a generated within a region equivalent to a Middleton pillar: = 0.76 m2/m3 for Set 1a for

this rock volume. (b) Traces of the set 1a fractures from (a) on planes corresponding to those mapped at

Middleton showing the clear anisotropy of the (m/m2) values.

21P

32P

21P

The process to determine follows the method described by Staub et al. (2002), and is

schematically illustrated in Figure 3.8. A DFN model is generated using the derived basic input

32P

- 36 -


parameters, which include fracture orientation, radius size distribution, shape and termination

probability. An initial estimated value completes the necessary input data required. For

each single fracture set several DFN models are generated; although there is no indication on

the specific minimum number of realisations required, the proposed methodology included a

value of ten realisations for each single fracture set. For each model a simulated sampling panel

is then used to determine the associated value. The process is repeated using different

estimated values and typically three values are sufficient to establish the correlation

described in Equations [3.12]. Note that scanline/boreholes and associated values could

also be used in conjunction with Equation [3.13].

32P

21P

32P 32P

10P

- 37 -


Figure 3.8: Schematic representation of the estimation process for a FracMan analysis. The points

A, B and C refer to different average levels of fracture intensity estimated on the sampling plane for

a given simulated .

32P

21P

32P

- 38 -


3.4 A case example - The development of a DFN model for Middleton mine

OMYA, an international white minerals company, provided access on several occasions to their

limestone mine at Middleton in Derbyshire (UK). Middleton mine is a classic square room-and-

pillar mining operation with drift access working mostly under a cover of about 100m. Pillars

are planned for nominal 16m x 16m dimensions in plan with rooms 14m wide. However

completed pillars are usually smaller, due to over-break. Rooms are created in single pass

operations up to 8m high, but double height rooms are created in suitable ground. Figure 3.9

shows a typical view in the mine.

Figure 3.9: Typical view of Middleton Mine, Derbyshire (UK).

3.4.1 Mapping considerations

In a research project considerable effort can be devoted to obtaining the best possible data sets

from mapping and boreholes, in excess of what would be practical in many other projects.

Whether a project is research-based or otherwise, it is important to be efficient in capturing

sufficient data for a meaningful model. The data collected at Middleton mine were obtained

within practical constraints of operating an active mine. Phasing of mining operations, lighting,

ventilation, access to mappable surfaces and supervision aspects of Health and Safety, were all

factors that had an influence on the ultimate data capture.

- 39 -


Discontinuity mapping at Middleton mine was undertaken by the author and Dr. Zara Flynn,

Research Assistant at Camborne School of Mines. The contribution of Dr. Zara Flynn to the

development of the FracMan DFN model for Middleton mine is gratefully acknowledged.

For the current research, the following approach was implemented:

i. Locations were selected for mapping which were considered representative of the rock

mass element to be modelled (i.e. within the same geological / geomechanical domain).

ii. Fracture orientations were measured directly with a suitable compass-clino over a wide

area of the domain to determine if there were any systematic or sudden (fault-induced)

trends.

iii. Mapping was undertaken within at least two windows with orientations approximately

at right angles (to ensure all fracture sets are adequately represented).

iv. Mapping locations were selected preferably where air quality was good enough to

permit flash photography.

v. For determination of fracture persistence, mapping was undertaken at two scales.

Windows of typically 2m high (within direct physical reach) and several metres long

were considered for detailed data. Longer fractures were recorded outside the original

window, and within range of flash photography, including also those that extended

beyond the initial 2m high window. In accordance with Sections 3.3.3 and 3.3.4, type

of fracture terminations were recorded, including one end, both ends, or neither ends

terminating within the window or truncated against other fractures. These data were

then used to determine the persistence trends and possible distributions, and also to

incorporate truncation percentages within the synthesised model.

vi. A minimum cut-off for fracture length during mapping was implemented as a practical

constraint for both the mapping effort and the subsequent geomechanical model run

times. A value of 0.5m was used initially for pillars with dimensions of typically 10 to

30 times this size.

3.4.2 Validation of the Enhanced Baecher DFN model for the Middleton mine case

Scanline surveys were incorporated into the fieldwork methodology in order to substantiate the

use of the Enhanced Baecher model as typical generation mode for the Middleton DFN model.

As discussed in Section 3.3.1 this model corresponds to a Poisson process leading to an

exponential distribution function for the fracture spacing along a sampling line. For a Poisson

process with intensity parameter λ , the spacing between the fractures is exponential and

expressed according to:

- 40 -


( ) xexf λλ −= with 1−= xλ [3.14]

where x is the mean value for the spacing between the fractures. Figure 3.10 shows the results

for Middleton mine: in the data captured the mapped spacing distribution was quite close to

negative exponential, therefore consistent with a model assumption of Poissonian distribution in

3D space. The intensity parameter λ was calculated as the average of the coefficients of the

exponential trendline.

Figure 3.10: (a) Data for Scanline F1 on Panel 1-Level 1 at Middleton mine and (b) test for negative

exponential fracture spacing.

3.4.3 Discontinuity mapping and orientation distributions for the Middleton mine case

Fracture orientation was determined using the code DIPS (Rocscience, 2005). Figure 3.11

shows a lower hemisphere stereoplot of the poles of fractures mapped at Middleton during the

2003 and 2004 site visits.

There are two main sub-vertical, moderately spaced sets of discontinuities: Set 1 strikes roughly

NE-SW and Set 2 NW-SE. Set 3 is a more widely spaced (2m to 4m) set striking West-East

and dipping at about 45° to both North and South. The near horizontal bedding planes are

widely spaced (2m to 4m). From mine wide surveys, the orientations of fractures appear

consistent with only localised effects at faults. Based on mapped tracelengths, Set 1 was

divided into two sub-sets: Set 1a comprises long fractures that extend over the full height of the

face whilst Set 1b comprises shorter fractures (maximum mapped tracelength 4.7m) of a similar,

but more dispersed, orientation. Set 2 was also divided into two sub-sets, based solely on

orientation. Similarly, Set 3 was divided into two sub-sets, based on dip direction of

discontinuity planes.

- 41 -


Set 1a

Set 2b

Set 2a

Set 3a

Set 3b

Set 1b

Figure 3.11: Stereoplot (lower hemisphere) of the fractures mapped at Middleton during the 2003 and

2004 site visits.

Among the orientation distributions currently available within the code FracMan, the Fisher

distribution was found to yield a better fit for all sets. The Fisher distribution is described by

the mean dip and dip-direction for the fracture set, plus a coefficient K (Fisher coefficient),

which is a measure of dispersion, i.e. a higher K value implies more clustered data. K values are

determined using the code DIPS (Rocscience, 2005) based on field data.

Table 3.3 lists the mean dip and dip-direction and coefficient K for the fracture sets identified

for the mapped pillars at Middleton mine and based on Figure 3.11.

Set Dip Dip direction Fisher Coefficient

1a 89 308 41.5

1b 84 323 8.3

2a 86 219 17.2

2b 89 269 28.2

3a 46 193 22.4

3b 44 016 12.6

Table 3.3: Mean dip and dip-direction and Fisher coefficient K for the fracture sets identified for the

pillars mapped at Middleton mine.

- 42 -


3.4.4 Fracture radius distribution for the Middleton mine case

Analytical solutions were used to derive fracture radius distributions from the mapped 2D

tracelengths, according to the method discussed in Section 3.3.4. As the windows in which the

fractures have been measured were finite in size, the mapped tracelength distribution was not

the true tracelength distribution, consequently four main biases were taken into consideration

when deriving true tracelength distributions:

i. Orientation bias: the probability of a fracture appearing in a window depends on the

relative orientations of the fracture and the window. Fractures striking perpendicular to

the window will be truly sampled, whilst all others will be under-represented. To

account for this, at least two 2 perpendicular windows were mapped. In the case of the

rectangular windows mapped at Middleton (14m long x 2m high), it was expected that

fractures striking parallel to long axis could be under-represented (e.g. if their average

spacing was greater than the window height), hence the superimposition of circular

windows within the mapped rectangular panels.

ii. Size (length) bias: longer fractures are more likely to intersect a sampling plane. This is

an important consideration when deriving radius distribution from tracelength

distribution. This is accounted for in the analytical method proposed by Zhang and

Einstein (2000) for determining the true trace length distribution.

iii. Truncation bias: trace lengths below a specific cut-off value are not recorded.

Tracelengths below 0.5m long were not recorded when mapping, therefore, when fitting

distributions, tracelengths less than 0.5m were discarded accordingly. However,

because ignoring smaller tracelengths could reduce the areal intensity , in FracMan

a minimum radius of 0.25m was set when generating fracture sets.

21P

iv. Censoring bias: if the end points of a trace cannot be seen, only a lower bound estimate

of its size can be made. The minimum size of the sampling window should be carefully

chosen to reflect the fracture tracelengths (however, with very long fractures, e.g. Set

1a, this is not always possible). The method of Zhang and Einstein (2000) described in

Section 3.3.4 takes account of censoring.

Figure 3.12 shows the histograms of trace lengths measured for Sets 1a, 1b, 2 and 3. The

fractures from Sets 2a and 2b were initially analysed separately, and appeared to conform to the

same radius distribution. Accordingly, data from Sets 2a and 2b were combined to give a better

distribution description. The same approach was followed for Sets 3a and 3b. While Sets 2 and

3 (combined) showed an essentially lognormal tracelength distribution, as often encountered in

the field, Sets 1a and 1b showed a bimodal distribution due to the relatively high number of

- 43 -


long traces (or low number of short traces) in Set 1a. The geological reason is not known but

this is a true reflection of the field observations.

02468

1012141618

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Tracelength (m)

Freq

uenc

y

Set 1aSet 1b

02468

10121416

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Tracelength (m)

Freq

uenc

y

Set 2 Combined

0123456789

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Tracelength (m)

Freq

uenc

y

Set 3 Combined

Figure 3.12: Histograms of trace lengths measured for each fracture set in two parallel windows that give

the maximum number of data points.

Using the analytical procedure proposed by Zhang and Einstein (2000) and the simplified

approximation method discussed in Section 3.3.4, the following distributions were estimated for

the fractures sets mapped at Middleton mine and used as input in the code FracMan.

- 44 -


Mapped traces length

distribution

( )lg

True trace length

distribution

( )lf

Discontinuity diameter

distribution

( )Dg

Log-Normal Log-Normal Log-Normal

Sets gµ gσ fµ fσ Dµ Dσ

1a 38.9 9.0 32.2 12.0 38.9 9

1b 3.3 0.6 2.6 0.9 3.3 1.6

2a 3.7 1.2 3.2 1.4 3.2 1.4

2b 3.7 1.2 3.2 1.4 3.2 1.4

3a 3.7 1.5 3.4 1.7 3.7 1.5

3b 3.7 1.5 3.4 1.7 3.7 1.5

Table 3.4: Fracture radius distributions for Middleton fracture sets, based on the analytical solution

proposed by Zhang and Einstein (2000) and the simplified approximation method discussed in Section

3.3.4.

3.4.5 Fracture intensity parameters for the Middleton mine case

Fracture intensity parameters for the Middleton mine case were derived based on the discussion

presented in Section 3.3.5. The volumetric intensity parameter (m2/m3) was assessed on a

set-by-set basis using the linear correlation with the areal intensity parameter (m/m2)

expressed by Equation [3.13] and measured for each mapped window. The results are presented

in Table 3.5, where P1, P2, P3 and P4 represent the four mapped faces of the pillar surveyed

during the field visit.

32P

21P

- 45 -


P1 P2 P3 P4 Notes

Mapped 21P 0.00 1.03 0.00 0.82 Use only P2 andP4 (a)

Estimated 21C 4.36 0.85 2.95 0.88 Average 32P St. Dev. 32PSet 1a

Calculated 32P 0.00 0.88 0.00 0.72 0.80 0.11

Mapped 21P 0.10 0.76 0.60 1.12 All panels

Estimated 21C 1.99 1.07 1.66 1.07 Average 32P St. Dev. 32PSet 1b

Calculated 32P 0.20 0.81 1.00 1.20 0.80 0.43

Mapped 21P 1.43 0.11 0.92 0.41 Use only P1, P3 andP4


Calculated 32P 1.37 0.32 0.91 1.17 1.15 0.23



Calculated 32P 0.95 0.62 0.29 0.76 0.66 0.28



Calculated 32P 0.09 0.58 0.11 0.28 0.27 0.23

Mapped 21P 0.00 0.14 0.00 0.34 Use only P2 andP4 (b)


Calculated 32P 0.00 0.19 0.00 0.47 0.33 0.20

Table 3.5: Results for the estimated (volumetric intensity) corresponding to each fracture set

mapped at Middleton mine (Level 1). Note: a) For Set 1a no fractures were measured on P1 and P3; b)

For Set 3b no fractures were measured on P1 and P3.

32P

- 46 -


3.4.6 FracMan DFN model for Middleton mine

To account for boundary effects, the DFN model representative of a pillar at Middleton mine

was generated within a Box Region whose dimensions are 100m x 100m x 100m. For

comparison and model validation purposes, a Pillar Region of equivalent dimension to the

mapped pillar at Middleton mine is selected within the Box Region, preserving a condition of

parallelism between the Box Region and Pillar Region faces (Figure 3.13).

The Box Region in FracMan can only have edges running North-South and East-West, therefore

requiring a rotational transformation for the orientation of the mapped pillar faces. Table 3.6

lists the original and corrected orientations.

Mapped orientations Corrected orientations

Pillar Face Dip Dip Direction Dip Dip Direction

P1 90 306 90 270

P2 84 034 84 358

P3 90 114 90 078

P4 82 035 82 359

Table 3.6: Middleton mine DFN model; original and corrected orientations for the pillar faces in

accordance with the condition of parallelism between Box Region and Pillar Region.

Figure 3.13: Box Region and Pillar region settings for the Middleton DFN model in FracMan.

Based on the discussion presented on the preceding sections, the final FracMan input

parameters for the Middleton model are listed in Table 3.7.

- 47 -


Set Orientation

Distribution

Assumed

Shape Terminations Radius Distribution

Volumetric

Intensity

1a

Fisher

Dip/Dip

direction

89/308

K=41.5

6 sides

polygons 0%

Log-Normal

Dµ =38.9

Dσ =9.0

Min. 2.0m

0.80

1b

Fisher

Dip/Dip

direction

84/323

K=8.3

6 sides

polygons 23%

Log-Normal

Dµ =3.3

Dσ =0.6

Min. 0.25m - Max 2.5m

0.80

2a

Fisher

Dip/Dip

direction

86/219

K=17.2

6 sides

polygons 50%

Log-Normal

Dµ =3.2

Dσ =1.2

Min. 0.25m

1.15

2b

Fisher

Dip/Dip

direction

89/269

K=28.2

6 sides

polygons 31%

Log-Normal

Dµ =3.2

Dσ =1.2

Min. 0.25m

0.66

3a

Fisher

Dip/Dip

direction

46/193

K=22.4

6 sides

polygons 29%

Log-Normal

Dµ =3.7

Dσ =1.5

Min. 0.25m

0.27

3b

Fisher

Dip/Dip

direction

44/016

K=12.6

6 sides

polygons 0%

Log-Normal

Dµ =3.7

Dσ =1.5

Min. 0.25m

0.33

Table 3.7: FracMan input parameters for the Middleton DFN model.

Figure 3.14 shows one realisation of the final 3D DFN model for a Pillar Region representing

one of the pillars at Middleton mine.

- 48 -


Figure 3.14: FracMan 3D model of a Middleton mine pillar; the dimensions of the pillar region are

approximately 14m x 14m x 7m high. Bedding planes were selected and added manually to the DFN

model based on the mapping data from Middleton mine.

3.4.7 Comparison between mapped and simulated traces on pillars

Validation of the DFN model generated in FracMan for one of the pillar at Middleton mine is

achieved by taking 2D slices through the model along planes corresponding to the mapped

windows and comparing the intensity and pattern of the fracture traces (from each set) with

those measured in the equivalent mapped panels.

Since the generation process in FracMan is stochastic, there are infinitely many possible

realisations of the 3D fracture system based on the mapped data. Indeed, the mapping process

is itself random by the nature of how fractures are presented in available windows. However,

except for very explicit modelling of an individual fracture or simplified sets of fractures, which

- 49 -


cannot be wholly representative, the stochastic approach provides the best option for creating

realistic geometric models of fracturing.

Figure 3.15 shows the comparison between mapped and simulated traces for Set 1a. Although

no fractures belonging to Set 1a were mapped for Face 1 and 3 of the Middleton pillar, the

stochastic FracMan model returned very sparse fractures. However, because of their limited

intensity, particularly for Face 3, and considering the good comparison between the simulated

and mapped traces for Set 1a on Face 2 and 4, the overall input data for Set 1a were considered

satisfactory.

Figure 3.15: Comparison between mapped and simulated traces in FracMan for Set 1a; the comparison

refers to the entire pillar faces (dimensions approximately 14m x 7m).

Figures 3.16 to 3.18 graphically illustrate the comparison between the synthesised and mapped

fracture traces on for Set 1b, Set 2a, Set 2b, Set 3a and Set 3b.

- 50 -


Figure 3.16: Comparison between mapped and simulated traces for Set 1b; the comparison refers to the

mapped windows at the base of each pillar face (window dimensions approximately 15m x 2m).

Figure 3.17: Comparison between mapped and simulated traces for Sets 2a and 2b combined; the

comparison refers to the mapped windows at the base of each pillar face (window dimensions

approximately 15m x 2m).

- 51 -


Figure 3.18: Comparison between mapped and simulated traces for Sets 3a and 3b combined; the

comparison refers to the mapped windows at the base of each pillar face (window dimensions

approximately 15m x 2m).

Table 3.8 lists the areal intensity ( ) values for the simulated and mapped windows. 21P

Set Mapped 21P Simulated 21P

Face 1 0.10 0.36

Face 2 0.76 0.98

Face 3 0.60 0.44 1b

Face 4 1.12 0.80

Face 1 2.03 1.60

Face 2 0.66 0.38

Face 3 1.06 2.05

2a and 2b

combined

Face 4 1.07 0.93

Face 1 0.09 0.41

Face 2 0.63 0.31

Face 3 0.12 0.53

3a and 3b

combined

Face 4 0.59 0.16

Table 3.8: Areal intensity ( ) values for the simulated and mapped windows for fracture Sets 1b, 2a

and 2b combined, 3a and 3b combined.

21P

- 52 -


Although the graphical comparison yields encouraging results, for different realisations of the

FracMan model the values obtained on a sample plane corresponding to a pillar face can

range from 0.8 to 2.3 m/m2 for a mapped of 1.5 m/m2. It is assumed that this variability for

is dependent on somewhat inconsistent results from the analyses of the Middleton data,

probably associated with the limited amount of field data available. In an operating mine there

are several practical considerations that can affect data capture; the mapping area in Middleton

was limited to the size of the pillars, with a maximum height of about 8m. However, only the

bottom 2m was physically accessible for detailed mapping. Above this, the extent of fracture

traces had to be inferred from photographs. Orientation data were indirectly measured above

the 2m windows. Often, due to poor lighting and air quality, and parallax issues occurring

during the photomontage, the condition of the photographs was such that only the most

prominent fractures could be conclusively identified. These were usually the longer fractures;

shorter fractures (less than about 4m) could not be determined in the top half of the face.

21P

21P

21P

In spite of these considerations, it is argued that it is sometimes necessary to use a best-estimate

intensity to produce a coherent rock mass model. The DFN model generated in FracMan for

one of the pillars at Middleton mine was ultimately considered as being reasonably adequate for

applications relative to the subsequent geomechanical modelling presented in Chapters 5 and 7.

3.4.8 Transfer of fracture geometry data from FracMan to ELFEN

The methodology for transfer of the fracture geometry data from FracMan to the hybrid

FEM/DEM code ELFEN includes the following essential steps:

i. The fracture geometry data are exported from the FracMan system in files defining

fracture planes within a rock mass on a full 3D basis. These planes may intersect at

arbitrary angles and do not normally traverse the entire region. This information is

imported to a specific solid modelling interface module in ELFEN in which the joints

are represented as lines for 2D and planar surfaces for 3D situations.

ii. Each fracture surface is then assigned specified contact properties. Normally this would

be on the basis of geologically defined sets identified in FracMan.

iii. Fracture entities are first constructed independently as a network, accounting for

intersections of lines or surfaces, including part intersection of surfaces in 3D and the

intersection with material region boundaries.

iv. Once the network has been constructed, this is embedded within the solid model of the

rock mass. Special cases need to be considered when fractures or faults run along

material interfaces or through rock strata with discontinuous properties, etc., to ensure

- 53 -


correct insertion of properties and assignment of operations. This solid model is then

discretised to provide a finite element mesh using automatic mesh generation

techniques employing triangular elements in 2D and tetrahedral for 3D problems.

During this meshing procedure, algorithms are employed to ensure that excessively

small elements are not formed, due to either (a) very closely spaced joints or (b)

termination of a joint in close proximity to another joint. In the former case, the two

joints are merged, whilst in the latter the nodes defining the near interception are

snapped together.

v. In such cases, one of the fractures have to be displaced; this is done preserving the

overall intensity of fractures ( or ) and maintaining the orientation of the

displaced fracture. These adjustments are considered to have negligible impact on the

overall geomechanical behaviour of a system of multiple fractures.

21P 32P

3.5 Summary

This Chapter reviewed the process of synthesis of a 3D DFN model in FracMan, which included

a consideration of the followings concepts:

i. Choice of a specific DFN model.

ii. Fracture orientation

iii. Fracture termination

iv. Fracture shape and size

v. Fracture intensity

A specific case study was considered, showing the applicability of the code FracMan to the

development of a realistic 3D fracture network model for an underground limestone pillar,

based on qualitative and quantitative field data collected at Middleton mine (Derbyshire, UK).

Using an appropriate numerical interface, the synthesised DFN model represents the input data

of 2D trace sections and 3D fracture planes for the subsequent geomechanical modelling as part

of the study on the mechanical behaviour of jointed rock masses.

Although it is relatively easy to generate many realisations of the FracMan DFN model for the

Middleton pillar, under current limitations of computing capacity, only a limited number can be

subsequently used for geomechanical modelling.

A notable result of the DFN synthesis for the Middleton pillar was that few complete blocks

were formed; this is considered to be undoubtedly true in real rock masses and has obvious

- 54 -


implications for geomechanical modelling. It is expected that for such a rock mass to fail under

induced stresses, the processes involved are likely to involve extension of existing fractures or

growth of fractures in the adjacent intact rock, or both. The failure process would not initially

involve the high magnitude of immediate displacement made possible by the presence of

complete blocks.

A module, FracBlock (Golder, 2005), is available within FracMan to determine the formation of

such complete rock blocks; although outside the scope of the current research, it is expected that

further work could see the combined use of FracMan-FracBlock to generate realistic 3D rock

mass models, which could in turn be analysed within dedicated geomechanical models, such as

ELFEN (Rockfield, 2005).

- 55 -

The use of a hybrid FEM/DEM method to model continuum to discontinuum transition

4

The use of a hybrid FEM/DEM method to model

continuum to discontinuum transition

4.1 Introduction

This Chapter introduces a new approach to the numerical modelling of rock mass behaviour,

which includes incorporation of both rock material behaviour and the effects of a pre-existing

fracture network. The hybrid FEM/DEM method used as part of the current research is based

on the proprietary code ELFEN (Rockfield, 2005), and the following sections review its

computational characteristics for the modelling of practical problems, which involve multi-

fracturing and discrete element behaviour. A discussion is presented on the use of the hybrid

code ELFEN to model the continuum/discontinuum transformation typical of rock brittle

failure, showing how the ELFEN fracture generation/insertion capability can be used to

simulate the process of fracture growth, extension and coalescence typical of rock brittle failure,

without assuming any form of degradation of the initial intact material properties. Numerical

examples are given of the applicability of the code ELFEN in terms of modelling the brittle

response of laboratory scale rock specimen, showing in particular the capability of the ELFEN

fracture generator algorithm for Mode I (tensile) fracturing, which can effectively reproduce

fracture propagation in a plane parallel to the maximum principal compressive stress. The

ELFEN data structure used to define the necessary parameters for the numerical analysis is also

introduced, with indication of the most important modelling options that may be combined to

create models for differing classes of problems.

It is noted that the ELFEN pre-processing version 3.7.0 and analysis version 3.3.31 were used as

part of the current research.

- 56 -


4.2 The use of a hybrid approach to model continuous/discontinuous transformations

Chapter 2 reviewed the use of continuum and discontinuum models to simulate multi-fracturing

phenomena and mechanical behaviour of discrete systems. It was stated how neither approach

could alone capture the interaction of existing discontinuities and the creation of new fractures

through fracturing of the intact rock material (Stead et al., 2004; Coggan and Stead, 2005). The

discussion also showed that, as reported by several authors (e.g. Pariseau, 1993; Hart, 2003 and

Owen et al., 2004b), the introduction of a hybrid continuum/discontinuum approach, linking

different numerical techniques such as FEM and DEM, could allow for large scale analysis and

locally large displacements along fracture planes. A hybrid approach could provide a better

description of the physical processes involved, accounting for diverse geometric shapes and

effective handling of large numbers of contact entities with specific interaction laws. The

implementation of specific fracture criteria and propagation mechanisms allows also the

simulation of the progressive fracture process within both the finite and discrete elements.

Consequently there are significant advantages in employing combined FEM/DEM solution

strategies to model discrete/discontinuous systems.

The current research concentrated on the analysis of the failure of brittle rock materials, with a

particular emphasis on the mechanical behaviour of a pre-fractured rock mass. Chapter 2

discussed how the degradation of quasi-brittle materials, represented by pre-peak inelasticity

corresponding to the initiation of micro-cracking and the mobilisation of micro-mechanical

mechanisms, may be invoked constitutively, in an average sense, at scales far in excess of the

micro-mechanical mechanisms involved. The cohesion-weakening/friction-hardening and

degradation index models proposed by several authors (e.g. Hajiabdolmajid et al., 2002 and

Fang and Harrison, 2002) represent constitutive models attempting to recreate the non-linear

macroscopic behaviour typical of brittle rock materials within the context of continuum

numerical analysis. However, to represent brittle fracture of rock in compression, a model

should not only be able to accommodate continuum degradation processes, but also to predict

the simultaneous development of fractures resulting from this micro-scale degradation. For

these reasons, the ability of a finite element methodology to accommodate the continuum to

discrete transition is of paramount importance in the modelling of post-failure interaction of

brittle rock material (Owen et al., 2004a).

The numerical modelling of continuous/discontinuous transformation includes the consideration

of the following key factors (Owen et al., 2004a, b):

i. Development of constitutive models that govern the material failure.

- 57 -


ii. Ability of the proposed numerical approach to introduce discontinuities such as shear

bands and cracks generated during the material failure and fracture process.

iii. Effective simulation of contact between the region boundaries and crack surfaces during

the failure process and particle behaviour motion of fragments in post-failure phases.

To accommodate point ii) above, a numerical methodology should incorporate fracture criteria

and propagation mechanisms within both the finite and discrete elements, which consequently

would require mesh adaptivity procedures for discretisation and introduction of fracture

systems. Point iii) involves the introduction of detection procedures for monitoring contacts

between discrete elements and interaction laws governing the response of contact pairs.

The combined FEM/DEM method presented in this thesis is based on the proprietary code

ELFEN (Rockfield, 2005), which is a hybrid 2D/3D numerical modelling package,

incorporating both Finite Element (FE) and Discrete Element (DE) analysis, and specifically

designed for application to complex non-linear finite element simulations. The code ELFEN

allows the simulation of crack growth, accumulation and coalescence, thus enabling the

investigation of the interaction between newly generated and pre-existing fractures, capturing

the subsequent displacement and/or rotation of independent blocks in a typical DEM fashion.

This research concentrated on the evaluation of the current capabilities of the ELFEN code, in

terms of fracture mechanics and use of defined constitutive models for the intact rock material

and discontinuity planes. Although the attention primarily focused on the 2D analysis of the

mechanical behaviour of pre-fractured mine pillars, the applicability of the hybrid FEM/DEM

code ELFEN was also analysed with respect to simpler laboratory-scale problems. The scope

was not only to investigate the effectiveness of the ELFEN code, but also to study its

applications in terms of rock mechanics problems, analysing the dependence of the modelling

results to specific numerical parameters. As discussed in Section 2.2, the aim was to introduce a

method that could help in representing complex mechanical interactions between blocks within

fractured rock masses, critically discussing its advantages and limitations, and providing a

framework for future work using a similar numerical approach.

4.3 A rock fracture mechanics approach to model rock brittle failure

Section 2.8.1 introduced the concept of rock brittle failure, presenting how, in a modelling

perspective, different approaches can be used to capture the process of brittle failure in rock

masses. The constitutive relationship for quasi-brittle materials in compression is generally

determined by performing conventional triaxial tests, extension tests and true triaxial tests on

- 58 -


laboratory-size rock specimens, making sure that due consideration is given to end-effects,

stiffness of the testing apparatus and loading rate. Figure 4.1 illustrates a typical generalised

quasi-brittle response obtained from conventional triaxial tests (Hallbauer et al., 1973 in Klerck,

2000).

Figure 4.1: Generalised quasi-brittle response obtained from conventional triaxial test (a) Maximum

stress difference vs. axial strain (b) Maximum stress difference vs. volumetric strain (c) Volumetric

strain vs. axial strain (d) Cylindrical specimen and principal stress orientations (after Klerck, 2000).

Zone I in Figure 4.1 corresponds to a non-linear elastic response attributed to the closing of

cracks and pores, a manifestation that disappears with increasing confinement (Scholz, 1968 in

Klerck, 2000); the linear-elastic response of region II is followed in region III by the onset of

pre-peak inelasticity corresponding to the initiation of micro-cracking and the mobilisation of

micromechanical mechanisms, a process that propagates further in region IV. This pre-peak

inelasticity is associated with a steady decrease in the loading modulus and positive dilatancy

and this process ultimately leads to some form of mechanical instability in the post-peak region

V, with the resulting formation of macroscopic failure planes through the coalescence and

complex interaction of micro-cracks.

Determining the degree of fracturing and the subsequent response of a fractured rock mass to

subsequent changes of the stress state is fundamental in any stability analysis; in this context,

Figure 4.2 illustrates the equivalence between laboratory tests and in-situ mining conditions.

- 59 -


Figure 4.2: Equivalence between laboratory experiments and in-situ mining conditions (after Klerck et

al., 2004).

It is known that the final failure planes observed in unconfined compressive strength (UCS)

tests or extension tests are generally parallel to the direction of maximum compressive stress

and similar observations can be made in the mining environment, where the absence of

confinement in rock adjacent to excavations or in mining pillars results in fracturing parallel to

the maximum compression direction (slabbing). In confined regions ahead of mining faces the

rock fails typically in a mechanism similar to that of a conventional triaxial test, with shear

deformation occurring along oblique failure planes making finite angles with the maximum

compressive stress direction (Klerck, 2000 and Klerck et al., 2004). To represent brittle fracture

of rock in compression, a model should therefore be able to predict the simultaneous

development of fractures due to all these different mechanisms.

Because the process of brittle failure for rock in compression is typically associated with the

accumulation and growth of stress-induced fractures growing parallel to the excavation

boundary, many attempts have been made to study the behaviour of fracturing rock through the

use of fracture mechanics theory.

- 60 -


Rock fracture mechanics can give an explanation of rock failure, crack occurrence and

propagation for a wide range of geological and engineering situations (Whittaker et al., 1992).

The concept of rock failure results from the in-situ state of stress being disturbed and hence the

induced state of stress exceeding the rock strength, as defined by a specific failure criterion (e.g.

Mohr-Coulomb and Hoek-Brown). Rock fracture mechanics can be helpful in explaining the

stress intensity at the tips of flaws that are inherently present in rock material, thus providing a

useful insight on the rock structural integrity and its subsequent response to changes in stress

and material properties. Under loading conditions, flaws or micro-cracks in rock materials act

as stress concentrators and as the source of further cracking/fracturing of the rock due to their

coalescence. From the point of view of mechanics, any material or geometrical discontinuities

may promote crack initiation and propagation in compression, therefore the micro-scale of the

problem translates to a macro-scale with the activation and propagation of pre-existing

discontinuities.

Figure 4.3 illustrates the evolution of micro-cracking and the formation of macroscopic failure

planes for a hypothetical cylindrical rock specimen under uniaxial compression, showing how

rock failure can be viewed as a process with distinct deformation stages, which include crack

initiation, crack propagation and coalescence. Extensive theoretical, experimental and

numerical studies on the failure process of intact rock exist and it is generally assumed that

micro-crack initiation starts at 0.3-0.5 times the peak value for uniaxial compressive strength

(Cai et al., 2004).

Figure 4.3: Evolution of micro-cracking and formation of macroscopic failure planes for a hypothetic

cylindrical rock specimen under uniaxial compression.

- 61 -


Although it is widely recognised that rock failure can be considered as a fracture mechanics

problem, since it results from the propagation of one or more cracks (Zhang, 2002), the

investigations of fracture development during rock mass failure in engineering are far from the

direct application of fracture mechanics knowledge (Liu et al., 2000). The same author argued

that there is still insufficient study of the characteristic behaviour of rock mass fracture as

distinguished from intact rock fracture.

The application of fracture mechanics to rock mechanics is essentially based on Irwins

modification and extension of the Griffith theory of fracture, which deals with crack

propagation and give an explanation of the mechanics of fracture. In this context, rock fracture

mechanics refers to the discrete initiation and propagation of cracks/discontinuities that are

common structural features of rock masses. A survey of the concepts, definitions and theory of

rock fracture mechanics can be found, among the others, in Irwin and de Wit (1983), Liu (1983)

Atkinson B.K. (1987) and Whittaker et al. (1992). A literature source on the subject of brittle

fracture in compression can be found in Wang and Shrive (1995). Recent studies on the

investigation of fracture behaviour include Chunlin Li et al. (1998) Liu et al. (2000), Ying Pin

Li et al. (2005).

Figure 4.4 illustrates the three fundamental modes of fracture, which are termed Mode I

(tensile), Mode II (in-plane shear) and Mode III (anti-plane shear) respectively; in problems

concerning crack loading, the most general case of crack tip deformation and stress field can be

described by the superimposition of these three basic models:

i. Mode I: the crack is subjected to a normal tensile stress and the faces separates

symmetrically with respect to the crack front resulting in displacements of the crack

surfaces that are perpendicular to the crack plane.

ii. Mode II: loading subjects the crack to an in-plane shear stress; crack surfaces slide

relative to each other and general displacement of the rock surfaces are in the crack

plane and perpendicular to the crack front.

iii. Mode III: loading subjects the crack to an anti-plane shear stress; as for mode II, crack

surfaces slide pass each other but general displacements are in the crack plane and

parallel to the crack front.

- 62 -


Figure 4.4: Fundamentals modes of fracture. Mode I Tensile, Mode II in-plane shear and Mode III Anti-

plane shear.

For a homogeneous, linear elastic medium, the stresses near the crack tip are proportional to the

distance measured from the crack, the constant of proportionality being named stress intensity

factor, . For plane strain and assuming linear elasticity, crack extension force G can be

defined based on stress intensity factor K, Poisson’s ratio

iK

ν and Young modulus E (Unit

dimensions: MPa m1/2):

( )E

KG I

I

22 1 υ−=

( )E

KG II

II

22 1 υ−= [4.1]

( )E

KG III

IIIυ−

=12

Note that crack extension force or strain energy release rate G for different modes of crack tip

displacement are additive. The extension of a fracture will occur once a critical value of

extension force, named , is reached. Under the condition of linear elasticity and plane strain,

is a measure of fracture toughness and the material is said to obey linear elastic fracture

mechanics (LEFM). Fracture toughness is therefore an important parameter in engineering

applications related to rock failure; values of fracture toughness for different rock types are

included in Whittaker et al. (1992), whilst empirical correlations between Mode I fracture

toughness and tensile strength have been proposed by Gunsallus and Kulhawy (1984), Bhagat

(1985), Whittaker et al. (1992), Bearman (1999) and Zhang (2002).

cG

cG

- 63 -


Whittaker et al. (1992) argued that in rock engineering pure Mode I seldom represents real

conditions and fracturing of rock structures more commonly occurs under mode II and

particularly mixed Mode I-II. However, most of the work on rock fracture mechanics has

concentrated on the simple case of Mode I loading; studies of fracture behaviour under Mode II,

Mode III or mixed Mode I-II loading in rock have received only limited attention (Whittaker et

al., 1992; Chunlin Li et al., 1998; Liu et al, 2000; Al-Shayea, 2005). The code ELFEN used as

part of this research allows fracturing to occur under Mode I (tensile) only; as shown in Section

4.5, however, the ELFEN coupled Rankine/Mohr-Coulomb compressive (fracturing) model is

capable of reproducing the typical progressive strain softening observed for rock specimen

under laboratory uniaxial conditions. Chapter 6 will discuss the effectiveness of such fracturing

model to reproduce a typical shearing mechanism: as discussed by many authors (e.g.

Handanyan, 1990; Pereira and de Freitas, 1993 and Grasselli, 2006), the breakage of joint

asperities during shearing could also be associated to tensile stress, with tensile cracking

developing at the base of asperities during shear deformation.

Klerck (2000) discussed whether the classical theory of linear elastic fracture mechanics

(LEFM) is applicable to the simulation of the quasi-brittle fracture process: citing Bazant and

Planas (1998), he reported how LEFM is a failure theory strictly applicable to homogeneous

brittle materials that exhibit a linear-elastic response prior to failure. Failure occurs with the

violation of an energy criterion and is necessarily associated with the propagation of a pre-

existing sharp crack. The absence of initiation criteria in unflawed homogeneous continua

severely restricts the predictive ability of LEFM in general systems and the dependency on

system specific parameters further restricts application to relatively simple geometries and

loading configurations. In this context, non-linear fracture mechanics (NLFM) is the extension

of LEFM specifically developed out of a need to account for non-linear effects during fracture.

The cohesive crack model was the first significant attempt to describe the non-linear fracture

process in full, although with simplifying assumptions (Dugdale, 1960 and Barenblatt, 1962; in

Klerck, 2000); however, this method still presupposes the existence of cracks and is therefore

not applicable to flawless homogeneous continua. Hillerborg et al. (1976) introduced an

extension of the cohesive crack model to include a crack initiation rule (fictitious crack model),

which was the first NLFM model to describe the complete fracturing process in arbitrary quasi-

brittle systems.

A rotating crack model is specifically implemented in the code ELFEN as part of its capability

of numerically simulating crack generation, extension and coalescence. A characteristic of the

smeared crack models is that they consider a cracked solid as an equivalent anisotropic

continuum with degraded properties in directions normal to crack orientations (Guzina et al.,

1995). More specifically, according to the smeared rotating crack model adopted in ELFEN, a

- 64 -


discrete crack is introduced at a point (Gauss point) in an isotropic, linear-elastic continuum

when the maximum principal stress 1σ surpasses the tensile yield strength tσ of the material.

Definitions and limitations of the smeared rotating crack model are discussed in detail in

William et al. (1987) and Klerck (2000). Further details on the smeared rotating crack model

and his implementation in the code ELFEN are given in Appendix I.

4.4 ELFEN Material constitutive models

In a general context, for an ELFEN analysis, the material properties assigned to the elements are

defined within the material data structure. The rock mass can be represented in ELFEN by a

number of constitutive behaviours, among which are a Rankine rotating crack model and a

coupled Mohr-Coulomb compressive model, in which the Rankine rotating crack model is

complemented with a capped Mohr-Coulomb criterion in order to attain a softening response

coupled to the tensile model. The Mohr-Coulomb compressive model was expressly designed

in ELFEN to model tension/compression stress states.

Klerck (2000) and Klerck et al. (2004) provide detailed descriptions of the Mohr-Coulomb

compressive model, in which fracturing due to dilation is accommodated by introducing an

explicit coupling between the inelastic strain accrued by the Mohr-Coulomb yield surface and

the anisotropic degradation of the mutually orthogonal tensile yield surfaces of the rotating

crack model. The compressive fracture model represents a phenomenological approach in

which micro-mechanical processes are only considered in terms of the average global response.

Local isotropy of strength in compression is justified by assuming uniform material

heterogeneity, while accumulation of inelastic strain and associated degradation of the tensile

strength is necessarily anisotropic and dependent on the loading direction. The yielding

surfaces defining the compressive fracture model are illustrated in Figure 4.5. The solution

procedure has been fully validated for 2D situations, such as borehole breakout problems in

both weak (limestone) and strong (granite) rocks where the failure mechanisms are distinctly

different (Klerck, 2000). Limited applications to 3D problems were examined as part of the

current research and discussed in Chapter 8.

- 65 -


Figure 4.5: The compressive fracture model, the isotropic Mohr-Coulomb yield surface with softening

anisotropic tensile planes (after Klerck, 2000).

It must be highlighted how the ELFEN compressive fracture model is based on the assumption

that quasi-brittle fracture is extensional in nature (Mode I fracture), i.e. any phenomenological

yield surface is divided into regions in which extensional failure can be modelled either directly,

as in the case of tensile stress fields or indirectly, as in the case of compressive stress fields

(Klerck et al., 2004). Figure 4.6 illustrates the dilation response of compressive failure in quasi-

brittle materials and clearly indicates the extensional inelastic strain directions associated with

fracturing.

Figure 4.6: a) compressive loading with confining stress, b) relationship between axial and volumetric

strain and c) compressive failure with associated lateral extensional inelastic strain causing fracture and

dilation (after Klerck et al., 2004).

- 66 -


Chapter 2 described the implementation within continuum models of so-called mobilised

parameters with plastic strain (Hajiabdolmajid et al., 2000 and 2002) and degradation

coefficients (Fang and Harrison, 2002) to numerically simulate the rock brittle response. The

Mohr-Coulomb compressive model with Rankine cut-off in ELFEN has the option to

accommodate mobilised material parameters (cohesion and friction) that realise

hardening/softening with respect to effective plastic strain such that the permissible elastic

domain depends on the current state of inelastic strain as well as the history of evolution (Klerck

et al., 2004).

The propagation of fractures, hence rock mass behaviour in 2D and 3D, is controlled in ELFEN

by stress intensity factors related to the scale of the existing fractures (extension of existing) or

flaws in intact rock (creation of new fractures). It is reasonable to assume that in weaker rocks

the influence of fractures is less important, and this can be accommodated by using lower values

of fracture toughness and tensile strength whilst at the same time recognising the greater

importance of Mohr-Coulomb parameters for the behaviour of the intact rock.

Table 4.1 lists the material parameters that have to be specified in ELFEN when using the

Mohr-Coulomb with Rankine tensile cut-off type material model.

Compulsory data Optional data

Constant parameters specified using:

Elastic_properties

Young’s modulus

Poisson’s ratio

Density

Initial values specified using:

Plastic_properties

Cohesion ci

Friction angle φ i

Dilatancy angle ψ

Tensile strength σt

Fracture energy Gf

Hardening/softening specified using:

Hardening_properties

Effective Plastic strain

Cohesion ci

Friction angle φ i

Dilation angle ψ

Table 4.1: Material parameters that have to be specified in ELFEN when using the Mohr-Coulomb with

Rankine tensile cut-off type material model. Also listed are the optional hardening/softening properties.

- 67 -


In Table 4.1 above, the fracture energy is typically derived from the value of the critical stress

intensity factor, according to:

EK

G Icf

2

= [4.2]

4.5 ELFEN modelling of rock brittle failure

The following numerical examples illustrate the applicability of the code ELFEN to the

modelling of brittle rock failure. In these examples the progressive strain softening typically

observed for rock specimen under uniaxial loading is captured by the specific form of the

coupled Mohr-Coulomb with Rankine cut-off material model. Post-initial yield, the Rankine

rotating crack formulation represents the anisotropic damage evolution by degrading the elastic

modulus in the direction of the major principal stress invariant; furthermore, indirect softening

does also result from the degradation of cohesion ensuring that a compressive normal stress

always exists on the failure shear plane (see also Appendix I). No direct degradation with

plastic strain of both the initial rock cohesion and initial rock friction is implemented in the

models presented as part of the current research. According to the specific constitutive model

used in the analysis, the numerical examples considered in this section reproduce intact rock

failure as a combination of fracture initiation, fracture growth and coalescence. This approach

recognises the importance of discrete fractures inserted within the initially intact media,

anticipating the discussion on the modelling of fractured rock pillars, in which rock mass failure

is modelled by a combination of shear along existing fracture planes and brittle failure resulting

from the accumulation and growth of stress-induced fractures.

The numerical models show also the fundamental characteristics of the ELFEN fracture

generator algorithm for Mode I (tensile) fracturing.

4.5.1 Modelling tensile fracturing in ELFEN

As introduced in Section 4.4, the ELFEN basic Rankine rotating crack numerical formulation,

together with the more advanced compressive fracture model have been rigorously applied and

validated for several problems (Klerck, 2000 and Klerck et al., 2004).

A significant point is the effective capability of the code ELFEN to reproduce fracture

propagation in a plane normal to the minor principal stress 3σ , i.e. parallel to the maximum

- 68 -


principal compressive stress, 1σ (Pure Mode I tensile fracturing). The current ELFEN version

accommodates also the option of considering pre-existing fractures, thus initial numerical

models have also attempted to reproduce the initiation of wing cracks at the tips of relative large

pre-existing cracks (Figure 4.7). Horii and Nemat-Nasser (1985), in order to understand the

fracture process within solids, carried out a series of uniaxial compression tests using resin

plates containing artificial pre-existing cracks. Their experimental results demonstrated that

wing cracks were firstly initiated at the tips of large pre-existing cracks. The wing cracks

propagated to some extent and then stopped at a certain stress level. After that, new wing cracks

were initiated on small pre-existing cracks.

Figure 4.7: Wing-crack initiation process for the 2D case (after Eberhardt et al., 2004).

Numerical tests were carried out in ELFEN using 2D representations of laboratory-scale rock

specimen as shown in Figure 4.8. Rock material properties are listed in Table 4.2, whilst Table

4.3 contains indication of the platen material properties and discrete (contact interactions)

parameters. The selected rock material properties were derived from a combination of

laboratory measurements on Hoptonwood (Middleton) limestone and typical values for similar

limestones from the literature (Stephen, 1987 and Bearman, 1999). All models under

consideration were uniaxially loaded using an applied displacement loading function, which

returns an axial closure of both platens and equivalent to 2% of the model height over a given

simulation time, typically 8 seconds. All models had an equivalent mesh element size of 1mm.

- 69 -


Figure 4.8: Geometric representation of the UCS model used in the analysis, with indication of principal

dimensions and loading directions. (Left) intact model and (Right) model containing a single fracture

oriented at a given angle α with respect to the direction of maximum applied compressive stress.

Rock material properties (Limestone) Unit Value

Unconfined compressive strength, σci MPa 48

Fracture energy, Gf Jm-2 19.47

Tensile strength, σt MPa 3.84

Young’s Modulus, E GPa 27.5

Poisson’s ratio, ν 0.23

Density, ρ kgm-3 2600

Internal cohesion, ci MPa 9

Intact rock material

Internal friction, φi degrees 40

Table 4.2: Rock material properties for the model illustrated in Figure 4.8. Note that ELFEN does not

require the use of the rock unconfined compressive strength as a direct input property, which was listed

here purely for reference purposes.

- 70 -


Platen material properties and discrete (contact) parameters Unit Value

Surface cohesion, cf MPa 0.1

Surface friction, φf degrees 30

Normal stiffness (Normal Penalty ) nP GPa/m 2.7 Rock fracture

Shear stiffness (Shear Penalty ) tP GPa/m 0.27

Young’s Modulus of platen GPa 200

Poisson’s ratio of platen 0.3 Platen properties

Density of platen kgm-3 7860

Rock / platen cohesion MPa 0

Rock / platen friction degrees 3

Rock / platen normal stiffness (Normal penalty ) nP GPa/m 27

Rock-platen

contacts

Rock / platen shear stiffness (Tangential penalty ) tP GPa/m 2.7

Table 4.3: Platen material properties and discrete contact parameters for the model illustrated in Figure

4.8. Rock fracture normal and shear stiffness are taken as being respectively equivalent to the ELFEN

normal and tangential penalties used in the analysis (see Section 4.6.2 for further detail on the penalty

parameters).

Figure 4.9 shows the results for an intact UCS model, with indication of measured axial-strain

and axial stresses, showing also fracture initiation and growth in a direction parallel to the

maximum applied stress. For the same UCS model, Figure 4.10 shows in more details the

process of fracture initiation and wing cracks development at the tips of previously generated

fractures.

0.15% 36MPa (Pre-peak) 0.18% 38MPa (Peak) 0.85% 20MPa (Failed)

Figure 4.9: Intact UCS model; fracture evolution at different stages of the simulation.

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Figure 4.10: Intact UCS model; fracture evolution at different stages of the simulation, showing details of

fracture initiation (a) and wing cracks development at the tips of previously generated fractures (b).

Similar UCS models, but containing a single inclined fracture, yielded the results illustrated in

Figure 4.11, which clearly shows the initiation of wing cracks at the tips of a large pre-existing

crack. Different models considered both varying crack length and crack inclination with respect

to the loading direction. As shown in Figure 4.11, newly generated fracture surfaces develop at

the tips of the pre-existing crack and then grow parallel to the loading direction. For longer and

more critically oriented pre-existing fractures, shearing along the fracture plane is considered

responsible for the observed change in growth direction. For very short and steeply dipping

fractures, the development of wing crack from the tips ultimately returns a mechanical response

typical of axial splitting.

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Pre-inserted

fracture

Figure 4.11: Examples of initiation of wing cracks for models with a pre-existing fracture.

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4.5.2 Modelling the continuum/discontinuum transition of rock brittle failure in ELFEN

The preceding numerical examples discussed the capability of the code ELFEN to reproduce

tensile fracturing within a compressive stress field. The current section presents a discussion on

the potential of the ELFEN hybrid FEM/DEM computational method of effectively reproducing

the process of brittle failure for rock in compression, which is typically associated with the

initiation and growth of stress-induced fractures.

No form of mobilised material parameters (rock cohesion and rock friction) with plastic strain

to realise hardening/softening behaviour was adopted in the analysis. The specific formulation

of the coupled Rankine/Mohr-Coulomb compressive model ultimately yielded the typical strain

softening behaviour associated with the deformation of rock in compression. In this context, it

was assumed that the initiation of discrete fractures within the initially intact rock specimen

accounted for the degradation of the material response to loading.

Extending the scale of the problem from laboratory rock specimen to large pre-fractured rock

masses, it was intended that the proposed modelling approach could reproduce rock mass failure

by combination of shear along existing fracture planes and brittle failure resulting from the

accumulation and growth of stress-induced fractures; ultimately, it is argued this may better

capture the continuum/discontinuum transition typical of jointed rock masses and better reflects

the key role of rock fractures on rock mass behaviour.

A series of simulations was performed in ELFEN using a UCS model similar to that in Figure

4.8 and with material and modelling parameters equivalent to those listed in Tables 4.2 and 4.3

respectively. The axial stress-strain response of the model is illustrated in Figure 4.12. Since

the analysis was carried out in 2D plane strain conditions, the indicated volumetric strain is not

the effective volumetric variations as such, but rather a modelling approximation calculated

considering a unit length in the out-of-plane direction. The roman numbers (II, III, IV, V and

VI) refer to the equivalent zones in Figure 4.1. Figure 4.12 does not contain any reference to

zone I in Figure 4.1, since the current UCS model did not have any pre-inserted flaw whose

closure might have yielded the relative non-linear elastic response. It is noted that, although

numerically feasible, the pre-insertion of very small fractures (length of fracture << specimen

width), is computationally very expensive due to the very fine (uniform) mesh required.

The linear-elastic response of region II is followed at point III by the onset of pre-peak

inelasticity, illustrated by the corresponding initiation of cracks, in a process that propagates

further at point IV. The process ultimately leads to the formation of macroscopic failure planes

through the coalescence and complex interaction of cracks (Points V and VI).

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Figure 4.12: UCS model. (a) Axial strain-stress response; (b) Simulated volumetric strain-axial stress

response; (c) axial strain-simulated volumetric strain response.

These results clearly show the potential of the constitutive material model implemented in

ELFEN and used as part of this research. However, as discussed by Jing (2003) the most

important step in numerical modelling is not running the calculations, but the conceptualisation

of the problem regarding the dominant processes, properties, parameters and their mathematical

presentations. In this context, it is important to evaluate the different modelling options

available, in an attempt to estimate their relations to the modelling results. The following

section presents a review of the fundamental ELFEN modelling options, introducing the

underlying necessity of carefully calibrating the model under consideration and its dependence

on the selected numerical parameters.

4.6 ELFEN - Principal numerical parameters for modelling applications

This research did not include any attempt by the author to further develop the current numerical

structure of the code ELFEN. The emphasis was on the development of correct input models

(e.g. using FracMan to process fracture data), running models suitable for validation against

empirical results for pillars and evaluating also the failure mode(s) in detail. The following

sections review the primary numerical parameters modelling options used as part of the explicit

solution scheme currently implemented in the code ELFEN. A comprehensive review of the

ELFEN solution scheme, including complex mathematical definitions, can be found in Klerck

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(2000), Klerck et al. (2004), Owen et al. (2004a,b) and in the ELFEN user’s manual (Rockfield,

2005). A short technical note reviewing the ELFEN explicit solution scheme is included in

Appendix I.

4.6.1 ELFEN loading data structure

The code ELFEN allows the definition of various sorts of loading functions, describing

particular loading curves, and imported in the neutral file as simple text files. Several loading

cases and time variations may be defined and activated at the same time.

Loading and its variation throughout each analysis phase is defined by three factors:

i. An activation flag to define whether the loading is to be applied (Specified within the

Load-case-control-data structure).

ii. The load type and the reference magnitude of the load.

iii. The load variation (Specified within the Load-curve-data structure).

4.6.2 ELFEN discrete element data structure

The ELFEN discrete data structure defines the fundamental parameters controlling the

interactions between discrete elements (i.e. pre-inserted fractures, discrete blocks, newly

generated fractures). The mechanical contact forces that govern these interactions can be

loosely defined as the forces that are required to prevent matter intersecting. Contact forces are

realised at contacting nodes and are evaluated by considering the relative kinematics of surface

entities. Contacting surface entities are known as contacting couples and generally occur as

node-node, node-facet and facet-facet pairings.

The normal contact constraints are expressed as (Klerck, 2000):

0≥ng , 0≤nσ , 0≥⋅ nn gσ [4.3]

where nσ is the normal contact stress and is the normal scalar gap. Note that in the

ELFEN notation the normal contact stress is negative if compressive. The two complementary

constraints are the kinematics condition of zero penetration and the static condition of

compressive normal stress.

ng

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Contacting surfaces may also be characterised by relative tangential constraints, which are

defined in ELFEN as:

i. Frictionless contact, i.e. there is no tangential force between contacting surfaces. tf

ii. Sticking contact, i.e. there is no relative tangential displacement between contacting

surfaces.

tg

iii. Slipping contact, i.e. a Coulomb friction model is assumed, according to which a critical

tangential force is defined and proportional to the normal force at which the relative

displacement will take place, according to:

nf

nt ff µ= [4.4]

Whereas both frictionless and sticking tangential contact constraints may exist as idealistic

limits of physical material behaviour, slipping contact in the form of a Coulomb friction model

is generally applicable to quasi-brittle materials and adopted in the current numerical analysis.

The enforcement of these constraints is established using a penalty method, based on which

proportionality between the degree of constraint violation and the degree of corrective measure

is assumed. Surface penetration that violates the impenetrability constraint invokes normal

penalty (contact) forces that prompt surface separation. Similarly, tangential penalty forces are

invoked by the relative tangential displacement between contacting surface entities. These

tangential penalty forces are set to zero in the case of frictionless contact or appropriately

relaxed in the case of slipping contact. It is noted that for both explicit and implicit solution

procedures, the penetration should be negligible to recover an accurate response (Klerck, 2000).

The current release version of the ELFEN code used as part of this research employs a node-to-

facet contact algorithm.

In the 2D node-to-facet contact algorithm, the intersection of discretised surfaces is interpreted

as the intersection of surface nodes and surface facets. A contacting node-facet couple consists

of a contacting node that exhibits a penetration (negative gap) with respect to a target facet

plane and has a normal projection that lies within the facet boundaries. The intuitive notion of

contact stiffness reveals the identical result by the insertion at the point of contact of springs of

stiffness nα and tα respectively, as illustrated in Figure 4.13.

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Figure 4.13: The penalty contacting couple in ELFEN as an equivalent spring system (after Klerck,

2000).

In Figure 4.13, nα and tα represent the penalty contacting couple and are numerically defined

in ELFEN as a modulus relating stress and displacement (i.e. each penalty has unit dimensions

of ). Note that this thesis adopts the notation and to indicate the normal and

tangential penalty coefficient respectively. Although Figure 4.13 may suggest a correlation

between penalty coefficients and joint stiffness, the penalty method implemented in ELFEN

corresponds only to a numerical artifice which controls penetration between contacting surfaces;

for these reasons, careful consideration should be given to the use of estimated joint stiffness

values as corresponding penalty parameters. Chapter 5 attempts to correlate the ELFEN normal

penalty coefficient to the simulated normal stiffness of a horizontal joint.

1mPa −⋅ nP tP

For the 3D case, the contact properties are derived from the 2D case and considering the

triangular facets of the tetrahedral elements. Accordingly, the derivation of the normal force

vector is analogous to the derivation for the 2D case, whilst the tangential force vector for

the 3D case has two components, and respectively.

nf

t1f t2f

4.6.2.1 The Mohr-Coulomb contact friction criterion

In the code ELFEN shear phenomena between contacting facets (i.e. fracture planes) are

considered based on the classical Coulomb theory, which predicts that surfaces will remain at

rest (tangentially) until a maximum shear stress is reached. This perfect dry behaviour exhibits

a frictional force proportional to the normal load and independent of the apparent area of contact

and other state variables (Peric and Owen, 1992), expressed in the typical form of:

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φστ tannc += [4.5]

Because, as recognised by Wriggers et al. (1990, in Klerck, 2000), contacting surfaces may

realise reversible tangential micro-displacements within the contact area before the sliding

process starts, the ELFEN numerical formulation allows the introduction of reversible elastic

tangential deformation prior to the inelastic slipping. The decomposition of tangential

displacement into reversible elastic and irreversible inelastic components permits constitutive

equations for contact with friction to take the simple format of the classical theory of

elastoplasticity.

Chapter 6 presents the results of the numerical simulation of simple shear tests using the code

ELFEN, showing how the use of Equation [4.5], coupled with the ELFEN fracture generation

capability, allows the capture of a more typical non-linear behaviour for rock joints, similar to

that proposed by Barton (1971). However it is noted that the current release of the ELFEN code

does not include the option of using an explicit Barton model. The mechanical modelling of a

fractured rock mass using the code ELFEN could in principle incorporate different shear

strength properties for different fracture planes and, with some limitations, the code could also

consider non-planar fracture surfaces, which are geometrically defined as a multi-line.

4.6.2.2 Contact damping for contacting surfaces

The introduction of displacement damping into the ELFEN explicit integration algorithm is

unsuitable for contact interactions as it affects rigid body motion even in the absence of contact

and is ineffective at high frequencies (Klerck, 2000). The purpose of the contact damping

algorithm implemented in the code ELFEN is to minimize high frequency oscillations which

may be introduced when contacting surfaces frequently alternates between contact and non-

contact states. The contact damping applies to contacting surfaces only; as discussed by

Munjiza (2004), the physical interpretation of such damping is to allow for energy dissipation

due to friction or plastic straining of contacting surfaces. Within the ELFEN code, the contact

damping acts as a modifier of the normal penalty coefficient , which is accordingly altered

using a specific damping factor that may be increased or decreased depending on the velocities

(vectors) of the contacting bodies defined in the analysis. The effect of contact damping

increases with increasing magnitude of the damping factor.

nP

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4.6.2.3 General limitations of the node-to-facet contact and penalty method

The node-to-facet contact couple requires the association of an equivalent surface area with

each contacting node; this association is part of the finite element discretisation process. As

recognised by Klerck (2000), these equivalent areas are used as approximations to the true

contact areas. As a result, poor contact area approximations would result in spurious contact

stresses when utilising stress based penalty coefficients.

These limitations could be overcome by the implementation of a penalty method for facet-to-

facet contacting couples, which naturally introduces the true contact area; Klerck (2000)

described this facet-to-facet method in his PhD thesis. Although the facet-to-facet contact

algorithm is not included in the current release version of ELFEN (Rockfield, 2005, personal

communication), it is available as a built-in experimental option and only limited tests were

performed as part of this research. The reader should assume that the release version of the

node-to-facet contacting algorithm was used in the analysis if not mentioned otherwise.

As discussed in more details in Chapter 5, the determination of appropriate constant penalty

coefficients can be a complex undertaking, which for large systems can be expensive in terms of

time and computational resources. Additionally, if a compressive contact system with constant

penalty coefficient is subject to an indefinite increase in applied loading (e.g. quasi-brittle

material specimens that are loaded to failure utilising applied displacement loading) there will

necessarily be a stress level achieved at which the penetration of surfaces is of an unacceptable

magnitude (Klerck, 2000). The same author proposed an explicit Lagrangian method for the

determination of the contact forces. This method considered the coupling between contact

constraints and required a minimum of user intervention by removing the need for penalty

definitions and contact damping. However, this experimental method was not available in the

current release version of ELFEN (Rockfield, personal communication, 2005).

It is apparent that the selection of specific penalty coefficients will ultimately influence the

overall deformation of the system, in analogy with the deformability of a rock mass being

dependent on the normal and shear stiffness values for the rock fractures. The current research

investigated the effects of varying normal and tangential penalties for a specific problem.

Simple pillar models intersected by a single set of horizontal (through-going) fractures were

analysed to verify the equivalence between the selected normal penalty coefficient and the

simulated normal stiffness for the rock joints; the results are included in Chapter 5.

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4.6.2.4 The ELFEN data structure for contact properties

Within the ELFEN neutral file for a discrete explicit analysis, the Contact-surface-properties

data structure defines the contact surface properties of the discrete bodies. The ELFEN user’s

manual (Rockfield, 2005) suggests the following relations for the definition of normal and

tangent penalty for a given problem:

i. Normal penalty value for the evaluation of the normal contact force: the value is

normally taken (in magnitude) in the range

nP

2EP0.5E n << , where E is the intact

material Young’s Modulus.

ii. Tangential penalty value for the evaluation of the tangential contact force: usually

taken as an order of magnitude less than , i.e.

tP

nP nt PP101

≅ .

These relationships have a general validity, which applies to all explicit modelling in ELFEN,

and they are not specific to modelling of brittle failure in compression. As anticipated in the

previous section, the current research attempted to verify these guidelines and to extend this

correlation to the normal stiffness of a joint surface.

Whereas the proprietary character of the ELFEN code made it difficult to obtain further details,

it is interesting to mention the physical interpretation of the normal penalty parameter given by

Munjiza (2004) for a penalty method for a general combined Finite-Discrete Element Method;

likewise, the previous author stated how, in order to limit penetration, it is enough to select a

penalty term to be proportional to the modulus of elasticity, though no exact figures are given.

Table 4.4 lists the parameters required as part of the Contact-surface-properties data structure to

define the contact surface properties of the discrete bodies/fracture planes. Pre-existing

fractures could in theory be assigned different properties, whilst newly generated discrete cracks

are automatically given properties equivalent to the selected discrete global properties.

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Discrete global properties Surface type properties (if different from global

properties)

Contact damping coefficient

Normal penalty coefficient Pn

Tangential penalty Pt

Cohesion fc

Friction fφ

Normal penalty coefficient Pn

Tangential penalty Pt

Cohesion fc

Friction fφ

Table 4.4: Material parameters which are required as part of the Contact-surface-properties data

structure to define the contact surface properties of the discrete bodies/fracture planes.

4.6.3 The use of displacement damping in ELFEN

Displacement damping can be implemented in the ELFEN analysis in order to dissipate

vibrational energy and to allow the system to converge at a steady state. A damping coefficient

in the x, y and rotation-z directions (specifically for a 2D analysis) can be applied to each

defined block surface. This type of damping applies velocity proportional damping to all nodes,

and the value specified is relative to the automatically estimated lowest frequency of vibration

for the application (Rockfield, 2005, personal communication). As discussed by Cundall

(1987), the use of velocity proportional damping introduces body forces that in some cases may

influence the mode of failure. Within this context, the current research evaluated the effects of

varying displacement damping on the fracture generation process in ELFEN, in order to verify

if the use of relative high damping coefficients could somehow inhibit both the extension of

existing fractures and the initiation of new fractures. The results are presented in Chapter 5.

4.6.4 The ELFEN critical time step concept

The ELFEN solution scheme for discrete analysis (i.e. analysis which involves fracturing of the

intact material) requires the definition of a critical time step that is imposed in the explicit time-

integration procedure. The critical time step in ELFEN is expressed as (Rockfield, 2005):

cl

=rc∆t where ρEc ≈ [4.6]

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where is the length of the smallest mesh element, is the wave speed (highest frequency of

the system),

l c

E is the material intact Young’s modulus and ρ the material density. However,

ELFEN automatically modifies this formula to take account of different element shapes, and the

fact that elements become distorted as the analysis progresses. The critical time step is usually

small, which results in the requirement that a very large number (e.g. millions) of time

increments have to be performed. According to Klerck (2000), the time step stability limit does

not ensure an acceptably accurate solution and a considerably small time step is typically

required.

It can be observed that the critical time step in Equation [4.6] is dependent on the minimum

mesh element size used in the analysis, thus a simulation would be respectively faster or slower

depending on the discretisation of the problem by using a larger or smaller mesh element size.

However, the choice of a specific mesh size has to be assumed taking into consideration the

scale of the problem under examination. For simulations which include fracturing, there is the

option to specify a given smallest mesh element , representing a minimum threshold for

fracturing elements. If the smallest mesh element is set as less than the minimum mesh size

used in the analysis, then fracturing will occur through elements; however, this will

consequently require a much lower time step. In the current research, the smallest mesh

element was limited to a specific fraction of the minimum finite mesh size used to discretise

the geometry under consideration; typically was taken as half of the minimum mesh size.

l

l

l

The computationally time step at time is further defined according as (Rockfield, 2005): nt

crn tft ∆=∆ [4.7]

where is a factor in the range f 0.10 << f . In order to ensure that simulation remains

stable, the response of the solution may dictate the use of a low coefficient . All the

numerical examples described in this thesis had a factor in the range 0.1 to 0.75. The choice

of a greater factor would certainly contribute to limit the time required to run a specific

model, but this may have detrimental effects on the convergence of the explicit solution.

f

f

f

4.7 Summary and discussion

This Chapter introduced the computational characteristics of a combined FEM/DEM approach

for modelling the mechanical behaviour of jointed rock masses. The proposed approach is

based on the use of the code ELFEN, whose mathematical definitions were introduced prior to

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presenting some numerical examples illustrating the functionality of the numerical code. The

Mohr-Coulomb compressive model with Rankine cut-off was then introduced as the primary

material constitutive model used in the analysis.

Numerical tests simulating rock specimen under 2D plane strain conditions showed how

ELFEN could simulate the continuum to discontinuum transition typical of brittle rock. Models

with a critically aligned pre-existing fracture, showed that initiation and development of wing

cracks was in accordance with the expected theoretical behaviour.

The numerical examples reproduced intact rock failure as a combination of fracture initiation,

fracture growth and coalescence. It was argued that this approach could better recognise the

importance of discrete fractures inserted within the initially intact media, anticipating the

discussion on the modelling of fractured rock pillars presented in the forthcoming sections, in

which rock mass failure is to be modelled by a combination of shear along existing fracture

planes and brittle failure resulting from the accumulation and growth of stress-induced fractures.

A description was also given of several ELFEN fundamental modelling parameters, including

loading functions, penalty method for contact properties and displacement damping.

As discussed in more detail in Chapter 5, these modelling options have a fundamental

importance. Due to its inherent complexity, the ELFEN code requires a careful calibration of

any model under consideration. This can only be achieved by performing a series of numerical

tests to evaluate the effects of the aforementioned parameters. For analysis involving fractured

rock masses, it was found that one of the major problems was to estimate the correct normal

penalty coefficient for the contacting surfaces, including pre-existing fractures. The mechanical

response was found to be clearly dependent on the selected normal penalty coefficient. In

addition, the use displacement damping apparently limits the relative displacement of

discretised blocks and the overall fracturing evolution; however, the use of displacement

damping is required in ELFEN at least in the initial stages of the modelling to dissipate

vibrational energy and to allow the system to converge at a steady state. Further inconsistencies

may derive from the use of different loading functions: the specified ELFEN time variable does

not correspond to real time, and thus it is difficult to define a correct loading rate.

The scope of the following Chapter is to address these potential inconsistencies and limitations,

and to describe their effects on the modelling results. Whereas a research project may

accommodate the undertaking of a large number of numerical models, this may not always be

the case for direct engineering applications. In this context, the current research attempts to

provide some form of guidelines for further analysis using the ELFEN code.

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Initial ELFEN modelling of fractured pillars and dependence of modelling results to numerical parameters

5

Initial ELFEN modelling of fractured pillars and

dependence of modelling results to numerical

parameters

5.1 Introduction

The preceding Chapter presented numerical examples illustrating the effectiveness of the crack

model implemented as part of the hybrid FEM/DEM code ELFEN (Rockfield, 2005) to model

the mechanical response typical of brittle rock failure. The numerical examples were followed

by an introduction to the specific ELFEN data structure, which defines the primary numerical

parameters used in the analysis.

In order to validate the recent ELFEN release version used as part of this research and assess its

suitability for modelling of rock behaviour it was important to devise a series of tests to

investigate the sensitivity of specific numerical parameters, particularly those associated with

the definition of the properties for newly generated and pre-inserted fracture planes. This

Chapter describes the methodology that was adopted to investigate the effects of various input

parameters on model behaviour using a series of numerical tests on a 2D model of a pre-

fractured rock pillar. The results of the sensitivity analysis were used to provide a modelling

strategy for future ELFEN applications in the field of rock-engineering.

5.2 Development of a strategy for a sensitivity analysis in ELFEN

Section 4.6 reviewed the principal numerical modelling parameters which are typically used in

an ELFEN analysis for rock mechanics applications. Whereas the ELFEN compressive fracture

model had been rigorously validated for several problems (e.g. Klerck, 2000 and Klerck et al.,

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2004) and successfully applied in Section 4.5.2 to model the typical mechanical response

associated with brittle rock failure, a certain degree of uncertainty is associated with the

definition of several numerical parameters which are required in the analysis. It is also noted

how the code ELFEN was initially developed for mechanical engineering applications, hence

the modelling of specific rock mechanics parameters (e.g. rock fracture contact properties) has

to be specified by means of general modelling options.

The ELFEN neutral file contains the modelling data associated with the application. The use of

the ELFEN code requires a specific knowledge of its neutral file data structure, since some

modelling options may have to be included in the form of add-ins to the automatically generated

data file. The reader is referred to the ELFEN user’s manual (Rockfield, 2005) for a more

detailed description. In simple terms, the neutral file contains several elements, which can be

grouped as follows:

i. System data

ii. Operation Assignment and Element Definition data

iii. Nodal data

iv. Material models

v. Constraint and boundary conditions

vi. Loading

vii. Discrete element data

Whilst it is reasonable to assume that points (i), (ii) (iii) do not have any influence on the final

modelling results, since they typically contain information on the geometrical arrangement of

the problem, the definition of points (v) to (viii) has to be carefully considered; these include the

description of the contact properties (e.g. properties assigned to the fracture planes), loading and

displacement damping. Point (iv) includes the selection of a given material model, which in this

specific case was the Mohr-Coulomb compressive model with Rankine tensile cut-off.

As part of the current research a sensitivity analysis was carried out on modelling parameters

defined in points (v) to (vii) above. The flowchart shown in Figure 5.1 illustrates the

methodology implemented in the current research to develop an adequate sensitivity analysis,

with the scope of limiting the required number of simulated models, hence reducing the

computational time required.

As a way to verify its effectiveness, the modelling strategy derived based on the results of the

sensitivity analysis on numerical parameters was then applied to models with varying material

properties (and in particular cohesion and angle of friction for the fracture planes).

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Figure 5.1: Flowchart showing the methodology implemented to develop the sensitivity analysis for

various ELFEN numerical parameters.

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5.3 Combined effects of displacement damping and loading rates on ELFEN analysis of

pillar behaviour

This analysis primarily concentrated on the evaluation of both the effects of displacement

damping and loading rates on the mechanical behaviour of a simulated 2D pre-fractured pillar.

A number of models with increasing applied displacement damping were also considered in the

analysis to examine the effects of varying loading rates.

As introduced in Sections 4.6.3, a displacement damping coefficient in the x, y and rotation-z

directions (specifically for a 2D analysis) can be applied to each defined block surface. This

type of damping applies velocity proportional damping to all nodes, and the value specified is

relative to the automatically estimated lowest frequency of vibration for the application

(Rockfield, 2005, personal communication).

Different loading rates can be implemented in the analysis by simply selecting different loading

functions, as discussed in Section 4.6.1. However, the time variable specified in the loading

function in ELFEN has to be intended as computational time, hence different from real time (in

the following pages the term effective time is used as a synonymous of real time). For instance,

by selecting a loading time of one second in a given ELFEN analysis will not necessarily result

in the analysis to be completed in one second (real time). The effective time required to

complete the analysis depends on the specified critical time step; for a given critical time step,

the effective time to complete the analysis is also dependent on the number of discrete elements

pre-inserted in the model and on the amount of fracturing occurring in the model throughout the

simulation. The ELFEN critical time step is expressed by Equation [4.6] in Section 4.6.4;

because ELFEN automatically modifies this formula to take account of different element shapes

and elements maybe being distorted as the analysis progresses, simulations having equivalent

loading functions but different complexity of elements discretisation may require different

effective times for completion.

For these reasons, it was found that it was not possible to assume a time variable in ELFEN

equivalent to real time loading conditions. Furthermore, the code seems to lack clear guidelines

in terms of selection of appropriate time variables for a given analysis. Whereas it might be

possible to calibrate a laboratory test against an ELFEN analysis to get an approximate

correlation between ELFEN time and real time, due to the influence of other modelling

parameters (such as damping and penalty coefficients) this correlation would then apply only to

the specific model under consideration.

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Based on these remarks, and in order to limit the required computational run-timea, relative fast

simulations (in terms of ELFEN loading time expressed in the analysis) had to be incorporated

to replicate pillar loading conditions. It was argued that faster loading functions might have

introduced undesired dynamic effects between contacting elements (platen-rock interaction and

pre-inserted fractures); hence the introduction of a specific velocity proportional damping to

limit these potential effects. Since no precise guidelines were available to establish the precise

amount of damping required, the analysis ultimately attempted to derive some form of

directives to be used also in subsequent ELFEN pillar models.

It is noted that a specific example (WPSD007, Rockfield, 2005) within the ELFEN user’s

manual quoted a displacement damping of 0.03 for the simulation of mechanical behaviour of

brittle rock under uniaxial compression, though no numerical justification was given. It was

assumed that this value was somehow linked to the material parameters described in the

example, and not a unique design value.

5.3.1 Model Set-up

The model consisted of a pre-fractured rock pillar, which was loaded as if it were subjected to

uniaxial laboratory loading conditions. In the model shown in Figure 5.2, the ends of the pillars

are considered as a continuous part of the floor and roof material with no jointing, with steel

equivalent platens producing a uniform lateral constraint. The presence of a layer of rock

between the steel platen and the rock material itself is considered to have had negligible effect

on the overall pillar behaviour, whilst at the same time limiting observed numerical instability

due to the contacting interaction between fractures and steel platen. It was observed that

excessive stress concentration could occur at fracture tips terminating against the rock-platen

facet boundary, the fractures being not allowed to extend in the steel platen by the geometrical

definition of the problem.

Considering the significance of in-situ tests on large-scale rock specimen, Bieniawski (1975)

argued that the constraining effects produced by the roof and floor on a coal pillar could be

simulated by the introduction of wood/steel shuttering on the top and bottom of the specimen.

Similar observations were also made by Babcock (1968), who discussed how experimental

models of pillars with their ends constrained by steel rings were better for predicting the

strength of mine pillars than either cylindrical or prismatic specimens. It is noted that actual a A standard PC with a Pentium IV 2.8 GHz and 512MB RAM memory was used to carry out these series

of tests. An 8 seconds simulation time for a pre-fractured slender pillar as in Figure 5.1 with a minimum

element size of 0.2m for the mesh, required in excess of 12 hours to be completed.

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mine pillars are an integral part of a structure and hence confined at both ends. The same

geometrical arrangement as shown in Figure 5.2 was implemented, with some minor

modifications to the zone of confined rock, in all pillar models considered in the current

research.

A specific realisation of the Middleton mine DFN model presented in Chapter 3 was

incorporated in the ELFEN model to represent a particular jointed geometry. The fracture

geometry data from the Middleton DFN model were exported from the FracMan code in files

defining fracture planes within a rock mass on a full 3D basis. This information was then

imported to a specific solid modelling interface module in ELFEN in which the joints are

represented as lines for the 2D case (planar surfaces for 3D situations).

Figure 5.2: ELFEN model for displacement damping analysis. (a) typical dimensions, (b) x-y model

constraints and (c) typical mesh definition.

Minimum mesh element size was taken as 0.2m for a 7m high pillar model. The top and bottom

platens were modelled using a greater element size of 0.4m.

An initial state of stress was initiated within the mesh by means of gravity loading applied in the

initial stage of the analysis. Alternatively, ELFEN allows initiating the model by specifying a

given state of stress in both x and directions (for a 2D analysis); this option requires the user

to manually modify the ELFEN neutral file data structure.

y

- 90 -


The analysis was carried out using a single-stage loading phase, through which the load was

applied as an applied-displacement to the top and bottom platens according to a given loading

function, as shown in Figure 5.3. A full applied-displacement of 0.14m (i.e. 2% of pillar height,

taken as ±0.07m) was set to have occurred by the end of the simulation time. Additionally,

gravity loading was instantaneously applied at time 0.

Figure 5.3: Different loading functions used in the analysis (e.g. for the 1sec curve, the maximum platens

closure of ±0.07m occurs in 1 second time). Note that the indicated loading rates are calculated based on

the selected ELFEN time variable; hence they do not represent real time loading rates.

It is noted that a pillar in a typical room-and-pillar mine is normally created by sets of

excavation stages, which would imply that a pillar is not immediately loaded in its entirety.

However, some specific mining operations stages might lead to a rapid increase in load/stress

due to the removal of adjacent rock support by blasting. In the same context, ISRM (1981)

guidelines for uniaxial laboratory tests quote values between for loading

rates (i.e. time to obtain peak strength of approximately 5 to 10 minutes). Taking into

consideration the non equivalence between real time and the loading time specified in the

ELFEN analysis, it was assumed that non feasible computing run-times would have been

required in order to replicate the loading rates cited in ISRM (1981). Faster loading functions

(1, 2, 4 and 8 seconds, corresponding to ELFEN loading rates of 2x10-2, 1x10-2, 5x10-3 and

25x10-4 sec-1 respectively) were therefore implemented in the current analysis.

1-45 sec10to10 −−

Loading rates were calculated according to:

Time height Pillar

closure pillar Maxrate Loading .= [5.1]

- 91 -


Rock properties for the intact rock used in the ELFEN model are shown in Table 5.1, whilst

Table 5.2 contains indication of the platen material properties and discrete (contact interactions)

parameters.

Displacement Damping Model - Rock Material properties Unit Value







Internal cohesion, ci MPa 7.0

Intact rock

material


Table 5.1: ELFEN displacement damping model; material properties for the intact rock material and rock

fractures. The unconfined compressive strength is listed here purely for reference purposes.

Platen material properties and discrete (contact) parameters Unit Value



Normal stiffness (Normal Penalty ) nP GPa/m 20 Rock fracture

Shear stiffness (Shear Penalty ) tP GPa/m 2


Poisson’s ratio of platen 0.3 Platen properties





Rock-platen

contacts

Rock / platen shear stiffness (Tangential penalty ) tP GPa/m 2

Table 5.2: ELFEN displacement damping model; platen material properties and discrete contact

parameters for the models illustrated in Figure 5.2. Rock fracture normal and shear stiffness are taken as

being equivalent to the ELFEN normal and tangential penalty respectively.

- 92 -


5.3.2 Analysis of results and discussion

As discussed in more details in Section 7.4.3, the estimated strength for a pillar with width to

height ratio of 0.4 and material properties as listed in Table 5.2 (and using an equivalent

RMR/GSI of 70 to characterise the pre-inserted fracture geometry), would be in the range

[8MPa, 12MPa], depending on the empirical method used in the analysis. In a qualitative sense,

Figure 5.4 shows an example of the expected stress-strain response for a slender rock pillar

based on observations by Pritchard and Hedley (1993).

Figure 5.4: Typical stress strain behaviour for a slender rock pillar, based on observations by Pritchard

and Hedley (1993).

The simulated axial stress-strain curves for the models under consideration are shown in Figure

5.5 for the models with respectively 1sec, 4sec and 8sec loading time. Axial stress values for

the simulated pillars were measured as the average stress of a given number of nodes across the

pillar section at mid-height. Axial strains were measured from the closure of the whole pillar.

Also indicated is the calculated range of pillar strength for a typical rock pillar with properties

equivalent to those listed in Table 5.2. Avi movie files showing the fracture pattern

development under increased loading conditions are included in Slide 1 in the attached DVD.

- 93 -


Figure 5.5: Simulated axial strain-axial stress curves for the ELFEN models corresponding to 1, 4 and 8

seconds loading time respectively, with indication of the associated displacement damping used in the

analysis (e.g. D 0.00 refers to zero displacement damping).

- 94 -


Cai and Kaiser (2004), discussing the results of Brazilian tests simulated in ELFEN, reported

that the non-smoothness of the simulated stress-strain curve might represent the influence of

loading rate on the solution; smoother curves could be obtained for a smaller loading rate, but at

the cost of a longer computation time. Indeed, a sharp oscillating behaviour in the stress-strain

curves was observed for the ELFEN models run at the relative higher loading rate of 2x10-2 sec-1

(1sec ELFEN loading time).

With the exception of specific combinations of loading rate and displacement damping (e.g.,

1sec-0.0 Damping and 8sec-0.01 Damping), the stress-strain curves represented in Figure 5.5

above do not yielded comparable values of pillar strength and in addition did not present a clear

pre-peak and post peak section, as it would be expected by comparison with Figure 5.4.

Figure 5.6 illustrates the observed behaviour in terms of estimated maximum axial stress for

different loading rates and corresponding to various displacement damping values. Referring to

Figure 5.6(a), it can be observed how the simulated model with no applied damping, when

loaded at higher loading rates, failed at higher values of peak stress. Although the ELFEN

models do refer to simulated rock pillars, the observed behaviour is similar to experimental

results for rock specimens loaded under uniaxial laboratory conditions reported by several

authors (e.g. Hoek and Brown, 1980: Blanton, 1981). Blanton (1981) reported a constant rate of

increase in compressive strength with increasing loading rates (log-scale) over a range of

(real time), though the same author argued how significant changes in this

trend could be found in the literature, describing three main type of behaviour:

134 sec10to10 −−

i. Constant strength or constant rate of increase in strength (For increasing loading rates)

ii. Sudden increase in strength above a certain loading rate.

iii. Apparent fluctuations in strength above a certain loading rate.

Whilst Figure 5.6(a) is in good agreement with the behaviour described above in (i), conversely

Figures 5.6(b, c and d) show a sudden decrease in the estimated peak stress for higher loading

rates, although the recorded peak stress values for the simulations run at 2x10-2 sec-1 (i.e. 1

second simulation time), independently of the chosen damping factor, were all comparable.

This behaviour suggested that the selected displacement damping used in the analysis had a

greater effect for relatively slower simulations.

- 95 -


Figures 5.6: Observed behaviour in terms of estimated maximum axial stress for different loading rates.

(a) 0.0, (b) 0.01, (c) 0.03 and (d) 0.1 displacement damping respectively.

Figure 5.7 shows the observed relationship between the estimated peak stress value and the

displacement damping factor for different combinations of loading rates. It is noted how faster

simulations (2x10-2 sec-1 rate) gave an almost linear increase in the measured pillar strength with

damping, whilst the simulations run at 5x10-3 and 25x10-4 sec-1 loading rates respectively were

characterised by a sharp increase in pillar strength with a given damping factor (a power-law

correlation was estimated to best-fit the associated curves), with the simulations run at the

intermediate rate of 5x10-3 seemingly being more influenced by the selected damping factor.

Figure 5.7: Effects of displacement damping and ELFEN loading rates for pillar models corresponding to

loading times of 1, 4 and 8 seconds respectively.

- 96 -


Figure 5.8 shows how relative higher ELFEN loading rates corresponded, for a given

displacement damping coefficient, to a greater degree of fracturing of the pillar models at peak

(or near peak).

1sec simulation 4sec simulation 8sec simulation

0.0

Dam

ping

7.84MPa 0.178 % 3.18MPa 0.079 % 2.54MPa 0.059 %

0.01

Dam

ping

4.96MPa 0.127 % 21.30MPa 1.282 % 11.30MPa 0.268 %

Increasing loading rate

Increasing degree of fracturing

Figures 5.8: Fracturing development at peak stress or near peak stress for the models with 0 and 0.01,

0.03 displacement damping and for loading functions of 1, 4 and 8 seconds.

As loading rates changes, it is reasonable to assume that the mechanical properties and fracture

behaviour of a rock mass can change significantly. Fracture growth in brittle rocks can occur at

velocities ranging over many orders of magnitude due to the change of loading rate, which

results in different fracture patterns and geometries (Wu and Pollard, 1993). Within this

context, the ELFEN output history processor was used to obtain variations of kinetic energy

- 97 -


with time, which were subsequently recalculated as a function of the measured axial strain

(Figure 5.9).

Figure 5.9: Kinetic energy variation with measured axial strain of the pillar. Models with (a) 0.0, (b)

0.01, (c) 0.03 and (d) 0.1displacement damping respectively. Note that the symbol-key is the same for all

figures.

By assuming that the basic kinetic energy level associated with the relative movement of the

platen should be constant (for a given loading function), the observed variations in kinetic

energy with axial strain were therefore related to the process of fracture initiation, growth and

extension, representing also an indication of the speed at which this process was occurring.

Figure 5.9(a) clearly shows how in the models with no applied displacement damping,

fracturing occurs very quickly and in an uncontrolled manner. The combined observation of

Figures 5.8 and 5.9 shows how the selected loading rate of 1 second was characterised by an

overall uncontrolled fracturing process, independently of the selected applied displacement

damping. Increasing the value of applied displacement damping yielded a more stable response

- 98 -


in terms of kinetic energy levels, particularly for loading rates of 4 and 8 seconds respectively.

However, as shown in Figure 5.8, no actual release of discrete blocks was yet observed.

Figure 5.10 illustrates how the process of fracture propagation in the current model is clearly

dependent on the damping factor, with simulations run over a time of 4 and 8 seconds showing

a significant inhibition of fracture extension and coalescence with increasing damping factor.

Faster simulations (1 second) seemed not to be greatly affected by the variation in damping

factor, in accordance with the results presented in Figure 5.7. The fracturing outputs shown in

Figure 5.10 refer to equivalent axial strain percentages.

1sec simulation 4sec simulation 8sec simulation

0.0

Dam

ping

4.24MPa 0.12 % 0.10MPa 0.13 % 2.54MPa 0.12 %

0.1

Dam

ping

5.44MPa 0.12 % 6.52MPa 0.13 % 7.57MPa 0.12 %

Incr

easi

ng d

ispl

acem

ent d

ampi

ng

Dec

reas

ing

degr

ee o

f fra

ctur

ing

Figure 5.10: Effects of displacement damping and different loading conditions on the fracture pattern and

geometry for the initial jointed pillar model.

In more general terms, the observed mechanical response to loading was not considered

realistic. Although extension and interaction of existing fractures was indeed recorded, the

absence of any significant relative block displacement, particularly for some of the pre-existing

- 99 -


blocks, was considered a major limitation of the current model. As discussed in Section 5.4,

this was later found to be specifically related to the contact definitions (i.e. penalty coefficients)

used in the model.

In conclusion, the analysis contributed to highlight the followings:

i. The time variable used within the ELFEN loading function does not correspond to real

time.

ii. Whereas relative faster loading rates (ELFEN time) would limit the effective

computational time needed to complete the analysis, models of jointed pillars loaded

using relative faster loading functions returned a non realistic mechanical behaviour,

both in terms of stress-strain response and fracturing evolution.

iii. In order to compensate for undesired dynamic effects (Figure 5.9) and oscillations in the

kinetic energy levels, particularly in the early modelling stages, a given displacement

damping has to be incorporated in the analysis.

iv. The use of displacement damping has, however, a fundamental effect on the fracturing

evolution of the modelled pillars, apparently inhibiting the extension of existing and

newly generated fractures.

The lack of clear guidelines for the use of a specific displacement damping value leaves the user

with no other option than to devise a form of calibration approach for the model under

consideration. Calibrating a model against experimental results might provide a more

reasonable answer; however, this may not always be possible for generic applications.

Based on the above remarks, it was concluded that, in order to simulate quasi-static loading

conditions for the pillar, the ELFEN loading time had to be chosen as longer as feasibly allowed

by the associated computational run-times. Because displacement damping had also to be

incorporated in the model (as a measure to compensate for potential dynamic effects associated

with fast loading conditions), a combined strategy was devised, according to which the pillar

was to be uniaxially loaded over a period of 8 seconds in two separate stages: an initial one

(relative short, i.e. 0.5 seconds) incorporating a value of displacement damping of 0.1, and a

subsequent longer second stage (7.5 seconds) with no applied displacement damping. It was

argued this was a reasonable option to avoid the observed inhibition of fracture initiation and

extension associated with the use of displacement damping.

- 100 -


5.4 Effects of the ELFEN contact penalty method on the analysis of jointed pillar

behaviour

Rock mass structure provides most of the deformability of typical rock masses. The mechanical

effects are caused by the much lower stiffness and strength of the discontinuities compared to

the solid matrix. In order to predict with some level of confidence the behaviour of jointed rock

masses under uniaxial stress, the numerical modelling approach used as part of the current

research required the use of specific material parameters for the joint surfaces.

Whereas joint strength parameters are defined in ELFEN in terms of cohesion ( ) and friction

angle (

fc

fφ ) based on a linear Mohr-Coulomb criterion, the deformability of the discontinuities

(intended either as interfaces between blocks or independent discretised fractures) is modelled

using prescribed force-displacement relations resulting in a so-called penalty method. Normal

( ) and tangential ( ) contact penalties are defined in ELFEN. Figure 5.11 shows the

analogy between the implementation of the penalty method in ELFEN and the concept of joint

normal and shear stiffnesses as understood in rock mechanics. However, it must be pointed out

how the penalty method implemented in ELFEN specifically corresponds to a numerical artifice

devised to control the penetration between contacting surfaces. No mention to a direct

equivalence with normal and shear stiffness for discontinuous rock surfaces was found in the

ELFEN user’s manual or in the literature with regards to ELFEN applications to rock mechanics

problems.

nP tP

Indeed, the following analysis included an attempt to verify the correlation between joint

stiffness parameters and penalty coefficients, in order to justify the use of given penalties with

reference to published values for normal and shear joint stiffnesses.

Figure 5.11: (a) Diagrammatic representation of a block containing a single discontinuity (after Bandis,

1993) and the node-to-facet penalty contacting couple in ELFEN as an equivalent spring system (after

Klerck, 2000).

- 101 -


5.4.1 Possible correlation between joint stiffness parameters and penalty coefficients

In order to investigate the relationship between penalty values and simulated joint stiffness, a

simple pillar model containing a single horizontal set of uniformly spaced discontinuities was

considered. This allowed the measurement of the joint normal stiffness, which was then

compared with the corresponding normal penalty value. The model geometry and mesh

definition is illustrated in Figure 5.12. The model did not include any steel-platen device and

the load was applied directly to the top and bottom part of the rock specimen. This specific

geometric arrangement was chosen in order to neglect the influence of the contact interaction

between rock material and the steel-equivalent platen.

Minimum mesh element size was 0.2m, equivalent to that used for all pillar models considered

in the current research.

Figure 5.12: Geometry and mesh definition for the model testing the equivalence between normal penalty

and normal joint stiffness.

Material properties are listed in Tables 5.3. It is noted that the analysis did not include any

attempt to correlate tangential penalty and joint shear stiffness. As stated in the ELFEN user’s

manual, the tangential penalty is usually taken as one order of magnitude (i.e. 1/10) less that the

corresponding normal penalty; accordingly, in the current model it was assumed that

. tn PP 10=

Contact damping for the fracture planes was set to 0.3. The ELFEN user’s manual suggests the

use contact damping in the range [0.1, 0.5]; hence the choice of mean value for the current

analysis.

- 102 -


Penalty Method Model - Rock Material properties Unit Value











Normal stiffness (Normal Penalty ) nP GPa/m 1 and 5 Rock fracture

Shear stiffness (Shear Penalty ) tP GPa/m tn PP 10=

Table 5.3: Normal penalty test model; material properties for the intact rock material and rock fractures.

The unconfined compressive strength is not an ELFEN material input parameter and it is indicated purely

for reference purposes.

The procedure to derive the normal stiffness of a horizontal joint has been described by Hudson

and Harrison (2001). The model assumes that the thickness of the discontinuities is negligible

in comparison to the model height. It is also assumed that the deformation observed in the

model consists of two components: one due to the deformation of the intact rock and the other

due to the deformability of the discontinuities. If the rock is regularly crossed by a single set of

uniformly spaced joints, the relationship between the deformation of the rock considered as

an equivalent continuum, its intact modulus of elasticity

mE

E and the joint normal stiffness can

be expressed as:

nk

SkEE nm

111+= [5.2]

where is the spacing between the discontinuities. S

The initial elastic stress-strain response for the models is shown in Figure 5.13, with indication

of the simulated rock mass deformation , which was estimated respectively as 1.07GPa for

the model with 1GPa/m penalty and as 4.48GPa for the model with 5GPa/m penalty .

mE

nP nP

- 103 -


Figure 5.13: Initial elastic stress-strain response for the models with 1GPa/m and 5GPa/m penalty

respectively.

nP

Based on Equation [5.2], the simulated values for the horizontal set of joints were

1.11GPa/m and 5.36GPa/m for the models with 1GPa/m and 5GPa/m penalty respectively.

Accordingly, the calculated ratio was approximately 1.11 and 1.06 for the first and

second case respectively, showing that the normal stiffness for the joint can effectively

considered as being equivalent, in magnitude, to the selected value. It is noted how stress

values for the simulated pillars were measured as the average stress of a given number of nodes

across the pillar section at mid-height; hence a more precise estimate would correspond to a

larger number of selected nodes. This could explain for the small discrepancies in the

calculated ratios.

nk

nP

nn P:k

nk

nP

nn P:k

It is noted how published data for joint normal stiffness values are limited; whereas direct shear

box tests could provide laboratory values for normal joint stiffness, these would not be

completely representative of in-situ conditions due to scale effects. The presence of infill

material may add other uncertainty to the exact value of to use in the modelling. Wines and

Lilly (2003), quoting the UDEC user’s manual, stated that the normal stiffness for rock joints

with clay-infilling can range from roughly 10 to 100MPa/m, while that for tight joints in granite

and basalt can exceed 100 GPa/m. Eberhardt et al. (2004) reported a normal stiffness of

10GPa/m for gneissic rock mass. It was assumed that the range [1GPa/m, 5GPa/m] for joint

normal stiffness could be considered as a reasonable approximation of in-situ conditions for

typical rock joints in limestone rock at Middleton mine. In view of the results discussed above,

nk

- 104 -


a normal penalty value within a similar range should be used accordingly in the analysis. The

following sections will investigate the effects of varying normal penalty on the modelling

results.

5.4.2 Analysis of the sensitivity of the model to varying the applied normal penalty

The analysis was carried out using a jointed pillar model similar to that shown in Figure 5.2,

with the initial fracture network derived from the FracMan model for Middleton mine. Rock

properties for the intact rock and rock fractures used in the ELFEN model are shown in Table

5.4. Based on the results presented in the Section 5.3, the analysis was carried out using a two-

stage loading phase, as illustrated in Figure 5.14. Gravity loading was also applied in the

analysis and implemented by means of an s-shaped loading function. A full applied-

displacement of the top and bottom platen corresponding to 0.14m (i.e. 2% of pillar height,

taken as ±0.07m) was set to have occurred by the end of the simulation time. A displacement

damping factor of 0.1 was applied during the first stage of the simulation, and then reverted to 0

in the second final stage.

Penalty Method Model - Rock Material properties Unit Value








Intact rock

material




Normal stiffness (Normal Penalty ) nP GPa/m See note Rock fracture

Shear stiffness (Shear Penalty ) tP GPa/m See note

Table 5.4: ELFEN penalty method model; material properties for the intact rock material and rock

fractures. Rock fracture normal and shear stiffnesses were varied throughout the analysis using the

combinations listed in Table 5.5. The unconfined compressive strength is not an ELFEN material input

parameter and it is indicated purely for reference purposes.

- 105 -


Figure 5.14: ELFEN penalty method model; loading functions and selected displacement damping

factors for each stage of the simulation.

Two separate models were considered in the analysis, the main difference between them being

represented by the applied contact damping. Model H had a contact damping of 0.5, whilst

model L was given a lower 0.3 value. In this way the analysis attempted to consider two

different ways of penalty variation, one direct (by use of specified coefficients), and one indirect

(by use of contact damping).

As discussed in Section 4.6.2.2, the contact damping augments the normal penalty force ;

hence it acts as a modifier of the normal penalty coefficient . The proprietary character of

the ELFEN code meant that specific details of the contact damping algorithm were not readily

available. It was understood, however, that the contact damping option in ELFEN has not

relation to any other form of damping and is purely optional (Rockfield, personal

communication, 2005). The contact damping parameter is non-dimensional and takes a value

between 0 and 1, with 1 being full damping. The ELFEN user’s manual suggests using values

in the range [0.1, 0.5] for general mechanical interactions.

nf

nP

Table 5.5 lists the different combinations of and used in models H and L respectively. nP tP

- 106 -


H Model L Model

Contact Damping 0.5 0.3

Penalty Coefficient nP (GPa/m) tP (GPa/m) nP (GPa/m) tP (GPa/m)

1 5 0.5 0.5 0.25

2 2.50 2.50 1 0.5

3 20 2 5 2.5

4 2 0.2 10 0.5

5 0.5 0.5 0.5 0.1

6 1.25 1.25 1 0.2

7 0.5 0.05 5 1

Model ID

8 20 20 10 2

Table 5.5: ELFEN penalty method model; combinations of and values for rock fracture used in

the analysis for models H and L respectively.

nP tP

The properties of the platen and the rock/platen contacts were similar to those listed previously

in Table 5.2, with the exception represented by the rock/platen normal stiffness (normal penalty

) and rock/platen shear stiffness (tangential penalty ). These were respectively taken as

being equivalent to the normal and tangential penalties and specified for the rock

fracture, and in accordance with the values listed in Table 5.5 above.

nP tP

nP tP

As discussed in Section 4.6.2.4, the ELFEN user’s manual suggests the use of normal penalty

coefficients (in magnitude) in the range of nP EPE n 25.0 << , where E is the intact

material Young’s Modulus. The tangential penalty value is usually taken as an order of

magnitude less than . Indeed, the analysis included various ratios of to , with

taken in the range

tP

nP nP tP nP

EPE n <<04.0 , with the minimum well below the stated values. The

evidence for such an approach was provided by early pillar models with higher values,

which yielded an unrealistic mechanical response (e.g. Figure 5.8). A lower value

equivalent to was also implemented in a similar pillar model verified by ELFEN

developers (Rockfield, 2005, personal communication), which gave a more realistic response in

terms of displacement and rotation of discrete blocks.

nP

nP

E04.0

- 107 -


5.4.3 Analysis of results

The simulated axial strain-axial stress curves for models H and L are shown in Figure 5.15 and

Figure 5.16 respectively. Axial stress values for the simulated pillars were measured as the

average stress of a given number of nodes across the pillar section at mid-height. Axial strains

were measured from the closure of the whole pillar. Figures 5.17 to 5.18 and 5.19 to 5.20 show

the fracturing development for models H and L respectively. Avi movie files showing the

fracture pattern development under increased loading conditions are included in Slide 2 and 3 in

the enclosed DVD.

As discussed in Section 5.3.2, the evaluation of each simulation was again quantitatively based

on estimated pillar strength derived from empirical methods (see also Section 7.4.3) and on a

qualitative comparison with the expected stress-strain response for a slender pillar represented

in Figure 5.4. The degree of lateral spalling and increased fracturing of the pillar core at peak

stress were both considered as supplementary (qualitative) marker points between realistic and

unrealistic behaviour.

Figure 5.15: H model axial stress-strain curves. Models with (a) tn 10PP = and (b) . tn PP =

Figure 5.16: L model axial stress-strain curves. Models with (a) tn PP 5= and (b) . tn 2PP =

- 108 -


Penalty values

Model ID

nP GPa/m

( )tn 10PP =

Pre-peak Peak Post-peak

H 7 0.5

0.31% 1.9MPa 0.70% 5.2MPa 0.765% 4.0MPa

H 4 2

0.08% 2.0MPa 0.11% 2.8MPa 0.12% 0.9MPa

H 1 5

0.03% 1.4MPa 0.06% 2.6MPa 0.085% 2.0MPa

H 3 20

0.06% 3.46MPa 0.07% 3.8MPa 0.09% 2.4MPa Figure 5.17: H model (contact damping 0.5), fracturing evolution at different stages of the simulation.

Cases with tn 10PP = .

- 109 -


Penalty values

Model ID

nP GPa/m

( )tn PP =


H5 0.5

0.511% 4.54MPa 0.827% 9.02MPa 0.844% 4.02MPa

H6 1.25

0.277% 7.39MPa 0.452% 9.93MPa 0.512% 7.50MPa

H2 2.50

0.139% 5.28MPa 0.239% 7.67MPa 0.453% 5.86MPa

H8 20

0% 0MPa 0.027% 2.60MPa 0.319% 1.20MPa Figure 5.18: H model (contact damping 0.5), fracturing evolution at different stages of the simulation.

Cases with tn PP = .

- 110 -


Penalty values

Model ID

nP GPa/m

tn 2PP =


L 1 0.5

0.560% 5.13MPa 0.903% 7.75MPa 1.098% 5.84MPa

L 2 1

0.311% 6.76MPa 0.535% 10.2MPa 0.976% 5.08MPa

L 3 5

0.02% 1.74MPa 0.08% 3.18MPa 0.13% 0.75MPa

L 4 10

0.02% 2.18MPa 0.04% 3.08MPa 0.07% 2.38MPa Figure 5.19: L model (contact damping 0.3), fracturing evolution at different stages of the simulation.

Cases with tn 2PP = .

- 111 -


Penalty values

Model ID

nP GPa/m

tn PP 5=


L 5 0.5

0.48% 3.0MPa 1.11% 5.63MPa 1.27% 3.82Mpa

L 6 1

0.20% 3.14MPa 0.46% 5.88MPa 0.61% 3.61MPa

L 7 5

0.02% 1.67MPa 0.04% 2MPa 0.13% 1.94MPa

L 8 10

0.02% 2.03MPa 0.03% 2.44MPa 0.10% 0.04MPa Figure 5.20: L model (contact damping 0.3), fracturing evolution at different stages of the simulation.

Cases with tn PP 5= .

- 112 -


Figure 5.15(a) clearly shows how the modelled jointed pillar with a lower applied

(0.5GPa/m) was characterised by a softer response, with estimated peak stress reached after the

onset of a greater deformation in terms of pillar closure. Figure 5.17 illustrates how the same

model (H7) was characterised by significant lateral spalling and increased fracturing of the

pillar core at peak stress. Modelled pillars H4, H1 and H3 (Figure 5.17) showed a stiffer

response, which resulted in a more brittle fracturing mechanism with very limited lateral

spalling.

nP

The H-models with (Figure 5.15b), with the exception of model H8, were characterised

by behaviour similar to model H7 in terms of deformation response. The estimated pillar

strength for H2, H5 and H6 were significantly greater compared to those of the models in Figure

5.15(a). Again, model H8 (

tn PP =

== tn PP 20GPa/m) yielded a very brittle response, considered not

to be typical of a fractured mine pillar, with a sudden explosive breakdown of the entire pillar

structure.

L-models L1 and L2 with tn 2PP = (Figure 5.16a) showed a response similar to some of the H-

models (H5, H6 and H2) in Figure 5.15(b), both in terms of estimated pillar strength and

deformation. These models (Ls and Hs) were characterised by to ratios of 2 and 1

respectively and contact damping factors of 0.3 and 0.5 respectively. The fracturing evolution

for these models was also similar, with some of the L-models (L1 and L2 in Figure 5.19) being

characterised by a greater degree of lateral spalling before the attainment of peak stress. The

loading response of models L3 and L4, both with higher values, was deemed as to be not

representative of typical pillar behaviour, since it terminated again in a form of sudden explosive

fracturing (Figure 5.19).

nP tP

nP

The results for the L-models with to ratios of 5 (Figure 5.16b) were comparable with

those of the previous L-models, with similar estimated deformation modulii but with reduced

pillar strengths. L7 and L8 showed a response equivalent to that of models L3 and L4. The

fracturing evolution for L5 and L6 (Figure 5.20) was almost identical to L1 and L2, with a more

realistic sliding and spalling of pre-existing discrete blocks.

nP tP

Figures 5.21 show the variation of pillar strength and deformation modulus with normal penalty

for the H and L models. nP

- 113 -


Figure 5.21: Variation of pillar strength and deformation modulus with normal penalty for the H and

L models.

nP

The following observations were made based on the results shown in Figure 5.21:

i. Both H and L models were characterised by higher pillar strength values for

intermediate values of in the range [0.5GPa/m, 2.5GPa/m], with higher strength

measured for decreasing : ratios. The lower the ratio, the stiffer and stronger

being the response of the modelled jointed material. Higher values, which lead to

the observed sudden explosive breakdown of the entire pillar structure, resulted in lower

nP

nP tP

nP

- 114 -


estimated pillar strengths. It was argued that the lower joint compliance associated with

higher normal penalty determined the pillar models to fail primarily through

fracturing of the intact rock material in a somehow uncontrolled manner, the reason of

which is, however, uncertain.

nP

ii. It was observed (Figure 5.21f) that the estimated deformation modulus was almost

independent of the ratio : for in the range [0.5GPa/m, 5GPa/m]. nP tP nP

iii. Varying contact damping apparently did not have significant effects on the simulated

deformation modulii (Figure 5.21f), with the exception of H-models with tn 10PP =

and greater than 5GPa/m. On the contrary, the effect of varying contact damping on

the simulated pillar strength was not clearly intuitive; for example:

nP

L-models

( ) tn PP 2=H-models ( ) tn PP =

Contact damping 0.3

≈

(Pillar strength - response) nPContact damping 0.5

Increasing contact damping corresponds to decreasing : ratio. nP tP

but also

L-models

( ) tn PP 5=

H-models

( ) tn 10PP =

Contact damping 0.3

≈

(Pillar strength - response) nPContact damping 0.5

Increasing contact damping corresponds to increasing : ratio. nP tP

However, both H-models ( tn 10PP = ) and L-models ( tn PP 5= ) were characterised by

simulated pillar strengths which were outside the estimated range calculated using an

empirical approach for an equivalent rock pillar (Figure 5.21e); hence, a first conclusion

could be that varying contact damping may have the same effects of decreasing the

: ratio. By comparing Figure 5.18 (H-models,nP tP tn PP = ) with Figure 5.19 (L-

models, ), it is apparent that the increased contact damping, for less than

2GPa/m, resulted in a greater degree of fracturing of the intact rock material, in a

similar trend to that observed when increasing normal penalty for a given contact

tn PP 2= nP

nP

- 115 -


damping value. Additionally, for greater than 2GPa/m the increasing contact

damping factor reduced the degree of lateral spalling and in more general terms the

relative displacement of larger blocks, which again resulted in the sudden explosive

breakdown behaviour discussed above. This may indicate a threshold value for normal

penalty above which the effects of varying contact damping are not significant in

terms of pillar response. Only for less than this apparent threshold value, contact

damping reasonably acts as a modifier of the normal penalty coefficient .

nP

nP

nP

nP

iv. Only for values lower than 2GPa/m the models showed what was interpreted as an

acceptable mechanical response in terms of discrete blocks displacement, lateral

spalling and pillar core fracturing. However, these models were characterised by

simulated deformation modulii in the range [2GPa, 3GPa]. The rock mass deformation

modulus of a fractured rock mass can be empirically derived by means of correlations

with rock mass classification schemes. Several empirical approaches for prediction of

the deformation modulus of rock masses are summarised in Table 5.6. Using RMR/GSI

equal to 60-70b, equal to 6-17c and

nP

Q ciσ (UCS) equal to 48MPa, Table 5.6 also lists

the deformation modulii calculated accordingly for the jointed pillar model used in the

ELFEN analysis.

Empirical Equation

Required parameters Equation ELFEN equivalent

jointed pillar model

Bieniawski (1978) RMR 1002 −= RMREM [20GPa, 40GPa]

Barton (2002) Q ( )100/UCSQEM = [2.8GPa, 8.1GPa]

Hoek and Brown (1997) GSI, UCS ( ) ( )[ ]40/10010100/ −= GSI

M UCSE [12.3GPa, 21.9GPa]

Table 5.6: List of some empirical equations suggested for estimating the modulus of

deformation of a fractured rock mass; also indicated are the corresponding estimates for a jointed

pillar model equivalent to that used in the current ELFEN analysis.

It is evident how different results could be obtained just by using different empirical

approaches; in some case (Bieniawski, 1978 and RMR=70), the estimated rock mass

b Based on field data collected at Middleton mine, see also Appendix I. c represents the Q-index (Barton et al., 1974). Based on field data collected at Middleton mine, see

also Appendix I.

Q

- 116 -


deformation modulus for the fractured rock mass is also greater than the Young’s

Modulus for the intact rock. The ELFEN models returned values of deformation

modulus for the rock mass approximately one-sixth to one-tenth of the corresponding

values estimated for a similar jointed rock pillar using empirical correlations with RMR

and GSI rock mass classification systems. On the contrary, a better comparison with

the ELFEN modelled values was obtained using field estimates for Q .

5.4.4 Conclusions on the effects of varying the applied normal penalty for the jointed

pillar models

Table 5.7 summarises the results of the analysis investigating the effects of varying normal

penalty (which also included varying : ratio and contact damping), contributing to

eliminate what were considered as unrealistic combinations of normal penalty , contact

damping and : ratio.

nP tP

nP

nP tP

Model ID - nP H5 - 0.5GPa/m H6 - 1.25GPa/m H2 - 2.5GPa/m H8 - 20GPa/m

tn PP =

Modelled pillar strength

is acceptable.

Limited lateral spalling

and excessive rock

fracturing.


is acceptable.

Limited lateral spalling

and excessive rock

fracturing.


is low.

Explosive breakdown.


is very low.


Model ID - nP H7 - 0.5GPa/m H4 - 2GPa/m H1 - 5GPa/m H3 - 20GPa/m

tn 10PP = Modelled pillar strength

is very low.


is very low.


is very low.



is very low.


Con

tact

dam

ping

0.5

Model ID - nP L1 - 0.5GPa/m L2 - 1GPa/m L3 - 5GPa/m L4 - 10GPa/m

tn PP 2=


is acceptable.

Realistic lateral spalling

and pillar core

fracturing.


is acceptable.

Realistic lateral spalling

and pillar core

fracturing.


is very low.



is very low.


Model ID - nP L5 - 0.5GPa/m L6 - 1GPa/m L7 - 5GPa/m L8 - 10GPa/m

Con

tact

dam

ping

0.3

tn PP 5= Modelled pillar strength

is low.


is low.


is very low.



is very low.


Table 5.7: Summary of results of the ELFEN analysis investigating the effects of varying normal

penalty, : ratio and contact damping. Shadowed boxes indicate what was assumed to be not a

completely realistic behaviour, both in terms of stress-strain response and fracturing evolution.

nP tP

- 117 -


The following conclusions were drawn relative to the effects of varying normal penalty : nP

i. The deformability of the fractured pillar models is directly related to the normal penalty

value used in the analysis, but apparently independent of the selected : ratio.

Figure 5.22 shows the axial stress-strain curve obtained for an intact pillar model with

material and contact properties equivalent to that of L2. The analysis yielded a

deformation modulus for the modelled intact rock mass equivalent to the input

parameter of 27GPa. The results for the intact material model were not influenced by

the penalty properties, since fracturing did not occur in the pre-peak region.

nP nP tP

Figure 5.22: Stress-strain curve for the L2 model with intact and pre-fractured geometries.

ii. The use of combined higher and values results in a mechanical response that

cannot be considered as completely representative of jointed pillar behaviour. The

simulated pillar strength for the pillar models with combined higher and values

were not in agreement with the calculated range (8 to 12MPa) based on empirical pillar

strength formulae (see also 7.4.3). The use of lower values yielded positive

estimates of pillar strength. As discussed by Brown (1993), a jointed rock mass

generally behaves in more plastic and less brittle manner than does the intact rock

material: the post-peak behaviour observed in the models with lower normal penalty

was also encouraging in that sense.

nP tP

nP tP

nP

nP

iii. Models with relative higher values yielded a very brittle response, considered not to

be typical of a fractured mine pillar, with a sudden explosive breakdown of the entire

nP

- 118 -


pillar structure. Conversely, the results showed how for lower values individual

blocks were free to rotate or translate.

nP

iv. As discussed in point (iii) above, relative lower values (0.5 to 2GPa/m) give a more

acceptable mechanical response in terms of discrete blocks displacement, lateral

spalling and pillar core fracturing. However, this is associated with relative low values

of deformation modulus for the modelled rock mass, and approximately one-sixth to

one-tenth of the corresponding values estimated for a similar jointed rock pillar using

empirical correlations with RMR and GSI rock mass classification systems (Bieniawski,

1978 and Hoek and Brown, 1997); a better comparison is obtained, on the contrary,

using field estimates for Q (Barton, 2002). To partly account for this, it is noted that

the strain in the simulated pillars is measured from the closure of the whole pillar. At

late stages of loading the cross section of the pillar would be much reduced, thus the

inferred local deformation modulus could be higher. Also, it may be that the

unconfined state of the modelled pillars means that there is more freedom for

deformation than is implied by rock mass rating approaches. Rock mass classification

systems generally do not account for potential blocks movement/displacement and

orientation of discontinuities with respect to loading direction; the relative low

deformation modulus estimated for the simulated pillar models with

may, in these circumstances, be a reflection of the specific geometry

used for the initial pre-inserted fracture network.

nP

2GPa/m<nP

v. Some locking-up of discretised elements were observed, whose nature was not assumed

to be a direct result of the normal penalty used in the analysis (Figure 5.23).

Figure 5.23: Example of the locking-up effects observed in the simulations, following fracturing

of the intact rock material and blocks ejection.

- 119 -


These effects were indeed considered as a potential result of the adopted contact

scheme. As also discussed by Hart (1993) and Munjiza (2004), an unrealistic response

can result in DEM models when block interaction occurs close to or at opposing

corners, which results in the blocks becoming blocked or hang-up. This is associated

with the modelling assumption that blocks corners are sharp or have infinite strength.

In reality, crushing of the corners would occur as a result of stress concentration,

although the explicit modelling of such an effect is impractical. Roundness of pre-

existing blocks may be incorporated in ELFEN at a geometry definition stage; however,

the problem could still persist due to the definition of new fractures (hence the

generation of new discretised blocks with sharp edges) while the simulation proceeds.

5.5 Sensitivity analysis for material parameters used in the modelling of jointed pillars

The preceding sections primarily investigated the sensitivity of the modelling results to varying

specific numerical parameters. The scope of the analysis was to eliminate unrealistic models, in

an attempt to define an explicit combination of numerical parameters which could then be used

in further (analogous) ELFEN models. Based on the previous results, the attention in the

current section is switched to the consideration of varying material parameters such as joint

cohesion and joint friction angle. It is reasonable to assume that the models should return a

coherent behaviour once a suitable combination of numerical parameters is selected.

5.5.1 Model set-up

The jointed pillar model described in the previous sections was chosen to perform a sensitivity

analysis on the shear strength parameters for the discretised fractures. The geometrical set-up

was similar to that shown in Figure 5.2, whilst material parameters and rock-platen contact

properties were equivalent to those listed in Table 5.2 and Table 5.3 respectively. Following the

analysis presented in Section 5.4, the normal and tangential penalty coefficients for the models

were set as 1GPa/m and 0.5GPa/m respectively.

Table 5.8 indicates the different combinations of joint cohesion ( ) and joint friction angle

(

fc

fφ ) used in the analysis. Similar values of joint cohesion and friction were equally applied to

pre-inserted joints and newly generated fractures. The QF-models listed in Table 5.8 were run

with constant and ic iφ for the intact rock of 7MPa and 40 degrees respectively.

- 120 -


Rock joint properties Model ID

fc fφ

QF1 0 0

QF2 0 10

QF3 0 30

QF4 0 60

QF5 0.22 0

QF6 0.22 10

QF7 0.22 30

QF8 0.22 60

QF9 1 0

QF10 1 10

QF11 1 30

QF12 1 60

Table 5.8: Different combinations of and fc fφ used in the analysis.


Stress-strain curves for the QF-models are shown in Figure 5.24 below.

Figure 5.24: (a, b, c) Stress-strain curves for the QF model and (d) variation of the estimated pillar

strength with fracture cohesion for a given value of friction fc fφ .

- 121 -


A typical brittle post-peak section characterised all models with the exception of QF10, QF11

and QF12, which presented a steady increase in the modelled axial strength; for these models,

which had relative higher joint shear properties, failure primarily occurred within the intact

rock, initially starting along critically oriented discontinuity planes and then, due to the increase

loading of the pillar, propagating to the intact material in the form of new fracturing (Figure

5.25). The oscillations which characterised the curves representing QF11, QF12 and in a more

limited manner QF10 (Figure 5.24c) were associated with locking-up of discretised blocks in

proximity of sharp edges, with the oscillation cycles corresponding to the breaking and forming

of these blocks. Figure 5.24(d) shows how the QF models presented a linear increase of the

estimated pillar strength with fracture cohesion for a given value of friction fc fφ . This result

could be explained taking into consideration the use of a typical Mohr-Coulomb constitutive

model for the shear strength of the fracture surfaces (see also Section 4.6.2.1).

The trends shown by the curves in Figure 5.24(d) are believed to reflect a credible prediction

highlighting how relative higher pillar strength estimates can be reasonably expected for jointed

rock pillars, depending on the joint surface conditions (i.e. joint cohesion) and joint orientation

(i.e. joint angle of friction versus joint inclination with respect to applied loading).

Fracturing evolution at peak stress is shown in Figure 5.25. Avi movie files showing the

complete fracturing process are included in Slide 4 (DVD). Figure 5.25 shows how the pre-

existing fractures affected the mechanical behaviour of the pillar models. Invoking 2D in-plane

continuity for the discretised fracture elements, the occurrence of critically oriented (with

respect to the loading direction) planes of weakness enabled failure to take place along the

larger fracture surfaces, i.e. the process was entirely dependent on the selected shear strength

parameters. The use of higher shear strength properties ( and fc fφ ) limited

displacement/rotation along existing fracture planes, thus explaining the greater degree of

fracturing observed within the intact rock material, which in some case resulted in the sudden

release of small block fragments.

- 122 -


f

fc

φ 0.0MPa 0.22MPa 1MPa

0

0.06% 0.82MPa 0.11% 2.14MPa 0.20% 8.98MPa

10

0.58% 3.89MPa 0.61% 5.70MPa 1.02% 15.38MPa

30

0.46% 6.76MPa 0.57% 11.29MPa 1.35% 18.17MPa

60

0.56% 8.21MPa 0.41% 11.82MPa 1.75% 23.13MPa Figure 5.25: Sensitivity analysis for material parameters. Fracture evolution at peak stress for the QF

model.

- 123 -


It was concluded that the simulated pillar models returned credible behavioural predictions and

in particular showed how the model response was clearly dependent on the selected

discontinuity shear strength parameters. For these reasons, it was also assumed that the results

contributed to indirectly verify the validity of the chosen numerical parameters set. As a general

aside, the QF-models highlighted the importance that fractures could have on the ultimate

behaviour of jointed rock masses, and the necessity of representing such structural elements

within a numerical model attempting to reproduce the fracturing evolution of a given fractured

rock mass.

5.6 Summary

One of the fundamental requirements for any applied numerical method is the reproducibility of

the expected simulated behaviour for various classes of problems. Within the context of the

hybrid FEM/DEM method discussed in this thesis, it is noted that some difficulties were

experienced with the definition of material behaviour at the contact points and the damping of

the system.

The novelty of the ELFEN release version used as part of the current research required the

verification of specific numerical parameters used in the analysis. A specific methodology was

devised in order to develop an adequate modelling strategy that could be used for future ELFEN

applications for comparable problems. Accordingly, a series of numerical tests were undertaken

using 2D models of a jointed rock pillar, with the pre-inserted fracture geometry derived from a

DFN model generated in FracMan. The selected numerical examples included the followings:

i. Analysis of the effects of varying displacement damping and loading rate on the

modelling results.

ii. Analysis of the equivalence between normal penalty coefficient in ELFEN and joint

normal stiffness; additionally, the effects of varying normal penalty were also

considered in terms of simulated pillar strength, pillar deformation and fracturing

evolution.

iii. Sensitivity of the modelling results to varying material parameters (shear strength

properties of the pre-inserted fractures). The scope was to verify the suitability of the

numeric parameters set derived based on points (i) and (ii) above.

The evaluation of each simulation was quantitatively based on estimated pillar strength derived

from empirical methods and on a qualitative comparison with the expected stress-strain

response for a slender pillar represented in Figure 5.4. The analysis also considered the degree

- 124 -


of lateral spalling and increased fracturing of the pillar core at peak stress as a supplementary

qualitative indicator of modelled behaviour.

The analysis conducted as part of point (i) above concluded that the time variable used within

the ELFEN loading function did not correspond to real time. It was also found that models of

jointed pillars loaded using relative faster loading functions returned a non realistic mechanical

behaviour, both in terms of stress-strain response and fracturing evolution. Whereas a given

displacement damping needs to be used in the analysis in order to compensate for undesired

dynamic effects, it was found that the use of displacement damping could have, however,

fundamental effects on the fracturing evolution of the modelled pillars. Consequently, the

following combined two-stages loading/displacement damping set-up was proposed:

Stage 1 Stage 2

Time: 0 to 0.5 seconds Time: 0.5 to 8 seconds

Displacement damping: 0.1 Displacement damping: 0

Table 5.9: Proposed ELFEN loading/displacement damping two-stages set-up.

As part of point (ii) above, the analysis showed how the normal stiffness for a modelled

joint surface could effectively be considered equivalent, in magnitude, to the selected normal

penalty coefficient . Additionally, it was discussed how the deformability of the modelled

pillars ultimately depended on the normal penalty coefficient, with the use of relative lower

normal penalty coefficients yielding a more realistic mechanical response in terms of discrete

block displacement, lateral spalling and pillar core fracturing. The use of relative lower

coefficients, however, led to potential underestimation of the rock mass deformation modulus

when compared with estimates derived based on correlation with rock mass classification

systems. It was argued that this could be explained by considering the way the deformation

modulus was actually estimated for the simulated pillars. The unconfined state of the modelled

pillars could have also played a major part, resulting in a more freedom for deformation than

would be implied by rock mass rating approaches. The relative low deformation modulus

estimated for the simulated pillar models with <2GPa/m may also reflect the specific

geometry used for the initial pre-inserted fracture network, in terms of fracture orientation with

respect to loading direction.

nk

nP

nP

nP

The following combination of normal penalty , contact damping and : ratio was

proposed for future analysis of jointed pillar models:

nP nP tP

- 125 -


Normal penalty nP Contact damping nP : ratio tP

[1GPa/m, 2GPa/m] 0.3 2

Table 5.10: Proposed ELFEN normal penalty , contact damping and : ratio combination. nP nP tP

As part of point (iii) above, the analysis considered varying material parameters such as joint

cohesion and joint friction angle. The scope was to prove that the models would return a

coherent behaviour once a suitable combination of numerical parameters was selected. Indeed,

the results for the simulated pillar models yielded credible behavioural predictions and in

particular showed how the model response was clearly dependent on the selected discontinuity

shear strength parameters.

Overall, Chapter 5 highlights how the use of the code ELFEN necessarily requires a careful and

lengthy calibration process, for which no clear guidelines exist. In this context, the modelling

approach implemented in the current research has attempted to provide a methodology for

assessing the impact of various input parameters on model behaviour that provides useful

guidance for future ELFEN applications. However, it must be pointed out how the proposed

combinations of numerical parameters are only strictly applicable to the specific pillar models

considered in the current research. Due to the intensive computational time necessary to

conduct such a calibration process, it was not possible to extend the validity of the proposed

numerical parameters set to a different class of geomechanics problems/case examples.

Based on these results, Chapter 7 will then include a detailed analysis of different jointed pillar

geometries in an attempt to evaluate the potential application of a combined FracMan-ELFEN

approach in terms of rock mass strength characterisation

- 126 -

Modelling of continuum/discontinuum transition using a combined FEM/DEM method

6

Numerical modelling of the progressive shear

behaviour of rock joints with tooth-shaped

asperities using a combined FEM/DEM method

6.1 Introduction

This Chapter presents the results of the numerical simulation of simple shear tests using the

hybrid FEM/DEM code ELFEN (Rockfield, 2005), with the objective to study the capability of

the proposed hybrid approach to simulate the progressive shear behaviour of rock joints with

tooth-shaped asperities. The numerical analysis was performed to evaluate potential limitations

of the ELFEN rotating crack model when simulating shear mechanism.

6.2 ELFEN constitutive criterion to model shear strength of discontinuities

The shear strength of a horizontal flat fracture with bonded surfaces (Figure 6.1) can be

described in terms of (Hoek, 2000):

[6.1] fnf tanσcτ φ+=

where is the cohesion along the discontinuity plane and fc fφ represents the basic angle of

friction. When considering the residual strength, the cohesion in Equation [6.1] drops to

zero and the relationship between

fc

rfφ (residual angle of friction) and nσ is represented by:

rfntanστ φ= [6.2]

- 127 -


Equations [6.1] and [6.2] refer to the Mohr-Coulomb criterion, which is widely used in rock

mechanics to define shear failure along fracture planes. In rock mechanics, true cohesion only

occurs when cemented surfaces are sheared; it is reasonable to assume that the shear behaviour

of existing fractures within a rock mass is governed by friction resistance only, with the value of

cohesion in Equation [6.1] set to zero. However, in many practical applications, the term

cohesion is used for convenience and it refers to a mathematical quantity related to surface

roughness (Hoek, 2000).

fc

Figure 6.1: Schematic representation of the Mohr-Coulomb criterion for shear strength of discontinuities

(after Hoek, 2000).

Several empirical shear strength criteria, mostly in a non-linear logarithmic or power-law form,

have been reported in the literature (Maksimovic, 1996). In addition to the widely used Mohr-

Coulomb, the Barton model, developed by Barton and his co-workers (1973, 1976, 1977, 1983

and 1990) has also found widespread application.

As introduced in Section 4.6.2.1, shear phenomena between contacting facets (i.e. fracture

planes) are considered in ELFEN based on the classical Coulomb theory: shear strength of pre-

existing and newly generated fractures is defined in terms of cohesion and friction angle, based

on a linear Mohr-Coulomb criterion. It is noted how the current version of the ELFEN code,

however, does not include the option of using an explicit Barton model. The mechanical

modelling of a fractured rock mass using the code ELFEN could in principle incorporate

- 128 -


different shear strength properties for different fracture planes and, with some limitations, the

code could also consider non-planar fracture surfaces, which are geometrically defined as a

multiline.

6.3 Effects of discontinuity geometry on shear strength and deformation

It is reasonable to assume, for a given stress state, that joint shear strength and deformation

depend on the joint geometry characteristics, including surface roughness and spatial geometry

of the contact area.

Hopkins (2000) described how two joints with the same total contact area can have substantially

different stiffness values depending on the spatial geometry of their contact areas; for instance,

modelling results indicate that joints with small contact areas uniformly distributed across the

surfaces can be nearly as stiff as a perfectly mated interface. Bandis et al. (1983) recognised

that joint normal stiffness can be influenced by different parameters, such as (i) the initial

contact area, (ii) fracture wall roughness, (iii) strength and deformability of the asperities along

the fracture plane and (iv) thickness, type and physical properties of any infill material.

Many studies have considered joint roughness only as a dilation parameter that effectively

increases the friction angle above the basic friction angle (Yang and Chang, 2000). Based on

work by Patton (1966), who categorized asperity into first-order (waviness) and second-order

(unevenness) categories, Yang et al. (2001) discussed the influence of joint roughness on the

shear behaviour of rock joints. They discussed how the behaviour of rock joints is controlled

primarily by the second-order asperity during small displacements, whilst first-order asperity

governs the shearing behaviour for large displacements. The same concepts were considered by

Barton (1973) and Hoek and Bray (1981), who stated that at low normal stress levels the

second-order asperity (with higher-angle and narrow base length) controls the shearing process;

as the normal stress increases, the second-order asperity is sheared off and the first-order

asperity (with longer base length and lower-angle) takes over as the controlling factor.

Although the shear strength criterion proposed by Barton for rock joints is widely adopted in

engineering practice, its use may be dependent on the subjectivity of the field parameter JRC.

Alternative methods have been proposed for JRC estimation; for example, correlations with the

profile geometry coefficient Z2 have been suggested by Tse and Cruden (1979). The parameter

Z2 is the root mean square of the tangents of the slope angles along the geometry profile.

Following the introduction of fractal geometry by Mandelbrot (1983), numerous researchers

have tried to interpret the JRC parameter using the concept of fractal dimensions (e.g. Turk et

- 129 -


al., 1987; Lee et al., 1990; Odling, 1994; Seidel and Haberfield, 1995; quoted in Yang et al.,

2001; Fardin et al., 2001).

Figure 6.2 illustrates some basic shear stress-displacement curves of progressive joint damage

under shear (Cundall et al., 1978). At low normal stress, the applied shear stress initially causes

elastic deformation of the asperity up to the yield deformation: the dilation of the joint with

sliding of one asperity over another then follows; at a certain shear displacement, the resistance

of the asperity is exceeded and shearing through the asperity occurs. Further shearing leads to

residual shearing resistance along the newly created surface. At high normal stress, no dilation

occurs and the asperities are immediately sheared off at the base (Yang and Chiang, 2000).

High Asperity Angle Low Asperity Angle

Low Normal Stress

High Normal Stress

Figure 6.2: Basic shear stress-displacement curves of joint asperity (after Cundall et al., 1978).

Several authors (Yang and Chiang, 2000; Yang et al., 2001 and Huang et al., 2002) have

recently investigated shear mechanisms and strength degradation in the shear stress-

displacement of rock joints as influenced by the progressive shearing of asperities. Using

artificially mixed plaster/sand materials to represent low strength brittle rock (Deere et al.,

1973), they have considered simple joint model specimens, as illustrated in Figure 6.3, to study

the transition between sliding and shearing through mode of failure.

- 130 -


A B

Angle θ

30 Degrees 15 Degrees

Figure 6.3: Example of simple joint specimen used in laboratory testing to study shear mechanisms and

strength degradation (after Yang and Chiang, 2000).

6.4 ELFEN modelling of the progressive shear behaviour of rock joints with tooth-

shaped asperities

The preceding section introduced a discussion on the effects of discontinuity geometry on shear

strength and deformation. This section presents the results of a series of numerical simulations

investigating the influence of surface roughness on the mechanical behaviour of a rock joint.

The aim is to evaluate the capability of the code ELFEN to numerically simulate the fracturing

of asperities for a rock joint with tooth-shaped asperities. As reported by Huang et al. (2002)

for a brittle and highly fractured rock mass, it is often found that the fracture of asperities has

significant effects on its shear strength and dilatancy behaviour.

The simple 2D single joint asperity model considered in the current analysis is useful to verify

the effectiveness of the ELFEN solution scheme to model shearing mechanism along fracture

planes, before it is extended to more complicated and realistic situations (e.g. slope stability

problems or underground structures in jointed rock masses). Based on the simulated results,

observations are made to identify mechanisms that have important effects on the deformation

behaviour of a rock joint. The model performance is evaluated by comparing the model

predictions with experimental results published in Huang et al. (2002).

6.4.1 Model set-up

In order to have a rock joint with well-defined triangular asperities, Huang et al. (2002)

undertook laboratory test using a material that was a mix of chalk, sand and water (ratios

1:0.25:0.85) so that they could easily control and duplicate the joint surface topology. A

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material database was created in ELFEN replicating the properties of the material used by

Huang et al. (2002), as shown in Table 6.1. The fracture toughness was derived based on

the empirical correlation proposed by Zhang (2002):

ICK

ICt K88.6=σ [6.3]

The fracture energy parameter was estimated using Equation [4.2]. fG

Shear Model - Rock Material properties Unit Value


Unconfined compressive strength, σci MPa 8.5



Young’s Modulus, E MPa 6500





Internal dilation, ψi degrees 5

Table 6.1: ELFEN shear model; material properties used in the analysis are based on the material

parameters for the rock specimen tested by Huang et al. (2002). Note: the unconfined compressive

strength of the material is not used as input parameter in ELFEN and it is here indicated purely for

reference purposes.

The intact rock material was modelled using the ELFEN elasto-plastic Mohr compressive model

with Rankine cut-off.

Using the simulated material, models of two types of interface topography were considered: 15

degrees asperity and 30 degrees asperity, as shown in Figure 6.4. Total length of the joint

specimen is 100mm. For rock joints with 15 degrees asperities, the base length of each

triangular tooth is 20mm and the height is 0.258 mm. For rock joints with 30 degrees asperities,

the base length of each triangular tooth is 20mm and the height is 0.577 mm.

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For each type of model, the simulations were also performed under five different normal

stresses: 0.1, 0.3, 0.5, 1.0, and 1.5MPa. The horizontal loading was simulated using the face-

loading option in ELFEN; this loading was applied to the opposite faces of the model, as shown

in Figure 6.5. The vertical stress was applied using an initial s-shaped loading function up to

0.01 seconds, and then kept constant throughout the simulation. The horizontal shear stress was

applied using a simple ramp-load function over a period of 0.09 second (starting at

0.01seconds), over which it was gradually increased up to 10MPa. A displacement damping of

0.1 was assigned for the vertical loading stage only, and then reverted to 0 in the second part of

the loading phase.

This specific loading time-displacement damping combination was based on the results of the

analysis discussed in Section 5.3. Relative fast simulation times were implemented in the

analysis in order to limit the necessary computing time required: a standard PC with a Pentium

IV 2.8 GHz processor required between 15 and 36 hours to complete each model, the fine mesh

setting incorporated in the model adding extra complexity.

Figure 6.4: Shear model; geometrical assembly and dimensions for the (a) 15 and (b) 30 degrees asperity

models respectively.

Figure 6.5: Shear model; (a) loading directions and (b) loading functions for the applied vertical and

horizontal stresses respectively.

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A graded meshing approach was incorporated in the analysis, smallest element size being

0.5mm for the both models (Figure 6.6).

Figure 6.6: Mesh set-up for the (a) 15 and (b) 30 degrees asperity model respectively.

The properties of the rock joint contacts are shown in Table 6.2. They were assumed based on

testing models, using a normal penalty coefficient that could guarantee a minimum level of

penetration between the two sliding blocks, and which was ultimately assumed to represent

wearing of the surfaces due to sliding. The tangential penalty value was of one-tenth of the

corresponding normal value, as recommended in the ELFEN user’s manual. Global discrete

properties for newly generated fractures were taken as being equivalent to the specified rock

joint contact properties.

Rock joint contact properties Unit Value

Joint cohesion MPa 0

Joint friction degrees 38

Joint normal stiffness (Penalty value ) nP MPa/mm 240

Joint shear stiffness (Penalty value ) tP MPa/mm 24

Table 6.2: Shear model. Properties of the rock joint contact and discretised fracture elements.


The simulated shear stress-displacement curves are shown in Figures 6.7 and Figure 6.8 for the

model with 15 and 30 degrees joint asperity respectively.

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Figure 6.7: Shear stress-displacement curves for joint models with 15 degrees asperities and varying

normal stress.

Figure 6.8: Shear stress-displacement curves for joint models with 30 degrees asperities and varying

normal stress.

Avi movie files showing the fracture pattern development under increased loading conditions are

included in Slide 5 (enclosed DVD). Figure 6.9 shows the comparison between the simulated

and experimental (Huang et al., 2002) shear stresses at failure. The comparison was positive for

lower applied normal stresses, whilst overall encouraging results were observed in terms of the

general trend of the curves. The specific nature of the material tested by Haung et al. (2002), a

mix of chalk, sand and water, probably might have added extra difficulty in terms of exact

reproducibility of the experimental results, since some parameters for the modelled material had

to be numerically derived from published sources (e.g. fracture toughness). Note that the

experimental results refer to 3D simulations, whilst the ELFEN models were carried out in 2D

plane-strain conditions. However, it was evident from the comparison between Figures 6.7 and

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6.8 with the experimental results (Figure 6.10) that the simulated models were characterised by

a much lower deformability (i.e. greater modulus of deformation). In view of the results of the

sensitivity analysis in terms of adequate penalty coefficients to represent normal and shear

stiffness of simulated joint surfaces (Section 5.4), it was argued that the difference could be

related to the choice of the specific penalty coefficients used in the analysis and describing the

contact properties of the two joint surfaces.

Figures 6.9: Comparison between simulated and experimental (Huang et al., 2002) maximum shear

stresses at failure for the (a) 15 degrees and (b) 30 degrees model respectively.

Figure 6.10: Shear stress-displacement curves for joint with (a) 15 and (b) 30 degrees asperities

respectively. Original test results from Huang et al. (2002).

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Typical failure modes observed from the simulated 15 and 30 degrees joint models are shown in

Figures 6.11 and 6.12 respectively. For comparison purposes, they also include the failure

modes observed from the specimens of the direct shear experiments conducted by Huang et al.

(2002).

Failure modes included sliding, shearing, separation and crushing, the occurrence of which

depended upon the magnitude of the applied normal stress. Sliding occurred, accompanied by

limited wearing, under a lower normal stress level. Under a higher vertical stress level, the

asperities were fractured by shear, and subsequently sliding took place. Vertical cracking

associated with flexural bending occurred in some of the models. This was considered as a

result of the specific geometrical definition of the ELFEN model, which considered the normal

stress as applied directly on the rock block, rather than through a stiffer interface (e.g. steel

platen).

Note that complete shearing for the models with 1.5MPa of applied vertical stress is not shown

in Figure 6.12 since the simulations stopped after attaining the specified maximum shear load of

10MPa at 0.1 seconds of simulation time. Either an increased applied shear load or a longer

simulation time would have been necessary in this case to obtain the complete shearing of the

tooth-shaped asperity.

As experimentally observed by several authors (Handanyan, 1990; Pereira and de Freitas, 1993

and Grasselli, 2006), the breakage of the asperity could be caused by either shear stress or

tensile stress, with tensile cracking developing at the base of asperities during shear

deformation. As described in Section 4.4, the current ELFEN crack model can only numerically

simulate crack generation, extension and coalescence under pure Mode I (tensile mode). The

asperity breakages observed in the simulated models are due to growth and coalescence of

tensile fractures, which ultimately result in the formation of a shearing surface. The orientation

of the shearing plane, defined as the angle between the fractured plane and the horizontal plane,

depended greatly on the magnitude of the applied normal stress, being nearly horizontal under a

high-applied normal stress, regardless of the joint geometry.

As shown in Figures 6.11 and 6.12, the ELFEN simulated models compared reasonably well

with the experimental results in terms of the failure modes observed under different conditions

of applied vertical stress.

- 137 -


Normal stress 15 Degrees asperity - ELFEN 15 Degrees asperity - Huang et al. 2002

0.1MPa

Sliding failure

0.3MPa

Sliding failure

0.5MPa

Shear failure

1.0MPa

Shear failure

1.5MPa

Shear failure, with significant tension cracks at the toe of the asperity. Crushing.

Figure 6.11: Typical failure modes observed from the simulated 15 degrees joint model at different

values of normal stress. Also shown are the failure modes observed from the specimens of the direct

shear experiments conducted by Huang et al. (2002).

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Normal stress 30 Degrees asperity - ELFEN 30 Degrees asperity - Huang et al. 2002

0.1MPa

Sliding failure

0.3MPa

Sliding - Shearing failure

0.5MPa

Shear failure, with significant tension cracks at the toe of the asperity

1.0MPa


1.5MPa


Figure 6.12: Typical failure modes observed from the simulated 30 degrees joint model at different

values of normal stress. Also shown are the failure modes observed from the specimens of the direct

shear experiments conducted by Huang et al. (2002).

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Although the use of lower penalty coefficients (Figure 6.13a) determined a better comparison

between simulated and experimental shear stress-displacement curves, the simulated models

with lower normal penalty showed an increased penetration between elements of the sliding

blocks (Figure 6.13b).

Figure 6.13: (a) Comparison between simulated and published (Huang et al., 2002) shear stress-shear

strain curves, with simulated curves being characterised by decreasing values of applied normal penalty

; (b) modes of failure for curves with of 240MPa/mm and 8MPa/mm respectively, showing the

greater degree of penetration observed for the latter case.

nP nP

6.5 Summary and discussion

This Chapter presented a series of numerical simulations of simple shear tests using the hybrid

FEM/DEM code ELFEN, evaluating the capability of the proposed hybrid approach to simulate

the progressive shear behaviour of rock joints with tooth-shaped asperities.

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Overall, the results showed how the ELFEN code could be used in analyses concerning more

typical shear mechanisms, with asperity breakages and shearing in the simulated models due to

growth and coalescence of tensile fractures.

The following limitations, however, should be accounted for:

i. The material used by Huang et al. (2002) in their experimental results refers to a mix of

chalk, sand and water. The value for the fracture toughness was not readily available

and had to be inferred based on empirical correlations. Potential effects in terms of

fracture initiation and propagation should therefore be considered.

ii. A more comprehensive analysis should have included simulating the triaxial response

for the experimental material, in order to comply with the relative behaviour presented

in Huang et al. (2002). However, due to the time-consuming nature of each numerical

simulation this was not a feasible option. The reader should note how the uniaxial

models described earlier in Chapter 4 required more than 36 hours for completion.

iii. In relation to points (i) and (ii) above, the numerical analysis considered the intact

Young’s Modulus of the simulated material as being equivalent to the published value.

Calibrating the simulated material parameters to the experimental results could have

contributed to define a more suitable value, which might have accounted for the lower

deformability observed in the simulated models.

iv. The analysis showed again the difficulty associated with the choice of a specific normal

penalty value. The use of a relative low normal penalty coefficient led to a more

realistic behaviour in terms of comparison with the experimental (shear) stress-strain

response, but the associated displacement/deformation of the upper block was not

acceptable due to the excessive degree of penetration between the upper and lower

blocks. Following the discussion presented in Section 5.4, it is argued that the use of a

penalty method to simulate contact interaction, hence joint behaviour in ELFEN, may

introduce extra complexity in the numerical analysis. Whereas normal penalty and

normal joint stiffness were found, in magnitude, to be equivalent (see Section 5.4.1), the

penalty method is primarily a numerical artifice implemented in the ELFEN code to

avoid physical penetration between objects. Additionally, the guidelines for calculating

normal and tangential penalties suggested in the ELFEN user’s manual do not necessary

apply to every problem under consideration and the user is left with no other option

than an intensive computational approach to determine suitable values.

v. The ELFEN Mohr compressive model with Rankine cut-off allows only the simulation

of Mode I fracturing: incorporating in the numerical solution a mix Mode I-II fracturing

may improve the simulation of the experimentally observed shear mechanism.

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In conclusion, both Chapter 5 and Chapter 6 have discussed how modelling results in ELFEN

clearly depend on a careful and lengthy calibration process, for which no clear guidelines exist.

The modelling approach implemented in the current research has attempted to provide a general

path for future ELFEN applications. The following Chapter will show the potential application

of a combined FracMan-ELFEN approach in terms of rock mass strength characterisation. The

encouraging results and contributions presented hereafter are believed to prove the relative

effectiveness of the chosen modelling strategy.

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A new modelling approach for determination of rock mass strength of jointed pillars

7

A new modelling approach for determination of

rock mass strength of jointed pillars and

implications for rock mass strength

characterisation

7.1 Introduction

Chapters 4 and 5 introduced the use of the hybrid FEM/DEM code ELFEN (Rockfield, 2005) as

a new modelling approach for the investigation of the mechanical response of fractured rock

masses. This Chapter illustrates the applicability of such a numerical method for the

determination of rock mass strength of jointed pillars.

The analysis was carried out in 2D plane strain conditions and initially included numerical

modelling of the failure mechanism of a pillar intersected by a single plane of weakness; the

intention was to investigate the validity of the model rationale through a comparison with the

theoretical strength reduction associated with the variation of the angle of orientation for a pre-

existing single discontinuity.

The computational analysis was then extended to investigate the effects of multiple pre-existing

discontinuities on the overall pillar strength. The study assumed 2D pillar models containing

varying fracture geometries, derived from discrete fracture networks systems generated in

FracMan (Golder, 2005) and based on actual field mapping data, as previously described in

Chapter 3. Modelled pillar strengths were compared with existing empirical mass strength

approaches for determination of pillar strength. The analysis also investigated the correlation

between mapped fracture intensity and simulated pillar strength, in an attempt to provide a

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stronger link between mapped fracture systems and rock mass strength than is possible with

existing rock mass classifications.

7.2 Engineering aspects of the design of hard-rock pillars

Rock pillars can be defined as the in-situ rock between two or more underground openings. In

the design of room-and-pillar or stope-and-pillar systems, the loading capacity of the pillar, i.e.

its strength, is equally as important as the stability of the roof and walls (Nordlund et al., 1995).

Studies on hard rock pillars have been undertaken by several authors (Hedley and Grant, 1972;

Von Kimmelman et al., 1984; Krauland and Soder, 1987; Hudyma, 1988; Potvin et al., 1989;

Sjoberg, 1992; Lunder and Pakalnis, 1997) and more recently the subject has been reviewed by

Martin and Maybee (2000).

7.2.1 Pillar stress analysis

Figures illustrating typical elastic stress distributions for pillars between rectangular openings,

with the width of the pillar set equal to the opening, can be found in Hoek and Brown (1980).

They demonstrate how stress distribution is dependent on pillar shape, expressed in terms of

pillar width to pillar height ratio. Slender pillars are typically characterised by stress

distributions across the centre of the pillar that can be approximated to uniaxial conditions.

Conversely, triaxial conditions are generated at the centre of squat pillars by the minor principal

stress acting across the centre of the pillar and increasing to a significant proportion of the

average pillar stress (Hoek and Brown, 1980).

Recent research has further confirmed that the stress distribution in pillars is a function of the

pillar geometry (Martin and Maybe, 2000). Following advances in computer technology, such

distributions can readily be determined through numerical computer programs. Lunder and

Pakalnis (1997), investigating the stress distribution in hard-rock pillars in Canadian mines,

show that the stress distribution in the pillar is a function of the ratio of 1σ to 3σ and the pillar

geometry. However, as reported by Maybee (1999), confinement in a pillar also depends on

the ratio between the far-field horizontal stress k 3σ and vertical stress 1σ ; the effects of

could be ignored only for pillars with width to height ratio of less than 1.

k

When considering pillar failure, it is important to remember that the propagation of failure from

an initiation point within the pillar does not necessarily result in the failure of the pillar in its

entirety. In highly stressed pillars, failure typically initiates at the corners and in the centre of

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the pillar sidewalls, resulting in the transfer of load from the failed material towards the core of

the pillar (Hoek and Brown, 1980); entire pillar failure eventually occurs when the strength to

stress ratio for the material forming the inner pillar core falls below unity.

7.2.2 Pillar strength

Mine pillars provide a clear example of the importance of determining accurate mass strength

values. Techniques for estimating pillar strength, defined as the ultimate load per unit area of a

pillar, use empirical formulae based on survey data from actual mining conditions

(Iannacchione, 1999). Empirical methods fail to consider specific failure mechanisms and

limitations exist associated with their intrinsic derivation from specific material properties (size,

shape and stress conditions). Numerical simulations offer a potentially useful means of

overcoming some of the limits of the empirical methods.

As reported by Lunder and Pakalnis (1997), empirical pillar formulae can be expressed

according to the following general expression:

⎟⎟⎠

⎞⎜⎜⎝

⎛+= β

α

σHWBAKp [7.1]

where pσ is the pillar strength and the term represents the strength of unit cube of the rock

material forming the pillar. W and

K

H are the pillar width and height respectively. The

constants A and are derived empirically, likewise the power coefficients B α and β . For

instance, in the formulae proposed by Krauland and Soder (1987) and Sjoberg (1992) the

constants A and are equal to 0.778 and 0.222 respectively, whilst in the empirical formulae

proposed by Hedley and Grant (1972) and Von Kimmelmann (1984) the constants

BA and are

equal to 0 and 1 respectively.

B

Pillar formulae can also be subdivided in two groups, influenced by shape and size (Lunder and

Pakalnis (1997). For a given rock type, shape effect formulae assume that a pillar of given

shape, expressed in terms of its width to height ratio, will have a constant strength independent

of changes in its size. Size effect formulae are based on the principle that, for a given rock type,

a pillar of given shape (pillar width to height ratio) will have reduced strength as its size

increases, since samples of increasing size are thought to contain more structural discontinuities.

However, Hoek and Brown (1980) have shown that for intact rock above a sample size, side

length of 1m to 1.5m, the resultant decrease in sample strength due to increasing sample size

- 145 -


becomes negligible. Figure 7.1 illustrates how the strength achievable in a pillar largely

depends on the confining stress developed across the centre of the pillar; the confinement effect

is ultimately related to the pillar width-to-height ratio.

Figure 7.1: Effect of confining stress on compressive strength of intact and fractured rocks (after Gale,

1999. In: www.cdc.gov/niosh/mining/pubs/publist.html).

A summary of the most common empirical pillar strength formulae was provided by Martin and

Maybee (2000); typical empirical mass strength and width-to-height ratio models are illustrated

in Figures 7.2 and 7.3.

Figure 7.2: Pillar width-to-height ratio models for different empirical pillar strength formulae.

- 146 -


Figure 7.3: Effect of rock quality on pillar strength (based on Hoek and Brown, 1980).

In the current research it was assumed that the basic strength relationship for square pillars

could be been taken as (Brady and Brown, 1993):

7.05.0 −= HWK masspσ [7.2]

where pσ is the ultimate pillar strength and is the strength of unit cube of the rock

material forming the pillar; W and

massK

H are the pillar width and height respectively. It is

recognised that this approach takes into consideration both material strength and pillar shape in

the calculation of pillar strength.

7.2.3 Numerical analysis of the failure of rock pillars

Progressive failure in hard-rock pillars typically occurs in the form of spalling or slabbing. As

discussed in Section 2.7.1, current numerical models cannot predict fracture behaviour with a

reasonable level of confidence; major difficulties are associated with the capture of the

transition from a continuum to a discontinuum failure process. When using the Hoek-Brown

failure criterion (Hoek et al., 2002), the strength of a rock mass can usually be expressed in

terms of a constant cohesive component (related to the parameter s ) and a normal-stress or

confinement-dependent component (related to the parameter ); hence for pillars with width

to height ratios greater than 1, the strength should increase as the confining stress increases

(Martin and Maybe, 2000). Hoek and Brown (1980) suggested that for a good quality rock

mass the progressive spalling and slabbing should be treated according to an elastic-brittle

constitutive criterion. However, Martin and Maybee (2000) showed how the conventional

bm

- 147 -


Hoek-Brown parameters used within the context of continuum analysis of hard-rock pillars

predicted failure envelopes that were not in agreement with the empirical results. With

reference to work by Hedley and Grant (1972), Martin and Maybee (2000), noted that beyond a

pillar width to height ratio of 1.5, the elastic-brittle response over predicted the pillar strength.

These results agreed with work by Lunder and Pakalnis (1997), who argued that the confining

stress-dependent frictional strength component contributes less to the overall pillar strength than

predicted by the conventional Hoek-Brown failure envelope. This has led to an improved

understanding of the nature of the failure of hard-rock pillars, and the suggestion that the pre-

peak stress-induced slabbing and spalling is fundamentally a cohesion loss process (Martin,

1997) and that Hoek-Brown brittle parameters should be used instead to characterise this

process (Martin et al., 1999).

Hoek-Brown brittle parameters are, however, not applicable to conditions where the frictional

component of the rock mass strength can be mobilised and dominates the behaviour of the rock

mass (Martin and Maybee, 2000). This suggests that a different numerical approach has to be

adopted to investigate the failure of naturally jointed pillars.

Figure 7.4 illustrates four different cases of naturally jointed pillars, showing how the specific

mechanical response is directly linked to the structural geology of the pillar.

Figure 7.4: Failure modes of naturally jointed pillars (after Nordlund et al., 1995).

Figure 7.4(a) shows how for a naturally jointed pillar, the occurrence of pre-formed blocks may

result in the lateral kinematic release of these blocks due to the increasing vertical load. The

effects of such type of failure would be clearly more evident for relative slender pillars.

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Similarly, as shown in Figure 7.4(b), a relative low width-to-height ratio may favour the

formation of inclined shear fractures transecting the pillar. Figure 7.4(c) shows how a pillar

with a set of transgressive fractures could be expected to fail if the angle of inclination of the

fractures to the pillar principal axis of loading exceeds their angle of friction. Failure will

typically occur in a buckling mode for a pillar with well developed foliation or schistosity

parallel to the principal axis of loading (Figure 7.4d).

Nordlund et al. (1995) discussed the use of the codes UDEC and 3DEC (Itasca, 2005) to

determine failure modes in jointed pillars. They considered pillars intersected by, or containing

a sub-vertical single joint and determined the failure modes of the pillars as a function of the

joint characteristics.

In 2001, Alber and Heiland, investigating pillar failure at shallow depth, observed that the

failure could not be explained by the typical rock engineering approach of comparing pillar

strength with stresses induced on the pillar by mining activities. They adopted a new approach

based on the applications of the principles of fracture mechanics. Using Stacey’s (1981) strain

extension criterion and the strain energy release rate criterion (Shen and Stephanson, 1993),

they were able to give a reasonable explanation for the observed pillar behaviour, concluding

that the formation of the observed fractures in the pillars could be associated with: (i) principal

stress related causes (i.e. causes related to the stress path history of the pillar and the rock strata

adjacent to the pillar as a result of geological and excavation history), (ii) strain related causes

and (iii) energy release-rate related fracture initiation and propagation.

This Chapter presents the results of a new approach to the modelling of progressive pillar

failure. The proposed approach emphasises the importance of representing structural elements

(i.e. discontinuities) within a numerical model in order to reproduce the fracture evolution of a

given jointed rock mass. The uniaxially loaded jointed pillar models (with ratio of

1:2.5) presented in Section 5.5 showed how their mechanical behaviour was characterised by

displacement and rotation of large pre-existing blocks. The base case model used was

developed as a result of the initial modelling strategy presented in Chapter 5. The analysis is

further extended in the present Chapter to consider different jointed pillar geometries with

increasing ratio, in an attempt to review the dependence of pillar behaviour on the

initial fracture network, in terms of ultimate strength and mechanical response.

H:W

H:W

As shown by the numerical examples presented in Chapter 4, the hybrid FEM/DEM code

ELFEN allows the analysis of rock mass failure as a combination of shear along or tensile

opening of existing fracture planes and brittle failure resulting from the accumulation and

- 149 -


growth of stress-induced fractures, hence capturing the continuum/discontinuum transition

typical of jointed rock masses and better reflecting the key role of rock fractures.

7.3 2D modelling of pillars with one single intersecting fracture

In hard rock and low stress environments existing fractures control the potential failure modes

and the associated extent of failure. In the specific context of mine pillars, natural fracture

planes provide less resistance to sliding, hence reducing the pillar loading capacity. This

section presents the results of 2D modelling of pillars with one single intersecting fracture,

discussing the influence that through-going fractures can have on pillar stability. The simulated

results are then compared with the typical variation of peak strength with angle of inclination of

a single intersecting joint predicted by analytical solutions.

7.3.1 Model set up

The model consisted of a 2.8m x 7m pillar uniaxially loaded and containing a single intersecting

fracture inclined at different angles with respect to the direction of the major principal stress

(Figure 7.5).

Figure 7.5: Model definition and loading directions for the pillar with a single intersecting fracture.

The shear strength of the joint and the pillar-roof/pillar-floor contacts are defined respectively

according to:

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fnf στ φtan= [7.3]

and

cnc στ φtan= [7.4]

where the parameters fφ and cφ represent the friction angle of the joint and the friction angle of

the pillar-roof/pillar-floor contacts respectively. Kinematic conditions of equilibrium can be

expressed in terms of the parameters fφ and cφ and the dip angle α of the joint:

fφα ≤

fφα > and cf φφα +≤ [7.5]

cf φφα +>

The first expression in Equation [7.5] indicates that the slip along the plane of weakness cannot

occur and the failure of the pillar is controlled by a combination of fracturing of the intact rock

and sliding along the joint; in this case the properties of the contacts (i.e. cφ ) do not affect the

pillar stability. The second expression states the condition in which slip along the joint is

kinematically possible; the nature of the contacts controls the sliding of the pillar ends with

respect to the roof and floor interfaces. The third expression describes pure slip along the plane

of weakness, with no influence of the pillar-roof (floor) contacts.

Materials parameters for the intact rock material and rock-platen contacts are included in Table

7.1. The intact rock material was modelled using the ELFEN Mohr-Coulomb compressive

model with a Rankine tensile cut-off.

No gravity loading was applied, and the loading function used in the analysis is illustrated in

Figure 7.6. Whereas Section 5.6 proposed the use of an 8 seconds loading function for

equivalent pillar models, slightly faster loading rates were implemented in the current analysis

in order to limit computational run-times. As discussed in Sections 5.3 and 5.6, a damping

factor of 0.1 was implemented in the first stage of the loading phase in order to dissipate

vibrational energy and to allow the system to converge at a steady state.

- 151 -


Figure 7.6: Loading function for the 2D modelling of pillars with a single discontinuity.

Jointed pillar Model - Material and contact properties Unit Value








Intact rock

material



Surface friction, φf degrees See Table 7.2

Normal stiffness (Normal penalty ) nP GPa/m 1

Rock

fracture

Shear stiffness (Tangential penalty ) tP GPa/m 0.5


Poisson’s ratio of platen 0.3 Platen


Rock / platen cohesion cc MPa 0

Rock / platen friction φc degrees See Table 7.2


Rock-Platen

contacts

Rock / platen shear stiffness (Tangential penalty ) tP GPa/m 0.5

Table 7.1: Jointed pillar model; material and contacts properties used in the analysis. The unconfined

compressive strength is indicated purely for reference purposes, since it does not represent a direct

ELFEN input parameter. See Table 7.2 for the different fracture friction angles and rock/platen friction

values used in the analysis.

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The model was simulated for different values of the dip angle α . The friction angle of the joint

plane ( fφ ) was kept constant at 30 degrees. Two cases (A and B) were then considered based

on the value of the angle of friction for the pillar-roof/pillar-floor contact ( cφ ), as presented in

Table 7.2.

Angle of friction for the

pillar-roof/pillar-floor

contacts cφ (degrees) Dip of joint α (degrees)

Friction of joint fφ (degrees)

Case A Case B

10

30

45

60

75

90

30 30 5

Table 7.2: Different fracture friction angles and rock/platen friction values used in the analysis.


Figure 7.7 shows the variation of maximum estimated pillar strength with dip angle α of the

joint. Progressive fracture evolution for the models described above is shown in Figures 7.8 and

7.9 for case A and B respectively. Slide 6 in the enclosed DVD contains movie files illustrating

the complete fracturing process.

Figure 7.7 reproduces the typical variation of peak strength with the angle of inclination of a

single intersecting joint predicted by the theory at a given constant confining pressure (e.g.

Hoek and Brown, 1980 and Brady and Brown, 1993). Note that in this specific case the pillar

models were uniaxially loaded, hence the confining pressure was zero. In accordance with the

theory (minimum strength occurring for 2/4/ fφπα += ), the models with yielded

the minimum estimated pillar strength, independently of the applied value of

o60=α

cφ , with failure

occurring as pure sliding along the joint surface.

- 153 -


Figure 7.7 clearly shows the effects of the different contact properties in terms of the value

of . For , the models were characterised by a lower shear resistance along the pillar

end-contacts. This contributed to a greater degree of freedom in terms of lateral displacement

accompanying the slip along the joint, and hence a lower estimated pillar loading capacity (e.g.

cases with and in Figure 7.7 and 7.9). For and , the pillar

end-contacts did not have any significant effect. Although theoretically the models with

and met the condition of slip along the plane of weakness (

cφ o5=cφ

o30=α o5=cφ o75=α o90=α

o75=α o90=α cf φφα +> ),

actual slip along the joint plane was prevented by the joint surface not day-lighting on the pillar

faces, with the failure of the pillar occurring as a result of fracturing of the intact rock material.

For fφα ≤ failure occurred as a combination of fracturing of the intact rock material and

limited sliding along the joint.

Figure 7.7: Modelled variation of maximum estimated pillar strength with dip angle α of the joint (a)

and theoretical curve (b).

The mechanism of failure included, for a certain critical value of the angle α (typically

fφα > ) and , separation between the pillar and the contacts, with the two parts of the

pillar rotating and simultaneously sliding along the joint. The model with and

was characterised by a failure mechanism similar to that illustrated in Figure 7.4(d),

with thin vertical slabs forming parallel to the pre-inserted vertical fracture, which eventually

failed by progressive buckling of the outer slabs. The reader is referred to Slide 6 in the

enclosed DVD for a more detailed illustration of the described mechanisms.

o5=cφ

o90=αo30=cφ

- 154 -


2D pillar with a single intersecting fracture. Case A ( ) o30=cφ

o10=α

Fracturing of intact rock

0.20% 20.6MPa 0.42% 35.2MPa 0.70% 32.0MPa 1.20% 15.9MPa

o30=α


0.40% 7.88MPa 0.58% 15.0MPa 1.17% 23.0MPa 1.50% 1.6MPa

-

o45=α

Sliding along joint plane

0.50% 0.49MPa 1.0% 1.09MPa 1.50% 2.18MPa

-

o60=α


0.01% 0.04MPa 0.02% 0.04MPa 0.10% 0.03MPa

o75=α


0.20% 11.8MPa 0.55% 14.7MPa 0.79% 18.3MPa 1.00% 10.73MPa

o90=α


0.04% 15MPa 0.11% 29.0MPa 0.50% 21.9MPa 1.00% 24.5MPa Figure 7.8: Progressive fracture evolution for Case A ( ). Strain measured as percentage of

initial pillar height.

o30=cφ

- 155 -


2D pillar with a single intersecting fracture. Case B ( ) o5=cφ

o10=α Fracturing of intact rock

0.09% 9.7MPa 0.23% 20.9MPa 0.30% 9.3MPa 0.89% 5.3MPa

o30=α

Combination of sliding along joint plane and fracturing of intact rock

0.50% 0.9MPa 1.00% 2.3MPa 1.50% 5.6MPa 1.67% 7.7MPa

-

o45=α Sliding along joint plane

0.25% 0.05MPa 0.50% 0.03MPa 1.00% 0.01MPa

- -

o60=α


0.01% 0.01MPa 0.04% 0.08MPa

o75=α


0.16% 10.3MPa 0.36% 20.7MPa 0.40% 2.3MPa 0.68% 1.1MPa

o90=α


0.05% 16.7MPa 0.12% 29.6MPa 1.00% 21.9MPa 1.50% 20.3MPa Figure 7.9: Progressive fracture evolution for Case B ( ).Strain measured as percentage of initial

pillar height.

o5=cφ

- 156 -


It is recognised that the selected relatively fast loading scheme might have had some effect on

the estimated ultimate pillar loading capacity; however, this series of tests served to demonstrate

the general validity of the modelling strategy described in Chapter 5. The modelled pillars

yielded results which agree with the mechanical response expected from theoretical analysis.

The results also demonstrated how, in the case of a pillar intersected by a major joint, the failure

of the pillar itself is controlled by the friction angle of the contacts (pillar-roof and pillar-floor)

and the friction angle of the plane of weakness. Overall, the analysis showed the effectiveness

of the proposed hybrid FEM/DEM approach to simulate pillar failure, since it captured both

failure of the rock material and displacement/rotation along the existing natural joints.

7.4 A 2D geomechanical model of naturally jointed pillars in ELFEN

The analysis assumed 2D jointed pillars containing a specified number of fracture traces based

on a realisation of the Middleton mine DFN model presented in Chapter 3. The results of the

numerical examples presented in Chapter 5 provided the basis for the use of the assumed

ELFEN modelling parameters in the definition of the model.

7.4.1 Model set-up

The computational model consists of a jointed pillar and relatively small sections of the

surrounding rock. The analysis considered pillars with different ratio of 1:2.5, 1:1 and

2:1 respectively. As an integral part of the current research, Rockfield Software developed an

interface enabling ELFEN to upload files generated within FracMan, which allowed definition

of the 2D arrangement in space (i.e. x and y coordinates) of the DFN fracture planes for the

Middleton model (assuming a specific sampling plane). For 3D simulations the interface would

upload 3D coordinates (x, y and z) of the planar surfaces.

H:W

Two-dimensional sampling planes of 14m (width) x 7m (height) and corresponding to simulated

Middleton pillar faces were selected from the FracMan model. Smaller sections of the sampling

planes were used to define the 2D fractures network arrangements for the 7m x 7m and 2.8m x

7m pillars respectively. Figure 7.10 shows an example of the various sampling planes

considered in the FracMan DFN model for one of the Middleton mine pillars.

- 157 -


Figure 7.10: Pillar region in FracMan with indication of the sampling planes used to define the 2D

fracture traces models for ELFEN.

Figures 7.11 to 7.13 illustrate the different fracture network geometries considered in the

analysis and the respective pillar dimensions in terms of ratio. Numerous simulations

were undertaken using slender 2.8m x 7m pillar models, as these demonstrated critical aspects

of the failure process at earlier stages than the wider pillars and computer run-times were also

significantly less.

H:W

- 158 -


Typical dimensions for the pillar model with

= 1:2.5 H:W

Models RA to RE sampled on the same FracMan

simulated pillar face. Models P13B and P13C

sampled on a simulated pillar face orthogonal to

that of models RA-RE.

RA =2.62 21P#23 Fractures

RB =1.80 21P#14 Fractures

RC =2.60 21P#19 Fractures

RD =2.63 21P #23 Fractures

The term denotes the fracture intensity, which is measured as total length of fractures divided by the area of the pillar face.

21P

RE =2.66 21P#18 Fractures

P13B = 2.95 21P#24 Fractures

P13C = 2.89 21P #23 Fractures

Figure 7.11: Typical pillar dimensions and fracture geometries for the simulated 2.8m x 7m pillar.

- 159 -


Typical dimensions for the pillar model

with = 1:1 H:W

The term denotes the fracture

intensity, which is measured as total

length of fractures divided by the area of

the pillar face.

21P

All models sampled on parallel FracMan

simulated pillar face.

RA-2 =2.37 #39 Fractures 21P RB-2 =2.58 #37 Fractures 21P

RC-2 =2.33 #35 Fractures 21P RD-2 =1.79 #24 Fractures 21P

Figure 7.12: Typical pillar dimensions and fracture geometries for the simulated 7m x 7m pillar.

- 160 -


Typical dimensions for the pillar model with

= 2:1 H:W

The term denotes the fracture intensity,

which is measured as total length of fractures

divided by the area of the pillar face.

21P

All models sampled on parallel FracMan

simulated pillar faces.

RA-3 =2.60 #77 Fractures 21P RB-3 =1.82 #48 Fractures 21P

P42A-3 =2.91 #80 Fractures 21P P42B-3 =2.37 #64 Fractures 21P Figure 7.13: Typical pillar dimensions and fracture geometries for the simulated 14m x 7m pillar.

It is emphasised that special cases have to be considered to ensure correct insertion of fractures

and assignments of properties and in order to facilitate the correct meshing of the discretised

elements. As this is a function of the selected minimum element mesh size, it was necessary to

verify that excessively small elements were not formed, due to either very closely spaced joints

or termination of joints in close proximity to each other. In the former case, this was resolved

by merging joints together, whilst the latter situation was remedied by snapping together the

nodes defining the near intersection. These adjustments did not affect the statistical basis of the

FracMan model and within the overall context of fracture development and mass strength it is

assumed they had a negligible influence.

- 161 -


Loading functions and displacement damping assumptions were similar to those shown in

Figure 5.14 in Section 5.4.1. Whereas in the current research the loading of the pillar models

was simulated as if they were subjected to uniaxial laboratory loading conditions, the analysis

could be extended with further models to investigate pillars created by staged excavation within

pre-stressed 2D regions containing the same fracture systems.

The intact rock material was modelled using the ELFEN Mohr-Coulomb compressive model

with Rankine tensile cut-off. The Mohr-Coulomb constitutive model (see Section 4.6.2.1) was

assumed for the discretised fractures. Rock properties for the intact rock, rock fractures and

rock-platen contacts are equivalent to those included in Table 7.1. These were derived from a

combination of laboratory measurements on the Hoptonwood (Middleton) limestone and typical

values for similar limestones from the literature (Stephen, 1987 and Bearman, 1999). The

friction angle fφ for the fracture surfaces was 30 degrees, whilst the angle of friction cφ for the

pillar-roof/pillar-floor contact was 5 degrees. For the Mohr-Coulomb model of the intact rock,

internal values of cohesion and friction ic iφ were determined from the value for ciσ of 48MPa

and Hoek-Brown parameters of 1=s and 312 ±=im (typical for limestone), over a limited

range of confining stress, using the program RocLab with GSI set to 100 (Hoek et al., 2002).

7.4.2 Modelling results

Stress-strain curves for the simulated pillar models are shown in Figure 7.14, whilst Figures

7.15 to 7.17 present the fracture evolution at peak stress for the 2.8m x 7m, 7m x 7m and 14m x

7m pillar models respectively. Avi movie files showing the complete fracturing process are

included in Slide 7 (enclosed DVD).

- 162 -


Figure 7.14: Stress-strain curves for the jointed pillar models. (a) 2.8m x 7m, (b) 7m x 7m and (c) 14m x

7m pillar models.

- 163 -


RA RB RC RD

0.42% 3.27MPa 0.75% 11.03MPa 0.77% 2.91MPa 0.77% 2.94MPa

RE P13B P13C

0.75% 4.40MPa 0.53% 0.65MPa 0.75% 3.65MPa

Figure 7.15: Fracturing evolution at peak stress for the 2.8m x 7m pillar models.

RA-2 RB-2

1.05% 11.57MPa 0.87% 7.03MPa

RC-2 RD-2

0.97% 11.37MPa 0.74% 15.15MPa

Figure 7.16: Fracturing evolution at peak stress for the 7m x 7m pillar models.

- 164 -


RA-3 RB-3

1.19% 22.1MPa 0.92% 27.1MPa

P42A-3 P42B-3

1.07% 9.7MPa 1.49% 23.0MPa

Figure 7.17: Fracturing evolution at peak stress for the 14m x 7m pillar models.

The combined observation of Figure 7.14(a) and Figure 7.15 clearly confirmed how the

mechanical response of the 2.8m x 7m pillar models was controlled by critically aligned planes

of weakness, as anticipated in Section 5.5. The estimated pillar strength ( pσ ) and modulus of

deformation ( ) were dependent on the fracture intensity , with higher values measured

for the models with the lower fracture intensity . Pillar models 7m x 7m (Figure 7.14b and

Figure 7.16) and 14m x 7m (Figure 7.14c and Figure 7.17) showed again a mechanical response

in terms of

mE 21P

21P

pσ and controlled by the fracture intensity of the pre-inserted fracture network.

However, as illustrated in Figure 7.16 and 7.17, pillars with increasing ratio were

characterised by lateral spalling occurring in the early stages of the simulations and additional

fracturing of the intact core as the loading of the pillar was increased. The increased

confinement for the pillars with higher ratio was responsible for the higher measured

pillar strengths and lower deformability (i.e. higher ), since lateral spalling did not

compromise the effective bearing capacity of the pillars.

mE

H:W

H:W

mE

Overall, the simulated progression of failure from the outside of the pillar towards the central

core, and the associated stress distributions (Figures 7.18 to 7.20) compares well with previous

- 165 -


observations and modelling of pillar behaviour (e.g. Hoek and Brown, 1980; Iannachione,

1999). Limited new fractures are generated in the early stages of loading, in addition to

extension of pre-existing fractures and the beginning of block ejection from the pillar surface.

A clear load-bearing core appears to be established in all simulations just before the peak stress

is accrued, with the outer layers of rock becoming detached. Within the core new fracturing

occurs in local positions where the longer pre-fractures appear to channel stress concentrations.

In the post peak region the remaining core is almost wholly consumed by local fracturing. It is

evident that, partly due to the initial pre-fracture pattern, the overall failure behaviour is

asymmetric. The author is not aware of significant work in the literature providing numerical

analysis of asymmetric behaviour for naturally jointed pillars; hence the current analysis clearly

constitutes a contribution to knowledge, demonstrating how relative slender pillars could be

more sensitive to the presence of inclined discontinuities, which ultimately control the overall

failure process.

RA RB RC RD

0.42% 3.27MPa 0.75% 11.03MPa 0.77% 2.91MPa 0.77% 2.94MPa

RE P13C P13B

0.75% 4.40MPa 0.75% 3.65MPa 0.53% 0.65MPa

All models except P13B: ELFEN negative convention for compressive fields (Pa). White areas within pillar material indicate stress values lower than 5kPa.

Model P13B: ELFEN negative convention for compressive fields (Pa). White areas within pillar material indicate stress values lower than 2kPa.

Figure 7.18: Axial stress contour plots at peak stress level for the 2.8m x 7m pillar models.

- 166 -


RA-2 RB-2

1.05% 11.57MPa 0.87% 6.99MPa

RC-2 RD-2

0.97% 11.37MPa 0.74% 15.15MPa

ELFEN negative convention for compressive fields (Pa). White areas within pillar material indicate stress values lower than 0.1MPa.

Figure 7.19: Axial stress contour plots at peak stress level for the 7m x 7m pillar models.

- 167 -


RA-3 RB-3

1.19% 22.1MPa 0.92% 27.1MPa

P42A-3 P42B-3

1.07% 9.7MPa 1.49% 23.0MPa

Key: ELFEN negative convention for compressive fields (Pa). White areas within pillar material indicate stress values lower than 0.2MPa.

Figure 7.20: Axial stress contour plots at peak stress level for the 14m x 7m pillar models.

7.4.3 Comparison with empirical pillar formulae

The simulated pillar strengths derived from the ELFEN modelling of jointed pillars were

compared with the empirical methods for the estimation of pillar strength introduced in Section

7.2.2.

Using a GSI/RMR value of 70 (based on the results of the field mapping at Middleton mine), a

ciσ value of 48MPa (Table 7.2) and an of 10, the RocLabim 1 program (Rocscience, 2005 and

Hoek et al., 2002) for the determination of rock mass strength gave a rock mass uniaxial

1 RocLab is a software package for rock mass strength analysis using the Hoek-Brown criterion.

- 168 -


compressive strength cσ of 9.0MPa, and a partially confined rock mass compressive strength

cmσ of 13.5MPa. The former is insensitive to the value of , and the latter varies between

12.4 and 15.8MPa for in the range 8 to15. The confining stress range for the latter is

between the tensile strength and ¼ of the compressive strength of the intact material (Hoek et

al., 2002). These values of rock mass compressive strength were used in Equation [7.6]:

im

im

-0.70.5

massp HW K σ = [7.6]

where pσ is the ultimate pillar strength and is the strength of unit cube of the rock

material forming the pillar; W and

massK

H are the pillar width and height respectively. The value

of pσ was adjusted for the effective width ( ) of a rib pillar (Brady and Brown, 1993): eW

PAWe 4= [7.7]

where A is the plan area of the pillar and P is the pillar circumference.

Two approaches can be followed which differ in the way the value is estimated. Using massK

cmσ directly from RocLab as a value for and putting massK WWW e 2== for rib pillars gave

the first column of results for pσ in Table 7.3. Using the rock mass uniaxial value of cσ from

RocLab as a pσ value for a square pillar with W = 2.8m and H = 7m in Equation [7.6] gave

of 21.3MPa. Again adjusting for massK WWW e 2== in equation [7.7] for rib pillars, gave

the second column of results for pσ in Table 7.3. The average results from the ELFEN

modelling are in the third column of pσ values in Table 7.3.

As discussed by Hoek et al. (2002), it is sometimes useful to consider the overall behaviour of a

rock mass rather than the detailed failure propagation; this is particularly true, for instance,

when considering the strength of a pillar. Using cmσ directly from RocLab as a value for

provides a better estimate of the overall strength of the pillar.

massK

- 169 -


Width W

(m)

Width

(m)

eW Height H

(m)

H:W

ratio

pσ (MPa)

massK =13.5

pσ (MPa)

massK =21.3

pσ MPa)

ELFEN

2.8 5.6 7 0.4 8.2 12.9 3 to 11

7 14 7 1.0 12.9 20.4 7 to 15

14 28 7 2.0 18.3 28.9 10 to 27

Table 7.3: Comparison of empirical with modelled average pillar strengths.

Comparison between empirically estimated and ELFEN modelled pσ values shows how the

ELFEN modelled pillar strength magnitudes provide a lower and upper bound range of values

respectively for the corresponding pillar strengths calculated using the RocLab approach and

cmσ as a value for (Figure 7.21). It is considered that this is associated with the

variability, in terms of fracture intensity/pattern, which is accounted for in the current DFN

based numerical models. In this context, the analysis suggests how the proposed numerical

approach could provide useful additional information concerning fracture development and rock

mass behaviour when considering the stability of excavations.

massK

The modelled pillar strengths are lower than those calculated using RocLab and cσ to derive

the value. It is noted that this approach is more appropriate to describe the overall failure

propagation process, with failure initiating at the boundary of an excavation when

massK

cσ is

exceeded by the stress induced on that boundary.

Figure 7.21: Potential implications of the numerical results in terms of rock mass strength

characterisation as a function of fracture intensity and failure mechanisms.

- 170 -


It is suggested that Figure 7.21 summarises an important novel contribution of the current

research, particularly in terms of rock mass strength characterisation as a function of fracture

intensity and failure mechanisms, which will be further discussed in the following sections.

Figures 7.22 and 7.23 allow a comparison between the results of the simulations described

above and empirical results for a variety of pillar types. All are presented as average capacity

(stress) normalised to the intact ciσ values against the pillar ratio. The simulations

included a range of values from 1.8 to 2.6 m/m

H:W

21P 2 and the empirical data showed trends for

comparable rock materials including metasediments and limestones. It is noticeable how the

mechanical response of slender pillars ( of 0.4) was dependent on the initial fracture

pattern and intensity, whilst wider pillars ( of 2) showed a similar behaviour independent

of the initial fracture intensity. Despite some relative uncertainty in material parameter values,

the magnitudes of the pillar strengths and shape effects appeared similar.

H:W

H:W

Figure 7.22: Comparison between the results of several simulations described in the text and empirical

results for a variety of pillar types. All are presented as average capacity (stress) normalised to the intact

ciσ values against the pillar ratio. The simulations include a range of values from 1.8 to

2.6 m/m

H:W 21P2. Note: [1] Von Kimmelmann et al., (1984); [2] Hedley and Grant (1972); [3] Sjoberg (1992)

and [4] Krauland and Soder (1987).

- 171 -


Figure 7.23: Comparison between simulated results and width-to-height ratio models for an equivalent

rock type (based on Hoek and Brown, 1980).

7.4.4 Analysis of the relationship between rock mass strength and fracture intensity for

the modelled pillars

Figure 7.24 shows the variation of simulated pillar strength with fracture intensity for the

different models described in the preceding sections; the assumed linear correlations apply to

in the range [1.8, 2.95].

21P

21P

Figure 7.24: Correlation between estimated pillar strength and fracture intensity . 21P

- 172 -


The results clearly show the effects of fracture intensity on the simulated pillar strength. The

correlations indicate how the ultimate pillar strength of a naturally jointed pillar could be

expressed as a function of fracture frequency and 21P HW : ratio, i.e.

( HWPfp :,21= )σ [7.8]

Additionally, since the current numerical analysis also accounts for fracturing of the intact

material and/or extension of existing fractures, Equation [7.8] could be re-written as:

( ) ( )( HWPyhxgfp :,,, 21= )σ [7.9]

where and indicate the dependence to the specific strength criteria controlling

respectively the fracturing through the intact rock material and the shear strength of the

discontinuities. In the specific context of the ELFEN code, is associated to the intact rock

tensile strength (

)(xg )( yh

)(xg

tσ ), rock fracture energy ( ), rock internal cohesion ( ), rock internal

friction (

fG ic

iφ ) and dilation (ψ ). Shear strength properties are described by in terms of

cohesion ( ) and friction (

)( yh

fc fφ ) for the joint surface based on the Mohr-Coulomb criterion (see

also Section 6.2).

At this stage of the research, the purely qualitative character of the formulation proposed in

Equation [7.9] has to be recognised; however, it is believed that Equation [7.9] is original in

combining intact rock behaviour, joint surface conditions (i.e. joint properties), fracture

intensity and shape effects in a single formulation characterising rock mass strength (Figure

7.25). It is noted the general validity of the proposed Equation [7.9], in which specific criteria

could be used in place of and ; hence its validity, in principle, is not limited to the

use of the ELFEN code.

)(xg )( yh

Validation of Equations [7.8] and [7.9] would require, however, extending the current numerical

analysis in order to reproduce the results shown in Figure 7.24 for different combinations of

material parameters and jointing conditions.

- 173 -


Figure 7.25: Diagrammatic explanation for the proposed general expression for rock mass strength

characterisation.

CPU computing time limited the number of simulations that could effectively be completed as

part of the current research for both the 7m x7m and 14m x 7m pillar models; however, a larger

number of models of the slender 2.8m x7m pillar were later added to the analysis in order to

further investigate the possible correlation between fracture intensity parameter , 21P HW : ratio

and rock mass strength suggested in Figure 7.24. In addition to the 5 different 2.8m x 7m pillar

models already described, a further 17 models were analysed. Various fracture patterns (and

associated fracture intensity ) were considered; all but 7 models were developed having a

similar ratio of approximately 0.4 (the FracMan term indicates the number of

fractures per unit area, see also Figure 3.1). The remaining 7 models had a higher ratio

in the range [0.57, 0.95]. Figures 7.26 and 7.27 illustrate the different geometrical definitions

for each model, with the grouping based on the preceding remarks.

21P

2120 P:P 20P

2120 P:P

- 174 -


Models with ratio =0.4 2120 P:P

P21 04C P21 04B P21 10A P21 10D

21P =0.43 21P =0.43 21P =1.03 21P =1.03

P21 10B P21 10C P21 20D P21 20A

21P =1.03 21P =1.03 21P =1.98 21P =2.00

P21 20C P21 20B

21P =2.00 21P =2.00

Figure 7.26: Different geometrical definitions for the models with ratio of 0.4. 2120 P:P

- 175 -


Models with varying ratio 2120 P:P

Mod. ratio P21 15A Mod. ratio P21 14B Mod. ratio P21 13C Mod. ratio P21 12D

21P =1.51

=0.57 2120 P:P

21P =1.41

=0.58 2120 P:P

21P =1.30

2120 P:P =0.59

21P =1.20

2120 P:P =0.60

Mod. ratio P21 10E Mod. ratio P21 09F Mod. ratio P21 07G

21P =1.00

2120 P:P =0.71

21P =0.90

=0.68 2120 P:P

21P =0.70

2120 P:P =0.95

Figure 7.27: Different geometrical definition for the models with varying ratio. 2120 P:P

Figure 7.28(a) shows the variation of the simulated pillar strength with fracture intensity for the

different models. By comparing Figure 7.28(a) with Figure 7.24 (2.8m x 7m pillar case), it was

suggested that the proposed correlation between the simulated pillar strength and the

parameter , within the wider range [0.4, 2.95], could be additionally sub-categorised with the

identification of two curves representing a form of lower and upper bound response respectively

(Figure 7.28b). The results suggest that the proposed pillar strength characterisation defined by

Equations [7.8] and [7.9] could be modified to also account for more specific jointing

conditions/effects; hence, the lower and upper bound curves shown in Figure 7.28(b) could be

considered as representing critical and less critical jointing conditions respectively. In this

context, the results also showed how the failure response for increasing ratio was

21P

2120 P:P

- 176 -


equivalent to that of a model with similar fracture intensity but with a potentially more

critical jointing condition and lower ratio.

21P

2120 P:P

Figure 7.28: (a) Variation of the simulated pillar strength with fracture intensity for the different 2.8mx

7m pillar models; (b) and (c) Correlation between fracture intensity and simulated pillar strength, based

on the proposed grouping into critical and non critical jointing conditions.

An important statement that was made at the end of Chapter 3 was that a notable result of the

DFN synthesis for the Middleton pillar was that few complete blocks were formed (in a 3D

space); this is considered to be undoubtedly true in real rock masses and has obvious

implications for geomechanical modelling. A similar statement could be made by observing the

results shown in Figure 7.28(c) and invoking 2D in-plane continuity for the fracture planes: the

critical and less critical jointing conditions introduced above could represent a measure of the

blockiness of the system. It is reasonable to expect that for such a rock mass to fail under

induced stresses, the processes involved are likely to include extension of existing fractures or

growth of fractures in the adjacent intact rock, or both. The failure process would not initially

involve the high magnitude of immediate displacement made possible by the presence of

complete blocks; hence this may explain the relative higher pillar strength observed for the

simulated pillar models with no initial pre-formed blocks.

- 177 -


Figure 7.29 below combines the previous results with those obtained for the pillar models

intersected by a single through-going fracture (see Section 7.3, case with fφ = 30 degrees and

cφ = 5 degrees). For these models, the ultimate strength was almost entirely dependent on the

relative orientation of the single through-going fracture with respect to the loading direction.

Figure 7.29: Results for the pillar models intersect by a single through-going fracture (case with fφ = 30

degrees and cφ = 5 degrees) and multi-fractured pillar models.

It is suggested that Figures 7.28 and 7.29 form a major and original contribution of the current

research, whose implications are important in terms of:

i. The coherence in the simulated mechanical response shown in Figure 7.29 is an indirect

verification of the effectiveness of the calibration strategy developed for the current

pillar model in Chapter 5.

ii. The strength of the approach is that the anisotropic and inhomogeneous distribution and

effects of the jointing are fully accounted for and the resulting deformation and failure

mechanisms are considered to be more realistic than for other methods.

iii. The results of the numerical analysis serve to prove how the strength of slender pillars

can be highly variable; the increased variability being due to a combination of fracture

intensity and jointing conditions. A major corollary is that a higher factor of safety

would be required when designing slender pillars to account for this variability.

iv. Slender pillars are highly sensitive to the presence of inclined discontinuities. In

relation to point (iii) above, appropriate consideration should be given to the potential

effects of prominent discontinuities when designing slender pillars.

- 178 -


v. Pillar strength equations developed from empirical studies cannot predict, for slender

pillars, these significantly different strength estimates.

vi. In relation to point (v) above, the proposed general Equations [7.8] and [7.9] could be

modified by introducing a specific factor to account for jointing variability and expected

mechanical response. For instance, corrections factors for joint orientations and failure

mechanisms are included in a similar fashion in the Stability Graph Method for open-

stope design (Mathews et al., 1981).

vii. As also discussed in Section 7.4.4, the results shown in Figure 7.29 are believed to

provide the basis for the formulation of a new criterion for rock mass strength

characterisation combining intact rock behaviour, joint surface conditions (i.e. joint

properties), fracture intensity and shape effects.

viii. By extending the analysis to consider staged excavation within pre-stressed solids

containing the same fracture systems, it is suggested that the proposed modelling

approach could have wider application for the design of most excavations in fractured

rock masses.

7.4.5 Implications of the proposed hybrid approach in terms of characterisation of rock

mass strength

The preceding sections discussed how the hybrid method presented in this thesis (and coupled

with the code FracMan) constitutes a new approach to the modelling of jointed rock masses

which maximises the representation of the character of existing rock jointing networks, both in

terms of style of jointing and shear properties of the discontinuity planes. The current section

discusses the potential implications of such an approach in terms of providing a stronger link

between rock mass strength and existing rock mass classification systems. Rock mass

classification systems can be used to provide an initial estimation of the strength of the rock

mass; amid the various rock mass classification systems currently available, the Geological

Strength Index (GSI), introduced by Hoek, Wood and Shah (1992), has the advantage of being

related to the Hoek-Brown failure criterion (Hoek et al., 2002) for rock masses, which is widely

accepted in geotechnical and rock engineering applications. Further details of the GSI system

and Hoek-Brown criterion are given in Appendix I.

Throughout Section 7.4, the numerical analysis demonstrated how stress distribution for the

modelled pillars was dependent on pillar shape, expressed in terms of pillar ratio; the

stress distribution across the pillar centre for slender pillars could be approximated to uniaxial

conditions, whilst triaxial conditions were observed at the centre of the larger pillars. The

estimated value of axial stress

H:W

1σ at failure represented the ultimate strength of the modelled

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pillars, and Figure 7.24 showed how this was related to the initial fracture pattern by means of

the parameter . 21P

Models of a 2.8m x 7m pillar are now investigated, by additionally including the effects of a

lateral confinement, representing in this case the horizontal stress component 3σ . Models with

3σ equal to 1, 2 and 4MPa were included in the study (in addition to the unconfined case

described in the preceding sections). The current analysis has concentrated on 2.8m x7m pillar

models with a fracture intensity of 1.8 and 2.6, which are equivalent respectively to the RB

and RA models in Figure 7.11. Using a combined GSI-RocLab approach for the determination

of rock mass strength, with = 12 and GSI in the range of 70-80 and 40-50 respectively,

produced the curves shown in Figures 7.30(a) and 7.30(b). A comparison between the ELFEN

modelled

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im

1σ - 3σ response and the RocLab-GSI curves is presented in Figures 7.30(c) and

7.30(d) for the models with fracture intensity of 1.8 and 2.6 respectively. 21P

Figure 7.30: Combined GSI-RocLab approach (Rocscience, 2005 and Hoek et al., 2002) for the

determination of rock mass strength, with = 12 and GSI in the range of 70-80 (a) and 40-50 (b)

respectively. Comparison between the ELFEN modelled response and the RocLab-GSI approach for the

models with fracture intensity of 1.8 (c) and 2.6 (d) respectively.

im

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Figures 7.30(c) and 7.30(d) suggest an encouraging similarity between the geomechanical

modelling results and the widely adopted RocLab-GSI approach. It is noted how the basic

aspect of the current research was to couple the representation of the geometry of existing rock

jointing with a geomechanical model in which fracturing of the intact rock material is controlled

by tensile strength and fracture energy parameters.

In relation to the findings presented in Section 7.4.4, the results shown in Figure 7.30 are

original in demonstrating how the mapped fracture intensity could be potentially used as a

readily measurable indicator of the structural character of the rock mass. Additionally, these

results could be interpreted as partly confirming the coupled GSI-RocLab approach as a reliable

measure of rock mass strength. The novelty of the current research is evident in the fact that

this verification is the result of numerical analysis of simulated rock pillars and not derived from

direct observations and empirical approaches.

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Further work is recommended to substantiate these results. Figures 7.30(c) and 7.30(d) only

include results from numerical modelling of limestone-equivalent pillars; hence further studies

should include both different rock materials and rock fracture properties.

7.5 Summary and conclusions

Chapter 7 investigated the use of the hybrid FEM/DEM code ELFEN to study the failure modes

of jointed pillars. The FracMan generated fracture network system for Middleton mine was

imported into ELFEN via a specific interface developed by Rockfield Software Ltd, which

allowed the definition of 2D pre-fractured models of pillars. Modelling specifications in terms

of applied displacement damping, contact penalty coefficients and loading functions were based

on the results of the numerical examples presented in Chapter 5. The analysis considered pillars

with different ratios of 1:2.5, 1:1 and 2:1 respectively. H:W

The following comments are made based on the results from the 2D ELFEN modelling of

jointed pillar presented in the preceding sections:

i. The overall simulated progression of failure and associated stress distributions appeared

to be realistic. There is, however, no direct evidence from an actual field scale pillar

failure at Middleton mine. It is noted that the debris which would result from any pillar

failure is mostly inaccessible and it would be likely impossible to distinguish between

small blocks resulting from overstressing in the pillar and small blocks arising from

larger ejected blocks that have been subsequently broken. It is anticipated that further

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detailed work would be needed to examine complete block creation and block sizes

during the failure process, and would be most realistic when undertaken in 3D.

ii. Comparison of Figures 7.14 to 7.22 suggested that the influence of the natural fractures

diminishes with increasing pillar width, consistent with the strength gain seen in Figure

7.22. This is also consistent with observations by Diederichs et al. (2002), who show

that in a specific mine the macroscopic pillar behaviour can be modelled with limited

consideration of rock structure. However, this is unlikely to be true in more tensile

regimes, e.g. hangingwalls and block caving (Pine et al., 2006). It is also clear from

Figure 7.22 that for slender pillars in particular the value of (fracture intensity) has

a critical influence on pillar strength. This influence diminishes with increasing pillar

width. Figure 7.22 also showed how the modelled behaviour indicated a rapidly

increasing average strength with increasing ratio. This type of behaviour is

more commonly predicted for squat or barrier type pillars with higher ratio,

(Madden, 1991) but was also seen at similar lower ratio with Hoek-Brown

brittle fracture modelling (Martin and Maybee, 2000).

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H:W

H:W

H:W

iii. The average axial stresses in the Middleton Mine pillars are estimated to be less than 15

MPa. This indicates that the pillar factors of safety could be as high as 2 from the

ELFEN model and in the range 1.2 to 2 for a ratio of 2 in Table 7.3. Factors of

safety values are at least 1.5 based on visual observation of pillar condition and

observations elsewhere on actual pillar behaviour (Diederichs et al., 2002; Lunder and

Pakalnis, 1997 and Roberts et al., 1998).

H:W

iv. As discussed in detail in Section 5.4.3, the ELFEN model typically yielded deformation

modulus values ( ) for the pillars that were significantly lower than what would be

predicted by empirical methods. The estimated low values could be partly

explained, as noted in Section 5.4.3, considering that the strain in the simulated pillars

was measured from the closure of the whole pillar, whilst the axial stress was measured

as the average stress of a given number of nodes across the pillar section at mid-height:

at late stages of loading the cross section of the pillar would be much reduced, thus the

inferred local deformation modulus could be higher. The unconfined state of the

modelled pillars also allowed a greater freedom of deformation than is implied by rock

mass rating approaches.

mE

mE

v. It is noted that, although the results are encouraging in comparison with previous

empirical mass strength and width to height ratio models (Table 7.3 and Figures 7.21,

7.22 and 7.23), the pillar modelling considered in this research has been mainly carried

out in 2D plane strain conditions. There are two effects, which probably partially

compensate, for which 3D modelling will also provide useful insight: (a), the fractures

in the 2D model cross-section are more continuous out of plane than will be the case in

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3D. The exaggerated out of plane continuity will tend to have led to an underestimate

of the pillar strength. However, in the particular case of Middleton mine there is

substantial real continuity in the major fractures in plane and out of plane, as evidenced

by the mapping. (b), in a 3D pillar model there is greater freedom for failure to occur

due to lack of confinement out of the 2D plane and this means that the 2D model will

tend to have led to an overestimate of the pillar strength.

It is believed that the numerical analysis described in this Chapter provided original

contributions to the study area in terms of:

i. The coherence in the simulated mechanical response shown in Figure 7.29 is an indirect

verification of the effectiveness of the calibration strategy developed for the current

pillar model in Chapter 5.

ii. The analysis of Figure 7.24 showed how the modelled pillar strength (for a given

ratio) varied depending on the fracture intensity of the initial fracture

pattern. These results clearly highlighted the potential contribution of the current

research in terms of rock mass strength characterisation. A qualitative formulation was

proposed in Equation [7.9] which is original in combining intact rock behaviour, joint

surface conditions (i.e. joint properties), fracture intensity and shape effects in

potentially one single expression characterising rock mass strength. The validity of the

proposed formulation, in principle, is not limited to the use of the ELFEN code.

H:W 21P

iii. By extending the analysis to the consideration of a larger number of different pre-

fractured (slender) pillar models it was concluded that the strength of slender pillars can

be highly variable; the increased variability was assumed to be a result of the

combination of fracture intensity and jointing conditions. Simulated slender pillars

were found to be particularly highly sensitive to the presence of inclined discontinuities.

It was concluded that pillar strength equations developed from empirical studies could

not predict, for slender pillars, these significantly different strength estimates.

iv. The current research discussed the possible implications of the proposed coupled

FracMan-ELFEN approach in terms of providing a stronger link between rock mass

strength and rock mass classification systems. The results shown in Figure 7.30

demonstrated that the mapped fracture intensity could be potentially used as an

indicator of the structural character of the rock mass. Additionally, the results were

interpreted as partially confirming the coupled GSI-RocLab approach as a reliable

measure of rock mass strength. It is noted that this verification is the result of

numerical analysis of simulated rock pillars and is not derived from direct observations

and empirical approaches.

21P

- 183 -


In conclusion, the results of the modelling demonstrated that the proposed approach could have

wider application for the design of most excavations in fractured rock. The analysis could be

extended with further models investigating pillars created by staged excavation within pre-

stressed 2D and 3D solids containing the same fracture systems.

Additional work, however, would be recommended, since the current analysis only included

results from numerical modelling of limestone-equivalent pillars; further work would be needed

to include different combinations of material parameters and jointing conditions.

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Evaluation of the applicability of the hybrid FEM/DEM code ELFEN for 3D analysis of jointed pillars

8

Evaluation of the applicability of the hybrid

FEM/DEM code ELFEN for 3D analysis of

jointed pillars

8.1 Introduction

This Chapter discusses the application of the hybrid FEM/DEM code ELFEN to three-

dimensional (3D) analysis, and in particular it reviews the effectiveness of the ELFEN

fracturing algorithm for the 3D case. Due to difficulties in terms of 3D mesh generation and the

associated large computational time required, the current research included only limited 3D

analysis. Numerical examples are presented, consisting of laboratory-scale rock specimen and

pillars intersected by, or containing a single fracture. It is noted that the 3D models presented in

this thesis required computing run-times which in cases were in excess of 30 days, hence

outside the logistical time frame of the current research.

8.2 Hardware considerations for 3D analysis

A PC with a 2.8GHz Pentium IV processor and 512MB RAM was the standard platform used as

part of this research. An important aspect that had to be considered when studying an ELFEN

3D model was access to the post-simulation results. The hardware specifications of the standard

PC used in the current research had a significant limitation in terms of fixed RAM memory and

maximum allocated virtual memory (i.e. temporary increase of the fixed RAM memory), which

eventually determined a series of fatal errors whilst attempting to view the fracture propagation

within the 3D models. The process of generation of a single movie file and relative capture of

the frames (screen-outputs) used to visualise the failure mechanism also required longer time for

the 3D models when compared to the 2D models.

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Some of the 3D models presented in this Chapter required in excess of 15 days to be completed.

The minimum element size of the mesh was taken as 1/10th of the model width. 3D numerical

tests with smaller mesh size and up to 1/50th of the model width and similar loading conditions

had to be necessarily aborted since the time for reaching 50% of the applied load-factor was

estimated in excess of 30 days. For these reasons, it is expected that realistic 3D complex

applications will be extremely computationally intensive, hence requiring the implementation of

parallel processing procedures.

8.3 ELFEN fracturing algorithm for 3D analysis

Chapter 4 introduced the capability of the ELFEN 2D solution scheme to simulate Mode I

(tensile) fracturing; wing-crack initiation from an existing (critically aligned) fracture was also

successfully simulated. The initiation of wing-cracks for the 3D case, whose typical mechanism

is illustrated in Figure 8.1, is a far more complex task, involving the definition of curved

surfaces within a 3D space. This would need to be accommodated within the 3D meshing

processor in ELFEN, which notably involves the definition of tetrahedral elements. The way

for the tetrahedral elements to reproduce these curved surfaces in 3D would necessarily be a

function of the specified mesh size; hence coarser meshes would result in a certain degree of

simplification. Very fine meshes would, at the same way, imply impractical computational run-

times.

Figure 8.1: Wing-crack initiation process for the 3D case (after Chunlin et al., 1998)

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The UCS laboratory-scale model presented in Section 4.5.1 was re-defined in a 3D space with

the insertion of a 20mm (in diameter) circular fracture oriented at 45 degrees with respect to the

applied vertical stress. As shown in Figure 8.2, it was not possible to realistically simulate

perfectly curved surfaces in the 3D model. This result was considered as being dependent on

the relative coarse mesh of 5mm used for the model and the tetrahedral meshing effect

discussed above. The implementation of a 1mm mesh size in the model might have improved

the way the process of fracture initiation was modelled in a 3D space; however, CPU time

restrictions strictly limited the possibility to validate such an assumption. Overall, it was argued

that the results still showed a good comparison with the wing-crack process in a 3D space.

The relative coarse tetrahedral mesh elements used in the analysis likely affected the final

results of a similar model but without the pre-inserted fracture, though crack initiation

apparently did took place in a direction parallel to the maximum principal stress 1σ (Figure

8.3).

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45 Degrees Fracture, Diameter 20mm

Figure 8.2: ELFEN results for the model attempting to generate wing crack extension at the tips of an

existing fracture in a 3D space.

- 188 -


Intact rock specimen

Figure 8.3: Fracturing evolution for an intact 3D model equivalent to the 2D UCS model described in

Section 4.5.1.

8.4 3D modelling of UCS rock specimen with and without the use of mobilised material

parameters with plastic strain

Having discussed the potential dependence of the fracturing process to the selected mesh size

for a 3D problem, this section describes now 3D numerical examples of a model consisting of a

laboratory-scale rock specimen (Figure 8.4). The ELFEN Mohr-Coulomb with Rankine cut-off

material option was used in the analysis; degradation of the initial material parameters with

plastic strain, in terms of rock internal cohesion, friction and dilation, was adopted in some of

the models. Material properties and mobilised parameters associated with plastic strain are

indicated in Table 8.1 and Table 8.2 respectively. For the current analysis, the mobilised values

for cohesion, internal friction and dilation were estimated from published data for Tyndall

limestone (Klerck et al., 2004), which has similar material parameters (e.g. intact Young

modulus, ciσ and ) to those listed in Table 8.1.

The 3D mesh had a minimum element size of 5mm. Loading was applied in the form of relative

applied-displacement of the top and bottom steel-equivalent platens, in an arrangement typical

of uniaxial laboratory conditions.

fG

- 189 -


UCS Model - Rock Material properties Unit Value




Young’s Modulus, E GPa 275








Normal stiffness (Normal Penalty ) nP GPa/m 20 Rock fracture

Shear stiffness (Shear Penalty ) tP GPa/m 2


Poisson’s ratio of platen 0.3





Platen properties Rock-platen contacts

Rock / platen shear stiffness (Tangential penalty ) tP GPa/m 2

Table 8.1: ELFEN 3D UCS model; material parameters and rock-platens contact properties. Note that

ELFEN does not require the use of the rock unconfined compressive strength as a direct input property,

which was listed here purely for reference purposes.

Plastic

strain (%) 0 0.0005 0.002 0.0055 0.011 0.02 0.022 0.03

Cohesion

(MPa) 9 9 8.4 7.1 5.1 2.6 2.6 2.6

Friction (°) 5.6 29.6 35.6 38.4 38.8 40 40 40

Dilation (°) 0 19.6 19.6 19.6 19.6 19.6 5 5

Table 8.2: ELFEN 3D UCS model; mobilised parameters with effective plastic strain for the intact

material.

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Figure 8.4: Geometrical definition for the ELFEN UCS model in 3D space.

The simulated axial stress strain-curves for the 3D UCS models are shown in Figure 8.5, which

also includes a comparison with the results obtained for equivalent 2D models. Also indicated

are specific strain levels corresponding to the screen-outputs showed in Figure 8.6.

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Figure 8.5: (a) Simulated axial stress strain-curves for the 3D UCS models and (b) comparison with the

results obtained for the equivalent 2D models.

- 192 -


Intact UCS Intact UCS with mobilised parameters

A

B

C

D

E

Figure 8.6: Fracture propagation steps for the 3D UCS models described in Figure 8.5(a).

- 193 -


Figure 8.5(a) clearly shows how the 3D models were characterised by a response in accordance

with the selected material model up to a critical strain level, beyond which the occurrence of a

sudden hardening was observed. From the observation of Figure 8.6, it was deduced that the

select (relative) large mesh size played a major role in this hardening behaviour, since it

prevented the formation in a 3D space of clear shear bands, which, on the contrary, were

observed in equivalent 2D models (e.g. Figure 4.12). This hardening response was also

associated with a confinement effect resulting from complete 3D blocks not being removed (i.e.

no observed kinematic release of blocks) once fracturing had occurred.

8.5 3D analysis of jointed pillars

Despite the limitations introduced in the previous sections, it was decided to evaluate the

capability of the ELFEN code to model the mechanical behaviour, in a 3D space, of a pillar

containing or intersected by a single fracture. Whereas the FracMan-ELFEN interface allowed

uploading a complete 3D fracture network representation in ELFEN, some meshing difficulties

were experienced, particularly in relation to the minimum mesh size required to represent the

small 3D tetrahedral elements resulting from tight interlocking and intersection of fractures in

3D. Accordingly, this would have required very expensive computational run-times, which

could not be accommodated within the current research time frame.

Figure 8.7 shows the typical 3D model of a pillar used in the current analysis. Rock material

properties were equivalent to those used for the modelling of the 2D pillars in Chapter 7. The

loading function implemented in the analysis was, however, sensibly faster (1 second simulation

time instead of 8 seconds) in order to limit the required run-times.

Figure 8.7: Typical 3D pillar model with a single intersecting fracture.

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The intact rock material was modelled using the ELFEN Mohr-Coulomb with Rankine cut-off

material option. The Mohr-Coulomb constitutive model (see Section 4.6.2.1) was assumed for

the discretised fractures, with joint friction angle and pillar-roof (floor) contact

friction . Figure 8.8 shows the stress-strain curves for some of the 3D simulated pillar

models, with indication of specific strain levels corresponding to the screen-outputs showed in

Figure 8.9. Movie files illustrating the complete fracturing process are included in Slide 8

(enclosed DVD).

o30=fφ

o30=cφ

Figure 8.8: Stress-strain curves for some of the 3D models. The stages A, B, C, D and E refer to the

fracture propagation steps showed in Figure 8.9.

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Intact 10o Intersecting Fracture

30o Intersecting Fracture

45o Contained Fracture

A

0.05% 14.0MPa 0.06% 6.5MPa 0.05% 6.5MPa 0.05% 9.5MPa

B

0.16% 39.7MPa 0.16% 20.2MPa 0.16% 7.9MPa 0.15% 28.9MPa

C

0.29% 38.7MPa 0.29% 31.7MPa 0.29% 6.6MPa 0.29% 38.8MPa

D

0.46% 38.4MPa 0.46% 39.2MPa 0.45% 8.5MPa 0.45% 49.0MPa

E

1.00% 47.2MPa 1.00% 54.2MPa 1.00% 13.1MPa 1.00% 143.0MPa Figure 8.9: Fracture propagation steps for the 3D models described in Figure 8.8.

- 196 -


The hardening response described by the curves in Figure 8.8 was considered somewhat

anomalous. The pillar models, in particular the intact one, again returned a typical elasto-plastic

response up to a certain critical strain level, and in accordance with the material model

employed in the analysis, beyond which a sudden increase in axial strength was observed. This

behaviour was less evident for the model with the single 30° intersecting fracture, since in this

case pillar failure consisted of a combined fracturing of intact material and sliding along the

joint plane (Figure 8.9). As it would be expected, fracture initiation occurred in a direction

parallel to the applied major principal stress, with the model containing a single fracture

oriented at 45° also showing what was interpreted as a typical wing crack initiation, although

the specified out-put time step was not sufficiently small to allow a better appreciation of the

mechanism. Fracturing occurring along the pillar-roof and pillar-floor contacts was considered

to be the combined result of relative shearing of the pillar ends and the use of possibly low

contact penalties, hence penetration of the pillar structure within the steel platen, with fracturing

initiating at interacting nodes. The initial generation and subsequent fracture interaction within

the pillar appears realistic, however, the origin of the rather unusual behaviour is uncertain.

It was argued that the selected mesh size (0.28m) was possibly too large to allow the definition

of a sufficient number of parallel fractures, whose interaction and coalescence might ultimately

have resulted in the formation of typical shear bands and the consequent failure of the pillar

models. It is noted that the 3D model with the single fracture oriented at 10° eventually reached

a potential peak stress level, which is probably associated with the presence of the pre-existing

fracture, which acted as a macroscopic shear band. As discussed above, technical problems (i.e.

long computational run-times associated with the need of using a relative small mesh size) in

addition to the 3D mesh generation of multiple intersecting fractures, did not allow the

investigation of more complex 3D pillar models. Further studies would therefore be required to

examine the mesh dependence of the results shown in Figure 8.8; however this would probably

require the introduction of parallel computing in order to reduce the time needed to complete the

simulations.

8.6 Summary and conclusions

The current Chapter discussed some of the constraints for 3D analysis in ELFEN. Although

limited 3D pillar models were effectively studied as part of this research, it was argued that the

somehow anomalous mechanical response observed for the simulated 3D pillar models was

dependent on the selected mesh size; in order to better represent multi-fracturing phenomena in

a 3D space, a relative smaller mesh size should have been incorporated in the models. The

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logistical feasibility of such model definition was however precluded by the long computational

run-times required when employing a standard PC with a 2.8 GHz processor.

Ultimately, due to the extremely intensive computations involved in simulations of realistic 3D

applications, parallelisation becomes an obvious necessity. Whereas significant advances in the

development of parallel computer hardware, particularly the emergence of commodity PC

clusters, may make such a parallel computing option feasible and attractive, for problems in

which progressive fracturing takes place the situation may be further complicated by a

continuing change in problem topology, resulting from the creation of new surfaces and objects

(Owen et al., 2004b). The same author argued that while parallel implementation on shared

memory machines may be tractable, considerable difficulties exist for distributed memory

platforms arising from such dynamically changing geometric configurations and boundary

conditions. Although some progress has been made in the development of appropriate

algorithms, considerable further work is required to provide general robust and efficient solution

procedures that account for the complexities encountered in industrial problems.

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Conclusions and recommendations for further work

9

Conclusions and recommendations for further

work

9.1 Introduction

This Chapter serves as a final summary and discussion, and provides recommendations for

further work.

9.2 The use of a hybrid FEM/DEM approach to model the behaviour of a fractured

rock mass, with a particular emphasis on jointed rock pillars

A new numerical modelling approach for naturally fractured rock masses has been applied to

the modelling of mine pillars. Numerical methods and computing techniques have nowadays

become integrated components in studies for rock mechanics and rock engineering; state-of-the-

art numerical techniques for rock mechanics include continuum methods (Finite difference

method, FDM, Finite element method, FEM and Boundary element method, BEM) and

discontinuum methods (Discrete elements method, DEM and Discrete fracture network method,

DFN). Continuum and discontinuum models are traditionally used to simulate multi-fracturing

phenomena and mechanical behaviour of discrete systems; however neither approach alone can

capture the interaction of existing discontinuities and the creation of new fractures through

fracturing of the intact rock material.

In this context, the current research evaluated a new recent hybrid continuum/discontinuum

approach for modelling fractured rock masses, which links different numerical techniques such

as FEM and DEM, allowing for large scale analysis of locally large displacements along

fracture planes. This hybrid approach was coupled with a discrete fracture network model that

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maximises the quality of representation of the geometry of existing rock jointing and fully

accounts for the style of naturally fractured rock masses.

Key aspects and contributions of the current research were:

i. Reviewing the process of developing a stochastic DFN model using the code FracMan

(Golder, 2005) from initial field data; this involved the introduction of an updated

methodology the use of a simplified method to derive synthetic fracture radius

distributions from mapped fracture traces.

ii. The novelty of the ELFEN release version used as part of the current research required

the verification of specific numerical parameters used in the analysis. A specific

methodology was devised in order to develop an adequate modelling strategy that could

be used for future ELFEN applications for comparable problems. An explicit

combination of numerical parameters was proposed specifically for the jointed pillar

models considered in the current research. Although the intensive computational time

required to conduct the calibration process did not allow the validation of extending the

validity of the proposed numerical parameters set to a different class of problems, it is

suggested that the proposed modelling strategy will provide a general template for

future ELFEN applications.

iii. Using an appropriate interface, the synthesised FracMan model represented the source

of 2D trace sections which were subsequently imported in the hybrid FEM/DEM code

ELFEN as part of the study of the mechanical behaviour of jointed rock masses. It was

recognised that the proposed modelling approach was capable of representing not only

complete blocks, as for example in UDEC (Itasca, 2005), but it also allowed inclusion

of networks of intersecting and isolated fractures.

Having given appropriate consideration to the modelling constraints represented by the

necessary careful calibration of the proposed ELFEN pillar model, the current research also

provided encouraging results and novel contributions to improved understanding of modelled

brittle behaviour of rock, in particular the modelling fractured pillars, in relation to the

following:

iv. Capturing the continuum to discontinuum transition typical of brittle failure as a result

of local degradation of the intact rock material due to the insertion of new fractures.

v. Modelling pre-fractured models of 2D pillars, with fracture geometries derived from

discrete fracture network systems generated in FracMan and based on actual field

mapping data, to capture the key mechanical role of pre-existing fractures and their

effects on the modelled pillar strength and deformation. The adopted approach

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modelled the failure behaviour of a rock mass as a combination of the interaction of

existing discontinuities and failure of the intact material deriving from the insertion of

new fractures.

vi. Demonstrating that, for the modelled 2D jointed pillars, the influence of the natural

fractures diminishes with increasing pillar width. The simulated results were also

encouraging when compared with previous empirical mass strength and width-to-height

ratio models. The pre-inserted fracture pattern typically resulted in overall asymmetric

failure behaviour, reflecting the anisotropic and inhomogeneous distribution and effects

of the fractures. Overall, the modelling demonstrated that the proposed approach could

have wider application for the design of most excavations in fractured rock, particularly

for rock masses that are too sparsely jointed to be treated as an equivalent continuum.

vii. Demonstrating how, for a given width-to-height ratio, the modelled pillar loading

capacity was ultimately related to the fracture intensity of the existing fracture network.

A qualitative formulation was proposed combining intact rock behaviour, joint surface

conditions (i.e. joint properties), fracture intensity and shape effects to characterise rock

mass strength. It was also suggested that the validity of such a formulation, in

principle, was not limited to the use of the ELFEN code.

viii. By extending the analysis to consider a larger number of different pre-fractured

(slender) pillar models it was shown that the strength of slender pillars could be highly

variable; the increased variability was assumed to be a result of the combination of

fracture intensity and jointing conditions. It was concluded that pillar strength

equations developed from empirical studies could not predict, for slender pillars, these

significantly different strength estimates.

ix. Discussing the implications of the proposed coupled FracMan-ELFEN approach in

context of providing a stronger link between rock mass strength and rock mass

classification systems. The analysis demonstrated how the mapped fracture intensity

parameter could potentially be used as a readily measurable indicator of the

structural character of the rock mass. Additionally, the results were interpreted to

partially confirm the coupled GSI-RocLab approach as a reliable measure of rock mass

strength. It is noted that this verification was the result of numerical analysis of

simulated rock pillars and was not derived from direct observations and empirical

approaches.

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x. Discussing the potential limitations of the ELFEN pure Mode I failure adopted in the

current release version. Numerical simulations investigating the influence of surface

roughness on the mechanical behaviour of a rock joint showed how the ELFEN code

could also be used in analyses concerning more typical shear mechanisms, with asperity

breakages and shearing in the simulated models due to growth and coalescence of

tensile fractures.

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9.3 Final conclusions on FracMan modelling

The present study presented a new methodology to assist in the development of a stochastic

model in FracMan. It was argued that a critical stage in the generation of a specific FracMan

model was the conversion of the measured fracture trace length data into statistically valid

fracture radius or diameter distributions. The analysis was carried out using an analytical

process (Zhang and Einstein, 2000), which involves the use of circular windows of a defined

diameter (within the rectangular mapped windows) and the relative proportions of fracture

terminations (none, one or two terminations per fracture) on a set-by-set basis. A simplified

method was then introduced to overcome some of the difficulties associated with the complexity

of the published analytical process. The proposed approach included the integration of complex

functions by means of a simple software tool, FNGraph (Minza, 2002).

The FracMan code was used to develop a realistic 3D fracture network model for an

underground limestone pillar based on qualitative and quantitative field data collected at

Middleton mine (Derbyshire, UK). The fact that mapping was carried out in an operating mine

provided several practical considerations that ultimately affected the quality and quantity of data

capture. The mapping area in Middleton was limited to the size of the pillars, which had a

maximum height of about 8m, but with only the bottom 2m physically accessible for detailed

mapping. The variability in estimated fracture intensity from the FracMan model, when

compared to the values mapped in-situ, was a result of the limited amount of field data

available. However, direct graphical comparison for different realisations of the FracMan

model yielded reasonable results.

Despite these considerations, and because current limitations of computing capacity somewhat

limited the number of FracMan realisations that could be subsequently used for geomechanical

modelling in ELFEN, it was argued that the Middleton model generated in FracMan was

ultimately reasonably adequate and represented a best-estimate intensity model for the mapped

pillar. Using an appropriate numerical interface, the synthesised FracMan model represented

the input data of 2D trace sections and 3D fracture planes for the subsequent geomechanical

modelling. The FracMan-ELFEN interface allowed uploading of a complete 3D fracture

network representation within a pillar model in ELFEN. However, due to mesh size restrictions

associated with intensive computing run-times, the analysis presented in this thesis was

primarily focused on 2D scenarios.

A notable result of the DFN synthesis for the Middleton pillar was that few complete blocks

were formed; this is considered to be undoubtedly true in real rock masses and has obvious

implications for geomechanical modelling. It is expected that for such a rock mass to fail under

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induced stresses, the processes involved are likely to involve extension of existing fractures or

growth of new fractures in the adjacent intact rock, or both. The failure process would therefore

not initially involve high magnitude, immediate displacement made possible by the presence of

complete blocks.

The development of the FracMan model for Middleton pillar data highlighted some of the

limitations of field mapping with standard techniques. These limitations include the number of

fractures that can be actually mapped within an achievable window diameter and the speed and

accuracy with which outcrops and excavation surfaces can be mapped. In this context, a

research project currently being undertaken at the Camborne School of Mines is investigating

the use of photogrammetric and laser scanning methods to capture field data, which after post-

processing can produce spatially accurate, detailed 3D bit-mapped representations of the rock

mass.

9.4 Final conclusions on the use of a hybrid FEM/DEM code with fracture insertion

capability to model rock mass behaviour

Current numerical methods are typically grouped into either continuum and/or discontinuum

models, based on the conceptualisation and modelling of the fractured rock mass and the

deformation that can take place in it. As discussed in Chapter 2, a continuum approach may

circumvent some of the difficulties associated with discontinuum models (e.g. DEM models),

since the former does not include the use of parameters such as damping and contact properties

of the discretised elements. The inherent complexity of a DEM model and impracticality of

modelling every fracture in a deterministic way may add to the advantage of using a continuum

modelling approach for particular rock types. However, the utilisation of a continuum approach

to modelling a process that is ultimately discontinuum has some intrinsic limitations. For

instance, for engineering problems in moderately fractured rock masses, a continuum model

may not adequately simulate the stress acting on a specific fracture, which cannot be

approximated to the overall stress, since it depends on the stiffness of the fracture itself and on

the stiffness of the fracture’s surrounding matrix (Cai and Horii, 1993). Trading material

complexity for geometrical simplicity, the continuum approach relies on the homogenisation of

the material properties, with the use of specific constitutive equations for the equivalent

continuum. In addition, blocks relative displacements and interlocking, with associated internal

moments produced by block rotations, cannot be adequately accounted for in a continuum

model. Ultimately, problem scale and fracture system geometry affect the choice between a

continuum and a discrete approach: a continuum model can effectively characterise a rock mass

with either no fractures or many fractures, whilst numerical modelling of moderately fractured

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rock masses, where displacements of individual blocks are possible, are better described by a

discrete approach.

By combining the above methods, a hybrid model can therefore overcome undesired

characteristics and at the same time retain as many advantages as possible. The introduction of

a hybrid continuum/discontinuum approach linking FEM and DEM techniques may, for

example, allow for large-scale analysis and locally large displacements along fracture planes.

More recently, the rapid progress in available computer power has allowed the development of

numerical methods that can simulate the progressive fracture process, including crack initiation,

propagation and interaction. Extension of existing discontinuities and initiation of new fractures

through fracturing of the intact rock material necessarily requires an appropriate topological

data structure update, which results from the process of inserting a new fracture into a

continuum based finite element mesh.

In this context, the current research was based on the use of the hybrid FEM/DEM code ELFEN

(Rockfield, 2005). ELFEN incorporates a compressive fracture model (Klerck, 2000 and

Klerck et al., 2004), which has been shown to recover the salient features of the brittle response

including dilation. The compressive fracture model consists of a Mohr-Coulomb failure

criterion coupled with a fully anisotropic tensile crack model, which can be used within the

ELFEN FEM/DEM framework to introduce physical cracking of the intact material.

2D and 3D models included in this thesis demonstrated the effectiveness of the code ELFEN to

reproduce fracture propagation in a plane normal to the minor principal stress 3σ , i.e. parallel to

the maximum principal compressive stress, 1σ . 2D and 3D studies have also shown that

ELFEN has the capability of modelling the initiation of wing cracks at the tips of a large pre-

existing crack (e.g. Sections 4.5.1 and 8.3). However, the representation of the initiation of

wing-cracks for the 3D case was found to be dependent on the topological update of the meshed

tetrahedral elements, since the selected (relative) coarse mesh resulted in an unavoidable degree

of over simplification.

The proposed approach was used to model the evolution of cracking and the formation of

macroscopic failure planes for a hypothetical 2D rock specimen under uniaxial compression,

demonstrating how rock failure could be viewed as a process with distinct deformation stages,

which include crack initiation, crack propagation and coalescence. Using this philosophy, the

proposed approach could effectively investigate brittle failure of rock masses, which is typically

associated with the accumulation and growth of stress-induced fractures.

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In current practice, continuum analysis techniques are typically used to simulate brittle failure of

rocks, and adopting constitutive criteria such as elastic-brittle-plastic or strain-softening.

Applications of continuum analysis in the modelling of brittle failure around underground

excavations include Fang and Harrison (2002), Hajiabdolmajid et al. (2000 and 2002) and

Martin and Maybee (2000). As proposed by Martin and Maybee (2000), continuum models

with mobilised parameters, although well-established, are not fully applicable to problems

where the frictional component of the rock mass strength can be mobilised and dominates the

behaviour of the rock mass, i.e. in the case of moderately fractured rock masses. From a

modelling perspective, the research presented in this thesis included the modelling of brittle

failure by using an alternative approach that included insertion of pre-existing fractures and

development of new fractures. This included modelling the transition from a continuum to a

discontinuum state. The fracture insertion capability of the code ELFEN could be exploited to

simulate the stress concentration at the tips of pre-inserted flaws, providing an indicator of the

rock structural integrity and its subsequent response to changes in stress and material properties.

Under either tensile or compressive loading conditions, pre-inserted flaws act as the source of

further cracking/fracturing of the rock due to their coalescence, returning the typical mechanism

of brittle failure.

A fundamental part of the current research was a review of the principal numerical input

parameters for the hybrid FEM/DEM code ELFEN and the investigation of their effects on the

modelling results. As clear guidelines for the use of the specific numerical parameters were not

always readily available, the analysis incorporated a sensitivity analysis process. The main

objective was the calibration of the basic pillar model which was then used in subsequent

research on the mechanical behaviour of jointed pillars.

9.4.1 ELFEN - The use of an appropriate displacement damping coefficient and suitable

loading rate

Damping is implemented in the ELFEN analysis in order to dissipate vibrational energy and to

allow the system to converge at a steady state. The ELFEN pre-processor function allows the

definition and assignment of two different types of numerical damping coefficients: (a) a

velocity proportional damping of all nodes (displacement damping) and (b) a contact damping

that alters the contact penalty force. Whereas the analysis in most cases assumed a constant

contact damping of 0.3 (based on the ELFEN user’s manual recommendations), by simply

varying displacement damping for a given loading rate, it was demonstrated how relative higher

damping coefficients tended to inhibit both the extension of existing fractures and the initiation

of new fractures. In a qualitative sense, the results were in agreement with observations made

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by Cundall (1987), who argued how the use of velocity proportional damping could introduce

body forces that in some cases may influence the mode of failure. Simulations with relatively

higher loading rates were less affected by the chosen damping value; this was related to the

dynamic effects introduced in the system by the use of higher loading rates. This ultimately led

to a faster velocity of propagation for the cracks and the subsequent breakdown of the pillar

models. The pillar models with no applied displacement damping produced an increase in axial

strength with increasing loading rate. Although the ELFEN models were established to

simulate rock pillars, the simulated behaviour compares well with observations from

experimental results for rock specimens loaded under uniaxial laboratory conditions reported by

several authors (e.g. Hoek and Brown, 1980, Blanton, 1981).

Considering the effects of loading rate, it is known that material characteristics can be

significantly influenced when loading rates become high, resulting in high strain rates within the

material. The numerical examples provided in Section 5.3 demonstrated how, the crack

initiation and crack velocity was apparently dependent on the loading rate, with higher loading

rates corresponding, for a given damping coefficient, to a greater degree of fracturing of the

pillar models.

In conclusion, the analyses highlighted the following:

i. Whereas relatively faster loading rates (ELFEN numerical time) would limit the

effective computational time needed to complete the analysis, models of jointed pillars

loaded using relative faster loading functions returned a non realistic mechanical

behaviour, both in terms of stress-strain response and fracture evolution.

ii. In order to compensate for undesired dynamic effects and oscillations in the kinetic

energy levels, particularly in the early modelling stages, a given displacement damping

has to be incorporated in the analysis.

iii. The use of displacement damping has, however, a fundamental effect on the fracturing

evolution of the modelled pillars, apparently inhibiting the extension of existing and

newly generated fractures.

A combined two-stage loading/displacement damping set-up was proposed for the pillar

modelling, as shown in Table 5.9. The ELFEN modelled loading time had to be chosen taking

into consideration the associated computational run-times. Because displacement damping also

had to be incorporated in the model (as a measure to compensate for potential dynamic effects

associated with fast loading conditions), a combined strategy was devised, according to which

the pillar was to be uniaxially loaded over a period of 8 seconds in two separate stages: an initial

stage (relative short, i.e. 0.5 seconds) incorporating a value of displacement damping of 0.1, and

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a subsequent longer second stage (7.5 seconds) with no applied displacement damping. The use

of a zero displacement damping in the second stage was deemed a necessary requirement in

order to minimize the effects of displacement damping on the observed fracturing process.

9.4.2 ELFEN - Use of specific contact properties for the discretised elements

Section 5.4 discussed how the deformability of the discontinuities (intended either as interfaces

between blocks or independent discretised fractures) is modelled in ELFEN using prescribed

force-displacement relations resulting in a so-called penalty method. Normal ( ) and

tangential ( ) contact penalties are defined in ELFEN rather than normal and tangential

stiffnesses. The sensitivity analysis therefore included an analysis of the correlation between

joint normal stiffness and normal penalty coefficient, in order to justify the use of given

penalties with reference to published or laboratory measured values.

nP

tP

The analysis showed how the normal stiffness for a modelled joint surface could effectively

be considered equivalent in magnitude to the selected normal penalty coefficient .

Additionally, it was shown that the deformability of the modelled pillars ultimately depended on

the normal penalty coefficient, with the use of relative lower normal penalty coefficients

yielding a more realistic mechanical response in terms of discrete block displacement, lateral

spalling and pillar core fracturing. The use of relatively lower coefficients resulted in a

potential underestimation of the rock mass deformation modulus when compared with estimates

derived from empirical correlations with rock mass classification systems. It was suggested that

this could be explained by considering the manner in which the deformation modulus was

actually estimated for the simulated pillars. The unconfined state of the modelled pillars may

also have been a contributing factor, resulting in a more freedom for deformation than would be

implied by rock mass rating approaches. The relatively low deformation modulus estimated for

the simulated pillar models with <2GPa/m was also considered a reflection of the specific

geometry used for the initial pre-inserted fracture network, in terms of fracture orientation with

respect to loading direction. Based on this work an explicit combination of normal penalty ,

contact damping and : ratio was proposed in Table 5.10 for future analysis of similar

jointed pillar models.

nk

nP

nP

nP

nP

nP tP

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9.4.3 ELFEN - Selection of the minimum element mesh size

The critical time step definition in ELFEN, which ultimately controls the amount of time

required to perform a discrete analysis, is directly proportional to the smallest mesh element .

If the smallest mesh element is set as less than the minimum mesh size used in the analysis, then

fracturing will occur through elements. Problems that require the use of a very small mesh size

will accordingly require a longer computational time, since the smaller the time step the longer

the time for the analysis to be completed. The minimum mesh element size should be selected

in order to optimise the computational requirements and the possibility to introduce, within the

intact continuum, a certain number of fractures.

l

FracMan realisations of 3D discrete fracture network systems were used as part of this research

to derive 2D fracture traces for the ELFEN jointed pillar models. Chapter 3 discussed how the

uploading of the FracMan fractures trace coordinates into ELFEN required careful consideration

when the model was discretised into a finite element mesh, due to the constraints associated

with the assumed minimum mesh element size (typically 0.1m to 0.2m for pillars with a height

of 7m). Specific procedures had to be followed to ensure that either excessively small elements

were not formed, due to either (a) very closely spaced joints or (b) termination of a joint in close

proximity to another joint.

Eventually, the analysis demonstrated that the selected mesh size could have a significant effect

on the process of fracture initiation and interaction, which ultimately resulted in the hardening

behaviour observed in some of the 3D models. ELFEN allows fracturing to occur under Mode I

(tensile) failure only. It was thus suggested that the selected mesh size implemented in the 3D

pillar models was possibly too large to allow the definition of a sufficient number of parallel

tensile fractures, whose interaction and coalescence might ultimately have resulted in the

formation of typical shear bands and the consequent failure of the models. The use of a smaller

mesh size allowed reasonable simulations of typical shear band failure in 2D models of

uniaxially loaded rock specimens. The models created to simulate the shearing of a tooth-

shaped asperity (Chapter 6) also incorporated a very fine mesh, in the order of 1/200 of the

maximum dimension of the problem under consideration (i.e. 0.5mm mesh for a model with a

width of 100mm). However, the models required larger computational run-times. Similar

considerations apply to the model simulating the typical wing-crack initiation in 3D, whose

results were partly affected by the degree of simplification implicit in the use of a relatively

larger mesh size.

It is recognised that the use of a graded mesh may improve the computational side of the

analysis; however, the choice of a particular graded mesh is ultimately dependent on the

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mechanism of the problem under consideration. For instance, a graded mesh was implemented

in the 2D joint asperity model, since the analysis focused on the shearing of the asperity, which

accordingly had a very high mesh density. Pillar models were studied using a uniform mesh

density for the pillar and the pre-inserted fractures, since in this case the analysis focused on the

failure of the entire pillar structure.

9.4.4 ELFEN - Influence of surface roughness on the mechanical behaviour of a rock

joint

Numerical simulations were undertaken in order to evaluate the capability of ELFEN to model

the fracturing of asperities for a rock joint with tooth-shaped asperities.

Overall, the results showed how the ELFEN code could be used in studies involving more

typical shear mechanisms, with asperity breakages and shearing in the simulated models due to

growth and coalescence of tensile fractures. Taking into consideration the limitations discussed

in Chapter 6, it was concluded that the mechanical behaviour observed for the ELFEN models

compared well with published experimental results by Huang et al. (2002) and was also in close

agreement with experimental observations (e.g. Handanyan, 1990, Pereira and de Freitas, 1993

and Grasselli, 2006) which confirmed that the breakage of joint asperities could be caused by

either shear stress or tensile stress.

9.4.5 ELFEN - Hardware specifications and computational limitations

The ELFEN simulations undertaken as part of this research were performed using a PC with a

2.8 GHz Pentium IV processor, with a fixed 512MB RAM memory, extended to a maximum of

1512MB by means of the virtual memory option available within Windows XP.

Some of the 3D numerical models presented in Chapter 8 required in excess of 20 days to be

completed. The 2D 14m x 7m jointed pillar models described in Chapter 7 took an average of

12 days to run, although it is noted that on several occasions two simulations were performed

simultaneously on one machine.

As discussed by Owen et al (2004b), for explicit solution schemes the overall computational

cost of an analysis depends crucially on the processor time spent with Gauss point level

calculations. Therefore, the efficiency of numerical integration schemes for integration of the

constitutive equations has a direct impact on the overall efficiency of the finite element

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framework. In this context, the use of more complex constitutive models could potentially

result in a dramatic increase in analysis time.

Although ELFEN includes a reduction of the return mapping to the solution of a single scalar

equation to improve the efficiency of the overall solution scheme and allows the introduction of

crack closure effects in damage evolution with little impact on the overall analysis costs (Owen

et al., 2004b), it was noted that complex applications were extremely computationally intensive,

hence requiring the future implementation of parallel processing solution procedures. As

discussed in Chapter 8, the implementation of parallel procedures is well established for

conventional finite element problems; however, for problems in which progressive fracturing is

allowed, the resulting topology update provides some additional complexity. Although some

progress has been made in the development of appropriate algorithms, considerable further

work is required to provide general robust and efficient solution procedures that account for the

complexities encountered in industrial problems (Owen et al., 2004a).

9.5 Modelling of naturally fractured pillars using the hybrid code ELFEN

The above discussion has highlighted that appropriate consideration must be given to the

modelling constraints and require careful calibration of the proposed ELFEN pillar model.

The modelling of naturally fractured pillars undertaken using ELFEN was carried out assuming

2D plane strain conditions and initially included numerical modelling of the failure mechanism

of a pillar intersected by a single plane of weakness. The modelled results compared well with

the mechanical response expected from theoretical analysis. The computational analysis was

then extended to investigate the effects of multiple pre-existing discontinuities on the overall

pillar strength, by simulating 2D pillar models containing different fracture geometries, which

were derived from discrete fracture networks systems generated in FracMan (Golder, 2005) and

based on field mapping data (Chapter 3).

The pillar models described in Chapter 7 realistically simulated progression of failure from the

outside of the pillar towards the central core, and the associated stresses distribution, as

described by several authors (e.g. Hoek and Brown, 1980; Iannachione, 1999 and Fang and

Harrison, 2002). The pre-inserted fracture pattern typically resulted in overall asymmetric

failure behaviour, hence reflecting the anisotropy in the response to loading determined by the

existence of one or more sets of discontinuities. The analysis showed how simulated slender

pillars were found to be particularly highly sensitive to the presence of inclined discontinuities.

Increased confinement due to higher ratio resulted in higher measured pillar strengths H:W

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and lower deformability; accordingly, it was found that for these models lateral spalling and

simulated blocks displacement/rotation did not compromise the effective core loading capacity

of the pillars. This is consistent with observations by Diederichs et al. (2002), who suggested

that the influence of the natural fractures diminishes with increasing pillar width.

Figures 7.24 and 7.29 in Chapter 7 clearly highlight the potential contribution of the current

research to the understanding and modelling of the fractured rock mass behaviour. A novel

application is the use of the modelling approach for rock mass strength characterisation. It was

demonstrated how the modelled pillar strength (for a given ratio) varied depending on

the initial fracture intensity . As a result, a qualitative formulation was proposed combining

intact rock behaviour, joint surface conditions (i.e. joint properties), fracture intensity and shape

effects. Significantly, it was noted that, in principle, the validity of the proposed formulation is

not limited to the use of the ELFEN code.

H:W

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Importantly, the numerical analysis showed how the strength of slender pillars could be highly

variable and this increased variability was assumed to be associated to fracture intensity and

jointing conditions. It was concluded that pillar strength equations developed from empirical

studies could not predict, for slender pillars, these significantly different strength estimates.

The current research discussed the possible implications of the proposed coupled FracMan-

ELFEN approach in terms of providing a stronger link between rock mass strength and rock

mass classification systems. Figure 7.30, in Section 7.4.5, demonstrated how the mapped

fracture intensity could be potentially used as an easily determined indicator of the

structural character of the rock mass; the specific fracture shear properties used in the analysis

accounting for surface conditions. Additionally, the modelled results compare favourably with

rock strength estimates derived from the conventional GSI-RocLab approach. A novel

component of the current research was that the above verification was the result of numerical

analysis of simulated (fractured) rock pillars and was not derived from direct observations or

empirical approaches.

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Overall, the results of the modelling demonstrated that the proposed approach could have wider

application for the design of most excavations in fractured rock, particularly for rock masses

that are too sparsely jointed to be treated as an equivalent continuum. The strength of the

approach is that the inhomogeneous distribution of the discontinuities and the anisotropic

effects of the jointing are fully accounted for and the resulting deformation and failure

mechanisms are considered to be more realistic than derived using alternative modelling

methods.

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Although at this stage of the research the loading of the pillar models was simulated as if they

were subjected to uniaxial laboratory loading conditions, it is anticipated that the analysis could

be extended to consider pillars created by staged excavation within pre-stressed 2D and 3D

solids containing the same fracture systems. The analysis of complete block creation and block

interaction during the failure process would certainly be more realistic when undertaken in 3D.

However, the large computational requirements for 3D analysis limited the number and

definition of 3D models completed as part of this research, with the observed results clearly

being dependent on the large selected mesh size.

9.6 Recommendations for further work

The results of the modelling demonstrates that a combined use of discrete fracture network

characterisation and geomechanical modelling of blocky rock masses has wider application for

the design of most excavations in fractured rock.

It is anticipated that the combined use of the FracMan and ELFEN codes will provide a stronger

link between mapped fracture systems and rock mass strength than is possible with current rock

mass classification systems. Based on the modelling results presented, it is a realistic objective

to examine the geometrical aspects of fracturing (spacing, continuity, orientation) in more detail

to provide more control for the rock mass classification approach to mass strength (e.g. RocLab)

and excavation stability.

Recent developments in the use of digital photography and photogrammetry and laser scanning

for remote rock mass characterisation provides an opportunity for more effective mapping of in-

situ jointing data where suitable exposures are available, both for underground and surface

applications. This can have a significant impact on the speed and quality of the fracture

network models created (Pine et al., 2006b). Research is currently being undertaken at the

Camborne School of Mines using the proprietary software Sirovision (CSIRO, 2005) to

remotely capture discontinuity data from digital photography.

The analysis could be extended with further pillar models created by staged excavation within

pre-stressed 2D structures containing the same fracture systems. From the initial modelling of

progressive pillar failure, it is expected that the modelling methodology could also be extended

to consider other design applications, such as tunnel roof behaviour and simulation of

discontinuity-controlled caving mechanics. For instance, research currently being undertaken at

the Camborne School of Mines, in collaboration with EPSRC and Rio Tinto involves the use of

FracMan to investigate the relationship between geological structure and the observed cavability

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of the rock mass. It is expected that as part of the proposed research, some 2D pre-stressed

staged excavation models with fracture traces derived from a relative FracMan model will be

examined in ELFEN.

In addition, further development of the ELFEN code is required to explicitly include effective

stress considerations in geomechanical modelling. In this context, a new project has been

initiated at the Camborne School of Mines in conjunction with EPSRC, the University of Wales

(Swansea) and Rockfield Software, whose main objective is to create an effective stress module

within ELFEN. This will incorporate fluid pressures within fractured rock masses in order to

model slope stability problems in fractured rock masses.

Faster codes and the use of parallel processing will provide further opportunities for more

realistic rock mass modelling, both in 2D and 3D. The implementation within ELFEN of a mix

Mode I-II fracturing processes could also provide further insight into design situations

influenced by more typical shear failure mechanisms. This is currently under development.

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Appendix I

Appendix I

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Appendix I - Discrete Fracture Network models in FracMan

Discrete Fracture Network models in FracMan

A.1 FracMan DFN models

The basis of the code FracMan (Golder, 2005) is to characterise each discontinuity set within a

single structural domain using statistical distributions to describe variables such as the

orientation, persistence and spatial location of the discontinuities. The approach taken

maximises the utility of discontinuity data from mapping of exposed surfaces and boreholes or

any other source of spatial information (e.g. geophysics). Discontinuity data sampled from

exposures in differently oriented outcrops (2D) and boreholes (1D) can be used to synthesize a

3D stochastic discontinuity model that shares the statistics of the samples and allows for

specific (deterministic) discontinuities to be incorporated.

The code FracMan allows the following DFN models to be used:

i. Enhanced Baecher

ii. Nearest neighbour

iii. Levy-Lee

A.1.1 The FracMan Enhanced Baecher model

The Enhanced Baecher model is a development of the original Baecher disk model (Baecher et

al., 1977) and accommodates the option to account for fracture termination at intersections with

pre-existing fractures and for more general fracture shapes (Geier et al., 1989). The geometric

parameters used to describe the fracture network are the density of the fractures (number of

fractures per unit volume), the orientation distribution of these fractures and the size and shape

of the fractures (Long and Billaux, 1987, in Staub et al., 2002). The same requirements as for

the ordinary Baecher model are needed for orientation and size statistics.

As a derivative of the ordinary Baecher model, in the Enhanced Baecher model fracture location

may be defined by a regular (deterministic) pattern or a stochastic process. The stochastic

approach assumes that the fracture centres are randomly located in space using a Poisson

process, although the number of fracture centres within equivalent volumes is kept uniform, i.e.

it realises a uniform distribution in space, as illustrated in the example of Figure A.1.

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(a) (b)

Figure A.1: Enhanced Baecher model. a) Example of no. 8 fractures centres randomly located in space

with a uniform distribution of their number within equivalent volumes and b) Example of 3D geometric

Enhanced Baecher model in FracMan, showing intersections of polygonal fractures.

A.1.2 The FracMan Nearest-Neighbour model

The Nearest-Neighbour model can be considered identical to the Enhanced Baecher model

except for its assumptions in terms of spatial distribution of fractures (Geier et al., 1989). The

Nearest-Neighbour model simulates the tendency of fractures to be clustered around major

points and faults by preferentially producing new fractures in proximity of earlier fractures

(Dershowitz et al., 1998). In addition to the basic input data similarly required by the Enhanced

Baecher model, the Nearest-Neighbour model necessitates information in terms of spatial

interrelationship of fractures and fracture terminations, in accordance with (Staub et al., 2002):

( ) bx CLxP −= [A.1]

Equation [A.1] expresses the probability of a fracture to be located at a point x in 3D space.

The term represents the distance from point L x to the nearest fracture of a previous, dominant

group, and b and are empirical constants. C

An example of Nearest-Neighbour model generated using the code FracMan is shown in Figure

A.2.

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Figure A.2: Example of 3D geometric Nearest-Neighbour model, showing cluster of fractures

preferentially located close to pre-determined fractures, shown in green and orange colours respectively.

A.1.3 The FracMan Levy-Lee model

The Levy-Lee model is a fractal model whose key feature is that fracture centres are created

sequentially by a Levy flight process in 3D and that the size of a fracture is related to its distance

from previous fractures (Geier et al., 1989). The Levy flight process is described by the

probability function:

( ) DL LLLP −=>′ [A.2]

where is the length of each step and L ′ D is the fractal dimension of the point field formed by

the fracture centres. According to equation [A.2], clusters of smaller fractures around widely

scattered larger fractures are produced as a result of the correlation of fracture size to step

length.

The Levy-Lee model requires the same data as the Enhanced Baecher model, plus data

describing the spatial structure of fracture population, e.g. correlation between fracture spacing

and size, correlation between fracture spacing and orientation, fractal dimension of the point

field defined by fracture centres (Geier et al., 1989, in Staub et al., 2002).

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Figure A.3: Example of 3D geometric Levy-Lee model. Different realisations with, respectively, fractal

dimension D and time step dimension of 0.8-0.4 (Grey cluster) and 1.0-0.5 (Green cluster).

A.1.4 Choice of a specific FracMan DFN model

Considering the relation between different fracture sets, two groups of models can be identified:

stationary models and non-stationary models. Whilst the Enhanced Baecher model is a

stationary model, the Nearest-Neighbour and Levy-Lee are non-stationary models and they

incorporate the option of generating spatial and temporal relations between different set of

fractures. This requires, however, a good knowledge of the geological and geomechanical

history for the specific area.

The definition of the fracture groups in the Nearest-Neighbour model allows more explicit

representation of geological observations than is allowed in the Levy-Lee model, for cases in

which the fractures can be classified into different groups in correspondence with a theory of

fracture genesis (Staub et al., 2002).

The quality of DFN models depends directly on the quality of the field mapping, thus is related

to factors such as available exposure areas of the mapping sites, limits of window/scanline

mapping regarding trace length biases and effect of cut-off assumptions in the mapping

methodology. The choice of a specific DFN model is therefore based on assumptions made

from field data and geological observations. In a general case the Enhanced Baecher model

provides an appropriate solution for different applications. This model corresponds to a Poisson

process leading to an exponential distribution function for the fracture spacing along a sampling

line. Generally, it can be expected that fractures will not have uniquely defined spacing values,

but will rather take a range of values, possibly according to some form of statistical distribution.

- 218 -


Based on measurements on a number of sedimentary rock masses, Priest and Hudson (1976)

found that in each case the distribution form for the fracture spacing could be approximated by a

negative exponential distribution. Brady and Brown (1993) have reported these findings to be

verified for a wider range of rock types. Scanline surveys may be incorporated into the

fieldwork methodology in order to validate these assumptions and to substantiate the use of the

Enhanced Baecher model as typical generation mode of the DFN model. Nearest-Neighbour

and Levy-Lee model should be used when Poisson processes are deemed as not being

satisfactory to simulate fracture patterns.

A.2 Fracture characterisation

Dershowitz (1984) introduced the concept of primary and secondary characteristics to describe

the geometry of fractures. Primary characteristics are used to define the fracture system (i.e. the

fracture assembling in a 3D space), whilst secondary characteristics provide a useful grade of

correlation between the primary characteristics. Considering work by La Pointe (1993), the

geometric characteristics of fractures that are necessary to describe the rock mass can be

grouped to consider individual fractures and fracture networks (Table A.1).

The grouping implicitly expresses the distinction between primary and secondary

characteristics, since individual fractures properties are used to define the fracture system

(which is ultimately represented by specific fracture sets), whilst fracture network parameters

provide the correlation between fractures in each set and between different sets.

Individual fracture Fracture networks

Orientation Degree of interconnection

Trace length or size Termination style

Aperture Spatial inhomogeneity

Planarity Anisotropy

Roughness Abutting relations

Surface morphology Fracture frequency

Location

Table A.1: Geometric characteristics of fractures and fracture networks (after La Pointe, 1993).

- 219 -


Observation and measurement data allow the definition of distribution forms that constitute the

basis for the development of the rock fracture system model. Typical distribution forms for the

principal fracture characteristics are summarised below.

i. Fracture Shape. This parameter is typically assumed to be constant within the model.

The code FracMan allows for specific distribution forms to be applied to the fracture

aspect ratio and to the fracture elongation.

ii. Fracture size. This parameter is expressed by the interpolation of 2D data (mapped

fracture traces) to 3D data. The three main distribution forms, as reported by

Dershowitz (1984) are: (a) Exponential, (b) Lognormal and (c) Hyperbolically shaped

distributional form. Within the code FracMan, fracture size is represented by the

distribution form for the radius of a fracture of equivalent area. Several types of

distribution forms are then available.

iii. Fracture location. This is typically extrapolated from 2D data based on a stochastic

process, the most common being a Poisson process. As discussed in section 5.3, a

Poisson process produces an exponential distribution of fractures along a sampling line

in any direction and the code FracMan accommodates a typical Poisson process model

(Enhanced Baecher) as well as probabilistic (Nearest Neighbour) and fractal (Levy-Lee)

models.

iv. Fracture orientation. Distributions such as Fisher, Bingham, Bivariate Fisher and

Bivariate Bingham are typically incorporated in FracMan to represent fracture

orientation.

- 220 -

Appendix I - Field mapping at Middleton mine

Field mapping at Middleton mine (Derbyshire,

UK)

B.1 General overview of Middleton Mine

OMYA, an international white minerals company, provided access on several occasions to their

limestone mine at Middleton in Derbyshire (UK). Middleton mine is a classic square room-and-

pillar mining operation with drift access working mostly under a cover of about 100m. Pillars

are planned for nominal 16m x 16m dimensions in plan with rooms 14m wide. However

completed pillars are usually smaller, due to over-break. Rooms are created in single pass

operations up to 8m high, but double height rooms are created in suitable ground. The mine

area is divided by several normal faults across which there can be significant vertical

displacement. The orientations of the major faults determine the layout of the rooms.

Middleton mine develops beneath Middleton Moor and the excavation is within the payable

Hoptonwood limestone, whose thickness in the mined area averages 80m (Stephen, 1987). The

Middleton mine orebody consists of strong, thickly bedded creamy-white crystalline limestone

with interbedded clay wayboards. All strata are very continuous with remarkably consistent

thickness and low dip (less than 5°). The unconfined compressive strength ranges from 40-

65MPa and the Young’s modulus is about 20GPa (Bearman, 1991). Rock mass classification

results in RMR values of about 60 to 70 and Q-index values of 6-17. The main differences from

location to location are due to the intensity (spacing) of fractures.

B.2 Data collection

Figure B.1 shows a copy of the survey sheet that was developed for collecting data when

carrying out site visits at Middleton mine. It was designed taking into consideration the

definitions given by the ISRM (1981) in terms of parameters needed to characterise the

discontinuities and allow their engineering properties to be established. The survey log fulfils

the requirements for both scanline and window mapping techniques, whilst also including

parameters from the NGI (Q-system) classification system (Barton et al, 1974). A specific

separate data sheet for the estimation of the NGI Q-index was designed based on a version

proposed by Barton (2002).

- 221 -


Figure B.1: Copy of the survey sheet that was developed for collecting data when carrying out site visits

at Middleton mine.

- 222 -


The parameters included in the survey log are as follows:

i. Dip and dip direction describe the attitude of the discontinuity in space.

ii. Type of discontinuity indicates the nature of the fracture being investigated (e.g. joint,

bedding plane, foliation, and fault).

iii. Aperture is the perpendicular distance between adjacent walls of a discontinuity.

iv. Infill describes the material that separates adjacent discontinuity walls. Presence of soft

clay or other highly weathered materials and rock particles was considered.

v. Spacing is in general terms the perpendicular distance between adjacent discontinuities.

For a scanline survey it represents the progressive distance along the scanline, whilst in

the case of window mapping it is usually given as the perpendicular distance between

discontinuities inferred to belong to the same set. The column reference indicates the

discontinuity that is used as reference point for spacing calculation.

vi. Persistence is the trace length of a discontinuity as observed in the mapped exposure.

The nature of the termination of each discontinuity trace is also measured (e.g.

discontinuity trace terminates in rock, against another discontinuity, or it extends

outside the mapped region).

vii. JRC is the roughness coefficient introduced by Barton (1973).

viii. Roughness class represent the roughness profiles as described in ISRM (1981).

ix. , , , and SRF are the parameters used in the NGI (Q-system) classification

system (Barton et al., 1974).

nJ rJ aJ wJ

x. RQD is the Rock Quality Designation developed by Deere (Deere et al., 1967).

Following is a brief summary on the characterisation of the joint surface roughness and on the

method implemented for the estimation of the parameter from field observations of

roughness of joint wall surfaces.

rJ

B.2.1 Estimation of the coefficient Jr (Q-index, Barton et al., 1974) based on field

observations

The characterisation of the joint surface roughness aspect is clearly scale dependent. As a

descriptive term, joint surface roughness is often described based on representative profiles such

as those proposed by ISRM (1981) for small and intermediate scales of observation. The joint

roughness coefficient JRC (Barton, 1973) is another empirical index generally used for joint

surface characterisation. JRC values vary between 0 (slickensided planar surfaces) to 20

(Rough stepped surfaces). Barton and Choubey (1977) indicated a representation of roughness

- 223 -


profiles with associated JRC values for small-scale features (0.1m scale) whilst Barton and

Bandis (1983) reported the intrinsic scale dependent nature of the coefficient JRC.

Barton and Bandis (1990) described guidelines for selecting appropriate JRC values for

different types of joints at a scale of 0.2m to 1.0m (JRC20 and JRC100 respectively), relating them

to the profiles included in the ISRM methods for visual description of joint roughness (Table

B.1). JRC20 and JRC100 values as indicated in Table B.1 have been considered to represent the

maximum roughness coefficient JRCx associated to the specific ISRM roughness class (i.e.

JRC20 and JRC100 relative to the roughness class profile no. 2 could not be greater than 14 and 9

respectively).

rJ is the joint roughness parameter from the Tunnelling Quality Index Q developed by Barton

et al. (1974). values are listed in Table B.2 in addition to the associated ISRM roughness

profiles. Note that values for stepped-like ISRM roughness profiles are not defined within

the NGI Q-index guidelines and were deduced by the author based on the linear correlations

shown in Figure B.2.

rJ

rJ

Roughness Profile JRC20 JRC100

1 20 11

2 14 9

3 11 8

4 14 9

5 11 8

6 7 6

7 2.5 2.3

8 1.5 0.9

9 0.5 0.4

Table B.1: JRC values for different types of joints at scales of 0.2m and 1.0m (after Barton and Bandis,

1990).

- 224 -


ISRM Roughness Profile Jr description Jr value

1 Rough or irregular, stepped 4.5

2 Smooth, stepped 3.0

3 Slickensided, stepped 2.5

4 Rough or irregular, undulating 3.0

5 Smooth, undulating 2.0

6 Slickensided, undulating 1.5

7 Rough or irregular, planar 1.5

8 Smooth, planar 1.0

9 Slickensided, planar 0.5

Table B.2: Parameters and associated ISRM roughness values. values for stepped-like ISRM

roughness profiles deduced based on the linear correlations shown in Figure B.2. Reference to specific

notes on the use of (Barton, 2002) is recommended.

rJ rJ

rJ

Figure B.2: Estimated values for parameter Jr relative to ISRM (1981) roughness profiles 1 - 9.

- 225 -


B.3 Middleton mine - Data capture and synthesis

Discontinuity mapping at Middleton mine was undertaken by the author and Dr. Zara Flynn,

Research Assistant at Camborne School of Mines. The contribution of Dr. Zara Flynn to the

development of the FracMan DFN model for Middleton mine is gratefully acknowledged.

Mapping was carried out in two separate visits to Middleton mine, including:

i. July 2003: Site familiarisation and mapping of four windows in proximity of the corners

of two adjacent pillars located on Level 2 (Figure B.3). Window dimensions were 5m

(wide) x 2m (high).

ii. February 2004: Mapping the entire four faces of a pillar located on Level 1 (Figure

B.4), including a detailed window mapping of 15m x 2m panels located at the base of

the pillar faces. The pillar was almost vertically above the pillars surveyed in the

previous visit, so to ensure that the mapped fractures belonged to the same geological

domain and that the two data sets could be thus combined.

Figure B.3: Middleton mine; location of mapped pillar faces on Level 2 indicated within the circle.

- 226 -


Figure B.4: Middleton mine; location of mapped pillar on Level 1. Also indicated is the corresponding

location of pillar faces mapped on Level 2.

Mapped panels and corresponding fracture traces for the July 2003 and February 2004 visits are

schematically shown in Figure B.5 and Figure B.6 respectively. Digital photographs of the

pillar faces for the February 2004 visit are shown in Figures B.7 to B.10, including the digitised

fracture traces and window outlines.

Figure B.5: Middleton mine Level 2. Mapped panels and digitised fracture traces.

- 227 -


Level 1, Panel 1 (35 fractures)




Figure B.6: Middleton mine Level 1. Mapped panels and digitised fracture traces.

- 228 -


Figure B.7: Middleton mine Level 1. Mapped panel No. 1 and digitised fracture traces for the entire

pillar face (Drawing by D. Elmo, 2004).


pillar face (Drawing by Z. Flynn, 2004).

- 229 -



pillar face (Drawing by D. Elmo, 2004).


pillar face (Drawing by Z. Flynn, 2004).

- 230 -

Appendix I – The ELFEN explicit solution scheme

The ELFEN explicit solution scheme

C.1 The ELFEN explicit solution scheme

The numerical analysis of fracturing processes in rock, beside its intrinsic

discrete/discontinuous nature, has also to consider that such problems are often highly dynamic

with rapidly changing domain configurations, requiring sufficient resolution and allowing for

multi-physics phenomena. Additionally, contact behaviour also gives rise to a very strong non-

linear system response. For these reasons, such problems are typically simulated employing

time integration schemes of an explicit nature (Owen et al., 2004a). Application of dynamic

explicit time integration schemes to multi-fracturing solids, particularly to those involving high

non-linearity and complex contact conditions, has increased notably in the recent years (Owen

et al., 2004b).

The discretisation of the equilibrium equations at time in the normal finite element manner

yields (Owen and Hinton, 1980; Bathe, 1986; Hughes, 1987 and Zienkiewicz, 2000; all referred

to in Owen et al., 2004b):

nt

( ) ( )nnext

nn uFt F uC uM int−=+ &&& [C.1]

where is the vector of nodal displacements at time , nu nt M and are, respectively, the

mass and damping matrices and

C

( )nuF int represents the internal force contribution from the

element stress field which satisfies the (non-linear) constitutive relations. The term ( )next tF is

the external force vector arising from applied boundary tractions and contact constraints. It is

apparent that at discrete time Equation [C.1] represents a coupled system of second order

linear differential equations with constant coefficients.

nt

The diagonal mass matrix M is defined by (Belytschko, 1983; Feenstra, 1993 and Owen et al.,

2004a):

[ dV,N,,NN ρ N M nnode

iV nnode

Ti

εnelem

εeA ∑∫

==

=1

211

K ] [C.2]

- 231 -


The diagonal damping matrix , which defines the viscous damping term with constant factor C

α , is expressed according to:

MC α= [C.3]

Central difference approximations for the velocity and acceleration in terms of displacements,

coupled with mass lumping techniques and proportional damping matrices, are used to derive

Equation [C.4], from which the nodal displacements 1+nu at time can be evaluated in terms

of the corresponding quantities at time stations , and (Klerck, 2000 and Owen et al.,

2004b):

1+nt

nt 1−nt

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−++⎟

⎠⎞

⎜⎝⎛ +=

−

+ 1int2

1

1 22 t

2 n-next

nnn u∆tM-CM uFF -∆∆tCM u [C.4]

Equation [C.4] permits the evaluation of displacement on an individual nodal basis with inter-

nodal coupling occurring only throughout the calculation of the internal force vector . The

usually small time step imposed in the explicit time-integration procedure results in the

requirement that a very large number (e.g. millions) of time increments have to be performed.

In addition to that, the introduction of new fractures and the consequent adaptive re-meshing at

both local and global levels adds another dimension of complexity. For these reasons, realistic

complex applications may be extremely computationally intensive, requiring the

implementation of parallel processing solution procedures.

intnF

C.2 The ELFEN tensile fracture model

Crack models are implemented in the code ELFEN as part of its capability of numerically

simulating crack generation, extension and coalescence.

Hillerborg et al. (1976) introduced the fictitious crack model, which represents an extension of

the cohesive crack model (Dugdale, 1960 and Barenblatt, 1962; in Klerck, 2000) and the first

Non Linear Fracture Mechanics (NLFM) model to describe the complete fracturing process in

arbitrary quasi-brittle systems. The fictitious crack model is defined by the tensile strength tσ

and the specific fracture energy , providing a continuous link between classical strength

based analysis of structures and the energy based classical Linear Elastic Fracture Mechanics

(LEFM). Cohesive cracks are realised by a strength criterion and naturally evolve towards an

fG

- 232 -


energetic criterion for large cracks. The inclusion of the specific fracture energy allows the

consideration of fracture mechanics size effects that result from the release of potential energy

into the fracture process zone (Klerck, 2000).

fG

Although conceptually different, the fictitious crack model is found to be mathematically

equivalent to the crack model proposed by Bazant (Bazant, 1976; Bazant and Cedolin, 1979;

Bazant and Planas, 1989; all in Klerck, 2000); the crack band model has an extra parameter in

the crack bandwidth , which is used in defining the specific fracture energy . ch fG

Smeared crack models include the fixed crack and rotating crack models. The smeared crack

models consider a cracked solid as an equivalent anisotropic continuum with degraded

properties in directions normal to crack orientations (Guzina et al., 1995). A crack band is

introduced at a point (Gauss point) in an isotropic, linear-elastic continuum when the maximum

principal stress 1σ surpasses the tensile yield strength tσ of the material. Definitions and

limitations of the fixed crack and rotating crack models are discussed in detail in William et al.

(1987) and Klerck (2000). In general terms, in the fixed crack models, the elastic properties

degrade or plastic strain accumulates across a pre-defined plane inducing a coupling between

the shear and normal stresses. The rotating crack concept has no memory of the crack direction

and damage accrues in the direction of the current principal stresses (William et al., 1987).

As discussed by Klerck et al. (2004), the fixed crack model is typically constrained with

induced shear stresses unable to invoke effective reorientation of crack directions, as is the

physical manifestation. Conversely, the rotating crack model is under constrained, exhibiting

much-reduced shear stresses (for coincident rotation of principal stress and strain); the crack

direction reflects the current state of damage and not the damage history. In oscillating stress

fields that realise large rotations of the principal directions, it is possible that the introduced

discrete fractures will not reflect the previous history of directions in which damage was

accumulated. The rotating crack model is an effective mechanism for eliminating stress locking

and excess shear stress and has been shown to yield a more reliable lower bound (conservative)

response compared to the fixed crack models (Klerck, 2000).

In the rotating crack model, post initial yield, anisotropic damage evolves by degradation of the

elastic modulus in the direction of the major principal stress invariant. The Rankine model is an

equivalent formulation of the rotating crack model; it uses the same softening criterion but is

applied within a continuum formulation rather than following the direction of cracks at a Gauss

point. For tension/compression stress states, the Rankine model is complemented with a capped

Mohr-Coulomb criterion in which the softening response is coupled to the tensile model.

- 233 -


The following sections review the definitions and mathematical formulations of the Rankine

rotating crack and Mohr-Coulomb models implemented in the code ELFEN. The reader is

referred to the ELFEN user’s manual (Rockfield, 2005) for a description of the material models

data structure.

C.2.1 Rotating crack and Rankine models

The rotating crack and Rankine models are designed for modelling the tensile failure of brittle

materials e.g. rock, ceramic and glass. Figure C.1 shows the yield surface and softening curve

for rotating crack and Rankine models. For Mode I dominated problems, the initial failure

surface for both models is defined by a tension failure surface:

0=−= ∆+∆+t

tti

ttg σσ [C.5]

where iσ represents the principal stress invariants and tσ is the tensile strength of the material.

Post-Initial yield, the rotating crack formulation represents the anisotropic damage evolution by

degrading the elastic modulus in the direction of the major principal stress invariant, according

to:

nnd

nn E ε=σ where ( )EE d ω−= 1 [C.6]

where ω is the damage parameter and the subscript refers to the local coordinate system

associated with the principal stresses.

n

Figure C.1: Yield Surface and Softening Curve for Rotating Crack and Rankine Models (after ELFEN

user’s manual, Rockfield, 2005).

- 234 -


The damage parameter is dependent on the fracture energy , which is defined as: fG

∫ ∫ εσ=σ= dssduG )( [C.7]

Integrating over a localisation bandwidth for a constant slope-softening model gives: cl

f

ctTG

lfE

2

2−= where ( )ec Afl = [C.8]

In Equation [C.8] is the area of the element. Note that the fracture energy is related to the

critical stress intensity factor by:

eA

EKG Ic

f

2= [C.9]

The unit dimensions of are . These formulations pertain both to the rotating crack

and the Rankine models, with the only difference that softening is applied in the Rankine model

within a continuum material formulation rather than by following the direction of cracks or

micro-cracks at a Gauss point.

fG 2mJ −⋅

C.2.2 The Mohr-Coulomb material model for compression states

Klerck (2000) and Klerck et al. (2004) provide detailed descriptions of the Mohr-Coulomb

model in the specific context of its implementation in the code ELFEN. The Mohr-Coulomb

yield criterion is a generalisation of the Coulomb friction failure law and is defined by:

φστ tannc += [C.10]

where τ is the magnitude of the shear stress, nσ is the normal stress, is the cohesion and c φ

is the friction angle. In principal stress space the yield surface is a six-sided conical shape. The

conical nature of the yield surface reflects the influence of pressure on the yield stress and the

criterion is applicable to rock, concrete and soil problems.

- 235 -


Expressing Equation [C.10] in terms of major and minor principal stresses 1σ and 3σ , it is

apparent that the Mohr-Coulomb failure criterion is independent of the intermediate principal

stress 2σ resulting in the failure plane manifesting itself necessarily parallel to this direction.

Some authors have justified the omission of the term 2σ on the basis that it does not generally

affect the failure mode and on the assumption that the order of influence is comparable to that of

the random variation of material properties exhibited by heterogeneous quasi-brittle materials

(Mogi, 1976 and Yumlu and Ozbay, 1995; all quoted in Klerck, 2000). The Mohr-Coulomb

yield surface in tension cannot reasonably represent the physically observed plastic flow

directions normal to the mutually orthogonal principal tensile planes. The numerical artifice

most widely adopted in the literature for the amelioration of the Mohr-Coulomb tensile response

is the so-called hydrostatic cut-off, illustrated in Figure C.2.

Figure C.2: Yield surface for the conventional Mohr-Coulomb model (after ELFEN user’s manual,

Rockfield, 2005).

This hydrostatic cut-off plane introduces additional return-mapping possibilities at the

intersection with the Mohr-Coulomb yield surface. However, although the inclusion of the

hydrostatic tensile cut-off can be considered as an improvement over the standard Mohr-

Coulomb yield surface, it is not suitable for application to quasi-brittle fracture due to the

arbitrary plastic flow directions (Klerck, 2000). The same author argued that only the mutually

orthogonal tensile planes of the isotropic Rankine yield surface are able to recover the correct

plastic flow directions in tension, thus as an approximation to the anisotropic softening response

of physical quasi-brittle materials, the isotropic non-hardening Rankine tensile cut-off emerges

as the only feasible cut-off formulation.

Figure C.3 shows the conventional Mohr Coulomb tensile yield surface in a 3D space, together

with the Mohr Coulomb tensile yield surface with Rankine tensile corner.

- 236 -


Figure C.3: a) Conventional Mohr Coulomb Tensile Yield Surface b) Mohr Coulomb with Rankine

Tensile Corner. Note that this figure specifically adopts the continuum mechanics stress convention, such

that 321 σσσ ≥≥ , where 1σ is the maximum tensile stress, 3σ is the maximum compressive stress and

2σ is the so-called intermediate principal stress (after Klerck, 2000).

Fracturing due to dilation is accommodated by introducing an explicit coupling between the

inelastic strain accrued by the Mohr-Coulomb yield surface and the anisotropic degradation of

the mutually orthogonal tensile yield surfaces of the rotating crack model. The Rankine tensile

corner introduces additional yield criteria defined by:

0=− ti σσ [C.11] 3,2,1=i

where iσ refers to each principal stress and tσ is the tensile strength. Although at present no

explicit softening law is included for the tensile strength, indirect softening does result from the

degradation of cohesion according to the following criteria, ensuring that a compressive normal

stress always exists on the failure shear plane:

( )φφ

σcos

sin1 −≤

ct [C.12]

C.2.3 Explicit coupling between degradation in compression and tension

The ELFEN compressive fracture model is based on the assumption that quasi-brittle fracture is

extensional in nature, i.e. any phenomenological yield surface is divided into regions in which

- 237 -


extensional failure can be modelled directly, as in the case of tensile stress fields and indirectly,

as in the case of compressive stress fields (Klerck et al., 2004).

Figure C.4 illustrates the dilation response of compressive failure in quasi-brittle materials and

clearly indicates the extensional inelastic strain directions associated with fracture.

Figure C.4: a) compressive loading with confining stress, b) relationship between axial and volumetric

strain and c) compressive failure with associated lateral extensional inelastic strain causing fracture and

dilation (after Klerck et al., 2004).

An explicit coupling is proposed between the extensional inelastic strain associated with the

dilation response of compressive failure and the tensile strength in the dilation direction, even in

the absence of tensile stress. The proposed implementation of an explicit coupling between

compressive stress induced extensional strain and tensile strength degradation permits the

realisation of discrete fracturing in purely compressive stress fields. The compressive fracture

model is defined by a composite yield surface consisting of the fully anisotropic rotating crack

model coupled with the isotropic, non-associative Mohr-Coulomb model, as illustrated in Figure

C.5.

- 238 -


Figure C.5: The compressive fracture model; a) The isotropic Mohr-Coulomb yield surface with

softening anisotropic tensile planes and b) Π-plane representation indicating possible return-mappings.

Note that this figure specifically adopts the continuum mechanics stress convention, such that

321 σσσ ≥≥ , where 1σ is the maximum tensile stress, 3σ is the maximum compressive stress and

2σ is the so-called intermediate principal stress (after Klerck, 2000).

The isotropic non-hardening Rankine tensile cut-off for the isotropic Mohr-Coulomb yield

surface constitutes a means of incorporating the return-mappings needed for the implementation

of the compressive fracture model.

The vector of principal increments of inelastic strain obtained from the single vector return-

mapping to the Mohr-Coulomb main-plane is given by (Klerck, 2000):

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

+∆=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∆∆∆

=∆1sin

0sin1

3

2

1

ψ

ψλ

εεε

ε ap

p

p

p [C.13]

where the increment of inelastic strain in the first principal stress direction is extensional.

Such increments of extensional strain must be associated with tensile strength degradation in the

parallel direction, giving:

p1ε∆

( )pititi εσσ = where [C.14] p

ipin

pi ε∆+ε=ε

where tiσ is the tensile strength in the i-th principal stress direction. In pure, triaxial, tension,

the strength degradation will occur in all three principal stress directions, whilst for a biaxial

state of stress with one direction in tension, and the orthogonal direction in compression, the

- 239 -


degradation will occur perpendicular to the tensile stress, with the possibility of additional

damage occurring in the same direction if the compressive stress exceeds the strength (Klerck et

al., 2004).

C.3 Use of mobilised material parameters to realise hardening/softening behaviour

Chapter 2 and Chapter 4 discussed the implementation within continuum models of so-called

mobilised parameters with plastic strain (Hajiabdolmajid et al., 2000 and 2002) and degradation

coefficients (Fang and Harrison, 2002) to numerically simulate rock brittle failure. The Mohr-

Coulomb material model with Rankine cut-off in ELFEN accommodates mobilised material

parameters that realise hardening/softening with respect to effective plastic strain ( pε ) such

that the permissible elastic domain depends on the current state of inelastic strain as well as the

history of evolution (Klerck et al., 2004).

The form of the hardening/softening curves for the mobilised cohesion ( )pεcc = and the

mobilised friction angle ( )pεφφ = are established by considering laboratory test data. For

cemented granular materials degradation occurs with the loss of inter-particle cementation or

cohesion and the simultaneous mobilisation of inter-particle friction. Cohesion is effectively

defined as the difference between the peak strength and the residual strength and the

dependency of residual strength on inter-particle friction accounts for increased ductility with

increased confinement.

A non-associative implementation of the Mohr-Coulomb elasto-plasticity model is required for

the recovery of the correct physical dilation response in compression. Replacing the friction

angle φ in the Mohr-Coulomb function (expressed in terms of the major and minor principal

stresses 1σ and 3σ ) by the so-called dilation angle ψ , such that φψ < , recovers the plastic

flow potential Ψ given by (Klerck, 2000):

( ) ( ) ( ) ψψσσσσεσ cossin21

21, 03131 cp −++−=Ψ [C.15]

where ( )pocc ε=0 and ( )pεψψ = . The relationship between the dilation angle ψ and

the observed dilatancy is defined by the constant dilatancy observed near peak stress. After the

formation and subsequent mobilisation of a macroscopic failure plane the dilation tends to zero.

- 240 -


In the cohesion weakening/friction strengthening model proposed by Hajiabdolmajid et al.

(2000 and 2002), the plastic strain limits at which the cohesional component of strength reaches

a residual value, and the frictional strength component mobilises, are two material properties

that in reality depend primarily on heterogeneity and grain characteristics. Hajiabdolmajid

(2001) argued that the plastic strain limit for cohesion loss can be considered a true material

property, whilst the strain limit under which the frictional strength reaches its full mobilization

depends to some extent on the loading system characteristics (i.e. geometry and loading rate).

Klerck (2000) presented several numerical examples to validate the use of mobilised material

parameters that realise hardening/softening with respect to effective plastic strain in the context

of the Mohr-Coulomb material model with Rankine cut-off.

C.4 Numerical damping

The viscous damped system represents the only type of damping mechanism for which

analytical solutions of the equations of motion can be obtained by different methods, including

the Fourier and Laplace transformations approach. Raleigh damping assumes that damping is a

linear combination of mass and stiffness according to:

KM C βα += [C.16]

where M is the mass matrix of a structure, is the stiffness matrix of a structure, K α and β

are constants to be determined from two given damping ratios ( )21 ,ξξ that corresponds to two

unequal frequencies of vibrations ( )21 ,ωω . A significant shortcoming of the Raleigh

damping regime is that within the frequency range [ ]21 ,ωωω ∈ the higher frequency modes

are considerably more damped than the lower frequency modes.

- 241 -


Figure C.6: Raleigh damping (after Klerck, 2000).

As introduced in Section C.1, the discretised dynamic equilibrium Equation [C.1] at time is

fully coupled when consistent mass and damping matrices are utilised. The central difference

integration scheme therefore requires the inversion of a global matrix at every timestep. The

computational cost of these matrix inversions is restrictive in light of the large number of

timesteps required to satisfy the timestep constraint imposed by the conditional stability of the

integration scheme (Klerck, 2000). As discussed by the same author, rendering the system of

equations uncoupled reduces the computational cost to that of evaluating a set of algebraic

expressions corresponding to the degrees of freedom of the system. This is achieved by

ensuring that the mass and damping matrices are diagonal matrices. Whilst nodal mass lumping

renders a diagonal mass matrix by effectively assigning fractions of element mass to each

element node, it is required that the damping matrix C be appropriately constructed such that:

nt

ijiijTi C δξωφφ 2= nji ,,2,1, K= [C.17]

where ijδ is the Kronecker delta; the eigenvector iφ , the free-vibration frequency iω and the

damping ratio iξ correspond to the equilibrium equation of the i-th degree of freedom.

If proportional damping is invoked, C cannot generally be constructed by element mass and

stiffness matrices to satisfy the relationship defined in Equation [C.17]. Damping is therefore

redefined in the system context as an approximation to global energy dissipation. It is observed

that the decoupled nature of the equilibrium equations renders the total energy dissipation as the

piecewise sum of the damping in each mode. The stiffness (tangent) matrix is not K

- 242 -


assembled in the explicit solution procedure, omitting the stiffness term in Equation [C.16]

giving:

MC α= or iiiii MC α= for ni ,,2,1 K= [C.18]

The uncoupled nature of the damping matrix permits differential damping throughout the

system by recognising that for the i-th degree of freedom iii ξωα 2= such that the target

frequency iω is damped exactly with respect to the damping ratio iξ . It is observed that higher

frequencies require higher damping ratios than lower frequencies in order to achieve equivalent

damping.

It is noted that both the Raleigh damping [C.16] and the explicit simplification [C.18]

necessarily damp rigid body motion and can thus introduce an undesirable viscous system

response.

C.5 Meshing objectivity and adaptive remeshing

The code ELFEN incorporates a continuous adaptive meshing tool, which allows potential

difficulties to be overcome associated with deformation-induced element distortion, and fine

scale features to be resolved in the solution.

Although the energy dissipation in the crack band model implemented within the ELFEN code

is rendered objective by normalising the softening curve with the specific fracture energy ,

the spatial localisation is necessarily arbitrary. Localisation occurs in individual elements,

resulting in the width of localisation and the crack band spacing depending on the discretisation,

i.e. mesh element size. However, this form of mesh dependence is realised in all local failure

models, but does not necessarily render their application spurious. Furthermore, because the

mesh orientation can result in directional bias of propagating crack bands due to the fact that

strain discontinuities exist at the element boundaries, a non-local averaging of the damage

measure is adopted in each orthotropic direction to ensure discretisation objectivity by

introducing a length scale to govern the width of the localisation zone (Owen et al., 2004b).

The use of relative finer (high density) unbiased meshes, generated by methods such as

Delauney triangulation prevents crack band propagation from following arbitrary rows of

aligned elements (Klerck, 2000).

fG

As introduced in Chapter 4, the failure of heterogeneous quasi-brittle material is associated with

the stable growth of an extensive non-linear process zone responsible for the dissipation of

- 243 -


energy and the widespread redistribution of stress. This stable fracture process ultimately

results in the formation of macroscopic fractures prior to the maximum load being reached.

Accordingly, the computational framework for the realisation of the continuum-discrete

transition requires a form of topological update of the meshed problem, through the definitions

of an efficient re-meshing procedure.

As part of the computational scheme of the code ELFEN, when the unloading process within a

localisation zone is complete, a discrete (and physical) crack is inserted as illustrated in Figure

C.7. The discrete crack is introduced when the tensile strength in a principal stress direction

reaches zero and is orientated orthogonal to this direction.

The insertion of discrete cracks into the quasi-brittle continuum follows three steps (Owen et al.,

2004b):

i. Create a non-local failure map (weighted nodal averages).

ii. Determine fracture feasibility and the order of discrete crack insertion.

iii. Perform the topological update (remeshing).

Figure C.7: Discrete crack insertion and the associated topological update. a) weighted average nodal

failure direction, b) intra-element fracture description and c) the inter-element fracture description (after

Owen et al., 2004b).

- 244 -

Appendix I - A brief review of the principal rock mass failure criteria

A brief review of the principal rock mass failure

criteria

D.1 Introduction

As addressed by Hoek in his keynote address at the U.S. Rock Mechanics Symposium (Vail,

1999), quantifying the post-peak response of rock masses is one of the biggest challenges facing

the rock mechanics community. The knowledge of the complete load-deformation curve for a

rock a mass is of fundamental importance in any rock engineering design, though various

uncertainties associated with the intrinsic DIANE nature (Harrison and Hudson, 2000) of the

rock mass make the modelling of the post-peak response of fractured rock masses a challenging

task. As reported by Fairhurst (1993) it is expected that the rock mass will exhibit a behaviour

similar to that of Figure D.1. Failure criteria provide formulae enabling a prediction of the level

at which failure will occur; they do not describe the processes taking place in the material in the

course of loading that can eventually lead to failure (Bieniawski, 1967). The failure criteria,

which can be used in the design process to obtain estimates of rock mass strength, are typically

in the form of stress dependent functions, including in several cases parameters based on rock

characterisation and classification systems.

Figure D.1: Example of complete load-deformation curve for the rock mass (after Fairhurst, 1993).

- 245 -


A comprehensive review of failure criteria is outside the scope of this research, and it was

decided, for reasons associated to the numerical analysis proposed in the current, to concentrate

on the most well known and most widely used failure criteria in geotechnical and rock

engineering applications, namely the Mohr-Coulomb and Hoek-Brown criteria. For a recent

and more comprehensive literature review of both intact rock and rock mass failure criteria and

classification systems the reader is refereed to Edelbro (2004).

D.2 Mohr-Coulomb criterion

Mohr related shear failure on an inclined material plane to the normal stress nσ acting on the

plane, according to some characteristic material function given by:

( nf )στ = [D.1]

where τ is the shear stress. The Coulomb criterion makes a first order approximation to this

function, equating the shear failure stress τ to the sum of the internal friction φ and the

inherent material cohesion , giving: c

φστ tannc += [D.2]

Equation [D.2] is often referred to as the Mohr-Coulomb criterion, which is used in rock

mechanics to investigate shear failure along fracture planes and in rock masses.

Applying stress transformation equations gives:

( ) ( ) βσσσσσ 2cos21

21

3131 −++=n [D.3]

( ) βσστ 2sin21

31 −= [D.4]

Combining Equations [D.2] and [D.4], gives the limiting stress condition on any plane defined

by β (Figure D.2):

( )[ ]( )βφβ

βφβσσ

2cos1tan2sin2cos1tan2sin2 3

1 +−−++

=c

[D.5]

- 246 -


The orientation of the critical plane is given by:

24φπβ += [D.6]

In terms of major and minor principal stresses, 1σ and 3σ , Equation [D.2] can be re-written as:

31 sin1sin1

sin1cos2 σ

φφ

φφσ

−+

+−

=c

[D.7]

Figure D.2: Mohr-Coulomb strength criterion: a) shear failure on a plane defined by β , b) strength

envelope of shear ad normal stress and c) strength envelope of principal stresses.

The uniaxial compressive strength ciσ and the uniaxial tensile strength tσ are related to and c

φ by:

φφσ

sin1cos 2

−=

cci [D.8]

and

φφσ

sin1cos 2

+=

ct [D.9]

- 247 -


The criterion assumes that failure occurs along a plane without any dilation and a tensile cut-off

is often included to account for the fact that rock cannot sustain large tensile stresses (Figure

D.3). Many books of rock mechanics contain a detailed discussion on the Mohr-Coulomb

strength criterion (e.g. Brady and Brown, 1993 and Hoek and Brown, 1980).

Figure D.3: Mohr-Coulomb criterion in terms of a) principal stresses and b) normal and shear stresses

with tension cut-off (after Edelbro (2004).

D.3 The generalised Hoek-Brown criterion

The Hoek-Brown failure criterion for rock masses has been applied to large numbers of projects

around the world and the traditional Hoek-Brown parameters have been proved appropriate for

estimating the strength of slopes and rock masses around underground excavation at shallow

depth. Since its introduction in 1980, the Hoek-Brown criterion has been modified several

times, most recently by Hoek and Brown (1997) and Hoek et al. (2002). Hoek-Brown criterion

typical analytical expression is reported below:

a

bci sm ⎟⎟⎠

⎞⎜⎜⎝

⎛++=

1

331 σ

σσσσ [D.10]

where , bm s and are constants for the rock mass, and a ciσ is the uniaxial compressive

strength of the intact rock. Mohr-Coulomb strength parameters and c φ can also be obtained

based on the Hoek-Brown criterion envelope (Hoek et al., 2002).

- 248 -


The expressions for , bm s and in equation [D.10] are as follows (Hoek et al., 2002): a

⎟⎠⎞

⎜⎝⎛

−−

= DGSI

ib emm 1428100

[D.11]

⎟⎠⎞

⎜⎝⎛

−−

= DGSI

es 39100

[D.12]

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

−−3

2015

61

21 eea

GSI

[D.13]

In Equations [D.11] and [D.12], D is a factor that depends upon the degree of disturbance of

the rock mass due to blast damage and stress relaxation; it varies from 0 for undisturbed in-situ

rock masses to 1 for very disturbed rock masses. Guidelines for the selection of D are

discussed in Hoek et al. (2002).

The Hoek-Brown criterion can be utilised for estimating the strength and deformability of

jointed rock masses, requiring an estimate of (i) the uniaxial compressive strength ciσ of the

intact rock pieces in the rock mass, (ii) Hoek-Brown constant for these intact rock pieces

and (iii) Geological Strength Index (GSI) for the rock mass. GSI was introduced by Hoek et al.

(1992, 1995) to provide a system for estimating the rock mass strength for different geological

conditions; GSI can be related to either the rock mass quality index Q (Barton et al., 1974) or

the rock mass rating RMR (Bieniawski, 1976 and 1989). Based upon practical observations and

back analysis of excavations behaviour, Hoek et al. (2002) presented a correlation between GSI

and rock mass modulus of deformation, which is expressed as:

im

4010

101002

1−

⋅⎟⎠⎞

⎜⎝⎛=

GSIci

mσD- (GPa) E [D.14]

4010

102

1−

⋅⎟⎠⎞

⎜⎝⎛=

GSI

mD- (GPa) E [D.15]

Equation [D.14] applies for ciσ ≤100MPa, whilst for ciσ >100MPa, Equation [D.15] should be

used instead.

- 249 -


The uniaxial compressive strength of the rock mass is expressed as cσ and is calculated by

setting 3σ in Equation [D.10] to zero; compressive failure at the boundary of an excavation is

initiated when the stress induced on that boundary exceeds cσ . Fracture propagation from the

initial failure location into a biaxial stress field stabilises when the local strength defined by

Equation [D.10] is higher than the induced stress 1σ . This approach enables consideration of

the failure propagation in details; however, in order to consider the overall behaviour of a rock

mass recognising the effect of some confinement, the concept of global rock mass strength has

been proposed by Hoek and Brown (1997). For the stress range tσ < 3σ < ciσ /4, with tσ

being the tensile strength, the global rock mass strength can be computed according to (Hoek et

al., 2002):

( )[ ]

( )( )aas

msmasm

ab

bb

cicm ++

⎟⎠

⎞⎜⎝

⎛+

⋅−−+=

−

2124

841

σσ [D.16]

- 250 -

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