The Grain-Based Model Numerical Simulation of Unconfined Compressive Strength ...hgycg.cdut.edu.cn/data/upload/1563259207193.pdf · 2019. 7. 16. · The Grain-Based Model Numerical

KSCE Journal of Civil Engineering (2018) 22(8):2764-2775

Copyright ⓒ2018 Korean Society of Civil Engineers

DOI 10.1007/s12205-017-1228-z

− 2764 −

pISSN 1226-7988, eISSN 1976-3808

www.springer.com/12205

Geotechnical Engineering

The Grain-Based Model Numerical Simulation of Unconfined Compressive

Strength Experiment Under Thermal-Mechanical Coupling Effect

Zhongyuan Xu*, Tianbin Li**, Guoqing Chen***, Chunchi Ma****, Shili Qiu*****, and Zhi Li******

Received May 17, 2015/Revised 1st: February 15, 2016, 2nd: August 11, 2017/Accepted September 25, 2017/Published Online January 8, 2018

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Abstract

PFC-GBM (Particle Flow Code-Grain Based Model) is the major and fundamental method to simulate rock behaviors in thispaper. Geo-materials are composed of micro-grains, the behavior of these grains and the interface between them influencemacroscopic behavior of materials. Traditional PFC simulation method could simulate the integral micro-behavior of material, andPFC-GBM simulation focuses on every different grains and interfaces to simulate micro-heterogeneity of material. In this paper, anattempt is made to investigate the strength of rock masses and the development of micro-cracks under thermal-mechanical couplingeffect. For this purpose, a numerical model is established based on mineral analysis and pre-existing mechanical experiments ofGranite from one of complex tunnels in Yunnan Province. After establishing the model, the specimen were first heated and thencompressed according to test sequence of laboratory experiments. The simulation results are calibrated to match the laboratory testresults including thermal behaviors and fracture development. The conclusion of simulations show that both of thermal behaviorsand fracture development are depended on the micro-heterogeneity of granite in UCS (Unconfined Compressive Strength)experiment, the characteristics of minerals influence the macroscopic fracture mechanism. The simulation reveals that in a certainrange of temperature (40oC ~ 90oC), temperature increasing enhance the brittle damage of granite. The situation of 130oC had theobvious thermal-crack before loading and then exhibited a much lower peak strength and failure strain. This numerical observationmay guide the underground construction in complex geo-environment.

Keywords: PFC-GBM, numerical simulation; thermal-mechanical coupling, micro-heterogeneity

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1. Introduction

A lot of experiment researches have focused on the micro-

mechanical processes of brittle rocks to investigate fracture-

mechanism. The traditional method of analyzing the macro-

mechanism has become not insufficient in the recent rock

mechanism investigation especially under the complex engineering

conditions like tunnel construction. Coulthart (1999) has

summarized several numerical approaches and analyzed the

limitations of previous simulation methods of underground

mining. A lot of scholars (Hallbauer et al., 1973; Olsson and

Peng, 1976; Kranz, 1983) have mentioned that stress distribution

is heterogeneity in the rock, and the crack development is influenced

by the micro-structure. To simulate the micro-structure of rock,

discrete modeling technics was used in the numerical simulation,

commonly referred to as the Discrete Element Method (DEM).

Based on the DEM, particle-based model which called Particle

Flow Code (PFC) was proposed to simulate the non-cohesive

media, such as soils and sands (Cundall and Strack, 1979), then

bonded-particle model which applies cohesive bonds between

particles was developed to simulate the behaviors of solid rocks

(Potyondy and Cundall, 2004). Some scientists has used PFC

modeling to investigate micro-structure (Bock, 2006), even

nanostructure (Penumadu, 2009) of geo-material, and material

mechanics under coupling effect (Lee, 2013).

Grain-based models (GBM) provide a synthetic material that

mimics deformable, breakable polygonal grains cemented along

their adjoining sides (Potyondy, 2010). This numerical method

could be simulated in UDEC (Universal Distinct Element Code)

(Itasca, 2004) and PFC2D (Itasca, 2008). UDEC-GBM has been

TECHNICAL NOTE

*Ph.D. Student, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu, PR

China, 610059; Dept. of Earth, Ocean and Atmosphere Science, Florida State University, Tallahassee 32306, FL, USA (E-mail: [email protected])

**Professor and Dean of College, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology,

Chengdu 610059, PR China (Corresponding Author, E-mail: [email protected])

***Professor, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, PR

China (E-mail: [email protected])

****Lecturer, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, PR

China (E-mail: [email protected])

*****Associate Researcher, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, PR China (E-mail: [email protected])

******Engineer, Sichuan Provincial Architectural Design and Research Institute, Chengdu, PR China (E-mail: [email protected])

The Grain-Based Model Numerical Simulation of Unconfined Compressive Strength Experiment Under Thermal-Mechanical Coupling Effect

Vol. 22, No. 8 / August 2018 − 2765 −

used in laboratory specimen under UCS experiment while the

grains in the rock are unbreakable (Lan et al., 2010). PFC2D-

GBM has been used in different simulations of brittle rock, such

as intact and granulated marble’s rock mass strength at various

confinement levels (Bahrani et al., 2011), Brazilian test of Lac

Du Bonnet Granite (Bahrani et al., 2012), shear rupture and rupture

mechanism of Lodève sandstone in direct shear experiment

(Bewick et al., 2014a; Bewick et al., 2014b).

The effect of micro-heterogeneity in compressive failure of

hard rock has been widely admitted, and Grain-based Model has

also been more and more popular in relative research area.

However, the effect of micro-heterogeneity with rock under

coupling effect such as thermal-mechanical environment has not

been investigated. In this article, the micro-rupture and the

energy process are investigated in using a calibrated synthetic

intact granite under high temperature UCS experiments, different

temperatures are considered in the simulations. Particle Flow

Code (PFC) is an idea numerical simulation tool and its embedded

Grain-based Model (GBM) is one of the best numerical method

for this purpose.

2. PFC2D-GBM of Granite

2.1 About PFC2D-GBM

PFC models the movement and interaction of circular particles

by the Distinct Element Method (DEM) (Itasca, 2008), it applies

cohesive bonds between the small circle particles, and the

movement of particles simulates the behaviors of solid body. The

resultant model is commonly referred to as the Bonded-particle

Model (BPM) for rock (Potyondy and Cundall, 2004). In a BPM,

crack nucleation is simulated through breaking of internal bonds

while fracture propagation is obtained by coalescence of multiple

bond breakages. Blocks of arbitrary shapes can form as a result

of the simulated fracturing process and can subsequently interact

with each other (Lisjak and Grasselli, 2014). There are two types

bonds in PFC, contact bond and parallel bond. Contact bond

(Fig. 1(a)) models adhesive over vanishingly small area of

contact point, it cannot resist moment. Parallel bond (Fig. 1(b))

which is adopted in this paper simulates additional material

deposited after balls are in contact, it can resist moment (Itasca,

2008). The total force and moment vector of parallel contact are

and M, which are initial to zero. The force vector can be

divided into normal and shear component,

(1)

The elastic force increment and moment increment over a

timestep of Δt are calculated,

(2)

(3)

(4)

With and

is normal stiffness of parallel bond; is shear stiffness of

parallel bond; ni is normal unit; A is area of the bond cross-

section; I is moment of inertia of the bond cross-section about an

axis through the contact point and in the direction of Δθ; ΔUi is

relative contact displacement increment; Vi is shear velocity

between two particles; is rotational velocity of particle i.

The traditional motion between particles in PFC2D is insufficient

to simulate the existing cracks in the rock, the motions of two

sides of these cracks are always influenced by the roughness of

ball contacts. Ivars et al. (2008) firstly proposed Smooth-Joint

Contact Model (SJM), which allowed overlapping of balls and

developed a smooth interface regardless of the local particle

topology (Fig. 2). The properties of smooth joint are inherited

from the properties of parallel contact between particles, the

normal force and shear force is updated via:

(5)

(6)

is normal stiffness of smooth joint; is shear stiffness of

Fi

n s

i i iF F F= +

( )n n ni i i

F k A U nΔ = − Δ

s s s

i iF k A UΔ = − Δ

n

M k I θΔ = − Δ

i iU V tΔ = Δ

[ ] [ ]( )B A tθ ω ωΔ = − Δ

kn

ks

ωi[ ]

n n n nF F k A U= + Δ

s s s sj sF F k A U= − Δ

kn ks

Fig. 1. Two Types of Bonds in PFC (Itasca, 2008): (a) Contact Bond,

(b) Parallel Bond

Fig. 2. (a) & (b) are the Particle Contact and Ball Movement of Standard Case, (c) & (d) are the Particle Contact and Ball Movement of

SJM (Ivars, 2010)

Zhongyuan Xu, Tianbin Li, Guoqing Chen, Chunchi Ma, Shili Qiu, and Zhi Li

− 2766 − KSCE Journal of Civil Engineering

smooth joint; Asj is area of the smooth-joint cross section; ΔUn,

ΔUs are displacement increment on normal and shear direction;

For bonded smooth joint model (Bond mode = 3), if ,

then the bond breaks in tension; if , then the bond

breaks in shear. σc and τc are bond normal and shear strength,

respectively. τc is calculated from cohesion (cb) and compression

stress.

The grain-structure generation procedure is described by Potyondy

(2010) to generate the arrangement and size of minerals which is

calculated by Voronoi model (Fig. 3). This method produces no

gaps between polygons, these polygons mimic minerals and

each interface mimics the boundary of minerals, the location data

of interface are recorded in this procedure. Then this polygons-

model is attached on the Bonded-particle Model (BPM), and

each interface consists of a collection of smooth-joint contacts

that effectively modify the surfaces of the contacting particles to

align with the interface (Fig. 4).

2.2 Grain Structure of a Granite

It is obvious that the real grain structure cannot be modeled

exactly in any numerical model (Lan et al., 2010). The numerical

model in the PFC2D-GBM contains no gaps in the rock specimens

before loading or heating, the grain size and arrangement simulated

is not intended to exactly match the real micro-structure of

granite, GBM is just a method which account for micro-

heterogeneity in the numerical simulation. The shape and

arrangement of minerals are random by the Voronoi model

calculation. The compositions of studied granite are determined

by using X-ray scattering techniques (Table 1). For reducing the

calculation burden, the major composites used for modeling are:

biotite (27%), quartz (25%), K-feldspar (15%) and Plagioclase

(33%). The size and arrangement of minerals observed in

polarized light thin section (Fig. 5(b) and 5(c)), the statistical

data shows grain size in the Table 2. While the real parameters of

grain size are too difficulties for simulation because an unimaginable

number of particles in a model results an exceeding computing

ability of computer. The major purpose of model generation is to

embody the micro-heterogeneity of grain structure and grain

properties, so the grain size need to be adjusted (Table 2).

Fig. 5(a) shows the final model general result of granite model,

n c sjF Aσ≤ −

s c sjF Aτ≥

Fig. 3. (a) Initial Disk Packing Showing Disks and Contacts, (b) Filled Circles at Internal-void Centroids, (c) Grain Structure Consisting of

Polygons, One for Each Internal Disk, with Nodes at Internal-void Centroids, (d) Generated Mesh (Potyondy, 2010)

Fig. 4. BPM Overlaid on a Grain Structure and PFC2D GBM Con-

sisting of Grains (bonded disks) and Interfaces (smooth-

joint contacts) (Potyondy, 2010)

Fig. 5. (a) the Simplified GBM Generated for Granite, (b) Close-up

View of Boxed in Area in a, (c) Mineral Grain Structure

Observed in Polarized Light thin Section

Table 1. The Major Composition of Studied Granite by X-ray Scattering Techniques

NumberPercentage (%)

Biotite Quartz K-feldspar Plagioclase Chlorite Tremolite Pyrite others

1 20 25 15 33 2 3 2


Vol. 22, No. 8 / August 2018 − 2767 −

for reducing the time of computing, the size of model is set to be

50 cm × 25 cm.

3. Methods

This numerical simulation is based on the corresponding

laboratory experiments. The specimens were collected from one

complex tunnel and made into a 100 mm × 50 mm cylinder

shape. For reducing computing time, 50 mm × 25 mm was used

in numerical simulation.

3.1 Laboratory Experiment

In the laboratory experiments, rock samples were tested in

MTS815 Testing System under 20oC, 40oC, 60oC, 90oC, 130oC

(these temperatures were set according to the field monitoring).

The thermo-mechanical stress path is temperature-controlled

mechanical loading. First, heating the rock samples to target

temperature and then maintaining this temperature; Second,

applying axial loading at the speed of 0.1 mm/min to yield stage

and then change the controlling speed to 0.03 mm/min. The

stress-strain curves were shown in Fig. 6.

3.2 Numerical Methods

For investigating rock thermos-mechanical behavior and

comparing with the laboratory results, the numerical simulation

adopted the same steps as laboratory tests: heating the numerical

model to target temperature (20oC, 40oC, 60oC, 90oC, 130oC),

and then loading under maintaining temperature. Mostly, samples

were experienced a heating phase, this process would be discussed

in 5.1 and 5.2. By using the method above, the calibration of

numerical model is performed in the next chapter.

4. Calibration of PFC2D-GBM

The property of material in the PFC is different from the

traditional macroscopic properties, the micro-properties and

thermal-properties of four minerals were compiled from literature

and are listed in Table 3.

Some micro-parameters in the DEM (i.e. stiffness ratio of

parallel bond, cohesion of smooth joint) must be calibrated to

match the macro-properties of material, such as peak strength,

strain-stress curves, loading displacement, elastic modulus and

Poisson’s ratio. Here, the calibration is first down with the micro

and thermal properties above in the 40oC UCS experiment to

match the peak strength, elastic modulus, Poisson’s ratio of

laboratory tests. Then, the calibration results would be used in

60oC numerical triaxial tests (confining pressure is 15 MPa) to

match the contemporary laboratory test results, the introducing

of confining pressure and different temperature could be applied

Table 2. Estimated Grain Sizes by Mineral Types

Mineral

Polarized light thin testAdjusted size

(used in simulation)

Dmin(mm)

Dmax(mm)

Major range of minerals

Rmax/RminRmin

(mm)

Biotite 0.07 1 0.5~0.9 1.8 0.5

Quartz 0.06 1.5 0.4~0.8 1.8 0.4

K-feldspar 0.25 3 1.0~1.9 1.9 1.0

Plagioclase 0.30 3 1.0~1.5 1.5 1.0

Table 3. Properties of Four Minerals in the Granite

Material-properties

Mineral Density (g/cc) Poisson’s Ratio Elastic Modulus (K,G) (GPa)

Biotite (Mavko et al., 2003) 3.05 0.36 33.8 (41.1,12.4)

Quartz (Mavko et al., 2003) 2.65 0.08 94.5 (37.0,44.0)

K-feldspar (Lan et al., 2010) 2.56 0.28 69.8 (53.7,27.2)

Plagioclase (Mavko et al., 2003) 2.63 0.35 68.6 (75.6,25.6)

E = 3K(1 − 2 μ), E = 2G(1 + 2 μ)

Thermal-properties

MineralCoefficient of linear thermal

expansion (10−6/oC) (Huotari and Kukkonen 2004)

Thermal conductivity (W/mK) (Clauser and Huenges 1995)

Specific heat at constant volume (J/KgoC)

Biotite 12.125(9.94 ~ 13.79) 3.14 760 (Frankin and Charles 1982)

Quartz 16.665(10 ~ 23.33) 6.15 800 (Liu et al., 2009)

K-feldspar 3.687(0.25 ~ 10.5625) 2.34 ± 0.08 630 (Liu et al., 2009)

Plagioclase 4.167(2.5 ~ 5) 2.04 700 (Liu et al., 2009)

Fig. 6. Stress-strain Curve of Laboratory Results



to verify elastic modulus and Poisson’s ratio of specimen since

the Poisson’s ratio of numerical simulation in Unconfined test is

unmatchable. The results will again be used in the UCS numerical

tests until getting the perfect property parameters. The calibration

procedure shows that the elastic modulus of specimen is major

influenced by stiffness of parallel bond and smooth joint, and the

tensile and coefficient play an important role in peak strength of

specimen. The set of laboratory data (UCS under 20oC, 60oC,

90oC, 130oC) are used to validate the results of calibration. The

comparison of laboratory and simulation is displayed in Table.4.

The calibration results are listed in the Table 5.

After calibration, UCS tests were started from heating process

and then compression in terms with the laboratory tests, the

temperature, stress and strain value, energy information, crack

development would be recorded by the software.

5. Analysis of Temperature Effect

It is extremely important to study and understand the deformation

behavior and strength characteristics of rocks under Thermal-

mechanical (TM) coupling effects (Zuo et al., 2012a). Thermal

effect originates micro-cracks and anisotropic stress distribution

in the granite. The granite components have different thermal-

properties (Table 3 show three major thermal-properties in the

PFC), thermal stress leads to various minerals properties changes,

and then the intergranular compressive and tensile forces become

complex and unpredictable. Previous studies have discussed the

coupling effect on the properties of rock, such as microstructure,

bulk density, effective porosity and P-wave velocity (Yavuz et

al., 2009), porosity and permeability (Chaki et al., 2008), crack

density and fracture state (David et al., 1999). Micro-mechanics

is also analyzed by investigating thermal expansion and conductivity

(Ferrero et al., 2014; Chen et al., 2012). The in-situ test also

simulated by GBM in the coupled thermo-mechanical loading

effect (Lan et al., 2013).

5.1 UCS Under Different Temperatures

According to the thermal condition of the certain tunnel, the

temperatures of this simulation are set to be 20oC, 40oC, 60oC,

90oC, 130oC. For simulating the laboratory tests, the loading

procedure is performed after the heating process. Fig. 7 shows

the strain-stress curves of different temperatures and mechanic

parameters calculated by PFC are listed in Table 6.

The results above show that the specimen of 60oC and 90oC

are inclined to brittle damage, they reveal a higher value of

Table 4. Comparison between Laboratory Test Results and PFC

Simulation

Peak Strength (σf, MPa)

Elastic Modulus (E, GPa)

Poisson’s Ratio (μ)

Laboratory GBM Laboratory GBM Laboratory GBM

Calibration

UCS (40°C) 107.0 100.8 42.2 39.8 0.17 0.24

Trixial (60°C) 228.9 228.2 52.3 52.1 0.28 0.33

Validation

UCS (20°C) 93 100.2 34.8 35.9 0.15 0.24

UCS (60°C) 102 98.0 34.6 41.2 0.12 0.25

UCS (90°C) 103 94.4 39.4 43.5 0.13 0.27

UCS (130°C) 96 77.1 44.1 45.8 0.13 0.27

Table 5. The Results of Calibration, Including Properties of Particles, Parallel Bonds between Particles and Smooth Joints (interfaces)

between Grains

Properties of particles

Property Symbol Biotite Quartz K-feldspar Plagioclase

Density ρball (Kg/m3) 3050 2650 2560 2630

Modulus Ec (GPa) 33.8 94.5 69.8 68.6

Stiffness ratio (normal/shear) kn/ks 2.5 1.0 2.0 2.5

Friction coefficient μ 0.5 0.5 0.5 0.5

Properties of parallel bonds

Radius multiplier 1.0 1.0 1.0 1.0

Modulus 33.8 94.5 69.8 68.6

Stiffness ratio (normal/shear) 3.0 1.0 3.0 2.5

Mean of normal strength σc (MPa) 300 380 350 350

s.d. of normal strength σcs (MPa) 0 0 0 0

Mean of shear strength τc (MPa) 400 500 380 380

s.d. of shear strength τcs (MPa) 0 0 0 0

Properties of smooth joints

Property Symbol Value Property Symbol Value

Radius multiplier 1.0Normal and shear

stiffness0.2 × inherited

Bond mode D 3 Tensile strength σc (MPa) 20

Bonded-system cohesion cb (MPa) 100Bonded-system friction angle

φb 35°

λ

Ec

kn

ks

⁄

λs kn ks,


Vol. 22, No. 8 / August 2018 − 2769 −

elastic modulus and peak strength, and a faster stress-dropping at

after-peak stage. The increasing temperature has major influence

on crack-initiation strength and elastic modulus, heating process

induce the inflation among minerals and then the elastic modulus

are calculated higher in the loading process. Within the 90oC, the

temperature has limited influence on peak strength, while the

strength of 130oC is very low compared to other conditions and

its crack-initiation strength suggests that many fractures has

appeared in the heating process.

5.2 Strain of Different Temperatures

Some researchers have investigated the thermal cracking

temperature of granite, Zhao et al. (2008) observed few micro-

cracks in the granite propagated under 200oC through micro-CT

(Computer Tomography) experimental system, Zuo et al. (2011)

defined granite threshold temperature of thermal cracking as

68oC ~ 88oC, Zhang and Wang (2009) suggests 260oC is the

threshold temperature of granite. Thermal stress transform through

the bonding between particles and is stored in the particles in the

PFC (Itasca, 2008), the particles expand with thermal effect

depends on the temperature value and thermal-properties. The

test results show that the thermal strain obviously increases with

the rising temperature, representing a linear relation (Fig. 8).

Strain energy is the energy stored by system undergoing

deformation. The deformation occurred in the heating process,

and the strain energy which was recorded could be used to

identify deformational degree and energy transformation. Strain-

energy also increases with the temperature increasing at the

negative direction of ε1, the higher temperature, the more strain-

energy accumulation at the beginning of the loading process

(Fig. 9). Thermal stress accumulated in the heating process,

inducing thermal cracking which appears in the specimen of

130oC.

Fig. 7. Strain-stress Curves of Various Temperatures

Table 6. The Results of Simulations Under Different Temperatures

T(oC) σf (MPa) σci (MPa) E (GPa) μ

20 100.2 25.2 35.9 0.24

40 100.8 18.3 39.8 0.24

60 98.0 7.4 41.2 0.25

90 94.4 1.2 43.5 0.27

130 77.1 0.1 45.8 0.27

T-temperature; σf-peak strength; σci-crack-initiation strength; E-elasticmodulus; μ-poisson’s ratio

Fig. 8. The Curves of Thermal Strain-temperature

Fig. 9. The Curves of Strain Energy-thermal Strain

Fig. 10. Micro-crack Propagation in the Heating Process (In these figures, gray refers to interfaces; orange refers to inter-grain tensile

crack)



Thermal cracking is also major controlled by the property-

heterogeneity of minerals. In this compacted model, thermal

expanding of one mineral may influence lateral minerals, providing

a tensile stress to the interface of two surrounded minerals. Different

temperature determines different crack initiation strength (crack

initiation strength is defined that the strength value of 1% gross

number of cracks in the PFC2D), Fig. 10 shows the micro-crack

propagation in the heating process, tensile crack is the major

failure mode, among these results, 130oC causes the most number of

tensile cracks and lowest crack initiation strength.

6. Micro-mechanics of Rock Failure

6.1 Fracture Development Under Coupling Effect

With the temperature changing, granite fracture mechanism

will change from the boundary (inter-granular) fracture mechanism

at low temperature to the couple mechanism of boundary (inter-

granular) and grain (trans-granular) fracture (Zuo et al., 2012b).

The temperature range of simulations is 20oC to 130oC, the

major micro-cracks are the tensile boundary fractures in this

range (Table 7). Rupture images also support this conclusion that

Table 7. The Fracture Development of Various Temperatures by Percentage of Peak Strength

T/oC 25%UCS 50%UCS 75%UCS UCS FailureFailure of Laboratory

specimenSketch of laboratory

result

20

40

60

90

130

In these figures, gray refers to interfaces; orange and dark gray refers to inter-grain tensile and shear crack, respectively; blue refers to intra-graincracks


Vol. 22, No. 8 / August 2018 − 2771 −

the tensile boundary fracture is the major failure mode (Table 6).

Under low normal stress (25%UCS, 50%UCS), the crack

development of five situations is different and depends on the

temperature value. But under high normal stress (75%UCS,

UCS), the crack propagation and coalescence of five situations

are similar, for the reason that the crack development is major

influenced by the grain structure and mechanical properties in

the compression loading test, cracking growth is always apt to

develop along the grain boundary direction, initial crack growth

has little effect on the axial transmission of stress and does not

significantly disturb the rest of the sample (Lan et al., 2010).

The laboratory results show that the failure mode transmit

from tensile shear failure to tension failure with the increasing

temperature, while the figures of specimen failure (Table 7) and

ratio of micro-tensile cracks to micro-shear cracks of simulations

(Table 8) show the similar thermal effect on the fracture

development only in the range of 40oC ~ 90oC.

6.2 Rupture Process

The effect of GBM is to help researchers investigate the micro-

crack development which is difficult to be observed in the

laboratory tests. The results of the crack development, stress

distribution and energy evolution could be used to analyze the

fracture mechanic of rocks. This paper choose the simulation

results of 60oC and 130oC to do the further analysis. At the

Table 8. Crack Number of Numerical Simulations at the Peak

Strength

T (oC) Bt Bs Bt/Bs Trans

20 1359 124 10.9 2

40 1442 150 9.61 2

60 1546 150 10.3 3

90 1754 143 12.2 6

130 2101 181 11.6 2

T-temperature; Bt-number of boundary tensile cracks; Bs-number ofboundary shear cracks; Trans-number of Trans-granular cracks

Fig. 11 Crack Development Images of 60oC: (a) the Location of AA area on the Specimen, (b) the Curve of Crack Number to Strain, (c) ε1= 0.10%, (d) ε1 = 0.15%, (e) ε1 = 0.20%, (f) ε1 = 0.25%, (g) Peak Strength, (h) Model Failure after Peak Strength (In this figure,

orange and dark gray refers to inter-grain tensile and shear crack, respectively; blue refers to intra-grain cracks)



beginning of the compression test, the tensile cracks occur

sporadic in the synthetic specimen (Fig. 11(c), Fig. 12(c)), this

number and arrangement of cracks is not only influenced by the

temperature, but also depends on the geometry of minerals,

especially the direction of interfaces. Only negative (tensile)

minor principal stresses develop along the plane of interfaces

which parallel with the direction of compression stress; on the

other hand, both positive and negative minor principal stress

magnitudes develop with positive minor principal stress developed

along other interfaces which is not parallel with the loading

stress (Fig. 13) (Bewick et al., 2012). The properties of each

interface and grain are also another important factor in the

initiate crack. The interfaces of quartz inherited a higher tensile

stiffness from mineral property, while the higher thermal

expanding of quartz may provide a higher tensile stress to the

interface of surrounded minerals.

As the strain increased, more tensile cracks growth, but no

fracture system developed in the specimen (Fig. 11(d) and 11(e)).

The continuous compression process leads the boundary shear

fractures along interface of two squeezing grains, these shear

fractures link the lateral tensile fractures to form the fracture

system (Fig. 11(f) and 11(g); Fig. 12(d) and 12(e)). Fracture

systems form the major failure plane under peak strength (Fig.

11(g); Fig. 12(e)), in this process, larger abundant grains would

likely form the dominant load-bearing skeleton within the granite

mass and could therefore dominate the stress development (Lan

Fig. 12. Crack Development Images of 130oC: (a) the Location of AA area on the Specimen, (b) the Curve of Crack Number to Strain, (c) ε1= 0.10%, (d) ε1 = 0.15%, (e) Peak Strength, (f) Model Failure after Peak Strength, the Major Fracture System Is Shown on (e) and

(f) (In this figure, orange and dark gray refers to inter-grain tensile and shear crack, respectively; blue refers to intra-grain cracks)


Vol. 22, No. 8 / August 2018 − 2773 −

et al., 2010). Under the temperature of 60oC, the angle of

beginning tensile fractures comparing with the angle of later

fracture system shows a change from tension to shear. Tensile

fracture systems which predominately formed pre-peak strength,

and shear fracture systems which are composed of linked tensile

fracture arrays which have shear displacement (Wibberley et al.,

2000). With the increasing strain, shear fractures control the

rupture development, tensile fractures turn into shear structures,

displacement vectors rotate and display antithetic direction along

tensile fracture which produces shear fracture system (Fig. 14).

It can be seen from Fig. 11 and Fig. 12, the different crack

development of 60oC and 130oC. Under same strain value and

same grain distribution, the crack number of 130oC is much

larger than 60oC and the major fracture plane is more obvious,

which means the accumulation of thermal cracks acts an

important role in the later crack developing process.

The number of cracks increases rapidly after crack coalescence

(Fig. 11(b); Fig. 12(b)) and major trans-granular cracks occur at

the same time, more micro-cracks created and linked to form

larger fractures which no longer follow the previous cracking

mode, different rupture mode exhibited under different temperatures

in Fig. 10. The curves of kinetic energy-strain which has an

identical effect with AE (Acoustic Emission) experiment (Ma et

al., 2015), shows a similar result that the major energy release

after the stage of crack coalescence, and several curves show

Fig. 13. (a) Development of Tensile Stress in the Preferential Tensile Geometric Arrangement with Increasing Applied Displacemen,

(b) Development of Tensile Stress in the Non-preferential Tensile Geometric Arrangement with Increasing Applied Displacement

(Bewick et al., 2012)

Fig. 14.The Images of Displacement Vectors: (a) and (b) Show the Same Area of 60oC, (c) and (d) Show the Same Area of 130oC. Dis-

placement Vectors Gradually Change Under Loading Process and Finally Show A Antithetic Direction, this Leads to the Produce

of Shear Fracture System: (a) ε1 = 0.15% 60°C, (b) ε1 = 0.30% 60°C, (c) ε1 = 0.15% 130°C, (d) ε1 = 0.30% 130°C

Fig. 15. The Curves of Kinetic Energy to Strain of Numerical Mod-

eling Under Different Temperature



foreshock-main shock type which means a brittle behavior

(Fig. 15). The comparison of five curves shows that from 40oC to

90oC, the main shock mode is more obvious and energy releasing

become more activate under increasing temperature, Fig. 6 also

show the same result which doesn’t follow the traditional

understanding that rock tends to be more plastic with increasing

temperature. A similar conclusion could also be found by Chen

et al. (2014): Temperature increasing enhance the brittle damage

of granite in the range of 60oC ~ 100oC.

7. Conclusions

The PFC-GBM was developed to simulate the micro-structure

and used in the UCS to analyze the failure of brittle rock, the

influence of temperature was considered in the simulation by

setting the thermal-properties of each grain. The numerical model

was calibrated by using the thermal-mechanics UCS laboratory

data and the best matched micro-properties could be confirmed.

In the UCS tests, the temperature, properties of minerals and

arrangement of grains were combined to impact the failure

mechanics. Micro-crack distribution in different process was

heavily influenced by the interface arrangement of grains. Thermal

effect major determined thermal-cracks before compression loading

and then had an effect on the later rupture process. During

compressed stage, the crack distribution was impacted by the

micro-properties and grain arrangement, different temperature

conditions produced similar crack development track because of

the same properties and arrangement.

To a certain specimen, the number of crack in the failure stage

is about 1500 to 2000, and more thermal-cracks would lead the

number of crack increasing to the stage of crack coalescence at a

low strain value. The situation of 130oC had the obvious thermal-

crack before loading and then exhibited a much lower peak

strength and failure strain compared to other situations. While

the threshold temperature of granite need more laboratory data

and simulations to be confirmed.

Fracture process developed from crack sporadic distribution to

coalescence, the micro-tensile crack was the major crack mode

in the specimen, then the micro-shear cracks became the key

connection of several tensile fracture systems, and the tensile

fracture structure changed to shear structure by viewing the

displacement vector. The fracture system development became

random and didn’t follow the previous cracking norm after peak

strength.

The laboratory data showed that the failure mode transmit

from tensile shear failure to tension failure with the increasing

temperature in the range of 20oC to 130oC. While the simulation

revealed a more accurate result that in a certain range of

temperature (40oC ~ 90oC), temperature increasing enhance the

brittle damage of granite.

The investigation of micro-heterogeneity is a practical research

direction in the rock mechanics. This paper took into account the

thermal effect in the UCS test could be used to conduct rock

engineering in the similar condition, especially in the long-large-

deep tunnels. The analysis and conclusion of this paper could be

the fundamental study of further research, such as the simulation

of rock burst and large deformation in the coupling effect. The

authors welcome further suggestion and criticism correction.

Acknowledgements

The author would like to express his great appreciation to the

guidance of Hengxing Lan from Institute of Geographic

Sciences and Natural Resources Research, Chinese Academy of

Sciences. The financial support from National Natural Science

Foundation of China (NO.41230635, NO.41172279, NO.41272330

and NO.51309218) and State Key Laboratory of Geohazard

Prevention and Geoenvironment Pretection Independent Research

Project (SKLGP2017Z001) are also gratefully acknowledged.

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