elastoplastic creep

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An Elasto-Viscoplastic Creep Model for Prediction ofDeformations Around Tunnels

Ahmad Fahimifar a

, Mohammad Ghasemi b*

, Mojtaba Karami c

a Professor, Amirkabir University of Technology, Tehran, Iran

bGraduate Student, Tafresh University, Tafresh, Iran

cGraduate Student, Amirkabir University of Technology, Tehran, Iran

*Corresponding Author’s E-mail: [email protected]

ABSTRACT

By excavating an underground space, the state of stress and displacement in the surrounding

medium are changed in comparison to the inital state. As time passes, the variation of the displacement

mainly depends on the creep behaviour of the hosting rock mass. In this paper, an elasto-viscoplasticcreep model is proposed. In the proposed model, the main purpose is to consider plastic deformations

increasing with time. The viscoplastic behaviour of rocks plays a key role in the tunnelling works,

especially for deep tunnels subjected to large in situ stresses. Using non-linear Hoek-Brown yield

criterion in a creep model is the other important aim of this paper, which eliminates estimating specific

equivalent Mohr-Coulomb strength parameters from the Hoek-Brown parameters. The equations

related to the proposed model are derived and then, to reach a numerical solution of the equations, the

finite difference software (FLAC2D-FISH Editor) is used. The application of the proposed model is

illustrated through an example analyzed numerically using the finite difference software FLAC2D.

Keywords: Creep, Rock Mass Behaviour, Tunnel, Elasto-Viscoplastic model, Hoek-Brown Criterion

1. INTRODUCTION

Time-dependent responses of rock materials should be taken into account in many engineering

problems, such as long-term stability analysis of underground constructions (Zhou et al., 2008). The

tunnels excavated in rocks would considerably experience deformations increasing with time, that

might lead to a delayed failure of structures. According to the results of experimental researches and

the experience from engineering practices, many creep models have been proposed to predict the time-

dependent behaviour of rock mass of many different types. These models are divided into: classic

viscoelastic models, classic viscoplastic models, etc. Equations related to the proposed models are

based on different assumpions and conditions.

In the distant past, for the analysis of time-dependent behaviour, especially creep, linear creep

models were used (Jaeger and Cook, 1969). However, with such models the experimental results donot agree with the theoretically obtained curves (Valsangkar and Gokhale, 1971). Nowadays, the

classic viscoelastic Burger model and the classic Burger-MC model are widely used by engineers to

predict time-dependent behaviour of rock especially in tunnelling works. Besides, these two models

are practical models which exist in the finite differenc software FLAC2D

. The classic viscoelastic

models can be represented by a series of springs and dashpots which are connected in parallel and/or

in series (Kontogianni et al., 2005; Dai, 2004; Sakurai, 1978). The constitutive laws in the classic

viscoplastic models try to relate the current strain rate to the current stress (and/or stress rate) directly.

Particularly, the relationship between the deviatoric strain rate and the deviatoric stress (and/or stress

rate) can be schematically represented by a series of spring, dashpot and a plastic slider that connected

in parallel and/or in series (Guan et al., 2007).

At great depth, where the rock mass is subjected to large in situ stresses, the stress redistribution

may lead to the so called squeezing conditions (Barla, 1995). These conditions yield irreversible

deviatoric creep strains that develop during time at constant and eventually increasing rate, namely the

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secondary and tertiary creep stages (Sterpi and Gioda, 2007). This behaviour usually characterizes

weak or altered rocks and, for very deep excavations, it can be experienced also in hard rocks

(Dusseault and Fordham, 1993; Malan, 2002). The viscoplastic behaviour of rocks has been widely

discussed on the basis of experimental findings from laboratory or field investigations, constitutive

modelling and numerical analysis (Ladanyi, 1993; Cristescu and Gioda, 1994; Cristescu and Hunsche,

1998; Pellet et al., 2005).

In this paper, with respect to powers and weaknesses of previous creep models and based on the

classic Burger-MC model, an elasto-viscoplastic creep model is proposed. In the proposed model both

viscoelastic and viscoplastic deformations are considered and non-linear Hoek-Brown yield criterion is

used. Although the proposed model is not included directly in the FLAC2D,

s constitutive laws library,

alternatively, the programmable language FISH (FLACish) helped to implement it indirectly in the

medium of the software. For verification of the proposed model, the model is implemented in a special

tunnel (engineering instance) and then, creep diagram of the model and results of numerical analysis

are compared with some practical models which exist in FLAC2D

.

2. THE ELASTO-VISCOPLASTIC MODEL

2.1 The classic Burger-MC modelThe classic Burger-MC model consists of a Burger body, which is able to present the primary and

seconary creep regions of rock masses (Fahimifar et al., 2010), is connected in series with a plastic

Mohr-Coulomb unit. The constitutive laws of the classic Burger-MC model are characterized by an

elastoplastic volumetric behaviour and a viscoelasto plastic deviatoric behaviour. The deviatoric

behaviour is schematically illustrated in figure 1.

Figure 1. Schematical representation of Burger-MC creep model

In the classic Burger-MC model a Kelvin unit is characterized by its shear modulus GK and

viscosity ηK , a Maxwell unit is characterized by its shear modulus GM and viscosity η

M and a plastic

Mohr-Coulomb unit, is characterized by its cohesion c, friction angle φ and dilation angle ψ . The

constitutive equations of the Burger-MC model can be worked out as follows:

Total strain rate:

1

Introducing the total strain rate , Maxwell strain rate , Kelvin strain rate and the plastic

strain rate

. The Maxwell and Kelvin units do not carry volumetric stress and they only participate

in carrying deviatoric stress.

The amount of the plastic strain rate is described by the flow rule governed by the plastic slider

(Sterpi and Gioda, 2007):

λ ∂Q∂σ 2

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The deviatoric behaviour of the Burger-MC model can be formulated as:

+ 3

Kelvin unit:

2 2 4

Maxwell unit:

2

2 5

Plastic Mohr-Coulomb unit:

3 6

And the volumetric behaviour is given by:

K 7

In the above equations, the symbols and are the deviatoric stress and the deviatoric strain,

respectively, K is the bulk modulus, Q is the plastic potential, is a multiplier and is the kronoker parameter which is equal to 1 in a condition of i=j, otherwise it is equal to 0. The superscripts K and M

refer to the Kelvin and Maxwell components of the corresponding variables, and the superposed dot

denotes time derivative.

and

are the deviatoric and volumetric plastic strain rates respectively,

and is the total stress.

For the plastic Mohr-Coulomb unit, the yield criterion f and the plastic potential Q are formulated

as:

2 8

Q σ σ 9

Where C is the material cohesion, is the friction, is the material dilation, σ and σ are the

minimum and maximum principal stresses (compression negative), 1/1,

1/1. (Itasca FLAC2D

, 2004)

The constitutive equations related to the classic Burger-MC model can predict deformation of rock

as reversible primary and secondary creep associated with Kelvin and Maxwell unit, respectively and

immediate plastic (irreversible) deformation by the Mohr-Coulomb unit. But, experimental studies on

creep behaviour of rocks show that irreversible creep deformation considerably icreases with time,

especially for deep tunnels subjected to large in situ stresses for which squeezing conditions may

develop (Sterpi and Gioda, 2007). So, it is necessary to consider modelling of viscoplastic

deformations. Inasmuch as, rock mass strength is a non-linear function of stress level, the empirical

Hoek-Brown failure criterion accounts for this observation and it is the criterion most commonly

employed to characterize failure of rock masses in tunnelling projects (Jimenez et al., 2008). So itwould be better to utilize non-linear Hoek-Brown criterion istead of Mohr-Coulomb criterion as a

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yield surface in a creep model.

2.2 The elasto-viscoplastic model

In order to model creep behaviour of rocks, many researchers have been used the slider element in

parallel with the viscous element to predict viscoplastic deformations of rocks (Tomanovic, 2006;

Sterpi and Gioda, 2007; Malan, 1999). In this paper, time-dependent irreversible deformations are

produced using a viscoplastic unit consists of a Hoek-Brown plastic slider and of a dashpot element

which are connected in parallel. For instance, D.F. Malan (Malan, 1999) used a viscoplastic unit

(consists of a Mohr-Coulomb plastic slider and of a dashpot element which are connected in parallel)

by simply connecting it in series with an elastic spring element so the final model was composed of

the spring element to represent the instantaneous deformation and of the viscoplastic unit to produce

time- dependent deformation.

Inasmuch as, creep deformation of rocks is considered to be both reversible and irreversible, in this

paper a new elasto-viscoplastic creep model is proposed which considers immediate deformations,

reversible and irreversible deformations increasing with time. In the proposed model the Burger body

(a Kelvin and a Maxwell unit in series) is connected in series with a viscoplastic unit. The viscoplastic

unit consists of a dashpot element which is connected in parallel with a plastic Hoek-Brown slider

element. The constitutive laws of the proposed model are characterized by an elastoplastic volumetric behaviour and an elasto-viscoplastic deviatoric behaviour. The deviatoric behaviour of the proposed

model is schematically illustrated in figure 2.

Figure 2. Schematical representation of the proposed model

In the proposed model a Kelvin unit is characterized by its shear modulus GK and viscosity η

K , a

Maxwell unit is characterized by its shear modulus GM and viscosity ηM and a viscoplastic unit

consists of a dashpot which is characterized by its viscosity ηvp

and a plastic Hoek-Brown slider which

is characterized by its strength parameters m b , S and the dilation angle ψ .

The constitutive equations of the proposed model can be worked out as follows:

Total strain rate:

10

Introducing the total strain rate , Maxwell strain rate , Kelvin strain rate and the

viscoplastic strain rate

. As mentioned in section 2.1, the Maxwell and Kelvin units do not carry

volumetric stress and they only participate in carrying deviatoric stress.

The amount of the viscoplastic strain rate is described by the flow rule governed by the plastic

slider (Sterpi and Gioda, 2007):

λ ∂Q

∂σ 11

The deviatoric behaviour of the proposed model can be formulated as:Total strain rate:

293



+ 12

The Maxwell and Kelvin strain rate incorporated on Eq.5 are equal to what described in Eq.1.

Dashpot:

2 13

Plastic Hoek-Brown slider:

3 14

And the volumetric behaviour is given by:

K 15

In the above equations, the symbols and are the deviatoric stress and the deviatoric strain,

respectively, K is the bulk modulus, Q is the plastic potential, is a multiplier and is the kronoker

parameter which is equal to 1 in a condition of i=j, otherwise it is equal to 0. The superscripts K and M

refer to the Kelvin and Maxwell components of the corresponding variables, and the superposed dot

denotes time derivative.

and

are the deviatoric and volumetric plastic strain rates respectively,

and is the total stress.

For the plastic Hoek-Brown slider, the yield criterion f and the plastic potential Q with respect to

non-associated flow rule are formulated as:

16

Q σ σ 17

Where m b and S are the strength parameters, is the uniaxial compressive strength of the intact

rock material, is the material dilation, σ and σ are the minimum and maximum principal stresses

(compression negative), 1/1.

The proposed model which is illustrated in figure 2, is implemented in finite difference software

FLAC2D

using the built-in FISH language for constitutive models. The programmable language FISHoffers an environment to user to introduce a new function or constitutive model that could work in

such a way that the built-in constitutive models could. The FISH function calculates the value of c and

φ for each zone in every step. Thus, as σ changes, the values of c and φ will also change. It is noted

that the instantaneous values of c and φ calculated in this way closely match those calculated using

Hoek’s (Hoek, 1990) expressions based on normal and shear stress (Itasca FLAC2D

, 2004).

Finally by application of the plastic yield condition (f = 0) , the parameter will be determined as:

1 1 2 C2 1

18

Where C and are the instantaneous values of and . Now that the is computed, the

viscoplastic strain rate component can be computed by the flow rule described in Eq.6:

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λ 19

0 20 λ 21

The governing equations of the constitutive model are implemented in numerical software FLAC2D

using the built-in language FISH function for the constitutive model. As long as the stress level is

below the threshold described by the slider element, the proposed model behaves like the viscoelastic

Burger model. For the higher stress level where the plastic limit condition is satisfied, the viscoplastic

unit calculates the plastic strain which is added to the elastic Burger strain resulting total elasto-

viscoplastic strain.

3. PRACTICAL APPLICATION OF THE PROPOSED MODEL TO A TUNNELLINGPROBLEM

The potential applicability of the proposed model is tested through an example (Sterpi and Gioda,

2007) in the medium of the finite difference software FLAC2D

. The problem concerns the full faceexcavation, at a rate of 6m/day, of a circular tunnel of radius R =4.5 m, in a homogeneous, isotropic

rock mass, subjected to an isotropic state of stress =10 MPa. This corresponds to a tunnel depth of

approximately 400 m.

Inasmuch as, strength parameters of the hosting rock mass (Sterpi and Gioda, 2007) are in terms of

Mohr-Coulomb strength parameters, the Hoek-Brown strength parameters of the hosting rock mass are

derived (Hoek and Brown, 1997). Creep parameters of the hosting rock mass and the strength

parameters of rock are presented in Table 1. and Table 2. respectively.

Figure 3 shows the geometric and boundary conditions of this tunnel. The numerical analysis

considers a creep time of 365 days.

Table 1. Creep parameters of rock mass used in the present study

K (MPa) G M (MPa) η

M (Mpa.day) G

K (MPa) η

K (Mpa.day) η

vp(Mpa.day)

1470 678 321555 5848 49470 1651

K : the Bulk modulus , G : the Maxwell shear modulus, η M

: the Maxwell viscosity , G K

: the Kelvin

shear modulus, η K

: the Kelvin viscosity, ηvp

: the viscoplastic viscosity.

Table 2. Strength parameters of rock used in the present study

GSI mi (MPa) mb S

50 12 13 2.01 0.004 15.4

GSI: Geological Strength Index, mi: the Hoek-Brown strength parameter of intact rock ,

: the

rock mass compressive strength, mb: the Hoek-Brown strength parameter of rock mass, S : the

Hoek-Brown strength parameter of rock mass, : the dilation angle.

295



Figure 3. Mesh and boundary conditions of tunnel

In figure 4, time-dependent behaviour of the surrounding rock mass implementing the Burger and

the Burger-MC models are compared to find out the differences and weaknesses of these two practical

models which exist in FLAC2D.

Figure 4. Comparison between the Burger and the Burger-MC model for time-dependent shear strain

of the tunnel

Figure 4 shows that the difference between the Burger and the Burger-MC models is in the

immediate plastic (irreversible) deformation which is considered by the Burger-MC model and causes

a gap between these two models. But, as time passes the shear strains of these two models increase

with an approximate same rate. Figure 5 shows the radial convergence of the tunnel face using the

Burger, Burger-MC and the proposed model.

296



Figure 5. Comparison between the Burger, Burger-MC and the proposed model for the radial

convergence of the tunnel face

It can be seen in figure 5 that at early times the radial convergence of the tunnel face produced by

the proposed model is near to the Burger’s model and as time passes by icreasing viscoplastic (time-

dependent irreversible) deformations it reaches to the Burger-MC’s model. Inasmuch as, elastic and

plastic deformation of rocks are both immediate and time-dependent features, the results of creep

behaviour of the proposed model are in proper agreement to the time-dependent behaviour of rock

masses in a real medium.

4. CONCLUSIONS

In this study, based on the well-known Burger-MC model, a new elasto-viscoplastic creep model is

proposed. The proposed model cosists of a Burger body which is connected in series with a

viscoplastic unit accounting for the irreversible creep deformation of rock. Besides, in the proposed

model the slider element satisfies the non-linear Hoek-Brown yield criterion which is used most

commonly to characterize failure of rock masses in tunnelling projects. The equations of the proposed

model are implemented in numerical software FLAC2D

using the built-in language FISH function for

the constitutive model. Comparative analysis between the creep deformaion of the proposed model

and that of the Burger and Burger-MC models indicates that the proposed model can describe elastic

and plastic creep deformation of rock as two stages: the primary elastic creep deformation and the

secondary creep including both elastic and plastic with constant strain rate. It is also obsereved that the

irreversible creep deformation is crucial in the prediction of time-dependent behaviour of rock

especially at high stress level where the rock gets its plastic phase.

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elastoplastic creep