Research ArticleA Coupled Elastoplastic Damage Model for Clayey Rock and ItsNumerical Implementation and Validation

Shanpo Jia ,1,2 Zhenyun Zhao ,3 Guojun Wu ,4 Bisheng Wu ,5 and Caoxuan Wen 3

1Institute of Unconventional Oil & Gas, Northeast Petroleum University, Daqing 163318, China2Key Laboratory of Oil & Gas Reservoir and Underground Gas Storage Integrity Evaluation of Heilongjiang Province,Daqing 163318, China3School of Urban Construction, Yangtze University, Jingzhou, 434023 Hubei, China4State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academyof Sciences, Wuhan, 430071 Hubei, China5State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University,Beijing 100084, China

Correspondence should be addressed to Shanpo Jia; jiashanporsm@163.com

Received 7 November 2019; Revised 10 February 2020; Accepted 20 February 2020; Published 20 March 2020

Academic Editor: Marcello Liotta

Copyright © 2020 Shanpo Jia et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a new constitutive model for describing the strain-hardening and strain-softening behaviors of clayey rock. Asthe conventional Mohr-Coulomb (CMC) criterion has its limitation in the tensile shear region, a modified Mohr-Coulomb (MMC)criterion is proposed for clayey rock by considering the maximal tensile stress criterion. Based on the results of triaxial tests, acoupled elastoplastic damage (EPD) model, in which the elastic and plastic damage laws are introduced to describe thenonlinear hardening and softening behaviors, respectively, is developed so as to fully describe the mechanical behavior of clayeyrock. Starting from the implicit Euler integration algorithm, the stress-strain constitutive relationships and their numericalformulations are deduced for finite element implementation in the commercial package ABAQUS where a user-defined materialsubroutine (UMAT) is provided for clayey rock. Finally, the proposed model is used to simulate the triaxial tests and the resultsvalidate the proposed model and numerical implementation.

1. Introduction

Clayey rock is always selected as the geological barriers forradioactive waste disposal and considered the effective cap-rock for CO2 sequestration and oil and gas reservoir [1–3].Clayey rock in its natural state exhibits good plastic deforma-tion ability and very low permeability as geological barriers.One of the main concerns is that, due to underground exca-vation or fluid injection, the favorable properties of clayeyformation change and the host rock loses part of its barrierfunction, thus negatively influencing the performance of arepository [4, 5]. In addition, the damage associated withexcavation or injection causes obvious degradation ofmechanical and hydraulic properties of clayey rock and thusplays an important role in changing the stress and seepagefields in the practical engineering.

The damage and integrity behaviors of clayey rock are thekey factors in studying the storage and environmental safetyof geological disposal in deep repository. The undergroundexcavation can redistribute the stress and generate irrevers-ible damage around a gallery. The safety evaluation of arepository in clayey rock requires accurate prediction of themechanical perturbations and damage zone, which is calledas excavation damaged zone (EDZ) [6–8]. Therefore, a suit-able constitutive model, which can characterize accuratelythe mechanical behavior of clayey rock, is very importantfor repository design [9–11].

Clayey rock is one type of sedimentary rock, and somecharacteristics of its mechanical behavior are similar to thoseof soft rock [4]. Laboratory experiments show that clayeyrock has significant strain-softening behavior and goodplastic deformation ability [2, 12, 13]. The strain-softening

HindawiGeofluidsVolume 2020, Article ID 9853782, 14 pageshttps://doi.org/10.1155/2020/9853782

behavior of clayey rock has been studied from micro- andmacromechanical viewpoints [2, 4]. The micromechanicalmodel can reproduce not only the crack opening, interactionbetween crack surfaces, and crack closure but also the processof strain localization or shear banding. However, the micro-mechanical model has limited use in practical engineeringdue to large number of excessive preoccupied cracks andcomplex modelling approaches. Therefore, the macrome-chanical model has been mainly used in underground engi-neering and can be easily applied to simulate engineeringproblems, for example, determining the distribution of EDZand extent of failure. As for plastic deformation and strain-softening behavior for clayey rock, the modelling approachesmainly include the following [7, 14–17]: (1) the strengthparameters are considered to be weakened during strain soft-ening based on laboratory tests and field observations and (2)the damage concept is introduced into the classical contin-uum theory to describe the degradation of stiffness andstrength of clayey rock.

A large number of models, such as Cam-Clay, Drucker-Prager, Mohr-Coulomb, Cap model, and bounding surfacemodel [18–21], have been developed to describe the failurebehavior of clayey rock. Among these models, the Mohr-Coulomb failure criterion has been widely used for geotech-nical applications due to its easy implementation. Manyresearchers have noted that modelling the failure behaviorof clayey rock should take into account the damage thatgoverns the complex mechanical behavior [21, 22]. The lab-oratory tests show that plastic deformation and damage arecoupled [2, 4, 11, 21, 23], indicating that the plastic yieldcriterion should be used simultaneously with the damagecriteria to consider the changes of mechanical propertiesof clayey rock. The damage of rock has obvious effect onthe stress, strain, elastic stiffness, strength, yield function,potential function, and softening parameters. In the coupleddamage model developed recently for clayey rock [4, 6, 8–10], in order to build the complex relationship betweenplastic flow and damage evolution, some assumptions aremade on the basis of laboratory tests. To the best of theauthors’ knowledge, there is little available research on pro-posing a coupled damage constitutive model for clayey rockbased on laboratory tests. This is also the motivation of thepresent work.

The paper is organized as follows. In Section 2, theMMC criterion, which combines the CMC criterion withthe maximal tensile stress criterion, is proposed to describethe yield and plastic flow behavior of clayey rock. Section3 presents a damage constitutive model, which considersthe stiffness degradation, strain-hardening, and plasticsoftening effect. Numerical implementation of the pro-posed model in finite element software ABAQUS is pre-sented in Section 4. Finally, in Section 5, numericalresults are provided to show the effectiveness of the pro-posed model.

2. Modified Mohr-Coulomb Criterion

The Mohr-Coulomb criterion is still widely used for a largenumber of routine design in the geotechnical engineering,

which is the lower limit of all the linear strength criterion[24]. According to many experimental studies [4, 11, 25,26], the Mohr-Coulomb criterion can well reflect the elas-toplastic mechanical behavior of clayey rock. However,among the popular finite element software, such as ABA-QUS, ANSYS, and COMSOL, the built-in Mohr-Coulombmodel cannot well describe the dilatancy characteristicsand the truly associated flow properties of geomaterialbecause of the different expressions of yield and potentialfunctions [4].

The CMC criterion is generally written in terms of stressinvariants [11, 27]:

F = σm sin ϕ + �σK ′ − c cos ϕ = 0, ð1Þ

where σm, �σ, c, and ϕ denote the average stress, equivalentstress, cohesion, and friction angle, respectively.K ′ is a func-tion of Lode angle θ and friction angle ϕ.

K ′ = cos θ −1ffiffiffi3

p sin ϕ sin θ: ð2Þ

In Equations (1) and (2), σm, �σ, and θ are defined by

σm =σ1 + σ2 + σ3

3,

�σ =ffiffiffiffiJ2

p,

θ =13sin−1 −

3ffiffiffi3

p

2J3�σ3

!,

−30° ≤ θ ≤ 30°,

ð3Þ

where σ1, σ2, and σ3 are the three principal stresses and J2and J3 are the second and third invariant, respectively, ofthe deviatoric stress.

The CMC criterion does not consider the real tensilestrength of rock, and it has the limitation for the actual tensileshear zone. In Figure 1, σ∗m is the three-dimensional tensilestrength by the CMC criterion, which is larger than the actualuniaxial tensile strength of rock f t. Assuming that the three-dimensional tensile strength of rock is equal to the uniaxialtensile strength, σt = f t, the tensile Mohr-Coulomb criterionis defined as [4]

σ1 ≥ f t, ð4Þ

where f t is the uniaxial tensile strength of geomaterials andσ1 = ð2/ ffiffiffi

3p Þ�σ sin ðθ + 120°Þ + σm.

Thus, Equation (4) is rewritten in terms of stressinvariants:

F =2ffiffiffi3

p �σ sin θ + 120°ð Þ + σm − f t = 0: ð5Þ

Because the proposed MMC criterion considers bothtensile strength and shear strength of geomaterials, theyield surface is a result of combination of two different

2 Geofluids

failure criteria: shear failure and tensile failure. The yieldfunction is defined as follows [11] (as shown in Figure 1):

F = σm sin ϕ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σK ′� �2

+m2c2 cos2ϕr

− c cos ϕ = 0, ð6Þ

where 0 ≤m ≤ 1 is a parameter reflecting the tensilestrength of geomaterials. When m = 0 indicates a highertensile strength, the MMC yield criterion in Equation (6)becomes the CMC criterion; m = 1 denotes that there isno tensile strength. In addition, the parameter m cansmooth the vertex of the yield surface and avoid thenumerical divergence and slow convergence.

The Mohr-Coulomb failure surface is a hexagonal conein the principal stress space with six sharp corners in theoctahedral plane [11, 28]. The six singularities occur at θ =±30° in the octahedral plane (as shown in Figure 2), makingthe numerical convergence difficult. To approach the CMCyield surface, the parameter K ′ is expressed by using as apiecewise function:

K ′ =�A − �B sin 3θ� �

,  θj j > θT,

cos θ −1ffiffiffi3

p sin ϕ sin θ

� �,  θj j ≤ θT,

8><>: ð7Þ

K ′ =�A − �B sin 3θ� �

,  θj j > θT,

cos θ −1ffiffiffi3

p sin ϕ sin θ

� �,  θj j ≤ θT,

8><>: ð8Þ

where

�A =13cos θT 3 + tan θT tan 3θTð

+1ffiffiffi3

p sign θð Þ tan 3θT − 3 tan θTð Þ sin ϕÞ,

�B =1

3 cos 3θTsign θð Þ sin θT +

1ffiffiffi3

p sin ϕ cos θT� �

,

sign θð Þ =+1, θ ≥ 0°,

−1, θ < 0°:

(ð9Þ

The variable θT denotes the transition angle in the vicin-ity of the six singularities, and its value is in the range of0° ≤ θT ≤ 30°. In the present calculations θT = 25° is adopted.The yield function rounded in both the meridional andoctahedral planes is modified by adjusting the parametersm and K ′ in Equations (6) and (7), where the yield surfaceis continuous and differentiable for all stress states.

The plastic potential function can be defined from theyield function as follows:

G = σm sin φ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σK ′� �2

+m2c2 cos2φr

, ð10Þ

where φ is the dilatancy angle. Similarly, K ′ is a function ofLode angle θ and dilatancy angle φ. Associated flow occursif φ = ϕ, while unassociated flow occurs if 0 ≤ φ < ϕ.

Average stress 𝜎m

Effec

tive s

tres

s 𝜎

Tensile Mohr-Coulomb yield surface (ft)Shear Mohr-Coulomb yield surface (c, 𝜙)Modified Mohr-coulomb yield surface (c, 𝜙, m)

ft𝜎m⁎

𝜙

Figure 1: Modified Mohr-Coulomb criterion in the meridionalplane.

m = 0.3

, 𝜃 T = 25°

𝜎2

𝜃 =

30°

𝜃 =

25°

𝜃 = 0°

m = 0.3

m = 0, 𝜃T = 25°

𝜎1

𝜎3

Figure 2: Smooth processing of modified Mohr-Coulomb criterionin the octahedral plane.

3Geofluids

3. Coupled Elastoplastic Damage Model

3.1. Model Description. Figure 3 schematically shows thestress-strain curve of clayey rock observed in laboratory tests.The mechanical properties of clayey rock are affected by frac-tures growth, cementation degree, and stress state [4]. Whenthe microcracks grow, the clayey rock undergoes nonlineardeformation, leading to deterioration of its macroscopic elas-tic properties and strength. The interaction between the non-linear deformation and the damage can be dealt with by adamage model. For the purpose of simplicity, the stress-strain curve is divided into four stages, i.e., elastic deforma-tion, strain hardening, strain softening, and plastic flow.Due to the complexity of stress-strain behavior under triaxialstress state, the conventional elastoplastic constitutive modelcannot accurately describe this process. Thus, the presentwork uses a piecewise method to fully characterize themechanical behavior of clayey rock. Based on the damagemechanics theory, an EPD constitutive model, which con-tains elastic law, elastic damage law for strain hardening,and plastic damage law for strain softening, is developed.According to the position of peak stress point, the completestress-strain curve is divided into two zones, i.e., elastic andplastic. For the plastic zone, the Mohr-Coulomb failure crite-rion is adopted to describe the progressive plastic deforma-tion and strain-softening process.

Although anisotropic stress-induced damage has beenobserved in some laboratory tests on clayey rock, it is notobvious in this study [4, 18, 21]. Therefore, a scalar isotropicdamage variable is used in the present coupled EPD model.Following the convention of continuum damage mechanics[29], the effective stress ~σ is defined as follows from theLemaitre strain equivalent hypothesis:

~σ = σ

1 −Ω, ð11Þ

where σ is the nominal stress and Ω is the damage variable.

According to the MMC failure criterion, the yield func-tion in terms of effective stress can be written as

F = ~σm sin ϕ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~�σK ′� �2

+m2c2 cos2ϕr

− c cos ϕ: ð12Þ

By substituting Equation (11) into Equation (6), theyield function in terms of nominal stress can be deter-mined by

F = σm sin ϕ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σK ′� �2

+ 1 −Ωð Þ2m2c2 cos2ϕr

− 1 −Ωð Þc cos ϕ,

ð13Þ

Similarly, the potential function can be defined interms of nominal stress as

G = σm sin φ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σK ′� �2

+ 1 −Ωð Þ2m2c2 cos2φr

: ð14Þ

3.2. General Stress and Strain Relationship. Generally, theassumption of small strain is suitable for clayey rock.The stress-strain relationship of clayey rock shows obviousnonlinear characteristics before peak strength [2, 4, 21].Although the strain during the hardening after unloadingcannot be restored completely, the plastic strain in thisstage is so small that it can be neglected. For the purposeof simplicity, the elastic damage occurs in the strain-hardening stage and the total strain dεij is decomposedinto the following two parts:

dεij = dεRij = dεeij + dεedij , ð15Þ

where dεRij is the reversible strain, dεeij is the elastic strain,

and dεedij is the elastic damage strain.

Elasticdamage

𝜎c0

𝜎cu

𝜎r 𝜎r

Elasticdamage Plastic damage

Lateral strain

Hardening So�ening Plasticflow

HardeningSo�eningPlasticflow

Axial strain

Axi

al st

ress

Plastic damage

Figure 3: The stress-strain curve in different stages.

4 Geofluids

The elastic damage strain dεedij is deduced from the changeof elastic parameters during the strain hardening [17]:

dεedij =∂Cijkl Ωð Þ

∂ΩdΩ ⋅ σkl , ð16Þ

where Cijkl is the elastic stiffness tensor and σkl is the stresstensor.

When the stress in clayey rock reaches the peak strength,the rock will undergo plastic deformation which is coupledwith the damage. For the stages of strain softening and plasticflow, the irreversible strain dεIRij is decomposed into plasticand damage parts:

dεIRij = dεpij + dεdij, ð17Þ

where dεpij is the plastic strain and dεdij is the plastic damagestrain.

According to the potential function in Equation (14), theirreversible strain dεIRij can be defined as follows [8, 10]:

dεIRij = dλ∂G σij,Ω� �∂σij

, ð18Þ

where dλ is the plastic multiplier.The general stress and strain relationship can be expressed

as follows:

dσij = Cijkl dεkl − dεIRkl� �

+ εkl − εIRkl� � ∂Cijkl

∂ΩdΩ: ð19Þ

3.3. Definition of Damage. According to stress-strain rela-tionship of clayey rock as shown in Figure 3, the curve is lin-ear in the elastic stage and nonlinear in the strain-hardeningstage. The transition point delineating these two stages isconsidered the elastic damage starting point. The elasticdamage stage ends and the plastic damage commences whenthe peak strength is reached. The evolution equation of elas-tic damage is

Ωe = β1 �e −�e0eð Þ, ð20Þ

where β1 is the elastic damage parameter and�e0e is the energyfactor of the elastic damage starting point. The energy factor�e is defined as [4]

�e =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεijC

0ijklεkl

q, ð21Þ

where C0ijkl is the undamaged elastic stiffness tensor.

The peak stress of the stress-strain curve is defined as thethreshold of plastic damage. Actually, plastic damage isdependent on the history of plastic deformation and its ratetends to stabilize gradually with the increase of accumulateddeformation. The evolution equation of plastic damage iswritten as follows:

Ωp =�e −�e0p

α2 + β2 �e −�e0p� � , ð22Þ

where �e0p is the energy factor corresponding to the plasticdamage starting point and α2β2 are the plastic damageparameters.

3.4. Evolution of Model Parameters. According to the contin-uum damage mechanics, the degradation of elastic modulusduring loading is defined as follows [4, 29]:

E = 1 −Ωð ÞE0, ð23Þ

where E is the damaged elastic modulus, E0 is the initialundamaged elastic modulus, and Ω is the sum of elasticand plastic damage variables, i.e., Ω =Ωe +Ωp.

Equation (23) shows that the elastic modulus decreasesgradually during the damage deformation. When the rockapproaches a complete damage, the damage variable Ω isclose to 1 and thus, the elastic modulus tends to be 0. How-ever, this is not in accordance with engineering application,where the elastic modulus of clayey rock retains its residualvalue even after a significant damage. In order to overcomethis problem, here the elastic modulus is expressed by usinga bilinear function of the damage variable:

E = E0 − E0 − Erð Þ Ω

Ωlim

� �, 0 ≤Ω ≤Ωlim,

E = Er, Ωlim ≤Ω ≤ 1,

8><>: ð24Þ

where Er is the residual elastic modulus of clayey rock andΩlim is the critical damage value and Ωlim = 0:99 is adoptedin this study.

During the strain-softening stage, the strength of clayeyrock decreases while the plastic damage variable and plasticdeformation increase. As the friction angle has small changeduring the strain-softening process [4, 30], it is assumed thatthe strength parameter evolution is only described by thecohesion, which is expressed as follows:

c = c0 − c0 − crð Þ ⋅Ωpη, ð25Þ

where c0 and cr denote the initial and residual cohesions,respectively; η ∈ ð0, 1� is a model parameter controlling theslope of the strain-softening curve.

4. Numerical Implementation

4.1. Implicit Constitutive Integration Algorithm

4.1.1. Backward Euler Implicit Integration (BEII) Algorithm.The fully BEII method is used in this study to return thestresses for predicting the yield surface when the stressesexceed the yield strength [31–33]. Given the responses attime tn, i.e., stress σn, damage variable Ωn, and a total strainincrement Δεn+1, the objective is to determine these state var-iables σn+1, εn+1, and Ωn+1 at time tn+1. The BEII method

5Geofluids

includes mainly three steps, i.e., elastic predictor, plastic cor-rector, and damage corrector.

In the elastic predictor step, the scalar damage Ωn isassumed to be a constant. A trial damage is defined as

Ωtrialn+1 =Ωn, ð26Þ

and a trial stress σtrialn+1 is defined based on the elastic predictor

σtrialn+1 = σn +CΔεn+1, ð27Þ

where C is the elastic stiffness matrix, which is a function ofthe trial damage Ωtrial

n+1.The onset of plastic flow and deformation is determined

by the loading condition, which is expressed as follows:

F σtrialn+1,Ωtrialn+1

� �= 0,

dλ ≥ 0,

F σtrialn+1,Ωtrialn+1

� �⋅ dλ = 0:

ð28Þ

The trial stress at point B, σtrialB , is obtained by the elasticpredictor (as shown in Figure 4)

σtrialB = σn +CΔεn+1 = σn + Δσe, ð29Þ

where Δσeis the increment of elastic stress.In order to return the trial stress σtrialn+1 for predicting the

yield surface, the stress increment is defined as

Δσn+1 =C Δεn+1 − ΔεIRn+1� �

=CΔεn+1 − ΔλCb = Δσe − ΔλCb,ð30Þ

where b = ð∂Gðσtrialn+1,Ωtrialn+1Þ/∂σÞn+1. The derivatives of the

potential function and yield function are given in theappendix.

Thus, the trial stress at point C is expressed as

σtrialC = σtrialB − ΔλCb: ð31Þ

The first-order Taylor expansion of the yield function atpoint B gives

F = FB +∂F∂σ

� �TΔσ + ∂F

∂c∂c∂Ωp

ΔΩp +∂F∂Ω

ΔΩ = 0: ð32Þ

In the plastic corrector step, the scalar damage Ωtrialn+1 is

regarded as constant. Then, Equation (32) reduces to be

F = FB +∂F∂σ

� �TΔσ = FB − ΔλaTBCbB = 0, ð33Þ

where a = ∂F/∂σ and FB = FðσtrialB ,Ωtrialn+1Þ.

Here the plastic multiplier Δλ can be determined explic-itly as

Δλ =FB

aTBCbB: ð34Þ

The backward Euler algorithm is based on the equation

σC = σB − ΔλCbC: ð35Þ

where σB is the elastic trial stresses and the variables withsubscript C are related to the current configuration. A start-ing estimate for σC is defined as

σC = σB − ΔλCbB: ð36Þ

Generally, this starting value of stress σC does not satisfythe yield function and further iterations will be requiredbecause the normal at the trial position B is not equal to thefinal normal (as shown in Figure 4). To control the iterativeloop, a vector r is used to represent the difference betweenthe current stresses and the backward Euler calculations, i.e.,

r = σ − σB − ΔλCbCð Þ = σ − σB + ΔλCbC: ð37Þ

The iterations continue until the norm of vector r is smallenough (almost zero) while the final stresses should satisfythe yield criterion.

With the trial stresses σB being kept fixed, a truncatedTaylor expansion can be applied to Equation (37). A newresidual vector rN is expressed as follows:

rN = r0 + _σ + _λCb + ΔλC ∂b∂σ _σ, ð38Þ

where r0 is the residual vector at point B, _σ is the change in σ,_λ is the change of Δλ, and N is the number of iteration.

𝜎

A

B

CN

C0

FB > 0

F = 0

Figure 4: Sketch map of the backward Euler integration algorithm.

6 Geofluids

Setting rN to zero gives

_σ = − I + ΔλC ∂b∂σ

� �−1r0 + _λCb� �

= −Q−1r0 − _λQ−1Cb,

ð39Þ

where I is the unit matrix; Q = I + ΔλCð∂b/∂σÞ.Similarly, a truncated Taylor series on the yield func-

tion is

FCN = FC0 +∂F∂σ

� �T_σ = 0: ð40Þ

By substituting Equation (39) into Equation (40), thechange of Δλ is determined by

_λ =FC − aTCQ−1r0aTCQ−1Cb

, ð41Þ

and the plastic multiplier after N times of iteration isobtained:

Δλ Nð Þ = Δλ N−1ð Þ + _λ N−1ð Þ: ð42Þ

The modified stress vector σC after N times of itera-tions is

σC Nð Þ = σB − Δλ N−1ð ÞCbC N−1ð Þ: ð43Þ

The strain vector εn+1 at time tn+1 is expressed as

εn+1 = εn + Δεn+1: ð44Þ

The third step in the BEII method is the damage cor-rector. The updated damage at time tn+1 is

Ωn+1 =Ωn + ΔΩ, ð45Þ

where ΔΩ = ΔΩe + ΔΩp.After the damage corrector is completed, the final stress

at time tn+1 is determined:

σn+1 = Cn+1 : εen+1 =Cn+1 : εen + Δεn+1 − ΔεIRn+1� �

=Cn+1Cn

σtrialn+1 − Cn : ΔεIRn+1� �

=1 −Ωn+11 −Ωn

σC Nð Þ:ð46Þ

4.1.2. Consistent Tangent Stiffness Matrix. The consistenttangent stiffness matrix is related to the convergence speedof global equilibrium iteration, and it does not influencethe final results of stress updating. By dropping the subscriptC in Equation (35), the standard back-Euler algorithm isexpressed as

σ = σB − ΔλCb: ð47Þ

It should be noted that the variables without subscript“C” in Equation (47) are related to the current configuration.

When the change of damage in plastic corrector is omitted,differentiation of Equation (47) gives

_σ = C_ε − _λCb − ΔλC ∂b∂σ _σ, ð48Þ

which is simplified to be

_σ = I + ΔλC ∂b∂σ

� �−1C _ε − _λb� �

=Q−1C _ε − _λb� �

= R _ε − _λb� �

,

ð49Þ

where R =Q−1C.To retain the current stress σ on the yield surface, _F

should be zero. The consistency conditions of yield functionF is expressed as follows:

_F =∂F∂σ

� �T_σ + ∂F

∂c∂c∂Ωp

_Ωp +∂F∂Ω

_Ω = aT _σ = 0: ð50Þ

By substituting Equation (49) into Equation (50), _λ isdetermined by

_λ =aTR_εaTRb : ð51Þ

The consistent tangent stiffness matrix,Dct, can beobtained by substituting _λ into Equation (49):

_σ = R −RbaTRT

aTRb

!_ε =Dct _ε: ð52Þ

4.2. Procedure of the Constitutive Integration Algorithm. Inthe fully implicit backward Euler algorithm [31, 34], theincrement of irreversible strain and damage variable is calcu-lated at the end of increment step n. The constitutive modelintegration algorithm can be expressed as

ε n+1ð Þ = ε nð Þ + Δε,

εIRn+1ð Þ = εIRnð Þ + Δλ n+1ð Þb n+1ð Þ,

Ω n+1ð Þ =Ω nð Þ + ΔΩ,

σ n+1ð Þ =C : ε n+1ð Þ − εIRn+1ð Þ� �

,

F n+1ð Þ = F n+1ð Þ σ n+1ð Þ,Ω n+1ð Þ� �

:

8>>>>>>>>>>><>>>>>>>>>>>:

ð53Þ

Equation (53) is a system of nonlinear equations ofεðn+1Þ, εIRðn+1Þ, and Ωðn+1Þ. At any time tn, a set of values ðεðnÞ,εIRðnÞ,ΩðnÞÞ and strain increment Δε = Δt _ε are given. The

updated variables come from the convergence value at theend of the previous time step, which achieves the effect ofavoiding nonphysical meaning. The solution of the

7Geofluids

nonlinear equations is solved by Newton-Raphson iterativemethod. For the updated variables, the superscript k repre-sents the number iteration and the subscript n representsthe increment step.

The stress update algorithm flow is as follows:

Step 1. Set the initial value. The initial value of irreversiblestrain and damage variable is the convergence value atthe end of the last load step. The incremental value of

plastic parameter is set to zero, and the elastic trial stressis calculated.

k = 0 : εIR 0ð Þn+1ð Þ = εIRnð Þ,

Ω0ð Þn+1ð Þ =Ω nð Þ,

Δλ0ð Þn+1ð Þ = 0,

σ 0ð Þn+1ð Þ =C : ε n+1ð Þ − εIR 0ð Þ

n+1ð Þ� �

:

ð54Þ

Step 2. Check the yield condition and convergence at theiteration number k.

F kð Þn+1ð Þ = F σ kð Þ

n+1ð Þ,Ωkð Þn+1ð Þ

� �: ð55Þ

For a given stress error tolerance TOL, if FðkÞðn+1Þ < TOL,

convergence; otherwise, go to Step 3.

Step 3.Calculate the increment of plastic parameters, then theincrement of stress and damage variables can be obtained.

δλkð Þn+1ð Þ =

F kð Þn+1 − a kð Þ

n+1ð Þ� �T

Q−1r0

a kð Þn+1ð Þ

� �TQ−1Cb

,

δσ kð Þn+1ð Þ = − I + ΔλC ∂b

∂σ

� �−1r0 + _λCb� �

,

δΩkð Þn+1ð Þ = δ Ω

kð Þe n+1ð Þ +Ω

kð Þp n+1ð Þ

� �:

ð56Þ

Increment step startingCall subroutine UMAT

Calculate elastictentative stress

Calculate yield functionjudging whether yield

Calculate Jacobian matrix, and update stress

Plastic flow rule

Stress update algorithm

Pullback of stress toyield surface

Exit subroutine UMAT, increment step end

No yield

Yield

Judge damage state according todamage equation, if 𝛺 = 0

Yes

Parameter correction of the model

No

Calculate elastic matrix

Calculate modified elastic matrix

Figure 5: Flow chart of the UMAT subroutine.

76 mm

38 mm

𝜎1

𝜎1

𝜎3𝜎3

Figure 6: Geometrical model for triaxial tests.

8 Geofluids

Step 4. Update plastic multiplier, strain, damage, and stress.

Δλk+1ð Þn+1ð Þ = Δλ

kð Þn+1ð Þ + δλ

kð Þn+1ð Þ,

Ωk+1ð Þn+1ð Þ =Ω

kð Þn+1ð Þ + δΩ

kð Þn+1ð Þ,

εIR k+1ð Þn+1ð Þ = εIR kð Þ

n+1ð Þ + C−1 : δσ kð Þn+1ð Þ,

σ k+1ð Þn+1ð Þ = σ kð Þ

n+1ð Þ + C : ε n+1ð Þ − εIR k+1ð Þn+1ð Þ

� �:

8>>>>>>>><>>>>>>>>:

ð57Þ

Let k = k + 1; go to Step 3.

4.3. Secondary Development of UMAT Subroutine. Thenumerical formulations discussed above are programmed asa UMAT subroutine in ABAQUS with FORTRAN language[28]. The material properties are defined in this UMAT sub-routine, in which the stress and other related state variablesare updated at the end of each time step. The UMAT subrou-tine can also work with a user subroutine to redefine fieldvariables at a material point (USDFLD) to update the fieldvariables [4, 21]. The stress, elastic, and plastic strain areobtained by iteration in the UMAT subroutine, and then,the state variables such as plastic parameters and damagevariable are updated. The change rate of _σ with respect to _εis provided by the constitutive Jacobian (DDSDDE) in theUMAT subroutine as the Jacobian matrix of the materialconstitutive model. The flow chart of the UMAT subroutinefor the proposed model is shown in Figure 5.

5. Application of the Proposed Model

5.1. Model Validation. The proposed constitutive model wasdeveloped and implemented in the UMAT subroutine ofABAQUS. In order to test the calculation ability, accuracy,and efficiency of this UMAT subroutine, the performanceof the constitutive formulation was evaluated for uniaxialtension and confined compression scenarios. In the special

case when no damage is mobilized, the proposed EPD modelreduces to an ideal elastoplastic model. For the purpose ofnumerical illustration, the numerical results using theUMAT subroutine based on the MMC criterion were com-pared with those by using the built-in Mohr-Coulomb crite-rion in ABAQUS.

Figure 6 shows the geometrical model for one cylinderrock sample with a diameter of 38mm and a height of76mm. The bottom of the sample is fixed, and a vertical dis-placement load is applied on the top surface of the sample.The elastic modulus E = 700MPa and Poisson’s ratio μ =0:25. The plastic properties of rock include the friction angle,cohesion, and dilatancy angle, whose values are φ = 18°, c =0:3MPa, and ϕ = 18°, respectively. For the nonassociatedflow rule, the dilatancy angle is ϕ = 0°. The parameter m =0:05 is used in the MMC criterion for avoiding the numericaldivergence in this study.

Figures 7 and 8 compare the stress-strain relationship foruniaxial and triaxial compression, respectively, between thepresent MMC model and the built-in Mohr-Coulomb modelin ABAQUS. It can be found that the numerical results ofdeviatoric stress and volumetric strain predicted from bothmodels are in good agreement. For a given axial strain undera specific confining pressure, the yield stress obtained fromthe MMC model (with the UMAT subroutine) is slightlysmaller than that obtained from the built-in model. This isdue to the fact that the yield surface of the MMC criterionis inside that of built-in Mohr-Coulomb criterion. In partic-ular, the volumetric strain predicted from both models agreevery well. This indicates that the proposed MMC model isfeasible to describe the ideal plastic response under compres-sion conditions.

The mechanical response of the sample under uniaxialtension is shown in Figure 9 for different values of mreflecting the tensile strength of rock. It can be seen thatwhen m = 0:05, the numerical results calculated from thepresent model are in agreement with those obtained by

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Axial strain (%)

–0.20

–0.15

–0.10

–0.05

0.00

0.05

0.10

0.15

Volumetric strain

Vol

umet

ric st

rain

(%)

Deviatoric stress

Dev

iato

ric st

ress

(MPa

)

UMATABAQUS

(a)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Axial strain (%)

Volumetric strain

Deviatoric stress

–4.5–4.0–3.5–3.0–2.5–2.0–1.5–1.0–0.50.00.5

UMATABAQUS

Vol

umet

ric st

rain

(%)

Dev

iato

ric st

ress

(MPa

)

(b)

Figure 7: Stress-strain relationship under uniaxial compression: (a) nonassociated flow and (b) associated flow.

9Geofluids

the built-in model in ABAQUS. In this case, the calculatedtension strength from the MMC model is 0.424MPa, whichis very close to the calculated value of 0.436MPa from thebuilt-in model. When m = 0:2, the tension strength obtainedby the MMC model is 0.353MPa, and it is significantlylower than that predicted by the built-in model. Therefore,

the actual tension strength of rock is dependent on theparameter m.

5.2. Triaxial Tests and Numeric Simulation on Clayey Rock.In this subsection, the mechanical characteristics of clayeyrock are studied by using the proposed EPD model.

The clayey rock in this study is a stiff clay. The undis-turbed samples (Φ38 × 76mm) used in the undrained triaxialtests are extracted from clayey rock formation at a depth of247m. Considering the low permeability of clayey rock (withan order of magnitude 10−19m2), these tests can be used toestimate the undrained shear strength by analogy with quickexcavation and tunneling for repository [18, 35].

The tests were performed in a triaxial apparatus with aconfining pressure between 0.89MPa and 5.42MPa accord-ing to following sequences [4, 36]. First, samples were loadedin increment of approximately 0.5MPa to the scheduled con-fined pressure, and then, a valve is opened to apply a backpressure (nominally 50% of cell pressure) allowing consoli-dation to commence. Once consolidation and pore pressure

–0.5 0.0 0.5 1.0 1.5 2.00.0

0.1

0.2

0.3

0.4

0.5

Lateral strain (%) Axial strain (%)

UMAT, m = 0.05UMAT, m = 0.2ABAQUS

Dev

iato

ric st

ress

(MPa

)

Figure 9: Stress-strain relation in uniaxial tensile cases.

Table 1: Initial damage points under different confining pressures.

Confining pressure (MPa) �e0e (MPa1/2) �e0p(MPa1/2)

0.89 0.273 0.354

2.50 0.227 0.413

2.85 0.197 0.322

5.42 0.191 0.553

Table 2: The known parameters for clayey rock.

ParametersE0

(MPa)Poisson’sratio μ

c0(MPa)

ϕ (°) k (m2)

Value 300 0.13 0.30 18 3 × 10−19

Table 3: The results of unknown parameters by back analysis.

Confiningpressure

β1 α2 β2cr

(KPa)η

Er(MPa)

φ (°)

0.89 1.381 0.569 0.990 58.132 0.608 77.418 7.290

2.50 0.860 0.583 1.243 97.946 0.404 95.593 2.501

2.85 0.939 1.190 1.886 44.399 0.693 66.270 0.804

5.42 0.590 2.008 0.951 49.008 0.699 117.196 0.921

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Volumetric strain

Deviatoric stress

Axial strain (%)

Vol

umet

ric st

rain

(%)

Dev

iato

ric st

ress

(MPa

)

UMATABAQUS

(a)

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5Deviatoric stress

Axial strain (%)

–3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0.0

0.5

1.0

Volumetric strain

Vol

umet

ric st

rain

(%)

Dev

iato

ric st

ress

(MPa

)

UMATABAQUS

(b)

Figure 8: Stress-strain relationship under triaxial compression with a confining pressure 3MPa: (a) nonassociated cases and (b) associatedflow.

10 Geofluids

dissipation were completed, the specimens were sheared bythe application of small increments in axial stress.

The finite element model is shown in Figure 6. Theboundary conditions and simulation sequences are the sameas those in laboratory tests. Simulation of undrained triaxialtest involves solving a coupled hydromechanical problem,and the computational precision depends mainly on themesh size and time step, especially the stage at the beginningof calculation. In order to ensure a uniform excess pore pres-sure in the sample, the initial time step Δt is estimated basedon the following formula [28, 37]:

Δt ≥γw6E′k

Δhð Þ2, ð58Þ

where E′ is the drained elastic modulus, γw is the density ofpore water, k is the permeability of clayey rock, and Δh isthe characteristic length of the mesh.

The elastic modulus, Poison’s ratio, peak cohesion, fric-tion angle, and two energy factors of the damage startingpoint can be obtained directly from tested results, which are

listed in Tables 1 and 2. The unknown parameters in the pro-posed EPDmodel can be determined by using a back analysis[4], which are listed in Table 3.

The stress-strain curves under varying confining pres-sures from 0.89 to 5.42MPa are shown in Figure 10. Theundrained triaxial tests show that clayey rock exhibits obviousstrain hardening and strain softening during the shearingdeformation, and the stress-strain curves can be classified intofour stages. Moreover, large plastic deformation during strainsoftening is the main feature of the mechanical response ofclayey rock. It can be observed that the numerical results fromthe proposed model are in good agreement with the testresults. This indicates that the proposed EPDmodel is capableof effectively capturing the strain-hardening and strain-softening characteristics of the clayey rock. When the defor-mation of clayey rock reaches the initial elastic damage point,the elastic damage increases slowly and then remains constantduring plastic deformation stage. From Figure 10, it can alsobe seen that the total damage variable increases graduallywith the increase of axial strain and the plastic damage behav-ior plays an important role in the irreversible deformation.

–4 –3 –2 –1 0 1 2 3 4 5 6 7 80.00.20.40.60.81.01.21.41.61.82.0

Lateral strain (%) Axial strain (%)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Dam

age

Dev

iato

ric st

ress

(MPa

)

ModellingTest dataDamage

(a)

–8 –6 –4 –2 0 2 4 6 8 10 12 14 160.0

0.5

1.0

1.5

2.0

2.5

Axial strain (%)Lateral strain (%)

Dam

age

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Dev

iato

ric st

ress

(MPa

)

ModellingTest dataDamage

(b)

–6 –4 –2 0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

2.5

Axial strain (%)Lateral strain (%)

Dam

age

ModellingTest dataDamage

0.0

0.1

0.2

0.3

0.4

0.5

Dev

iato

ric st

ress

(MPa

)

(c)

–10 –8 –6 –4 –2 0 2 4 6 8 10 12 14 16 180.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Axial strain (%)Lateral strain (%)

Dam

age

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Dev

iato

ric st

ress

(MPa

)

ModellingTest dataDamage

(d)

Figure 10: Comparison of the stress-strain curves under different confining pressures between the numerical analysis and experiments: (a)confining pressure 0.89MPa, (b) confining pressure2.5MPa, (b) confining pressure 2.85MPa, and (d) confining pressure 5.42MPa.

11Geofluids

6. Conclusions

In the present work, a new coupled elastoplastic damage (EPD)model was developed by using a modified Mohr-Coulomb cri-terionwhich considers themaximal tensile strength. Elastic anddamages were introduced to describe the strain-hardeningand strain-softening processes, respectively.

Based on the implicit Euler stress integration algorithm,the constitutive integrating formulations and the consistentstiffness matrix for the coupled EPD model were deduced.For engineering application, the proposed model was codedas a subroutine UMAT in ABAQUS.

To validate the EPD model, the triaxial compression testand uniaxial tension test were first simulated without consid-ering the damage effect. The comparison of the numericalresults between the EPDmodel (with the UMAT subroutine)and built-in Mohr-Coulomb model in ABAQUS indicatesthat the proposed model is capable of effectively describingthe tensile and compression responses of rocks. Then, thecoupled EPDmodel is used to simulate the undrained triaxialtests of clayey rock. Comparisons between numerical simula-tions and experimental data show that the proposed modelcan characterize the strain hardening, strain softening, andplastic flow of clayey rock very well.

Further experimental studies are necessary to obtainmore data for a better understanding of the plastic anddamage mechanical behavior of clayey rock, especiallywhen some parameters in the proposed model have strongdependence on the confining pressure. In the future, theproposed model will be extended to consider the effect ofconfining pressure.

Appendix

Derivatives of Yield Function andPotential Function

Numerical implementation of the backward Euler algorithmrequires the first derivative of yield and potential functionsand the second derivative of potential function.

The first derivative of the yield function with respect tostress is

a = ∂F∂σ = C1′

∂σm∂σ + C2′

∂�σ∂σ + C3′

∂J3∂σ , ðA:1Þ

where C1′, C2′, C3′, ∂σm/∂σ, ∂�σ/∂σ, and ∂J3/∂σ are definedas follows:

C1′ =∂F∂σm

= sin ϕ,

∂σm∂σ =

131, 1, 1, 0, 0, 0½ �T,

C2′ =∂F∂�σ

−tan 3θ

�σ

∂F∂θ

= α′ K ′ − tan 3θdK ′dθ

!,

∂�σ∂σ =

12�σ

sx, sy, sz , 2τxy , 2τxz , 2τyz T,

C3′ = −ffiffiffi3

p

2 cos 3θ�σ3∂F∂θ

= α′ −ffiffiffi3

p

2 cos 3θ�σ2dK ′dθ

!,

∂J3∂σ =

sysz − τ2yz ,

sxsz − τ2xz ,

sxsy − τ2xy,

2 τyzτxz − szτxy� �

,

2 τxyτyz − syτxz� �

,

2 τxzτxy − sxτyz� �

,

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

+�σ2

3

1,

1,

1,

0,

0,

0,

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

ðA:2Þ

where

α′ = �σK ′ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σ2K ′2 +m2c2 cos2ϕ

q ,

dK ′/dθ =−3�B cos 3θ,  θj j > θT,

−sin θ −1ffiffiffi3

p sin ϕ cos θ,  θj j ≤ θT:

8><>:

ðA:3Þ

The first derivative of potential function with respectto stress, denoted by vector b, is similar to that of theyield function ∂F/∂σ and can be obtained by replacingϕ in the formula of K ′ðθÞ, C1, C2, C3, α′, and dK ′/dθwith φ:

b = ∂G∂σ = C1

∂σm∂σ + C2

∂�σ∂σ + C3

∂J3∂σ , ðA:4Þ

where C1 = ∂G/∂σm, C2 = ∂G/∂�σ − ðtan 3θ/�σÞð∂G/∂θÞ andC3 = −ð ffiffiffi

3p

/2 cos 3θ�σ3Þð∂G/∂θÞ. They are simplified to be

C1 = sin φ, ðA:5Þ

C2 = αCmc2 = α K ′ − tan 3θ

dK ′dθ

!,

C3 = αCmc3 = α −

ffiffiffi3

p

2 cos 3θ�σ2dK ′dθ

!,

ðA:6Þ

where α = �σK ′/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σ2K ′2 +m2c2 cos2φ

q.

The second derivative of the plastic potential function isdetermined by

∂b∂σ =

∂2G∂σ2 =

∂C2∂σ

∂�σ∂σ + C2

∂2�σ∂σ2 +

∂C3∂σ

∂J3∂σ + C3

∂2 J3∂σ2 , ðA:7Þ

where ∂C2/∂σ = αð∂Cmc2 /∂σÞ + Cmc

2 ð∂α/∂σÞ and ∂C3/∂σ = αð∂Cmc

3 /∂σÞ + Cmc3 ð∂α/∂σÞ. ∂Cmc

2 /∂σ, ∂Cmc3 /∂σ,∂α/∂σ, ∂2�σ/∂

σ2, and ∂2 J3/∂σ2 are defined as follows:

12 Geofluids

where ∂θ/∂σ = −ffiffiffi3

p/2�σ3 cos 3θð∂J3/∂σ − ð3J3/�σÞð∂�σ/∂σÞÞ.

In the special cases when jθj = 300, C2, C3, ∂θ/∂σ, ∂α/∂σ,∂C2/∂σ, and ∂C3/∂σ become infinite, which makes numericalimplementation difficult. These singularities can be dealtwith appropriately through setting the values of C2, C3, ∂θ/∂σ, ∂α/∂σ, ∂C2/∂σ, and ∂C3/∂σ as zero.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

There is no conflict of interests that exists in this paper.

Acknowledgments

The authors gratefully acknowledge the support of theOpen Research Fund of State Key Laboratory of Geomecha-

nics and Geotechnical Engineering (Grant No. Z013007),the Oil and Gas Reservoir Geology and Exploitation (GrantNo. PLN1507), the Natural Science Foundation of HubeiProvince (Grant No. 2015CFB194), and the National Natu-ral Science Foundation of China (Grant No. 41102182).

References

[1] D. L. Katz andM. R. Tek, “Storage of natural gas in saline aqui-fers,” Water Resources Research, vol. 6, no. 5, pp. 1515–1521,1970.

[2] H. D. Yu, W. Z. Chen, S. P. Jia, J. J. Cao, and X. L. Li, “Exper-imental study on the hydro-mechanical behavior of Boomclay,” International Journal of Rock Mechanics and Mining Sci-ences, vol. 53, pp. 159–165, 2012.

[3] J. G. Wang and Y. Peng, “Numerical modeling for the com-bined effects of two-phase flow, deformation, gas diffusionand CO2 sorption on caprock sealing efficiency,” Journal ofGeochemical Exploration, vol. 144, pp. 154–167, 2014.

[4] S. P. Jia, “Hydro-mechanical coupled creep damage constitu-tive model of Boom clay, back analysis of model parameters

∂Cmc2

∂σ =∂θ∂σ

∂K ′∂θ

+∂2K ′∂θ2

tan 3θ − 3∂K ′∂θ

sec23θ !

,

∂Cmc3

∂σ =ffiffiffi3

p

2 cos 3θ�σ2∂θ∂σ

∂2K ′∂θ2

− 3∂K ′∂θ

tan 3θ !

+2�σ

∂K ′∂θ

∂�σ∂σ

" #,

∂α∂σ =

1 − α2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�σ2K ′2 +m2c2 cos2φ

q ∂�σ∂σK

′ + �σ∂K ′∂θ

∂θ∂σ

!,

∂2�σ∂σ2 =

1�σ

13−

s2x4�σ2

−16−sxsy4�σ2

13−

s2y4�σ2

sym

−16−sxsz4�σ2

−16−sysz4�σ2

13−

s2z4�σ2

−τxysx2�σ2

−τxysy2�σ2 −

τxysz2�σ2

1 −τ2xy�σ2

−τxzsx2�σ2

−τxzsy2�σ2 −

τxzsz2�σ2

−τxzτxy�σ2

1 −τ2xz�σ2

−τyzsx2�σ2

−τyzsy2�σ2 −

τyzsz2�σ2

−τyzτxy�σ2

−τyzτxz�σ2

1 −τ2yz�σ2

2666666666666666666666664

3777777777777777777777775

,

∂2 J3∂σ2 =

13

sx − sy − sz

2sz sy − sx − sz sym

2sy 2sx sz − sx − sy

2τxy 2τxy −4τxy −6sz2τxz −4τxz 2τxz 6τyz −6sy−4τyz 2τyz 2τyz 6τxz 6τxy −6sx

2666666666664

3777777777775,

ðA:8Þ

13Geofluids

and its engineering application,” PhD Thesis, Wuhan Instituteof Rock & Soil Mechanics, Chinese Academy of Sciences,China, 2009.

[5] O. Kolditz, H. Shao, W. Q. Wang, and S. Bauer, Thermo-Hydro-Mechanical-Chemical Processes in Fractured PorousMedia: Modelling and Benchmarking, Springer, Switzerland,2015.

[6] A. Millard, J. Maßmann, A. Rejeb, and S. Uehara, “Study of theinitiation and propagation of excavation damaged zonesaround openings in argillaceous rock,” Environmental Geol-ogy, vol. 57, no. 6, pp. 1325–1335, 2009.

[7] B. Pardoen, S. Levasseur, and F. Collin, “Using local secondgradient model and shear strain localisation to model the exca-vation damaged zone in unsaturated claystone,” Rock Mechan-ics and Rock Engineering, vol. 48, no. 2, pp. 691–714, 2015.

[8] J. C. Zhang, W. Y. Xu, H. L. Wang, R. B. Wang, Q. X. Meng,and S. W. du, “A coupled elastoplastic damage model for brit-tle rocks and its application in modelling underground excava-tion,” International Journal of Rock Mechanics and MiningSciences, vol. 84, pp. 130–141, 2016.

[9] A. S. Chiarelli, J. F. Shao, and N. Hoteit, “Modeling of elasto-plastic damage behavior of a claystone,” International Journalof Plasticity, vol. 19, no. 1, pp. 23–45, 2003.

[10] M. R. Salari, S. Saeb, K. J. Willam, S. J. Patchet, and R. C.Carrasco, “A coupled elastoplastic damage model for geo-materials,” Computer Methods in Applied Mechanics andEngineering, vol. 193, no. 27-29, pp. 2625–2643, 2004.

[11] J. J. Muñoz, “Thermo-hydro-mechanical analysis of soft rock:application to a large heating test and large scale ventilationtest,” PhD Thesis, Universitat Politecnica De Catalunya, 2006.

[12] A. Bouazza, W. F. van Impe, and W. Haegeman, “Somemechanical properties of reconstituted Boom clay,” Geotechni-cal & Geological Engineering, vol. 14, no. 4, pp. 341–352, 1996.

[13] N. Sultan, Y.-J. Cui, and P. Delage, “Yielding and plasticbehaviour of Boom clay,” Géotechnique, vol. 60, no. 9,pp. 657–666, 2010.

[14] S. L. Wang, H. Zheng, C. Li, and X. Ge, “A finite elementimplementation of strain-softening rock mass,” InternationalJournal of Rock Mechanics and Mining Sciences, vol. 48,no. 1, pp. 67–76, 2011.

[15] Z. H. Zhao, W. M. Wang, and X. Gao, “Evolution laws ofstrength parameters of soft rock at the post-peak consideringstiffness degradation,” Journal of Zhejiang University ScienceA, vol. 15, no. 4, pp. 282–290, 2014.

[16] H.-C. Wang, W.-H. Zhao, D.-S. Sun, and B.-B. Guo, “Mohr-Coulomb yield criterion in rock plastic mechanics,” ChineseJournal of Geophysics, vol. 55, no. 6, pp. 733–741, 2012.

[17] K. Zhang, H. Zhou, and J. Shao, “An experimental investiga-tion and an elastoplastic constitutive model for a porous rock,”Rock Mechanics and Rock Engineering, vol. 46, no. 6, pp. 1499–1511, 2013.

[18] J. D. Barnichon, “Contribution of the bounding surface plas-ticity to the simulation of gallery excavation in plastic clays,”Engineering Geology, vol. 64, no. 2-3, pp. 217–231, 2002.

[19] W. H. Wu, X. Li, R. Charlier, and F. Collin, “A thermo-hydro-mechanical constitutive model and its numerical modelling forunsaturated soils,” Computers and Geotechnics, vol. 31, no. 2,pp. 155–167, 2004.

[20] N. Conil, I. Djeran-Maigre, R. Cabrillac, and K. Su, “Poroplas-tic damage model for claystones,” Applied Clay Science, vol. 26,no. 1-4, pp. 473–487, 2004.

[21] Z. Gong, W. Z. Chen, H. D. Yu, Y. S. Ma, H. M. Tian, and X. L.Li, “A thermo-mechanical coupled elastoplastic damage modelfor Boom clay,” Rock and Soil Mechanics, vol. 37, no. 9,pp. 2433–2442, 2016.

[22] D. Hoxha, A. Giraud, F. Homand, and C. Auvray, “Saturatedand unsaturated behaviour modelling of Meuse-Haute/Marneargillite,” International Journal of Plasticity, vol. 23, no. 5,pp. 733–766, 2007.

[23] P. Kolmayer, R. Fernandes, and C. Chavant, “Numericalimplementation of a new rheological law for argilites,” AppliedClay Science, vol. 26, no. 1-4, pp. 499–510, 2004.

[24] M. H. Yu, M. Yoshimine, H. F. Qiang et al., “Advancesand prospects for strength theory,” Engineering Mechanics,vol. 21, no. 6, pp. 1–20, 2004.

[25] V. Labiouse and A. Giraud, “Analytical solutions for theundrained response of a poro-elasto-plastic medium arounda cylindrical opening,” in Poromechanics, J. F. Thimus, Ed.,pp. 439–444, Balkema, Rotterdam, 1998.

[26] X. L. Li, W. Bastiaens, and F. Bernier, “The hydromechanicalbehaviour of the Boom clay observed during excavation ofthe connecting gallery at Mol site,” in Proceedings of EURO-CK2006—multiphysics coupling and long term behaviour inrock mechanics, pp. 467–472, Balkema, Lie’ge, Belgium, 2006.

[27] A. J. Abbo and S. W. Sloan, “A smooth hyperbolic approxima-tion to the Mohr-Coulomb yield criterion,” Computers &Structures, vol. 54, no. 3, pp. 427–441, 1995.

[28] D. Hibbit, B. Karlsson, and P. Sorensen, ABAQUS AnalysisUser’s Manual; Version 6.5, vol. 3, HKS Inc., Pawtucket, RI,2004.

[29] J. Lemaitre, “How to use damage mechanics,” Nuclear Engi-neering and Design, vol. 80, no. 2, pp. 233–245, 1984.

[30] H. D. Yu, “Study on long-term hydro-mechanical coupledbehavior of Belgium Boom clay,” Ph. D. Thesis, WuhanInstitute of Rock and Soil Mechanics, Chinese Academy ofSciences, Beijing, China, 2010.

[31] M. A. Crisfield, Non-linear Finite Element Analysis of Solidsand Structures, John Wiley & Sons Ltd., 1997.

[32] X.Wang, L. B.Wang, and L. M. Xu, “Formulation of the returnmapping algorithm for elastoplastic soil models,” Computersand Geotechnics, vol. 31, no. 4, pp. 315–338, 2004.

[33] C. S. Zhang, J. Ji, S. Q. Yang, and J. Kodikara, “Implicit integra-tion of simple breakage constitutive model for crushable gran-ular materials: a numerical test,” Computers and Geotechnics,vol. 82, pp. 43–53, 2017.

[34] S. Wang, W.Wu, C. Peng, X. He, and D. Cui, “Numerical inte-gration and FE implementation of a hypoplastic constitutivemodel,” Acta Geotechnica, vol. 13, no. 6, pp. 1265–1281, 2018.

[35] Y. Li, G. Matej, and W. Isabelle, Boom clay hydraulic conduc-tivity: a synthesis of 30 years of research report, Mol: SCK•CEN,2011.

[36] S. T. Horseman, M. G. Winter, and D. C. Enwistle, Geotech-nical characterization of Boom clay in relation to the disposalof radioactive waste (No. EUR–10987), Commission of theEuropean Communities, 1987.

[37] P. A. Vermeer and A. Verruijt, “An accuracy condition forconsolidation by finite elements,” International Journal forNumerical and Analytical Methods in Geomechanics, vol. 5,no. 1, pp. 1–14, 1981.

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