International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102

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International Journal ofRock Mechanics & Mining Sciences

1365-16

http://d

n Corr

E-m

xpyuan

journal homepage: www.elsevier.com/locate/ijrmms

An interacting crack-mechanics based model for elastoplasticdamage model of rock-like materials under compression

X.P. Yuan n, H.Y. Liu, Z.Q. Wang

College of Engineering and Technology, China University of Geosciences (Beijing), Beijing 100083, China

a r t i c l e i n f o

Article history:

Received 30 June 2011

Received in revised form

20 June 2012

Accepted 24 September 2012Available online 17 November 2012

Keywords:

Crack-interaction

Self-consistent

Plasticity

Quasi-static crack growth

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ijrmms.2012.09.007

esponding author.

ail addresses: yuanxiaoping007@126.com,

1@hotmail.com (X.P. Yuan).

a b s t r a c t

A micro-mechanical elastoplastic damage model for rock-like materials under compressive loading is

proposed based on the growth of pre-existing flaws. Interaction among the cracks is included through

the self-consistent approach. The evolution of damage is quantified by the spatial flaw density and the

density of the quasi-static spherical region, enclosing the flaw and its wings. The flaw density is defined

by the absolute volume strain in the two-parameter Weibull statistical model. Mixed-mode fracture

model is adopted to calculate the wing crack length by the strain energy density (SED) criterion.

Drucker–Prager yield criterion and Voyiadjis’ strain hardening function under compression are

employed to represent the equivalent plastic behavior of such materials. This self-consistent scheme

is implemented numerically with an implicit updated and a prediction–correction decomposition.

Numerical simulations are carried out, and the factors of friction coefficient, confining pressure and

initial flaw size are analyzed.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Brittle materials such as rocks, concrete and ceramics, exhibita complex non-linear effective response when subjected tocompressive loading. The dominant micro-mechanism that com-monly characterizes damage in brittle materials is attributed tothe presence of intrinsic flaws such as microcracks and pores.

Plasticity theory is often proposed to characterize the non-linearbehavior of rock-like materials [1–5]. The main characteristic of thesemodels is the pressure sensitivity, the loading path sensitivity, thenon-associative flow rule, and the strain hardening prior to ultimatefailure. Continuum damage mechanics is necessary to capture thedegradation of the material elastic properties due to anisotropicdamage effects due to, for example, crack closure and irreversiblesliding [6,7]. Plasticity and continuum damage mechanics are usuallyused together to represent the mechanical behavior of such materials[8–17], as it is done in this contribution.

Damage in rock-like materials is primarily caused by thepropagation and coalescence of microcracks. In this way, slidingcrack models were proposed to analyze the generation, growthand coalescence of microcracks occurring via tension cracks or‘‘wing cracks’’ from pre-existing flaws. The conditions underwhich an array of such cracks interacts in a complex stress field

ll rights reserved.

[18–21]. Ravichandran et al. [22] proposed a micromechanicalapproach for biaxial dynamic compressive loading, based on non-interacting, randomly distributed sliding microcracks which acti-vate when the stress intensity factor reaches its critical value. Thecracked spatial distribution could be characterized by Refs.[23,24] for high strain-rate loading.

Experimental observations [25] have demonstrated that theinteraction among the growing microcracks has a profoundinfluence on the failure behavior and macroscopic stress–strainresponse of brittle materials. One of the reasons is that crack-interaction leads to an effective stress intensity factor at the cracktips which is different from that which would develop if thecracks were isolated when crack density exceeds a certainamount. Ashby et al. [17,18] included a tensile field that affectsgrowth to account for interaction. Instead of considering theinteraction effect through the additional field, Paliwal et al. [26]developed a strain-rate dependent constitutive model for brittlefailure under compressive loading with an explicit account ofcrack-interactions, modeled by means of a crack-matrix-effective-medium approach.

In the present work, we develop an elastoplastic damagemodel that incorporates pre-existing flaws which are assumedto be randomly distributed in space and permitted only overselected orientations and certain size. The crack-interaction isaccounted for with the self-consistent approach to represent thebrittle failure process under predominantly compressive loading.The plastic yield criterion of the homogenized medium combinedwith the micro-mechanical damage model to simulate the

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102 93

inelastic deformation. The Drucker–Prager criteria and Voyiadjis’strain hardening function under compression are employed todescribe plastic behaviors of such materials. The growth ofdamage is quantified by spatial flaw density and quasi-staticspherical cavity region calculation. Fracture mechanics-basedmodels are typically employed to account for the quasi-staticwing crack length by the SED criterion for mixed-mode fractureunder loading condition. The density of flaws is mainly deter-mined through the two-parameter Weibull statistical model.

The micro-mechanical constitutive model captures the plasticdeformations up to failure occurs in these materials, particularly athigh-confining pressures, and it is also able to account for mixed-mode fracture of interaction among cracks or fully static crackpropagation (further developed based on Ref. [27]). In addition, theproposed constitutive model is fully three-dimensional and could beimplemented in any finite element code. An algorithm for theincremented stress–strain update is proposed based on a prediction–correction split.

The following direct tensor notations [28] are used in the paper:

i� dijei � ej

I � 12ðdikdjlþdildjkÞei � ej � ek � el

u� v� uivjei � ej

A� B� AijBklei � ej � ek � el

uUv� ukvk

AUu� Aikukei

AUB� AikBkjei � ej

T : B� TijklBklei � ej

C : D� CijklDklmnei � ej � em � en

trA¼ i : A¼ Aii

A : B� AikBki

where i is the second-order identity tensor; I is the fourth-order(symmetric) identity tensor; dij is the Kronecker delta; {ei} (i¼1, 2, 3)is an arbitrary orthonormal basis; � denotes the tensor product; u, vare vectors; A, B are symmetric, second-order tensors; T, C, D arefourth-order tensors;: denotes the scalar product of second-ordertensors; and tr denotes the trace of a second-order tensor.

2. Development of the constitutive model and algorithm

2.1. Wing crack growth from initial flaws under compressive loading

The pre-existing flaws in rocks are assumed to be an initialpenny-shaped crack, radius a, randomly distributed in space, asshown in Fig. 1(a). We shall also assume that although frictionalsliding of the pre-existing flaws also causes inelastic deformation,the significant damage is due to the nucleation and growth ofopen wing cracks, and the model is considered to be free fromdamage before the propagation of these microcracks.

Under compressive load, the activation of frictional sliding istypically attributed to the shear stress over the faces of cracks.Since the cracks are closed, the tendency for two crack surfaces toslide due to the shear stress is opposed to friction stress. Slidingwill occur when the resolved shear stress along the main cracksexceeds the threshold shear friction, which causes wing cracks tonucleate and propagate, from each tip, at an angle of about70.51 [16–19]. After a short initial curving, these wing cracks

align themselves in the direction of the maximum macroscopicprincipal compressive stress and become rather straight. Weapproximate these wing cracks as open straight cracks, causingdamage evolution in our model, and the length of each wing crackis assumed to be l (see Fig. 1(d)).

Interactions among these cracks will generally cause theeffective stress field around them to be different from that if theywere isolated, shown in Fig. 1(b) and Fig. 1(c). However, it is not ingeneral feasible to account for the stress field for each individualmicrocrack and account for the interaction with all the othermicrocracks at large flaw densities. We develop a methodology bymeans of stress concentration tensor approach to obtain anapproximate local effective stress field as a manifestation ofcrack-interactions, which will be proposed in next section.The effective shear stress tc and the normal stress pc on the crackface are given by

tc ¼ tðrc ,y1Þ ¼sc

yy�scxx

2sin 2y1þsc

xycos 2y1

pc ¼ pðrc ,y1Þ ¼sc

xxþscyy

2þsc

yy�scxx

2cos 2y1�sc

xysin 2y1 ð1Þ

where y1 is the orientation angle of the cracks and rc is theeffective stress tensor around isolated cracks.

Then the effective stress teff on the crack surface can beexpressed as

tef f ðrc ,y1Þ ¼ tc�mpc ð2Þ

where m is the friction coefficient on surface of cracks. The forcesFI, FII on the penny-shaped crack surface that cause the wing cracktension and shearing are, respectively, shown as follows:

FI ¼ pa2tef f cosy1

FII ¼ pa2tef f siny1 ð3Þ

The function teff(rc, y1) attains its maximum value when

y1¼(1/2)tan�1(1/m), the most favorable orientation of the closedsliding cracks for the nucleation of the wing cracks [17–19].Although the orientation y1 is expected to be random in realrocks, it is assumed to be the constant y1E451 (when m-0þ) asan approximation throughout this paper for simplicity.

Several analytical models have been proposed to formulatemode I stress intensity factor KI at the wing crack tip (for details,see Refs. [20,29,30]). To consider shear stress and the direction ofwing crack growth under biaxial compressive loading, the effec-tive mode I stress intensity factor (SIF) KI and mode II SIF KII at thetip of the wing cracks are given by

KI ¼�FI

pðlþ lnÞ½ �3=2 þ

2p s

cxx

ffiffiffiffiffiplp

KII ¼�FII

pðlþ lnÞ½ �3=2

ð4Þ

where ln¼0.27a was introduced in Ref. [19] to render KI and KII

non-singular at l¼0.The effective strain energy density at wing crack tip can be

expressed as follows [31]:

S¼ a11K2I þ2a12KIKIIþa22K2

II ð5Þ

where

a11 ¼1þn8pE 3�4n�cosy3ð Þ 1þcosy3ð Þ� �

a12 ¼1þn8pE

2siny3ð Þ cosy3� 1�2nð Þ� �

a22 ¼1þn8pE

4 1�cosy3ð Þ 1�nð Þþ 1þcosy3ð Þ 3cosy3�1ð Þ��

ð6Þ

and y3¼0, charactering the strain energy density at the extendeddirection of wing crack. The minimum strain energy density can

Fig. 1. Schematic of self-consistent model: (a) undamaged microstructure consisting of penny-shaped cracks; (b) damage develops in axially compressed material, cracks

are assumed to grow in direction of loading; (c) isolated undamaged region around a single crack and local stresses around individual crack and (d) penny-shaped crack.

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–10294

be expressed as follows [31]:

SC ¼ð1þnÞð1�2nÞ

2pEK2

IC ð7Þ

where KIC is the rock fracture toughness.The wing cracks propagate from each tip and follow a path so

as to maximize the mode I stress intensity factor, KI. Ashby et al.[17,18] considered that the wing crack growth was mainly due tothe extension induced by the pre-existing flaws, and the max-imum tensile stress (MTS) criterion was employed to calculate thewing crack length as well as dynamic crack length [21,22,26,27].

Since the crack propagation is mixed-mode fracture model inthis work, SED criterion [31] is employed to calculate the quasi-static wing crack length using the Newton iteration method.The quasi-static wing crack length at time t is obtained as thefollowing expression:

lt ¼ 0, if ðSÞt rSC

increase lt , until ðSÞt ¼ SC , if ðSÞt 4SC

(ð8Þ

where (S)t is the strain energy density of wing crack tip at time t.

2.2. Interaction among cracks: self-consistent model

We also note that the level of damage-induced stress field thatdepends in general on the distribution of the pre-existing flawsand the interaction among the cracks as the damage progresses.Some of these cracks are activated as a result of the load-induceddamage and have straight open wing cracks at their tips, and theproperties of the surrounding medium begin to differ from that ofthe matrix. Due to the difference in the properties of the mediumand the matrix, interaction can cause the effective stress fieldaround a crack to be different from that of isolated cracks.

In this section, we shall employ the self-consistent scheme as anapproximation model to consider the effect of the crack-interaction.The local stress fields around every flaw or every flaw-wing cracksystem (an initial crack with the wing cracks at its tips) in a realmaterial are sensitive to the spatial distribution and the character-istics of the neighboring cracks. Assume that a flaw with two wingcracks at its tips is completely embedded in a spherical matrix ofpristine material, which in turn is embedded in an effective mediumwith the effective properties of the cracked solid, shown in Fig. 1(b).

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102 95

The effective compliance tensor of spherical matrix andeffective medium take the following forms [32], respectively:

Mc ¼1

3Kc

1

3i� iþ

1

2GcI

M ¼1

3K

1

3i� iþ

1

2GI ð9Þ

where I¼I�1/3i�i, effective elastic modulus K¼(1�D)Kc, G

¼(1�D)Gc, D is the scalar-valued damage parameter, and Kc, Gc

are bulk and shear modulus of spherical matrix, respectively.Consequently, the spherical matrix becomes effectively ortho-

tropic and the Eshelby tensor [33] can be described as thefollowing form:

S ¼ a013i� iþb0I ð10Þ

where a0 ¼1/3(1þn)/(1�n), b0 ¼2/15(4–5n)/(1�n), as a compro-mise and given insufficient data, we assume effective Poisson’sratios to stay constant, i.e. n¼n.

Combining Eq. and Eq. (10), the elastic and compliance con-straint tensors of effective medium are given, respectively:

Ln¼ LðS

�1�IÞ ¼

3Kð1�a0Þa0

1

3i� iþ

2Gð1�b0Þb0

I

Mn¼ L

n�1¼ ðS

�1�IÞ�1M ¼

a0

3Kð1�a0Þ1

3i� iþ

b0

2Gð1�b0ÞI ð11Þ

where L is the equivalent elastic tensor of effective medium.Hence, using Eqs. (9–11), stress concentration tensor of thematrix [32] can be described as follows:

Bc¼ ðM

nþMcÞ

�1ðM

nþMÞ

¼1

Da0 þ 1�Dð Þ

� �1

3i� iþ

1

Db0 þ 1�Dð Þ

� �I ð12Þ

Crack-interactions can cause cracks to be under the influenceof the effective stress field sc different from the macroscopicstress r. The effective stress field acting on isolated cracks can bedescribed from macroscopic stress by stress concentration tensoras follows:

rc ¼ Bc : r ð13Þ

The effective SIF KI and KII at the wing crack tips, discussed inthe section above, are obtained by means of the effective stressfield sc around a crack.

2.3. Flaw density growth law and damage definition

For the rock damage, the weakening of the elastic modulus is afunction of the number of active flaws. Walsh et al. [34] proposedrock damage scalar as follows:

Dt ¼NtVt ð14Þ

where Nt is the number of flaws per unit volume that arefavorable for growth and Vt¼4plt

3/3 is the quasi-static sphericalregion surrounding a wing crack of radius lt, which approximatesthe stress relieved volume due to the traction-free boundary ofthe wing crack.

Since the initial flaws are stochastically distributed in thespecimen, the well-established Weibull statistic model is adoptedto provide a satisfactory description of the inherent flaw distribu-tion. The flaw distribution is described by a two-parameterWeibull function of elastic volume strain parameter [21,22]. Theabsolute volume strain e9V9 is introduced to include plastic strainin the distribution function for finite element analysis in thispaper. The Weibull distribution function and absolute volumestrain can be defined as the following expression:

Zt ¼ kðeV Þmt ð15Þ

ðeV Þt ¼ ðeeV Þt

�� ��þ ðepV Þt

�� �� ð16Þ

where Zt (spatial flaw density) is the number of flaws per unitvolume which can activate at or below a absolute volume strainlevel of e9V9, constant k is used to describe the nucleation of themicrocracks at a certain strain level in Weibull distributionfunction, and m is material constant which determines the staindependence of crack density increment. The terms ee

V

� t

�� �� andep

V

� t

�� �� are the absolute elastic volume strain and plastic volumestrain, respectively. With continued loading, new flaws willbecome available for activation. Because of previous damage,however, a volume fraction D of the rock would have been stressrelieved. Thus, the number of flaws that will actually activate isgiven by [21,22]:

Nt ¼ Zt 1�Dtð Þ ð17Þ

Substituting Eq. (17) into Eq. (14), the damage scalar is derivedas follows:

Dt ¼ZtVt

1þZtVtð18Þ

Eq. (18) clearly shows that the damage accumulation increaseswith the increase of the absolute volume strain and wing cracklength. The flaws will not propagate if the strain energy density ofwing crack tip at the time t satisfies (S)trSC, namely, the wingcrack length lt¼0 and damage scalar Dt¼0; while (S)t4SC, theflaws will propagate and damage scalar Dt40.

2.4. Plasticity in the rock-like materials

For the sake of simplicity the pressure-sensitive Drucker–Prager model is employed to describe the equivalent plasticbehavior, where the plastic loading function and plastic potentialfunction are as follows:

Fðr,k,DÞ ¼ aI1þffiffiffiffiJ2

p�ð1�DÞk

Gðr,k,DÞ ¼ bI1þffiffiffiffiJ2

p�ð1�DÞk ð19Þ

where I1¼tr(r)¼rii denotes the first invariant of the ‘nominal’stress tensor r, J2¼1/2s:s¼1/2sij: sij, is the second invariant of thedeviatoric stress tensor, s¼(r� I1/3i), k is the hardening functionof rock cohesion. The two Drucker–Prager parameters a¼2 sin j/[ffiffiffi3p

(3�sin j)] and b¼2 sin c/[ffiffiffi3p

(3�sin c)] are a measure ofinternal rock friction and dilation, and j, c are internal frictionangle and dilation angle, respectively.

The hardening function k of rock cohesion can be expressed asthe following form:

k¼ kðr,kcÞ ¼ s0þkc ð20Þ

where s0¼6c cos j/[ffiffiffi3p

(3�sin j)], kc is the compressive isotro-pic hardening scalar.

Since the rock behavior in compression is more of a ductilebehavior, the evolution of the compressive hardening scalar kc isdefined by the Voyiadjis’ form [13,14]:

kc ¼Q 1�expð�beeqÞ� �

deeq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3deq : deq

rð21Þ

where Q and b are rock constants characterizing the saturatedstress and the rate of saturation, respectively, which are obtainedin the effective configuration of the compressive uniaxial stress–strain diagram.

To determine the direction of plastic strain-rate, the form ofthe flow rule is expressed as follows:

eq:

¼ g: @G

@r¼ g: biþ

s

2ffiffiffiffiJ2

p !

ð22Þ

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–10296

where g is the plastic multiplier, and the plasticity consistencycondition is applied by taking the time derivative of the plasticityfailure function, F : ¼ 0, such that the Kuhn–Tucker plasticityconsistency conditions are satisfied [35]:

Fðr,k,DÞr0,g: Z0,Fðr,k,DÞg: ¼ 0 ð23Þ

2.5. Integration algorithm of elastoplastic damage model

Numerical implementation of the model requires integratingthe rate form of the constitutive relations in the finite time stepDtnþ1

¼tnþ1�tn. Given the rock response at time tn and a finite

strain increment Denþ1, the objective is to determine theunknown external and internal state variables rnþ1, enþ1, ep

nþ1

and Dnþ1 at time tnþ1. The implicit Euler backward integrationmethod is utilized where the stress tensor at time tnþ1 is updated.The return mapping algorithm [36,37], characterized by elasticprediction, plastic correction and damage correction, is employedin this paper. The above formulations can be summarized in thefollowing steps:

A. In order to solve the local problem, an initial approximationfor the unknowns is also needed. The standard choice is theelastic trial state, which is

rtrnþ1 ¼ rnþðc

edÞn : Denþ1 ð24Þ

where ced¼l(D)i�iþ2m(D)I is elastic damage tensor, l(D), m(D) are

damage Lame constants, which can be described by Lame constantl0, m0 as the form l(D)¼(1�D)l0, m(D)¼(1�D)m0, hence, thedamage elastic tensor can be expressed as ce

d¼(1�D)ce0.

In the elastic prediction stage, the damage and plastic vari-ables are assumed to be unchanged, namely:

gtrnþ1 ¼ gn

ktrnþ1 ¼ kn

ðe Vj jÞtrnþ1 ¼ ðe Vj jÞn

ltrnþ1 ¼ ln

Dtrnþ1 ¼Dn ð25Þ

B. Substituting Eqs. (24) and (25) into F(rtrnþ1,ktr

nþ1,Dtrnþ1), if the

stress state at the trial stress state are inside or on the damagesurface, i.e., Fr0, then the step is indeed purely elastic and the

trial state is the final solution, rnþ1 ¼rtrnþ1, return to Step A;

otherwise, F40, go into Step C, solving the macroscopic stress

rnþ1, plastic strain epnþ1and damage scalar Dnþ1 at time tnþ1.

C. In the plastic correction process, the damage scalars remainunchanged. We adopt the following standard return mappingalgorithm in stress space:

rnþ1 ¼ rtrnþ1�Dgðc

edÞn :

@G

@r

� �nþ1

ð26Þ

We also assume an evolution equation for the internal plasticvariables of the form:

gnþ1 ¼ gnþDg

knþ1 ¼ s0þQ 1�exp �beeqnþ1

�h iDeeq

nþ1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23Deq

nþ1 : Deqnþ1

qð27Þ

Substituting Eqs. (26) and (27) into the failure function yields:

Fðrnþ1,knþ1,DnÞ ¼ 0 ð28Þ

In a strain-driven problem we are generally given a fixed set ofpredictor values rtr

nþ1 and the goal is to find the correspondingvalues of rnþ1, knþ1 and Dg satisfying Eqs. (26), (27) and (28).We can readily solve the non-linear problem using a standardNewton–Raphson iteration, defining the vector of unknowns x

and the construct residual vectors r as

x¼ r k Dgn oT

8�1ð29Þ

rðxÞ ¼ðceÞ

�1 : ðr�rtrnþ1ÞþDg@G=@r

k�k eeqnþ1

�Fðr,k,DÞ

8<:

9=;

8�8

ð30Þ

Solution to the local system of generally nonlinear equations isachieved when r(x)¼0 and the rate of convergence is intimatelydependent on the consistent local tangent (Jacobian) such that

r0ðxÞ ¼ðceÞ

�1þDgG,rr

�DgG,rk G,r

0 1 �k,Dg

F ,rT F ,k

T 0

24

35

8�8

ð31Þ

The smaller system can now be solved using followingNewton–Raphson iteration:

xk0 þ1

nþ1 ¼ xk0

nþ1� r0ðxÞð Þk0nþ1

h i�1rðxÞk

0

nþ1 ð32Þ

where k0 is the number of iteration, the solving will stop andstress rnþ1 together with plastic variable Dgnþ1 and knþ1 iscalculated if relative errors meet the requirements within k0 steps.

D. Compute the effective stress on isolated crack and updatethe damage scalars. The effective stress field rc

nþ1 acting onisolated cracks can be derived from macroscopic stress rnþ1 asfollows:

rcnþ1 ¼ Bc : rnþ1 ð33Þ

Then the trial stress intensity factor at nþ1 step:

KIð Þtrnþ1 ¼

�FtrI

p ltrnþ 1þ lnð Þ½ �

3=2 þ2p ðs

c11Þ

trnþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffipltr

nþ1

qKIIð Þ

trnþ1 ¼

�FtrII

pðltrnþ 1þ lnÞ½ �

3=2

ð34Þ

where FtrI ¼ pa2ttr

ef f cosy1, FtrII ¼ pa2ttr

ef f siny1, and ttref f ¼ tef f ðr

cnþ1,y1Þ

is the trail effective stress on the crack surface, and we assume thetrial wing crack length ltr

nþ1¼ ln at beginning of nþ1 step.Then the trial strain energy density factor at nþ1 step as

follows:

ðSÞtrnþ1 ¼ a11 KIð Þ

trnþ1

� �2þ2a12ðKIÞ

trnþ1ðKIIÞ

trnþ1þa22 ðKIIÞ

trnþ1

� �2ð35Þ

If ðSÞtrnþ1rSC, then the trial wing crack length ltr

nþ1¼0; and if

Sð Þtrnþ14SC , increase the trial wing crack length ltr

nþ1until

ðSÞtrnþ1¼SC using Newton iteration.

The wing crack length lnþ1 at nþ1 step can be updated to

lnþ1 ¼max ltrnþ1,0, max

tA 0,n½ �lt

� ð36Þ

The trial absolute volumetric strain ðeV Þtrnþ1:

ðeV Þtrnþ1 ¼ ðe

eV Þnþ1

�� ��þ ðepV Þnþ1

�� �� ð37Þ

Then the absolute volumetric strain (e9V9)nþ1 at nþ1 step canbe updated to

ðe Vj jÞnþ1 ¼max ðe Vj jÞtrnþ1,0, max

tA 0,n½ �ðe Vj jÞt

� ð38Þ

From Eqs. (36) and (38), the damage scalar at nþ1 step can beupdated to as follows:

Znþ1 ¼ kðe Vj jÞmnþ1

Vnþ1 ¼43pl3nþ1

Dnþ1 ¼Znþ1Vnþ1

1þZnþ1Vnþ1ð39Þ

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102 97

E. The corresponding stress tensor the stress tensor at nþ1step is updated to

rnþ1 ¼ ðcedÞnþ1 : e

enþ1 ¼ ðc

edÞnþ1 : ðe

enþDenþ1�Dep

nþ1Þ

¼1�Dnþ1

1�Dnðrtr

nþ1�ðcedÞn : Dep

nþ1Þ ¼1�Dnþ1

1�Dnrnþ1 ð40Þ

The total stress rnþ1 is therefore calculated by correcting thedamaged trial stress for the plastic deformation increment andbecause of the progressive stress degradation from rnþ1 to rnþ1,the trial stress reduces by a factor of (1�Dnþ1)/(1�Dn).

Fig. 2. Plots illustrates the stress–strain curves (a), and wing crack evolutions (b),

for non-interaction and interaction cases at uniaxial compression (initial flaw size

a¼800, 1000 um).

3. Numerical results and discussion

For the purpose of illustration, the elastoplastic damage modelbased on micromechanics was implemented in finite elementanalysis code. The performance of the constitutive formulationmodel was evaluated from aspects of friction coefficient, confin-ing pressure, initial flaw size and the flaw density parameter k

(one of the two Weibull parameters).In the numerical simulations below the present work, the

model parameters besides flaw data are as follows: the unda-maged elastic state of the isotropic rock is specified by the elasticmodulus E¼2.0�104 MPa, Poisson’s ratio n¼0.2, rock bulk den-sity g¼25 kN/m3, cohesion value c¼3.0�104 kN/m2, the angle ofinternal friction j¼c¼401, as the associated flow law, plasticparameters b¼274, Q¼30 MPa.

The sliding crack model together with its plastic deformationwill be the elastoplastic damage model if the initial flaw size isshort enough and the applied loading together with the strainaccumulation increases to a certain amount, conversely, it will bethe elastic damage model without plastic deformation.

3.1. Basic performance of the model

Flaw parameters are as follows: flaws oriented at an angley1¼451, wing crack propagation angle y2¼451, friction coefficientm¼0, Weibull parameter k¼4.0�1023/m3, m¼5, rock fracturetoughness KIC¼0.5 MPam1/2.

The stress–strain curves and wing crack evolution of theelastic damage model (initial flaw size a¼800, 1000 um) aregenerated for comparison of non-interaction and interactionamong cracks cases, shown in Fig. 2.

Fig. 2(a) shows that decreasing the initial flaw size causes anincrease of the strength with a concurrent increase in the strain tofailure. Note that the compressive strength and accumulatedstrain for larger initial flaw (a¼1000 um) and interaction casesis relative smaller than shorter flaw (a¼800 um) and non-interaction cases, respectively. We also note that the damageaccumulation path O–A–B–C (for a¼1000 um, interaction case)accelerates at a fast rate with shorter accumulating strain com-pared to path O–A–B–D (for a¼1000 um, non-interaction case),indicating that wing crack for interaction case has a longer lengthinduced by the larger effective stress.

As shown in Fig. 2(b), the wing crack length (for a¼1000 um)with damage evolution cased by crack propagation initiates frompoint A0, however, the macroscopic damage is noted to occurbeyond point A, indicating that crack growth reflected fromdamage accumulation is not obvious at the beginning.

In Fig. 2(b), path O0–A0–B0–C0 shows the wing crack growthconsidering the interaction among cracks, and path O0–A0–B0–D0

represents non-interaction (a¼1000 um). Note that prior to ultimatefailure strength, the growth rate of wing crack (stage B0–C0) accel-erates for the interaction case, while the wing crack evolution(stage B0–D0) for the non-interaction case becomes flat when exceedspoint B0. The deviation of crack evolution reveals that, based on Eqs.

(12) and (13), the stress response considering interaction will show agreater effective stress around cracks for the reason of damageaccumulation (initiated from point A), and hence have a larger wingcrack size (deviated from point B0) which leads to the greater damagewith the lower compressive strength, shown in Fig. 2(a).

Fig. 3 shows that the elastoplastic damage stress–strain curvesand wing crack evolution for the non-interaction and interactionamong short microcrack cases (initial flaw size a¼50, 100 um).With short flaw size, plastic strain will appear within the range ofaccumulating strain in the elastoplastic damage model which isdifferent from the elastic damage model.

Fig. 3(a) shows that the influence of considering interaction onstress responses is much less obvious than that of larger initialflaw case, shown in Fig. 2(a), revealing that interaction amongcracks for larger flaw has greater impact on effective stress field.In Fig. 3(b), compared with path F0–G0–H0–K0–L0 representing non-interaction among the cracks (a¼50 um), path F0–G0–H0–I0–J0

shows the wing crack evolution in which interaction amongcracks is taken into account. Note that the wing crack accumula-tion of non-interaction and interaction cases almost coincide witheach other prior to point H0. We also note that the wing crack size(stage H0 � I0 � J0) is slightly larger than that (stage H0 �K0 �L0) ofnon-interaction case after the point H0, indicating that surround-ings of microcracks have larger effective stress for interaction

Fig. 3. Plots illustrates the stress–strain curves (a), and wing crack evolutions (b),

for non-interaction/interaction cases at uniaxial compression (initial flaw size

a¼50, 100 um).

Fig. 4. Compressive strength as function of initial flaw size a for non-interaction/

interaction cases at uniaxial compression (Weibull function parameter

k¼4.0�1023/m3).

Fig. 5. Comparison of stress–strain curves for mode I and mixed-mode fracture

cases with accounting of crack-interactions among the cracks (Weibull parameter

k¼4.0�1023/m3, initial flaw size a¼50 um).

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–10298

case. According to the Eqs. (12) and (13), the effective stressaround microcracks for interaction case is greater than that ofnon-interaction case, consequently, it has a larger wing crack sizewhen considering the non-interaction among the cracks. For thereason of updating the wing crack size calculated from Eq. (13)and (36), and decreasing of effective stress, the wing crack sizesremain unchanged beyond point K0 and point I0.

The compressive strength as function of initial flaw size fornon-interaction and interaction cases at uniaxial compression isshown in Fig. 4. Sni and Si are non-interaction compressivestrength and interaction compressive strength, respectively, and(Sni–Si)/Si is differential strength rate for non-interaction andinteraction cases. The result suggests that the strength decreasesin a non-linear mode with the increase of initial flaw size, and thecompressive strength decreases rapidly when flaw varies from 50to 200 um. We also note that increasing the initial flaw results inan increase of the differential strength rate.

Ashby et al. [17,18] considered that the wing crack growth ismainly induced by the extension of pre-existing flaws, and thewing crack length is calculated using maximum tensile stress(MTS) criterion. Furthermore, the MTS criterion is employed tocount dynamic wing crack length for micro-mechanism model ofsolids [21,22,26,27].

Here we discuss the influence of mixed-mode fracture on thestress–strain relationship with accounting of crack-interactionsamong the cracks. Fig. 5 is comparison of stress–strain curvesbetween mode I fracture and mixed-mode fracture, it shows thatcompressive strength for mode I fracture case is obviously greaterthan that of mixed-mode fracture, indicating that the shear factorof cracks has great effect on the stress response.

3.2. Effect of the friction coefficient m

Flaw parameters are as follows: flaw orientation y1¼451 andwing crack propagation angle y2¼451, half-flaw size a¼50 um,Weibull parameter k¼4.0�1023/m3, m¼5, rock fracture tough-ness KIC¼0.5 MPam1/2.

During the deformation process, crack surfaces slide against eachother causing the stress concentration at their tips. Friction existingon these surfaces resists the sliding motion and prevents unlimitedcrack growth. Figs. 6 and 7 illustrate the influence of frictioncoefficient on the stress responses and induced wing crack evolutionwith the friction coefficient m is varied between 0.0 and 0.8.

Fig. 6(a) shows that compressive strength increases with thefriction coefficient m varies from 0.0 to 0.6. As shown in Fig. 6(a),

Fig. 6. Plots illustrates the stress–strain curves (a), and wing crack evolutions (b),

considering interaction effect for different friction coefficient cases at uniaxial

compression (initial flaw size a¼800 um).

Fig. 7. Plots illustrates the stress–strain curves (a), and wing crack evolutions (b),

considering interaction effect for different friction coefficient cases at uniaxial

compression (initial flaw size a¼50 um).

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102 99

also according to Eq. (18), damage parameter D is zero beforewing crack propagates and it starts to increase initially with aslower rate, and then rapidly as the wing crack propagates fasterbeyond a critical amount. Note that stage I–J is placed in elasticstage, while stage J–K is in a process of elastic damage state forcoefficient m¼0.0. A corresponding state for wing crack evolution,in Fig. 6(b), is stage R–S–T representing wing crack growth causedby elastic stress, and the increase in the friction coefficientrequires higher accumulating strain (and hence higher stress)for the initiation of the wing cracks from the initial flaws. Alsonote that the growth rate of wing crack decreases as the increas-ing of friction coefficient, and critical damage level for obtainingthe compressive pressures is very small (�0.10).

Fig. 7(a) shows that compressive strength increases with theincrease in friction coefficients as m varies from 0.0 to 0.8, and thedash dot lines show the stress responses for the non-interaction case.Note that stage L–M is in elastic state, stage M–N is in a process ofelastoplastic state and stage N–O–P is the elastoplastic damageprocess. It is observed that the stress response of non-interactioncase is relatively close to that of interaction case for different frictioncoefficients if the initial flaw is short (e.g. a¼50 um).

In Fig. 7(b), wing crack growth rate decreases with theincreasing of friction coefficient, and the greater friction coeffi-cient requires higher accumulating strain for the initiation of the

wing cracks from the initial flaws. We also note that the wingcrack initially propagates with a faster rate, and then slowlyevolves to be constant value (the greater friction coefficientcorresponds to shorter final wing crack length), as the uniaxialstrain beyond a critical amount. Note that increase in the frictioncoefficient requires greater strain for the initiation of the wingcracks from the flaws. Therefore, the load bearing capacity of therock increases with the increase in frictional resistance as thedamage tends to accumulate over a longer duration.

The compressive strength as function of friction coefficient munder uniaxial compression is shown in Fig. 8. The fracturestresses for non-interaction/interaction cases increase with theincreasing friction coefficient, this phenomenon is also shown inFig. 7(a). However, comparing with the influence of initial flawsize on compressive strength, revealed in Fig. 4, the differentialstrength rate (Sni�Si)/Si for the non-interaction and interactioncases evolves irregularly as friction coefficient m increases.

3.3. Effect of Weibull parameter k (spatial flaw density parameter)

The flaw parameters are as follows: flaws oriented at an angley1¼451, wing crack propagation angle y2¼451, half-flaw sizea¼50 um, friction coefficient m¼0, Weibull parameter m¼5, rockfracture toughness KIC¼0.5 MPam1/2.

Fig. 9. Plots illustrates the stress–strain curves (a), differential stress ratio (b), and

wing crack evolutions (c), for different Weibull parameter k cases at uniaxial

compression (initial flaw size a¼50 um).

Fig. 8. Compressive strength as function of friction coefficient m for non-interac-

tion/interaction cases at uniaxial compression (initial flaw size a¼50 um).

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102100

Fig. 9 denotes the stress responses, damage accumulation andwing crack evolution with the different Weibull parameter k

(1.0�1020 and 1.0�1022/m3). Fig. 9(a) shows that increasing theparameter k results in the decrease of compressive strength. Notethat with the damage accumulation (increase of flaw density), aspath E–F–G–H shown in Fig. 9(a), the differential stress ratio(rni�ri)/ri for the non-interaction/interaction cases increases, thisphenomenon can be clearly revealed by the rate in Fig. 9(b). Theinitiation of macroscopic damage for lower parameter k requireshigher strain, but the initiation of the wing cracks from initial flawsneeds the same accumulating strain for different parameter k,shown in Fig. 9(c). Fig. 9(c) also shows that the wing crack remainsunchanged before a critical strain initiates, and then it starts toincrease rapidly initially and then slowly as the uniaxial strainbeyond a critical amount (point O0). Note that the wing crack lengthsfor different Weibull parameter k cases nearly coincide with eachother before point O0. We also note that the wing crack accumula-tion (stage O0–R0–S0) for Weibull parameter k¼1022/m3 is slightlysmaller than that parameter k¼1020/m3 after the point O0. Thisphysical phenomenon can be explained as follows. According to thehypothesis discussed in Section 2.1, material is considered to be freefrom damage before the propagation of these cracks. From theEqs. (12) and (13), the number of flaws has no influence on theinitial growth of wing crack, consequently, wing crack initiatesunder the same strain amount (point N0, in Fig. 9(c)) for differentWeibull parameters. Subsequently, the wing crack evolution almostcoincides with each other beyond a critical damage (point O0).

3.4. Effect of confining pressures

In this section, the effects of confining pressure on the stress–strain relationship and damage accumulation are illustrated. Flawparameters are as follows: flaws oriented at an angle y1¼451, wingcrack propagation angle y2¼451, half-flaw size a¼50 um, frictioncoefficient m¼0, m¼5, rock fracture toughness KIC¼0.5 MPam1/2.

The stress–strain curves for non-interaction/interaction casesinfluenced by confining pressure are shown in Fig. 10(a), and(b) illustrates the influence of confining pressure on the wingcrack evolution.

From Fig. 10(a), the stress responses increase with the increas-ing of confining pressure. Note that the differential stresses forthe non-interaction and interaction cases increase with thedamage accumulates under no or low confining pressure (e.g.1st marker in Fig. 10(a)). However, the non-interaction andinteraction stress responses almost overlap with each other when

applied a higher confining pressure (2nd marker in Fig. 10(a)),indicating that higher pressure causes a shorter crack growth, andthen leads to reduce damage evolution and diminishes the stressdifference between non-interaction and interaction case.

Fig. 10(b) shows that higher confining pressure results in decreas-ing rate of wing crack growth beyond point V0, and wing crack endsup with a shorter crack size with the greater accumulating strain.

Fig. 10. Plots illustrates the stress–strain curves (a), and wing crack evolutions (b),

for non-interaction and interaction cases at confining pressures (initial flaw size

a¼50 um, k¼1.0�1018/m3).

Fig. 11. The stress–strain curves for different Weibull parameter k cases at

confining pressures (initial flaw size a¼50 um).

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102 101

The wing cracks tend to grow faster and the damage accumulatesrapidly in a short duration leading to complete fragmentation ifconfining pressure is low. While the modest or larger confiningpressures limit the growth of individual cracks and the damage tendsto accumulate over a longer duration leading to increased failurestress (as point Y0 and point W0 shown in Fig. 10(b)).

Fig. 11 illustrates the stress responses and induced damageevolution for different Weibull parameter k cases under confiningpressures from 0 to 20 MPa. The damage, shown in Fig. 11,accumulates in a short duration under low confining pressuresbut takes significantly longer when the pressure is increased,implying that when the confining pressure increases from a lowto a higher value, the mode of failure is transformed from brittlefracture to a pseudo-ductile fracture as has been experimentallyobserved [17]. We also note that when Weibull parameter k

decreases from a high to a low amount, the rock deforms from thebrittle fractures to the pseudo-ductile fracture.

4. Conclusions

A micro-mechanical elastoplastic damage model for rockunder compressive loading is developed based on frictionalsliding of pre-existing flaws in this work. These pre-existing flaws

are assumed to be randomly distributed in space but withselected orientations and certain size. Crack-interaction effectsare introduced by means of a self-consistent approach in whicheach crack experiences a stress field different from that acting onisolated cracks. The growth of damage is quantified by spatialflaw density and quasi-static spherical region calculation, whichis derived by the wing crack length. This plastic yield criterion isused simultaneously with the micro-mechanics damage model tosimulate the inelastic deformation of rocks. It embodies Drucker–Prager yield criterion and Voyiadjis’ strain hardening functionunder compression to define plastic behaviors of such materials.The numerical algorithm is proposed and the code of model isimplemented by using the return mapping method. The influ-ences of friction coefficient, confining pressure and the crackdensity parameters k on stress responses in the present model areanalyzed. The results can be summarized as follows:

(1)

For the larger initial flaw length, the growth rate of wing crackis still increasing for the case of interaction, while the wingcrack evolution for non-interaction case becomes flat beforeattaining the strength.

(2)

It shows that compressive strength for mode I fracture case isobviously greater than that of mixed-mode fracture, indicat-ing that the shear factor of cracks has great impact on thestress response.

(3)

The initiation of macroscopic damage for lower crack densityparameter k requires higher strain, but the initiation of thewing cracks from initial flaws needs the same accumulatingstrain for different parameter k.

(4)

The differential stress rate of the non-interaction and inter-action cases increases with the damage accumulates underthe low confining pressure. However, the stress responses fornon-interaction and interaction cases almost overlap witheach other if applied a higher confining pressure. When theconfining pressure increases to a higher value, the mode offailure is transformed to the pseudo-ductile fracture.

Acknowledgment

The project under which this paper was prepared was supportedby the Natural Science Foundations of China (No. 41002113; No.41162009; No. 10902111), the China Scholarship Council (No.201206400002) and the Fundamental Research Funds for the Central

X.P. Yuan et al. / International Journal of Rock Mechanics & Mining Sciences 58 (2013) 92–102102

Universities (No. 2011PY0190; No. 2010ZY45; No. 2010ZY33; No.2012069) which are gratefully acknowledged. The first author isgrateful for the technical discussions with Z.Q. Wang, and Yves M.Leroy (Ecole Normale Superieure, Paris) for useful comments. Theauthors also express their gratitude to the IJRMMS anonymousreviewers for the valuable comments.

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