A two-scale model for subcritical damage propagation

Cristian Dascalu *, Bertrand François, Oumar KeitaLaboratoire Sols Solides Structures – Risques, UJF, INPG, CNRS UMR 5521, Domaine Universitaire, B.P. 53, 38041 Grenoble Cedex 9, France

a r t i c l e i n f o

Article history:Received 23 July 2009Received in revised form 14 October 2009Available online 23 October 2009

Keywords:Micro-cracksSubcritical propagationHomogenizationTime-dependent damageRelaxationCreepSize effect

a b s t r a c t

The failure behaviour of quasi-brittle materials is often time-dependent. This dependence is due tophysical processes taking place at the level of the micro-structure. For a rigorous modeling of thetime-dependent behaviour of that kind of solids, a two-scale approach is well suited. This paper investi-gates time-dependent damage which microscopic origin is the subcritical micro-crack growth. Wepresent a two-scale time-dependent damage model completely deduced from small-scale descriptionsof subcritical micro-crack propagation, without any macroscopic assumptions. The passage from themicro-scale to the macro-scale is done through an asymptotic homogenization approach. At themicro-scale, the tensile failure due to the subcritical propagation of cracks is the dominant mechanismof creep observed at the macro-scale. We consider microstructures with cracks evolving in differentsubcritical regimes. We assume a complex propagation law that considers three characteristic regimesof subcritical crack growth, corresponding to different physical processes at the crack tip level. Numericalsimulations of constant strain rate, relaxation and creep tests illustrate the ability of the developed modelto reproduce different regimes of time-dependent damage response.

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1. Introduction

The problem of distributed subcritical failure must be accu-rately treated for many applications dealing with the mechanicsof quasi-brittle materials such as glasses, rocks, ceramics or cera-mic composites. A subcritical criterion of crack propagation consid-ers that crack may grow for energy lower than the critical limit offracture. The rate of crack propagation is usually related to thestress intensity factor at the crack tips, under mode I (tensilemode). At the micro-scale, the tensile failure due to the subcriticalpropagation of cracks may represent the main micro-mechanismof creep observed at the macro-scale. This effect can be observedin many different materials such as rocks, ceramics or glasses(Miura et al., 2003; Main, 2000; Nara and Kaneko, 2006; Muntand Fett, 2001).

The constitutive modeling of the time effect on the mechanicaldamage behaviour of materials (Betten, 2002; Cristescu and Hun-sche, 1998; Lemaitre and Desmorat, 2005) can be addressedaccording to two distinct approaches: (i) the phenomenologicalmodels are based on internal variables, having empirical charactersfor most of them, that are calibrated to fit with experimentalobservations (Challamel et al., 2005; Pietruszczak et al., 2004),(ii) the micromechanics-based approaches are able to capture thephysical micro-mechanisms that produce damage and irreversibledeformations which are partially or totally time-dependent. From

such micro-scale considerations, Abou-Chakra Guery et al. (2009)developed a model for argillite assuming it as a three-phase com-posite subject to viscous and damage effects. Also Nadot et al.(2006) (see also Dartois et al. (2009)) developed a viscoelastic dam-age model via a scale transition approach in which damage occursby grain/matrix debonding.

Experimental observations of fracture propagation (e.g. Ander-son and Grew, 1977; Meredith and Atkinson, 1985) indicate thatthe dependence of fracture propagation rate on the stress intensityfactor in mode I can generally be approximated by a trimodalbehaviour (Fig. 1). In region I, the rate of stress corrosion reactioncontrol the velocity of the crack growth. The plateau of region II ismainly determined by the rate of transport of reactive species tocrack tips. In region III, the drastic increase of the velocity of crackgrowth up to failure is relatively independent of the chemical envi-ronment and is controlled by mechanical rupture (Freiman, 1984;Atkinson and Meredith, 1987). This typical three-mode regime ischaracteristics of glasses, ceramics or rocks but is less marked inmetals. In most of the approaches studying the subcritical propaga-tion of cracks, this complex trimodal behaviour is often approxi-mated by a single mode. This approximation may be done if thebehaviour of the material is studied in a restricted range of stressintensity factor, limiting the study in one of the three regions. Onthe contrary, for a complete modeling of the problem, from crackinitiation to coalescence of cracks upon failure, the three regionsmust be considered.

Previous attempts to make a link between subcritical propaga-tion of micro-cracks and the macroscopic damage behaviour are

0020-7683/$ - see front matter ! 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijsolstr.2009.10.018

* Corresponding author.E-mail address: Cristian.Dascalu@hmg.inpg.fr (C. Dascalu).

International Journal of Solids and Structures 47 (2010) 493–502

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very few, e.g. the micromechanical analysis in Okui and Horii(1997) and Miura et al. (2003). The novelty of the present approachis many-fold: (i) the new subcritical damage model accounts,through a compact form of the damage law, for the three regimesof failure described previously; (ii) it appears to be the firstsubcritical model of damage deduced through the mathematicalhomogenization method based on asymptotic developments; and(iii) the subcritical damage laws allow for predictions of sizeeffects.

Experimental evidences on the trimodal character of the sub-critical crack propagation are, most of the time, found at the labo-ratory scale (through Double Torsion Test, for instance). Forapplications to geomechanics and geophysics, this scale may beconsidered as the microscopic scale. As observed by Scholz(1972), the macroscopic rock creep is related to the growth ofmicrofissures by stress corrosion cracking that may create delayedtriggering of earthquakes. In aluminium alloys, Ruckert et al.(2006) have shown that, within a limited range of stress intensityfactor, very good similarity has been obtained between micro- andmacro-scopic rates of crack growth. One may expect that subcriti-cal growth take place for micro-cracks at smaller scales. Theseobservations motivate the consideration of the subcritical growthof microscopic cracks as one of the origins of the time-dependentdamage behaviour.

A two-scale approach for damage was deduced in Dascalu et al.(2008) and Dascalu and Bilbie (2007) for brittle damage. A moregeneral formulation of this model, including non-brittle behav-iours, was recently given in Dascalu (2009) and a model accountingfor mixed-mode fracture in compression was developed in Franç-ois and Dascalu (submitted for publication). The main objectiveof the present contribution is to deduce a damage model from amicromechanical description of subcritical crack propagation intension and that accounts for the three regimes of failure.

We assume that micro-crack propagation follows an exponen-tial-type multi-regime subcritical law and we use homogenizationto obtain a macroscopic time-dependent damage model. The pas-sage from the micro-scale to the macro-scale is done through theasymptotic homogenization approach (e.g. Benssousan et al.,1978; Sanchez-Palencia, 1980).

The paper is organized as follows. First, the three-regimesubcritical propagation law is discussed. Then, the mathematicalformulation of the two-scale problem is presented and themacroscopic damage equations are deduced through the asymp-totic homogenization procedure. Finally, the ability of the modelto reproduce known time-dependent behaviours for constantstrain rate, relaxation and creep tests is demonstratedby means of numerical simulations of the local homogenizedresponse.

2. Subcritical crack growth

Classical fracture mechanics postulates that in a linear elasticbody an isolated crack under tensile loading will propagate inthe medium once a critical mode I stress intensity factor, K IC, hasbeen reached or exceeded, while for lower values of K I the propa-gation is not possible. The subcritical criterion considers that crackpropagation is time-dependent and may occur for stress intensityfactor lower than the critical limit of fracture (Atkinson and Mere-dith, 1987).

The most commonly used equations to describe the relation be-tween subcritical crack growth and stress intensity factor in modeI, K I , are the power law (Charles, 1958)

v ! V1"K I#n "1#

and the exponential law (Wiederhorn and Bolz, 1970)

v ! V2 exp"$c % bK I# "2#

where V1, V2, n, b and c are positive constants. The distinction be-tween the performance of the predictions of Eqs. (1) and (2) to de-scribe subcritical crack growth is often impossible. The range ofvariation of K I in the experimental data are generally limited whichmakes that both power and exponential laws fit with observations.However, outside of the range of experimental observations thepredictions of these two constitutive equations may diverge signif-icantly. Moreover, the previous equations are only valid in a givenregion of the trimodal response of the material (Fig. 1), generallyin region I, which makes impossible the simulation of advancedstages of material rupture (characteristic of region III).

From consideration of atomic theories and reaction rate theo-ries of crack propagation (including stress corrosion, dissolution,diffusion, ion-exchange and microplasticity), it is possible to relatethe empirical coefficients of Eqs. (1) and (2) with physical charac-teristics of the microstructure. Considering crack growth as a suc-cession of water vapor enhanced rupture of small material elementimmediately adjacent to the crack tips, Salganik et al. (1997)showed that the three regimes of crack propagation can be pre-dicted, respectively, by the three following equations:

v "1# !v0/

n0

nexp $

U001 $X0

1K I

!!!!!!!!!!!!2pdm

p.

kT

0

@

1

A "3#

v "2# !b0p

NAn!!!!!!!!!!!!!!!!2pmkT

p "4#

v "3# !dm

s0exp $

U003 $X0

3K I

!!!!!!!!!!!!2pdm

p.

kT

0

@

1

A "5#

where T is the absolute temperature, k is Boltzmann’s constant, X01

is a stress sensitivity factor, s0 & 10$12—10$13 s is a typical period ofatomic fluctuations, /0 is the relative concentration of water in thegas next to the crack tips, n is the number of molecules of water re-quired for the water-assisted rupture of a single bridging bond, dm isthe length of the material structure, b0 is the bridging bond length,NA is the Avogadro number, v0 is the mean velocity of diffusion,m isthe molecular mass of water, p is the partial pressure of watervapor, U0

01 and U003 are the zero stress activation energy in regions

I and III, respectively, and X01 and X0

3 are stress sensitivity factorsin regions I and III. Numerical values of those parameters are re-ported in Table 1.

Grouping the material and environmental constants by posing:

a! v0/n0

n exp $U001kT

" #; b! b0p

NAn!!!!!!!!!!!2pmkT

p ; c! dms0exp $U0

03kT

" #; S1 !

X01!!!!!!!!

2pdmp

kTand

S3 !X03!!!!!!!!

2pdmp

kT, we obtain the following expressions:

K !

Log v

! !! !!!

KT K m

v(2)

Fig. 1. Velocity of the crack propagation versus the stress intensity factor. The threeregions are separated by the two characteristic stress intensity factors: KT and Km .The bold line corresponds the composite law (Eq. (11)).

494 C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502

v "1# ! a exp"S1K I# "6#v "2# ! b "7#v "3# ! c exp"S3K I# "8#

The three regions are delimited by two characteristic stress inten-sity factors KT and Km that can be analytically determined accord-ing to the K I—v equations (6)–(8) of each region:

KT !1S1

lnba

$ %"9#

Km ! 1S3

lnbc

$ %"10#

The three distinct velocities (Eqs. (6)–(8)) corresponding to thethree regimes of crack propagation can be composed in a uniquerate of propagation through the following approximate relationship(Salganik et al., 1997):

v !v "1#v "2#

v "1# % v "2#% v "3# "11#

According to the value of K I with respect to KT and Km, v takes thevalue v "1# when K I < KT < Km (i.e. v "3# ' v "1# ' v "2#), the value v "2#

when KT < K I < Km (i.e. v "3# ' v "2# ' v "1#) or the value v "3# whenKT < Km < K I (i.e. v "2# ' v "1# ' v "3#). A generic form of the compos-ite velocity given by the relation (11) is represented in Fig. 1.

This trimodal behaviour of the micro-crack propagation hasseveral consequences in term of the predicted macroscopic time-dependent response of materials. (i) In region I, the relation be-tween velocity of crack propagation (v) and the stress intensity fac-tor "K I# is logarithmic. It means that v can never be equal to zero,even for low value of K I. So, in theory, for periods of time tendingto infinity, the material damage may increase even at very lowstress (or strain) level. (ii) The plateau of region II induces a ductilebehaviour of material. In that region, the stress intensity factor (re-lated to the loading level) may increase without producing anacceleration of crack propagation. (iii) On the contrary, region III,attainable for high loading level, is characterized by a brittle rup-ture of materials.

3. Two-scale problem

In this paper, we consider the quasi-static elasto-damage evolu-tion of the material body. At a given instance of time, the problemis that of an elastic body containing a large number of micro-cracks. We suppose that the micro-crack distribution is locallyperiodic. Each crack is assumed to be horizontal (parallel to thex1-axis), straight and of length l. The damage variable d is definedas the ratio d ! l

e between the crack length l and the distance be-tween the centers of neighbor micro-cracks e or, equivalently,the size of a periodicity cell (see Fig. 2a).

3.1. Equilibrium problem

We consider the instantaneous equilibrium of the initial heter-ogeneous medium, that we assume to be a two-dimensional iso-tropic elastic medium containing a locally periodic array ofmicro-cracks. In the solid part Bs ! B n C, where B is the wholebody and C the union of all micro-cracks inside B, the momentumequilibrium is

oreij

oxj! 0; in Bs "12#

and the linear elasticity constitutive relation is

reij ! aijklexkl"ue# "13#

where aijkl is the elasticity tensor. reij is the stress field and u! the

displacement field from which the strain tensor is deduced in thesmall deformation hypothesis

exij"ue# ! 12

oueioxj

%ouejoxi

$ %"14#

On the crack faces, traction free conditions are assumed:

reN ! 0 "15#

where N is a unit normal vector on the crack faces.

d

a b

y1

y2

"

"l

1

1x1

x2

l

Fig. 2. (a) Micro-fissured mediumwith locally periodic microstructure, e is the size of a period and l is the local micro-crack length. (b) Unit cell with rescaled crack of length d.

Table 1Material parameters used in the simulations.

E (Pa) m e (m) T (K) n /0 U001 "J#

2( 109 0.3 10$2 325 1 6:32( 10$10 2( 10$19

dm (m) s0 (s) U003 (J) v "2# (m/s) v0 (m/s) X0

1 (J) X03 (J)

25( 10$4 10$12 2( 10$18 10$8 9( 104 1:27( 10$23 8:28( 10$24

C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502 495

3.2. Asymptotic homogenization

The locally periodic microstructure is constructed from a refer-ence unit cell Y (Fig. 2b) referred to microscopic coordinates"y1; y2#. Rescaled with the small parameter e, the unit cell becomesthe physical period of the material eY . We assume that the micro-structural length e is small enough with respect to the characteris-tic dimension of the whole body, so that to distinguish betweenmicroscopic and macroscopic variations. These variations ofmechanical fields at different scales are represented by distinctvariables: x the macroscopic variable and y ! x=e the microscopicvariable.

The unit cell Y contains the rescaled crack CY and Ys ! Y n CY itssolid part. Following the method of asymptotic homogenization(e.g. Benssousan et al., 1978; Sanchez-Palencia, 1980), we lookfor expansions of ue and re in the form

ue"x; t# ! u"0#"x; y; t# % eu"1#"x; y; t# % e2u"2#"x; y; t# % ) ) ) "16#

re"x; t# ! 1e r

"$1#"x; y; t# % r"0#"x; y; t# % er"1#"x; y; t# % ) ) ) "17#

where u"i#"x; y; t#; r"i#"x; y; t#; x 2 Bs; y 2 Y are smooth functionsand Y-periodic in y.

Based on previous works (Leguillon and Sanchez-Palencia,1982; Sanchez-Palencia, 1980), it can be shown that the substitu-tion of Eqs. (16) and (17) in the set of expressions (12)–(15), leadsto boundary value problems for the different orders of e, formu-lated on the unit cell Y. It can be proved that the functionu"0# ! u"0#"x; t# is independent of y variable, representing the mac-roscopic displacement field.

For given u"0#"x; t#, for traction-free cracks, we deduce (Leguillonand Sanchez-Palencia, 1982; Sanchez-Palencia, 1980) the followingboundary-value problem for the function u"1#:

ooyj

aijkleykl u"1#& '& '! 0; in Ys "18#

aijkleykl u"1#& 'Nj ! $aijklexkl u"0#& '

Nj; on CY "19#

and with periodicity boundary conditions on the external boundaryof the cell.

The microscopic correction u"1# has a linear dependence of themacroscopic deformations: u"1#"x; y; t# ! npq"y#expq u"0#

& '"x; t#. Here,

the characteristic functions npq"y# are elementary solutions of (18)and (19), for the particular macroscopic deformationsexpq u"0#

& '! dpq.

By introducing the mean value operator h)i ! 1jYj

RYs)dy, where

jYj is the area of Y, we can prove (Leguillon and Sanchez-Palencia,1982; Dascalu et al., 2008) that the macroscopic stress is

R"0#ij * r"0#

ij

D E! Cijkl"d#exkl u"0#& '

"20#

as a function of the macroscopic strain ex u"0#& '

wherer"0#

ij ! aijkl exkl u"0#& '

% eykl u"1#& '& '

and

Cijkl"d# !1jY j

Z

Ys

aijkl % aijmneymn nkl" #" #

dy "21#

are the homogenized coefficients. The effective constitutive relation(20) should be used in the macroscopic equilibrium equation whichcan be deduced in the form (e.g. Sanchez-Palencia, 1980):

ooxj

Cijkl"d#exkl u"0#& '& '! 0 "22#

Although it is not our intention to consider the influence of thecrack orientations but to focus on the determination of a complexdamage law for a particular crack direction, we note that the pre-vious formulae are valid for every crack orientation in the period-icity cell. If we denote by h the angle made by the crack line with

the horizontal direction, then the effective coefficients are func-tions of d and h : Cijkl ! Cijkl"d; h#. The couple "d; h# completely char-acterizes the state of damage at a given macroscopic point.

These coefficients Cijkl can be computed by solving the unit cellproblems (18) and (19) for every d, the crack line being assumedhorizontal. In what follows, we consider that the elastic solid ma-trix is isotropic, of elastic constants E and m. For given couple "E; m#,the coefficients can be computed for a large number of d 2 +0;1,and, by interpolation, we can have polynomial expressions ofCijkl"d#. For this, polynomials of degree 5 have been used.

The FEAP finite element code (Taylor, 2008) has been used forthe computation of these homogenized coefficients. They are rep-resented in Fig. 3 as a function of the damage variable d. The pres-ence of the micro-cracks leads to induced anisotropy, the resultingeffective elastic response being orthotropic. We also note the non-linear dependence of the homogenized coefficients on the damagevariable d. With an horizontal crack line (direction 11), the loss ofrigidity with the damage variable increase is maximum when theunit cell is loaded in the vertical direction (22), i.e. perpendicularto the crack (coefficients C2222 and C1122) while the rigidity is muchless affected when loaded in the horizontal direction (11), i.e. par-allel to the crack (coefficient C1111). This is characteristic of thedamage-induced anisotropy observed at the macro-scale.

For d ! 1, the residual value of C2222 and C1122 is not zero be-cause the micro-crack tips are assumed to remain in contact, evenfor fully damaged state. It produces a residual rigidity of the unitcell. It explains that, in the numerical simulations (Section 5), thestrain-controlled tests end up with a residual stress value.

4. Damage equations

The effective constitutive law presented in the previous sectionenables us to compute the stress–strain behaviour of the materialat a given state of non-evolving damage. In this section, we con-sider the evolution of micro-cracks and we deduce the correspond-ing macroscopic evolution of damage.

Our objective is to deduce the equations characterizing the elas-to-damage evolution around a macroscopic point. Under a givenloading of the macroscopic structure, the resulting local state ofstress leads to the activation of particular families of micro-cracks.In what follows, we place ourselves in such a macroscopic pointand we assume that in a small vicinity a family of straight micro-cracks is activated and they are propagating in mode I, symmetri-cally with respect to their middle-point. Their evolution will be de-scribed by the composed subcritical law of propagation (11).

To deduce the macroscopic damage equations we will followthe method developed in Dascalu et al. (2008) and Dascalu(2009). Let us denote the microscopic energy-release rate Ge,which depends on the crack length l. The micro-crack propagationis described by the following relation:

_l ! !v"Ge# "23#

In our particular case, !v is given by (11), when expressed in terms ofthe energy release rate. This relation should be completed with thereduced dissipation inequality:

Df * Ge_l P 0 "24#

Eqs. (23) and (24) present the general criterion of time-dependentcrack propagation at the micro-scale. For brittle fracture, the Grif-fith criterion can be considered: _l ! 0 when Ge < Gf and _l P 0 whenGe ! Gf , where Gf is the critical fracture energy of the material. Thiscase was considered in Dascalu et al. (2008) where a brittle damagemodel has been deduced. For more complex fracture evolutions, Gf

may be a function of the crack length l and the crack speed _l (e.g.Freund, 1998). Such general fracture laws have been considered in

496 C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502

Dascalu (2009), where a general damage model has been obtainedby homogenization. In our particular case, the relation (23) corre-sponds to a critical fracture energy Gf that depends on the crackvelocity _l.

Our subcritical propagation law _l ! v"K I#, where the functionv"K I# is given by (11), can be written in terms of fracture energyby using the relation between the stress intensity factor and theenergy release rate, for the mode I propagation:

Ge !1$ m2

E"K I#2 "25#

It was proved in Dascalu et al. (2008) that, for evolving micro-cracks, we have

Ge

e ! Yd * $12dCijkl"d#

ddexkl u"0#& '

exij u"0#& '"26#

where Yd is the (macroscopic) damage energy release rate. Thisrelation is entirely deduced from microstructural assumptions,without any assumptions on the scaling of energy. This scaling withe is naturally appearing in the derivation of the damage equation(26). For evolving damage, the previous relation shows that themicrostructural length emakes the link between the surface energydissipated during micro-crack propagation and damage energy dis-sipated per unit volume. This energy scaling property will assurethe presence of the internal length e in the damage laws.

Using (26) from the micro-crack evolution laws (23) and (24),we deduce the damage law

_d ! 1e!v"eYd# "27#

Dd * Yd_d P 0 "28#

We remark the presence of the material length in the damage mod-el given by the relations (27) and (28).

To complete the proof of the thermodynamic compatibility forthe damage model obtained by homogenization, we note thatone can introduce the macroscopic free energy function

W"exij; d# *12Cijkl"d#exkl u"0#& '

exij u"0#& '"29#

which becomes a potential for the associated thermodynamical‘‘forces” (Lemaitre and Desmorat, 2005):

Rij !oWoexij

; Yd ! $oWod

"30#

Combination of (25) and (26) yields to the expression of thestress intensity factor

K I ! $ eE2"1$ m2#

oCijkl"d#od

exkl u"0#& 'exij u"0#& '$ %1

2

"31#

Substituting K I of Eq. (31) in the composite subcritical law (11) andconsidering the definition of the damage variable d ! l

e gives us thefollowing macroscopic damage law in the form of (27):

_d ! 1e

ab exp"S1K I#a exp"S1K I# % b

% c exp"S3K I#$ %

"32#

where K I is expressed according to macroscopic variables throughEq. (31). In this way, Eq. (32) becomes a macroscopic damage equa-tion which is coupled with the equilibrium equation (22).

This damage equation accounts not only for the rate effect butalso for the size effect. Indeed, according to Eq. (31), the stressintensity factor included in the damage equation (32) depends onthe size of the periodic structure, e, and on the damage variable d(being the ratio between the crack length l and e). So, the devel-oped model reflects the nonlinear, time- and size-dependent effecton the damage behaviour of material, as commonly observed inexperiment.

5. Numerical simulations

The time-dependent behaviour of materials can be underlinedby means of various laboratory tests. In particular, quasi-staticcompression tests, creep tests or relaxation tests may point outthe same time-dependent properties under different stress andstrain conditions. In this section, numerical results are presented

0 0.5 10

1

2

3 x 109

Damage variable (!)

C11

11 (P

a)

0 0.5 10

1

2

3 x 109

Damage variable (!)

C22

22 (P

a)

0 0.5 10

5

10

15 x 108

Damage variable (!)

C11

22 (P

a)

0 0.5 12

4

6

8 x 108

Damage variable (!)

C12

12 [P

a]

0 0.5 1!1

!0.5

0

0.5

1

Damage variable (!)C

1112

(Pa)

Fig. 3. Homogenized coefficients for horizontal crack orientation for elastic parameters E = 2 GPa and m = 0.3.

C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502 497

for these tests. We simulate the local macroscopic elastic damageresponse. The set of material parameters that have been used inthe simulations are reported in Table 1. The simulations have beenmade considering tension loading under strain- or stress-con-trolled conditions in the vertical direction while the horizontaldirection is free of stress. Plane-strain condition is considered inthe third direction. All the simulations start from an undamagedmaterial "d0 ! 1( 10$9 & 0#.

5.1. Numerical integration method

For the analysis of the homogenized response in a macroscopicpoint, the input of this system of Eqs. (20) and (32) is the macro-scopic stress R"0#

ij or strain exkl u"0#& ', depending on the physical

problem to be studied. Due to the dependency of the elastic mod-ulus Cijkl on the damage variable d and the form of the damage en-ergy release rate (26) used to calculate the stress intensity factors,the problem is highly non-linear. For each time step, the stress-controlled problem is solved by an iterative procedure as follows:

1. Initialization "n ! 1#; dt0 ! dt$1 (if t ! 0; d ! d0).

2. Cijkl ! Cijkl"dtn$1#.

3. Prediction of the macroscopic strain: exij;n$1 ! C$1ijkl dt

n$1

" #R"0#

ij .

4. Determination of the stress intensity factor: K I !K I Cijkl dt

n$1

" #" #.

5. Update of the damage: _d ! f exij;n$1;Cijkl dtn$1

" #" #;

dtn ! dt$1 % _dDt.

6. Update of the homogenized coefficient: Cijkl ! Cijkl dtn

" #.

7. Calculation of the updated strain: exij;n ! C$1ijkl dt

n

" #R"0#

ij .

8. The convergence of the solution is tested: CONV ! ken$en$1k2

kenk2.

(i) If CONV 6 Tol: Return to point 1 with a new time step"t ! t % 1#.

(ii) If CONV > Tol: Return to point 2 with n ! n% 1.

where n is the iteration step, t is the time step number, Dt is thesize of the time step, d0 is the initial damage variable and Tol isthe tolerance taken as 10$5.

We note that, because the damage energy release rate (26) iswritten in term of macroscopic strain, this procedure becomes triv-ial in the strain control case.

5.2. Loading at constant strain rate

Under tension tests at constant strain rate, the micro-crackingdepends not only on the strain level, that controls the stress inten-sity factor at the crack tips, but also on time. The response of themodel under a vertical tension rate of 10$8 s$1 is illustrated inFig. 4 in terms of stress, damage and stress intensity factor evolu-tions with time. Also plotted is the velocity of crack propagation re-lated to the stress intensity factor through the subcritical law. Aslong as the vertical strain increases, the stress intensity factor in-creases. So doing, the regime of crack propagation starts in regionI until K I reaches KT . At the beginning of this region, the rate ofcrack propagation is low, i.e. the damage increases slowly andthe loss of rigidity is relatively limited. So, at the beginning ofthe test, the behaviour appears quasi linear. Bold lines reproducethe response of the material for the material parameters reportedin Table 1. For such parameters, when the stress intensity factor

0 0.5 1 1.5 2 2.5x 106

0

0.2

0.4

0.6

0.8

1

Time [s]

Dam

age

varia

ble

[!]

0 0.5 1 1.5 2 2.5x 106

0

0.5

1

1.5

2

2.5

3 x 107

Time [s]

Verti

cal s

tress

[Pa]

0 0.5 1 1.5 2 2.5

x 106

0

1

2

3

4

5x 106

Time [s]

Stre

ss in

tens

ity fa

ctor

[Pa

m1/

2 ]

0 0.5 1 1.5 2 2.5 3 3.5 4

x 106

10!30

10!20

10!10

100

1010

Stress intensity factor [Pa m1/2]

Velo

city

of c

rack

pro

paga

tion

[m/s

]

B1

B2

C1C2A1=A2

B1

B2

C1C2A3

A3

B1

B2

C1C2A3

B1B2

C1C2A3

A1=A2

A1=A2

A1=A2

a b

c d

Fig. 4. Numerical results of a vertical tension test at constant strain rate. Evolution of (a) stress, (b) damage variable, (c) stress intensity factor with time and (d) velocity ofcrack propagation with respect to the stress intensity factor. The three damage regimes are separated by letters A, B and C. The different curves correspond to differentamplitudes of the plateau of region II.

498 C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502

overpasses KT (in point A1), the rate of crack propagation becomesindependent of the loading level, as seen in Fig. 4d between pointsA1 and B1. That regime is characterized by a constant rate of dam-age propagation inducing a progressive loss of material rigidity,that creates the peak and the post-peak softening behaviour ob-served in Fig. 4a. Finally, when K I reaches Km (at point B1), a suddenfailure appears up to coalescence of micro-cracks (d = 1, point C1).This numerical simulation clearly shows the trimodal behaviour ofthe crack growth rate with respect to the stress intensity factor atthe crack tips. So, the problem is treated completely from crack ini-tiation (usually in region I) toward constant rate of crack propaga-tion (region II) leading to failure (in region III).

The region II induces a ductile behaviour. Indeed, that regime ismainly controlled by the diffusion rate of the reactive species tocrack tips which cannot diffuse faster than a given value. It avoidsa dramatic increase of the crack velocity, producing a ductile mac-roscopic response of the material. On the contrary, if the diffusionrate of the reactive species is unlimited, v "2# tends to infinity andthe behaviour remains in region I until reaching a brittle failure(point A3 in Fig. 4). Also, the extension of the region II can be lim-ited by a lower critical value of the stress intensity factor whichcontrol the behaviour in region III. Decreasing the zero stress acti-vation energy in region III from 2( 10$18 to 1( 10$18, the region IIIappears (point B2 in Fig. 4) for lower value of the stress intensityfactor and the extension of the plateau of region II is lower. As aconsequence, the failure is also more brittle (point C2).

Because of the time-dependent behaviour of materials, thestrength is affected by the strain rate of loading _ex22. Usually, fasteris the loading and higher is the strength. When _ex22 is sufficientlylow, the subcritical micro-cracking in the material has enough timeto develop inducing a decrease of the material strength. On thecontrary, a fast loading avoid the development of delayed crackingwhich enhances the strength. The developed model reproducesthis effect as shown in Fig. 5. In this example, a loading 10 timesslower decreases the strength of about 15% (from 29 MPa to24.5 MPa).

The variation of the zero stress activation energy of region I, U001,

translates the curve logv "1#—K I& '

without changing its slope. Forincreasing value of U0

01, the velocity of crack propagation decreasesfor a same K I and the transition between regions I and II "KT# isshifted to the right. So, when U0

01 decreases, KT is tending to Km

and region II is progressively disappearing. The distance between

0 0.005 0.01 0.015 0.02 0.0250

0.5

1

1.5

2

2.5

3 x 107

Axial strain [!]

Axi

al s

tres

s [P

a]

.ex22 = 1 10-8

.ex22 = 5 10-9

.ex22 = 1 10-9

Fig. 5. Effect of the strain rate of loading on the behaviour of materials subject to avertical tension test at constant strain rate.

0 0.5 1 1.5 2 2.5 3 3.5

x 106

0

0.2

0.4

0.6

0.8

1

Time [s]

Dam

age

varia

ble

[!]

0 0.5 1 1.5 2 2.5 3 3.5

x 106

0

1

2

3

4

5

6

7 x 107

Time [s]

Verti

cal s

tress

[Pa]

U’01 = 2 10 !19

U’01 = 3 10 !19

U’01 = 4 10 !19

a b

A1 A2 A3

C3C2C1

B3

B2

B2

B1

A1

A2

A3

C3C2C1

B3

B1

(1)

(2)

(3)

Fig. 6. Effect of the zero stress activation energy of region I on numerical results of a vertical tension test at constant strain rate. Evolution of (a) stress, (b) damage variableand with time.

0 0.5 1 1.5 2 2.5x 106

0

0.5

1

1.5

2

2.5

3

3.5x 107

Time [s]

Verti

cal s

tress

[Pa]

0 0.5 1 1.5 2 2.5

x 106

0

0.2

0.4

0.6

0.8

1

Time [s]

Dam

age

varia

ble

[!]

a b" = 1 10!2

" = 2 10!2

" = 3 10!2

(1)

(3)(2)

A1

C1

B1

A2

A3

B2

B3

C2C3

A1

C1

B1

A2A3

B2

B3

C2C3

Fig. 7. Effect of the internal length on numerical results of a vertical tension test at constant strain rate. Evolution of (a) stress and (b) damage variable with time.

C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502 499

points A and B decreases which makes the behaviour more brittle(Fig. 6).

In Fig. 7, the role of internal length variation e on the responseof the tensile test at constant strain rate is shown. The effect of thisparameter is twofold. (i) Under a given strain field and a givendamage level, the stress intensity factor is proportional to e1=2(Eq. (31)). So, the smaller is the microstructure, the lower is thestress concentration at the micro-crack tips. In that sense, the de-crease of e postpones the crack propagation, by increasing theresistance to failure. Fig. 7b shows that the time (e.g. the verticalstrain) needed to reach region II (characterized by points A) isincreasing when the internal length decrease. (ii) However, thesame velocity of micro-crack propagation produces an higher dam-

age rate for smaller microstructure dddt !

ve

& '. So, the two effects of e

are opposite. The first one postpones the failure while the secondone accelerates it which makes that the effect of e on the peakstrength is a complex one.

5.3. Relaxation test

The relaxation test is obtained by keeping a constant strain le-vel. Even if the sollicitation is not evolving, the micro-crack maypropage under subcritical conditions until complete failure of thematerial. Under constant tension loading, the subcritical micro-crack growth produces a progressive decrease of the rigidity aslong as the damage state increases. As a consequence, the vertical

0 0.5 1 1.5 2 2.5

x 106

0

0.2

0.4

0.6

0.8

1

Time [s]

Dam

age

varia

ble

[!]

U’01= 2 10!19

U’01= 4 10!19

U’01= 4.1 10!19

0 0.5 1 1.5 2 2.5

x 106

0

1

2

3

4

5

6

7x 107

Time [s]

Verti

cal s

tress

[Pa]

A2 A3

C3C2C1

B3B2B1

A2

C3C2C1

B3B2

B1

A3

(1)

(2)

(3)

a b

Fig. 8. Effect of the zero stress activation energy of region I on numerical results of relaxation test. Evolution of (a) stress and (b) damage variable with time.

Dam

age

varia

ble

[!]

" = 0.7 10!2

" = 1 10!2

" = 2 10!2

Verti

cal s

tress

[Pa]

B2(1)(2)(3)

x 105

0

1

2

3

4

5

6

7x 107

Time [s]

C1

B1

B2

B3

C2C3

0 1 2 3 4 5 6 0 1 2 3 4 5 6x 105

0

0.2

0.4

0.6

0.8

1

Time [s]

C1

B1

B3

C2C3a b

Fig. 9. Effect of the internal length on numerical results of relaxation test. Evolution of (a) stress and (b) damage variable with time.

0 0.5 1 1.5 2 2.5 3x 106

0

0.2

0.4

0.6

0.8

1

Time [s]

Dam

age

varia

ble

[!]

0 0.5 1 1.5 2 2.5 3x 106

0

0.005

0.01

0.015

0.02

0.025

0.03

Time [s]

Verti

cal s

train

[!]

U’o3= 0.8 10!18

U’o3= 1 10!18

U’o3= 2 10!18

A1=A2=A3

C1

B1B2

B3

C2 C3

A1=A2=A3

C1

B1

B2

B3

C2 C3(1)

(2)

(3)

a b

Fig. 10. Effect of the zero stress activation energy of region III on numerical results of creep test. Evolution of (a) stress and (b) damage variable with time.

500 C. Dascalu et al. / International Journal of Solids and Structures 47 (2010) 493–502

stress is gradually relaxing with time, towards the ultimate state,when the micro-cracks coalesce and the rigidity tends to zero. Un-der a constant vertical strain ex22 u"0#

& '! 0:0273, Figs. 8 and 9

show, respectively, the effect of the variations U001 and e on the ver-

tical stress decrease in parallel with the damage increase. Instanta-neously, the applied vertical strain generates a vertical stress ofaround 60 MPa that is the stress produced by the applied strainfor the undamaged material. For U0

01 ! 2( 10$19, the zero stressactivation energy of region I is so low that the vertical strain ofex22 u"0#

& '! 0:0273 produces a stress intensity factor at the crack

tips, K I, higher than KT which makes that the crack propagationstarts in region II. However, for higher U0

01, KT is shifted towardhigher K I and the crack propagation starting in region I and thethree distinct regions are now visible with transition in points A,B and C (Fig. 8).

As explained in the previous section, the increase of e decreasesthe rate of damage propagation in region II (before points B inFig. 9b) and pushes forward the transition between region II andIII (points B).

5.4. Creep test

As for relaxation tests, under the condition of creep tests (i.e.keeping a constant stress level), the failure is not governed bythe maximal stress that the material may sustain but rather bythe time needed for the micro-cracks to propagate under subcriti-cal conditions. Figs. 10 and 11 depict the effect of the variations ofU0

03 and e on the vertical creep strain in parallel with the damageincrease under a constant tensile vertical stress R"0#

22 ! 15 MPa.After an instantaneous vertical strain corresponding to the short-term response of the material, the time effect makes damage var-iable increase, successively in regions I, II and III. The lower is U0

03and the faster is the transition between regions II and III (points Bin Fig. 10). The modification of U0

03 does not affect region I. The ef-fect of the variation of e (Fig. 11) is similar to the ones explained inthe two previous sections.

6. Conclusions

The subcritical growth of micro-cracks is responsible of thetime-dependent behaviour of many quasi-brittle materials. Thesubcritical propagation criterion has been applied for cracks atthe micro-scale and up-scaled by the asymptotic homogenizationprocedure. A time-dependent macroscopic damage model has beendeduced and the stress–strain response of materials, depending ontime, has been obtained in a macroscopic point. The damage lawcontains a microstructural length allowing for the prediction ofsize effects.

The subcritical law considers trimodal behaviour of crack prop-agation with time. The behaviour within those three regions havebeen interpreted in terms of physical processes at the crack tip le-vel, according to Salganik et al. (1997), through microstructuralmaterial parameters.

As long as the micro-cracks grow due to the combined effect oftime and high stresses at the crack tips, the damage increases, theglobal rigidity of the material decreases and the subcritical crackpropagation crosses the three regions of failure behaviour, fromcrack initiation to coalescence of cracks upon rupture. With theassumption of trimodal behaviour of crack propagation, there isno stress (or strain) threshold for the activation of the subcriticalcrack growth. As a consequence, in theory, the material creep dam-age may occur even at very low stress (or strain) level. However,for infinitely slow strain rate (for controlled strain rate test) or infi-nitely low constant stress or strain level (for creep or relaxationtests), the time of failure become infinite.

The evolution of rigidity with respect to damage has been quan-tified by the means of a series of finite element simulations on aunit cell with different crack lengths. Numerical simulations ofthe effective elasto-damage response have shown the ability ofthe model to reproduce the time-dependent behaviour of materi-als. In particular, tensile test at constant strain rate, creep andrelaxation tests have been simulated. A parametric studies of theeffect of the size of the micro-structure and the level of activationenergies has been carried out. They show the capacity of the modelto be adapted for a large range of different material behaviours. Foreach simulation, the trimodal behaviour has been emphasized.

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0 0.5 1 1.5 2 2.5 3x 106

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0.2

0.4

0.6

0.8

1

Time [s]

Dam

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varia

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