a plasticity and anisotropic damage model for

7/30/2019 A Plasticity and Anisotropic Damage Model For

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A plasticity and anisotropic damage model forplain concrete

Umit Cicekli, George Z. Voyiadjis *, Rashid K. Abu Al-Rub

Department of Civil and Environmental Engineering, Louisiana State University,

CEBA 3508-B, Baton Rouge, LA 70803, USA

Received 23 April 2006; received in final revised form 29 October 2006Available online 15 March 2007

Abstract

A plastic-damage constitutive model for plain concrete is developed in this work. Anisotropicdamage with a plasticity yield criterion and a damage criterion are introduced to be able to ade-

quately describe the plastic and damage behavior of concrete. Moreover, in order to account for dif-ferent effects under tensile and compressive loadings, two damage criteria are used: one forcompression and a second for tension such that the total stress is decomposed into tensile and com-pressive components. Stiffness recovery caused by crack opening/closing is also incorporated. Thestrain equivalence hypothesis is used in deriving the constitutive equations such that the strains inthe effective (undamaged) and damaged configurations are set equal. This leads to a decoupled algo-rithm for the effective stress computation and the damage evolution. It is also shown that the pro-posed constitutive relations comply with the laws of thermodynamics. A detailed numericalalgorithm is coded using the user subroutine UMAT and then implemented in the advanced finiteelement program ABAQUS. The numerical simulations are shown for uniaxial and biaxial tensionand compression. The results show very good correlation with the experimental data.

2007 Elsevier Ltd. All rights reserved.

Keywords: Damage mechanics; Isotropic hardening; Anisotropic damage

0749-6419/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2007.03.006

*

Corresponding author. Tel.: +1 225 578 8668; fax: +1 225 578 9176.E-mail addresses: [email protected] (G.Z. Voyiadjis), [email protected] (R.K. Abu Al-Rub).

International Journal of Plasticity 23 (2007) 18741900

www.elsevier.com/locate/ijplas
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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1. Introduction

Concrete is a widely used material in numerous civil engineering structures. Due to itsability to be cast on site it allows to be used in different shapes in structures: arc, ellipsoid,

etc. This increases the demand for use of concrete in structures. Therefore, it is crucial tounderstand the mechanical behavior of concrete under different loadings such as compres-sion and tension, for uniaxial, biaxial, and triaxial loadings. Moreover, challenges indesigning complex concrete structures have prompted the structural engineer to acquirea sound understanding of the mechanical behavior of concrete. One of the most importantcharacteristics of concrete is its low tensile strength, particularly at low-confining pres-sures, which results in tensile cracking at a very low stress compared with compressivestresses. The tensile cracking reduces the stiffness of concrete structural components.Therefore, the use of continuum damage mechanics is necessary to accurately model thedegradation in the mechanical properties of concrete. However, the concrete materialundergoes also some irreversible (plastic) deformations during unloading such that thecontinuum damage theories cannot be used alone, particularly at high-confining pressures.Therefore, the nonlinear material behavior of concrete can be attributed to two distinctmaterial mechanical processes: damage (micro-cracks, micro-cavities, nucleation and coa-lescence, decohesions, grain boundary cracks, and cleavage in regions of high stress con-centration) and plasticity, which its mechanism in concrete is not completely understoodup-to-date. These two degradation phenomena may be described best by theories of con-tinuum damage mechanics and plasticity. Therefore, a model that accounts for both plas-ticity and damage is necessary. In this work, a coupled plastic-damage model is thus

formulated.Plasticity theories have been used successfully in modeling the behavior of metals wherethe dominant mode of internal rearrangement is the slip process. Although the mathemat-ical theory of plasticity is thoroughly established, its potential usefulness for representing awide variety of material behavior has not been yet fully explored. There are many research-ers who have used plasticity alone to characterize the concrete behavior (e.g. Chen andChen, 1975; William and Warnke, 1975; Bazant, 1978; Dragon and Mroz, 1979; Schreyer,1983; Chen and Buyukozturk, 1985; Onate et al., 1988; Voyiadjis and Abu-Lebdeh, 1994;Karabinis and Kiousis, 1994; Este and Willam, 1994; Menetrey and Willam, 1995; Grasslet al., 2002). The main characteristic of these models is a plasticity yield surface that

includes pressure sensitivity, path sensitivity, non-associative flow rule, and work or strainhardening. However, these works failed to address the degradation of the material stiffnessdue to micro-cracking. On the other hand, others have used the continuum damage theoryalone to model the material nonlinear behavior such that the mechanical effect of the pro-gressive micro-cracking and strain softening are represented by a set of internal state vari-ables which act on the elastic behavior (i.e. decrease of the stiffness) at the macroscopic level(e.g. Loland, 1980; Ortiz and Popov, 1982; Krajcinovic, 1983, 1985; Resende and Martin,1984; Simo and Ju, 1987a,b; Mazars and Pijaudier-Cabot, 1989; Lubarda et al., 1994 ).However, there are several facets of concrete behavior (e.g. irreversible deformations,inelastic volumetric expansion in compression, and crack opening/closure effects) that can-

not be represented by this method, just as plasticity, by itself, is insufficient. Since bothmicro-cracking and irreversible deformations are contributing to the nonlinear responseof concrete, a constitutive model should address equally the two physically distinct modesof irreversible changes and should satisfy the basic postulates of thermodynamics.

U. Cicekli et al. / International Journal of Plasticity 23 (2007) 18741900 1875


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Combinations of plasticity and damage are usually based on isotropic hardening com-bined with either isotropic (scalar) or anisotropic (tensor) damage. Isotropic damage iswidely used due to its simplicity such that different types of combinations with plasticitymodels have been proposed in the literature. One type of combination relies on stress-based

plasticity formulated in the effective (undamaged) space (e.g. Yazdani and Schreyer, 1990;Lee and Fenves, 1998; Gatuingt and Pijaudier-Cabot, 2002; Jason et al., 2004; Wu et al.,2006), where the effective stress is defined as the average micro-scale stress acting on theundamaged material between micro-defects. Another type is based on stress-based plastic-ity in the nominal (damaged) stress space (e.g. Bazant and Kim, 1979; Ortiz, 1985; Lublineret al., 1989; Imran and Pantazopoulu, 2001; Ananiev and Ozbolt, 2004; Kratzig and Poll-ing, 2004; Menzel et al., 2005; Brunig and Ricci, 2005), where the nominal stress is definedas the macro-scale stress acting on both damaged and undamaged material. However, it isshown by Abu Al-Rub and Voyiadjis (2004) and Voyiadjis et al. (2003, 2004) that coupledplastic-damage models formulated in the effective space are numerically more stable andattractive. On the other hand, for better characterization of the concrete damage behavior,anisotropic damage effects, i.e. different micro-cracking in different directions, should becharacterized. However, anisotropic damage in concrete is complex and a combinationwith plasticity and the application to structural analysis is straightforward (e.g. Yazdaniand Schreyer, 1990; Abu-Lebdeh and Voyiadjis, 1993; Voyiadjis and Kattan, 1999; Carolet al., 2001; Hansen et al., 2001), and, therefore, it has been avoided by many authors.

Consequently, with inspiration from all the previous works, a coupled anisotropic dam-age and plasticity constitutive model that can be used to predict the concrete distinctbehavior in tension and compression is formulated here within the basic principles of ther-

modynamics. The proposed model includes important aspects of the concrete nonlinearbehavior. The model considers different responses of concrete under tension and compres-sion, the effect of stiffness degradation, and the stiffness recovery due to crack closure dur-ing cyclic loading. The yield criterion that has been proposed by Lubliner et al. (1989) andlater modified by Lee and Fenves (1998) is adopted. Pertinent computational aspects con-cerning the algorithmic aspects and numerical implementation of the proposed constitu-tive model in the well-known finite element code ABAQUS (2003) are presented. Somenumerical applications of the model to experimental tests of concrete specimens under dif-ferent uniaxial and biaxial tension and compression loadings are provided to validate anddemonstrate the capability of the proposed model.

2. Modeling anisotropic damage in concrete

In the current literature, damage in materials can be represented in many forms such asspecific void and crack surfaces, specific crack and void volumes, the spacing betweencracks or voids, scalar representation of damage, and general tensorial representation ofdamage. Generally, the physical interpretation of the damage variable is introduced asthe specific damaged surface area (Kachonov, 1958), where two cases are considered: iso-tropic (scalar) damage and anisotropic (tensor) damage density of micro-cracks andmicro-voids. However, for accurate interpretation of damage in concrete, one should con-

sider the anisotropic damage case. This is attributed to the evolution of micro-cracks inconcrete whereas damage in metals can be satisfactorily represented by a scalar damagevariable (isotropic damage) for evolution of voids. Therefore, for more reliable represen-tation of concrete damage anisotropic damage is considered in this study.

1876 U. Cicekli et al. / International Journal of Plasticity 23 (2007) 18741900


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The effective (undamaged) configuration is used in this study in formulating the damageconstitutive equations. That is, the damaged material is modeled using the constitutivelaws of the effective undamaged material in which the Cauchy stress tensor, rij, can bereplaced by the effective stress tensor, rij (Cordebois and Sidoroff, 1979; Murakami and

Ohno, 1981; Voyiadjis and Kattan, 1999):

rij Mijklrkl 1where Mijkl is the fourth-order damage effect tensor that is used to make the stress tensorsymmetrical. There are different definitions for the tensor Mijkl that could be used to sym-metrize rij (see Voyiadjis and Park, 1997; Voyiadjis and Kattan, 1999). In this work thedefinition that is presented by Abu Al-Rub and Voyiadjis (2003) is adopted:

Mijkl 2dij uijdkl dijdkl ukl1 2where dij is the Kronecker delta and uij is the second-order damage tensor whose evolutionwill be defined later and it takes into consideration different evolution of damage in differ-ent directions. In the subsequence of this paper, the superimposed dash designates a var-iable in the undamaged configuration.

The transformation from the effective (undamaged) configuration to the damaged onecan be done by utilizing either the strain equivalence or strain energy equivalence hypoth-eses (see Voyiadjis and Kattan, 1999). However, in this work the strain equivalencehypothesis is adopted for simplicity, which basically states that the strains in the damagedconfiguration and the strains in the undamaged (effective) configuration are equal. There-fore, the total strain tensor eij is set equal to the corresponding effective tensor eij (i.e.

eij

eij, which can be decomposed into an elastic strain ee

ij (=

e

e

ij and a plastic strainepij(=epij such that:eij eeij epij eeij epij eij 3

It is noteworthy that the physical nature of plastic (irreversible) deformations in con-crete is not well-founded until now. Whereas the physical nature of plastic strain in metalsis well-understood and can be attributed to the generation and motion of dislocationsalong slip planes. Therefore, in metals any additional permanent strains due to micro-cracking and void growth can be classified as a damage strain. These damage strainsare shown by Abu Al-Rub and Voyiadjis (2003) and Voyiadjis et al. (2003, 2004) to be

minimal in metals and can be simply neglected. Therefore, the plastic strain in Eq. (3)incorporates all types of irreversible deformations whether they are due to tensilemicro-cracking, breaking of internal bonds during shear loading, and/or compressive con-solidation during the collapse of the micro-porous structure of the cement matrix. In thecurrent work, it is assumed that plasticity is due to damage evolution such that damageoccurs before any plastic deformations. However, this assumption needs to be validatedby conducting microscopic experimental characterization of concrete damage.

Using the generalized Hooks law, the effective stress is given as follows:

rij Eijkleekl 4

where Eijkl is the fourth-order undamaged elastic stiffness tensor. For isotropic linear-elas-tic materials, Eijkl is given by

Eijkl 2GIdijkl KIijkl 5



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where Idijkl Iijkl 13 dijdkl is the deviatoric part of the fourth-order identity tensorIijkl 12 dikdjl dildjk, and G E=21 m and K E=31 2m are the effective shearand bulk moduli, respectively, with E being the Youngs modulus and m is the Poissonsratio which are obtained from the stressstrain diagram in the effective configuration.

Similarly, in the damaged configuration the stressstrain relationship in Eq. (4) can beexpressed by:

rij Eijkleekl 6such that one can express the elastic strain from Eqs. (4) and (5) by the following relation:

eeij E1ijklrkl E1ijklrkl 7where E1ijkl is the inverse (or compliance tensor) of the fourth-order damaged elastic tensorEijkl, which are a function of the damage variable uij.

By substituting Eq. (1) into Eq. (7), one can express the damaged elasticity tensor Eijklin terms of the corresponding undamaged elasticity tensor Eijkl by the following relation:

Eijkl M1ijmnEmnkl 8Moreover, combining Eqs. (3) and (7), the total strain eij can be written in the followingform:

eij E1ijklrkl epij E1ijklrkl epij 9By taking the time derivative of Eq. (3), the rate of the total strain, _eij, can be written as

_eij _eeij _epij 10where _eeij and _e

pij are the rate of the elastic and plastic strain tensors, respectively.

Analogous to Eq. (9), one can write the following relation in the effective configuration:

_eij E1ijkl _rkl _epij 11However, since Eijkl is a function of uij, a similar relation as Eq. (11) cannot be used.Therefore, by taking the time derivative of Eq. (9), one can write _eij in the damaged con-figuration as follows:

_eij E1ijkl _rkl _E1ijklrkl _epij 12

Concrete has distinct behavior in tension and compression. Therefore, in order to ade-quately characterize the damage in concrete due to tensile, compressive, and/or cyclicloadings the Cauchy stress tensor (nominal or effective) is decomposed into a positiveand negative parts using the spectral decomposition technique (e.g. Simo and Ju,1987a,b; Krajcinovic, 1996). Hereafter, the superscripts + and designate, respec-tively, tensile and compressive entities. Therefore, rij and rij can be decomposed asfollows:

rij rij rij ; rij rij rij 13where rij is the tension part and r

ij is the compression part of the stress state.

The stress tensors rij and rij can be related to rij byrkl Pklpqrpq 14rkl Iklpq Pijpqrpq Pklpqrpq 15



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such that Pijkl Pijkl Iijkl. The fourth-order projection tensors Pijkl and Pijkl are definedas follows:

Pijpq X

3

k1

H

rk

n

ki n

kj n

kp n

kq ; P

klpq

Iklpq

Pijpq

16

where Hrk denotes the Heaviside step function computed at kth principal stress rk ofrij and n

ki is the kth corresponding unit principal direction. In the subsequent develop-

ment, the superscript hat designates a principal value.Based on the decomposition in Eq. (13), one can assume that the expression in Eq. (1)

to be valid for both tension and compression, however, with decoupled damage evolutionin tension and compression such that:

rij Mijklrkl; rij Mijklrkl 17where Mijkl is the tensile damage effect tensor and M

ijkl is the corresponding compressive

damage effect tensor which can be expressed using Eq. (2) in a decoupled form as a func-tion of the tensile and compressive damage variables, uij and uij , respectively, as follows:

Mijkl 2dij uij dkl dijdkl ukl1; Mijkl 2dij uij dkl dijdkl ukl1

18Now, by substituting Eq. (17) into Eq. (13)2, one can express the effective stress tensor

as the decomposition of the fourth-order damage effect tensor for tension and compressionsuch that:

rij Mijklrkl Mijklrkl 19

By substituting Eqs. (14) and (15) into Eq. (19) and comparing the result with Eq. (1),one can obtain the following relation for the damage effect tensor such that:

Mijpq MijklPklpq MijklPklpq 20Using Eq. (16)2, the above equation can be rewritten as follows:

Mijpq Mijkl Mijkl

Pklpq Mijpq 21One should notice the following:

Mijkl 6 Mijkl Mijkl 22or

uij 6 uij uij 23It is also noteworthy that the relation in Eq. (21) enhances a coupling between tensile

and compressive damage through the fourth-order projection tensor Pijkl. Moreover, forisotropic damage, Eq. (20) can be written as follows:

Mijkl Pijkl

1 u Pijkl

1 u 24

It can be concluded from the above expression that by adopting the decomposition of

the scalar damage variable u into a positive u+ part and a negative u

part still enhances adamage anisotropy through the spectral decomposition tensors Pijkl and P

ijkl. However,

this anisotropy is weak as compared to the anisotropic damage effect tensor presentedin Eq. (21).



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3. Elasto-plastic-damage model

In this section, the concrete plasticity yield criterion of Lubliner et al. (1989) which waslater modified by Lee and Fenves (1998) is adopted for both monotonic and cyclic load-

ings. The phenomenological concrete model ofLubliner et al. (1989) and Lee and Fenves(1998) is formulated based on isotropic (scalar) stiffness degradation. Moreover, thismodel adopts one loading surface that couples plasticity to isotropic damage throughthe effective plastic strain. However, in this work the model of Lee and Fenves (1998) isextended for anisotropic damage and by adopting three loading surfaces: one for plastic-ity, one for tensile damage, and one for compressive damage. The plasticity and the com-pressive damage loading surfaces are more dominate in case of shear loading andcompressive crushing (i.e. modes II and III cracking) whereas the tensile damage loadingsurface is dominant in case of mode I cracking.

The presentation in the following sections can be used for either isotropic or anisotropicdamage since the second-order damage tensor uij degenerates to the scalar damage vari-able in case of uniaxial loading.

3.1. Uniaxial loading

In the uniaxial loading, the elastic stiffness degradation variables are assumed asincreasing functions of the equivalent plastic strains eeq and e

eq with e

eq being the tensile

equivalent plastic strain and eeq being the compressive equivalent plastic strain. It shouldbe noted that the material behavior is controlled by both plasticity and damage so that,

one cannot be considered without the other (see Fig. 1).For uniaxial tensile and compressive loading, rij and rij are given as (Lee and Fenves,

1998)

r 1 uEee 1 uEe ep 25r 1 uEee 1 uEe ep 26

The rate of the equivalent (effective) plastic strains in compression and tension, eepandeep, are, respectively, given as follows in case of uniaxial loading:

E

(1 )E

uf

of

p

e

p

+ e

+

+

0 uf+ +=

+

E

(1 )E+

Fig. 1. Concrete behavior under uniaxial (a) tension and (b) compression.



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_eeq _ep11; _eeq _ep11 27such that

eeq Zt

0

_

eeq dt; eeq Zt

0

_

eeq dt 28Propagation of cracks under uniaxial loading is in the transverse direction to the stress

direction. Therefore, the nucleation and propagation of cracks cause a reduction of thecapacity of the load-carrying area, which causes an increase in the effective stress. Thishas little effect during compressive loading since cracks run parallel to the loading direc-tion. However, under a large compressive stress which causes crushing of the material, theeffective load-carrying area is also considerably reduced. This explains the distinct behav-ior of concrete in tension and compression as shown in Fig. 2.

It can be noted from Fig. 2 that during unloading from any point on the strain soften-

ing path (i.e. post peak behavior) of the stressstrain curve, the material response seems tobe weakened since the elastic stiffness of the material is degraded due to damage evolution.Furthermore, it can be noticed from Fig. 2a and b that the degradation of the elastic stiff-ness of the material is much different in tension than in compression, which is more obvi-ous as the plastic strain increases. Therefore, for uniaxial loading, the damage variable canbe presented by two independent damage variables u+ and u. Moreover, it can be notedthat for tensile loading, damage and plasticity are initiated when the equivalent appliedstress reaches the uniaxial tensile strength f0 as shown in Fig. 2a whereas under compres-sive loading, damage is initiated earlier than plasticity. Once the equivalent applied stressreaches f0 (i.e. when nonlinear behavior starts) damage is initiated, whereas plasticity

occurs once fu is reached. Therefore, generally f0 fu for tensile loading, but this isnot true for compressive loading (i.e. f0 6 fu . However, one may obtain f0 % fu in caseof ultra-high strength concrete.

3.2. Multiaxial loading

The evolution equations for the hardening variables are extended now to multiaxialloadings. The effective plastic strain for multiaxial loading is given as follows (Lublineret al., 1989; Lee and Fenves, 1998):

fG

+

p

0f

+

f

G

p

0f

Fig. 2. Schematic representation of the fracture energy under the uniaxial stressplastic strain diagrams for(a) tension and (b) compression.



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_eeq rrij _epmax 29_eeq 1 rrij _epmin 30

where _epmax and _epmin are the maximum and minimum principal values of the plastic strain

tensor _epij such that _ep1 > _ep2 > _ep3 where _epmax _ep1 and _epmin _ep3 .Eqs. (29) and (30) can be written in tensor format as follows:

_jpi Hij _epj 31

or equivalently

_eeq0

_eeq

8>:

9>=

>;

H 0 0

0 0 0

0 0 H

264

375

_ep1

_ep2

_ep3

8>:

9>=

>;32

where

H rrij 33H 1 rrij 34

The dimensionless parameter rrij is a weight factor depending on principal stresses and isdefined as follows (Lubliner et al., 1989):

rrij P3

k1hrki

P3k1jrkj

35

where h i is the Macauley bracket, and presented as hxi 12jxj x, k 1; 2; 3. Note that

rrij rrij. Moreover, depending on the value of rrij,

in case of uniaxial tension rkP 0 and rrij 1, in case of uniaxial compression rk6 0 and rrij 0

3.3. Cyclic loading

It is more difficult to address the concrete damage behavior under cyclic loading; i.e.transition from tension to compression or vise versa such that one would expect that undercyclic loading crack opening and closure may occur and, therefore, it is a challenging taskto address such situations especially for anisotropic damage evolution. Experimentally, itis shown that under cyclic loading the material goes through some recovery of the elasticstiffness as the load changes sign during the loading process. This effect becomes more sig-nificant particularly when the load changes sign during the transition from tension to com-pression such that some tensile cracks tend to close and as a result elastic stiffness recoveryoccurs during compressive loading. However, in case of transition from compression totension one may thus expect that smaller stiffness recovery or even no recovery at all

may occur. This could be attributed to the fast opening of the pre-existing cracks thathad formed during the previous tensile loading. These re-opened cracks along with thenew cracks formed during the compression will cause further reduction of the elasticstiffness that the body had during the first transition from tension to compression. The



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consideration of stiffness recovery effect due to crack opening/closing is therefore impor-tant in defining the concrete behavior under cyclic loading. Eq. (21) does not incorporatethe elastic stiffness recovery phenomenon as well as it does not incorporate any couplingbetween tensile damage and compressive damage and, therefore, the formulation of Lee

and Fenves (1998) for cyclic loading is extended here for the anisotropic damage case.Lee and Fenves (1998) defined the following isotropic damage relation that couples

both tension and compression effects as well as the elastic stiffness recovery during transi-tion from tension to compression loading such that:

u 1 1 su1 u 36where s0 6 s 6 1 is a function of stress state and is defined as follows:

srij s0 1 s0rrij 37where 0 6 s0 6 1 is a constant. Any value between zero and one results in partial recovery

of the elastic stiffness. Based on Eqs. (36) and (37):

(a) when all principal stresses are positive then r = 1 and s = 1 such that Eq. (36)becomes

u 1 1 u1 u 38which implies no stiffness recovery during the transition from compression to tensionsince s is absent.

(b) when all principal stresses are negative then r = 0 and s s0 such that Eq. (36)becomes

u 1 1 s0u1 u 39which implies full elastic stiffness recovery when s0 0 and no recovery when s0 1.

In the following two approaches are proposed for extending the Lee and Fenves (1998)model to the anisotropic damage case. The first approach is by multiplying uij in Eq. (18)1by the stiffness recovery factor s:

Mijkl 2dij suij dkl dijdkl sukl1 40such that the above expression replaces M

ijklin Eq. (21) to give the total damage effect

tensor.Another approach to enhance coupling between tensile damage and compressive dam-

age as well as in order to incorporate the elastic stiffness recovery during cyclic loading forthe anisotropic damage case is by rewriting Eq. (36) in a tensor format as follows:

uij dij dik suikdjk ujk 41which can be substituted back into Eq. (2) to get the final form of the damage effect tensor,which is shown next.

It is noteworthy that in case of full elastic stiffness recovery (i.e. s = 0), Eq. (41) reduces

to uij uij and in case of no stiffness recovery (i.e. s = 1), Eq. (41) takes the form ofuij uij uik uikujk such that both uij and uij are coupled. This means that duringthe transition from tension to compression some cracks are closed or partially closedwhich could result in partial recovery of the material stiffness (i.e. s > 0) in the absence



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of damage healing mechanisms. However, during transition from compression to tension,the existing and generated cracks during compressive loading could grow more whichcauses further stiffness degradation and such that no stiffness recovery is expected. Thisis why the parameter s only affects the tensile damage variable uij . However, one may

argue that during tensile loading a minimal stiffness recovery could occur due to geomet-rical constraints set up by the interaction between the cracks and the microstructure ofconcrete. This is what is suggested in the theory manuals of ABAQUS (2003) for theimplemented plastic-damage (isotropic) model of Lubliner et al. (1989). However, the for-mer approach is followed in this work where elastic stiffness recovery occurs only duringthe transition from tensile to compressive state of stress.

Now from Eq. (1), one can write the following:

rij M1ijklrkl 42where M1

ijkl

can be written from Eq. (2) as follows:

M1ijkl 1

2dij uijdkl dijdkl ukl 43

By substituting Eq. (41) into Eq. (43), one gets a coupled damage effect tensor in terms ofuij and u

ij as follows:

M1ijkl 1

2dim suimdjm ujmdkl dijdkm sukmdlm ulm 44

It can be noted that the above expression couples tensile damage and compressive damage.Moreover, it takes into account the stiffness recovery during the transition from tension to

compression. For full elastic stiffness recovery (s = 0), Eq. (44) yields Eq. (18)2 such thatMijkl Mijkl. However, for no elastic stiffness recovery (s = 1), Eq. (44) reduces to

M1ijkl 1

2dim uimdjm ujmdkl dijdkm ukmdlm ulm 45

which shows a coupling between tensile damage and compressive damage.

3.4. Plasticity yield surface

For the representation of concrete behavior under tensile and compressive loadings, ayield criterion is necessary. It is known that concrete behaves differently in tension andcompression, thus, the plasticity yield criterion cannot be assumed to be similar. Assumingthe same yield criterion for both tension and compression for concrete materials can leadto over/under estimation of plastic deformation (Lubliner et al., 1989). The yield criterionof Lubliner et al. (1989) that accounts for both tension and compression plasticity isadopted in this work. This criterion has been successful in simulating the concrete behav-ior under uniaxial, biaxial, multiaxial, and cyclic loadings (Lee and Fenves, 1998 and thereferences outlined there). This criterion is expressed here in the effective (undamaged)configuration and is given as follows:

f ffiffiffiffiffiffiffi

3J2p

aI1 bjpi Hrmaxrmax 1 a _ceeq 6 0 46where J2 sijsij=2 is the second-invariant of the effective deviatoric stresssij rij rkkdij=3, I1 rkk is the first-invariant of the effective stress rij, jpi

Rt0_j

pi dt is



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the equivalent plastic strain which is defined in Eq. (31), Hrmax is the Heaviside stepfunction (H= 1 for rmax > 0 and H=0 for rmax < 0, and rmax is the maximum principalstress.

The parameters a and b are dimensionless constants which are defined by Lubliner et al.

(1989) as follows:

a fb0=f

0 12fb0=f0 1

47

b 1 a ceeq

ceeq 1 a 48

where fb0 and f

0 are the initial equibiaxial and uniaxial compressive yield stresses, respec-tively. Experimental values for fb0=f

0 lie between 1.10 and 1.16; yielding a between 0.08

and 0.12. For more details about the derivation of both Eqs. (47) and (48), the reader is

referred to Lubliner et al. (1989).Since the concrete behavior in compression is more of a ductile behavior, the evolutionof the compressive isotropic hardening function _c is defined by the following exponentiallaw:

_c bQ c_eeq 49where Q and b are material constants characterizing the saturated stress and the rate ofsaturation, respectively, which are obtained in the effective configuration of the compres-sive uniaxial stressstrain diagram. However, a linear expression is assumed for the evo-lution of the tensile hardening function _c such that:

_c h_eeq 50where h is a material constant obtained in the effective configuration of the tensile uniaxialstressstrain diagram.

3.5. Non-associative plasticity flow rule

The shape of the concrete loading surface at any given point in a given loading stateshould be obtained by using a non-associative plasticity flow rule. This is important forrealistic modeling of the volumetric expansion under compression for frictional materials

such as concrete. Basically, flow rule connects the loading surface and the stressstrainrelation. When the current yield surface f is reached, the material is considered to be inplastic flow state upon increase of the loading. In the present model, the flow rule is givenas a function of the effective stress rij by

_epij _kp

oFp

orij51

where _kp is the plastic loading factor or known as the Lagrangian plasticity multiplier,which can be obtained using the plasticity consistency condition, _f 0, such that

f 6 0;_

k

pP

0;_

k

p

f 0;_

k

p _

f 0 52The plastic potential Fp is different than the yield function f (i.e. non-associated) and,therefore, the direction of the plastic flow oFp=orij is not normal to f. One can adoptthe DruckerPrager function for Fp such that:



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Fp ffiffiffiffiffiffiffi

3J2p

apI1 53where ap is the dilation constant. Then the plastic flow direction oFp=orij is given by

oFp

orij 3

2

sijffiffiffiffiffiffiffi3J2

p apdij 543.6. Tensile and compressive damage surfaces

The anisotropic damage growth function proposed by Chow and Wang (1987) isadopted in this study. However, this function is generalized here in order to incorporateboth tensile and compressive damage separately such that:

g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2YijL

ijklY

ijr K

ueq 6 0 55

where the superscript designates tension, +, or compression, , K is the tensile or com-pressive damage isotropic hardening function, K K0 when there is no damage, K0 isthe tensile or compressive initial damage parameter (i.e. damage threshold), Lijkl is afourth-order symmetric tensor, and Yij is the damage driving force that characterizes dam-age evolution and is interpreted here as the energy release rate (e.g. Voyiadjis and Kattan,1992a,b, 1999; Abu Al-Rub and Voyiadjis, 2003; Voyiadjis et al., 2003, 2004). In order tosimplify the anisotropic damage formulation, Lijkl is taken in this work as the fourth-orderidentity tensor Iijkl.

The rate of the equivalent damage _ueq is defined as follows (Voyiadjis and Kattan,

1992a,b, 1999; Abu Al-Rub and Voyiadjis, 2003; Voyiadjis et al., 2003, 2004 ):

_ueq ffiffiffiffiffiffiffiffiffiffiffiffi_uij _u

ij

qwith ueq

Zt0

_ueqdt 56

The evolution of the tensile damage isotropic hardening function K+ is assumed to havethe following expression (Mazars and Pijaudier-Cabot, 1989):

_K K

B K0

K

expB1 K

K0 _ueq 57

whereas the evolution of the compressive damage isotropic hardening function K is as-

sumed to have a slightly different form (Mazars and Pijaudier-Cabot, 1989):

_K K0

Bexp B 1 K

K0

_ueq 58

where K0 is the damage threshold which is interpreted as the area under the linear portionof the stressstrain diagram such that:

K0 f202E

59

The material constant B is related to the fracture energy Gf , which is shown in Fig. 2

for both tension and compression, as follows (Onate et al., 1988):

B Gf E

f20 1

2

" #1P 0 60



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where is a characteristic length scale parameter that usually has a value close to the sizeof the smallest element in a finite element mesh. Of course, a local constitutive model withstrain softening (localization) cannot provide an objective description of localized damageand needs to be adjusted or regularized (see Voyiadjis et al., 2004). As a remedy, one may

use a simple approach with adjustment of the damage law (which controls softening)according to the size of the finite element, , in the spirit of the traditional crack-bandtheory (Bazant and Kim, 1979). Therefore, the parameters B are used to obtainmesh-independent results. A more sophisticated remedy is provided by a localization lim-iter based on weighted spatial averaging of the damage variable, which will be presented ina forthcoming paper by the authors.

The model response in the damage domain is characterized by the KuhnTucker com-plementary conditions as follows:

g 6 0; _kdg 0; and _g< 0 ) _kd 0 0 ) _kd 0 0 ) _kd > 0

8>: 9>=>; ()effective

undamaged state

damage initiationdamagegrowth

8>:61

With the consideration of the above equation, one writes specific conditions for tensileand compressive stages: when g < 0 there is no tensile damage and ifg > 0 there is ten-sile damage in the material; and when g < 0 there is no compressive damage in the mate-rial and ifg > 0, it means there is compressive damage.

4. Consistent thermodynamic formulation

In this section, the thermodynamic admissibility of the proposed elasto-plastic-damagemodel is checked following the internal variable procedure ofColeman and Gurtin (1967).The constitutive equations listed in the previous section are derived from the second law ofthermodynamics, the expression of Helmholtz free energy, the additive decomposition ofthe total strain rate in to elastic and plastic components, the ClausiusDuhem inequality,and the maximum dissipation principle.

4.1. Thermodynamic laws

The Helmholtz free energy can be expressed in terms of a suitable set of internal statevariables that characterize the elastic, plastic, and damage behavior of concrete. In thiswork the following internal variables are assumed to satisfactory characterize the concretebehavior both in tension and compression such that:

w weeij;uij ;uij ;ueq;ueq; eeq; eeq 62where ueq and u

eq are the equivalent (accumulated) damage variables for tension and com-

pression, respectively, which are defined as ueq R

t

0_ueq dt. Similarly, e

eq and e

eq are the

equivalent tensile and compressive plastic strains that are used here to characterize the

plasticity isotropic hardening, eeq Rt0 _eeqdt.One may argue that it is enough to incorporate the damage tensor uij instead of incor-

porating both uij and ueq. However, the second-order tensor u

ij introduces anisotropy

while the scalar variable ueq introduces isotropy characterized by additional hardening.



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The Helmholtz free energy is given as a decomposition of elastic, we, plastic, wp, anddamage, wd, parts such that:

w weeeij;uij ;uij wpeeq; eeq wdueq;ueq 63It can be noted from the above decomposition that damage affects only the elastic prop-

erties and not the plastic ones. However, for more realistic description, one should intro-duce the damage variables in the plastic part of the Helmholtz free energy (see AbuAl-Rub and Voyiadjis, 2003; Voyiadjis et al., 2003, 2004). However, these effects arenot significant for brittle materials and can, therefore, be neglected.

The elastic free energy we is given in term of the second-order damage tensors uij asfollows:

we 12eeijEijkluij ;uij eekl

1

2rije

eij

1

2rij rij eeij 64

Substituting Eq. (17) along with Eqs. (1) and (20), considering rij Pijklrkl andrij Pijklrkl into Eq. (64) and making some algebraic simplifications, one obtains the fol-lowing relation:

we 12M1ijpqrpqe

eij

1

2rije

eij 65

such that one can relate the nominal stress to the effective stress through Eq. (42) with M1ijklgiven by

M1ijpq M1

ijkl Pklpq M

1ijkl P

klpq 66

This suggests that the inverse of Eq. (66) yields Eq. (20) if the projection tensors Pklpq andPklpq from the effective stress rij coincide with that from the nominal stress rij, which is notnecessarily true for anisotropic damage. Similarly, for isotropic damage, one can write Eq.(66) as follows:

M1ijkl 1 uPijkl 1 uPijkl 67The ClausiusDuhem inequality for isothermal case is given as follows:

rij _eij q _wP 0 68

where q is the material density. Taking the time derivative of Eq. (63), the followingexpression can be written as:

_w owe

oeeij_eeij

owe

ouij_uij

owe

ouij_uij

owp

oeeq_eeq

owp

oeeq_eeq

owd

oueq_ueq

owd

oueq_ueq 69

By plugging the above equation into the ClausiusDuhem inequality, Eq. (68), and mak-ing some simplifications, one can obtain the following relations for any admissible statessuch that:

rij

qowe

oe

e

ij 70

and

rij _epij Yij _uij Yij _uij c _eeq c _eeq K _ueq K _ueq P 0 71



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where the damage and plasticity conjugate forces that appear in the above expression aredefined as follows:

Yij

qowe

ouij

72

Yij q

owe

ouij73

K q owd

oueq74

K q owd

oueq75

c

qow

p

oeeq 76

c q ow

p

oeeq77

Therefore, one can rewrite the ClausiusDuhem inequality to yield the dissipation energy,P, due to plasticity, Pp, and damage, Pd, such that

P Pd Pp P 0 78with

P

p

rij_

e

p

ij c_

eeq c_

eeqP

0 79Pd Yij _uij Yij _uij K _ueq K _ueq P 0 80

The rate of the internal variables associated with plastic and damage deformations areobtained by utilizing the calculus of functions of several variables with the plasticity anddamage Lagrangian multipliers, _kp and _kd , such that the following objective function canbe defined:

X P _kpFp _kdg _kdg P 0 81Using the well-known maximum dissipation principle (Simo and Honein, 1990; Simo

and Hughes, 1998), which states that the actual state of the thermodynamic forces (rij,Yij , c, K) is that which maximizes the dissipation function over all other possible admis-

sible states, hence, one can maximize the objective function X by using the necessary con-ditions as follows:

oX

orij 0; oX

oYij 0; oX

oc 0; oX

oK 0 82

Substituting Eq. (81) along with Eqs. (79) and (80) into Eq. (82) yield the followingthermodynamic laws:

_e

pij

_

kp oF

p

orij 83

_uij _kdog

oYij84



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_uij _kdog

oYij85

_eeq

_kpoFp

oc

86

_eeq _kp

oFp

oc87

_ueq _kdog

oK88

_ueq _kdog

oK89

4.2. The Helmholtz free energy function

The elastic part of the Helmholtz free energy function, we, as presented in Eq. (64) canbe substituted into Eq. (70) to yield the following stressstrain relation:

rij Eijkleekl Eijklekl epkl 90Now, one can obtain expressions for the damage driving forces Yij from Eqs. (64), (72),

and (73) as follows:

Yrs 1

2eeij

oEijkl

ourseekl 91

By taking the derivative of Eq. (8) with respect to the damage parameter uij one obtainsoEijkl

ours oM

1ijmn

oursEmnkl 92

Now, by substituting Eq. (92) into Eq. (91), one obtains the following expression for Yij :

Yrs 1

2eeij

oM1ijmnours

Emnkleekl 93

where from Eq. (66), one can write the following expression:

oM1ijmnours

oM1ijpq

oursPpqmn 94

One can also rewrite Eq. (93) in terms of the effective stress tensor by replacing eekl from Eq.(4) as follows:

Yrs 1

2E1ijabrab

oM1ijpqours

rpq 95

The plastic part of the Helmholtz free energy function is postulated to have the follow-

ing form (Abu Al-Rub and Voyiadjis, 2003; Voyiadjis et al., 2003, 2004):

qwp f0 eep

1

2heeq2 f0 eeq Q eeq

1

bexpbeeq

96



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Substituting Eq. (96) into Eqs. (82) and (83) yields the following expressions for the plas-ticity conjugate forces c+ and c:

c f0 heeq 97c f0 Q 1 expbeeqh i 98

such that by taking the time derivative of Eqs. (97) and (98) one can easily retrieve Eqs.(49) and (50), respectively.

The damage part of the Helmholtz free energy function is postulated to have the follow-ing form:

qwd K0 ueq 1

Bf1 ueq ln1 ueq ueqg 99

where K0 is the initial damage threshold defined in Eq. (59) and B are material constants

which are expressed in terms of the fracture energy and an intrinsic length scale, Eq. (60).Substituting Eq. (90) into Eqs. (74) and (75), one can easily obtain the following expres-

sions for the damage driving forces K:

K K0 1 1

Bln1 ueq 100

By taking the time derivative of the above expression one retrieves the rate form of thedamage hardening/softening function K presented in Eqs. (57) and (58) such that:

_K

K0

Bexp

B

1

K

K0 _ueq

101

It is noteworthy that the expression that is presented in Eq. (58) for tensile damage is a

slightly different than the one shown in Eq. (101). However, in the remaining of this study,Eq. (57) is used. This is attributed to the better representation of the stressstrain diagramunder tensile loading, when Eq. (58) is used instead of Eq. (101).

5. Numerical implementation

In this section, the time discretization and numerical integration procedures are pre-sented. The evolutions of the plastic and damage internal state variables can be obtainedif the Lagrangian multipliers _kp and _kd are computed. The plasticity and damage consis-tency conditions, Eqs. (52) and (61), are used for computing both _kp and _kd . This is shownin the subsequent developments. Then, by applying the given strain incrementDeij en1ij enij and knowing the values of the stress and internal variables at the begin-ning of the step from the previous step, n, the updated values at the end of the step,n1, are obtained.

The implemented integration scheme is divided into two sequential steps, correspondingto the plastic and damage parts of the model. In the plastic part, the plastic strain epij andthe effective stress rij at the end of the step are determined by using the classical radial

return mapping algorithm (Simo and Hughes, 1998) such that:

rij rtrij EijklDepkl rtrij DkpEijkloFp

ornkl

102

http://-/?-http://-/?-


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where rtrij rnij EijklDekl is the trial stress tensor, which is easily evaluated from the givenstrain increment. If the trial stress is not outside the yield surface, i.e.frtrij ; cnc 6 0, the stepis elastic and one setsDkp 0, rtrij rn1ij , epn1ij epnij , cn1 cn. However, if the trial

stress is outside the yield surface

rn

1

ij , ep

n

1

ij , and cn

1

are determined by computingDk

p

.In the damage part, the nominal stress rij at the end of the step is obtained from Eq.(42) by knowing the damage variables uij , which can be calculated once Dk

d are computed

from the damage consistency conditions.

5.1. Evolution of the plastic multiplier

From the effective consistency condition, one can write the following relation at n + 1step:

fn

1

fn

Df 103where

Df oforij

Drij oformax

Drmax ofoeeq

Deeq of

oeeqDeeq 0 104

fn1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3Jn12

q aIn11 bn1rn1max 1 acn1 0 105

bn1 Hrn1max bn1 106Making few arithmetic manipulations, one can obtain the plastic multiplier Dkp from

the following expression:

Dkp f

tr

H107

where ftr and H are given as

ftr ffiffiffi

3

2

rkstrijk aItr1 brtrmax 1 acn 108

H

3G

9Kapa

bZ

1

r

of

oeeq

oFp

or

trmin

rof

oeeq

oFp

ortrmax

109

with

Zffiffiffi

6p

Grtrmaxkstrijk

3Kap ffiffiffi

2

3

rG

Itr1kstrijk

110

oFp

ortrmin;max

ffiffiffi3

2

rrtrmin;max 13Itr1

kstrijk

ap 111

of

oeeq 1 aQb expbeeq 1 h

rmaxic

112of

oeeq hrmaxi c

1 ahc2 113



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It should be noted that, in the case of the DruckerPrager potential function, a singularregion occurs near the cone tip unless ap 0. The reason of the occurrence of the singularregion is that any trial stress that belongs to the singular region cannot be mapped back toa returning point on the given yield surface.

5.2. Evolution of the tensile and compressive damage multipliers

In the following, the damage multipliers, _kd , are obtained using the consistency condi-tions in Eq. (61). The incremental expression for the damage consistency condition can bewritten as

gn1 gn Dg 0 114where g+ is the damage surface function given in Eq. (55) and Dg is the increment of thetensile damage function which is expressed by

Dg og

oYijDYij

og

oKDK 115

However, since Yij is a function ofrij and u

ij one can write the following:

DYij oYijorkl

Drkl oYijoukl

Dukl 116

where Dukl is obtained from Eqs. (84) and (85) such that:

Dukl Dkdog

oYkl 117and Drkl can be obtained from Eq. (17) as follows:

Drkl oM1

klrs

oumnDumnr

rs M1

klrs Dr

rs 118

By substituting Eqs. (116)(118) into Eq. (115) and noticing that _k _ueq ffiffiffiffiffiffiffiffiffiffiffiffi_uij _u

ij

q, one

can obtain the following relation:

Dg

og

oYij

oYij

orkl

oM1

klrs

oumn

rrsog

oYmnDk

d

og

oYij

oYij

orkl M1

klrsDrrs

og

oYij

oYij

oukl

og

oYklDkd

og

oKoK

oueqDkd 119

Substituting the above equation into Eq. (114), one obtains Dkeq by the following relation:

Dkeq gn

Heq120

where Heq is the tensile or compressive damage hardening/softening modulus and is given

as follows:

Heq og

oYij

oYijorkl

oM1

klrs

oumnrrs

og

oYmn og

oYij

oYijoukl

og

oYkl og

oKoK

oueq121

http://-/?-http://-/?-


21/27

where og=oK 1 and the expressions for og=oYij, oYij=orkl, oM1klrs=oumn, and oYij=ouklcan be found in Abu Al-Rub and Voyiadjis (2003), Voyiadjis et al. (2003, 2004), whereasthe expression ofoK=oueq can be obtained from Eq. (100).

It is noteworthy that the consistent (algorithmic) tangent stiffness is used to speed the

rate of convergence of the NewtonRaphson method. The expression of this stiffnesscan be simply obtained using the guidelines in Simo and Hughes (1998).

6. Comparison with experimental results

The use of the present model in solving several problems is conducted using the finiteelement program, ABAQUS (2003). The proposed constitutive model and the numericalalgorithm presented in the previous section are implemented in ABAQUS via the user sub-routine UMAT. Comparison of the results is done using experimental data for uniaxialloadings (Karsan and Jirsa, 1969) and biaxial loading (Kupfer et al., 1969) for normalstrength of concrete (NSC). The material constants used in conducting these comparisonsare listed in Table 1. These material constants are obtained from the experimental resultsin Fig. 4. The values for E, m, fu , f

u , ap, and a are taken from the work ofLee and Fenves

(1998). The values for the fracture energies Gf are obtained by establishing the stress-plas-tic strain diagram and then calculating the area under the curve (see Fig. 2). A materiallength scale of = 50 mm is assumed as a typical measure of the size of the localized dam-age zone in frictional materials, which is used along with Gf to calculate the damage con-stants B from Eq. (60). Furthermore, it is assumed that the onset of plasticity,characterized by f0 , coincides with the onset of damage, characterized by K

0 in Eq.

(59), such that f0 fu whereas f0 is obtained from Fig. 4(b) beyond which nonlinearitybecomes visible. Generally, h, Q, and b can be obtained by constructing the effective stress,r r=1 u11, versus the plastic strain diagram such that u11 can be obtained by mea-suring the reduction in the material stiffness from loading/reloading uniaxial cyclic test(one point is extracted for each unloading cycle). However, due to the lack of such exper-imental data, the material constants h, Q, and b are obtained by taking the best fit of theexperimental data in Fig. 4a and b.

A single-element mesh of size 200 200 mm and with one Gauss integration point isused in testing the proposed elasto-plastic-damage model (see Fig. 3). Simulations are per-formed using one step and 100 iterations. Convergence is obtained quickly. Numerical

results show a good agreement between the stressstrain curves compared to the experi-mental results as shown for uniaxial and biaxial tensile/compressive loadings in Figs. 4and 5, respectively. Although the experimental data in Fig. 4a shows a very steep softeningcurve, the proposed model can predict to a large extent such a large negative stiffness. Thesimulated compressive stressstrain curve in Fig. 4b agrees very well with the experimentaldata.

Table 1The material constants for normal strength concrete

E

31:7 GPa m

0:2 fu

3:48 MPa fu

27:6 MPa

f0 3:48 MPa f0 20 MPa Gf 42 N=m Gf 1765 N=map 0:2 a 0:12 50 mm h 25 GPaQ 50 MPa b = 2200 B 0:54 B 0:12K0 191 N=m2 K0 6310 N=m2



22/27

The same material constants as listed in Table 1 are used to test the model under biaxialloading. It can be seen that the dilatancy in concrete, which is mainly controlled by the apparameter in the plastic potential function, can be modeled well by the present model.

2

00mm

200 mm

Fig. 3. (a) Uniaxial tension, (b) uniaxial compression, (c) biaxial tension, and (d) biaxial compression.

Fig. 4. Stressstrain curves for (a) uniaxial tension and (b) uniaxial compression. Comparison of the modelpredictions with experimental results (Karsan and Jirsa, 1969).

Fig. 5. Stressstrain curves for (a) biaxial tension and (b) biaxial compression. Comparison of the modelpredictions with experimental results (Kupfer et al., 1969).



23/27

Figs. 6 and 7 show the damage evolution versus strain for the numerical uniaxial andbiaxial curves in Figs. 4 and 5. It can be noted from Fig. 6a that the maximum tensile dam-age is about 0.88 which corresponds to a strain of 0:6 103, whereas Fig. 6b shows amaximum compressive damage of 0.62 at a strain of 5

103. As expected the material

under compressive loading damages less and sustains more load than under tensile load-ing. Moreover, Fig. 7a shows a maximum biaxial tensile damage of 0.65 at 0:7 103,whereas Fig. 7b shows a maximum compression damage of 0.75 at a strain of 7 103.This indicates that in biaxial compression the material sustains more strains and as a resultof that, higher values of damage occur than in the corresponding biaxial tension.

It is noteworthy to compare the damage evolution in both uniaxial and biaxial tensile/compressive loadings as shown in Fig. 8. It can be seen that the largest evolution of dam-age occurs during uniaxial tension. Moreover, generally one can conclude that damageevolution during uniaxial tensile loading is higher than biaxial or multiaxial tensile load-ing. This can be attributed to geometrical constraints set by the evolution of cracks in dif-ferent directions during biaxial or multiaxial loadings such that crack propagation can besuppressed by other propagating cracks. Interaction between cracks may cause the mate-rial to harden such that a stronger response is obtained. However, an interesting behavior

Fig. 6. Damage evolution in (a) uniaxial tension and (b) uniaxial compression.

Fig. 7. Damage evolution in (a) biaxial tension and (b) biaxial compression.



24/27

can also be noticed such that the previous conclusion cannot be inferred from the evolu-tion of damage during uniaxial and biaxial compression. One can notice from Fig. 8 thatthe amount of cracks generated during biaxial compression is higher than during uniaxialcompression. This can be attributed to the amount of generated dilation and shear cracks(crushing) being larger during biaxial compression.

7. Conclusions

An elasto-plasticity-damage model for plain concrete is presented in this work. Basedon continuum damage mechanics, anisotropic damage evolution is considered. The plas-ticity and damage loading surfaces account for both compressive and tensile loadings suchthat the tensile and compressive damage are characterized independently. The plasticityyield surface is expressed in the effective (undamaged) configuration, which leads to adecoupled algorithm for the effective stress computation and the damage evolution. Theconstitutive relations are validated using the second law of thermodynamics. Numericalalgorithm is presented for the implementation of the proposed model in the well-knownfinite element code ABAQUS via the material user subroutine UMAT.

The overall performance of the proposed model is tested by comparing the model pre-

dictions to experimental data. Experimental simulations of concrete response under uniax-ial and biaxial loadings for both tension and compression show well agreement with theexperimental data. This shows that the proposed model with corporation of plasticityand damage provides an effective method for modeling the concrete behavior in tensionand compression. The damage evolution for both uniaxialbiaxial and tensilecompressiveloadings shows a clear representation of the failure of concrete.

The proposed model can be applied to numerous structural engineering applications.This will be the focus of forthcoming papers by the current authors. Moreover, the pre-sented local constitutive relations result in ill-posed initial boundary value problem formaterial softening (Voyiadjis et al., 2004). Consequently, uniqueness of the solution for

the problem is not guaranteed such that mesh dependent results in the finite element anal-ysis may be obtained. However, to overcome this problem a localization limiter such as thestrain gradient theory (Voyiadjis et al., 2004) can be used. This will be presented in detailin a separate publication.

Fig. 8. Comparison between the damage evolution during uniaxial and biaxial tensile and compressive loadings.



25/27

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