ansiotropic damage plasticity

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Anisotropic damage–plasticity model for concrete

George Z. Voyiadjis *, Ziad N. Taqieddin, Peter I. Kattan

Department of Civil and Environmental Engineering, Louisiana State University, Taylor Hall 3508-B, Baton Rouge, LA 70803, USA

a r t i c l e i n f o

Article history:

Received 17 December 2007

Received in revised form 30 March 2008

Available online 13 April 2008

Dedicated to Jean-Louis Chaboche

Keywords:

Anisotropic continuum damage mechanics

Concrete plasticityThermodynamics

Strain energy equivalence

Damage–elasto-plastic tangent operator

a b s t r a c t

A material model for concrete is proposed here within the frame-

work of a thermodynamically consistent elasto-plasticity–damage

theory. Two anisotropic damage tensors and two damage criteria

are adopted to describe the distinctive degradation of the mechan-

ical properties of concrete under tensile and compressive loadings.

The total stress tensor is decomposed into tensile and compressive

components in order to accommodate the need for the above men-

tioned damage tensors. The plasticity yield criterion presented in

this work accounts for the spectral decomposition of the stress ten-sor and allows multiple hardening rules to be used. This plastic

yield criterion is used simultaneously with the damage criteria to

simulate the physical behavior of concrete. Non-associative flow

rule for the plastic strains is used to account for the dilatancy of

concrete as a frictional material. The thermodynamic Helmholtz

free energy concept is used to consistently derive dissipation

potentials for damage and plasticity and to allow evolution laws

for different hardening parameters. The evolution of the two dam-

age tensors is accounted for through the use of fracture-energy-

based continuum damage mechanics. An expression is derived

for the damage–elasto-plastic tangent operator. The theoretical

framework of the model is described here while the implementa-

tion of this model will be discussed in a subsequent paper.

Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Concrete prevails as one of the most widely used materials in numerous civil engineering applica-

tions due to its workability and formability into various structural components. Understanding and

subsequently modeling the mechanical/material behavior of concrete under different loading states

0749-6419/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2008.04.002

* Corresponding author.

E-mail address: [email protected] (G.Z. Voyiadjis).

International Journal of Plasticity 24 (2008) 1946–1965

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is an essential yet challenging task for standard civil engineering applications, not to mention complex

concrete structures that require further understanding in terms of the prediction of failure patterns.

The distinctive behavior of concrete under tensile or compressive loading has also increased the com-

plexity of the constitutive modeling of its behavior (see Fig. 1).

One of the challenging characteristics of concrete is its low tensile strength, particularly at low-

confining pressures, which results in tensile cracking at a very low stress compared with compressive

stresses. The tensile cracking reduces the stiffness of concrete structural components. Therefore, the

use of continuum damage mechanics is necessary to accurately model the degradation in the mechan-

ical properties of concrete. However, the concrete material undergoes also some irreversible (plastic

and damage) deformations during loading such that the continuum damage theories cannot be used

alone, particularly at high-confining pressures. Therefore, the nonlinear material behavior of concrete

can be attributed to two distinct material mechanical processes: damage and plasticity.

Continuum damage mechanics – a combination of the internal state variable theory and the phys-

ical considerations of the irreversible thermodynamic processes – provides a powerful framework for

the derivation of consistent constitutive models suitable for many engineering problems. Several stud-

ies have been performed using continuum damage mechanics to better describe the behavior of var-

ious materials under different loading conditions (e.g. Cordebois and Sidoroff, 1979; Chaboche, 1981;Krajcinovic, 1983; Lemaitre, 1985; Chow and Wang, 1987; Simo and Ju, 1987a; Chaboche, 1988; Lu-

barda and Krajcinovic, 1993; Hansen and Schreyer, 1994; Doghri and Tinel, 2005; Menzel et al., 2005;

Voyiadjis and Kattan, 1999, Voyiadjis et al., 2008; just to mention a few).

Damage in concrete is primarily caused by the propagation and coalescence of micro-cracks, a phe-

nomenon often treated as strain softening in structural analysis. The modeling of crack initiation and

propagation is the intended role of damage mechanics in this model.

Isotropic damage models (scalar based) with one or two (tension and compression) damage vari-

ables have been extensively studied by numerous authors (e.g. Krajcinovic, 1983, 1985; Mazars and

Pijaudier-Cabot, 1989; Lubliner et al., 1989; Lubarda et al., 1994; Faria et al., 1998; Lee and Fenves,

1998; Peerlings et al., 1998; Jason et al., 2004; Bonora et al., 2005; Jason et al., 2006; Pirondi et al.,

2006; Bonora et al., 2006; Wu et al., 2006; Celentano and Chaboche, 2007). In addition, anisotropic(tensor based) damage models (e.g. Dragon and Mroz, 1979; Murakami and Ohno, 1981; Chaboche,

1981; Sidoroff, 1981; Krajcinovic and Fonseka, 1981; Ortiz, 1985; Simo and Ju, 1987a,b; Ju, 1989,

1990; Valanis, 1991; Ramtani et al., 1992; Lubarda and Krajcinovic, 1993; Chaboche, 1993; Voyiadjis

and Abu-Lebdeh, 1994; Yazdani and Schreyer, 1990; Govindjee et al., 1995; Chaboche et al., 1995;

Halm and Dragon, 1996; Fichant et al., 1999; Carol et al., 2001; Hansen et al., 2001; Gatuingt and

Pijaudier-Cabot, 2002; Brunig, 2003; Menzel et al., 2005; Cicekli et al., 2007; Hammi and Horstemeyer,

2007) have been also proposed and investigated.

σ −

ε −

E

(1 ) E ϕ −

−

u f

−

o f

−

eε

−

pε

+

eε

+

σ +

0 u f f + +

=

ε +

E

a b

(1 ) E ϕ +

−

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

G.Z. Voyiadjis et al. / International Journal of Plasticity 24 (2008) 1946–1965 1947



Plasticity theories have been used successfully in modeling the behavior of metals where the dom-

inant mode of internal rearrangement is the slip process. Although the mathematical theory of plas-

ticity is thoroughly established, its potential usefulness for representing a wide variety of material

behavior has not been yet fully explored. There are many researchers who have used plasticity alone

to characterize the concrete behavior (e.g. Chen and Chen, 1975; William and Warnke, 1975; Bazant,

1978; Dragon and Mroz, 1979; Schreyer, 1983; Chen and Buyukozturk, 1985; Oñate et al., 1988; Voy-

iadjis and Abu-Lebdeh, 1994; Karabinis and Kiousis, 1994; Este and Willam, 1994; Menetrey and Wil-

lam, 1995; Grassl et 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 strain hardening.

However, these works did not address the degradation of the material stiffness due to micro-cracking.

On the other hand, others have used the continuum damage theory alone to model the material non-

linear behavior such that the mechanical effect of the progressive micro-cracking and strain softening

are represented by a set of internal state variables 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 cannot be rep-resented by this approach, which renders it, just as plasticity by itself, to be insufficient. Since both

micro-cracking and irreversible deformations are contributing to the nonlinear response of concrete,

a constitutive model should address equally and in a coupled form the two physically distinct modes

of irreversible changes and should satisfy the basic postulates of thermodynamics.

Plasticity and damage are usually used together to represent the mechanical behavior of concrete

(see Fig. 1). One type of model relies on stress-based plasticity formulated in the effective (undam-

aged) space (e.g. Simo and Ju, 1987a,b; Ju, 1989; 1990; Yazdani and Schreyer, 1990; Lee and Fenves,

1998; Gatuingt and Pijaudier-Cabot, 2002; Jason et al., 2004; Wu et al., 2006; Cicekli et al., 2007),

where the effective stress is defined as the average micro-scale stress acting on the undamaged mate-

rial between micro-defects. Another type of model is based on stress-based plasticity in the nominal

(damaged) stress space (e.g. Bazant and Kim, 1979; Ortiz, 1985; Lubliner et al., 1989; Imran and Pan-tazopoulu, 2001; Ananiev and Ozbolt, 2004; Kratzig and Polling, 2004), where the nominal stress is

defined as the macro-scale stress acting on both the damaged and undamaged materials. Ju (1990) ad-

dressed the characterization of the concrete damage behavior that incorporates anisotropic damage

effects, i.e. different micro-cracking in different directions. Anisotropic damage in concrete yet re-

mains complex and the coupling with plasticity and the application to structural analysis is not

straightforward (e.g. Ju, 1989, 1990; Yazdani and Schreyer, 1990; Voyiadjis and Abu-Lebdeh, 1994;

Meschke et al., 1998; Voyiadjis and Kattan, 1999; Carol et al., 2001; Hansen et al., 2001; Cicekli

et al., 2007), and consequently, it has been avoided by many authors.

In this work, a coupled elasto-plastic–anisotropic damage constitutive model for concrete is formu-

lated based on sound thermodynamics principles. The proposed model includes important aspects of

the concrete nonlinear behavior. It considers different responses of concrete under tension and com-pression. A brief description of anisotropic damage is provided first, followed by the application of the

spectral decomposition technique to the stress and damage effect tensors. Thermodynamically consis-

tent derivations are then presented to introduce the proper evolution laws for the model and to define

the plastic and damage dissipation potentials. The plastic and damage yield criteria are then defined,

followed by the derivation of the damage–elasto-plastic tangent operator. Specific forms of the elastic,

plastic, and damage parts of the Helmholtz free energy function are then specified.

2. Modeling anisotropic damage

Material damage can be used to model specific void and crack surfaces, specific crack and void vol-umes, or the spacing between cracks or voids. It could be represented by scalar damage variables or

the general tensorial representation of damage. Generally the physical interpretation of the damage

variable is introduced as the specific damaged surface area (Kachanov, 1958). In particular, two cases

can be considered: isotropic (scalar) damage and anisotropic (tensorial) damage density of

1948 G.Z. Voyiadjis et al. / International Journal of Plasticity 24 (2008) 1946–1965



micro-cracks, micro-voids, and other microscopic defects. Scalar damage representation is widely

used because of its simplicity in dealing with finite element code implementation. However, for the

proper interpretation of damage in concrete, one should consider the anisotropic damage case. Aniso-

tropic damage is considered in this work.

The effective (undamaged) configuration is used in this study in order to facilitate the formulation

of the damage constitutive equations. That is, the damaged material is modeled first within the effec-

tive elastic–plastic framework, followed by the introduction of the degradation processes that corre-

late the effective undamaged material properties to those of the damaged material through the use of

the following mapping relation (Cordebois and Sidoroff, 1979; Murakami and Ohno, 1981; Voyiadjis

and Kattan, 1999, 2006):

rij ¼ M ijklrkl; ð1Þ

where M ijkl is the fourth-order damage effect tensor. This tensor M ijkl can be of a general form, yet it is

desirable to have a form of M ijkl that will maintain the stress tensor symmetrical after mapping. There

are different definitions for the tensor M ijkl that could be used to symmetrize rij (see Voyiadjis and

Park, 1997a; Voyiadjis and Kattan, 1999, 2006). In this work, the definition presented by Cordebois

and Sidoroff (1982), Lee et al. (1986) and further elaborated by Voyiadjis and Kattan (1999), Voyiadjisand Kattan (2006) is adopted (throughout this work, the following notation for inverse and transpose

will be used for the sake of expediency, ( X À1Þij ¼ X À1ij and ( X TÞij ¼ X Tij):

M ijkl ¼1

2ðdilðdkj À ukjÞ

À1þ ðdil À uilÞ

À1dkjÞ; ð2Þ

where dij is the Kronecker delta and uij is the second-order damage tensor, the evolution of which will

be defined later taking into account the different damage behaviors under different loading patterns.

In this work, the superimposed dash designates a variable in the effective (undamaged) configuration.

The transformation from the effective (undamaged) to the damaged configuration in the elastic re-

gime can be accomplished by utilizing the elastic strain energy equivalence hypothesis (Cordebois and

Sidoroff, 1979; Chow and Wang, 1987; Hansen and Schreyer, 1994; Voyiadjis and Kattan, 1999, 2006

and others.), which basically states that the elastic strain energy for a damaged material is equivalentin form to that of the undamaged (effective) material. This hypothesis can be expressed as follows:

1

2rije

eij ¼

1

2rije

eij: ð3Þ

The total strain tensor in the effective configuration and its damaged counterpart, eij (eij), can be addi-

tively decomposed into elastic eeij (ee

ij) and plastic epij (ep

ij) parts such that:

eij ¼ eeij þ e

pij; eij ¼ ee

ij þ epij : ð4a;bÞ

Using the generalized Hook’s law, the effective stress is given as follows:

rij ¼ E ijkleekl; ð5Þ

where E ijkl is the fourth-order undamaged elastic stiffness tensor. For isotropic linear-elastic materials,

E ijkl is given by:

E ijkl ¼ 2GI dijkl þ KI ijkl; ð6Þ

where I dijkl ¼ I ijkl À 13dijdkl is the deviatoric part of the fourth-order identity tensor I ijkl ¼ 1

2ðdikd jl þ dild jkÞ,

and G ¼ E =ð2ð1 þ mÞÞ and K ¼ E =ð3ð1 À 2mÞÞ are the effective shear and bulk modulii, respectively, with

E being the Young’s modulus and m is the Poisson’s ratio of the initially linear isotropic elastic material.

In the damaged configuration, and similar to the stress–strain relationship in Eq. (5), one writes the

elastic constitutive relation as follows:

rij ¼ E ijkleekl; ð7Þ

where E ijkl is the fourth-order damaged elastic tensor, which is a function of the damage variable uij or

M ijkl. The elastic strain tensors can be expressed using Eqs. (5) and (7) by the following relations:

eeij ¼ E À1

ijklrkl; eeij ¼ E À1

ijklrkl: ð8a;bÞ




By substituting Eqs. (1) and (8)a into Eq. (3), one can express the damaged elasticity tensor E ijkl in

terms of the corresponding undamaged elasticity tensor E ijkl by the following relation:

E ijkl ¼ M À1ijmnE mnpqM ÀT

pqkl: ð9Þ

Using Eqs. (1), (5) and (7), the following expression relating the elastic strain tensors in the effective

and damaged configurations can be obtained:

eeij ¼ M ÀT

ijpqee

pq: ð10Þ

Moreover, combining Eqs. (8)a, b into Eqs. (4)a,b, respectively, the total strain tensors eij and eij can be

written in the following form:

eij ¼ E À1ijklrkl þ e

pij; eij ¼ E À1

ijklrkl þ epij: ð11a;bÞ

By taking the time derivative of Eqs. (4)a, b, the rate of the total strain tensor in the effective and dam-

aged configurations, _eij and _eij, respectively, can be written as follows:

_eij ¼ _eeij þ _e

pij ; _eij ¼ _ee

ij þ _epij; ð12a;bÞ

where _eeijð_e

eijÞ and _e

pijð_e

pijÞ are the rate of the elastic and plastic strain tensors in the effective and damaged

configurations, respectively.

Analogous to Eq. (11)a, one can write the following relation in the effective configuration

_eij ¼ E À1ijkl

_rkl þ _epij: ð13Þ

However, since E ijkl (the damaged counterpart of E ijkl) is a function of uij, a similar relation to Eq. (13)

cannot be used to write the total strain rate tensor _eij in the damaged configuration. Therefore, an

alternative methodology will be used to obtain that in the following sections.

In this section, anisotropic continuum damage formulations were presented in a general format. No

distinction was made to address the difference in the material response due to various loading behav-

iors. In the following section, the spectral decomposition technique is used to illustrate the adaptationof the above mentioned formulations to the study of concrete materials.

3. Spectral decomposition of the stress tensor

Concrete has different behaviors in tension than in compression. In order to adequately character-

ize the damage in concrete due to tensile and compressive loadings, the Cauchy stress tensor (nominal

or effective) is decomposed into positive and negative parts using the spectral decomposition tech-

nique (e.g. Simo and Ju, 1987a,b; Chaboche, 1992; Voyiadjis and Abu-Lebdeh, 1994; Hansen and

Schreyer, 1995; Krajcinovic, 1996). Hereafter, the superscripts ‘‘+” and ‘‘À” designate, respectively,

tensile and compressive entities. Therefore, rij and rij can be decomposed as follows:

rij ¼ rþij þ rÀij ; rij ¼ rþij þ rÀij ; ð14a;bÞ

where rþij ðrþ

ij Þ and rÀij ðrÀ

ij Þ are the tension and compression parts of the stress in the effective and dam-

aged states, respectively.

The tensile rþij and compressive rÀ

ij parts of the stress tensor rij can be related by:

rþkl ¼ P þklpqr pq; ð15Þ

rÀkl ¼ ½I klpq À P þijpqr pq ¼ P Àklpqr pq; ð16Þ

such that P þijkl þ P Àijkl ¼ I ijkl and P þijkl and P Àijkl are the fourth-order projection tensors defined, respectively

as

P þijpq ¼ X3

k¼1

H ðrðkÞÞnðkÞi nðkÞ

j nðkÞ p nðkÞ

q ; P Àklpq ¼ I klpq À P þijpq; ð17a;bÞ

where H ðrðkÞÞ denotes the Heaviside step function computed at the kth principal stress rðkÞ of rij and

nðkÞi is the kth corresponding unit principal direction. In the subsequent development, the superscript

hat designates a principal value.




Based on the decomposition in Eq. (14), one can assume the expression in Eq. (1) to be valid for

both tension and compression components of the stress tensor, however, with decoupled damage evo-

lution in tension and compression, one can write:

rþij ¼ M þijklr

þkl; rÀ

ij ¼ M ÀijklrÀkl; ð18a;bÞ

where M þijkl and M Àijkl are the tensile and compressive damage effect tensors which can be expressed,

similar to Eq. (2), in a decoupled form as a function of the tensile and compressive damage variables,

uþij and uÀ

ij , respectively, as follows:

M þijkl ¼1

2ðdilðdkj À uþ

kjÞÀ1 þ ðdil À uþ

il ÞÀ1dkjÞ;

M Àijkl ¼1

2ðdilðdkj À uÀ

kjÞÀ1 þ ðdil À uÀ

il ÞÀ1dkjÞ:

ð19a;bÞ

By substituting Eqs. (18) into Eq. (14)a, one can express the effective stress tensor in terms of the

fourth-order damage effect tensor for tension and compression and their corresponding stresses such

that:

rij ¼ M þijklrþkl þ M Àijklr

Àkl: ð20Þ

Furthermore, substituting Eqs. (15) and (16) into Eq. (20) and comparing the result with Eq. (1), one

can obtain the following relation for the damage effect tensor such that:

M ijpq ¼ M þijklP þklpq þ M ÀijklP Àklpq: ð21Þ

Using Eq. (17)b, the above equation can be rewritten as follows:

M ijpq ¼ ðM þijkl À M ÀijklÞP þklpq þ M Àijpq: ð22Þ

It is important to notice that:

M ijkl 6¼ M

þ

ijkl þ M

À

ijkl ð23Þand

uij 6¼ uþij þ uÀ

ij : ð24Þ

It is also noteworthy that the relation in Eq. (22) enhances a coupling between tensile and compressive

damage through the fourth-order projection tensor P þijkl. Moreover, for isotropic damage, Eq. (21) can

be written as follows:

M ijkl ¼P þijkl

1 À uþþ

P Àijkl

1 À uÀ: ð25Þ

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

variable u into a positive u+ and negative uÀ parts still enhances damage anisotropy through the spec-tral decomposition tensors P þijkl and P Àijkl, respectively. However, this anisotropy is weak when com-

pared to the anisotropic damage introduced by a tensorial damage variable as shown in Eq. (22).

4. Consistent thermodynamic formulation

In this section, a general framework for the elasto-plastic–anisotropic damage formulation for con-

crete is developed. Isothermal conditions and rate independence are assumed throughout this work.

Irreversible thermodynamic following the internal variable procedure of Coleman and Gurtin (1967)

and Chaboche (1988) will be applied. The internal variables and potentials used to describe the ther-

modynamic processes are introduced. The Lagrange minimization approach (Calculus of functions of

several variables) is used next to derive the evolution equations for the proposed model.The constitutive equations of the model are derived from the second law of thermodynamics, the

expression of the Helmholtz free energy, the additive decomposition of the total strain rate into elastic

and plastic components, the Clasius–Duhem inequality, and the maximum dissipation principle.




The Helmholtz free energy can be expressed in terms of a suitable set of internal state variables

that characterize the elastic, plastic, and damage behavior of concrete. In this work the following inter-

nal variables are assumed to satisfactorily characterize the concrete behavior in tension and compres-

sion: the elastic strain tensor eeij, the tensile/compressive equivalent (accumulated) plastic strains,e+ep,

eÀep, the tensile/compressive anisotropic damage tensors, uþij ;u

Àij , and the tensile/compressive equiva-

lent (accumulated) damage variables, uþeq;u

Àeq, such that:

w ¼ wðeeij; e

þep; eÀep;uþij ;u

Àij ;u

þeq;u

ÀeqÞ: ð26Þ

The second-order tensors uÆij characterizes anisotropic damage in the material while the scalars uþ

eq

and uÀeq, defined as uÆ

eq ¼R t

0_uÆ

eqdt , are used to characterize isotropic damage hardening which repre-

sents the generation and propagation of micro-defects in the material, i.e., they cause micro-cracks

and micro-surfaces to grow (Hansen and Schreyer, 1994). Similarly, e+ep and eÀep are the equivalent

tensile and compressive plastic strains that are used here to characterize the plasticity isotropic hard-

ening, eÆep ¼R t

0_eÆepdt .

The Clausius–Duhem inequality for the isothermal case is given as follows:

rij_eij À q

_

wP 0: ð27Þ

Taking the time derivative of Eq. (26), the following expression can be written:

_w ¼ow

oeeij

_eeij þ

ow

ouþij

_uþij þ

ow

ouÀij

_uÀij þ

ow

oeþep_eþep þ

ow

oeÀep_eÀep þ

ow

ouþeq

_uþeq þ

ow

ouÀeq

_uÀeq: ð28Þ

Substituting Eq. (28) into Eq. (27), and making some simplifications, one can obtain the following rela-

tions for any admissible states such that:

rij ¼ qow

oeeij

ð29Þ

and

rij _epij þ Y þij _uþ

ij þ Y Àij _uÀij À c þ _eþep À c À _eÀep À K þ _uþ

eq À K À _uÀeq P 0; ð30Þ

where the damage and plasticity conjugate forces that appear in the above expression are defined as

follows Chaboche (1989):

Y þij ¼ Àqow

ouþij

; Y Àij ¼ Àqow

ouÀij

; K þ ¼ qow

ouþeq

;

K À ¼ qow

ouÀeq

; c þ ¼ qow

oeþep; c À ¼ q

ow

oeÀep:

ð31a — f Þ

Therefore, one can rewrite the Clausius–Duhem inequality to yield the dissipation energy, P, due to

plasticity, Pp, and damage, Pd, such that

P ¼ Pd þPpP 0 ð32Þ

with

Pp ¼ rij _e

pij À c þ _eþep À c À _eÀep

P 0; ð33Þ

Pd ¼ Y þij _uþ

ij þ Y Àij _uÀij À K þ _uþ

eq À K À _uÀeq P 0: ð34Þ

The rate of the internal variables associated with plastic and damage deformations are obtained by

utilizing the calculus of functions of several variables with the plasticity and damage Lagrange multi-

pliers, _kp and _kÆd , and the plastic (F p) and damage ( g +, g À) dissipation potentials such that the following

objective function can be defined:

X ¼ PÀ _kpF p À _kþd g þ À _kÀ

d g À P 0: ð35Þ

Using 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, Y Æij ), K ±) is that which




maximizes the dissipation function over all other possible admissible states, hence, one can maximize

the objective function by using the necessary conditions as follows:

oX

orij

¼ 0;oX

oc Æ¼ 0;

oX

oY Æij¼ 0 and

oX

oK Æ¼ 0: ð36a—dÞ

Substituting Eq. (32) along with Eqs. (33) and (34) into Eqs. (36) yields the following thermodynamic

evolution laws:

_epij ¼ _kp oF p

orij

; _uþij ¼ _kþ

d

o g þ

oY þij; _uÀ

ij ¼ _kÀd

o g À

oY Àij; _eþep ¼ _kp oF p

oc þ;

_eÀep ¼ _kp oF p

oc À; _uþ

eq ¼ _kþd

o g þ

oK þ; _uÀ

eq ¼ _kÀd

o g À

oK À:

ð37a — gÞ

In what follows, specific forms and functions of the generalized thermodynamically consistent formu-

lations are provided. This includes: the yield and dissipation potentials for damage and plasticity, the

elastic, plastic, and damage parts of the Helmholtz free energy function, and the evolution law for the

state variables of the model.

5. Elasto-plastic–damage model

In this work, the concrete plasticity yield criterion of Lubliner et al. (1989) is adopted. The phenom-

enological concrete model of Lubliner et al. (1989) which was later modified by Lee and Fenves (1998),

is formulated based on isotropic (scalar) stiffness degradation. However, in this work the model of

Lubliner et al. (1989) is extended to anisotropic damage using three loading surfaces: one for plastic-

ity, one for tensile damage, and one for compressive damage. The plasticity and the compressive dam-

age loading surfaces are more dominant in the case of shear loading and compressive crushing (i.e.

modes II and III cracking) whereas the tensile damage loading surface is dominant in the case of mode

I cracking.

The presentation in the following sections can be used for either isotropic or anisotropic damagesince the second-order damage tensor uij degenerates to the scalar damage variable in the case of uni-

axial loading.

5.1. Uniaxial loading

It should be noted that the material behavior concrete is controlled by both plasticity and damage.

In this section, scalar variables will be used.

In the work of Lee and Fenves (1998), which is based on the work of Lubliner et al. (1989), the uni-

axial tensile and compressive stresses, rþij and rÀ

ij , in the damaged configuration are given as (plus and

minus signs cannot be interchanged):

rÆ ¼ ð1 À uÆÞE eÆe ¼ ð1 À uÆÞE ðeÆ À eÆpÞ: ð38a;bÞ

The rate of the equivalent (effective) plastic strains in compression and tension, eÀep and eþep, are,

respectively, given as follows in case of uniaxial loading:

_eþep ¼ _ep11; _eÀep ¼ À_e

p11 ð39a;bÞ

such that

eÀep ¼

Z t

0

_eÀepdt ;eþep ¼

Z t

0

_eþepdt : ð40a;bÞ

Propagation 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 the capacity of the load-car-rying area, which causes an increase in the effective stress. This has little effect during compressive

loading since cracks run parallel to the loading direction. However, under a large compressive stress

which causes crushing of the material, the effective load-carrying area is also considerably reduced.

This explains the distinct behavior of concrete in tension and compression as shown in Fig. 1.




It can be noted from Fig. 1 that during unloading from any point on the strain softening path (i.e.

post peak behavior) of the stress–strain curve, the material response seems to be weakened since the

elastic stiffness of the material is degraded due to damage evolution. Furthermore, it can be noticed

from Figs. 1a and b that the degradation of the elastic stiffness of the material is much different in ten-

sion than in compression, which is more obvious as the plastic strain increases. Therefore, for uniaxial

loading, the damage variable can be represented by two independent damage variables u+ and uÀ.

Moreover, it can be noted that for tensile loading, damage and plasticity are initiated when the equiv-

alent applied stress reaches the uniaxial tensile strength f þo as shown in Fig. 1a whereas under com-

pressive loading, damage is initiated earlier than plasticity. Once the equivalent applied stress reaches

f Ào (i.e. when nonlinear behavior starts) damage is initiated, whereas plasticity occurs once f Àu is

reached. Therefore, generally f þo ¼ f þu for tensile loading, but this is not true for compressive loading

(i.e. f Ào 6¼ f Àu ).

5.2. Multiaxial loading

The evolution equations for the hardening variables are extended now to multiaxial loadings. The

effective plastic strain for multiaxial loading is given as follows (Lubliner et al., 1989; Lee and Fenves,

1998):

_eþep ¼ wðrijÞ_ep

max; _eÀep ¼ Àð1 À wðrijÞÞ _epmin; ð41a;bÞ

where _e

pmax and _

epmin are the maximum and minimum principal values of the plastic strain rate tensor _e

pij

such that _e

p1 > _

ep2 > _

ep3, where _

epmax ¼ _e

p1 and _

epmin ¼ _e

p3.

Eqs. (41)a, b can be written in tensor format as follows:

_eepij ¼ H im

_e

pmj ð42Þ

or in matrix equivalent form as:

_eþep

0

_eÀep

8><>:

9>=>; ¼

H þ 0 0

0 0 0

0 0 H À

264

375

_ep1

_e

p2

_e

p3

8>><>>:

9>>=>>;; ð43Þ

where

H þ ¼ wðrijÞ;H À ¼ Àð1 À wðrijÞÞ: ð44a;bÞ

The dimensionless parameter wðrijÞ is a weight factor depending on principal stresses and is defined as

follows (Lubliner et al., 1989):

wðrijÞ ¼

P3k¼1 rk P3k¼1jrkj ; ð45Þ

where h i is the Macauley bracket, and presented as h xi ¼ 12

ðj xj þ xÞ, k = 1, 2, 3. Note that it can be as-

sumed that wðrijÞ ¼ wðrijÞ. Moreover, depending on the loading state, value of wðrijÞ can be equal to

unity in case of uniaxial tension ( rk P 0, wðrijÞ ¼ 1Þ or equal to zero in the case of uniaxial compres-

sion (rk 6 0;wðrijÞ ¼ 0).

5.3. Plasticity yield surface

To represent the response of concrete subjected to tensile or compressive loadings, a yield criterion

that takes into account the different behaviors of concrete under tension and compression is neces-

sary. Assuming the same yield criterion for both tension and compression for concrete materialscan lead to over/under estimation of plastic deformation (Lubliner et al., 1989). The yield criterion

of Lubliner et al. (1989), that accounts for both tension and compression plasticity, is adopted in this

work. This criterion has been successful in simulating the concrete behavior under uniaxial, biaxial,




multiaxial, and cyclic loadings (Lee and Fenves, 1998 and the references outlined therein). This crite-

rion is expressed in the effective (undamaged) configuration and is given as follows:

f ¼

ffiffiffiffiffiffiffi3 J 2

q þ aI 1 þ bH ðrmaxÞrmax À ð1 À aÞc ÀðeÀepÞ ¼ 0; ð46Þ

where J 2 ¼sijsij=2 is the second-invariant of the effective deviatoric stress

sij ¼

rij À

rkkdij=3, I 1 ¼

rmm is

the first-invariant of the effective stress rij, eÀep ¼R t

0_eÀepdt is the equivalent plastic strain in the effec-

tive configuration which is defined in Eq. (42), HðrmaxÞ is the Heaviside step function (H ¼ 1 for

rmax > 0 and H ¼ 0 for rmax < 0Þ, and rmax is the maximum principal stress. The parameters a and b

are dimensionless constants which are defined by Lubliner et al. (1989) as follows:

a ¼ð f b0= f À0 Þ À 1

2ð f b0= f À0 Þ À 1; ð47Þ

b ¼ ð1 À aÞ f À0 f þ0

À ð1 þ aÞ; ð48Þ

where f b0, f À0 and f þ0 are the initial equibiaxial, uniaxial compressive and uniaxial tensile yield stresses,

respectively. For more details about the derivation of Eqs. (47) and (48), the reader is referred to Lub-liner et al. (1989).

Since the concrete behavior in compression is more of a ductile behavior, the evolution of the com-

pressive isotropic hardening function c À is defined by the following exponential law:

_c À ¼ bðQ À c Ài Þ_eÀep; ð49Þ

where Q and b are material constants characterizing the saturated stress and the rate of saturation,

respectively, which are obtained in the effective configuration of the compressive uniaxial stress–

strain diagram, and c Ài ¼ c À À f Ào as will be shown in Section 6.2. The rate of the compressive equiva-

lent plastic strain _eÀep is expressed as shown in Eq. (31) and will be further elaborated. A linear

evolution will be assumed for the tensile isotropic hardening c þ.

It should be noted that the above yield surface when mapped in the damaged configuration resultsin an anisotropic function. This is elaborated later in Section 5.7 in Eq. (90).

5.4. Non-associative plasticity flow rule

The shape of the concrete loading surface at any given point in a given loading state should

be obtained by using a non-associative plasticity flow rule. This is important for realistic mod-

eling of the volumetric expansion under compression for frictional materials such as concrete.

Basically, flow rule connects the loading surface and the stress–strain relation. When the current

yield surface f is reached, the material is considered to be in a plastic flow state upon increase

of the loading. In the present model, the flow rule is given as a function of the effective stress

rij by:

_epij ¼ _kp oF p

orij

; ð50Þ

where _kp is the plastic loading factor or known as the Lagrangian plasticity multiplier. The damage

configuration counterpart of Eq. (50) is shown later in Section 5.7.

The plastic potential F p is different than the yield function f (i.e. non-associative) and, therefore, the

direction of the plastic flow oF p=orij is not normal to f . One can adopt the Drucker–Prager function for

F p such that:

F p ¼

ffiffiffiffiffiffiffi3 J 2

q þ apI 1; ð51Þ

where ap is the dilation constant. The corresponding plastic flow direction oF p=orij is then given by:

oF p

orij

¼3

2

sij ffiffiffiffiffiffiffi3 J 2

q þ apdij: ð52Þ


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5.5. The consistency condition and the plastic multiplier

The plasticity consistency condition is applied by taking the time derivative of the plasticity yield

function, _ f ¼ 0, such that the Kuhn–Tucker plasticity consistency conditions are satisfied:

f 6 0;_kp P 0;

_kp f ¼ 0; and

_kp

_

f ¼ 0: ð53Þ

In Eq. (46), the first invariant of the effective stress tensor is defined as I 1 ¼ rmm , while the effective

maximum principal stress rmax can be written in terms of the effective stress tensor and the prin-

cipal directions associated with the effective maximum principal stress according to the spectral

decomposition technique, such that rmax ¼ n pr pqnq. Applying these two changes back into Eq.

(46), one obtains:

f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi3

2S ijS ij

r þ armm þ bH ðrmaxÞn pr pqnq À ð1 À aÞc À ¼ 0: ð54Þ

By referring to Section 5.3, one can see that yield criterion, Eq. (54), is a function of the following

parameters:

f ¼ f ðrij;eÀepÞ ¼ 0: ð55Þ

Accordingly the plasticity consistency condition can be written as:

_ f ¼o f

orij

_rij þo f

oeÀep_eÀep ¼ 0; ð56Þ

where o f oeÀep

_eÀep can be expressed as o f oc À

oc À

oeÀep_eÀep.

Differentiating the yield function with respect to the effective stress tensor and again with respect

to the effective compressive equivalent plastic strain gives:

o f

orij ¼

3

2

sij

req þ a

p

dij þ bH ð

rmaxÞn pnqd pidqj ¼

3

2

sij

req þ a

p

dij þ bH ð

rmaxÞnin j; ð57Þo f

oeÀep¼

o f

oc Àoc À

oeÀep¼ Àð1 À aÞbðQ À c Ài Þ: ð58Þ

The rate of the effective compressive equivalent plastic strain _eÀep was defined in Eq. (41)b. By using

the non-associative flow rule (Eq. (50)), _eÀep can be written as follows:

_eÀep ¼ Àð1 À wÞ _epmin ¼ Àð1 À wÞ _kp oF p

ormin

; ð59Þ

where oF p

ormincan be defined as

oF p

ormin¼

3

2

ðrmin À I 1=3Þ ffiffiffiffiffiffiffi3 J 2

q þ ap: ð60Þ

Substituting the additive decomposition of the strain tensor used in defining _rij ¼ E ijkl_ee

kl ¼ E ijklð_ekl À _epklÞ,

the non-associative flow rule, Eq. (50) and Eq. (59) into Eq. (56), one obtains:

_ f ¼o f

orij

E ijkl _ekl À _kp o f

orij

E ijkl

oF p

orkl

À ð1 À wÞ _kp o f

oeÀep

oF p

ormin

¼ 0: ð61Þ

Rearranging Eq. (61) gives the following expression for the effective plastic multiplier _kp:

_kp ¼1h

o f

orij

E ijkl_ekl; ð62Þ

where h is defined as follows:

h ¼o f

orij

E ijkl

oF p

orkl

þ ð1 À wÞo f

oeÀep

oF p

ormin

: ð63Þ


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5.6. Tensile and compressive damage surfaces

The anisotropic damage growth function proposed by Chow and Wang (1987) is adopted in this

study. However, this function is generalized here in order to incorporate both tensile and compressive

damages separately such that:

g Æ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi1

2Y Æij LÆ

ijklY Æij

r À K ÆðuÆ

eqÞ 6 0; ð64Þ

where the superscript ± designates tension, +, or compression, À, K ± is the tensile or compressive dam-

age isotropic hardening functions, K Æ ¼ K Æ0 when there is no damage, K Æ0 is the tensile or compressive

initial damage parameter, Lijkl is a fourth-order symmetric tensor (Bielski et al., 2006; Lee et al., 1985;

Hansen and Schreyer, 1994), and Y ij is the damage driving force that characterizes damage evolution

and is interpreted here as the energy release rate (e.g. Voyiadjis and Kattan, 1992a,b, 1999, 2006; Voy-

iadjis and Park, 1997b).

The rate of the equivalent damage _uÆeq is defined as follows (Voyiadjis and Kattan, 1992a,b, 1999;

Voyiadjis and Park, 1997b):

_uÆeq ¼

ffiffiffiffiffiffiffiffiffiffiffiffi_uÆ

ij _uÆij

q with uÆ

eq ¼

Z t

0

_uÆeqdt : ð65Þ

Several types of functions have been proposed in the literature to represent the evolution of the tensile

and compressive damage isotropic hardening K ±. Examples can be found in (Faria et al., 1998; Wu

et al., 2006; Cicekli et al., 2007; just to mention a few). The functions used by Cicekli et al., 2007 for

the evolution of the tensile and compressive damage isotropic hardening K ± are given as:

_K þ ¼K þ

Bþ þK þoK þ

exp ÀBþ 1 ÀK þ

K þo

_uþ

eq ð66Þ

and

_K À ¼K ÀoBÀ exp ÀBÀ 1 À

K À

K Ào

_uÀ

eq: ð67Þ

In these functions, parameters such as K Æo and B± are defined, where K Æo is the initial damage threshold

which is interpreted as the area under the linear portion of the stress–strain diagram such that:

K Æo ¼f Æ2

o

2E ð68Þ

and B±, a material constant, is related to the fracture energy GÆf (shown in Fig. 2 for both tension and

compression) and defined as follows (Oñate et al., 1988):

BÆ ¼ GÆ

f E ‘ f Æ2

0

À 12

" #À1

P 0; ð69Þ

where ‘ is a characteristic length scale parameter that usually has a value close to the size of the small-

est element in a finite element mesh. Therefore, the parameters B± are used in order to reduce mesh-

sensitivity.

The model response in the damage domain is characterized by the damage consistency condition

similar to the Kuhn–Tucker plasticity consistency as follows:

g Æ 6 0; _kÆd g Æ ¼ 0; and _ g Æ

< 0 ) _kÆd ¼ 0

¼ 0 ) _kÆd ¼ 0

¼ 0 ) _kÆ

d> 0

8><>:

9>=>;

()

effectiveðundamaged stateÞ

damage initiation

damage growth

8><>:

: ð70Þ

One should note that using the above damage criteria, it is possible to write specific conditions for ten-

sile and compressive stages; i.e., tensile damage can be initiated independent from compressive dam-

age or parallel to compressive damage. The same applies to compressive damage. In order to avoid




repetition, however, the following derivations were performed using parameters with superscripts

(±), indicating that these formulations refer to tensile damage (+) or compressive damage (À)

independently.

Taking the time derivative of the damage criteria shown in Eq. (64), one obtains:

_ g Æ ¼o g Æ

oY Æmn

_Y Æmn þo g Æ

oK ÆoK Æ

ouÆeq

_uÆeq ¼ 0; ð71Þ

where the derivates o g Æ

oY Æmnand o g Æ

oK Æare defined as:

o g Æ

oY Æmn

¼LÆ

mnklY Ækl

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi

12 Y ÆabLÆ

abcdY Æcd

q ;o g Æ

oK Æ¼ À1 ð72a;bÞ

and oK Æ

ouÆeq

are nonlinear functions in terms of uÆeq.

Using Eqs. (65) and (37)b, c, along with Eq. (72)a, the following expressions can be derived:

_uÆeq ¼ b _kÆ

d ; b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiLÆ

mnklY ÆklLÆmnpqY Æ pq

2Y ÆabLÆabcdY Æcd

s ; ð73a;bÞ

where b is a scalar parameter-function of the damage conjugate force (tension or compression).

Substituting Eq. (73)a into Eq. (71) and rearranging terms to obtain expression for the tension and

compression damage multipliers_

kÆ

d , one obtains:

_kÆd ¼

o g Æ

oY Æmn

_Y Æmn

Àb o g Æ

oK ÆoK Æ

ouÆeq

: ð74Þ

Applying Eqs. (74) and (72)a into Eqs. (37)b, c, yields the following expression:

_uÆij ¼

o g Æ

oY Æmn

_Y Æmno g Æ

oY Æij

Àb o g Æ

oK ÆoK Æ

ouÆeq

: ð75Þ

By considering the definition of the damage conjugate forces Y Æmn given in Eq. (106), the time deriva-

tive of these forces can be obtains as follows (Voyiadjis and Guelzim, 1996; Voyiadjis and Kattan,

1999, 2006):

_Y Æmn ¼oY Æmn

orkl

_rkl þoY Æmn

ouÆkl

_uÆkl: ð76Þ

f G+

pε

σ

0 f

+

a b

f G−

σ

pε

0 f

−

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

compression.




Substituting Eq. (76) back into Eq. (75), and rearranging terms in order to obtain expressions for the

tensile or compressive damage tensors _uÆij , one obtains:

_uÆrs ¼ X rskl _rkl; X rskl ¼ BÀ1

rsij Aijkl; ð77a;bÞ

where

Bijpq ¼ d pidqj Àþ o g Æ

oY Æmn

oY Æmn

ouÆ pq

o g Æ

oY Æij

Àb o g Æ

oK ÆoK Æ

ouÆeq

; Aijkl ¼

o g Æ

oY Æmn

oY Æmn

orkl

o g Æ

oY Æij

Àb o g Æ

oK ÆoK Æ

ouÆeq

: ð78a;bÞ

Eq. (77)a show incremental relations between damage (in tension or compression) and the stress ten-

sor. These equations can be used to derive an incremental relation between the effective and damaged

stress tensors as follows:

The time derivative of the damage effect tensor M ijkl (shown in Eq. (2)) can be written as:

_M ijkl ¼oM ijkl

ouÆrs

_uÆrs: ð79Þ

This can be performed by using the spectral decomposition of the damage effect tensor shown in Eq.

(21). Accordingly _uÆrs in Eq. (79) can be substituted for by using Eq. (77)a to yield:

_M ijkl ¼ Gijklmn _rmn; Gijklmn ¼oM ijkl

ouÆrs

X rsmn: ð80a;bÞ

Eq. (80)a can then be substituted into the time derivative of Eq. (1), which is given as:

_rij ¼ M ijkl _rkl þ _M ijklrkl ð81Þ

to result into the following expression defining the incremental relation between the effective and

damaged stress tensors:

_

rij ¼ mijkl_

rkl; mijkl ¼ M ijkl þ Gijpqklr pq: ð82a;bÞThis incremental relation will be used here to drive the damage–elasto-plastic tangent operator. This

relation shows the stress path dependence of damage in this work.

5.7. Damage–elasto-plastic tangent operator

Starting with the rate of the constitutive equation, _rij ¼ E ijkl_ee

kl ¼ E ijklð_ekl À _epklÞ, in the effective config-

uration, and using the flow rule given in Eq. (50) along with the expression for the plastic multiplier

given by Eq. (62), one can obtain the following constitutive relation in the effective configuration:

_rij ¼ Dijkl_ekl; ð83Þ

where Dijkl is the elastic–plastic tangent operator in the effective configuration given by the following

expression:

Dijkl ¼ E ijkl À1h

E ijrs

oF p

orrs

o f

ormn

E mnkl: ð84Þ

All the parameters in Eq. (84) were previously defined. The objective now is to obtain a relation similar

to Eq. (84) except that it represents the elasto-plastic tangent operator in the damaged configuration.

The damage–elasto-plastic tangent operator is used at a later stage in the course of a finite element

formulation in order to obtain a new stress or strain increment (global nonlinear iterations, e.g., global

Newton–Raphson iterations). At that stage, all the parameters and the tensors within the current

increment have already been determined.

In order to derive an expression for the damage–elasto-plastic tangent operator, an incremental

relationship of the elastic strains in the effective and damaged configurations, _eeij and _ee

ij, respectively,

should exist. The same applies to the incremental plastic strains in the effective and damaged config-

urations, _epij and _e

pkl, respectively. Considering the increments of the elastic strains first, the use of a




relation similar to Eq. (10) to describe the increments of elastic strains, i.e., _eeij ¼ M ÀT

ijpq _ee

pq, would be

incompatible and therefore, inadmissible (Hansen and Schreyer, 1994). Therefore, the incremental

elastic strain energy equivalency is assumed here (Cordebois and Sidoroff, 1979; Voyiadjis and Guel-

zim, 1996; Voyiadjis and Deliktas, 2000; Voyiadjis and Kattan, 1999, 2006), such that

12

_rij _eeij ¼ 1

2_rij

_eeij: ð85Þ

Substituting Eq. (82)a into Eq. (85), the following relation between the elastic strain rates in the

effective and damaged configurations can be obtained:

_eeij ¼ mÀT

ijkl _eekl: ð86Þ

As for the plastic strain rates, the hypotheses of plastic dissipation equivalence ( Voyiadjis and Thiag-

arajan, 1997; Voyiadjis and Kattan, 1999, 2006) is used here as follows:

rij_e

pij ¼ rij _e

pij : ð87Þ

Substituting Eq. (1) into Eq. (87) results in the following expression that relates the rates of the plastic

strains in the effective and damaged configurations (Lee et al., 1985; Hansen and Schreyer, 1994; Voy-iadjis and Deliktas, 2000; Voyiadjis and Kattan, 1999, 2006):

_epij ¼ M ÀT

ijkl _epkl: ð88Þ

It should be noted here, that the spectral decomposition of the stress tensor can be taken into account

here by realizing that the damage effect tensor M ijkl can be written in terms of its tensile and compres-

sive components as shown in Eq. (21).

Applying Eqs. (12)b, (86), (88), and (82) into Eq. (83), one obtains:

mijkl _rkl ¼ DijklmÀT klmnð_emn À _ep

mnÞ þ DijklM ÀT klmn _e

pmn: ð89Þ

The rate of the plastic strain in the damaged configuration, _epmn, in Eq. (89) can be derived as follows:

the yield function, f , given in Eq. (46) can now be mapped into the damaged configuration (since all the

parameters required for this mapping have already been calculated at this stage) as follows:

f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2H klmnrklrmn

r þ aM mmpqr pq þ bH ðrmaxÞn pM pqmnrmnnq À ð1 À aÞc ÀðeÀepÞ ¼ 0; ð90Þ

where H klmn is a fourth order tensor defined by Voyiadjis and Kattan (1999, 2006) as:

H klmn ¼ N ijklN ijmn; N ijkl ¼ M ijkl À1

3M ppkldij;sij ¼ N ijklrkl ð91a; cÞ

such that

o f

orij ¼

3H ijmnrmn

2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

32

H klrsrklrrs

q þ aM mmij þ bH ð

^rmaxÞn pM pqijnq ð92Þ

and the compressive hardening parameter in the effective configuration is mapped to the damaged

configuration such that:

c ÀðeÀepÞ ¼ ð1 À uÀeqÞc ÀðeÀepÞ: ð93Þ

Therefore, the rate of the plastic strain can be defined in terms of the flow rule in the damaged con-

figuration as follows:

_epij ¼ _kp oF p

orij

; ð94Þ

where

oF p

orij

¼3H ijmnrmn

2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

32

H klrsrklrrs

q þ apM mmij ð95Þ




and

_kp ¼1

h

o f

orij

E ijkl _ekl; h ¼o f

orij

E ijkl

oF p

orkl

þð1 À wÞ

ð1 À uÀeqÞ

oF p

ormin

; ð96a;bÞ

where

oF p

ormin

¼3ðrmin À I 1=3Þ

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi

32

H klmnrklrmn

q þ ap; ð97Þ

which is the damage configuration equivalent to Eq. (60). Note that the norms given by Eqs. (52) and

(95), and Eqs. (60) and (97) are identical, given the assumption in Eq. (85).

Substituting Eq. (94) into Eq. (89) and rearranging terms results in the following expression for the

constitutive relation in the damaged configuration:

_rij ¼ Dijkl _ekl; ð98Þ

where

Dijkl ¼ mÀ1ijmnDmnpqmÀT

pqkl À 1h

mÀ1ijmn o f

ortuE tuklðDmnpqmÀT

pqrs À DmnpqM ÀT pqrsÞ oF

p

orrs

: ð99Þ

In the following sections, specific forms of the Helmholtz free energy function required to develop the

damage energy release rates and the evolution laws for the concrete material model are introduced.

6. The Helmholtz free energy function

Here, we will assume that the Helmholtz free energy function w consists of three parts: elastic,

plastic, and damage parts. The elastic Helmholtz free energy is defined to establish the plastic–damage

constitutive relation with the internal variables, which results in the damage thermodynamic release

rates in tension and compression (damage conjugate forces Y Æij ) used in the damage criteria g ±. The

plastic part introduces the mechanism defining the evolution law for the plastic strains and the mate-rial hardening. The damage part allows the evolution of the damage scalar parameters which repre-

sents the generation and propagation (possibly intersection) of micro-cracks in the concrete material.

6.1. The elastic part of the Helmholtz free energy function

The principle of elastic energy equivalence, which was introduced by Cordebois and Sidoroff

(1979), is used here to define the elastic part of the Helmholtz free energy function. In the damaged

configuration, it is given as a function of the degraded stress and the damage tensor:

qweðee

kl;uÆij Þ ¼

1

2rijE À1

ijklðuÆij Þrkl ¼

1

2rije

eij ¼

1

2ee

ijE ijkleekl; ð100Þ

whereas, in the effective configuration, it is given as a function of the effective stress:

qweðeeij;0Þ ¼

1

2ee

ijE ijkleekl: ð101Þ

The principle of elastic energy equivalence implies that the two preceding forms for the stored elastic

energy are equivalent (Hansen and Schreyer, 1994). Applying Eq. (1) into Eq. (101) and using the prin-

ciple of elastic energy equivalence, one can retrieve the following expression for E À1ijkl (which is the in-

verse of E ijkl given in Eq. (9)):

E À1ijkl ¼ M T

ijmnE À1mnpqM pqkl: ð102Þ

Now, one can obtain expressions for the damage driving forces Y

Æ

ij from Eqs. (31)a, b and (100) asfollows:

Y Ærs ¼ À1

2rij

oE À1ijkl

ouÆrs

rkl: ð103Þ




By taking the derivative of Eq. (102) with respect to the damage parameters uÆij , one obtains:

oE À1ijkl

ouÆrs

¼oM T

ijmn

ouÆrs

E À1mnpqM pqkl þ M T

ijmnE À1mnpq

oM pqkl

ouÆrs

: ð104Þ

Now, by substituting Eq. (104) into Eq. (103), one obtains the following expression for Y Æij

:

Y Ærs ¼ À1

2rij

oM T ijmn

ouÆrs

E À1mnpqM pqkl þ M T

ijmnE À1mnpq

oM pqkl

ouÆrs

!rkl: ð105Þ

For a symmetric damage effect tensor M ijkl, the expression between the parentheses can be simplified

to yield the following (Lee et al., 1985; Voyiadjis and Guelzim, 1996; Voyiadjis and Kattan, 1999,

2006):

Y Ærs ¼ Àrij

oM ijmn

ouÆrs

E À1mnpqM pqklrkl: ð106Þ

The equivalent to Eq. (106) in terms of the effective stresses can be derived in a similar way to yield:

Y Ærs ¼ Àrvw

oM À1vwij

ouÆrs

M ijtuE À1tumnrmn: ð107Þ

6.2. The plastic part of the Helmholtz free energy function

The plastic part of the Helmholtz free energy function given in terms of the effective plastic strain

in order to derive the evolution of the hardening parameters used in the yield function in Eq. (39) is

given as follows:

qwp ¼ f þo eþep þ1

2hðeþepÞ2 þ f Ào eÀep þ Q eÀep þ

1

bexpðÀbeÀepÞ

: ð108Þ

Substituting Eq. (108) into Eqs. (31)e, f yields the following expressions for the plasticity conjugateforces c þ and c À:

c þ ¼ f þo þ heþep; ð109Þ

c À ¼ f Ào þ Q 1 À expðÀbeÀepÞ½ : ð110Þ

By taking the time derivative of Eqs. (109) and (110) one can easily retrieve Eq. (49) and its tensile

counterpart (presented below), representing the evolution equations for the hardening parameters

in tension and compression.

_c þ ¼ h_eþep: ð111Þ

6.3. The damage part of Helmholtz free energy function

The damage part of the Helmholtz free energy function is postulated to have the following form:

qwd ¼ K Æo uÆ

eq þ1

BÆð1 À uÆ

eqÞ lnð1 À uÆeqÞ þ uÆ

eq

n o ; ð112Þ

where K Æo is the initial damage threshold defined in Eq. (68) and B± are material constants which are

expressed in terms of the fracture energy and an intrinsic length scale, Eq. (69).

Substituting Eq. (112) into Eqs. (31)c, d, one can easily obtain the following expressions for the

damage driving forces K ±:

K

Æ

¼ K

Æ

o 1 À

1

BÆ lnð1 ÀuÆ

eqÞ

: ð113Þ

By taking the time derivative of the above expression one retrieves the rate form of the damage hard-

ening/softening function K ± presented in Eqs. (66) and (67) such that (with minor modification in Eq.

(66)):




_K Æ ¼K ÆoBÆ exp ÀBÆ 1 À

K Æ

K Æo

!" #_uÆ

eq: ð114Þ

This concludes the derivation of all the components constituting the elasto-plastic–anisotropic dam-

age material model for concrete.

7. Conclusions

A new model to predict the elasto-plastic–anisotropic damage behavior of concrete is derived in

this work by applying sound thermodynamic principles. Tensile and compressive damage in concrete

are characterized and accounted for by using: two damage tensors (tension/compression) that are

used to define two fourth order damage effect tensors for mapping the stresses from the effective

to the damaged configurations, different tensile and compressive damage hardening rules character-

izing damage evolution, and two (tensile/compressive) damage conjugate forces resulting from the

application of the strain energy equivalence hypothesis. These two damage release rates are used to

define two damage criteria, one for tension and the other for compression. The damage tensors are

defined at both the total and incremental forms and are stress path dependent.Concrete plasticity is formulated within the effective stress space with multiple hardening rules to

allow for different performances under tension and compression. The spectral decomposition tech-

nique is used here to resolve the stress tensor into tensile and compressive components in order to

facilitate the derivation of the above mentioned tensors. The verification of the model and its material

parameters, as well as its numerical aspects and coding algorithms, will be forthcoming.

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