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Journal ArticleDOI

A new thermodynamically consistent continuum model for hardening plasticity coupled with damage

01 Dec 2002-International Journal of Solids and Structures (Pergamon)-Vol. 39, Iss: 25, pp 6241-6271

AbstractA phenomenological model for hardening-softening elasto-plasticity coupled with damage is presented. Specific kinematic internal variables are used to describe the mechanical state of the system. These, in the hypothesis of infinitesimal changes of configuration, are partitioned in the sum of a reversible and an irreversible part. The constitutive equations, developed in the framework of the Generalised Standard Material Model, are derived for reversible processes from an internal energy functional, postulated as the sum of the deformation energy and of the hardening energy both coupled with damage, while for irreversible phenomena from a dissipation functional. Performing duality transformations, the conjugated potentials of the complementary elastic energy and of the complementary dissipation are obtained. From the latter a generalised elastic domain in the extended space of stresses and thermodynamic forces is derived. The model, which is completely formulated in the space of actual stresses, is compared with other formulations based on the concept of effective stresses in the case of isotropic damage. It is observed that such models are consistent only for particular choices of the damage coupling. Finally, the predictions of the proposed model for some simple processes are analysed.

Topics: Elastic energy (54%), Internal energy (53%), Phenomenological model (53%), Dissipation (53%), Constitutive equation (52%)

Summary (3 min read)

Introduction

  • The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
  • The main characteristics of the proposed model are summarised in the following.
  • First, an unified format for the evolution laws of any internal variable is obtained, and they all derive from a single potential of the driving forces.

2.1. State variables and ambient spaces

  • A simple material is considered and the constitutive relations are developed within the framework of the Generalised Standard Material Model (Germain, 1973; Halphen and Nguyen, 1975).
  • The assumption of local state is kept valid, i.e. the equilibrium state of a material point is assumed independent of the state of the neighbouring elements.
  • The state of the system is phenomenologically described assigning a set of internal variables and the related mechanisms for energy exchange, distinguishing the reversible phenomena that modify the stored energy and the irreversible ones that cause energy dissipation.
  • The last term in (2) represents the heat exchange while the other terms correspond to the mechanical power.
  • Since for any closed system the power of the internal variables a, x must be zero, their total value vanishes, i.e. their elastic and plastic parts are opposite (Kluitenberg, 1962; Ziegler, 1977).

2.2. Internal energy functional

  • The internal energy, dependent only on the reversible part of the kinematic variables, is postulated as the sum of a deformation energy /ðee;xe; seÞ and of a hardening energy wðae;xe; seÞ coupled with damage: eðee; ae;xe; seÞ ¼ /ðee;xe; seÞ þ wðae;xe; seÞ þ InXðxeÞ ð6Þ.
  • In the sequel, as isothermal processes are considered, the dependence on entropy or temperature will be dropped.
  • Furthermore, the function EðxeÞ can be assumed convex in xe.
  • The form (8) has been often used in mechanics, and is based on the idea of the ‘‘effective stress’’ r̂, first proposed by Kachanov (1958).
  • The operators M1, M2 are in general fourth order tensor (Hansen and Schreyer, 1994).

2.3. The dissipation functional

  • The irreversible behaviour is ruled by the dissipation potential d, that has to comply with the second principle of thermodynamics, stating the irreversibility of entropy production (Chaboche, 1999): q_sþ div q T r T P 0 ð22Þ where q is the material density, r the density of the internal heat production and q the heat flux for unit area.
  • Eq. (34) implies that the dissipation functional is equal to the support function of the convex domain K : dð _gpÞ ¼ suppK ð35Þ.
  • Furthermore, forms of the internal energy or of the dissipation potential different from those used in the paper can be applied, subjected only to the thermodynamic restrictions (28).
  • Additional degrees of freedom can be obtained if the exponent n of the damage law is assumed to be different for the elastic and the plastic moduli.

3. The admissible domain

  • In paragraph 2 it has been shown that the maximum dissipation principle (34) implies the existence of an elastic domain K of the generalised stresses that can be described by means of the yield functional (43).
  • Aim of this paragraph is to examine and compare some of them with the present model.
  • All the convex domains having their borders between the solid and the dashed lines in Fig. 4, corresponding to different choices of the dissipation d, are admissible.
  • Two distinct Lagrangian multipliers are thus introduced.
  • The admissible domain (53) in the uniaxial case is examined in Fig. 7 where the undamaged elastic path for uniaxial stress is represented.

4.1. The case of isotropic damage

  • The functions g1 and g2 are the classical expressions of the Mises and Drucker-Prager criteria with the addition of two variables: the isotropic hardening variable v, that rules the homothetic expansion of the plastic surface and the isotropic damage energy f, dual of the damage variable x, that describes the contraction of the domain when damage is active.
  • A functional dependency on the stresses can also be introduced to reproduce coupling phenomena of plasticity and fracture.
  • It is observed that the considered yield functions are all positively homogeneous, contrarily to the expressions obtained in the case an effective stress plastic criterion is used (see Eq. (53)).
  • The model is completed by the elastic relations Eq. (17).

4.2. Predictions of the model

  • The damage response of the model presented in Sections 2 and 4.1 is analysed considering some simple uniaxial processes.
  • The loading unloading behaviour is considered in Fig. 9b.
  • Due to the presence of damage the application of this model to structural problems will cause strain localisation with the consequent mesh-dependency of the numerical results.
  • The parameters c and n particularly influence the degradation of the elastic modulus.

4.3. Multiaxial compression processes: comparison with experimental results

  • The predictions of the model examined in Section 4.2 are compared to experimental data obtained from tests on confined concrete (Van Mier, 1984).
  • In order to avoid the occurrence of localisation, only the data in the hardening phase, before the peak, are used.
  • (Van Mier, 1984), from which the experimental relation elim1 rlim1 has been derived for comparison with the prediction of the model given by Eq. (65).
  • Fig. 18. Envelope of limit uniaxial elastic states for hardening isotropically damaged material.
  • Constant n. Fig. 21. Experimental degradation of the elastic modulus and model prediction for the test of Fig. 20.

5. Numerical algorithm

  • As it has been previously observed, the numerical implementation of the presented coupled plastic model is a generalisation of the standard algorithm for elastoplasticity with internal variables.
  • De, it can be found the algorithmic consistent tangent operator (Cuomo, submitted for publication) UepðeÞ ¼ sup s ½hg; si ecðsÞ dcðsÞ hg0p; si ð66Þ where the vector g0p collects the values of accumulated irreversible kinematic variables at the beginning of the step.
  • The second form of (67) derives from an augmented Lagrangian Regularisation (Bertsekas, 1982) that turns the inequality constraint gi6 0 into the equality constraint gl ¼ 0 and has several advantages.
  • This form is particularly useful in the case of corner points.
  • The coupled Mises criterion (56) has been used for the material with different values of the material constant c. Fig. 25 shows the total reaction versus the imposed vertical displacement for three different values of the damage parameter c (c ¼ 0 is equivalent to absence of damage) in plane strain.

6. Conclusions

  • An internal variable model of plasticity coupled with damage has been formulated in the framework of GSMM.
  • Various damaging material behaviours have been modelled, as plastic-hardening or cohesive fracturelike behaviour, depending on the choice of the dissipation functional, and therefore of the generalised elastic domain.
  • Consequently the evolution laws of any internal variable are obtained in a unified way.
  • These can be easily identified by means of usual experimental tests.
  • Several mechanical behaviours have been modelled, both in monotonic and in cyclic processes.

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A new thermodynamically consistent continuum model
for hardening plasticity coupled with damage
Loredana Contrafatto, Massimo Cuomo
To cite this version:
Loredana Contrafatto, Massimo Cuomo. A new thermodynamically consistent continuum model for
hardening plasticity coupled with damage. International Journal of Solids and Structures, Elsevier,
2002, 39, pp.6241-6271. �hal-00878757�


Nomenclature
State variables
Kinematic
e strain tensor
a hardening internal variable
x damage internal variable
s specific entropy
g ¼ðe; a; xÞ vector of mechanical kinematic variables
Dual static
r stress tensor
v hardening internal force
f damage internal force
T temperature
s ¼ðr; v; fÞ vector of mechanical static variables
ðÞ
e
reversible component of the generic kinematic variable
ðÞ
p
irreversible component of the generic kinematic variable
ðÞ
:
rate of the generic variable
r
0
deviator of the stress tensor
e
0
e
deviator of the elastic strain tensor
^
rr,
^
vv effective stress and thermodynamic force
Energy symbols
eðe
e
; a
e
; x
e
; s
e
Þ specific internal energy
f ðe
e
; a
e
; x
e
; s
e
Þ specific Helmoltz free energy
/ðe
e
; x
e
; s
e
Þ deformation potential
wða
e
; x
e
; s
e
Þ hardening potential
^
//ðe
e
; x
e
; s
e
Þ effective deformation potential
^
wwða
e
; x
e
; s
e
Þ effective hardening potential
dð
_
ee
p
;
_
aa
p
;
_
xx
p
;
_
ss
p
Þ specific dissipation
d
c
ðr; v; f; T Þ specific conjugated dissipation
P
vi
internal virtual power
r specific internal heat production
q heat flux
q material density
c dissipated heat power
Constitutive symbols
E elasticity tensor
H hardening tensor
M
1
effective stresses operator
M
2
effective hardening forces operator
E
0
, G
0
initial Young and shear moduli
Kðr; v; fÞ generalised elastic domain
Xðx
e
Þ set of the admissible values of the damage variable x
e
R
set of non-positive real numbers
6242 L. Contrafatto, M. Cuomo / International Journal of Solids and Structures 39 (2002) 6241 6271

Object of this contribution is the analysis of a fully coupled phenomenological theory of plasticity with
hardening and damage. Damage is phenomenologically understood as a degradation both of the elastic
stiffness of the material (and, eventually, of its micro-structure) and of its strength. However, as it will be
shown, the theory of damage can also include phenomena such as fracture evolution, as it has been recently
suggested (Paas et al., 1993). The proposed model follows the phenomenological approach because the
material behaviour is described through a suitable set of internal variables, whose relation to micro-
mechanical processes is not exactly defined. Particularly, a subset of internal variables is considered
responsible of the damage evolution. The nature of these variables is generally tensorial (Hansen and
Schreyer, 1994; Zhu and Cescotto, 1995), although they can be reduced to scalar quantities if only isotropic
damage is considered.
Coupling between damage and plasticity is usually introduced redefining the plastic relations for the so
called effective stresses (Kachanov, 1958; Cordebois and Sidoroff, 1982; Krajcinovic, 1984; Lemaitre and
Chaboche, 1985; Ju, 1989). Flow rules for the plastic strain rates are consequently obtained including the
damage variable as a parameter, which, in turn, is determined through a specific evolution law independent
of the plastic potential (Hansen and Schreyer, 1994; Simo and Ju, 1987; Klisinski and Mr
ooz, 1988; Marotti
de Sciarra, 1997). It is then needed to solve a parametric optimisation problem, whose convergence
properties are strongly dependent on the form chosen for the two potentials. A different thermodynamical
model has been recently formulated by Armero and Oller (2000); this uses a partition of the strain in elastic,
plastic, plus damage components, and employs the damaged stiffness directly as damage internal variables.
In this paper a new phenomenological model for a class of elastic plastic damaging materials, that
deviates from those mentioned above is presented. The main characteristics of the proposed model are
summarised in the following. The framework of the Generalised Standard Material Model (Germain, 1973;
Halphen and Nguyen, 1975) is adopted, so that the model is defined through the specification of two
functionals of the kinematic variables, ruling the reversible and irreversible phenomena respectively. A
general scheme is developed, based on duality. All internal variables are consistently decomposed in a
reversible and an irreversible component, the first being responsible for the stored internal energy, the
second generating the internal dissipation. The second principle of thermodynamics is satisfied a priori
thanks to the hypotheses introduced for the structure of the dissipation functional. Full coupling with
damage is allowed both in the free energy and in the dissipation. Applying duality transformations
(Rockafellar, 1970), conjugated potentials of the mechanical variables dual to the kinematic variables are
obtained, resulting in the generalised complementary elastic energy and in the complementary dissipation
functional. The latter one is of fundamental importance, since its differential yields the evolution laws for
the rate of internal variables. According to the choice of the dissipation functional, and therefore of the
generalised elastic domain, coupling between plasticity and damage can be easily modelled and various
kinds of behaviour, such as hardening softening transition, cohesive fracture-like behaviour, etc., can be
recovered, with no need of introducing ad-hoc evolution laws.
Two major consequences of this approach are stressed. First, an unified format for the evolution laws of
any internal variable is obtained, and they all derive from a single potential of the driving forces. Secondly,
R
þ
set of non-negative real numbers
gðsÞ yield function
n real exponent of isotropic damage law
p real coupling damage parameter
k plastic multiplier
l penalty parameter
L. Contrafatto, M. Cuomo / International Journal of Solids and Structures 39 (2002) 6241 6271 6243

the model implies the existence of a single elastic domain in the extended space of the actual stresses and of
the thermodynamic forces dual to the hardening and damage internal variables. A natural extension of
DruckerÕs principle applies to the generalised elastic domain, so that the postulate of maximum dissipation
is fulfilled. Therefore satisfaction of the evolution laws is obtained solving a single optimisation problem,
which turns out to be a straightforward generalisation of the one used in perfect plasticity.
The main characteristics of the model are first introduced and discussed without specifying any par-
ticular form for the governing functionals nor for the internal variables. Subsequently, as an exemplifi-
cation, a very simple isotropic damage model is introduced similar to the one proposed by Lemaitre (1996)
and some peculiar features of this model are presented, underlying the differences with the traditional
damage models. It will be shown how the present proposal needs very few material constants, and that their
experimental determination is based on the response of the material to simple fundamental tests. This result
is due to the derivation of the coupled model from properly defined thermodynamic potentials, so that the
constitutive evolution laws are not postulated but derived as a consequence of energy balances. The paper is
concluded by an application to a structural problem.
As any damage/softening local model, also the proposed one suffers for the strain localisation problem,
due to the loss of ellipticity of the tangent operator. Some of the strategies that have been presented in the
literature to overcome this problem can be easily adapted to the proposed model, but the matter is beyond
the scopes and the limits of the present paper.
2. Constitutive model
2.1. State variables and ambient spaces
A simple material is considered and the constitutive relations are developed within the framework of the
Generalised Standard Material Model (Germain, 1973; Halphen and Nguyen, 1975). The assumption
of local state is kept valid, i.e. the equilibrium state of a material point is assumed independent of the state
of the neighbouring elements. The state of the system is phenomenologically described assigning a set of
internal variables and the related mechanisms for energy exchange, distinguishing the reversible phenomena
that modify the stored energy and the irreversible ones that cause energy dissipation.
The following kinematic variables and the associated dual mechanic variables, which are defined in the
adjoint spaces (denoted by a prime), are considered:
e 2 D macroscopic strain
r 2 D
0
stress
a 2 I hardening
v 2 I
0
hardening internal forces
x 2 C damage
f 2 C
0
damage driving forces
s 2 R entropy
T 2 R temperature
ð1Þ
Besides the deformation e, two sets of internal variables are introduced, acting at the micro-structural
level: the variables a which describe the hardening mechanisms and the variables x which measure the
degradation of the material integrity and account for the decay of the elastic and hardening stiffness and of
the strength of the material. Although some physical interpretation is possible, no link of these variables to
any mechanism is attempted, leaving it to a specific implementation of the general model.
To each pair of conjugated variables a duality product is associated, which generates the following
bilinear form of the internal virtual power:
P
vi
¼hr;
_
eeiþhv;
_
aaiþhf;
_
xxiþhT ;
_
ss2Þ
6244 L. Contrafatto, M. Cuomo / International Journal of Solids and Structures 39 (2002) 6241 6271

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References
More filters

Book
23 Sep 2014

3,794 citations


"A new thermodynamically consistent ..." refers background in this paper

  • ...…ð67Þ where l is an arbitrary fixed positive (penalty) parameter and gl ¼ max gi; ki l l 2 R ð68Þ The second form of (67) derives from an augmented Lagrangian Regularisation (Bertsekas, 1982) that turns the inequality constraint gi6 0 into the equality constraint gl ¼ 0 and has several…...

    [...]

  • ...(41), only one multiplier is needed, so that efficient algorithms can be used for overcoming the numerical difficulties at corner points (Bertsekas, 1982)....

    [...]


Book
01 Aug 1992
Abstract: 1 Phenomenological Aspects of Damage.- 1.1 Physical Nature of the Solid State and Damage.- 1.1.1 Atoms, Elasticity and Damage.- 1.1.2 Slips, Plasticity and Irreversible Strains.- 1.1.3 Scale of the Phenomena of Strain and Damage.- 1.1.4 Different Manifestations of Damage.- 1.1.5 Exercise on Micrographic Observations.- 1.2 Mechanical Representation of Damage.- 1.2.1 One-Dimensional Surface Damage Variable.- 1.2.2 Effective Stress Concept.- 1.2.3 Strain Equivalence Principle.- 1.2.4 Coupling Between Strains and Damage Rupture Criterion Damage Threshold.- 1.2.5 Exercise on the Micromechanics of the Effective Damage Area.- 1.3 Measurement of Damage.- 1.3.1 Direct Measurements.- 1.3.2 Variation of the Elasticity Modulus.- 1.3.3 Variation of the Microhardness.- 1.3.4 Other Methods.- 1.3.5 Exercise on Measurement of Damage by the Stress Amplitude Drop.- 2 Thermodynamics and Micromechanics of Damage.- 2.1 Three-Dimensional Analysis of Isotropic Damage.- 2.1.1 Thermodynamical Variables, State Potential.- 2.1.2 Damage Equivalent Stress Criterion.- 2.1.3 Potential of Dissipation.- 2.1.4 Strain-Damage Coupled Constitutive Equations.- 2.1.5 Exercise on the Identification of Material Parameters.- 2.2 Analysis of Anisotropic Damage.- 2.2.1 Geometrical Definition of a Second-Order Damage Tensor.- 2.2.2 Thermodynamical Definition of a Fourth-Order Damage Tensor.- 2.2.3 Energetic Definition of a Double Scalar Variable.- 2.2.4 Exercise on Anisotropic Damage in Proportional Loading.- 2.3 Micromechanics of Damage.- 2.3.1 Brittle Isotropie Damage.- 2.3.2 Ductile Isotropie Damage.- 2.3.3 Anisotropie Damage.- 2.3.4 Microcrack Closure Effect, Unilateral Conditions.- 2.3.5 Damage Localization and Instability.- 2.3.6 Exercise on the Fiber Bundle System.- 3 Kinetic Laws of Damage Evolution.- 3.1 Unified Formulation of Damage Laws.- 3.1.1 General Properties and Formulation.- 3.1.2 Stored Energy Damage Threshold.- 3.1.3 Three-Dimensional Rupture Criterion.- 3.1.4 Case of Elastic-Perfectly Plastic and Damageable Materials.- 3.1.5 Identification of the Material Parameters.- 3.1.6 Exercise on Identification by a Low Cycle Test.- 3.2 Brittle Damage of Metals, Ceramics, Composites and Concrete.- 3.2.1 Pure Brittle Damage.- 3.2.2 Quasi-brittle Damage.- 3.2.3 Exercise on the Influence of the Triaxiality on Rupture.- 3.3 Ductile and Creep Damage of Metals and Polymers.- 3.3.1 Ductile Damage.- 3.3.2 Exercises on the Fracture Limits in Metal Forming.- 3.3.3 Creep Damage.- 3.3.4 Exercise on Isochronous Creep Damage Curves.- 3.4 Fatigue Damage.- 3.4.1 Low Cycle Fatigue.- 3.4.2 Exercise on Creep Fatigue Interaction.- 3.4.3 High Cycle Fatigue.- 3.4.4 Exercise on Damage Accumulation.- 3.5 Damage of Interfaces.- 3.5.1 Continuity of the Stress and Strain Vectors.- 3.5.2 Strain Surface Energy Release Rate.- 3.5.3 Kinetic Law of Debonding Damage Evolution.- 3.5.4 Simplified Model.- 3.5.5 Exercise on a Debonding Criterion for Interfaces.- 3.6 Table of Material Parameters.- 4 Analysis of Crack Initiation in Structures.- 4.1 Stress-Strain Analysis.- 4.1.1 Stress Concentrations.- 4.1.2 Neuter's Method.- 4.1.3 Finite Element Method.- 4.1.4 Exercise on the Stress Concentration Near a Hole.- 4.2 Uncoupled Analysis of Crack Initiation.- 4.2.1 Determination of the Critical Point(s).- 4.2.2 Integration of the Kinetic Damage Law.- 4.2.3 Exercise on Fatigue Crack Initiation Near a Hole.- 4.3 Locally Coupled Analysis.- 4.3.1 Localization of Damage.- 4.3.2 Postprocessing of Damage Growth.- 4.3.3 Description and Listing of the Postprocessor DAMAGE 90.- 4.3.4 Exercises Using the DAMAGE 90 Postprocessor.- 4.4 Fully Coupled Analysis.- 4.4.1 Initial Strain Hardening and Damage.- 4.4.2 Example of a Calculation Using the Finite Element Method.- 4.4.3 Growth of Damaged Zones and Macrocracks.- 4.4.4 Exercise on Damage Zone at a Crack Tip.- 4.5 Statistical Analysis with Microdefects.- 4.5.1 Initial Defects.- 4.5.2 Case of Brittle Materials.- 4.5.3 Case of Quasi Brittle Materials.- 4.5.4 Case of Ductile Materials.- 4.5.5 Volume Effect.- 4.5.6 Effect of Stress Heterogeneity.- 4.5.7 Exercise on Bending Fatigue of a Beam.- History of International Damage Mechanics Conferences.- Authors and Subject Index.

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01 Jan 1996
Abstract: Keywords: Rheologie ; Milieux continus ; Plasticite ; Fissuration Reference Record created on 2004-09-07, modified on 2016-08-08

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Journal ArticleDOI
Abstract: Continuum elastoplastic damage models employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress] -based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence , and plastic flow is introduced by means of an additive split of the stress [strain] tensor . In a strain -based formulation we redefine the equivalent strain , usually defined as the J 2 -norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress -based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain -based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain-based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.

1,074 citations


"A new thermodynamically consistent ..." refers background in this paper

  • ...…the plastic strain rates are consequently obtained including the damage variable as a parameter, which, in turn, is determined through a specific evolution law independent of the plastic potential (Hansen and Schreyer, 1994; Simo and Ju, 1987; Klisinski and Mr oz, 1988; Marotti de Sciarra, 1997)....

    [...]

  • ...The form (17) of the generalised elastic relation recovers the one used by many authors (Simo and Ju, 1987; Ju, 1989)....

    [...]


Journal ArticleDOI
J.W. Ju1
Abstract: Novel energy-based coupled elastoplastic damage theories are presented in this paper. The proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials. At variance with Lemaitre's work on damage-elastoplasticity, the present formulation renders rational thermodynamic potential and damage energy release rate. In contrast to previous work by Simo and Ju (featuring an additive split of the stress tensor), current formulation assumes an additive split of the strain tensor. It is shown that the “strain split” damage-elastoplasticity formulation leads to more robust tangent moduli than the “stress split” formulation. The plastic flow rule and hardening law are characterized in terms of the effective quantities; viz. the effective stress space plasticity . This mechanism is both physically well-motivated and computationally efficient. Further, a fourth-order anisotropic damage mechanism is proposed for brittle materials. Rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain-rate dependency of microcrack growth. Efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the “operator splitting” methodology. In particular, new three-step operator split algorithms are presented Application is made to a class of inviseid and rate-dependent cap-damage models for concrete and mortar. Experimental validations are also given to illustrate the applicability of the proposed damage models.

809 citations