The Numerical Analysis of the Intrinsic Anisotropic Microdamage Evolution in Elasto-Viscoplastic...

http://ijd.sagepub.com

Mechanics International Journal of Damage

DOI: 10.1177/1056789508097543 Nov 21, 2008;

2009; 18; 205 originally published onlineInternational Journal of Damage MechanicsAdam Glema, Tomasz Lodygowski, Wojciech Sumelka and Piotr Perzyna

Evolution in Elasto-Viscoplastic SolidsThe Numerical Analysis of the Intrinsic Anisotropic Microdamage

http://ijd.sagepub.com/cgi/content/abstract/18/3/205 The online version of this article can be found at:

Published by:

http://www.sagepublications.com

at:can be foundInternational Journal of Damage Mechanics Additional services and information for

http://ijd.sagepub.com/cgi/alerts Email Alerts:

http://ijd.sagepub.com/subscriptions Subscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.co.uk/journalsPermissions.navPermissions:

http://ijd.sagepub.com/cgi/content/refs/18/3/205 Citations

by Wojciech Sumelka on April 6, 2009 http://ijd.sagepub.comDownloaded from

http://ijd.sagepub.com/cgi/alerts

http://ijd.sagepub.com/subscriptions

http://www.sagepub.com/journalsReprints.nav

http://www.sagepub.co.uk/journalsPermissions.nav

http://ijd.sagepub.com/cgi/content/refs/18/3/205


The Numerical Analysis of theIntrinsic Anisotropic Microdamage

Evolution in Elasto-Viscoplastic Solids

ADAM GLEMA, TOMASZ LODYGOWSKI AND WOJCIECH SUMELKA

Poznan University of Technology, Institute of StructuralEngineering ul. Piotrowo 5, 60–965 Poznan, Poland

PIOTR PERZYNA*

Institute of Fundamental Technological Research, PolishAcademy of Sciences Swie, tokrzyska 21, 00–049 Warsaw, Poland

ABSTRACT: The objective of the present article is to show the formulation forelastic-viscoplastic material model accounting for intrinsic anisotropic microdamage.The strain-induced anisotropy is described by the evolution of the intrinsic micro-damage process – defined by the second-order microdamage tensor. The first stepof the possibility of identification procedure (calibration of parameters) are alsoaccounted and illustrated by numerical examples.

KEY WORDS: microdamage, anisotropy.

INTRODUCTION

IN RECENT YEARS, there has been great activity in the field of continuummechanics to describe the influence of anisotropic effects on fracture

phenomena. This trend in research has been particularly valuable in thecontinuum damage mechanics, cf. Boehler (1990) and Voyiadjis et al. (1998).Analysis of recent experimental results concerning investigations of fracturephenomena under dynamic loading processes suggests that there are threemain reasons for anisotropic effects: (i) The strain-induced anisotropy iscaused by the residual type stresses which result from the heterogeneous

International Journal of DAMAGE MECHANICS, Vol. 18—April 2009 205

1056-7895/09/03 0205–27 $10.00/0 DOI: 10.1177/1056789508097543� SAGE Publications 2009

Los Angeles, London, New Delhi and Singapore

*Author to whom correspondence should be addressed. E-mail: [email protected] 6–9 appear in color online: http://ijd.sagepub.com



nature of the plastic deformation in polycrystalline materials. Experimentalevidence indicates that yield surfaces exhibit anisotropic hardening.Subsequent yield surfaces are both translated and deformed in stress space.In the phenomenological description this kind of anisotropy is modeled bythe shift of the yield surface in stress space. This shift of the yield surfacemight be described by the residual stress tensor (the back stress) a introduceas an internal state variable, Perzyna (2008). (ii) The anisotropy caused by theformation of localized shear bands and the evolution of microdamage processalong the localized regions. (iii) The anisotropy induced by the evolutionof the intrinsic microdamage process – described by the second-ordermicrodamage tensor m.

Experimental observations have shown that in the microdamage processthe generated anisotropy is a consequence of rather random phenomenaconnected with some directional property of the formation of microcracks.This anisotropic effect is very much affected by the crystallographicstructure of a material as well as by small fluctuations (due to waveinteractions) of the applied stress tensor at a particular point of a bodyduring the dynamic process.

The main objective of the present article is to show the formulation of theelastic-viscoplastic material model accounting for intrinsic microdamageanisotropy represented by second-order tensor. The first step of thepossibility of identification procedure (calibration of parameters) are alsoaccounted and illustrated by numerical examples. In the procedure, thestrain-induced anisotropy (the influence of the back stress a) as well asthermal effects are omitted. This procedure is based on experimentalobservations (cf. the review articles by Curran et al. (1987) and Meyers andAimone (1983)) for OFHC copper.

The model is developed within the thermodynamic framework of the ratetype covariance constitutive structure with a finite set of the internal statevariables. A set of internal state variables consists of one scalar and twotensors, namely equivalent inelastic deformation 2p, the second ordermicrodamage tensor m with the physical interpretation that ðm : mÞ1=2 ¼ �defines the volume fraction porosity and the residual stress tensor (the backstress) a. The equivalent inelastic deformation 2p describes the dissipationeffects generated by viscoplastic flow phenomena, the microdamage tensorm takes into account the anisotropic intrinsic microdamage mechanismsand the back stress tensor a aims at the description of dissipativeeffects caused by the kinematic hardening. To describe suitably the influenceof both induced anisotropy effects and the stress triaxiality observedexperimentally the new kinetic equations for the microdamage tensor m andfor the back stress tensor a are proposed (cf. Perzyna and Voyiadjis, 2005;Glema et al., 2005; Perzyna, 2008).

206 A. GLEMA ET AL.



The relaxation time is used as a regularization parameter. By assuming thatthe relaxation time tends to zero the thermo-elasto-plastic (rate independent)response of the damage material is obtained. A fracture criterion based onthe evolution of the anisotropic intrinsic microdamage is formulated.Particular attention is focused on the identification of the materialfunctions and constants involved in the evolution (growth) equation for themicrodamage tensor m.

It seems that this thermo-elasto-viscoplastic model with intrinsicanisotropic microdamage may be used for the description of importantproblems in modern manufacturing processes, and particularly for meso andnanomechanical issues. Other reserchers have also worked on coupleddamage and viscoplastic models (cf. Voyiadjis and Dorgan, 2004; Voyiadjiset al., 2004; Voyiadjis and Abu Al-Rub, 2005, 2006; Voyiadjis et al., 2003).

EXPERIMENTAL AND PHYSICAL MOTIVATION

For our considerations of anisotropic effects, the metallurgical aspects offracture and spalling have great importance. Effects of parameters such asgrain size and substructure, crack/void initiation sites, micromechanicalaspects of growth, and coalescence can influence fracture and spallingmechanisms. To show the complexity of the anisotropy phenomena in themicrodamage process we shall discuss several available experimental results.

The most popular dynamical experiment1 in the investigation of thefracture phenomenon in metals is a plate-impact configuration system.This experimental system consists of two plates, a projectile plane plateimpacts against a target plane. This is a good example of a dynamicdeformation process. If impact velocity is sufficiently high the propagationof a plastic wave through the target is generated. The reflection andinteraction of waves result in a net tensile pulse in the target plate. If thisstress pulse has sufficient amplitude and sufficient time duration, it willcause separation of the material and a spalling process.

The reason for choosing this particular kind of dynamical experiment isthat postshot photomicrographic observations of the residual porosity areavailable, and the stress amplitude and pulse duration can be performed suffi-ciently great to produce substantial porosity and the spall of the target plate.

The experimental data presented by Seaman et al. (1976) illustrate thedamage phenomena and provide a common basis for considering damagecriteria. They have used a plate-impact configuration system. Following thecompression waves resulting from the impact, rarefaction waves have

1For a thorough discussion of the experimental and theoretical works in the field of dynamic fracture andspalling of metals please consult the review papers by Meyers and Aimone (1983) and Curran et al. (1987),cf. also Meyers (1994) and Grebe et al. (1985).

Intrinsic Anisotropic Microdamage in Elasto-Viscoplastic Solids 207



intersected near the middle of the target plate to cause damage in the form ofnearly spherical voids. The heaviest damage is localized in a narrow zone,which is called the spall plane. Both the number and the size of voids decreasewith distance from this zone. This type of damage is termed ductile fracturebecause of high ductility (ability to flow) required of the platematerial (Figure 1).

0.236cm

154.2m/sec

0.635cm

Sectioning plane

Impactdirection

(a)

(b)

(c)

200m

625m

Figure 1. A cross-section of an aluminum target plate that has undergone a planar impact byanother aluminum plate (after Seaman et al., 1976).

208 A. GLEMA ET AL.



The final damage of the target plate (aluminium 1145) for a constantshot geometry but for different impact velocities has been performed byBarbee et al. (1972). The results suggest dependence of spalling process onthe pulse amplitude, Figure 2.

Spalling by ductile void formation in nickel is characterized byellipsoidal cracks with large axes perpendicular to the direction of theapplied tensile stress. A more advanced microdamage process has been

200m

Impact velocity–423 fps Impact velocity–433 fps

Impact velocity–468 fps Impact velocity–506 fps

Impact velocity–668 fps

GP-7456-23

Figure 2. The final damage of the aluminum 1145 target plate for a constant shot geometrybut for different impact velocities (after Barbee et al., 1972).




shown in Figure 3, where crack nucleation growth, and coalescencemechanisms in nickel have been observed.

An example of brittle fracture for Armco iron is presented in Figure 4(cf. Curran et al., 1981). It shows the polished cross-section through theplate impact specimen with very well visible cleavage (penny shape)microcracks. The damage, which appears as randomly orientedplanar microcracks, depends on the impact velocity as well as onthe duration of the tensile wave. The second property is directly observedfrom the results presented in Curran et al. (1977). Use of a taperedflyer results in longer tensile impulses at the thicker end. These longer pulseslead to greater damage in the Armco iron target (the inset givesto approximate durations of the tensile pulses). The damage observed inthis experiment is termed brittle, although the microcrack growth is muchslower than elastic crack velocities, indicating considerable plastic flowat micro crack tips.

Adiabatic shear band localization during dynamic process can be anothergood example of anisotropic effects. Shear bands form in one direction,which is determined by the state of stress and the properties of the materialof a body, as well as by the boundary conditions.

Grebe et al. (1985) conducted ballistic impact experiments on 12.5mmthick commercial purity titanium and T6Al4V alloy plates using steelprojectiles with 10.5mm diameter. The impact velocities in their experimentsvaried between 578 and 846m/s. The microstructural damage mechanisms

100μm

Figure 3. Void nucleation, growth, and coalescence observed in nickel (after Shockeyet al., 1973).

210 A. GLEMA ET AL.



associated with shear band formation, shock wave, and dynamic fracturewere investigated by optical and scanning and transmission electronmicroscopy. The shear bands were found along the two sides of thecross-section passing through the axis of the projectile. The measuredshear band width in T6A14V varied between 1 and 10 mm. Observationsof the onset of fracture along the shear band were also conducted. Sphericaland ellipsoidal microcracks in T6A14V were found along the bands,Figure 5. The mechanism of final failure in T6A14V is a simple propagationof a macrocrack along the damaged material within the shear band region.In the explanation of the phenomenon of fracture along the shear banda very important role has the microdamage process which consists of thenucleation, growth and coalescence of microvoids.

The investigations reported byGrebe et al. (1985) indicated that in dynamicprocesses the shear band regions behave differently than adjacent zones.Within the shear band region the deformation process is characterized by verylarge strains (shear strains over 100%) and very high strain rates (103–105 s�1).

Figure 4. Internal cleavage (penny shape) microcracks caused by shock loading in thepolished cross-section of an Armco iron specimen (after Curran et al., 1981).




The strain rate sensitivity of a material becomes a very important featureof the shear band region and the microdamage process is intensified.

THERMO-ELASTO-VISCOPLATIC MODEL OF A MATERIAL

Basic Assumptions and Definitions

Let us assume that a continuum body is an open bounded set B � IR3,and let � : B! S be a C1 configuration of B in S. The tangent of � isdenoted F ¼ T� and is called the deformation gradient of �.

Let fXAg and fxag denote coordinate systems on B and S, respectively.Then we refer to B � IR3 as the reference configuration of a continuum

50mm25mm

25mm

10mm

(a)

(c)(d)

(b)

Figure 5. Shear band in Ti6Al4V target impacted at 846 m/s (after Grebe et al., 1985):(a) single shear band; (b) microcracks in the shear band region; (c) elongated macrocracksalong the shear band; (d) characteristic dimples observed in spall region.

212 A. GLEMA ET AL.



body with particles X 2 B and to S ¼ �ðBÞ as the current configuration withpoints x 2 S. The matrix FðX, tÞ ¼ @�ðX, tÞ=@X with respect to the coordinatebases EAðXÞ and eaðxÞ is given by

F aAðX, tÞ ¼

@�a

@XAðX, tÞ, ð1Þ

where a mapping x ¼ �ðX, tÞ represents a motion of a body B.We consider the local multiplicative decomposition

F ¼ Fe � Fp, ð2Þ

where ðFeÞ�1 is the deformation gradient that releases elastically the stress

on the neighborhood �ðNðXÞÞ in the current configuration.Let us define the total and elastic Finger deformation tensors

b ¼ F � FT, be ¼ Fe � FeT , ð3Þ

respectively, and the Eulerian strain tensors as follows

e ¼1

2ðg� b�1Þ, ee ¼

1

2ðg� be

�1

Þ, ð4Þ

where g denotes the metric tensor in the current configuration.By definition2

ep ¼ e� ee ¼1

2ðbe

�1

� b�1Þ ð5Þ

we introduce the plastic Eulerian strain tensor.To define objective rates for vectors and tensors we use the Lie derivative.3

Let us define the Lie derivative of a spatial tensor field t with respect to thevelocity field � as

L�t ¼ ��@

@tð��tÞ, ð6Þ

where �� and �� denote the pull-back and push-forward operations,respectively.

The rate of deformation tensor is defined as follows

d[ ¼ L�e[ ¼

1

2L�g ¼

1

2gac�

cjb þ gcb�cjað Þea � eb, ð7Þ

2For precise definition of the finite elasto-plastic deformation see Perzyna (1995).3The algebraic and dynamic interpretations of the Lie derivative have been presented by Abraham andMarsden (1978), cf. also Marsden and Hughes (1983).




where the symbol [ denotes the index lowering operator and � the tensorproduct,

�ajb ¼@�a

@xbþ �abc�

c, ð8Þ

and �abc denotes the Christoffel symbol for the general coordinate systemsfxag. The components of the spin u are given by

!ab ¼1

2gac�

cjb � gcb�cjað Þ ¼

1

2

@�a@xb�@�b@xa

� �: ð9Þ

Similarly

de[

¼ L�ee[ , dp

[

¼ L�ep[ : ð10Þ

We assume the additive form of deformation (Lee, 1969)

d ¼ de þ dp: ð11Þ

Let � denote the Kirchhoff stress tensor related to the Cauchy stress tensor p by

s ¼ Jr ¼�Ref

�r, ð12Þ

where the Jacobian J is the determinant of the linear transformationFðX, tÞ ¼ ð@=@XÞ�ðX, tÞ, �RefðXÞ and �ðx, tÞ denote the mass density in thereference and current configuration, respectively.

The Lie derivative of the Kirchhoff stress tensor s 2 T2ðSÞ (elements ofT2ðSÞ are called tensors on S, contravariant of order 2) gives

L�s ¼ ��@

@tð��sÞ

¼ F �@

@t½F�1 � ðs � �Þ � F�1

T

� � FT

� �� 1

¼ _s� ðdþ uÞ � s� s � ðdþ uÞT,

ð13Þ

where � denotes the composition of mappings. In the coordinate system (13)reads

ðL�sÞab¼ F a

A

@

@tðF �1c

A� cd F �1d

BÞF b

B

¼@� ab

@tþ@� ab

@xc�c � � cb @�

a

@xc� � ac @�

b

@xc:

ð14Þ

214 A. GLEMA ET AL.



Equation (14) defines the Oldroyd rate of the Kirchhoff stress tensor q(cf. Oldroyd, 1950).

Constitutive Postulates

Let us assume that: (i) conservation of mass, (ii) balance of momentum,(iii) balance of moment of momentum, (iv) balance of energy, (v) entropyproduction inequality hold.

We introduce the four fundamental postulates:

1. Existence of the free energy function. It is assumed that the free energyfunction is given by

¼ ðe,F,#; kÞ, ð15Þ

where e denotes the Eulerian strain tensor, F is deformation gradient,# temperature, and k denotes a set of the internal state variables. Toextend the domain of the description of the material properties andparticularly to take into consideration different dissipation effects wehave to introduce the internal state variables represented by the vector k.

2. Axiom of objectivity (spatial covariance). The constitutive structureshould be invariant with respect to any diffeomorphism (any motion)� : S! S (cf. Marsden and Hughes, 1983). Assuming that � : S! S is aregular, orientation preserving map transforming x into x0 and T� is anisometry from TxS to Tx0S, we obtain the axiom of material frameindifference (cf. Truesdell and Noll, 1965).

3. The axiom of the entropy production. For any regular motion of a bodyB the constitutive functions are assumed to satisfy the reduced dissipationinequality

1

�Refs : d� ð _#þ _ Þ �

1

�#q � grad# 0, ð16Þ

where �Ref and � denote the mass density in the reference and actualconfiguration, respectively, q is the Kirchhoff stress tensor, d the rate ofdeformation, is the specific (per unit mass) entropy, and q denotesthe heat flow vector field. Marsden and Hughes (1983) proved thatthe reduced dissipation inequality (16) is equivalent to the entropyproduction inequality first introduced by Coleman and Noll (1963) in theform of the Clausius–Duhem inequality. In fact the Clausius–Duheminequality gives a statement of the second law of thermodynamics withinthe framework of mechanics of continuous media, cf. Duszek andPerzyna (1991a,b).




4. The evolution equation for the internal state variable vector k is assumedin the form as follows

L�k ¼ mðe,F,#, kÞ, ð17Þ

where the evolution function m has to be determined based on carefulphysical interpretation of a set of the internal state variables and analysisof available experimental observations.The determination of the evolution function m (in practice a finite set

of the evolution functions) appears to be the main problem of the modernconstitutive modeling.

For our practical purposes it is sufficient to assume that the internal statevector k has the form

k ¼ ð2p, m, aÞ, ð18Þ

where 2p is the equivalent viscoplastic deformation, i.e.,

2p¼

Z t

0

2

3dp : dp

� �1=2

dt, ð19Þ

m denotes the microdamage second order tensor and a denotes the residualstress (the back stress) – aims at the description of the kinematic hardeningeffects.

We assume the physical interpretation that ðm : mÞ1=2 ¼ m, which defines theclassical volume fraction porosity. Such an assumption allows for accountingthe complexity of possible porosity anisotropic evolution.When we introducethe second order tensorial measure for microstructure m, we obtain additionaldata comparing with previous scalar measure �. Now we can trace the pathofmicrodamage evolution, and also the velocity of microdamage.We are ableto measure in experiment the volume fraction porosity (norm of themicrodamage tensor) by using the conservation of mass law.

Let us introduce the plastic potential function f ¼ fðJ1, J2,#, kÞ, where J1,J2 denote the first two invariants of the Kirchhoff stress tensor ~s ¼ s� a.

Let us postulate the evolution equations as follows

dp ¼ �P, up ¼ �:, L�m ¼ ., L�a ¼ A, ð20Þ

where for an elasto-viscoplastic model of a material we assume (cf. Perzyna,1963, 1966, 1971, 1995)

� ¼1

Tm�

f

� 1

� �� , ð21Þ

216 A. GLEMA ET AL.



Tm denotes the relaxation time for mechanical disturbances, the isotropicwork-hardening–softening function is

¼ ð2p,#, mÞ, ð22Þ

� is the empirical overstress function, the bracket h�i defines the ramp function,

P ¼@f

@s

��m¼const

@f

@s

��

��

� ��1, ð23Þ

:, ., and A denote the evolution functions that has to be determined.

Constitutive Assumptions for the Plastic Spin

Let us postulate that : has the form4

: ¼ �ða � P� P � aÞ ð24Þ

where * denotes the scalar valued function of the invariants of the tensorsa, m, and P, and may depend on temperature #.

Intrinsic Microdamage Mechanisms

To take into consideration experimentally observed time-dependenteffects it is advantageous to use the proposition of the description of theintrinsic microdamage process presented by Perzyna (1986a,b) andDuszek-Perzyna and Perzyna (1994).

Let us assume that the intrinsic, anisotropic microdamage process consistsof the nucleation and growth mechanisms.5

Physical considerations (cf. Curran et al., 1987; Perzyna, 1986a,b) haveshown that the nucleation of microcracks in dynamic loading processeswhich are characterized by very short time duration is governed by thethermally-activated mechanism. On the other hand the growth mechanismof microcracks is determined by viscoplatic flow phenomena (cf. Johnson,1981; Perzyna, 1986a,b; Perzyna and Drabik, 1989, 2008; Dornowski andPerzyna, 2000, 2002). Based on this heuristic suggestions and taking into

4For a thorough discussion of the concept of the plastic spin and its constitutive description inphenomenological theories for macroscopic large plastic deformations please consult the critical reviewpaper by Van der Giessen (1991).5Recent experimental observation results (cf. Shockey et al., 1985) have shown that coalescence mechanism canbe treated as nucleation and growthprocess on a smaller scale. This conjecture simplifies verymuch the descriptionof the intrinsic microdamage process by taking account only of the nucleation and growth mechanisms.




account the influence of the stress triaxiality6 and anisotropic effects on thenucleation and growth mechanisms we assume as follows (Perzyna (2008)):

1. There exist two material functions

h ¼ hðs,#, 2p, m, aÞ and g ¼ gðs,#, 2p, m, aÞ; ð25Þ

2. There exist two threshold stresses that are approximated from experiments(Dornowski and Perzyna, 2006)

�n ¼ �nðm, a,#,2pÞ and �eq ¼ �eqðm, a,#, 2

pÞ, ð26Þ

for microcrack nucleation and growth, respectively;

3. The evolution equation for the microdamage tensor � is postulated inthe form

L�m ¼@h�

@s

1

Tm�

In�nðm,#, 2pÞ

� 1

� �þ@g�

@s

1

Tm�

Ig�eqðm,#, 2pÞ

� 1

� �: ð27Þ

We use the denotations as follows (functions h* and g* are introduced assupportive denotation)

@h�

@s¼

@h

@s

* +@h

@s

* +��1

and@g�

@s¼

@g

@s

� �@g

@s

� ��1

: ð28Þ

The tensorial function @h�=@s ¼ @h�=@sðhÞ describes the microcrack interac-tion for nucleation, while the tensorial function @g�=@s ¼ @g�=@sðgÞ representsthe microcrack interaction for growth process. We assume the denotations

In ¼ a1J1 þ a2ðJ02Þ

1=2þ a3ðJ

03Þ

1=3, ð29Þ

Ig ¼ b1J1 þ b2ðJ02Þ

1=2þ b3ðJ

03Þ

1=3, ð30Þ

for the stress intensity invariants for nucleation and growth, respectively,where ai and bi ði ¼ 1, 2, 3Þ are the material constants, J 02, J

03 are second and

third invariants of deviatoric part of the Kirchhoff stress tensor, respectively.The right-hand side of the Equation (27) determines the evolution

function ..

6An analysis of the experimental observations for cycle fatigue damage mechanics at high temperature ofmetals performed by Sidey and Coffin (1979) suggests that the intrinsic microdamage process does very muchdepend on the strain rate effects, the wave shape effects as well as on the stress triaxiality.

218 A. GLEMA ET AL.



Kinematic Hardening

To determine the evolution function A we shall follow some resultsobtained by Duszek and Perzyna (1991). The kinematic hardening evolutionlaw takes the form

L�a ¼1

Tm�

f

� 1

� ��

r1Pþ r2P : Q

~s : Qþ r3m : Q~sþ r3

P : Q

ð~s : QÞ=r2 þ ðm : QÞr3=r2m

,

ð31Þ

where r1, r2, and r3 are the material coefficients and

Q ¼@f

@sþ

@f

@m�@

@m

� �:@m

@s

@f

@sþ

@f

@m�@

@m

� �:@m

@s

��1

: ð32Þ

The kinetic law (31) represents the linear combination of the Prager andZiegler linear kinematic hardening rules and additionally depends linearlyon the microdamage tensor m.

Thermodynamic Restrictions and Rate Type Constitutive Equations

Suppose the axiom of the entropy production holds. Then the constitutiveassumption (15) and the evolution Equations (20) lead to the results as follows

s ¼ �Ref@

@e, ¼ �

@

@#, �

@

@k� L�k�

1

�#q � grad# 0: ð33Þ

The rate of internal dissipation is determined by

#i ¼ �@

@k� L�k ¼ �

@

@2p

ffiffiffi2

3

r !��

@

@m: .: ð34Þ

Operating on the stress relation (33)1 with the Lie derivative and keeping theinternal state vector constant, we obtain (cf. Duszek-Perzyna and Perzyna, 1994)

L�s ¼ Le : d� Lth _#� ðLe þ gsþ sgþWÞ : P½ �1

Tm�

f

� 1

� �� , ð35Þ

where

Le ¼ �Ref@2

@e2, Lth ¼ ��Ref

@2

@e@#, W ¼ � gs� sgð Þ : ag� gað Þ½ �: ð36Þ




Substituting _ into the energy balance equation and taking into account theresults (33)3 and (34) gives

�# _ ¼ �divqþ �#i: ð37Þ

Operating on the entropy relation (33)2 with the Lie derivative andsubstituting the result into (37) we obtain

�cp _# ¼ �divqþ #�

�Ref

@s

@#: dþ ��s : dp þ ��K : L�m, ð38Þ

where the specific heat

cp ¼ �#@2

@#2ð39Þ

and the irreversibility coefficients �* and �** are determined by

�� ¼ �@

@2p� #

@2

@#@2p

! ffiffiffi2

3

r1

s : P,

��K ¼ �@

@m� #

@2

@#@m

!:

ð40Þ

So, a set of the constitutive equations of the rate type has the form as follows

L�s¼Le : d�Lth _#� ðLeþgsþ sgþWÞ :P½ �1

Tm�

f

�1

� �� ,

�cp _#¼�divqþ#�

�Ref

@s

@#: dþ��s :Pþ��K :L�m,

L�m¼@h�

@s

1

Tm�

In�nðm,a,#,2pÞ

�1

� �þ@g�

@s

1

Tm�

Ig�eqðm,a,#,2pÞ

�1

� �:

ð41Þ

All the material functions and the material constants should be identifiedbased on available experimental data.

Fracture Criterion Based on the Evolution of Microdamage

We base the fracture criterion on the evolution of the microdamagetensor m. Let us assume that for

ðm : mÞ1=2 ¼ � ¼ �F ð42Þ

220 A. GLEMA ET AL.



catastrophe takes place (cf. Perzyna, 1984, 1986a,b), that is

¼ ð2p ,#, mÞ��ðm:mÞ1=2¼�F ¼ 0: ð43Þ

It means that for ðm : mÞ1=2 ¼ �F the material loses its carrying capacity.The condition (43) describes the main feature observed experimentallythat the load tends to zero at the fracture point.

It is noteworthy that the isotropic hardening–softening material function proposed in Equation (22) should satisfy the fracture criterion (43).

Assumption of the Material Functions for an Adiabatic Process

For the proper identification procedure we first make an assumptionof the material functions. The plastic potential function f is assumed in theform (cf. Perzyna, 1984a,b; Shima and Oyane, 1976)

f ¼ J 02 þ n1ð#Þ þ n2ð#Þðm : mÞ1=2 �

J21� �1=2

, ð44Þ

where

n1ð#Þ ¼ 0, n2ð#Þ ¼ n ¼ const: ð45Þ

The isotropic work-hardening–softening function is postulated as(cf. Perzyna, 1986a; Nemes and Eftis, 1993)

¼ ð2p ,#, mÞ ¼ sð#Þ � sð#Þ � 0ð#Þ½ � exp ��ð#Þ2p½ ��

1�ðm : mÞ1=2

�F

� � ð#Þ" #,

ð46Þ

where

sð#Þ ¼ �s �

��s #, 0ð#Þ ¼

�0 �

��0 #,

�ð#Þ ¼ �� #, ð#Þ ¼ � � ��#, # ¼#� #0#0

:ð47Þ

The overstress function �ððf=kÞ � 1Þ is proposed as

�f

� 1

� �¼

f

� 1

� �m

, ð48Þ




where due to the m assumed as the odd number the McCauley bracket canbe replaced by simple parenthesis.

For practical applications it would be reasonable to assume that we knowthe initial porosity

ðm : mÞ1=2jt¼0 ¼ �0, ð49Þ

and we take into consideration only growth mechanism for the evolutionof microcraks. Then the evolution equation for the microdamage tensor� has the form

L�m ¼@g�

@s

1

Tm�

Ig�eqðm,#, 2pÞ

� 1

� �, ð50Þ

where

@g�

@s¼

@g

@s

� �@g

@s

��

� ��1, ð51Þ

and the function of microdamage growth evolution and stress threshold takethe forms7

g ¼1

2fsgT½G�fsg,

�eqðm,#, 2pÞ ¼ c2ð#Þð1� ðm : mÞ1=2Þln

1

ðm : mÞ1=2

2sð#Þ � sð#Þ � 0ð#Þ½ �Fð�0, m,#Þ� �

,

c1ð#Þ ¼ const, c2ð#Þ ¼ const,

Fð�0, m,#Þ ¼�0

1� �0

1� ðm : mÞ1=2

ðm : mÞ1=2

� �2=3�

þ1� ðm : mÞ1=2

1� �0

� �2=3�

,

�Ig

�eqðm,#, 2pÞ� 1

� �¼

Ig�eq� 1

� �m

,

ð52Þ

where G is the material matrix that should be identified. For further detailsplease see Dornowski (1999) in which the author has studied the separatesingle void evolution, see also Dornowski and Perzyna (2000, 2002a,b, 2006)and Glema et al. (2006).

7For function g Voigt notation is used.

222 A. GLEMA ET AL.



The attention is paid on the growth function g. The simplified assumptionwas done, that the evolution of microdamage tensor takes place in theprincipal direction of stress state – in general case the evolution takes placein an arbitrary direction. This assumption enables us to write the explicitform of the growth function g as follows

g ¼1

2g11�

2I þ g22�

2II þ g33�

2III

� �, ð53Þ

and the gradient of g with respect to the stress field gives us the followingmatrix representation of a tensor

@g

@s¼

g11�I 0 0

0 g22�II 0

0 0 g33�III

264

375, ð54Þ

where �I, �II, �III are the principal values of Kirchhoff stress tensor.It is important to notice that the proposed form of the scalar-valued

function of the tensorial argument g (see Equation (52)1) has two mainfeatures:

. preserves the sign of stress component (‘direction’ of damage),

. is twice differentiable – so it is possible to obtain nontrivial gradient(@g=@s 6¼ 0).

The final form of the directional term of the microdamage evolutionfunction, in the Equation (51), indicates, that the evolution is induced bypositive stresses component only. Furthermore, the evolution is active whenthe process is active, what causes, that the microvoid closing-openingphenomenon of the microdamage is not included into the model.

The elastic range is assumed to be free of the influence of the initiallyexisting microdamage. As in the infinitesimal theory of elasticity we assumelinear properties of the material, i.e.,

Le ¼ 2�Iþ �ðg� gÞ ð55Þ

where � and � denote the Lame constants, and the thermal expansionmatrix is postulated as

Lth ¼ ð2�þ 3�Þ�g, ð56Þ

where � is the thermal expansion constant.




NUMERICAL EXAMPLES

Computational Algorithm

To present the idea, the numerical simulations presented below are made atthe level of the material point. The evolution of the intrinsic anisotropicdamage, modeled as a microdamage tensor m, is observed. All material func-tions, presented in the proceeding sections, which are functions of the volumefraction porosity � ¼ ðm : mÞ1=2, have undergo the anisotropic evolution.

The detailed flowchart of computational algorithm is shown in Box 1,where the following notations are used:

ð �� Þ – denotes values referred to the initial (t¼ 0) coordinate system,ð ~� Þ – denotes values referred to the coordinate system that coincidenceswith the principal direction of the stress tensor,

Each example is calculated for two cases, to indicate the differencesbetween the isotropic and anisotropic evolution of the microdamage growth.

First the analysis is prepared with artificial restriction that the evolutionof microdamage tensor is isotropic – simply by assumption that the term@g=@s is represented by scalar matrix during the analysis (the notion ‘scalarmatrix’ means the diagonal matrix with all components on diagonal equalone the other, but different from 1 – unit matrix). Such simplification allowsus to obtain the isotropy of microdamage tensor at all stages of materialmodification, so it was equivalent to scalar damage.

In the second case of analysis the numerical simulations were done strictlyby the flow chart defined in Box 1, so all anisotropy effects were observed.

Box 1. The flow chart of the evolution algorithm of the microdamage tensor m.

(i) Assume the distribution of the microdamage tensor m at the initial stage (isotropic oranisotropic)

m0 ¼

��11 ��12 ��13��21 ��22 ��23��31 ��32 ��33

24

35,

(ii) Assume the stress state at the material point and the method of the stress stateevolution,(iii) Compute the microdamage tensor increment

� ~mk ¼ �t 2symð~m �@ ~�

@xÞ þ

@ ~g�

@q

1

Tm

Ig�eq� 1

� �m ��k

,

(iv) Compute new microdamage tensor

~mk ¼ ~mk�1 þ� ~mk :

224 A. GLEMA ET AL.



It is also important to mention that independently of analysis type, theinitial microdamage tensor m0 was assumed to be isotropic at the initialstage, hence independent of the coordinate system.

Uniaxial Tension for an Isothermal Process

This example concerns the uniaxial tension test. The grate importanceof this numerical test is due to the fact, that the similar stress conditionsoccurs in a plate-impact configuration system.

The stress state was described by the following tensor

�k � s ¼600 0 00 0 00 0 0

24

35MPa,

where parameter � was incrementally changed during the analysis from 0 to1 (we can say that the process was controlled by force). As mentionedbefore, the initial microdamage tensor was assumed to be isotropic with thematerial porosity equal to �0 6� 10�4, Eftis et al. (1991), thus

�m0 ¼0:00034 0 0

0 0:00034 00 0 0:00034

24

35:

The results are shown in Figures 6 and 7 for material data (Table 1) foran AISI 4340 steel (taken partially from, Dornowski and Perzyna(2002a,b)). The parameters were chosen in the way that gives the solutionqualitatively similar to the results discussed in literature (cf. Dornowski,1999; Eftis et al., 1990, 1991a,b; Nemes et al., 1993). This similarity concernsthe shape of some characteristic curves e.g., �eq � � or � � �.

E+08(a) (b) (c)

6.8

6.4

6.0

5.6

5.2

E+08

IG8.0

6.0

4.0

2.0

0.0

E+08

E−02

5.2

5.2

5.2

5.2

5.20.0 0.2 0.4 0.6 0.8 1.0

λ0.0 0.2 0.4 0.6 0.8 1.0 0.05 0.1 0.15 0.20 0.25 0.30 0.35

λ ξ

teqteqteq

Figure 6. The material functions for uniaxial tension.




In Figure 6, the evolution of material functions is shown. Up to the stressstate when the evolution of microdamage tensor does not start yet (� � 0:8)it clear that �eq is constant. When loading process go further, hence � isbeing increased (functions �eq and Ig are crossing each other) the process ofthe evolution of the microdamage is started.

The plot in Figure 6(c) shows the evolution of �eq as a function ofprogressing microdamage. The character of this curve is in agreement withthe experimental results obtained by Chakrabarti and Spretnak (1975).

In Figure 7 the evolution of the components of the microdamage tensor min principal directions is shown. In Figure 7(a) the evolution of componentsand norm of the microdamage tensor for isotropic and anisotropic case arepresented. For isotropic case, after the initiation of the evolution processof material intrinsic structure, all the components grow proportionally.For anisotropic case only one component growths due to the fact that the

E−02

(a)

(c) (d)

(b)

E−03 E−03

E−05

0.4

0.4

0.0

−0.4−0.4

0.0

0.4 0.4

0.0

0.8 0.8

0.0

−0.8

0.0

−0.4

0.3

0.0 0.2 0.4 0.6

ComponentsNorm

AnisotropicIsotropic

0.8 1.0l

x

b c d

0.2

0.1

0.0

0.8

−0.8−0.8

0.0

2.0

−2.0−2.0

0.0

2.0 2.0

0.0

−2.0

0.0

Figure 7. The evolution of the components of the microdamage tensor � – principaldirections.

Table 1. An AISI 4340 steel.

s¼809 Mpa n1¼ 0 c2¼0.067 �F¼ 0.350¼ 598 MPa n2¼ 0.25 b1¼0.7 g11¼ 1�¼ 14 Tm¼ 2.5 ms b2¼0.7 g22¼ 1 ¼ 9 m¼ 1 b3¼0.25 g33¼ 1

226 A. GLEMA ET AL.



uniaxial stress state is enforced. Those observations are presented inFigure 7(b)–(d) were the principal values of microdamage tensor arecompared. At the end of the process the differences in evolution modes areconsiderably different.

It is important to mention that the comparison given in Figure 7 ispossible in one coordinate system, because the eigendirections of micro-damage tensor for isotropic case are arbitrary. Thus, for any analysis we canassume that they coincide to the eigendirection of anisotropy case.

Complex Stress State for an Isothermal Process

The second example presents the results obtained for the complex stressstate described by the tensor

lk � s ¼550 165 �330165 275 55�330 55 �165

24

35MPa:

As previously, the initial microdamage tensor was assumed to be isotropicwith material porosity equal to �0 6� 10�4, and material parameters werealso taken from Table 1. Once more two cases were calculated, for theisotropic and anisotropic microdamage growth.

In Figure 8, the evolution of material functions is shown. The starting pointof the process of the evolution of the microdamage is once more clearlyrecognized, and the characteristic shape of the function �eq � � is confirmed.

The anisotropic nature of the evolution of damage is more distinct in thiscase. FromFigure 9(b)–(d) one can notice that the microdamage takes its owndirections and components of microdamage tensor are changing dependenton stress state.

E+08(a)

6.5

5.5

4.5

3.50.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.20 1.60 2.00

λ λ ξ

(b)E+088.0

6.0

4.0

2.0

0.0

(c)E+08

E−02

6.5

5.5

4.5

3.5

IGteqteqteq

Figure 8. The material functions for complex stress state.




One should notice that in Figure 9(c) and (d) the dashed and continuoustriples could be presented, as in Figure 7(c) and (d), in one coordinatesystem, because the dashed one presents the eigenvalues of the isotropictensor. The reason of such presentation lays in conviction that thisdistinction better maps the directional nature of microdamage.

CONCLUSIONS

In this article, the complete thermo-elasto-viscoplastic constitutivemodel of a material including the effects of anisotropy induced by theevolution of intrinsic microdamage is formulated. The new internal statevariable that describes the directional nature of microdamage is proposed,and is described by the second-order tensor called microdamage tensor.The microdamage tensor is a generalization of the previously consideredscalar state variable – the volume fraction porosity.

The preliminary stage of the identification of the material functionsand constants involved in the constitutive equation were described.The identified material functions and constants had enabled to make thenumerical implementation of the proposed model.

The numerical simulations assumed the growth of the microdamageonly without any thermomechanical effects. The analysis was done at the

E−050.4

0.0

−0.4−0.4

E−040.4

0.0

−0.4−0.4

−0.4

0.0

0.4 0.4

0.0

0.0

0.4 0.4

0.0

−0.4

E−02

(a)

(c) (d)

(b)

1.6

1.2

0.0 0.2 0.4 0.6

Components

Norm

AnisotropicIsotropic

0.8 1.0l

x

b c d

0.8

0.4

0.0

E−034.0

−4.0−4.0

−4.0

0.0

0.4 4.0

0.0

0.0

Figure 9. The evolution of the components of the microdamage tensor � – principaldirections.

228 A. GLEMA ET AL.



level of the material point, and the directional (anisotropic) evolutionof microdamage for active process was shown. The computations hadconfirmed the nonuniform growth of the components of the microdamagetensor indicating the heterogenous nature of progressive microdamage.

ACKNOWLEDGMENTS

The support of Polish Ministry of Higher Education and Science undergrant N N519 419435 ‘The evolution of properties and failure criteria of mate-rials and structures under fast dynamic loadings’ is kindly acknowledged.

REFERENCES

Abraham, R. and Marsden, J.E. (1978). Foundations of Mechanics, 2nd edn, Addison–Wesley,Reading Mass.

Barbee, T.W., Seaman, L., Crewdson, R. and Curran, D. (1972). Dynamic Fracture Criteriafor Ductile and Brittle Metals, J. Mater., 7: 393–401.

Boehler, J.P. (ed.) (1990). Yielding, Damage, and Failure of Anisotropic Solids, In: Proc.IUTAM/ICM Symposium, Villard–de Lans, 24–28 August 1987, Mech. Eng. Public.Limited, London.

Chakrabarti, A.K. and Spretnak, J.W. (1975). Instability of Plastic Flow in the Directionsof Pure Shear, II Experimental, Metallurgical Transactions, 6A: 733–736.

Coleman, B.D. and Noll, W. (1963). The Thermodynamics of Elastic Materials with HeatConduction and Viscosity, Arch. Rational Mech. Anal., 13: 167–178.

Curran, D.R., Seaman, L. and Shockey, D.A. (1977). Dynamic Failure in Solids, Physics Today,January, pp. 46–55.

Curran, D.R., Seaman, L. and Shockey, D.A. (1981). Linking Dynamic Fractureto Microstructural Processes, In: Meyers, M.A. and Murr, L.E. (eds), Shock Wavesand High–Strain Rate Phenomena in Metals : Concepts and Applications, pp. 129–167,Plenum Press, New York.

Curran, D.R., Seaman, L. and Shockey, D.A. (1987). Dynamic Failure of Solids, PhysicsReports, 147: 253–388.

Dornowski, W. (1999). Influence of Finite Deformations on the Growth Mechanismof Microvoids Contained in Structural Metals, Archives of Mechanics, 51(1): 71–86.

Dornowski, W. and Perzyna, P. (2000). Localization Phenomena in Thermo–ViscoplasticFlow Processes Under Cyclic Dynamic Loadings, Computer Assisted Mechanics andEngineering Sciences, 7: 117–160.

Dornowski, W. and Perzyna, P. (2002a). Analysis of Various Effects in Dynamic FatigueDamage, Archive of Applied Mechanics, 72: 418–438.

Dornowski, W. and Perzyna, P. (2002b). Numerical Analysis of Macrocrack PropagationAlong a Bimaterial Interface Under Dynamic Loading Processes, Int. J. Solids andStructures, 39: 4949–4977.

Dornowski, W. and Perzyna, P. (2006). Numerical Investigation of Localized Fracture Pheno-mena in Inelastic Solids, Foundations of Civil and Environmental Engineering, 7: 79–116.

Duszek, M.K. and Perzyna, P. (1991a). On Combined Isotropic and Kinematic HardeningEffects in Plastic Flow Processes, Int. J. Plasticity, 7: 351–363.

Duszek, M.K. and Perzyna, P. (1991b). The Localization of Plastic Deformation inThermoplastic Solids, Int. J. Solids Structures, 27: 1419–1443.




Duszek-Perzyna, M.K. and Perzyna, P. (1994). Analysis of the Influence of DifferentEffects on Criteria for Adiabatic Shear Band Localization in Inelastic Solids,In: Batra, R.C. and Zbib, H.M. (eds), Material Instabilities: Theory and Applications,ASME Congress, Chicago, 9–11 November 1994, AMD–Vol. 183/MD–Vol. 50, ASME,pp. 59–85, New York.

Eftis, J., Nemes, J.A. and Randles, P.W. (1991a). Viscoplastic Constitutive Modelingof High Strain-Rate Deformation, Material Damage, and Spall Fracture, Journal ofApplied Mechanics, Transactions of the ASME, 57(2): 282–291.

Eftis, J., Nemes, J.A. and Randles, P.W. (1991b). Viscoplastic Analysis of Plate-ImpactSpallation, International Journal of Plasticity, 7: 15–39.

Glema, A., Lodygowski, T., Nowak, Z., Perzyna, P. and Voyiadjis, G.Z. (2005).Thermo-elasto-Viscoplastic Model of a Material with Non-Local and AnisotropicIntrinsic Microdamage, In: Proceedings of McMat2005 Joint ASME/ASCE/SESConference on Mechanics and Materials, 1–3 June, p. 5, Baton Rouge, USA.

Glema, A., Lodygowski, T., Perzyna, P. and Sumelka, W. (2006). Constitutive AnisotropyInduced by Plastic Strain Localization, In: 35th Solid Mechanics Conference, 4–8September, pp. 139–140, Krakow, Poland.

Grebe, H.A., Pak, H.R. and Meyers, M.A. (1985). Adiabatic Shear Band Localizationin Titanium and Ti-6PctAl-4PctV Alloy, Met. Trans., 16A: 761–775.

Johnson, J.N. (1981). Dynamic Fracture and Spallation in Ductile Solids, J. Appl. Phys.,52: 2812–2825.

Lee, E.H. (1969). Elastic-plastic Deformations at Finite Strains, J. Appl. Mech., 36: 1–6.

Marsden, J.E. andHughes, T.J.R. (1983).Mathematical Foundations of Elasticity, Prentice–Hall,Englewood Cliffs, New York.

Meyers, H.C. (1994). Dynamic Behaviour of Materials, John Wiley, New York.

Meyers, M.A. and Aimone, C.T. (1983). Dynamic Fracture (spalling) of Metals, Prog. Mater.Sci., 28: 1–96.

Nemes, J.A. and Eftis, J. (1993). Constitutive Modeling of the Dynamic Fracture of SmoothTensile Bars, International Journal of Plasticity, 9: 243–270.

Oldroyd, J. (1950). On the Formulation of Rheological Equations of State, Proc. R. Soc. Lond.,A200: 523–541.

Perzyna, P. (1963). The Constitutive Equations for Rate Sensitive Plastic Materials, Quart.Appl. Math., 20: 321–332.

Perzyna, P. (1966). Fundamental Problems in Viscoplasticity, Advances in Applied Mechanics,9: 343–377.

Perzyna, P. (1971). Thermodynamic Theory of Viscoplasticity, Advances in Applied Mechanics,11: 313–354.

Perzyna, P. (1984a). Stability of Flow Processes for Dissipative Solids with InternalImperfections, J. Appl. Maths and Physics (ZAMP), 35: 848–867.

Perzyna, P. (1984b). Constitutive Modelling of Dissipative Solids for Postcritical Behaviourand Fracture, J. Eng. Materials and Technology, 106: 733–748.

Perzyna, P. (1986a). Internal State Variable Description of Dynamic Fracture of Ductile Solids,Int. J. Solids Structures, 22: 797–818.

Perzyna, P. (1986b). Constitutive Modelling for Brittle Dynamic Fracture in Dissipative Solids,Arch. Mechanics, 38: 725–738.

Perzyna, P. (1995). Interactions of Elastic–Viscoplastic Waves and Localization Phenomena inSolids, In: Wegner, J.L. and Norwood, F.R. (eds), IUTAM Symposium on NonlinearWaves in Solids, 15–20 August, pp. 114–121, ASME 1995, Victoria, Canada.

Perzyna, P. and Drabik, A. (1989). Description of Micro-Damage Process by PorosityParameter for Nonlinear Viscoplasticity, Arch. Mechanics, 41: 895–908.

230 A. GLEMA ET AL.



Perzyna, P. and Voyiadjis, G.Z. (2005). The Thermodynamic Theory of Elasto-ViscoplasticityAccounting for Induced Anisotropy Effects, In: Proceedings of McMat2005 JointASME/ASCE/SES Conference on Mechanics and Materials, 1–3 June, p. 6, BatonRouge, USA.

Perzyna, P. and Drabik, A. (2008). Micro-damage Mechanism in Adiabatic Processes,Engineering Transactions (in print).

Perzyna, P. (2008). The Thermodynamical Theory of Elasto-Viscoplasticity Accountingfor Microshear Banding and Induced Anisotropy Effects, In Mechanics (in print).

Seaman, L., Curran, D.R. and Shockey, D.A. (1976). Computational Models for Ductile andBrittle Fracture, J. Appl. Phys., 47: 4814–4826.

Shima, S. and Oyane, M. (1976). Plasticity for Porous Solids, Int. J. Mech. Sci., 18: 285–291.

Shockey, D.A., Seaman, L. and Curran, D.R. (1973). The Influence of MicrostructuralFeatures on Dynamic Fracture, In: Rohde, R.W., Butcher, B.M., Holland, J.R. andKarnes, C.H. (eds), Metallurgical Effects at High Strain Rates, p. 473, Plenum Press,New York.

Shockey, D.A., Seaman, L. and Curran, D.R. (1985). The Microstatistical Fracture MechanicsApproach to Dynamic Fracture Problem, Int. J. Fracture, 27: 145–157.

Sidey, D. and Coffin, L.F. (1979). Low-cycle Fatigue Damage Mechamisms atHigh Temperature, In: Fong, J.T. (ed.), Fatigue Mechanisms, Proc. ASTM STP 675Symposium, May 1978, pp. 528–568, Kansas City, Mo., Baltimore.

Truesdell, C. and Noll, W. (1965). The Non-Linear Field Theories of Mechanics, In: Fliigge, S.(ed.), Handbuch der Physik III/3, Springer-Verlag, Berlin.

Van der Giessen, E. (1991). Micromechanical and Thermodynamic Aspects of Plastic Spin,Int. Journ. Plasticity, 7: 365–386.

Voyiadjis, G.Z., Ju, J.-W. and Chaboche, J.-L. (eds). (1998). Damage Mechanics in EngineeringMaterials, Elsevier, Amsterdam.

Voyiadjis, G.Z., Abu Al-Rub, R.K. and Palazotto, A.N. (2003). Non-Local Coupling ofViscoplasticity and Anisotropic Viscodamage for Impact Problems Using the GradientTheory, Archives of Mechanics, 55: 39–89.

Voyiadjis, G.Z. and Dorgan, R.J. (2004). Bridging of Length Scales through GradientTheory and Diffusion Equations of Dislocations, In: Liu, W.K., Qian, D. andHorstemyer, M.F. (eds), Computer Methods in Applied Mechanics and Engineering,Special Issue on Multiple Scale Methods for Nanoscale Mechanics and Materials, Vol. 193,pp. 1671–1692, Elsevier B.V.

Voyiadjis, G.Z., Abu Al-Rub, R.K. and Palazotto, A.N. (2004). ThermodynamicFramework for Coupling of Non-Local Viscoplasticity and Non-Local AnisotropicViscodamage for Dynamic Localization Problems Using Gradient Methods, InternationalJournal of Plasticity, 20(6): 981–1038.

Voyiadjis, G.Z. and Abu Al-Rub, R.K. (2005). Gradient Plasticity Theory with a VariableLength Scale Parameter, International Journal of Solids and Structures, 42(14): 3998–4029.

Voyiadjis, G.Z. and Abu Al-Rub, R.K. (2006). A Finite Strain Plastic-Damage Model for HighVelocity Impact Using Combined Viscosity and Gradient Localization Limiters: II.Numerical Aspects and Simulations, International Journal of Damage Mechanics, 15(4):335–373.




The Numerical Analysis of the Intrinsic Anisotropic Microdamage Evolution in Elasto-Viscoplastic...

Documents

Transcript of The Numerical Analysis of the Intrinsic Anisotropic Microdamage Evolution in Elasto-Viscoplastic...