A Finite Strain Plastic-damage Model for High Velocity Impacts...

A Finite Strain Plastic-damage Model forHigh Velocity Impacts using CombinedViscosity and Gradient Localization

Limiters: Part II – NumericalAspects and Simulations

GEORGE Z. VOYIADJIS* AND RASHID K. ABU AL-RUB

Department of Civil and Environmental EngineeringLouisiana State University, Baton Rouge, LA 70803, USA

ABSTRACT: In this companion article, we present within the finite element contextthe numerical algorithms for the integration of the thermodynamically consistentformulation of geometrically nonlinear gradient-enhanced viscoinelasticity derivedin the first part of the article. The proposed unified integration algorithms areextensions of the classical rate-independent return mapping algorithms to the rate-dependent problems. An operator split structure is used consisting of a trial statefollowed by the return map by imposing the generalized viscoplastic and visco-damage consistency conditions simultaneously. Furthermore, a trivially incremen-tally objective integration scheme is established for the rate constitutive relations.The proposed finite deformation scheme is based on hypoelastic stress–strainrepresentations and the proposed elastic predictor and coupled viscoplastic–viscodamage corrector algorithm allows for the total uncoupling of geometricaland material nonlinearities. A simple and direct computational algorithm is also usedfor calculation of the higher-order gradients. This algorithm can be implementedin the existing finite element codes without numerous modifications as compared tothe current numerical approaches for integrating gradient-dependent models. Thenonlinear algebraic system of equations is solved by consistent linearization and theNewton–Raphson iteration. The proposed model is implemented in the explicit finiteelement code ABAQUS via the user subroutine VUMAT. Model capabilities arepreliminarily illustrated for the dynamic localization of inelastic flow in adiabaticshear bands and the perforation of a 12mm thick Weldox 460E steel plates bydeformable blunt projectiles at various impact speeds. The simulated shear bandresults well illustrated the potential of the proposed model in dealing with the well-known mesh sensitivity problem. Consequently, the introduced implicit and explicit

International Journal of DAMAGE MECHANICS, Vol. 15—October 2006 335

1056-7895/06/04 0335–39 $10.00/0 DOI: 10.1177/1056789506058047� 2006 SAGE Publications

*Author to whom correspondence should be addressed. E-mail: [email protected]

length-scale measures are able to predict size effects in localization failures.Moreover, good agreement is obtained between the numerical simulations andexperimental results of the perforation problem.

KEY WORDS: finite element method, adiabatic shear bands, perforation, meshdependence, material length-scales.

INTRODUCTION

EXPERIMENTAL OBSERVATIONS OF the inelastic behavior of variousmaterials reveal the existence of localization phenomenon. This

phenomenon is observed on a wide class of engineering materials includingmetals, concrete, rock, and soil, which is a characteristic feature of theinelastic deformations. Strain localization is a notion describing a deforma-tion mode, in which the whole deformation of a material structure occurs inone or more narrow bands, while the rest of the structure usually exhibitsunloading. The width and direction of localization bands depend on thematerial parameters, geometry, boundary conditions, loading distribution,and loading rate.

Owing to the lack of classical local continuum formulation toproduce physically meaningful and numerically converging results withinstrain localization computations, a thermodynamically motivated coupledplasticity-damage formulation is presented in the companion article of thefirst part of this study for rate- and temperature-dependent problems. Thismodel uses combined viscosity and nonlocal gradient localization limiters toregularize the dynamic strain localization problems. The enhanced nonlocalgradient-dependent theory formulates a constitutive framework on thecontinuum level that is used to bridge the gap between micromechanicaltheories and classical (local) continuum theories. They are successful inexplaining the size effects encountered at the micron scale and in preservingthe well-posedeness of the (initial) boundary value problem governingthe solution of material instability triggering strain localization. Moreover,viscosity (rate dependency) allows the spatial difference operator in thegoverning equations to retain its ellipticity and the initial boundary valueproblem is well-posed. This is due to the incorporation of either explicit(via nonlocal theory) or implicit (via viscosity) intrinsic material length-scaleparameter in the constitutive description.

In the first part of this study, a general theoretical framework for theanalysis of heterogeneous media that assesses a strong coupling betweenrate-dependent plasticity and anisotropic rate-dependent damage ispresented for high velocity impact related problems. This framework isdeveloped based on thermodynamic laws, nonlinear continuum mechanics,

336 G. Z. VOYIADJIS AND R. K. ABU AL-RUB

and nonlocal gradient-dependent theory. The proposed formulationincludes hypoelasto-thermo-viscoplasticity with anisotropic thermovisco-damage; a dynamic yield criterion; a dynamic damage growth criterion;non-associated flow rules; thermal softening; nonlinear strain hardening;strain-rate hardening; strain hardening gradients; strain-rate hardeninggradients; and an equation of state. The idea of bridging length-scalesis made more general and complete by introducing spatial higher-ordergradients in the temporal evolution equations of the internal state variablesthat describe hardening in coupled viscoplasticity and viscodamage models.The model is mainly developed to regularize the ill-posedness caused bystrain-softening material behavior and to incorporate the size effect (sizeand spacing of microdefects) in material behavior for high velocity impactdamage related problems.

In this part of the study, we are concerned about the numerical imple-mentation and verification of the highly nonlinear constitutive equationsdeveloped in the first part of this article. The development of computationalalgorithms that are consistent with the proposed theoretical formulation isgiven in detail in this article. Also, the problem of numerically integratingthe constitutive equations in the context of the finite element methodis addressed. The standard return mapping algorithms of rate-independentproblems are extended to rate-dependent problems. Moreover, since thenumerical implementation of the nonlocal gradient-dependent constitutiveequations is not a direct task, because of the higher order of the governingequations, a direct and simple computational algorithm for the gradientapproach is utilized. This algorithm can be implemented in the existing finiteelement codes without numerous modifications as compared to currentnumerical approaches for gradient-enhanced models (see e.g., de Borst andMuhlhaus, 1992; de Borst et al., 1993; Pamin 1994; de Borst and Pamin,1996; de Borst et al., 1999; and the references quoted therein).

Furthermore, a trivially incrementally objective integration scheme isestablished for the rate constitutive relations. The proposed elastic predic-tor and coupled viscoplastic–viscodamage corrector algorithm allows fortotal uncoupling of geometrical and material nonlinearities. The nonlinearalgebraic system of equations is solved by consistent linearization and theNewton–Raphson iteration. The proposed model is implemented in theexplicit finite element code ABAQUS (2003) via the user subroutineVUMAT. We present complete details of the implementation of theproposed novel numerical treatment of the problem.

Model capabilities are preliminarily illustrated for the dynamic localiza-tion of inelastic flow in adiabatic shear bands and the perforation of 12mmthick Weldox 460E steel plates by deformable blunt projectiles at variousimpact speeds.

Finite Strain Plastic-damage Model for High Velocity Impacts – Part II 337

The numerical integration of the constitutive model developed in Part Iof this article is given in the next section along with the descretized formof the equations. In addition, the formal steps involved in the numericalimplementation of the model within the framework of the finite elementmethod are considered. Two numerical examples of the formation ofadiabatic shear bands are presented and the problem of a projectilepenetrating a steel target is simulated. Conclusions are also presented at theend.

NUMERICAL ASPECTS

In this section, the numerical integration of the nonlocal geometricallynonlinear thermoviscoinelastic model developed in Part I of this work aredeveloped. Table 1 summarizes the key ingredients of the model.The subsection on ‘General Formulation’ presents the equations ofmotion. The subsequent subsection describes in detail the return mappingalgorithm considered in the integration of the proposed nonlocalconstitutive model. The return mapping algorithm is based on theextension of the classical predictor–corrector structure of the rate-independent problems to rate-dependent ones. An operator split structureis developed consisting of a trial state followed by the return map byimposing the generalized viscoplastic and viscodamage consistencyconditions simultaneously. Furthermore, a trivially incrementally objectiveintegration scheme is established for the rate constitutive relations. Theproposed finite deformation scheme is based on hypoelastic stress–strainrepresentations and the proposed elastic predictor and coupled viscoplas-tic–viscodamage corrector algorithm allows for total uncoupling ofgeometrical and material nonlinearities. In addition, a simple and directcomputational algorithm is also used for the calculation of high-ordergradients. The nonlinear algebraic system of equations is solved byconsistent linearization and the Newton–Raphson iteration. Representativenumerical simulations illustrating the performance of the proposednumerical formulation are included in the section on ‘NumericalApplications’. Moreover, note that we use here the same notation as inPart I of the companion article, unless specifically stated otherwise.

General Formulation

Let t0, t1, . . . , tn, tnþ1¼ tnþ�t, . . . be convenient time instances alongthe time interval over which the dynamic response of the body is sought.Consider the time step �t ¼ tnþ1 � tn: at t¼ tn where all quantities are


known, which are the converged values of the previous step, and the

Table 1. Constitutive model.

1. Stress–strain rate relationship in the spatial and damaged configuration

sr¼ C : d� d vp

� d vd� �

� A : /r

�b _TT, where C ¼ M_ �1

: �CC : M_ �1

,

M_

¼ 2�1� /

_�� 1 þ 1�

�1� /

_�h i�1

,

�CC ¼ Ke1� 1þ 2GeIdev; and Aijkl ¼@M

_�1

@ /_ : M

_

: sþC : M_

:@M

_�1

@/_

: C�1 : sþ b�Tð Þ

2. Nonlocal viscoplasticity condition in the undamaged configuration

f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2�ss0 � �XX

_� �: �ss0 � �XX

_� �r�

h�YYyp þ

�R_

R_ih1þ �vp

_�p_

p_�p_

p_

� �1=m1i1�

T

Tm

� �n� � 0

3. Nonlocal viscodamage condition

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY_

�H_

� �: Y

_

�H_

� �r�l þ K

_�1þ

��vd

_r_

r_�1=m2

� 1�

T

Tm

� �n� � 0

4. Flow rules

dvp¼ _��vp

@f

@s; dvd

¼ _��vd@g

@s; /

r

¼ _��vp@f

@Yþ _��vd

@g

@Y

5. Nonlocal plasticity and damage laws

Plasticity

Isotropic hardening _�RR�RR ¼_�RR�RRþ

1

2‘21r

2 _�RR�RR with _�RR�RR ¼a1 _��vp�1� r

_�2 �1� k1�R_

R_�

Kinematic hardening �XX_r

¼ �XXr

þ1

2‘22r

2 �XXr

with �XXijr

¼ a2 _��vpM

_

:M_

:�@f@�ss

� k2 �XX_�

1�T

Tm

� �n� �

Damage

Isotropic hardening_K_

K_

¼ _KK þ1

2‘23r

2 _KK with _KK ¼ a3 _��vd�1� h1K

_�

Kinematic hardening H_r

¼ Hr

þ1

2‘24r

2 Hr

with Hr

¼ a4 _��vd @g

@Y� h2H

_� �

1�T

Tm

� �n� �

Strain energy release rate Y_

¼1þ a

2s� b�Tð Þ : M

_

:@M

_�1

@/_ : C�1 : sþ b�Tð Þ

6. Generalized Kuhn–Tucker conditions for rate-dependent problems

_��vp � 0, f � 0 , _��vpf ¼ 0; _��vd � 0, g � 0 , _��vdg ¼ 0

(Continued )


solution must be computed at tnþ 1 for a given body load increment, �b,and surface load increment, �t.

Let the dynamic evolution of a hypoelastic-thermo-viscoplastic andthermoviscodamaged body of volume V and surface S be governed atstep time nþ 1, by the derived constitutive relations in Table 1 and by thefollowing momentum, initial, and compatibility relations:

LTsnþ1 þ �0bnþ1 ¼ �0 _��nþ1 in V; tnþ1 ¼ snþ1 n on St ð1Þ

u ¼ u0, t ¼ t0 at t ¼ t0 ð2Þ

lnþ1 ¼ rtnþ1 ¼ C tnþ1 in V ð3Þ

unþ1 ¼ ~uu on Su; tnþ1 ¼ ~tt on S�;

Tnþ1 ¼ ~TT on ST; qnþ1 ¼ ~qq on Sq

ð4Þ

where ð�Þnþ1 ¼ ð�Þn þ�ð�Þ is the additive decomposition of each of theinternal variables. For algorithmic convenience, we have shifted to matrixvector notation in this section. Equations (1) express the discrete dynamicmotion in the volume V and equilibrium on the free part of the boundary St

at nþ 1. Viscohypoelasticity is not considered in this study; viscous dampingeffects are neglected. L is the differential operator, b and t are the bodyforce and the surface traction vectors, respectively, u is the three-componentdisplacement vector, and n denotes the outward normal to the surface S.The initial conditions on displacements and velocities are given by

Table 1. Continued.

7. Temperature evolution (adiabatic condition)

�ocp _TT ¼ �s0 : dvpþ dvd

� �� JePðde : 1Þ � Tb : de þ

X10k¼3

Vk @k

r

; where Vk ¼ T@�k

@T��k

8. Equation of state (thermodynamic pressure)

P ¼ 1� �ð ÞcvTig"e with "e ¼ 1=Je � 1

T ig ¼ Tr exp �� rð Þ=cv½ � 1þ "e �ð��1Þ

exp � � 1ð Þ 1= 1þ "eð Þ � 1ð Þ �

9. Failure criteria

�_

�� ¼

ffiffiffiffiffiffiffiffiffiffiffi�_

ij�_

ij

q�

��cand=or P � Pcutoff


Equations (2). The compatibility relation in volume V is given byEquation (3). The boundaries Su, S�, ST, and Sq are parts of the boundarywhere the displacement ~uu, the velocity ~tt, the temperature ~TT, and the heatflux ~qq is prescribed, respectively. It is clear that St [ Su [ S� [ ST [ Sq ¼ S,St \ Su ¼ 0, and ST \ Sq ¼ 0.

In the context of the finite element method, the discrete problem can beobtained via a spatial Galerkin-type (displacement-based) projection ofthe semidiscrete (i.e., discrete in space and continuous in time) probleminto a finite dimensional subspace of admissible continuous shape functions.Consequently, in the following sections the procedure for solving thederived set of governing equations using the finite element method areoutlined. To integrate the set of constitutive equations in Table 1, a returnmapping algorithm is developed in the subsequent section.

Numerical Integration of the Constitutive Model

Considering a given configuration of a known set of positions X at timetn, the problem is now to update all state variables to a new configurationdefined by its respective set of positions x (which are supposed to be known)at time tnþ 1. This situation typically arises in a nonlinear finite elementproblem where the new positions x are determined from the discretizedversion of the momentum equation, Equation (1).

In this section, a new semi-implicit stress integration algorithm for rate-dependent problems is developed. This stress update algorithm treatsthe rate-independent and rate-dependent problems in a unified way. It isunified in the sense that the same routines are able to integrate both rate-independent and rate-dependent models by simply setting the viscosityparameters, �vp and �vd, to zero. Moreover, in this article we extend thisalgorithm to fully nonlocal coupled viscoplastic–viscodamage constitutiveequations with a two-step predictor–corrector structure: hypoelasticpredictor and coupled viscoplastic–viscodamage corrector (Zhu andCescetto, 1995). In the model presented in this article, there exists twocoupled surfaces, and for each iteration, the viscoplastic surface and theviscodamage surface should be corrected simultaneously (see Figure 1). Thedifferent steps of the integration algorithm are detailed here.

If the variables at iteration i, such as si, X_

i, R_

i, /_

i, H_

i, K_

i, Y_

i, Ti, etc., areassumed to have been determined and the values of d and �t are given, thensiþ1 that satisfies the discretized constitutive equations can be obtained. Inthe following, a hypoelastic predictor and coupled viscoplastic–viscodamagecorrector is proposed. In the first step, the hypoelastic predictor problem issolved with initial conditions that are the converged values of the previousiteration i while keeping irreversible variables frozen. This produces a trial


stress state, trs, which, if outside the viscoplastic surface f and theviscodamage surface g is taken as the initial conditions for the solution ofthe viscoplastic–viscodamage corrector problem. The scope of this secondstep is to restore the generalized (i.e., rate-independent to rate-dependentproblems) consistency condition by returning back the trial stress to theviscoplastic surface f and the viscodamage surface g simultaneously asconceptually represented in Figure 1.

However, one of the major challenges while integrating the constitutiveequations in a finite deformation context is to achieve the incrementalobjectivity, i.e., to maintain correct rotational transformation properties allalong a finite time step. A procedure that has now become very popular isfirst to rewrite the constitutive equations in a corotational moving frame.Given a skew-symmetric tensor W ¼ �WT (e.g., W ¼ x where x is the spintensor; W ¼ _RRR

T; or the relative spin tensor W ¼ x� _RRR

T) such that it

satisfies the relation QQT¼ QTQ ¼ I, which if differentiated with respect to

time yields

W ¼ _QQQT¼ �Q _QQT

¼ ��_QQQT

�T¼ �WT ð5Þ

Therefore, to ensure the orthogonality of the rotation matrix, we maygenerate a group of rotations Q, by solving for

_QQ ¼ W Q with Qðt ¼ tnÞ ¼ 1 ð6Þ

Hypoelasticpredictor

Viscoplastic-viscodamagecorrector

(k)fi+1 = constant

trti+1

ti+1(1)

ti+1

ti+1

ti

(k)

gi+1 = constant(k)

gi+1 = 0

gi = 0

fi = 0

fi+1 = 0

Figure 1. Conceptual representation of the hypoelastic predictor and coupled viscoplastic–viscodamage corrector algorithm.


It is now possible to generate a change of frame from the fixed Cartesianreference axes to the corresponding rotating axes (corotational axes). TheKirchhoff stress tensor s can then be transformed by Q as

sc ¼ QTsQ ð7Þ

Differentiating the above equation with respect to time, we obtain

_ssc ¼ QT _ss�W sþ sWð ÞQ ¼ QT srQ ð8Þ

where sr¼ _ss�W sþ sW is a corotational objective rate of the Kirchhoff

stress.In the literature many objective rates are introduced, such as Jaumann,

Truesdell, and Green–Naghdi rates. From Equation (8) we can obtain theJaumann rate ifW ¼ x and the Green–Naghdi rate ifW ¼ _RRRT. Moreover,Equation (8) indicates that a somewhat complicated expression as anobjective derivative becomes a rather simple time derivative under theappropriate change of coordinates. This suggests that the entire theory andimplementation will take on canonically simpler forms if transformed to theQ-system. In the corotational frame, the rate of the corotational Kirchhoffstress is objective (frame-invariant), so that the stress rate equation takes thesimpler form

_ssc ¼ Cc�d c

� d vpc�d vdc�� Ac _//c

� bc _TT ð9Þ

The second-order tensors d, /r

, and b can then be rewritten in the unrotatedframe as:

d c¼ QTdQ; _//c

¼ QT /r

Q; bc ¼ QTbQ ð10Þ

The fourth-order tensors C and A are also expressed in terms of the corota-tional components as:

Cc¼ Q QTCQ

� �QT

¼ C; Ac¼ Q QTAQ

� �QT

¼ A ð11Þ

ELASTIC PREDICTORAssuming that the variables of the model at iteration i and the

displacement field u ¼ xiþ1 � xi at iteration iþ 1 are known, the trial elasticstress can then be given in the corotational frame by

trsciþ1 ¼ sci þ Cci : d

c�t ð12Þ


or, in the Cartesian frame, the trial stress can be written as follows

tr�iþ1 ¼ Qiþ1tr�ciþ1 Q

Tiþ1 ¼ Qiþ1 QT

i �i Qi þ Cci : d

c�t� �

QTiþ1 ð13Þ

Using the polar decomposition F ¼ RU, we can write

d c¼ QTdQ ¼

1

2QTR

�_UUU�1 þU�1 _UU

�RTQ ð14Þ

Equation (13) can then be simplified in the following way. Let us assumethat the reference configuration is the configuration at i (update Lagrangianformulation), which implies that Qi ¼ 1. Moreover, if we assume Qiþ1 ¼ R,this reduces Equation (14) to

d c¼

1

2

�_UUU�1 þU�1 _UU

�ð15Þ

A simple expression for the trial stress, Equation (13), can be obtainedby adapting the following assumptions. Let us assume the followingexponential map of the right stretch tensor, UðtÞ, such that (Simo andHughes, 1998):

U tð Þ ¼ expt� tn�t

E� �

ð16Þ

where E is a constant tensor to be determined. Upon time differentiation ofEquation (16), we obtain

_UUðtÞ ¼E

�texp

t� tn�t

E� �

ð17Þ

Substituting the above equation into Equation (15), yields

d c¼

E

�tð18Þ

The tensor E is simply determined using the following compatibilityconditions:

(a) in the reference configuration ðX, tnÞ: UðtnÞ ¼ expð0Þ ¼ 1

(b) in the current configuration ðx, tnþ1Þ: Uðtnþ1Þ ¼ expðE Þ

From these two conditions, we obtain

E ¼ lnU ¼1

2lnU 2 ¼

1

2ln FTF� �

ð19Þ


which implies that E is the (incremental) natural strain tensor between thereference configuration and the current one. Hence, the trial elastic stresstensor in Equation (13) can be evaluated by:

trsiþ1 ¼ R si þ Ci :Eð ÞRT ð20Þ

The final mapped stress is given in Equation (22).In the above procedure, it is essential to realize that:

. F¼RU are incremental tensors;

. it can be seen from Equation (19) that the proposed procedure istrivially incrementally objective. In the case of rigid body motion,Unþ1 ¼ Un and d c

¼ 0 or lnU ¼ 0, thus the stress tensor will be updatedexactly by the relation snþ1 ¼ R snR

T, whatever the amplitude ofthe rotation;

. the rotation tensor R is directly and exactly computed from the polardecomposition and not from the (approximate) numerical integration ofthe rate equation _QQ ¼ WQ over the time interval ½tn, tnþ1�;

. in the proposed procedure, R only needs to be evaluated once per timestep. This is different from the schemes proposed in Simo and Hughes(1998), where it needs to be evaluated twice per time step;

. all kinematic quantities are based on the deformation gradient F overthe considered time step, a quantity that is readily available in a nonlinearfinite element code, such as ABAQUS.

COUPLED VISCOPLASTIC–VISCODAMAGE CORRECTORNow if f ðtr�iþ1,X

_

i,R_

i,_p_p_,Ti,/

_

iÞ � 0 and gðtrY_

iþ1,H_

i,K_

i,_r_r_

i,Ti,/_

iÞ � 0, theprocess is clearly undamaged elastic and the trial stress is in fact thefinal state. On the other hand, if f>0 and g>0, the Kuhn–Tucker loading/unloading conditions are violated by the trial stress, which now liesoutside the generalized viscoplastic and viscodamge surfaces (Figure 1).Consistency, is restored by a generalization of the classical return mapp-ing algorithm to rate-dependent problems. The coupled viscoplastic–viscodamge corrector problem may then be rephrased as (the objectiverates reduce to a simple time derivative due to the fact that the globalconfiguration is held fixed):

_ss ¼ �C : d vpþ d vd

� �� A : _//� b _TT

¼ �C : _��vp@f

@sþ _��vd

@g

@s

� �� A : _��vp

@f

@Yþ _��vd

@g

@Y

� �� b _TT

ð21Þ


where _TT is calculated from the heat balance equation, while _��vp and _��vd

are calculated, in a later section, using a nonlocal computational algorithm.The hypoelastic predictor and coupled viscoplastic–viscodamage correc-

tor step yields the final stress as:

siþ1¼trsiþ1 � Ciþ1 : _��vp

@f

@sþ _��vd

@g

@s

� �� Aiþ1 : _��vp

@f

@Yþ _��vd

@g

@Y

� �� biþ1

_TT

ð22Þ

In the following sequence, we will show the procedure how to calculatethe inelastic multipliers _��vp and _��vd using the adiabatic heating condition,the generalized viscoplastic consistency condition, and the generalizedviscodamage consistency condition.

To establish the actual heat generation that occurs during the highlytransient impact events of the thermomechanically coupled finite element,discretization of the heat equation is imperative. Since the whole impactprocess lasts a few hundreds of microseconds, the effect of heat conductionis negligible over the domain of the specimen. By only considering adiabaticheating with no external heat source along with d e

¼ d� d vp� d vd, the

heat balance equation can then be rewritten as:

�ocp _TT ¼ �s0 þ J eP1þ Tibð Þ : d vpþ d vd

� �� J eP1þ Tbð Þ : dþ

X10k¼3

Vk @k

r

ð23Þ

where Vk ¼ T@�k=@T��k (k¼ 3, . . . , 10) and their expressions aresimilar to those outlined in Table 4 of Part I, but instead of # we substitute~## ¼ Tð@#=@T Þ � #, and their magnitudes are obtained from the previousiteration i.

By making use of the evolution equations for d vp, d vd, and @k

r

(k¼ 3, . . . , 10) given in Part I of the companion article, i.e., _pp, r2 _pp, ar,

r2ar

, _rr, r2 _rr, Cr, and r2C

r, the temperature evolution from Equation (23) can

then be reduced to:

_TT ¼ �1

�ocpJ eP1þ Tbð Þ : dþQtp

1_��vp þQtp

2 r2 _��vp þQtd1

_��vd þQtd2 r2 _��vd

ð24Þ

where Qtpk and Qtd

k (k¼ 1, 2) are obtained from the previous iteration i andare given by Equations (A-1)–(A-4) in the Appendix.


Moreover, we require the satisfaction of the generalized viscoplasticityconsistency condition, _ff, at the end of iteration iþ 1, such that

@f

@s: srþ

@f

@pþ

1

�t

@f

@ _pp

� _ppþ

@f

@r2pþ

1

�t

@f

@r2 _pp

� r2 _ppþ

@f

@a: arþ

@f

@r2a: r2a

r

þ@f

@/: /r

þ@f

@r2/: r2/

r

þ@f

@T_TT ¼ 0 ð25Þ

Since we are applying the local iteration process within the time step tþ�t(i.e., at step nþ 1) and the updated Lagrangian formulation is used, _pp ¼ €pp�tand r2 _pp ¼ �tr2 €pp are used in obtaining Equation (25). Substitution of thestress rate equation, s

r, from Table 1 and the evolution equations for _pp, r2 _pp,

ar, r2a

r, _rr, r2 _rr, /

r, and r2/

rfrom Part I of the companion article along with

Equation (24) into Equation (25) yields the following expression:

@f

@s: Cþ Z p

� �: dþQ

p1_��vp þQ

p2r

2 _��vp þQp3_��vd þQ

p4r

2 _��vd ¼ 0 ð26Þ

where Z p and Qpk (k¼ 1, . . . , 4) are obtained from the previous i iteration

and are given by Equations (A-5)–(A-9) in the Appendix.Similarly, the generalized viscodamage consistency condition _gg needs

to be satisfied. Since the viscodamage driving forces Y_

is a function of s

and /_

, we can then express _gg around (Y_

, r,r2r, _rr,r2 _rr,C,r2C,/,r2/,T ) asfollows:

_gg �@g

@s: srþ

@g

@rþ

1

�t

@g

@_rr

� �_rrþ

@g

@r2rþ

1

�t

@g

@r2 _rr

� �r2 _rrþ

@g

@CCr

þ@g

@r2C: r2C

r

þ@g

@/: /r

þ@g

@r2/: r2/

r

þ@g

@T_TT ¼ 0 ð27Þ

Substitution of the stress rate equation, sr, from Table 1 and the evolu-

tion equations for _rr, r2 _rr, Cr, r2C

r, /r, and r2/

rfrom Part I of the companion

article along with Equation (24) into Equation (27) yields the followingexpression:

@g

@s: Cþ Z d

� �: dþQd

1_��vp þQd

2 r2 _��vp þQd

3_��vd þQd

4 r2 _��vd ¼ 0 ð28Þ


where Z d and Qdk (k¼ 1, . . . , 4) are obtained from the previous i iteration

and are given by Equations (A-10)–(A-14) in the Appendix.Owing to the higher order of the governing equations, Equations (24),

(26), and (28), in the inelastic region, considerable difficulties are experi-enced with their numerical implementation. The consistency conditionof inelasticity is no longer an algebraic equation but is a differential onebecause of r2 _��vp and r2 _��vd, which needs to be calculated along with _��vp and_��vd. Another complication is the higher-order boundary conditions that arenecessary from the mathematical point of view and have to be prescribed onthe moving elasto-inelastic boundary (Figure 2). These internal boundariesare not always easy to interpret physically. The computational techniqueusually followed for integrating the gradient-enhanced constitutive relationswas first proposed by de Borst and coworkers (e.g., de Borst and Muhlhaus,1992; de Borst et al., 1993, 1999; Pamin 1994; de Borst and Pamin, 1996;Peerlings et al., 1996; Pamin et al., 2003; and the references quoted therein).In their works, the plasticity yield and/or damage growth conditions dependon the Laplacian of an equivalent kinematic measure (hardening/softeningstate variables), and the consistency conditions result in differential equa-tions with respect to the plastic/damage multipliers. The multipliers, _��vp

and/or _��vd, are considered as fundamental unknowns (additional degrees offreedom) having a role similar to that of displacements and are discretized inaddition to the usual discretization of the displacements. In addition, theconsistency condition is written in a weak form and solved simultaneouslywith the equation of motion. However, because of the presence of high-order derivatives in the weak form of the (initial) boundary value problem,

S

St

t

f < 0; g = 0

f < 0; g < 0

lvp = 0; lvd

> 0

V

Vlvp

f > 0; g > 0lvp

> 0; lvd > 0

lvp = lvd

= 0

f < 0; g < 0lvp

= lvd = 0

f = 0; g < 0lvp

> lvd = 0

Vlvd

Slvd

Slvp

Figure 2. Schematic representation of elastic, viscoplastic, and viscodamage boundaries.


there is a need for numerically expensive C1-continuous conditions on theshape functions or penalty-enhanced C 0 class functions for the interpolationof the plastic/damage multipliers (or other field variables, such as stress andstrain) in the finite element context. Moreover, C 2 and higher continuity arealso needed if fourth-order or higher-order gradient terms are incorporated,otherwise the gradient terms lose their presence. Therefore, in the de Borstapproach Hermitian or mixed formulations are unavoidable for a consistentfinite element formulation and, for the inelastic process, a standard returnmapping algorithm is performed, in which the values of the kinematicfields at an integration point are interpolated from their nodal values.This approach has been discussed thoroughly by Voyiadjis et al. (2001, 2003,2004) and used intensively by many other authors (e.g., Comi and Perego,1996; Mikkelsen, 1997; Ramaswamy and Aravas, 1998a,b; Svedberg andRunesson, 2000; Nedjar, 2001; Zervos et al., 2001; Chen and Yuan, 2002;Matsushima et al., 2002; Liebe et al., 2003; and the references quotedtherein). The disadvantage of this approach is that it gives rise to manynumerical difficulties that require considerable modifications to the exist-ing finite element codes, which make their implementation not an easy ordirect task.

We are looking for an alternative, simple, and robust numerical algorithmfor the computation of the higher-order gradient terms, r2 _��vp and r2 _��vd,without the need for large modifications of the finite element procedure,which makes the gradient approach comparatively easy and attractive toimplement in an existing finite element program, such as ABAQUS. Themain purpose of the next section is, therefore, to describe a new algorithmicimplementation of the gradient-based models.

COMPUTATION OF THE LAPLACIAN OF VISCOPLASTICITYAND VISCODAMAGE MULTIPLIERS

In this section, we will show how we can compute the Laplacian termsr2 _��vp and r2 _��vd using a simple and robust approach that is proposed byAbu Al-Rub and Voyiadjis (2005).

In the approach followed in this article, the nonlocal consistencycondition is transformed into a linear set of equations that depends onthe material parameters and on the current coordinates of the integrationpoints. These sets of linear equations are solved by any numerical iterativemethod for the inelastic multipliers of the integration points that exist in aglobal (nonlocal) superelement of eight adjacent local elements in a nonlocalsense (Figure 3). The gradient terms, r2 _��vp and r2 _��vd, at each integrationpoint in the local element are evaluated from the derivatives of a polynomialthat interpolates the value of the inelastic multiplier, _��vp and _��vd, in thesuperelement with classical integration points. In addition, this procedure


enforces the generalized consistency conditions, _ff and _gg, in the sense ofdistributions, i.e., f and g are satisfied at the end of the loading step.However, in this approach there in no need to consider the inelasticmultipliers as additional degrees of freedom. Therefore, by using thisapproach, we do not need to introduce high-order continuous shapefunctions (e.g., C1 class or penalty-enhanced C 0 class functions) for theinterpolation of the multiplier fields in the finite element context. Inconsequence, a straightforward one-field C 0-continuous finite elementimplementation can be easily used. Furthermore, this algorithm has themajor advantage that it avoids boundary conditions on the moving elasto-inelastic boundary since the resulting partial differential equations hold inthe whole body.

The inelastic multipliers are not considered here as independent globalvariables but as local internal variables. In classical inelasticity, the inelasticmultipliers are calculated by restoring the consistency condition iteratively.However, for the nonlocal formulation, it is not possible because it dependson high-order gradients. To evaluate the gradients r2 _��vp and r2 _��vd atintegration point m, we need the values of _��vp and _��vd at point m as well asthe values at the neighboring points (nonlocality). The gradient at eachintegration point m is evaluated from the derivatives of a polynomialfunction that interpolates the values of the inelastic multiplier at theneighboring points. Therefore, the gradient terms, r2 _��vp and r2 _��vd, can be

AB

Superelement for point B

Superelement for point A

Figure 3. A schematic illustration for the computation of the Laplacian terms from a regularfinite element mesh.


expressed in terms of _��vpn and _��vdn with n 2 1, . . . ,NGPf g using the followingrelations:

r2 _��vpm ¼XNGP

n¼1

�ggmn_��vpn ; r2 _��vdm ¼

XNGP

n¼1

�ggmn_��vdn ð29Þ

where NGP is the number of Gauss integration points. The computation ofcoefficients �ggmn is explained in what follows.

Figure 3 shows a schematic illustration for the computation of theLaplacian terms from a regular finite element mesh, where r2 _��vp and r2 _��vd

are needed at the integration points of each element. For two-dimensionalproblems four-noded element with nine integration points (full integration)is assumed. Eight elements (superelement) are used to compute r2 _��vp andr2 _��vd at each integration point. This means that 81 integration points areused to calculate the gradients at each integration point. Except for eachcorner and mid-boundary elements, their nine integration points are usedto calculate the gradients. This illustration is valid for any element withany number of integration points. However, with more integration points,higher accuracy is achieved. This illustration is valid for one-dimensional aswell as for three-dimensional mesh discretizations. However, the disadvan-tage of this approach is that regular meshes are required and with largeprecision loss for complicated problems, which limits the applicabilityof this approach. More elaborate studies are needed to generalize thisapproach to non-regular mesh discretizations. This is not the subject of thisstudy but it will be presented in a forthcoming article by the authors.

To determine the coefficients �ggmn, a complete second-order poly-nomial function is used to evaluate the inelastic multipliers aroundpoint m, such that

_�� ¼ aTm ð30Þ

where _�� could be _��vp or _��vd, a is the coefficients vector, and m is thevariables vector. For example in two-dimensional problems, we have thefollowing expressions for a and m: aT ¼ a1 a2 a3 a4 a5 a6

�and mT ¼

1 x y xy x2 y2 �

.To obtain the coefficients vector a, a minimization method by least

squares is used. Moreover, the interpolation is made in the globalcoordinate system (x, y, z) of the generated mesh with NGP integrationpoints. The coefficients vector a can be expressed in the following form:

, ¼ MTa ð31Þ


For a two-dimensional mesh, the matrix M and the inelastic multipliersvector , are defined by

M ¼

1 1 � � � 1

x1 x2 � � � xNGP

y1 y2 � � � yNGP

x1y1 x2y2 � � � xNGPyNGP

x21 x22 � � � x2NGP

y21 y22 � � � y2NGP

2666666666664

3777777777775

ð32Þ

and , ¼ _��1 _��2 � � � _��NGP

�T.

Multiplying both sides of Equation (31) by M, we can write

M, ¼ Ha ð33Þ

with H ¼ MMT is a symmetrical square matrix and can be written fortwo-dimensional problems as:

H ¼XNGP

n¼1

1 xn yn xnyn x2n y2n

x2n xnyn x2nyn x3n xny2n

y2n xny2n x2nyn y3n

x2ny2n x3nyn xny

3n

Symm x4n x2ny2n

y4n

266666666664

377777777775

ð34Þ

It is obvious that H needs to be updated at each loading increment for finitedeformation problems.

From Equations (30) and (33), we can then compute the inelasticmultiplier vector and its Laplacian, respectively, as follows:

_�� ¼ aTm ¼ H�1M,� �T

m ¼ H�1XNGP

n¼1

_��nmn

!T

m ð35Þ

r2 _�� ¼ H�1XNGP

n¼1

_��nmn

!T

rm ð36Þ


For the integration point m, we can write expressions for

r2 _��m ¼ rxx_��m þ ryy

_��m þ rzz_��m ð37Þ

as follows

r2 _��m ¼XNGP

n¼1

mTnH�1rxxmm þ mTnH

�1ryymm þ mTnH�1rzzmm

� �_��n ð38Þ

Comparing Equations (29) with Equation (38), we can then compute thecoefficients �ggmn using the following expression:

�ggmn ¼ mTnH�1rxxmm þ mTnH

�1ryymm þ mTnH�1rzzmm ð39Þ

The coefficients �ggmn depend only on the x, y, z updated coordinates of theGauss integration points. These coefficients are computed at each loadingincrement for finite deformations.

Using Equations (29) and (39) we can now determine nonlocally _��vp and_��vd at each integration point m from the generalized consistency conditions,Equations (26) and (28). This is shown in the subsequent development.

NONLOCAL COMPUTATIONAL ALGORITHMBy substituting Equations (29) into Equations (26) and (28), we can then

rewrite the generalized viscoplasticity consistency condition, Equation (26),and the generalized viscodamage consistency condition, Equation (28),at each integration point m, respectively, as follows:

@f

@sðiÞðmÞ

: CðiÞðmÞ

þ ZpðiÞðmÞ

!: dðmÞ þQ

pðiÞ1ðmÞ

_��vpðmÞ

þQpðiÞ2ðmÞ

XNGP

n¼1

�ggðmnÞ_��vpðnÞ þQ

pðiÞ3ðmÞ

_��vdðmÞ

þQpðiÞ4ðmÞ

XNGP

n¼1

�ggðmnÞ_��vdðnÞ ¼ 0 ð40Þ

@g

@sðiÞðmÞ

: CðiÞðmÞ

þ ZdðiÞðmÞ

!: dðmÞ þQ

dðiÞ1ðmÞ

_��vpðmÞ

þQdðiÞ2ðmÞ

XNGP

n¼1

�ggðmnÞ_��vpðnÞ þQ

dðiÞ3ðmÞ

_��vd

þQdðiÞ4ðmÞ

XNGP

n¼1

�ggðmnÞ_��vdðnÞ ¼ 0 ð41Þ

Note that (m) indicates the integration point and (i) indicates the previousiteration number.


Let us define the following expressions:

RpðiÞðmÞ

¼@f

@sðiÞðmÞ

: CðiÞðmÞ

þ ZpðiÞðmÞ

!: dðmÞ ð42Þ

RdðiÞðmÞ

¼@g

@sðiÞðmÞ

: CðiÞðmÞ

þ ZdðiÞðmÞ

!: dðmÞ ð43Þ

GppðiÞ¼

QpðiÞ1ð1Þ þQ

pðiÞ2ð1Þ �gg11 Q

pðiÞ2ð2Þ �gg12 � � � Q

pðiÞ2ðNGPÞ

�gg1NGP

QpðiÞ2ð2Þ �gg21 Q

pðiÞ1ð2Þ þQ


pðiÞ2ðNGPÞ

�gg2NGP

..

. ... ..

. . .. ..

.

QpðiÞ2ðNGPÞ

�ggNGP1 QpðiÞ2ðNGPÞ

�gg2NGP� � � Q

pðiÞ1ðNGPÞ

þQpðiÞ2ðNGPÞ

�ggNGPNGP

266666664

377777775ð44Þ

GpdðiÞ¼

QpðiÞ3ð1Þ þQ

pðiÞ4ð1Þ �gg11 Q


pðiÞ4ðNGPÞ

�gg1NGP

QpðiÞ4ð2Þ �gg21 Q

pðiÞ3ð2Þ þQ


pðiÞ4ðNGPÞ

�gg2NGP

..

. ... ..

. . .. ..

.

QpðiÞ4ðNGPÞ

�ggNGP1 QpðiÞ4ðNGPÞ


pðiÞ3ðNGPÞ

þQpðiÞ4ðNGPÞ

�ggNGPNGP

266666664

377777775ð45Þ

GdpðiÞ¼

QdðiÞ1ð1Þ þQ

dðiÞ2ð1Þ �gg11 Q

dðiÞ2ð2Þ �gg12 � � � Q

dðiÞ2ðNGPÞ

�gg1NGP

QdðiÞ2ð2Þ �gg21 Q

dðiÞ1ð2Þ þQ


dðiÞ2ðNGPÞ

�gg2NGP

..

. ... ..

. . .. ..

.

QdðiÞ2ðNGPÞ

�ggNGP1 QdðiÞ2ðNGPÞ


dðiÞ1ðNGPÞ

þQdðiÞ2ðNGPÞ

�ggNGPNGP

266666664

377777775ð46Þ

GddðiÞ¼

QdðiÞ3ð1Þ þQ

dðiÞ4ð1Þ �gg11 Q


dðiÞ4ðNGPÞ

�gg1NGP

QdðiÞ4ð2Þ �gg21 Q

dðiÞ3ð2Þ þQ


dðiÞ4ðNGPÞ

�gg2NGP

..

. ... ..

. . .. ..

.

QdðiÞ4ðNGPÞ

�ggNGP1 QdðiÞ4ðNGPÞ


dðiÞ3ðNGPÞ

þQdðiÞ4ðNGPÞ

�ggNGPNGP

266666664

377777775ð47Þ


,vp¼ _��vp1

_��vp2 � � � _��vpNGP

j kTð48Þ

,vd¼ _��vd1

_��vd2 � � � _��vdNGP

j kTð49Þ

RpðiÞ ¼ RpðiÞ1 R

pðiÞ2 � � � R

pðiÞNGP

j kTð50Þ

RdðiÞ ¼ RdðiÞ1 R

dðiÞ2 � � � R

dðiÞNGP

j kTð51Þ

We can then write Equations (40) and (41), respectively, as follows:

GppðiÞ,vpþ GpdðiÞ,vd

¼ RpðiÞ ð52Þ

GdpðiÞ,vpþ GddðiÞ,vd

¼ RdðiÞ ð53Þ

Combining Equations (52) and (53), we obtain

GppðiÞ GpdðiÞ

GdpðiÞ GddðiÞ

" #,vp

,vd

� �¼

RpðiÞ

RdðiÞ

( )ð54Þ

The above linear system of equations can be solved for ,vp and ,vd usinga numerical iterative scheme such as the Gauss–Jordan iterative method.The viscoplastic and viscodamage multipliers are obtained when the visco-plasticity and viscodamage conditions, f and g, are fulfilled at the end ofthe loading step for a suitable tolerance, such that:

XNGP

n¼1

fn � TOL andXNGP

n¼1

gn � TOL ð55Þ

where TOL could be set to a very small value in the order of 10�5.Note that in the undamaged-elastic elements _��m ¼ 0; however, for

spreading of the inelastic zone it is important that the numerical solutionallows r2 _��m>0 at the elastic–inelastic boundary. If inelastic integrationpoints appear in the structure, then in the elastic integration points adjacentto the inelastic zone we have non-zero , (Figure 2). Therefore, the proposedalgorithm has the feature, that these elastic integration points have _��m � 0and r2 _��m>0. As a result, the yield strength is increased/decreased and thedamage at these elastic points is delayed (hardening) or enters the softeningregion.

The above algorithm for gradient-dependent inelasticity appears to haveseveral advantages over the standard algorithm by de Borst and hiscoworkers with regard to the incorporation of the gradient-dependentmodel in a finite element code. The proposed computational algorithm can


be implemented in the existing finite element codes without large modifica-tions as compared to the computational approach of de Borst and hiscoworkers. In contrast to the later approach, for calculation of the gradientterms we do not need to introduce shape functions of the C1 class orpenalty-enhanced C 0 class functions for the interpolation of the gradientterms. This is because the governing constitutive equations in the proposedapproach are replaced directly by the difference equations of the fieldvariables and no interpolation functions are needed for the gradient terms.

NUMERICAL APPLICATIONS

In this section, some numerical examples are presented to verify theimplementation of the proposed constitutive model using the commercialfinite element program ABAQUS (2003). The algorithmic model presentedin the previous section is coded as a VUMAT user material subroutine ofABAQUS/Explicit. The ABAQUS/Explicit is mainly used for high transientdynamic problems and it uses explicit integration algorithms. For informa-tion about writing the VUMAT subroutine, consult the reference manualsof ABAQUS/Explicit (2003).

The first two examples are related to the nonlocal hypoelasto-thermo-viscoplastic (without damage) to show the efficiency of the model inobtaining mesh objective simulations. The first example demonstrates theuse of the viscosity as a localization limiter, while the second exampledemonstrates the use of the gradient approach as a localization limiter.The third example is related to the whole model used to solve perforation ofa steel target by a blunt projectile for different impact speeds.

Strip with a Fixed Edge

The role of the viscosity localization limiter in setting the character ofthe governing differential equations and in introducing a length-scale isillustrated by considering shear band development in a simple plane strainstrip subjected to low impact loading. Of particular interest are ill-posedinitial boundary value problems in hypoelastic and thermoviscoplastic solids(without damage).

Now let us consider a two-dimensional initial boundary value problem fora specimen of 100mm length and 20mm width (Figure 4(a)). The bottomside of the specimen is fixed and the topside is movable. The loadingis enforced by a velocity profile shown in Figure 4(a) that acts at the freeend of the specimen. Three mesh discretizations of 15 50, 25 70,and 30 100 meshes are used with eight-noded rectangular elements. Theconstitutive parameters used in the computation are listed in Table 2 for the


target material (Weldox 460 E steel). Time increments of the order 10�8 s areused to satisfy the stability criteria. The results presented below are focusedon the distribution of the effective viscoplastic strain at the final state oflocalization. Figure 4(b) shows clearly the localized regions of intense shearat the end of localization (tf¼ 700 ms). We can easily observe the intenseequivalent viscoplastic strain distributions that show the width and thelocation of the shear band development. Note that due to the inhomogeneityintroduced by the clamped boundary conditions, no geometric imperfec-tions are needed to initiate localized deformation modes. For other cases,not reported here, for large viscosity the viscoplastic deformations arediffused over the whole specimen and the localization does not manifestitself.

The deformed configurations shown in Figure 4(b) indicate the formationof a neck and a pronounced shear band of almost a mesh independent band

ttfto

Vo

V

V

a

bFixed edge

(a) (b)

Vo = 30 m/s

to = 100 ms

tf = 700 ms

a = 100 mm b = 20 mm

Figure 4. Study of mesh sensitivity: (a) problem description of a strip with a fixed edge underimpact tensile loading and (b) deformation patterns and contour plots of the effectiveviscoplastic strain for 1550, 2570, 30100 finite element discretizations.


width for the three finite element discretizations. Moreover, Figure 4(b)shows that the magnitude and the distribution of the equivalent viscoplasticstrain are almost independent of the mesh refinement. It can be seen that aswe refine the mesh identical results are obtained.

It is noteworthy that from results not reported here, we observe that withincreasing viscosity, the probability at which the results become meshindependent is decreased. Also, for very high viscosity values no localizationis observed, but necking behavior is observed. Moreover, as we increase thedeformation to very high strain values, we observe initially the developmentof shear bands, which is followed at a later stage by the evolution ofa necking failure mode.

Strip with a Geometrical Imperfection

We notice that in the absence of a physically motivated length-scale,which would govern the width of the shear band, a scale is inherentlyintroduced in the problem by the mesh size when using the finite elementmethod. To remedy the spurious mesh sensitivity of the numerical results,an internal length-scale needs to be incorporated in the continuumdescription. The following examples demonstrate the potential of thegradient approach (without viscosity and damage) in solving the meshsensitivity problem in quasi-static problems.

The simulation of a strip with a geometrical imperfection under lowimpact tensile loading is carried out. The geometric definition of thespecimen is depicted in Figure 5. The strip is constrained at the bottom,while a tensile velocity of 20mm/s is imposed at the top. The constitutive

Table 2. Material constants for target and projectile materials.

Target material of Weldox 460E steel

E¼200GPa a1¼400MPa a¼98mm2 h1¼0�¼0.33 b1¼0.005N � ¼ 0:9 h2¼0�YYo ¼ 490MPa k1¼0.1MPa�1 ‘1 ¼ 5 mm �vp ¼ 0:01 sTo¼295K a2¼20GPa ‘2 ¼ 5 mm �vd ¼ 0:01 sTm¼ 1800K b2¼0.25N ‘3 ¼ 0 b ¼ 0�¼ 7850kg/m3 k2¼15GPa�1 ‘4 ¼ 0 m1¼ 0.94cv¼ 266J/kgK ld¼0 ‘5 ¼ 14 mm m2¼ 0.94cp¼ 452J/kgK a3¼0 n¼1 �c

eq ¼ 0:30�¼ 1.7 a4¼0 �r¼0 Pcutoff¼ 160GPa

Projectile material of hardened arne tool steel

E¼204GPa �¼0.33 �¼ 7850kg/m3 Yo¼ 1900MPaEt¼15GPa "f¼ 2.15%


parameters used in the computation are listed in Table 2 for the targetmaterial (Weldox 460 E steel). Two uniformmeshes (mesh 1: 10 20, mesh 2:20 40) are used with four-noded quadrilateral plane-strain elements. Toavoid a homogeneous solution, we slightly increase the width of thespecimen toward the top, so that the shear band will be initiated at thebottom left and develops with an inclination angle of 45.

In Figure 6(a) the deformation patterns for both meshes are plotted fora zero length-scale, ‘ ¼ 0. We observe that the width of the shear band isdetermined by the element size. Deformation is localized along a line ofintegration points. Mesh dependence is also obvious from the effectiveplastic strain plots in Figure 7(a). When the mesh is refined, the dissipatedenergy decreases. In Figure 8(a), the overall load–deflection responses aredifferent for the two resolutions. Moreover, it can be shown from theseresults that the inclusion of the geometrical nonlinearity in the dissipation ofthe strain softening material cannot solve the discretization dependence.

To solve the mesh-sensitivity problem, a length-scale parameter of2.5mm (‘ ¼ 2:5mm) is assumed to keep the field equations well-posed.In Figure 6(b), the deformation patterns are plotted for the two consideredmeshes. By comparison with the results of Figure 6(a), we observe that the

20 mm

10 mm

10.5 mm

υ = 20 mm/s

Figure 5. Problem description of a strip with geometrical imperfection and subjected to lowtensile impact velocity.


shear band has a finite width, which is almost independent of the finiteelement size. The inclination of the shear band is close to 45. Also, a similardistribution of the effective plastic strain contours can be observed inFigure 7(b) for the two meshes, which corroborates the mesh objectivity ofresults. In Figure 8(b), the overall load–deflection responses is essentiallyidentical for the two resolutions. Apparently, no visible discretizationsensitivity is present in the results.

Since the formulation of the gradient dependent theory includes aninternal length-scale as a material parameter, numerical solutions employinga strain-softening model are no longer dependent on numerical discretiza-tion. However, one limitation to the discretization remains. The width of thelocalization zone needs to be many times (roughly 6) larger than the elementsize for proper calculation of the plastic strain.

Projectile Penetrating a Target

The recent advances in aerospace and war capabilities have madenecessary the modification of the design of structures so that they can resistpenetration and perforation by projectiles with much higher impactenergies. In this respect, high performance materials need to be developedso that they can offer significant advantages over the currently usedmaterials. Specific mechanical properties are targeted by the aid of thesenew materials, such as high specific strength, high stiffness, and lowcoefficient of thermal expansion. Therefore, the high velocity impactingmechanism needs to be understood properly to be able to design materials ofhigh ballistic resistant response. However, the exact mechanism by whichthe impacting target materials undergo fracture and ablation is a relatively

Figure 6. The deformation patterns for 1020 and 2040 meshes: (a) results when ‘ ¼0and (b) results when ‘ ¼2.5.


complex process (Zukas, 1990). Generally, strong shock wave–materialinteractions are generated and propagated along both the projectile andthe target, which can lead to fracture at low global inelastic strains.Phenomenologically, as illustrated in Figure 9, the penetration can beviewed as a process to generate a cone-shaped macrocrack in the material, inwhich, the kinetic energy of the penetrator is dissipated.

The computational model presented in the previous sections is used hereto model a blunt projectile impacting a target. This computational modelis implemented in ABAQUS/Explicit (2003) using VUMAT user materialsubroutine. The objective of this numerical example is to investigate if theproposed constitutive equations are able to describe the structural response

Figure 7. The effective plastic strain contours for 1020 and 2040 meshes: (a) resultswhen ‘ ¼ 0 and (b) results when ‘ ¼2.5mm.


to projectile impact damage when different failure modes are expected tooccur. This is carried out by conducting a numerical simulation of theexperimental tests presented by Borvik et al. (2002) for a blunt projectilemade of hardened Arne tool steel impacting a circular plate made of Weldox460 E steel. For more detailed information regarding the numerical scheme,such as the contact-impact algorithm, the automatic mesh generator, theArbitrary-Lagrangian-Eulerian (ALE) adaptive meshing, and the element

0

2

4

6

8

10

Vtop/H

F/(

Bs y

)

10 x 20 Mesh

20 x 40 Mesh

(b)

(a)

0

2

4

6

8

10

0 0.3 0.6 0.9 1.2 1.5

0 0.3 0.6 0.9 1.2 1.5

Vtop/H

F/(

Bs y

)

10 x 20 Mesh

20 x 40 Mesh

Figure 8. The load–deflection diagrams for 1020 and 2040 meshes: (a) results when‘ ¼ 0 and (b) results when ‘ ¼2.5mm.


erosion algorithm, the reader is referred to the manuals of ABAQUS/Explicit (2003).

For simplicity, the projectile is modeled as a bilinear elastic-plasticstrain rate-independent von Mises material with isotropic hardening, whichis already implemented in ABAQUS/Explicit code. The model materialconstants of the target material of Weldox 460 E steel and of the projectilematerial of hardened Arne tool steel are listed in Table 2. These materialconstants are determined based on the following considerations.

Although there has been tremendous efforts to understand the physicalrole of the gradient theory, this research area is still in a critical state withcontroversy. This is to some extent due to the difficulty in calibration of thedifferent material properties associated with the gradient-dependent models,which are impossible to obtain for certain cases. More important is thedifficulty of carrying out truly definitive experiments on critical aspects ofthe evolution of the dislocation, crack, and void structures. Furthermore,the calibration of a gradient-dependent model should not only be based onstress–strain behavior obtained from macroscopic mechanical tests, butshould also draw information from micromechanical, gradient-dominanttests, such as micro/nano-indentation tests, microbending tests, andmircotorsion tests. Abu Al-Rub and Voyiadjis (2004) and Voyiadjis andAbu Al-Rub (2005) presented a physically based approach for identificationof the material length-scales, ‘1 and ‘2, from microindentation, micro-bending, and/or microtorsion tests. Therefore, the values for ‘1 and ‘2 listedin Table 2 are identified using this approach. The values of ‘3 and ‘4 areassumed zeros and the value of ‘5 is identified based on the one that givesthe closest agreement with the experimental results of impact velocity versusthe projectile residual velocity. The values for E, �, To, Tm, �, cv, cp, �, �vp,

Existing cracks and voids as damageenhancement

Grain translation and rotation

Zones of high irrecoverable deformations asshock buffer, interface

Shock wave, compressionwave is reflected astensile wave at freeboundary

Shock wave

Projectile

v(t) Target

Shear/tension zone

Figure 9. Phenomenological illustration of the penetration process.


m1, n, and �ceq are as reported by Borvik et al. (2002). The material constants

a1 and k1 associated with the plasticity isotropic hardening are identifiedfrom a tensile test, while the material constants a2 and k2 associated withthe plasticity kinematic hardening are identified from a strain control cyclichardening test (Voyiadjis and Abu Al-Rub, 2003). Damage initiation isassumed to occur at the beginning of loading such that ld¼ 0. The damageisotropic and kinematic hardening is assumed negligible such that a3¼ a4¼ 0and h1¼ h2¼ 0, respectively. The material constants b1, b2, and a areidentified as b1 ¼ ð1=2Þa1‘

21, b2 ¼ ð1=2Þa2‘

22, and a ¼ ð1=2Þ‘23. We also

assumed that �vp¼�vd and m1¼m2. The material constants used for thehardened Arne tool steel are as those reported by Borvik et al. (2002). Thematerial constants listed in Table 2 are used to conduct the followingsimulations.

In these simulations, a four-node two-dimensional axisymmetric elementwith one integration point and a stiffness based on hourglass control is used.The plot of the initial configuration, showing a part of the target plate andthe blunt projectile just prior to impact, is shown in Figure 10. The targetplate has a nominal thickness of 12mm and a diameter of 500mm, while thenominal length and diameter of the hardened projectile are 80 and 20mm,respectively. In each run, the target plate is fully clamped at the support,while the projectile is given an initial velocity similar to the one used inthe corresponding experiment conducted by Borvik et al. (2002). The initialsize of the smallest element in the impact region is 0.25 0.2mm2 in all

Projectilehardened Arne tool-steel

TargetWeldox 460 E steel

Figure 10. Finite element mesh plot of the axisymmetric initial configuration just beforeimpact.


simulations, giving a total of 60 elements over the target thickness. Toreduce the computational time, which is affected both by the element sizeand number, the mesh was somewhat coarsened towards the boundary.Owing to this coarsening, the total number of elements in the target plateis not more than about 10,000 in the simulations. Contact was modeledusing an automatic two-dimensional single surface penalty formulationavailable in ABAQUS/Explicit. Frictional effects are neglected for the bluntprojectile. Time increments of order 10�8 s are used to satisfy the stabilitycriteria.

An adaptive meshing technique has received the most attention during thelast decade to solve high-speed impact damage problems. Coupled ALEis used to extend the domain of application in the Lagrangian codes.The advantages of using adaptive meshing in high-speed impact damagepenetration problems are many. It enables the simulation of large inelasticflow in Lagrangian framework. It may also include the possibility to obtaina solution of comparable accuracy using much fewer elements, and henceless computational resources than with a fixed mesh. In addition, it preventssevere mesh distortions and unacceptable small time steps in the simula-tions. However, the major disadvantage of this method is the possibleintroduction of inaccuracies and smoothening of the results during mappingof the history variables. Adaptive meshing is used in the following simula-tions, where 10 adaptive refinements are used in each run.

Borvik et al. (2001) simulations indicate the problem involving shearlocalization and plugging for blunt projectiles to be mesh size sensitive.However, the numerical solution using the present model is mesh sizeindependent and converges monotonically towards a limit solution when thenumber of elements over the target thickness becomes sufficiently large.Therefore, the numerical results improve as the element size is reduced untilit stabilizes at some value, i.e., the mesh size dependency is not pathological.This is expected since the width of the shear band is known to be in the orderof 10–100 mm (Bai and Dodd, 1992).

A direct comparison between the numerical and experimental residualvelocity curves for a blunt projectile is shown in Figure 11. As seen, there isgood agreement with the experimental results. Moreover, Figure 12 showsthat the perforation times obtained from the different numerical simula-tions are similar and close to the experimental values estimated from high-speed camera images obtained by Borvik et al. (2002). The plots showingperforation of the target plate by a blunt projectile at an impact velocityclose to the ballistic limit reported by Borvik et al. (2002) of 210m/s areshown in Figure 13. Here, the contours of accumulated viscoinelastic strainare plotted on the deformed mesh. It can be seen that limited inelasticdeformation occurs outside the localized shear zone. These plots clearly


demonstrate that the numerical model qualitatively captures the overallphysical behavior of the target during penetration and perforation. Noticealso that in these plots, only a part of the complete target plate is shown.

The elements in the impacted area are significantly distorted but stableresults are obtained. This delayed the damage evolution process and

0

100

200

300

400

500

100 200 300 400 500 600

Initial velocity (m/s)

Res

idua

l vel

ocit

y (m

/s)

Numerical

Experimental (Borvik et al., 2002)

Figure 11. A comparison between numerical and experimental results for the initial impactvelocity vs residual projectile velocity by a blunt projectile.

20

40

60

80

100

120

100 200 300 400 500 600

Initial projectile velocity (m/s)

Per

fora

tion

tim

e (m

s)

Numerical

Experimental (Borvik et al., 2002)

Figure 12. Comparison between perforation times vs initial impact velocities by a bluntprojectile.


consequently the erosion of damaged elements. However, the upper nodes ina critical element do not penetrate the lower nodes, giving a stable solutionand no error termination is encountered. Figure 14 shows the elementdistortion in the target plate just after impact.

The final cross section of a target plate perforated by a blunt projectile atan impact velocity close to the ballistic limit is shown in Figure 15, whichis very similar to the experimental cross section obtained by Borvik et al.(2002).

In general, it can be seen that close agreement between the numerical andexperimental results is achieved. Hence, the computational methodologypresented in this article seems to work well for ductile targets perforatedby deformable blunt projectiles. a More elaborate study using the current

Figure 14. Penetration of the target plate by a blunt projectile of initial impact velocity of300m/s using ALE meshing, plotted as contours of accumulated viscoinelastic strain andshowing details of element meshes just after adaptive remeshing.

Figure 13. Perforation of the target plate by a blunt projectile of initial impact velocityof 210m/s using ALE meshing, plotted as contours of accumulated viscoinelastic strainat different times. The lighter region indicates an accumulated damage between 0.25and 0.30.


gradient-dependent viscoplastic and viscodamage model is needed forsimulating high-speed impacts of different target thicknesses and also usingdifferent projectile nose shapes (e.g., hemispherical and conical). This is notthe subject of this article but will be presented in detail in a forthcomingarticle by the current authors.

CONCLUSIONS

In the first part of this study, it was recognized that the initiationand propagation of the microdefects depend on both the amplitude anddistribution of the inelastic strains/stress in the vicinity around the micro-defect. It means that evolution of the material inelasticity is practicallya nonlocal process. These observations imply that there is a need for sucha micro-mechanical approach that incorporates material length-scaleparameters into the classical constitutive relations. The incorporationof spatial higher-order gradient terms allows us to introduce a ‘missing’length-scale into the classical continuum theories allowing the size effect tobe captured.

A gradient-enhanced (nonlocal) coupled rate-dependent plasticity anddamage model is developed on the continuum level to bridge the gapbetween the micromechanical and classical continuum inelasticity. Thethermodynamical consistency of the gradient-dependent plasticity/damageconstitutive relations is discussed thoroughly in Part I of the companionarticle of this work. Implicit and explicit incorporation of material length-scales is achieved through the development of coupled viscoinelasticity (rate-dependent plasticity/damage) and nonlocal gradient-dependent constitutive

Figure 15. Final cross section of the target plate perforated by a blunt projectile of initialimpact velocity of 300m/s using ALE meshing and plotted as contours of accumulatedviscoinelastic strain. The lighter region part indicates an accumulated damage between 0.25and 0.30 (critical value).


relations, respectively. Therefore, the resulting model incorporates in avery modular fashion the damage and plasticity effects in solids, and mostimportantly in a highly physically motivated way.

Several numerical algorithms are developed in this part of the article,to integrate the highly nonlinear differential equations effectively and tooptimize the computational performance. We have presented a returnmapping algorithm for the integration of the proposed model in Part I,sharing remarkably the exact same structure as the classical return mappingemployed in the integration of rate-independent plasticity/damage andgradient-independent plasticity/damage. An operator split structure con-sisting of a trial state followed by the return map is developed by imposinga generalized (rate-independent to rate-dependent) viscoplastic and visco-damage consistency conditions simultaneously. This allows us to integrateboth rate-dependent and rate-independent models in a similar fashion.Moreover, a simple and direct computational algorithm is also used forcalculation of the Laplacian gradients allowing us to integrate both thegradient-dependent and gradient-independent models in a similar way.Furthermore, a trivially incrementally objective integration scheme isproposed. The proposed finite deformation scheme is based on hypoelasticstress–strain representations and the proposed hypoelastic predictor andcoupled viscoplastic–viscodamage corrector algorithm that allows for totaluncoupling of the geometrical and material nonlinearities. The nonlinearalgebraic system of equations is solved by consistent linearization and theuse of the Newton–Raphson iteration. The numerical implementation theninvolves a series of coupled routines providing the stresses and updates ofthe corresponding internal variables.

The proposed model and numerical integration algorithms are imple-mented in the well-known explicit finite element code ABAQUS/Explicitusing the material subroutine VUMAT. The first and second numericalsimulations presented herein illustrate the numerical performance of theproposed numerical algorithms. They also showed that the proposed modelintroduces properly the localization limiting of rate-dependent and nonlocalcontinua: the results of finite element simulations are almost insensitive tomesh refinement, since the width of the shear bands (fracture process zone)is determined by the internal length-scale incorporated in the proposedmodel.

Moreover, we have presented in this article the application of theproposed model to the numerical simulation of localization and formationof shear bands in structures subjected to high-speed impact loadingconditions. Good agreement is obtained between the numerical simulationsand experimental results. Therefore, the material length-scale has thepotential to predict the size effects in material failure.


We believe that the aforementioned advantages provided by the proposedmodel do lead to an improvement in the modeling and numerical simula-tion of high velocity impact related problems. We plan to explore thismodel further. The study of strain localization and its regularization onimpact damage related problems needs further attention to allow us toproperly simulate impact damage problems without conducting extensiveexperiments.

ACKNOWLEDGMENTS

We are indebted to Professor Anthony N. Palazotto for the helpfuldiscussions through the different stages of this work. Financial support forthis research has been provided by the AFSOR through the Air ForceInstitute of Technology at WPAFB, Ohio, under grant number F33601-01-P-0343. This support is gratefully acknowledged. We also acknowledgethe financial support under grant number M67854-03-M-6040 providedby the Marine Corps Systems Command, AFSS PGD, Quantico, Virginia.We thankfully acknowledge our appreciation to Howard ‘Skip’ Bayes,Project Director.

APPENDIX A

From the heat balance equation we have

Qtp1 ¼

1

�0cp�s0 þJeP1þTibð Þ :

@f

@sþ

V3

1� r_þ

V7r2r

ð1� r_Þ2

� ��1�k1

�R_

R_��k1V7

1� r_r

2 �RR

�

� V4 :M_

þV8 :r2M

_� �

:@f

@ �XXþk2

�X_

X_

� ��k2V8 : M

_

:r2 �XX

ðA-1Þ

Qtp2 ¼

1

�ocp

V7

1� r_

�1�k1

�R_

R_��V8 :M

_

:@f

@ �XXþk2

�X_

X_

� �� ðA-2Þ

Qtd1 ¼

1

�ocp�s0 þJeP1þTibð Þ :

@g

@sþV5 1�h1K

_� �

�h1V9r2K

�

�V6 :@g

@Hþh2H

_� �

�h2V10 :r2H

ðA-3Þ

Qtd2 ¼

1

�ocpV9 1�h1K

_� �

�V10@g

@Hþh2H

_� ��

ðA-4Þ


From the viscoplasticity generalized consistency condition we obtain

Z p ¼ �@f

@T�

@f

@s: b

� �� J eP1þ Tbð Þ ðA-5Þ

Qp1 ¼ Qtp

1 �@f

@s: C :

@f

@sþ A :

@f

@Y

� �þ

@f

@pþ

1

�t

@f

@ _pp

� �1� k1�R_

R_�

1� r_

þ@f

@r2pþ

1

�t

@f

@r2 _pp

�

r2r

ð1� r_Þ2

�1� k1

�R_

R_��

k1

1� r_r 2 �RR

�

�@f

@a: M

_

:@f

@ �XXþ k2

�X_

X_

� ��

@f

@r2a: r2M

_

:@f

@ �XXþ k2

�X_

X_

� �þ k2M

_

: r2 �XX

�

þ@f

@/:@f

@YðA-6Þ

Qp2 ¼ Qtp

2 � a@f

@s: A :

@f

@Yþ

@f

@r2pþ

1

�t

@f

@r2 _pp

� �1� k1�R_

R_�

1� r_

�@f

@r2a: M

_

:@f

@ �XXþ k2

�X_

X_

� �þ

@f

@r2/:@f

@YðA-7Þ

Qp3 ¼ Qtd

1 �@f

@s: C :

@g

@sþ A :

@g

@Y

� �þ

@f

@/:@g

@YðA-8Þ

Qp4 ¼ Qtd

2 � a@f

@s: A :

@g

@Yþ

@f

@r2/:@g

@YðA-9Þ

From the viscodamage generalized consistency condition we have

Zd ¼ �@g

@T�@g

@s: b

� �� J eP1þ Tbð Þ ðA-10Þ

Qd1 ¼ Qtp

1 �@g

@s: C :

@f

@sþ A :

@f

@Y

� �þ

@g

@/:@f

@YðA-11Þ

Qd2 ¼ Qtp

2 � a@g

@s: A :

@f

@Yþ

@g

@r2/:@f

@YðA-12Þ


Qd3 ¼ Qtd

1 �@g

@s: C :

@g

@sþ A :

@g

@Y

� �þ@g

@r1� h1K

_� �

� h1@g

@r2rr 2K þ

1

�t

@g

@_rr

�@g

@C:

@g

@Hþ h2H

_� �

� h2@g

@r2C: r2Hþ

@g

@/:@g

@YðA-13Þ

Qd4 ¼ Qtd

2 � a@g

@s

��TA @g

@Yþ

@g

@r2r1� h1K

_� �

þ1

�t

@g

@r2 _rr�

@g

@r2C

@g

@Hþ h2H

_� �

þ@g

@r2/

��T @g@Y ðA-14Þ

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