Journal of the Mechanics and Physics of Solids 56 (2008) 1624–1650

Modelling of dynamic ductile fracture and application to the

simulation of plate impact tests on tantalum

C. Czarnotaa, N. Jacquesb, S. Merciera,�, A. Molinaria

aLaboratoire de Physique et Mecanique des Materiaux, UMR CNRS 7554, Universite Paul Verlaine - Metz, Ile du Saulcy,

57045 Metz Cedex, FrancebLaboratoire de Mecanique des Structures Navales, ENSIETA, 2 rue Franc-ois Verny, 29806 Brest Cedex 9, France

Received 16 March 2007; received in revised form 11 July 2007; accepted 18 July 2007

Abstract

A model of dynamic damage by void nucleation and growth is proposed for elastic–viscoplastic materials sustaining

intense loading. The model is dedicated to ductile materials for which fracture is caused by microvoiding. The material

contains potential nucleation sites where microvoids are generated when the local pressure overcomes the nucleation

pressure. A probability density function is adopted to describe the fluctuation of the nucleation pressure within the

material. The void growth is described by using a hollow sphere model where micro-inertia effects are accounted for. The

matrix weakening due to void growth is also included.

The model has been first tested under uniaxial deformation. When the strain rate is assumed constant, the pressure inside

the material has nearly a linear response up to a maximum. An analytical expression for the maximum pressure is

proposed.

Finite element simulations of plate impact tests have been carried out and compared to experiments on tantalum. From

simulations based on the proposed model, an increase of the spall strength is observed with higher shock intensities.

Therefore, the relationship between the velocity pullback and spall strength usually assumed in the literature (based on the

acoustic approach) seems to be inadequate. Velocity profiles are simulated for different flyer thicknesses and different

impact velocities with close agreement with experiments.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Spall; Plate impact; Dynamic ductile fracture; Inertia effects; Tantalum

1. Introduction

Spall fracture is a dynamic fracture process that occurs in several fields such as: armor penetration, collision

of particles against satellite, laser induced shock. In laboratories, the spall fracture is conventionally obtained

when a target plate is impacted by a flyer plate. During impact, compressive shock waves are generated in the

two plates. As they encounter the free surface of the target and of the flyer, they reflect back in rarefaction

waves. In the spall plane, where these waves interact, a large tensile stress is generated and spalling is initiated.

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0022-5096/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jmps.2007.07.017

�Corresponding author. Tel.: +33 03 87 31 54 09; fax: +33 03 87 31 53 66.

E-mail address: mercier@lpmm.univ-metz.fr (S. Mercier).

More details are available in Meyers and Aimone (1983) and Antoun et al. (2003) who provide an overview on

various aspects of spalling.

Numerous experimental studies have been performed on different materials and for different flyer plate

velocities. Kanel et al. (1997) performed plate impact tests on aluminium and magnesium over a wide range of

loading duration and temperature; see also Seaman et al. (1976). Other materials have been tested, see for

example Boidin et al. (2006) for titanium alloys, Cochran and Banner (1977) for uranium, Steinberg and Sharp

(1981) for beryllium, Seaman et al. (1976) and Johnson (1981) for copper and Zurek et al. (1996), Roy (2003)

and Llorca and Roy (2003) for tantalum. Since no direct damage measurement can be done in situ during the

short duration of the impact test (order of a microsecond), the fracture mechanism is rather deduced from

external measurements such as the free-surface velocity of the target. VISAR interferometry is generally used

to get the free-surface velocity profile of the sample. From this measurement, the spall strength is generally

related to the amplitude of the velocity pullback (difference between maximum and minimum velocities of the

free surface). Zurek et al. (1996) noticed that a sharp velocity pullback indicates that the material has

undergone a complete separation. Eftis et al. (1991) in a first attempt to model spall experiments on OFHC

copper, mentioned that the amplitude of the pullback signal is related to the amount of tensile stress relaxation

and then to the corresponding level of damage. They considered that the damage is cumulated very rapidly in

0.05–0:06ms for the following conditions: flyer thickness 0.6mm, target 1.6mm and impact velocity 160m/s. In

fact, the fracture mechanism is complex. Kanel and co-workers have performed numerous spall experiments

(these results are presented in Antoun et al., 2003). From free-surface velocity measurements and

metallographic examinations of spalled specimens, Kanel and co-workers have observed that a velocity

pullback occurs even when a very low damage is observed in the spall plane. The stress relaxation due to the

initiation of the damage is responsible for the pullback. Due to relaxation, the subsequent damage is therefore

cumulated at lower stress magnitude and complete fracture occurs latter during the test. The dynamic damage

model developed in the present paper agrees with these observations.

In ductile materials, damage occurs by nucleation and growth of microvoids. Lots of theoretical models

have been devoted to couple void growth to elastic–(visco)plastic material properties. Under quasi-static

conditions, McClintock (1968), Rice and Tracey (1969) and Huang (1991) have analysed the evolution of a

spherical void in an infinite or finite matrix. Lee and Mear (1994) have determined the evolution of ellipsoidal

cavities in a finite matrix. Since in impact experiments, the level of pressure is high with respect to the tensile

yield stress, voids are observed to remain spherical, Antoun et al. (2003) and Roy (2003). As a consequence, no

deviation from spherical shape will be considered in our analysis.

The dynamic expansion of a single spherical void in an unbounded solid under hydrostatic tension was

analysed by Ortiz and Molinari (1992). These authors found that inertia dominates the late stage of cavity

growth whereas the early stage is controlled by strain rate and strain hardening effects. Inertia has a significant

role when voids reach a characteristic size. For dynamic evolution of a void in a bounded matrix, pioneering

works are those of Knowles and Jakub (1964) and Carroll and Holt (1972), see also Tong and Ravichandran

(1995). These authors investigated the dynamic response of a hollow sphere under spherical tensile or

compressive loading. Inertial effects are shown to be an important feature for high loading rates. In Carroll

and Holt (1972), the matrix material is elastic perfectly plastic. Johnson (1981) extended the previous work by

considering a rate-dependent material, the flow stress of the matrix being an affine function of the strain rate.

Inertia effects are shown to be negligible and void growth is shown to be controlled by viscosity. These results

could be due to the large strain rate sensitivity considered in this work (roughly equal to one), value which is

unrealistic for metallic materials. Perzyna (1986) revisited the Johnson (1981) contribution by including strain

hardening. Cortes (1992) generalized the two approaches by considering various viscoplastic descriptions of

the flow stress. Recently, Molinari and Mercier (2001) have developed a hollow sphere model, which accounts

for the viscoplastic behaviour of the matrix material and for inertia effects associated to the rapid void growth.

They found that the macroscopic stress has two contributions which are both porosity dependent: a quasi-

static part, which can be derived from any existing quasi-static potential, and an inertia-dependent part, see

also Wang and Jiang (1997), Leblond and Roy (2000) and Weinberg et al. (2006).

All previous models consider that the material is initially voided. For a pertinent physical description of

spalling, void nucleation has to be accounted for. Cavitation occurs when the applied pressure reaches a

critical value above which void growth is unbounded. Ball (1982) was the first to examine cavitation in

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nonlinear elastic solids. In elastic–plastic solids, cavitation is driven by the elastic energy of the surrounding

material. Early works on cavitation instabilities are summarized in Hill (1950) and other developments are

found in Horgan and Polignone (1995), Huang et al. (1991) and Tvergaard et al. (1992). In a recent work, Wu

et al. (2003) have clearly analysed the dependence of the nucleation pressure upon elastic and thermoplastic

properties of the matrix material.

From observations on specimens recovered after shock, the distribution of void radii can be measured.

Bontaz-Carion and Pellegrini (2006) have quantified the damage in the target after plate impacts, using X-ray

microtomography. By means of this technique, they obtained three-dimensional images for the pore volume

distribution. Roy (2003) observed on tantalum that the microvoids are located near the spall plane with a

heterogeneous size distribution. In addition, the number of microvoids per unit volume increases strongly with

the shock pressure. Seaman et al. (1976) have observed, from metallographic examinations, that the number of

voids (per unit volume) with radii greater that a given radius R is decreasing exponentially with R. From tests

at different stress levels and durations, the nucleation rate is found to increase exponentially with pressure. In

addition, the void growth, based on the expansion of a void in a viscous fluid, is assumed to be an affine

function of the pressure level. Based on this nucleation and growth scheme (NAG), Seaman et al. (1976) were

able to estimate the total volume of voids and the porosity in the plate during impact tests. Note that micro-

inertia is not included in their approach. Recently, Molinari and Wright (2005) have considered the case of a

material initially free of voids. Potential sites for nucleation are present in the material and a void nucleates at

a potential nucleation site when the local pressure overcomes a critical nucleation pressure. Nucleation

pressures are assumed to follow a statistical distribution. The growth of nucleated voids is governed by micro-

inertia. In the model of Molinari and Wright (2005), each void is considered as isolated and is therefore

embedded in an infinite matrix. Thus, inertia effects are overestimated. Czarnota et al. (2006) revisited the

work of Molinari and Wright (2005) by considering that void growth is based on the dynamic response of a

hollow sphere and by accounting for the matrix softening due to the increase of porosity. To some extent, void

interactions could be accounted for in this approach. The spall strength of a high purity grade tantalum was

then predicted when assuming the loading in the spall plane to be a pressure ramp.

To simulate plate impact experiments and to account for wave propagation, damage models based on the

above theoretical works have been implemented in different numerical codes. Johnson (1981) carried out one-

dimensional finite difference calculations of the plate impact test and compared results with experiments

performed on copper. The material response was assumed elastic–plastic, the yield strength being porosity

dependent. The Mie–Gruneisen equation of state was adopted to describe the relation between pressure and

volume strain. Since the Mie–Gruneisen approach was developed for solid response with no permanent

inelastic volume strain, a modified equation was proposed which integrates the inelastic volume strain

generated by the porosity. Inertial effects during the rapid expansion of microvoids were taken into account.

Velocity profiles obtained with the numerical model were in agreement with experiments on copper. Eftis and

Nemes (1991), based on the Perzyna (1986) model, performed one-dimensional finite element calculations of

the plate impact problem, for two plates made of OFHC copper. The void growth rate has two contributions:

a nucleation term governed by thermally activated mechanisms (Seaman et al., 1976; Curran et al., 1987) and a

growth rate term. As in Perzyna (1986), the void growth is governed using a quasi-static approximation and

micro-inertia is disregarded. The justification of this approximation is based on Johnson (1981) who observed

that the void growth is dominated by plastic deformation rather than by micro-inertia. From one-dimensional

finite difference simulations, the authors evaluated the evolution of the axial stress in the plate after impact.

They observed a softening of the tensile wave in the spall plane as the damage is cumulated. They showed that

the entire spall process, from the beginning of damage development to the occurrence of the spallation

(obtained when a critical void volume fraction was reached), took approximately 0:06ms. Recently, Eftis et al.

(2003) investigated the hypervelocity projectile-target impact. For the projectile velocity considered (6 km=s),the material remains in the solid state. Contrary to plate impact at moderate velocities, the effect of

temperature cannot be disregarded. The deviatoric stress tensor evolution is governed by the Perzyna (1963)

model. The mean stress is related to the total volume strain and energy through a Mie–Gruneisen type

equation. Note that Johnson (1981) was using some similar approach for the mean stress but has modified this

relation in order to take into account the softening due to the inelastic volume change generated by void

growth. This is not the case in Eftis et al. (2003) approach. Void growth is governed by a relationship where

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micro-inertia is present. Using the proposed approach, these authors simulated the impact of spherical glass

projectiles on an aluminium 1100 rectangular target plate. The simulated and experimental damages were

comparable. Numerous other approaches, not related to micro-inertia effects, have also been developed to

simulate spall experiments, Cochran and Banner (1977), Curran et al. (1987), Rajendran et al. (1989), Feng et

al. (1997), Lin et al. (2004), Suffis (2004), Campagne et al. (2005). One may refer to Chen et al. (2005) who

carried out one-dimensional simulations of impact experiments. They mentioned that the wave profile in the

target from the spall plane to the free surface is affected seriously by the stress relaxation due to damage

evolution. They concluded that the traditional determination of the spall strength (related to the pullback

velocity) without accounting for stress relaxation has to be used with caution. Note also the work of Ikkurthi

and Chaturvedi (2004) for a comparison of different models used for plate impact simulations.

In the present paper, a new constitutive damage model is proposed for the dynamic fracture of ductile

materials. The model of void nucleation and growth proposed by Czarnota et al. (2006) is extended to account

for elastic–viscoplastic material response (see also Wright et al., 2005). In the next section, the model is

presented and the general equations are formulated. The proposed model has been implemented in the finite

element software ABAQUS/Explicit. By simulations of plate impact tests, the damage evolution near the spall

plane can be analysed in details. Special attention is paid to the explanation of the pullback signal observed on

the free-surface velocity and the correlation with damage evolution. Finally, a comparison of the simulated

free-surface velocity profile with experimental data of Roy (2003) is carried out for a high purity grade

tantalum. The present approach integrates physical concepts such as progressive nucleation and void growth

governed by micro-inertia. Most of the model parameters are obtained from experimental data. In the

literature, numerous works have been dedicated to spallation and were compared to a restricted range of

experimental data. In the present contribution, simulations for different impact velocities and different target

thicknesses have been conducted, with the same set of material parameters. Results are found to be accurate

for all test configurations. To the authors point of view, a physically based model with such predictive

capability is quite new in the context of dynamic damage and spallation.

2. Constitutive modelling

2.1. Basic equations

The material is elastic–viscoplastic and the total strain rate tensor d is split into an elastic part de and a

viscoplastic part dvp

d ¼ de þ dvp. (1)

During high velocity plate impact experiments, voids are nucleated and their growth induces that inelastic

deformation has a spherical part. Therefore, the viscoplastic strain rate tensor dvp can be written as follows:

dvp ¼ dvp0 þ dvpm I , (2)

where the superscript ð0Þ refers to deviatoric quantities. dvpm ¼ 1

3trðdvpÞ is the mean viscoplastic strain rate and I

represents the second order identity tensor.

In the following, isotropic linear elasticity is considered, and the effect of damage (microvoiding) on elastic

stiffness is accounted for. Thus, the elastic law takes the following form, see Becker (2002) and Adam (2003):

~s ¼ C : de þ _C : Cÿ1: s , (3)

where C is the fourth order tensor of moduli for isotropic elasticity, _C represents the time derivative of the

tensor of elastic properties. _C is not frame dependent since only isotropic elasticity is considered in the present

approach. ~s is an objective time derivative for the stress tensor. Several definitions of ~s can be used, for

instance the Jaumann rate or the Green–Naghdi rate.

Eq. (3) can be decomposed into deviatoric and spherical components:

~s0 ¼ 2Gde0 þ_G

Gs0 and _p ¼ 3Kde

m þ_K

Kp, (4)

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where G is the current shear modulus of the voided material. p ¼ 13trðsÞ represents the mean stress or pressure

(here the pressure is positive for tension, negative for compression). dem ¼ 1

3trðdeÞ is the mean elastic strain

rate. K is the current elastic bulk modulus.

The presence of microvoids softens the elastic properties of the material. The elastic moduli G and K are

degraded according to the hollow sphere model proposed by Mackenzie (1958) and modified by Johnson

(1981):

G ¼ G0ð1ÿ f Þ 1ÿ6K0 þ 12G0

9K0 þ 8G0

f

� �

; K ¼ K0

4G0ð1ÿ f Þ

4G0 þ 3K0f, (5)

where f represents the volume fraction of voids (porosity). G0 and K0 are the elastic shear and bulk moduli of

the void-free material.

2.2. Deviatoric viscoplastic flow

The non-voided material is assumed to be elastic viscoplastic. A powerlaw equation is adopted to

characterize the flow stress sy, where strain hardening and strain rate sensitivity are taken into account:

sy ¼ kðd0 þ deqÞmð�0 þ �eqÞ

n, (6)

where m is the strain rate sensitivity, n the strain hardening exponent. d0 and �0 are reference strain rate and

strain, k a scaling factor for the flow stress level. deq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

23ðdvp0

: dvp0Þ

q

is the equivalent plastic strain rate and

�eq ¼R

deq dt is the cumulative plastic strain. (:) denotes the twice contracted product of second order tensors.

In our approach, the flow stress sy is only depending upon plastic strain rate and plastic strain, and therefore

the effect of temperature on the viscoplastic behaviour seems not to be accounted for. However the effect of

heating can be evaluated indirectly by considering the adiabatic stress–strain curves obtained from

experiments conducted at high strain rates. Then the effective strain hardening coefficient n is lower than

its isothermal counterpart as it integrates the effect of thermal softening. In Section 3, we have indirectly

evaluate the temperature effects by comparing calculations using an adiabatic strain hardening exponent to

calculations based on the isothermal material response.

The deviatoric viscoplastic strain rate tensor dvp0 is related to the deviatoric Cauchy stress tensor s0, using a

J2 flow law:

dvp0 ¼3

2deq

s0

seq, (7)

where seq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

32ðs0 : s0Þ

q

represents the equivalent stress. The yield function F, based on isotropic strain

hardening, is supposed to account for the softening due to porosity:

F ¼seq

1ÿ af bÿ syðdeq; �eqÞ, (8)

where sy is given by Eq. (6). Different values of the parameters a and b have been proposed in the

literature. Eftis et al. (1991) adopted the following set: a ¼ 1=ðf cÞ0:5, b ¼ 0:5, f c being the critical porosity for

complete loss of stress carrying capacity of the material. Eftis et al. (2003) proposed another set of values:

a � 1=f c, b ¼ 1. Relationships (7) and (8) reduce to the usual associated J2 flow theory for a non-voided material.

2.3. Dynamic damage evolution

2.3.1. Description of the material

The main features of the dynamic model recently proposed by Czarnota et al. (2006) are presented here.

To each material point is associated a representative volume element (RVE), which contains potential

nucleation sites. The total number of sites per unit volume is noted N and can be related to a characteristic

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length b0 defined by

N ¼1

ð4p=3Þb30. (9)

Initially, the RVE is free of voids (but the case of a material with initial voids could be as well treated). A void

nucleates at a potential site when the local pressure p overcomes a cavitation pressure pc. The nucleation

pressure pc is assumed to vary site by site and to follow a statistical distribution governed by a Weibull

probability density function:

W ðpcÞ ¼bwZw

pc ÿ p0cZw

� �bwÿ1

exp ÿpc ÿ p0c

Zw

� �bw" #

for pcXp0c, (10)

where Zw and bw are, respectively, the scale and shape parameters, and p0c stands for the location parameter. It

is worth noticing that, below the cutoff value p0c, there is no potential nucleation site (the probability of

nucleation is W ðpcÞ ¼ 0 if pcpp0c). Note also that the shape parameter has a marked effect on the density

distribution function W ðpcÞ. In the present approach, the continuous distribution (10) is discretized. N f

families of potential nucleation sites are defined. Each family i has an appearance frequency Pri (related to the

Weibull function (10)) and a nucleation pressure pci. Note thatPN f

i¼1Pri ¼ 1.

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.

.

.

.

.

.

..

.

.

.

.

.

..

Potential nucleation sites

Nucleated voids from potential nucleation sites

.

.

.

.

.

.

..

.

.

.

.

.

..

Nucleated voids from

Potential nucleation sites

2a

2b

Void just nucleated

potential nucleation sites

Fig. 1. (a) Schematic view of the representative volume element with potential nucleation sites and nucleated voids. (b) When a new family

of voids becomes active, the matrix material is distributed over all voids so that the outer radius is identical for all unit cells.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1629

When a potential site nucleates, a void of null volume is created. Later in the deformation process, the void

is assumed to remain spherical with radius a. As seen in Fig. 1, the void is embedded into a shell of outer

radius b made of matrix material. The corresponding hollow sphere (inner radius a and outer radius b) defines

a unit cell. The evolution of this unit cell is based on the dynamic response of the hollow sphere subjected to a

pressure loading, see Knowles and Jakub (1964), Carroll and Holt (1972) for the foundation of this approach.

In the present paper, we follow the model proposed by Czarnota et al. (2006).

Plastic volume strain is generated by void growth. Assuming that the elastic compressibility of the material

does not contribute to the porosity evolution, the rate of void volume fraction _f is related to the mean

viscoplastic strain rate dvpm , owing to plastic incompressibility of the matrix material:

_f ¼ 3ð1ÿ f Þdvpm . (11)

The mean distance between nucleated sites decreases when new sites are activated by a pressure increase.

This feature was not taken into account in Czarnota et al. (2006) but is now introduced in the present analysis.

The determination of the mean distance is based on volume conservation of the matrix material (elastic

deformation neglected). When a new potential site is nucleated, the volume of matrix is distributed over all

nucleated sites so that each void (including the new one which has just been created) will be embedded into a

spherical shell of matrix material, the outer radius of all spherical shells being identical for all unit cells, see

Fig 1. In that way, the mean distance decreases with increasing pressure, see Appendix A.

To summarize, it is assumed that potential sites for nucleation are distributed in the material domain. When

the local pressure overcomes the cavitation pressure, a microvoid is created with radius a equal to zero at

nucleation. The matrix material is equally distributed around all nucleated sites. Each void is embedded into a

spherical shell (inner radius a and outer radius b). The corresponding hollow sphere defines the unit cell. The

evolution law for the unit cell is given in the next section, see also Czarnota et al. (2006). When new sites are

activated, unit cells are re-defined, keeping the current radius of voids and adjusting the shell outer radius b in

order to preserve the volume of matrix material.

2.3.2. Void evolution in the unit cell (hollow sphere)

Consider a given family i associated to potential nucleation sites having the same cavitation pressure pci . Upon

loading, when the pressure exceeds the value pci , voids of the family i are nucleated. To each void is associated a

unit cell consisting of a hollow sphere of inner radius ai and outer radius bi. The local porosity in the cell is given by

f i ¼a3i

b3i. (12)

According to the analysis of Czarnota et al. (2006), the evolution of the void radius ai is governed by the following

differential equation:

pi ÿ pthresholdi ¼ rfai €aið1ÿ f1=3i Þ þ 3

2_a2i ð1ÿ

43f1=3i þ 1

3f4=3i Þg, (13)

where _ð Þ denotes the time derivative and pi is the pressure applied to the boundary of the unit cell. pthresholdi

represents the matrix resistance first to nucleation and later to void growth and is defined as in Czarnota et al.

(2006) by

pthresholdi ¼ minðpci ; pGiÞ, (14)

where pGirepresents the growth threshold stress and is therefore a decreasing function of the local porosity f i of

the unit cell. Different estimations of pGican be used to describe the loss of stress carrying capacity of the unit cell.

The following expression for pGi, derived from Gurson’s potential, is adopted, see Appendix B:

pGi¼

2k

3q2ln

1

q1f i

� �

�0 þ �prei þ2

3q2ln

ðq1f iÞÿf i=ð1ÿf iÞ

1ÿ f i

" #( )n

� � � d0 þ2

3q2ln

1

q1f i

� �

_f i

ð1ÿ f iÞ2

( )m

. ð15Þ

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�prei represents the equivalent plastic strain cumulated before the nucleation of the void i. In the present work,

q1 ¼ q2 ¼ 1. The evolution of the porosity f i is governed by plastic incompressibility of the matrix material within

the unit cell:

_f i ¼ 3f ið1ÿ f iÞ_ai

ai. (16)

During the deformation process, elastic unloading may occur and then the void growth will stop. Nevertheless, the

family i is considered as active, since the corresponding nucleation sites of this family has led to the occurrence of

microvoids with non-zero radius. The family i contributes to the overall porosity f.

2.3.3. Second level of homogenization

The material domain can be viewed as an assembly of unit cells: each unit cell corresponding to a hollow

sphere. Let us note the mean total strain rate and the overall pressure at the boundary of the material domain

by dm and p. At the local level inside the material domain, each unit cell with label i will face some pressure pi.

Different homogenization schemes can be used to link local quantities at the unit cell level to macroscopic

quantities defined at the remote boundary of the material domain. In the present contribution, two

homogenization schemes have been adopted named p-model or d-model.

For the p-model, the pressure loading applied to the outer boundary of the unit cell i is identical to the

remote pressure applied to the material domain, i.e. pi ¼ p, see Fig. 2. For the d-model, the mean macroscopic

strain rate dm is applied to all unit cells of the material domain, see Fig. 2. In that case, the unit cells in the

material domain do not face the same pressure. For both schemes, the response of each hollow sphere is

governed by the relationship (13).

2.4. Numerical implementation

The present model has been implemented in the commercial finite element software ABAQUS/Explicit

through the VUMAT subroutine. The objective rate of stress used in Eq. (3) is the Green–Naghdi rate. At

each time step, the increment of the total strain tensor D� is given. One has to find a way to update all other

mechanical quantities.

First, we update the pressure, void radii and porosity. When the porosity of the material domain is known,

the deviatoric quantities are defined via a radial return algorithm. This two steps way is possible because the

evolution of the porosity is controlled only by the spherical strain increment. Note that the porosity affects the

plastic deviatoric behaviour of the material through the relationship (8) and also the elastic moduli.

For the update of the pressure and related quantities, different algorithms are adopted for each of the

homogenization procedures. When the p-model is adopted, the update is organized as follows. The pressure at

the material domain level evolves according to Eq. (4). The evolution of ai (or f i) follows Eq. (13). The mean

viscoplastic strain increment D�vpm can be therefore calculated based on the evolution of the porosity f in the

material domain. Indeed f is linked, by volume averaging, to the local porosities f i and radii ai in all active unit

cells. Finally, the increment D�vpm is found using Eq. (11). Note that a fixed point algorithm has been used to

determine p, D�vpm and ai at the end of each step. For more details concerning the procedure of integration of

the constitutive equations, see Appendix C.

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dmdm

p p

dmdm

Unit cell

..

..

.

...

..

...

...

pp

Fig. 2. Two strategies of multiscale modelling. The p-model assumes that a uniform pressure is applied to all unit cells. In the d-model, a

uniform strain rate is prescribed on unit cells.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1631

When the d-model is used, the pressure acting on the unit cell i is given by Eq. (4) :

_pi ¼ 3K ideim þ

_K i

K i

pi, (17)

where K i is the bulk modulus of the unit cell i given by Eq. (5). The new void radius ai is evaluated using Eqs.

(13) and (17). The pressure over the material domain is obtained by volume averaging over all potentially

active families: p ¼ hpii. More details concerning the numerical procedure is presented in Appendix D.

3. Results

3.1. Description of the tantalum behaviour

The material is a high purity grade tantalum. The parameters of Eq. (6) have been identified based on

compression tests carried out by Roy (2003) at different strain rates, up to a strain of 0.1. The coefficients a

and b adopted for Eq. (8) have been proposed by Eftis et al. (1991) with a ¼ 1=ðf cÞ0:5 and b ¼ 0:5. Roy (2003)

has observed that the coalescence at the end of the ductile fracture process in tantalum occurs by direct

impingement. This author has evaluated the critical volume fraction of voids based on observations of void

density in the spall plane: f c ¼ 0:3. The material is initially free of voids. The statistical information on

cavitation pressure of potential nucleation sites is provided by the Weibull law (10). The parameters, listed in

Table 2, are evaluated according to trends observed by Roy (2003). For instance, no void were nucleated when

the shock pressure was below 3:2� 0:2GPa. Thus, the value of p0c is set to 3GPa. So far, three parameters

remain to be determined: bw, Zw and N. These parameters are chosen in order to reproduce the experimental

particle velocity profile of the free surface obtained by Roy (2003). Only the results with a flyer thickness of

ef ¼ 3mm have been used. The number of potential nucleation sites per unit volume is set to

N ¼ 1:96� 1013mÿ3. The values of bw, Zw are given in Table 2. The probability density function W ðpcÞ

adopted in the work, presented in Fig. 3, has a large distribution of nucleation pressures. This choice is guided

by experimental data. From Roy (2003), the number of nucleated voids increases strongly with the shock

pressure. The maximum shock pressure in plate impact tests conducted by Roy (2003) was 30GPa. No

saturation of the number of voids is observed at 30GPa. To rationalize the fact that new voids can be

nucleated for shock pressures over 30GPa, a narrow distribution is not adequate for this tantalum. With the

adopted value of bw, Zw, the mean cavitation pressure is pc ¼ 48:4GPa. Such a value enables us to describe a

continuous nucleation process with no saturation of the number of voids up to large shock pressures.

ARTICLE IN PRESS

nucleation pressure (GPa)

dis

trib

uti

on o

f pote

nti

al n

ucl

eati

on s

ites

(1/G

Pa)

0 20 40 60 80 100 120 1400

0.005

0.01

0.015

0.02

Fig. 3. Weibull probability density law of the cavitation pressures with parameters given in Table 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501632

However, since we have simulated plate impact tests with moderate shock pressures (less than 10GPa), only

the shape of the density function for low nucleation pressures is really needed. The slope of the density

function for pressures close to p0c is a relevant parameter for the present modelling.

The continuous distribution of nucleation pressures is discretized. Thus, N f families of potential nucleation

sites are introduced. The probability of existence of nucleation sites with nucleation pressure pc4p0c þ 9sw is

negligible, with sw being the standard deviation of the Weibull probability density function. Therefore, the

interval ½p0c; p0c þ 9sw� is equally divided into N f subintervals of length Lc ¼ 9sw=N f . The family F i

corresponds to the group of sites with nucleation pressure belonging to the interval ½p0c þ ði ÿ 1ÞLc; p0c þ iLc�.

The associated appearance frequency is Pri ¼R p0cþiLc

p0cþðiÿ1ÞLcW ðpcÞdpc. The nucleation pressure of all sites

belonging to the family F i is set to pci ¼ p0c þ ði ÿ 1=2ÞLc. No effect of the discretization scheme has been

observed, for the Weibull distribution adopted in the present work, when Lc is below 0.2GPa.

3.2. Uniaxial deformation

Before simulating complex experiments, as plate impact tests, where wave propagation has a strong effect

on the structural response, the model developed in Section 2 has been first investigated in uniaxial

deformation. This loading path is representative of the deformation state during impact loading when the

lateral extent of the plates is at least height times the thickness, see Meyers and Aimone (1983). The applied

strain rate d is

d ¼ D11

1 0 0

0 0 0

0 0 0

2

6

4

3

7

5. (18)

In the elastic regime, Eq. (3) simplifies into the classical Hooke’s law. The stress components are related to the

logarithmic longitudinal strain �11 ¼ D11t by

s11 ¼ ðK þ 43GÞ�11; s22 ¼ ðK ÿ 2

3GÞ�11; p ¼ K�11. (19)

The yield condition is satisfied when

js11 ÿ s22j ¼ syield (20)

with syield ¼ ð1ÿ af bÞsyðdeq; �eqÞ, see Eq. (8). Therefore, during plastic yielding, the longitudinal stress s11 is

related to p by

s11 ¼ pþ 23syield with p40 ðtensionÞ, ð21Þ

s11 ¼ pÿ 23syield with po0 ðcompressionÞ. ð22Þ

In the following, the evolution of the pressure p is analysed. The longitudinal stress s11 is obtained from p by

an appropriate translation of � 23syield during elastic–plastic loading.

The pressure within the material is presented in Fig. 4 for an overall imposed uniaxial straining. Numerical

tests have been conducted for three different strain rates. The results in Fig. 4 have been obtained by means of

the two different homogenization schemes introduced above, uniform pressure (p-model) or uniform strain

rate (d-model) applied on all unit cells. The initial stage of the deformation process is not affected by the

adopted scheme. The same maximum pressure is found. This is obviously related to the material homogeneity

before microvoiding. Differences are observed after the first oscillation. At the beginning of the loading, the

pressure evolves in a linear manner; the slope corresponding to the bulk elastic modulus. In uniaxial

deformation, when the material is free of voids, the pressure response is purely elastic since no volume plastic

strain can be generated. Therefore, one can easily show that, even if deviatoric plasticity is triggered, one

obtains the following pressure evolution:

p ¼ K0�11 ¼ K0D11t. (23)

When the pressure p overcomes p0c, voids are nucleated. Due to micro-inertia effects, void growth is very

limited, the volume plastic strain being still negligible. Thus the pressure evolution remains almost elastic even

when p overcomes p0c and can be approximated by Eq. (23) with good accuracy up to the maximum. It is

ARTICLE IN PRESS

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1633

observed, that when the maximum pressure is reached, the porosity remains low (less than 0.01 at 106 sÿ1), see

Fig. 5. As soon as voids start to grow significantly, the pressure decreases because of elastic relaxation

produced by the large increase of the plastic strain generated by void growth. During the pressure drop, the

porosity is cumulated and becomes significant. An additional pressure relaxation occurs because of the

softening of the bulk modulus with porosity. Note that void growth may continue even when the pressure

becomes negative. Nevertheless, the driving force for the void growth decreases. At a given time, no porosity is

further cumulated, see Fig. 5. Subsequently, the pressure increases elastically for the second time. The slope

ARTICLE IN PRESS

uniaxial strain,

pre

ssu

re, p

(G

Pa

)

0 0.05 0.1 0.15 0.2-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

ε =

d-model

p-model

D .t11

=10 s6 -1

D11=10 s

4 -1

D11=10 s

5 -1

11

11D

Ko

Fig. 4. Pressure evolution during uniaxial deformation. Simulations have been conducted for three different strain rates and for the two

homogenization schemes of Fig. 2. The material is a high purity grade tantalum, see Tables 1 and 2. The Weibull distribution law of Fig. 3

is adopted in the calculations.

porosity, f

pre

ssu

re, p

(G

Pa

)

0 0.05 0.1 0.15 0.2-8

-4

0

4

8

12

16

D =10 s6 -1

D11=10 s

4 -1

D11=10 s

5 -1

11

d-model

p-model

Fig. 5. Evolution of the pressure versus porosity. The maximum pressure occurs for limited porosity: less than 0.01 for D11 ¼ 106 sÿ1.

Parameters used in the calculations are listed in Tables 1 and 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501634

defining the pressurization rate is lower than K0 because of the softening of the elastic properties by porosity,

see Eq. (5). The slope is smaller for larger strain rates, indicating that a larger porosity is cumulated during the

pressure decrease at large strain rate. For D11 ¼ 104 sÿ1, the value of f during the second increase of the

pressure is 0:027. For D11 ¼ 105 sÿ1, f ¼ 0:059 and for D11 ¼ 106 sÿ1, f ¼ 0:165, see Fig. 5.

The maximum pressure in the material during uniaxial deformation increases strongly with the prescribed

strain rate. This is due to enhanced micro-inertia for faster loading. Up to the maximum, the rate of

pressurization is almost constant and given by _p ¼ K0D11, the porosity remaining limited. Therefore, voids

can be considered as isolated in an infinite matrix. During the early growth of voids, the radius evolution is

given by (see Molinari and Wright, 2005):

aðtÞ ¼

ffiffiffiffiffi

8

33

r

ffiffiffi

_p

r

s

pÿ pc_p

� �3=2

for p4pc. (24)

Per unit volume of matrix material, the void volume generated by the increase of pressure from pc to

pc þ dpc is

dV ¼4p

3Na3W ðpcÞdpc. (25)

The total void volume generated per unit initial volume at the pressure pXp0c becomes

V ¼4p

3N

Z p

p0c

a3W ðpcÞdpc. (26)

The porosity f defined as

f ¼V

1þ V(27)

is approximated by f ¼ V , since the volume of voids cumulated up to the maximum of pressure is small. For

moderate shock pressures, the term pÿ p0c is smaller than Zw and f is estimated as

f ¼8

33

� �3=21

ðb0K0D11Þ3r3=2

bw

Zbww

Z p

p0c

ðpÿ pcÞ9=2ðpc ÿ p0cÞ

bwÿ1dpc. (28)

When bw is an integer number, a closed form solution for f is obtained:

f ¼8

33

� �3=22bwbw!

Qbwj¼1ð9þ 2jÞ

1

Zbww ðb0K0D11Þ

3r3=2ðpÿ p0cÞ

ð9þ2bwÞ=2. (29)

The maximum pressure pmax can be defined by Eq. (4) with _p ¼ 0. Since dm ¼ D11=3 and according to Eq. (11),

the pressure rate (4) becomes

_p ¼ K D11 ÿ_f

1ÿ f

!

þ_K

Kp. (30)

The time derivative of the bulk modulus _K can be expressed in terms of the rate of porosity, using Eq. (5):

_K

K¼ ÿ _f

1

1ÿ fþ

3K0

4G0 þ 3K0f

� �

. (31)

The combination of Eqs. (30) and (31) together with the fact that the porosity is negligible up to the maximum

pressure, provides the following condition for the occurrence of the maximum pressure:

_f ¼D11

1þ pmax=K0 þ 3pmax=4G0

. (32)

ARTICLE IN PRESS

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1635

By time derivative of Eq. (29) and using Eqs. (23) and (32), the following implicit equation is derived for pmax:

pmax ¼ p0c þ333=2

Qbwÿ1j¼0 ð9þ 2jÞ

83=2 � 9 � 2bwÿ1bw!

!2=ð7þ2bwÞ

Zbww b30r

3=2K20D

311

1þ pmax=K0 þ 3pmax=4G0

!2=ð7þ2bwÞ

. (33)

For moderate shock pressures, pmax is smaller than the value of the elastic moduli, and the last equation can be

simplified to provide a closed form expression for pmax:

pmax ¼ p0c þ333=2

Qbwÿ1j¼0 ð9þ 2jÞ

83=2 � 9 � 2bwÿ1bw!

!2=ð7þ2bwÞ

ðZbww b30r3=2K2

0D311Þ

2=ð7þ2bwÞ. (34)

Note that, a closed form solution exists also when all sites have the same nucleation pressure p0c:

pmax ¼ p0c þ1331

384

� �1=7

ðb60r3K4

0D611Þ

1=7. (35)

Since the pressure rate is assumed constant during the loading, pmax will be overestimated by Eq. (34). Indeed,

when the pressure reaches the maximum, the damage rate begins to be significant, so that a pure elastic

response like Eq. (23) is no more valid. Despite the above simplifying assumptions, the analytical solution is in

good agreement with results obtained by numerical calculations for moderate strain rates (below 105 sÿ1 for

tantalum), see Fig. 6. One observes that pmax is close to p0c when the uniaxial deformation rate D11 or the mass

density r are small. Note that for tantalum, micro-inertia effects are significant for D114103 sÿ1. For less

dense materials, micro-inertia effects will be relevant for higher strain rates, all other parameters being

identical.

For uniaxial deformation at D11 ¼ 105 sÿ1, the effect of different parameters on the overall pressure is

analysed in Fig. 7. The p-model is adopted. This choice does not affect strongly the results since in Fig. 4, it

has been shown that during the early development of microvoids, pressures are identical for both p- and d-

models. The reference curve is obtained with parameters listed in Tables 1 and 2 (henceforth denoted as

reference parameters). When the number of potential nucleation sites per unit volume N decreases

(N ¼ Nref=8 or equivalently b0 ¼ 2b0ref ), the number of nucleated voids at a given pressure decreases, the

distribution law being unchanged. Thus, the volume of matrix material surrounding each void increases.

ARTICLE IN PRESS

max

imum

pre

ssure

, p

ma

x (G

Pa

)

108

109

1010

1011

1012

1013

1014

1015

1016

1017

0

2

4

6

8

10

12

14

poc=3GPa

ρ-2

kg.m .s11

Analytical solution

Numerical results

D ( )2 -3

Fig. 6. Evolution of the maximum pressure pmax versus the parameter rD211. An approximate solution for pmax has been derived, Eq. (34)

which is in agreement with numerical results. Parameters used in the calculations are listed in Tables 1 and 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501636

Micro-inertia is therefore enhanced and induces an increase of pmax, see Fig. 7. The parameters of the Weibull

law influence also the pressure evolution in the material. The effect of p0c is clear, since delaying the nucleation

of voids will generate a larger pressure in the material. A smaller value of Zw modifies the shape of the Weibull

distribution (see Fig. 8), leading to a larger number of nucleated voids for a prescribed pressure. Therefore, the

volume of matrix surrounding each void is lowered, and so is micro-inertia. The maximum pressure is then

decreased. An increase of bw induces the nucleation of fewer voids, for moderate pressure, as compared to the

reference value bwrefof Table 2, see Fig. 8. Thus, a larger maximum pressure is predicted, see Fig. 7.

3.3. Plate impact experiments

The model has been implemented in the explicit finite element software ABAQUS/Explicit using the user-

subroutine VUMAT. The predictions of the proposed model are compared to experiments of Roy (2003) in

tantalum. A cylindrical flyer plate of thickness ef (value given later) and diameter 50mm collides a target plate

of thickness et ¼ 4:95mm. Both plates are made of tantalum. The contact between the flyer and the target is

ARTICLE IN PRESS

time, t (µs)

pre

ssu

re, p

(G

Pa

)

0 0.25 0.5 0.75 1-4

-2

0

2

4

6

8

10

5 -1s=10

b =2b00 ref

η=η /1.5

reference

oc

ref

β=β 1.5ref

p =p +2GPaoc

ref

D11

x

Fig. 7. Effect of the number of potential nucleation sites per unit volume (N ¼ 1=ðð4p=3Þb30Þ) and of the Weibull parameters on the

pressure evolution. The loading rate is D11 ¼ 105 sÿ1. The reference curve is obtained with parameters listed in Tables 1 and 2.

Table 1

Material parameters representative of a high purity grade tantalum

K0 (GPa) G0 (GPa) r ðkg=m3Þ k (IS) e0 n d0 ðsÿ1Þ m f c

200 68.3 16600 528� 106 0.055 0.155 10ÿ4 0.058 0.30

Table 2

Parameters used to describe the statistical distribution of nucleation pressure in the tantalum material. The corresponding probability

density function, given by Eq. (10), is represented in Fig. 3

p0c (GPa) bw Zw (GPa) b0 ðmmÞ

3 2 51.19 23

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1637

governed by a Coulomb friction law with a friction coefficient of 0.3 and takes place over the whole surface,

see Fig. 9. The velocity of the flyer, before impact is V impact whereas the target plate is at rest. The density of

potential nucleation sites in the whole material is assumed homogeneous. Therefore, an axisymmetric

approach is adopted. The mechanical analysis is performed on a two-dimensional section of the flyer and

target plates. Four-node bilinear elements with reduced integration (named CAX4R) have been adopted.

Velocity profiles at the target free surface are obtained with simulations and are compared to experimental

measurements. In our approach, those profiles correspond to the velocity of the material point P located on

the rear face along the axisymmetric axis, see Fig. 9. The numerical analysis enables to capture information

such as pressure, strain rate and porosity evolution inside the target during tests. Such data cannot be obtained

during experiments. Therefore, a clear and complete picture of the plate impact test can be proposed.

ARTICLE IN PRESS

0 5 10 150

0.002

0.004

0.006

0.008

0.01

nucleation pressure (GPa)

dis

trib

uti

on

of

po

ten

tial

nu

clea

tio

n s

ites

(1

/GP

a)

0 20 40 60 80 100 1400

0.005

0.01

0.015

0.02

0.025

0.03 η=ηref

/1.5

reference

β=βref

x1.5

poc=pocref+2GPa

120

Fig. 8. Various Weibull distributions of the nucleation pressure. The reference curve is obtained with values defined in Table 2.

Vimpact

axis of revolution

CAX4Rmeasurement of thefree-surface velocity

V impact

50 mm

3 mm

4.95 mm

P

Fig. 9. Configuration of the plate impact experiments. An axisymmetric approach is adopted. Only a two-dimensional cross-section is

meshed. Four-node bilinear elements with reduced integration have been used for the simulations. The point P on the target indicates the

node where the free-surface velocity profile is captured.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501638

At impact, compressive waves propagate into the plates and are reflected into rarefaction waves at the rear

surfaces. The superposition of the rarefaction waves produces a large tensile mean stress and localized

damage. The effect of the mesh density on the size of the damaged zone has been investigated. A mesh

sensitivity study has been conducted for the velocity V impact ¼ 306m=s and for a flyer thickness of ef ¼ 3mm.

Four mesh configurations have been tested: from 100mm� 100mm for the element size in the coarse mesh to

15mm� 75mm for the dense mesh. Note that the length in the direction of impact is smaller than in the lateral

direction since dynamic wave propagation induces a strong gradient in the impact direction. Fig. 10 presents

the location and the extent of the damaged zone at the end of the test (for t ¼ 5:31ms), when spall has been

observed. In the simulations, an element is numerically deleted when a critical value of porosity is reached:

f c ¼ 0:3 for this tantalum. No mesh dependency is observed when the mesh size is below 25mm� 75 mm. This

element size 25mm� 75mm has been adopted in the following. In Fig. 10, the damaged zone, where the local

porosity is 0.3, is 225mm large and contains nine elements. With the same element size, the results are mesh

independent for the other impact velocities tested in the present work. The same trend has been encountered

for the velocity profile at the free surface. Note that in the present contribution, the material is a high purity

tantalum. Parameters adopted in Fig. 10 as in the following figures, are listed in Tables 1 and 2.

Fig. 11 presents the porosity distribution in the target at three moments during impact simulations

(V impact ¼ 306m=s and ef ¼ 3mm). Time t ¼ 0 corresponds to the instant when the flyer collides the target.

Fig 11(a) (for t ¼ 2:31ms) shows the initiation of the porosity inside the target. Due to the interaction of

rarefaction waves coming from boundaries, microvoiding occurs near the lateral boundary. One can mention

that voids are initiated earlier in the flyer near the lateral boundary but its development is limited. In Fig. 11(b)

(t ¼ 2:98ms), the damaged zone appears clearly. A larger porosity is first cumulated near the lateral boundary.

The porosity has already reached the value 0.3 so that some elements have been deleted and a crack has been

initiated. At the end of the test (t ¼ 6ms), see Fig. 11(c), the crack has propagated in the whole spall plane and

the spall fracture is met.

Fig. 12 presents the time evolution of pressure in the spall plane along the axisymmetric axis, for a velocity

V impact ¼ 306m=s and for a flyer thickness ef ¼ 3mm. As in Fig. 11, the flyer collides the target at time t ¼ 0.

The target thickness is 4.95mm, thus the spall plane is located at 3mm from the free surface or at 1.95mm

from the impact surface. The pressure is zero during the time necessary for the compression waves (created by

the shock) to propagate to the spall plane. The longitudinal sound velocity in tantalum being CL ¼ 4187m=s,

ARTICLE IN PRESS

distance x from the free surface (mm)

po

rosi

ty,

f

0 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

impacted planefree surface

(x=4.95mm)(x=0)

100 100 µm

50 100µm

25 75µm

15 75µm

2

2

2

2x

x

x

x

1 2 3 4

Fig. 10. Effect of the mesh size on the porosity distribution inside the target, at t ¼ 5:31ms, for the impact velocity V impact ¼ 306m=s. The

geometry of the plate impact test is shown in Fig. 9 and material parameters used for the simulation are listed in Tables 1 and 2. The p-

model is adopted. Note that the location and the width of the damaged zone is mesh independent when the 25mm� 75mm element size is

considered. This element size is used in simulations presented in the following.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1639

the pressure goes to negative values after 0:46ms. Afterwards, the pressure reaches the shock pressure:

pshock ¼ ÿ8:68GPa. The pulse duration tpulse in Fig. 12, from t ¼ 0:46ms to the end of the plateau can be

approximated by tpulse ¼ 2ef=CL ¼ 1:43ms. At the end of the plateau, the pressure increases up to large

positive values, due to the interaction of the rarefaction waves reflected at the target and flyer free surfaces.

ARTICLE IN PRESS

Fig. 11. Porosity evolution within the target during the impact test. Parameters are identical to those of Fig. 10. The material is a high

purity grade tantalum. (a) t ¼ 2:31ms: the damage initiation is located where release waves coming from lateral boundaries interact.

(b) t ¼ 2:98ms: the initiation of fracture is clearly observed in the spall plane. (c) t ¼ 6ms: spall fracture is completed.

time, t (µs)

pre

ssure

, p (

GP

a)

0 0.75 1.5 2.25 3.75 4.5-10

-8

-6

-4

-2

0

2

4

6

8

10

12

d-model

with no damage

p-model

fracture of theelement

3

pulset

Fig. 12. Pressure history in the spall plane calculated for the impact velocity V impact ¼ 306m=s. The maximum pressure corresponds to the

initiation of rapid void growth. Note that fracture occurs at a very low stress level. When damage is not taken into account (thick dashed

line), the pressure increases up to a level corresponding to the shock pressure. The test geometry is defined in Fig. 9. The material is a high

purity grade tantalum. Calculations are conducted with material parameters given in Tables 1 and 2 and for the two homogenization

schemes of Fig. 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501640

As mentioned previously, the pressure relaxation is produced by the rapid development of void growth.

Predictions based on the two homogenization schemes are reported in Fig. 12. The pressure responses in the

spall plane are similar up to the second peak. A porosity of 0.3 is encountered in the material domain at nearly

the same time for the p- and d- models. When damage is disregarded (thick dashed line), it is observed that the

tensile pressure reaches a higher value. Due to plasticity, dissipation and wave dispersion during propagation,

the shock amplitude is limited to 8GPa, whereas in compression the pressure was ÿ8:68GPa. In plate impact

experiments, the use of the longitudinal stress is more common. The maximum tensile longitudinal stress (in

the impact direction) smax11 is related to the maximum pressure pmax by

smax11 ¼ pmax þ 2

3syield, (36)

where smax11 represents the incipient spall strength of the material, see Antoun et al. (2003). The numerical

simulation provides the following value for the spall strength: smax11 ¼ 7:11GPa with pmax ¼ 6:65GPa.

The simulation of the particle velocity at the free surface is shown in Fig. 13 for the two homogenization

schemes. The velocity remains zero during 1:18 ms, due to wave propagation of the pulse. When the

compression waves arrive at the free surface and reflect into tension waves, the velocity of the particle reaches

first a velocity VHEL corresponding to the Hugoniot elastic limit. Later with the arrival of the elastic–plastic

waves, the velocity of the material point becomes V impact. The duration of the pulse corresponds to the

duration of the negative pressure pulse in Fig. 12 with a time delay due to wave propagation. The pressure

drop in the spall plane due to the incipient void growth induces the pullback signal on the free surface.

Without any damage (thick dashed line), the pullback signal does not exist. The velocity pullback is not

relevant of the fracture, at least for ductile materials, see Antoun et al. (2003). The period of the velocity

oscillation (after the pullback) can be related to the thickness of the scab. From calculations, the period is

almost tpulse. As a consequence, the thickness of the scab is close to ef , distance from the free surface to the

spall plane, Antoun et al. (2003). In Fig. 13, the velocity profiles are not affected by the prescribed

homogenization scheme, up to the pullback signal. A better agreement with experiments is obtained with

p-model for the late evolution of the free-surface velocity (subsequent velocity oscillations). The spall fracture

is observed inside the target when a critical porosity of 0.3 is reached. It is difficult to observe directly this

event on the velocity profile. From calculations, the time to fracture has been estimated and is presented in

Fig. 13 using an arrow.

ARTICLE IN PRESS

time (µs)

vel

oci

ty (

m/s

)

0 60

100

200

300

400

spall fracture

p-model

experimental data (Roy, 2003)

d-model

∇∇∇∇∇∇∇∇

with no damage

pullbackvelocity

pulset

1 2 3 4 5

Fig. 13. Simulations of free-surface velocity profiles and comparison with experimental data of Roy (2003) on tantalum. The impact

velocity is V impact ¼ 306m=s. The test geometry is defined in Fig. 9. The maximum pressure observed in Fig. 12 induces the velocity

pullback on the free-surface velocity profile. The thick dashed line corresponds to the case where damage is not accounted for.

Calculations are conducted for the two homogenization schemes of Fig. 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1641

In Fig. 13, calculations are carried out with material parameters listed in Tables 1 and 2. The hardening

behaviour has been identified on compression tests up to a strain of 0.1. For such a strain level, the

temperature increase generated during the deformation is limited. Therefore, parameters of Table 1 adopted

for the present work, are able to describe the isothermal response of tantalum. Recently, Rittel et al. (2007)

have performed dynamic testing, on a different pure grade tantalum. They carried out shear testing so that

they were able to reach large plastic strain (up to 0.5). As a result, they found that their tantalum has a little

strain hardening. The reduction of the strain hardening coefficient at high strain rate is due to adiabatic

heating. Based on these observations, the effect of temperature can be indirectly estimated in the calculations,

by adopting a lower value of the strain hardening coefficient. For that purpose, plate impact simulations have

been conducted with different material parameters (k ¼ 385� 106 IS, n ¼ 0, other parameters involved in

Eq. (6) being kept constant). k has been adjusted to preserve the same flow stress level for the tantalum.

Predicted free-surface velocities, presented in Fig. 14, are almost unchanged. As a consequence, it can be

concluded that the effect of thermal softening can be neglected for moderate shock pressures.

The effect of the impact velocity has been investigated in Fig. 15. The thickness of the flyer is ef ¼ 3mm and

the material parameters are unchanged and listed in Tables 1 and 2. Simulated velocities are compared to

experiments of Roy (2003) and Llorca and Roy (2003). The results are accurate, when the p-model is

prescribed, and remains in good agreement with the d-model. Fig. 16 presents the pressure in the spall plane

for the three impact velocities used for Fig. 15. In Fig. 16, time t ¼ 0 corresponds to the moment where the

pressure becomes positive. The simulations confirm that the material is subjected to uniaxial deformation in

the spall plane. As the impact velocity increases, the loading rate increases. But the rate of pressurization and

the total strain rate are not constant up to the maximum of pressure. The calculated slope at the origin

indicates that the initial rate of pressurization varies from _p ¼ 46GPa=ms for V impact ¼ 212m=s to _p ¼

132GPa=ms at V impact ¼ 412m=s. A higher rate of pressurization enhances micro-inertia effects and leads to a

larger maximum pressure: pmax ¼ 4:85GPa at V impact ¼ 212m=s and pmax ¼ 8:55GPa at V impact ¼ 412m=s.The maximum pressure is larger when the flyer collides the target at higher velocity, but the amplitude of the

velocity pullback is almost identical for the three velocities. This result agrees with experiments made on many

materials showing that increasing the shock amplitude does not lead to a change in the pullback magnitude,

see Antoun et al. (2003) for aluminium and titanium alloy, Roy (2003) and Llorca and Roy (2003) for

tantalum, see Fig. 15. In the literature, the longitudinal spall strength is linked to the amplitude of the

ARTICLE IN PRESS

time (µs)

vel

oci

ty (

m/s

)

0 60

100

200

300

400

zoom

reference (n=0.155, k=528.106 IS)

no strain hardening (n=0, k=385.106 IS)

1 2 3 4 5

Fig. 14. Predicted free-surface velocities for a plate impact test at moderate shock pressure: impact velocity V impact ¼ 306m=s and

thickness ef ¼ 3mm. The reference curve (solid line) is obtained with parameters of Table 1. The hardening coefficient is n ¼ 0:155. Forthe second curve (dashed line), the material response presents no strain hardening n ¼ 0 due to adiabatic heating. The parameter k has

been adjusted to keep the same flow stress level of the tantalum.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501642

pullback, by using for example the acoustic approach. Such a formula leads to spall strengths unaffected by

the impact velocity.From our calculations, it appears that the pressure in the spall plane is strongly affected by

the impact velocity, so does the maximum longitudinal stress which corresponds to the spall strength, see

Eq. (36). Therefore, spall strengths, calculated by means of the acoustic formula, are not able to reproduce the

value of the maximum pressure in the material. Stress relaxation due to damage is an important feature that

needs to be integrated to convert velocity pullback into spall strength.

Roy (2003) and Llorca and Roy (2003) have performed experiments with flyer thickness varying from ef ¼ 2

to 4mm, see Fig. 17. The calculations have been conducted for two impact velocities V impact ¼ 247m=s and

ARTICLE IN PRESS

time (µs)

vel

oci

ty (

m/s

)

0 60

100

200

300

400

500

experimental data (Roy, 2003)

d-model

∇∇∇∇∇∇∇∇

V =212impact

V =412impact

V =306impact

1

3

2

m/s

m/s

m/s

1

3

2

1 2 3 4 5

p-model

Fig. 15. Simulations of free-surface velocity profiles and comparison with experiments of Roy (2003) on tantalum, performed at different

impact velocities: V impact ¼ 212, 306 and 412m/s. As in experiments, the simulated velocity pullback is not affected by the impact velocity.

The plate impact geometry is defined in Fig. 9. Results are presented for the two homogenization schemes of Fig. 2.

time, t (µs)

pre

ssu

re,

p (

GP

a)

0 0.25 0.5 0.75 1-2

0

2

4

6

8

10

1

3

2

V =212impact

V =412impact

V =306impact

1

3

2

m/s

m/s

m/s

p-model

d-model

Fig. 16. Simulated pressure history in the spall plane for three impact velocities: V impact ¼ 212, 306 and 412m/s. The maximum pressure

increases strongly with the impact velocity, whereas it has been observed in Fig. 15 that the velocity pullback is not affected by the flyer

velocity. The geometry of the plate impact is indicated in Fig. 9. Results are presented for the two homogenization schemes of Fig. 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1643

V impact ’ 306m=s. Material parameters are unchanged and given in Tables 1 and 2. Note that parameters of

the Weibull law were calibrated based on experiments with a flyer thickness ef ¼ 3mm. The predictions based

on the two homogenization schemes are also presented. A close agreement with experiments is obtained for

the three thicknesses, whatever the adopted schemes. Nevertheless, as already mentioned, more accurate

results are given by the p-model. The duration of the pulse, from t ¼ 1:18ms to the end of the plateau increases

with the flyer thickness since it is proportional to the thickness of the flyer and approximated by 2ef=CL where

CL is the longitudinal elastic sound velocity. Considering tests carried out at V impact ’ 306m=s, it is observedthat the magnitude of the velocity pullback is larger for thinner flyer. Since the same impact velocity is

prescribed to the flyer, the magnitude of the maximum pressure in the spall plane is identical for the three tests

(results not presented here). The thickness of the flyer affects the location of the spall plane. When the flyer

thickness is below half of the target thickness, the spall plane is located near the free surface. On the contrary,

when the flyer thickness is over half of the target thickness, the spall plane is located near the impact surface.

Note that when the target thickness is twice the flyer’s one, the spall plane is exactly located in the middle of

the target. As maximum pressures are unaffected by the flyer thickness, the larger relaxation of tension waves

observed for a thicker flyer can be attributed to the longer propagation distance towards the free surface. The

larger damping generates a smaller pullback, as observed in Fig. 17. As in Fig. 13, the velocity oscillations

observed after pullback have periods linked to the scab extent when the spallation has been observed or to the

location of the damaged zone when the fracture is not complete. As expected, the duration of the oscillation

period for the three tests provides a value of the scab thickness close to the flyer extent ef and corresponds to

the location of the damaged zone or spall plane in the target.

To clarify the relative influence of the different physical concepts included in the proposed model, some

calculations have been done with simplified versions of the p-model, in which some phenomena (and their

mathematical descriptions) have been neglected. The corresponding free-surface velocities are presented in

Fig. 18. The reference free-surface velocity corresponds to a plate impact test at V impact ¼ 306m=s for a flyer

thickness ef ¼ 3mm. When the dependence of porosity on the yield stress is neglected (the a value in Eq. (8) is

set to zero), no effect is expected and observed on the velocity pullback. Indeed, the velocity pullback begins

with the pressure release observed in the spall plane at the initiation of microvoiding for which f � 0. The

effect of f on the yield function (8) is nevertheless necessary to capture accurately the velocity level during the

oscillations (after the pullback signal).

ARTICLE IN PRESS

time (µs)

vel

oci

ty (

m/s

)

0 1 2 3 4 5 60

100

200

300

400

V =303impact

13 2

1

2

4

e =4i m/smm

V =247impacte =2i m/smm

V =306impacte =3i m/smm

d-model

∇∇∇∇∇∇∇∇

3 V =307impacte =2i m/smm

4

p-model

experimental data (Roy, 2003)

Fig. 17. Simulated free-surface velocity profiles compared to experiments of Roy (2003) on tantalum. The target thickness is 4.95mm. The

flyer thickness is varied in the range (2–4)mm. Simulations are conducted for two impact velocities: V impact ¼ 247m=s and

V impact ’ 306m=s. Note that the velocity pullback is depending on the flyer thickness. Material parameters are given in Tables 1 and

2. Results are presented for the two homogenization schemes of Fig. 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501644

Fig. 18 clearly shows the necessity of accounting for the porosity dependence (at the unit cell level) of the

stress carrying capacity of the matrix pthresholdi , Eq. (14). When this term remains equal to the cavitation

pressure pci during all the deformation process, it is observed that the velocity pullback is not well predicted. In

addition, the velocity oscillations after the pullback are not reproduced. From calculations, the porosity level

cumulated at the end of the experiment is 0:18, value below f c. So, for pthresholdi ¼ pci , no spall fracture is met in

the simulations whereas in experiments, the fracture is clearly apparent with a well defined scab.

Our modelling is based on two major physical concepts: micro-inertia and continuous nucleation of voids.

Fig. 18 illustrates clearly that without those keypoints, the velocity profile cannot be simulated with good

accuracy.

When the mass density used in Eq. (13) is reduced to a low value r ¼ 0:3 kg=m3, micro-inertia effects almost

disappear. Note that the mass density is not modified in the equation of motion, so that the macroscopic

inertia is not changed. When the pressure reaches the lowest cavitation pressure p0c in the spall plane, the

porosity is immediately cumulated since micro-inertia effects are too weak to refrain void growth. Thus, the

pressure peak in the spall plane is underestimated, and so is the velocity pullback. In addition, without micro-

inertia, the velocity level during the subsequent oscillations are overestimated. Note that p0c is not an

adjustable parameter since it has been determined from experiments.

The importance of continuous nucleation of voids is illustrated also in Fig. 18. If all nucleation sites have

the same cavitation pressure p0c, it is observed that the magnitude of the velocity pullback can be reproduced

for V impact ¼ 306m=s by increasing the characteristic length b0 (or equivalently by decreasing the number of

potential sites N). But the velocity oscillations are not well captured. In addition, the evolution of the velocity

pullback for different impact velocities cannot be reproduced.

4. Conclusion

In this paper, a model of void nucleation and growth is proposed and used to characterize dynamic damage

in metals subjected to extreme loading conditions. The material is elastic–viscoplastic and initially void free

(but the presence of initial voids can be as well considered). Microvoids are generated at potential nucleation

sites when the nucleation pressure is overcome by the local pressure loading. Owing to microstructural

ARTICLE IN PRESS

time, t (µs)

vel

oci

ty (

m/s

)

2 3 4 5 650

150

250

350

1

b =237µm0

in Eq.(13)threshold

pci

=p4

3

5

1

2

2

3

4

5

for all unit cells ,

α=0 in Eq.(8)

without microinertia (ρ∼0 in Eq.(13))

reference

ip =pci

oc

Fig. 18. Effect of different parameters on the predicted free-surface velocity. The following conditions are considered: target thickness

4.95mm, flyer thickness 3mm, impact velocity: V impact ¼ 306m=s. Simulations are based on the p-model. Material parameters are given in

Tables 1 and 2.

C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1645

heterogeneities and to the presence of residual stresses, the nucleation pressure varies from site to site.

A Weibull probability density function is adopted to describe the fluctuation of nucleation pressures within the

material. When the local pressure increases, more voids are nucleated. The growth of a void is described by

using a hollow sphere model where micro-inertia effects are accounted for. The matrix weakening due to void

growth is also included.

To simulate plate impact experiments, our dynamic damage model has been implemented in ABAQUS/

Explicit. All the model parameters have been identified from independent experiments, except for three

parameters characterizing the statistical distribution of the nucleation pressures. These parameters have been

deduced from impact experiments at three different velocities with the same flyer size. The model is then used

as a predictive tool to simulate experiments with various flyer sizes and velocities. The free-surface velocity

profiles measured by Roy (2003) and Llorca and Roy (2003) are reproduced with excellent accuracy for all

experimental configurations.

These simulations bring a new insight into the mechanism of spalling and into the interpretation of

experimental measurements. From observations, it is generally deduced in the literature (acoustic assumption)

that for aluminium, titanium and tantalum, the maximum stress sustained by the material (spall stress) is not

affected by the impact velocity. In our model, the maximum stress in the spall plane strongly increases with the

impact velocity. Nevertheless, the velocity pullback at the target free surface is shown, as in experiments, to be

independent of the impact velocity for a given flyer geometry. In addition, the effect of the flyer size on the

velocity pullback is exactly captured. These results bring some confidence on the physical mechanisms

introduced in the present dynamic damage model. The stabilizing effect of micro-inertia is of particular

importance, as it controls the level of the maximum stress and the development of damage during stress

relaxation in the spall plane.

Appendix A. Evolution of the mean distance between active sites

In the present approach, the mean spacing between nucleated sites decreases when the pressure and the number

of nucleated voids increase. Different methods can be selected to describe the mean spacing evolution during the

loading. In the present paper, we have adopted the following concept: when new voids are nucleated, the total

volume of the matrix material is distributed so as to have the same external radii for all unit cells.

Let us consider the initial volume of the material domain Vdomain ¼ Ns=N ¼ ð4=3ÞpNsb30. Ns represents the

total number of potential sites and b0 is the characteristic length defined in Eq. (9). As the pressure overcomes

pc1 , NsPr1 voids are nucleated. Assuming that potential sites in the first family are randomly distributed in the

material domain, the mean spacing b between nucleated sites (only the first family being activated), is given by

NsPr1

4

3pb3 ¼ Vdomain ) b ¼

b0

P1=3r1

. (A.1)

Consider now that J families are active defining J unit cells. All unit cells of the family i have the same internal

and external radii ai and bi. At a given time, the pressure loading increases up to reach pcJþ1. Therefore, NsPrJþ1

voids nucleate, with zero void radius. For the considered time, we affect to all unit cells the same outer radius

b, keeping all current radii ai unchanged. Owing to volume preserving of the matrix material, the new mean

distance between voids satisfies:

Vdomain ¼X

J

i¼1

4

3pNsPri ðb

3 ÿ a3i Þ þ4

3pNsPrJþ1

b3. (A.2)

This leads to the following definition of the external radius b of all unit cells when the ðJ þ 1Þth family

nucleates:

b ¼b30 þ

PJi¼1Pria

3i

PJþ1i¼1 Pri

" #1=3

. (A.3)

If nucleation sites have the same cavitation pressure pc, the initial outer radius of the unit cell is b ¼ b0.

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C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501646

Note that with the present formalism, the local porosity in each unit cell is not continuous. Nevertheless, the

porosity of the material domain remains continuous. In addition, the rearrangement of the matrix material

over all nucleated sites, when a new family becomes active, induces a change in the total kinetic energy of the

whole representative volume element. The variation exists only during the nucleation stage. It has been

checked that the loss in kinetic energy is limited. The cumulated loss of kinetic energy at the end of the

nucleation stage represents less than 100th (10ÿ2) of the total kinetic energy in the representative volume

element.

Appendix B. Definition of the threshold pressure pGi

In pure hydrostatic loading, the potential developed first by Gurson (1977) and modified by Tvergaard

(1981) is

O ¼ 2q1f cosh3q22

Sm

s

� �

ÿ ð1þ ðq1f Þ2Þ ¼ 0, (B.1)

where Sm designates the macroscopic pressure (acting at the boundary of the hollow sphere). Therefore, Sm

represents exactly the threshold pressure PG, present in Eq. (14). s denotes an average strength of the matrix

material. One may refer to Duva (1986) for a precise definition of s related to particular matrix behaviour.

The matrix behaviour is governed by Eq. (6):

s ¼ kðd0 þ dÞmð�0 þ �Þn (B.2)

where d is a measure of the average microscopic strain rate in the matrix and � ¼R

d dt is an average

microscopic plastic strain.

From Eq. (B.1), the effective stress s becomes

s ¼3

2q2jSmj

1

lnð1=q1f Þ. (B.3)

For hydrostatic loading, neglecting elasticity, the equivalence of plastic work rate results in

3jSmjjDmj ¼ ð1ÿ f Þsd. (B.4)

The mean macroscopic plastic strain rate Dm is linked to the rate of porosity by

Dm ¼_f

3ð1ÿ f Þ. (B.5)

Combining Eqs. (B.3)–(B.4), the mean microscopic strain rate d is given by

d ¼2

3q2ln

1

q1f

� �

_f

ð1ÿ f Þ2. (B.6)

The average microscopic strain � is obtained by time integration of d:

� ¼2

3q2

lnð1=q1f Þ

1ÿ fþ ln

f

1ÿ f

� �� �

þ Cte. (B.7)

The constant term is found using the initial condition. When the void nucleates, the matrix has already faced a

microscopic plastic strain, noted �pre. Therefore, one obtains

� ¼ �pre þ2

3q2ln

ðq1f Þÿf =ð1ÿf Þ

1ÿ f

" #

. (B.8)

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C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–1650 1647

Finally, using Eqs. (B.2), (B.3), (B.6) and (B.8), the threshold pressure is obtained:

PG ¼2k

3q2ln

1

q1f

� �

�0 þ �pre þ2

3q2ln

ðq1f Þÿf =ð1ÿf Þ

ð1ÿ f Þ

" #( )n

� d0 þ2

3q2ln

1

q1f

� �

_f

ð1ÿ f Þ2

( )m

. (B.9)

Appendix C. Update with the p-model

In Abaqus, as in most Finite Element software, the total mean strain increment D�m is given at the beginning

of each calculation step. All quantities are defined at the beginning of the time increment (time t). Let us note

pðtÞ, f ðtÞ, f iðtÞ, aiðtÞ the pressure, porosities and void radius of the ith family at time t. Knowing D�m, one hasto update all other mechanical fields (pressure, void radii, porosity, mean viscoplastic strain increment). In this

Appendix, the p-model is adopted for the determination of the void radius in each unit cell in the material

domain. Therefore, for each active site, pi ¼ p.

The pressure rate is defined by Eq. (4). Thus, the pressure increment pðtþ DtÞ ÿ pðtÞ can be related to the

increment of elastic strain D�em:

pðtþ DtÞ ÿ pðtÞ ¼Kðf ðtþ DtÞÞ ÿ Kðf ðtÞÞ

Kðf ðtÞÞpðtÞ þ 3Kðf ðtÞÞD�em, (C.1)

where Dt is the time increment and f ðtþ DtÞ the porosity of the material domain at the end of the step time.

The total strain increment is the sum of the elastic and viscoplastic part, see Eq. (1). Thus the pressure

pðtþ DtÞ can be expressed in terms of total and viscoplastic strain increments:

pðtþ DtÞ ¼ pðtÞKðf ðtþ DtÞÞ

Kðf ðtÞÞþ 3Kðf ðtÞÞðD�m ÿ D�vpm Þ. (C.2)

The mean viscoplastic strain increment D�vpm is unknown. This quantity is related to the porosity in the

material domain which depends on the void radii in all active families. But void evolution is strongly

depending upon pressure, see Eq. (13). Thus, difficulties arise because pressure, void radii and inelastic strain

increment are strongly coupled. A fixed point algorithm is adopted to solve this coupling.

First, a guess for D�vpm is proposed. An elastic prediction may be adopted where D�vpm ¼ 0. Therefore,

according to Eq. (C.2), a guess of the pressure at the end of the step pðtþ DtÞ is obtained. Then, the void radius

ai of all active families is evaluated, using Eq. (13):

pðtþ DtÞ ÿ pthresholdi ðf iðtþ DtÞÞ ¼ r aiðtþ DtÞ €aiðtþ DtÞð1ÿ f1=3i ðtþ DtÞÞ

�

þ3

2_a2i ðtþ DtÞ 1ÿ

4

3f1=3i ðtþ DtÞ þ

1

3f4=3i ðtþ DtÞ

� ��

, ðC:3Þ

where

f iðtþ DtÞ ¼a3i ðtþ DtÞ

b3i ðtþ DtÞ. (C.4)

The external radius of the unit cell associated to the ith family is obtained owing to plastic incompressibility of

the matrix material inside the ith unit cell: b3i ðtþ DtÞ ÿ a3i ðtþ DtÞ ¼ b3i ðtÞ ÿ a3i ðtÞ. The time derivatives of ai are

defined using Newmark equations (with g ¼ 12and b ¼ 1

4):

_aiðtþ DtÞ ¼ 2aiðtþ DtÞ ÿ aiðtÞ

Dt

� �

ÿ _aiðtÞ,

€aiðtþ DtÞ ¼ 4aiðtþ DtÞ ÿ aiðtÞ

Dt2ÿ

_aiðtÞ

Dt

� �

ÿ €aiðtÞ. ðC:5Þ

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C. Czarnota et al. / J. Mech. Phys. Solids 56 (2008) 1624–16501648

The estimation of the radius aiðtþ DtÞ is derived from a single-variable algebraic equation. The porosity of the

material domain f ðtþ DtÞ is obtained by volume averaging over all active families:

f ðtþ DtÞ ¼

PJi¼1Pria

3i ðtþ DtÞ

PJi¼1Prib

3i ðtþ DtÞ

, (C.6)

where J is the number of active families at the beginning of the time increment. From the relationship (11), a

new guess for the mean viscoplastic strain increment D�vpm ðtþ DtÞ is derived:

D�vpm ðtþ DtÞ ¼f ðtþ DtÞ ÿ f ðtÞ

3ð1ÿ f ðtÞÞ. (C.7)

A new iteration is carried out. The procedure is continued until convergence on the mean viscoplastic strain

increment and on pressure. In the calculation, a relative precision of 10ÿ5 between two subsequent guesses (for

both pressure and viscoplastic strain increment) is adopted as convergence criterium.

Appendix D. Update with the d-model

In the d-model, for each active site, the total volume strain increment acting on the boundary D�mi is given by

D�mi ¼ D�m. Owing to Eq. (17), the pressure over a given unit cell is obtained in a similar way as for Eq. (C.2):

piðtþ DtÞ ¼ piðtÞK iðf iðtþ DtÞÞ

K iðf iðtÞÞþ 3K iðf iðtÞÞðD�m ÿ D�vpmiÞ (D.1)

with Dt is the time step. The mean strain increment D�m is known. The viscoplastic volume strain increment D�vpmi is

sought. The void radius ai is governed by Eq. (13):

piðtþ DtÞ ÿ pthresholdi ðf iðtþ DtÞÞ ¼ r aiðtþ DtÞ €aiðtþ DtÞð1ÿ f1=3i ðtþ DtÞÞ

�

þ3

2_a2i ðtþ DtÞ 1ÿ

4

3f1=3i ðtþ DtÞ þ

1

3f4=3i ðtþ DtÞ

� ��

ðD:2Þ

f iðtþ DtÞ and biðtþ DtÞ are defined as in Appendix C. A predictor–corrector scheme based on the Newmark

equations is applied to compute the quantities aiðtþ DtÞ, biðtþ DtÞ, f iðtþ DtÞ and D�vpmi for all potentially active

families (their number is denoted J). Finally, the pressure over the material domain is obtained by volume

averaging over all potentially active families:

pðtþ DtÞ ¼

PJi¼1 Prib

3i ðtþ DtÞpiðtþ DtÞ

PJi¼1 Prib

3i ðtþ DtÞ

. (D.3)

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