A Physical Model for Nucleation and Early Growth of Voids in Ductile Materials Under Dynamic Loading

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Journal of the Mechanics and Physics of Solids

53 (2005) 1476–1504

0022-5096/$ -

doi:10.1016/j

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A physical model for nucleation and earlygrowth of voids in ductile materials under

dynamic loading

A. Molinaria,�, T.W. Wrightb

aLaboratoire de Physique et Mecanique des Materiaux, ISGMP, Universite de Metz,

Ile du Saulcy, Metz 57045, FrancebUS Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA

Received 22 November 2004; received in revised form 9 February 2005; accepted 12 February 2005

Abstract

Spall fracture and other rapid tensile failures in ductile materials are often dominated

by the rapid growth of voids. Recent research on the mechanics of void growth clearly

shows that void nucleation may be represented as a bifurcation phenomenon, wherein a

void forms spontaneously followed by highly localized plastic flow around the new void.

Although thermal, viscoplastic, and work hardening effects all play an essential role in the

earliest stages of nucleation and growth, the flow becomes dominated by spherical

radial inertia, which soon causes all voids to grow asymptotically at the same rate, regardless

of differences in initial conditions or constitutive details, provided only that there is the

same density of matrix material and the same excess loading history beyond the cavitation

stress.

These two facts, initiation by bifurcation at a cavitation stress, at which a void first appears,

and rapid domination by inertia, are used to postulate a simple, but physically realistic, model

for nucleation and early growth of voids in a ductile material under rapid tensile loading. A

reasonable statistical distribution for the cavitation stress at various nucleation sites and a

simple similarity solution for inertially dominated void growth permit a simple calculation of

the initiation and early growth of porosity in the material.

see front matter r 2005 Elsevier Ltd. All rights reserved.

.jmps.2005.02.010

nding author. Tel.: +33387 315369; fax: +33 3 8731 53 66.

dress: [email protected] (A. Molinari).

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A. Molinari, T.W. Wright / J. Mech. Phys. Solids 53 (2005) 1476–1504 1477

Parametric analyses are presented to show the effect that loading rate, peak loading stress,

density of nucleation sites, physical properties of the material, etc. have on the applied

pressure and distribution of void sizes when a critical porosity is reached.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Dynamic void growth; Inertial effects; Statistics of void nucleation; Spall

1. Introduction

A metal subjected to large and rapid tensile stresses generally undergoes internaldamage at the microscopic scale in the form of micro-voids, grain cleavage,intergranular fracture, and possibly interconnection of damaged sites by adiabaticshear. In this paper we focus our attention only on void formation.Although the process of nucleation of micro-voids does not seem to be completely

understood from the point of view of dislocation dynamics (however, see Belak,2002; Rudd and Belak, 2002; Seppala et al., 2004; Lubarda et al., 2004), considerableinsight may still be gained from the point of view of continuum mechanics. Somevoids may be generated at hard inclusions (such as carbide or oxide particles or otherprecipitates) by debonding of the inclusion-matrix interface or cracking of theparticle due to large tensile stresses, and others can form within soft inclusions (suchas sulfides). These forms of damage are commonly seen in quasi-static or other formsof testing with relatively low rates of applied stress. However at high loading rates, thenumber of voids observed at the surface of a ruptured sample may exceed the totalnumber of potential nucleation sites offered by the existing inclusions. Therefore,nucleation sites other than inclusions are also thought to be activated in metalssubjected to high loading rates. An extensive discussion and many examples of dynamicnucleation at inclusions and other sites are given by Meyers and Aimone (1983).Potential sites for nucleation may be related to any microscopic material

heterogeneity that can generate a stress concentration. Because of their inherentlyheterogeneous nature, polycrystalline metals and alloys would seem to offer ampleopportunity for void nucleation. The bulk material, whether polycrystalline or not,offers an average background cavitation stress even in the absence of any structuralfeature that might cause a local stress concentration. Where some feature does causea stress concentration, the material may be expected to experience cavitation whenthe actual background stress is somewhat less than the bulk cavitation stress, asnoted by Meyers and Aimone (1983, p. 48). Two specific examples, showing higherspall strength in the single crystal than in the polycrystal, have been given by Antounet al. (2003) for Cu and Mo in their Figs. 5.13 and 5.16.A rough order of magnitude estimate may be made for the number of potential

nucleation sites, besides inclusions, to be expected per unit volume in a typicalpolycrystalline material. These sites may be categorized as lying in bulk crystallinematerial or in structural discontinuities at surfaces, lines, or points. Nucleationoccurring by bifurcation within the bulk crystalline material of a grain probablyreflects the inherent cavitation strength of the parent material most closely.

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A. Molinari, T.W. Wright / J. Mech. Phys. Solids 53 (2005) 1476–15041478

Two adjacent grains meet in a surface where their crystal lattices are misaligned,and impurities tend to collect. The addition of one more grain to the first pair willcreate two more intergranular surfaces, and the addition of each grain after that willusually create three more intergranular surfaces. Therefore, in a polycrystallinematerial there will be approximately three intergranular surfaces per grain. In amicrograph, which shows a plane cutting through the material, these surfaces willalso be cut and show as lines outlining the grains. Meyers and Aimone (1983,Fig. 37a) show an example of voids that have formed on sites of this kind.Furthermore, when the third grain meets the first two, in addition to new surfaces

of contact their common junction will be a line. Each grain added after that will tendto form three new lines of contact with three pairs of adjacent grains so that therewill be approximately three multigrain lines of contact per grain of material. Theselines would show up as triple points in the plane of a micrograph. Finally, when afourth grain is added to the first three, four grains will meet at a point. Every grainadded after that will create one new quadruple junction on average, which wouldrarely show up in a micrograph. In two-dimensional elasticity it is known that wheredissimilar or misoriented materials meet at a point there will be a stressconcentration for any condition of loading, Ting (1996), but the strength of thestress concentration will depend on the extent of the mismatch in properties. Thecorresponding problem in three-dimensional elasticity for multiple materials meetingat a point does not seem to have been solved, but it is safe to conjecture that a strongstress concentration will form there, as well.From the above enumeration of structural discontinuities of various kinds, it

would seem to be reasonable to estimate that in addition to inclusions there might beten or so structural features per grain where either stress concentrations or localweaknesses might create candidate sites for void nucleation. It is impossible to knowthe potential strength of each site, but it seems strongly suggested that the number ofcandidate sites per unit volume is proportional to the number of grains per unitvolume. Of course, grain size, precipitates, etc., also affect basic strength andcharacteristics of plastic flow, which will be reflected in the overall tensile yieldstress and work hardening exponent, and thus in the fundamental quasi-staticcavitation stress.An important premise of this work is that the characteristics of the final

configuration of voids are largely determined by the characteristics of their earlynucleation and growth, and the first step in the understanding of the nucleationprocess is provided by analysis of an isolated void in an infinite elastic–plastic matrixsubjected to a remote tensile pressure, p. A crucial concept is the existence of acavitation pressure, pc; beyond which the void sustains continuous and unboundedgrowth. Hill (1950) discussed growth of a spherical cavity under internal pressure,which is closely related to void growth under remote stress. It may be possible toinfer cavitation at a finite stress from this example, but a distinct bifurcation stress isnot immediately apparent. The concept of a cavitation pressure seems first to havebeen demonstrated by Ball (1982) for a non-linear elastic material and extended laterby Huang et al. (1991) for an elastic–plastic material. Wu (2002) made acomprehensive study for an elastic–plastic material, by accounting also for rate

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sensitivity, thermo-mechanical coupling, heat conduction, and plastic strain gradienteffects, as well as work hardening and inertia. This work has also been reported byWu et al. (2003a,b,c). It is important to note that pc is not only the pressure thatcauses unbounded growth in a preexisting void under quasi static loading, but it alsois the pressure at which a void can appear spontaneously (see Eq. (2) below). In otherwords it is the pressure at which a bifurcation can appear in the solution for auniform body in hydrostatic tension.As with all bifurcation problems, it now becomes important to consider questions

of imperfection sensitivity and post bifurcation response. Our approach will be toassume that there is a statistical distribution of cavitation strengths, correspondingto the microstructure, and that, once nucleated, a void grows according to a simplelaw that emphasizes the dominant role played by inertia in dynamic void growth. Itis tempting to assume a Gaussian distribution for cavitation pressures within thematerial because of its relative simplicity and well understood characteristics.However, a Weibull distribution may be more appropriate physically because, unlikethe Gaussian, it has a definite lower limit for cavitation pressure below whichbifurcation cannot occur, but like the Gaussian it decreases exponentially afterachieving a peak. Sample calculations will be made with both distributions, and itwill be seen that many trends in the results are the same for either choice.In the present work, damage by nucleation and evolution of micro-voids is

analysed in the context of metallic materials subjected to rapid loading rates andhigh tensile pressures. A typical example is provided by the dynamic failure of ametallic plate subjected to rapid loading in uniaxial strain. In the vicinity of the spallplane, material particles are subjected to an abrupt and severe tensile loading. Highstrain rate spall experiments show that for increasing loading rates in the tensileregion preceding spall, the spall strength increases with the rate of loading, seeChapter 5 in Antoun et al. (2003). However, each experiment must be carefullyinterpreted because only tensile pressures can nucleate voids. Examination of Fig. 1,which schematically shows a typical Lagrangean x–t plot of waves in a plate impactexperiment, shows steady shock waves emanating away from the plane of impact,but reflecting as expanding simple waves after reflection from free surfaces. It is theintersection of these reflected waves that produces locally intense tensile stresses, butthe rate of tensile pressurization at any point within the region of intersecting waveswill clearly depend on the structure of the expanding waves and their distance oftravel from the free surfaces, as has been well recognized in the literature (eg., seeKanel’ et al., 1996). Explosive loading, like plate impact, also gives rapid initialcompression, but in contrast to plate impact, tends to produce much slower tensilepressurization because of a ‘‘triangular’’ loading pulse, e.g., see Antoun et al. (2003).Much of the older work relied on explosive loading, and recent work in Russia hasalso emphasized explosive loading, Kanel’ et al. (1996). Recently investigators in theUnited States have also begun to reexamine the material response with ‘‘triangular’’loading pulses, eg., see Johnson et al. (1999), Gray et al. (2004), and Hixson et al.(2004). In all of these last works, the range of rates of tensile pressurization is rathernarrow (only about one decade or less) and is located toward the lower end of ratesconsidered in this paper and in the Russian literature. Consequently, and considering

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the ordinary scatter in experimental data, it may be difficult to identify the effect ofloading rate in these works.As will be shown, the rate of tensile pressurization is expected to have a profound

effect on the size and distribution of voids. As a result, it is clear that the geometry ofthe experiment and the characteristics of waves within the material (not just the peakimpact stress and the duration of the pulse) must be analyzed in order to understandthe results of each experiment. It is the scope of the present work to present a simplemodel accounting for these and other experimental observations.

2. Nucleation of an isolated void

In the analysis reported by Wu et al. (2003a,b,c) for a tensile pressure with a finiterise time followed by a constant value that exceeds the cavitation pressure, it wasfound that soon after the ramped loading, the growth of a void is dominated by theinertia of the material surrounding the void, and soon thereafter, all voids grow atthe same rate no matter what the value of the initial void radius. As a consequence,after a few hundred nanoseconds of growth all initially small voids in an otherwisehomogeneous material will appear to be nearly the same size, if subjected to the samestress history. This is true no matter what the constitutive behavior of the materialmay be because the constitutive response primarily determines only the cavitationpressure. Although rate dependence does have a delaying effect on the earliestgrowth of a void, the same effect may be mimicked by assigning a higher cavitationstress, thus delaying initiation. In any case even with rate dependence the growtheventually becomes dominated by inertia and has the same limiting rate of growth.Viscous or rate effects will dominate void growth only in extremely short pulses.In fact, there are two simple bounds for the cavitation stress (see Wu et al., 2003a).

An upper bound is found by assuming isothermal behavior, and a lower bound isfound by assuming adiabatic behavior, with all plastic work being converted to selfheating and thermal softening. Both bounds, as well as the separation between them,increase with increasing work hardening exponent.If finite thermal conductivity is considered in addition, it is found that nucleation

begins at the isothermal or upper limit (see Wu et al., 2003b). As the void grows,what may be thought of as an ‘‘effective cavitation stress’’ decays toward theadiabatic or lower limit. The existence of a constitutive response due to plastic straingradients would produce an effect similar to that from thermal conductivity (see Wuet al., 2003c). In this case nucleation occurs at an elevated stress (higher than theusual isothermal limit), but again the ‘‘effective cavitation stress’’ decays toward theadiabatic limit as the void grows, see the references for further details. In anyevent, for present purposes such refinements as a variable ‘‘effective cavitationstress’’ will be ignored.Based on handbook data, Wu et al. (2003b) reported estimates for both isothermal

and adiabatic bounds for a variety of materials, which were then compared withspall strengths, as found in the literature. Whenever both types of data are available,the spread of published spall strengths for a given material (obtained largely from

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Table 3.6 in Davison and Graham, 1979, and Table 3 in Grady, 1988) is much largerthan the difference between bounds for the critical stress, which in turn always lie atthe lower end of observed spall strengths, and usually just below them. A naturalworking hypothesis, therefore, is to assume that there is a quasi-static cavitationstress, below which void nucleation can in no circumstances occur, and whichfurthermore may be estimated from known laboratory quasi-static data for thepolycrystalline alloy in question.Now consider any potential nucleation site. When a void nucleates at the site, its

growth initiates when the local macroscopic pressure, pðtÞ; exceeds the valueof the cavitation pressure, pc; whose nature is described below. The evolution law,reported by Wu et al. (2003a) for an isolated site in an infinite plastic matrix, may bewritten as

rða €a þ 3 _a2=2Þ ¼ hpðtÞ � pci; or alternatively

rd2

dt2ða5=2Þ ¼

5

2a1=2hpðtÞ � pci (1)

with the convention hxi ¼ 12ðx þ jxjÞ: The left-hand side of Eq. (1) accounts for the

integrated effect of inertia, assuming incompressibility, as deduced fromthe acceleration potential of Carroll and Holt (1972); r is the mass density of thematrix material, and a; _a; and €a are respectively the radius, the radial velocity, andthe radial acceleration of material particles located at the void boundary. Theassumption of incompressibility allows the total inertial effect of all materialsurrounding the void to be expressed simply in terms of the void radius and itsderivatives. The right-hand side of Eq. (1) is obtained from integration of the radialequation of motion, together with the plastic constitutive response. The term pc isobtained from the limiting value of the following integral:

pc ¼ lima=A!1

Z 2 lnða=AÞ

0

se

expð3�=2Þ � 1d� ¼

2

3sY þ

Z 1

�Y

se

expð3�=2Þ � 1d�. (2)

Material properties are accounted for by the initial tensile yield stress and strain,sY and �Y ; and by the von Mises effective stress, se; which contains all the plasticconstitutive information, including work hardening and thermal–mechanicalcoupling. As mentioned above, additional effects such as heat conduction and ratedependence will cause the critical stress to be effectively time dependent, but sucheffects will be ignored in this paper. In any event, the infinite integral alwaysconverges because of the exponential in the denominator. If the material is perfectlyplastic with no thermal–mechanical coupling, the effective flow stress is equalto the tensile yield stress, sY ; and the final integral has the exact value 2

3sY lnð1�

e�3�Y =2Þ�1

� 23sY lnð2=3�Y Þ; where �Y is the tensile yield strain corresponding to sY :

(For further details, refer to Wu et al., 2003a.)The fact that the integral has the same value when either the void radius, a, grows

to infinity or the initial size, A, tends to zero indicates that void nucleation may bethought of as a bifurcation in the homogeneous solution. Furthermore, because allvoids eventually grow at the same rate for a given overstress above the cavitation

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stress (see Eq. (9) below and the accompanying discussion), it is a greatsimplification, but only a minor error, to ignore preexisting voids and to assumethat all voids nucleate at the local cavitation stress.As with all bifurcation problems, it is necessary to give some account of the effect

of imperfections (imperfection sensitivity) and of the post bifurcation response. Inthe following sections we construct a simple model that addresses both the statisticalnature of imperfections and the initial response after nucleation of individual voids.

3. Statistical characterization of the potential nucleation sites

An infinite material domain is considered with the pressure, p, applied to theremote boundaries. Potential nucleation defects are assumed to be randomlydistributed in the material. A representative domain is considered with a density N ofpotential nucleation sites per unit volume. The mean separation of these potentialsites, b; may be roughly estimated by b � N�1=3: Each site is characterized by its owncavitation pressure, pc: Interactions between nucleation sites and interactionsbetween growing voids are to be neglected. The initial porosity, which could easily beincluded in the model, is assumed to be zero for the clarity of the presentation andfor consistency with the notion of nucleation as a bifurcation. Because interactionsbetween voids are neglected, each site will be considered to be isolated in an infinitematrix with the tensile pressure, p, applied at infinity. It will be assumed that whenthe value of p exceeds the cavitation pressure, pc; for a particular nucleation site, avoid starts to grow there.Due to material heterogeneities, as discussed above, and fluctuations in the

internal residual stress field, the critical pressure varies from site to site. Thesevariations are accounted for by a probability density function. In this paper weconsider a Weibull law and compare its effects with those of a Gaussian law.According to these two laws, the probability density function, gðpcÞ; is given by

gðpcÞ ¼

gWðpc; p0c; a; gÞ ¼ghpc � p0ci

g�1

agexp �

hpc � p0ci

a

� �g� �: Weibull;

gGðpc; pc;sÞ ¼1ffiffiffiffiffiffi2p

psexp �

ðpc � pcÞ2

2s2

� �: Gaussian:

8>>><>>>:

(3)

Note that the probability for having a defect with a cavitation pressure in the rangedpc; on pc; is given by gðpcÞdpc: For the Weibull distribution, gX1; and gWðpcÞ40 ifpc4p0c; but gWðpcÞ ¼ 0 if pcpp0c: The lower limit for having a positive probabilityfor nucleation of a void is p0c; the probability increases to a maximum at p ¼

p0c þ að1� g�1Þ1=g; and then decreases exponentially. The mean critical pressure fornucleation is pc ¼ p0c þ aGðg�1 þ 1Þ; and the standard deviation, sW ; as determinedfrom the second moment about the mean, is calculated from ðsW=aÞ2 ¼ Gð2g�1 þ1Þ � ½Gðg�1 þ 1Þ�2; where Gð�Þ indicates the gamma function. In contrast, for theGaussian distribution gGðpcÞ40 for all values of its argument, the maximum and the

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mean both lie at pc; and the standard deviation is s: Because the cumulativeprobability for all possible defects must be 1, the integral of gðpcÞ over its full rangemust be equal to 1 for either distribution.In order that the Weibull and Gaussian distributions have the same mean, pc; the

lower Weibull cutoff value must satisfy p0c ¼ pc � aGðg�1 þ 1Þ: In order that the twodistributions have the same standard deviation, we must have s ¼

affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGð2g�1 þ 1Þ � ½Gðg�1 þ 1Þ�2

p: Combining these two formulas yields the Weibull

scale factor, a; and the lower Weibull cutoff stress, p0c; as a function of the Gaussianparameters, pc and s; and the Weibull exponent, g:

a ¼sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Gð2g�1 þ 1Þ � ½Gðg�1 þ 1Þ�2p ,

p0c ¼ pc � sGðg�1 þ 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Gð2g�1 þ 1Þ � ½Gðg�1 þ 1Þ�2p . (4)

Of course, pc and s could also be expressed as functions of p0c and a for anygiven g:Strictly speaking, the Gaussian distribution has no lower limit, but in the

calculations one will be introduced for simplicity in the numerical approximations.This will be denoted as pG0c:In either case the number per unit volume, dn; of potential nucleation sites in the

interval, dpc; centered on pc is

dn ¼ NgðpcÞdpc. (5)

For purposes of the analysis, it is convenient to consider a discrete family ofpotential nucleation sites. For the Weibull distribution the lowest value of thediscretization is p0c; and the upper value is taken well out into the region ofdecreasing exponential where there is little probability of void formation. For theGaussian distribution the interval for discretization is centered on pc with both lowerand upper limits lying well into the regions where the probability of finding anucleation site is also negligible. However, the discretization introduces a lower limit,pG0c; which in effect is chosen arbitrarily.

4. A model of dynamic damage at constant loading rate

In a plate impact experiment the loading history, roughly speaking, appears as a‘‘square wave’’ with a rapid rise on the leading edge and usually a somewhat slowerdecrease on the trailing edge. When the rapid rise or shock wave reflects from a freeboundary, the reflection usually will behave as a ramp wave due to the effects ofnonlinear compressibility, and as a result, when two unloading reflections intersectso as to move the stress state into tension, the tensile stress rate will not rise assharply as in the original compressive shock wave. This has been shownschematically in Fig. 1.

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x

t

C

T T

Spal

l

Fig. 1. Lagrangean x–t diagram showing schematically shock waves and reflected rarefaction waves

within a flyer plate on the left and a specimen on the right. C indicates a zone in compression and T

indicates a zone that would be in tension if the material did not fail. Other regions are essentially

unstressed. Void growth, resulting in spall and failure, occurs where the rarefaction waves intersect.


Many essential features can be illustrated by considering a particular loadinghistory where the applied pressure grows linearly with time, t, to mimic the actualrate of pressurization:

pðtÞ ¼ _pðt � t0Þ, (6)

where _p is a positive constant that characterizes the loading rate, and t0 is anarbitrary reference time.Nucleation at each potential site has been characterized by a constant cavitation

stress, pc; which will be reached at time, tc; say. The early growth of the void may begiven by a simple similarity solution of Eq. (1),

a ¼

ffiffiffiffiffi8

33

r ffiffiffi_p

r

sht � tci

3=2, (7)

where tc is the time of nucleation for the site in question, and as before the bracketsindicate the function hxi ¼ 1

2ðx þ jxjÞ: This solution was given by Wright et al. (2004)

with the approximationffiffiffiffiffiffiffiffiffiffi8=33

p� 1=2: Solution of (1) with a given initial radius or

with an arbitrary pressure history could be found numerically, but many essentialfeatures are illustrated using only Eqs. (6) and (7). It should also be noted that therate of growth for the pressurization in Eq. (6) is

_a ¼

ffiffiffiffiffi6

11

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_pht � tci

r

s¼

ffiffiffiffiffi6

11

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihp � pci

r

s. (8)

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If the pressure becomes constant at some fixed value after nucleation, then after ashort transient, according to Eq. (1) the rate of growth asymptotically becomes

_a ¼

ffiffiffi2

3

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihp � pci

r

s, (9)

which is only about 10% larger than the previous value in Eq. (8). It thereforeappears that only a relatively small error would be introduced by switching to thesecond velocity to approximate the void growth after the pressure has becomeconstant, provided that the assumption of isolated voids is still appropriate.Because _p in Eq. (6) is constant, expression (7) can be rewritten as

a ¼

ffiffiffiffiffi8

33

r1

_pffiffiffir

p hp � pci3=2. (10)

This result provides a simple characterization of void evolution, in terms of thematerial characteristics, pc (nucleation pressure) and r (mass density of the matrixmaterial). Note that inertial effects restrain the void growth, since for a givenpressure history, a larger value of r results in a smaller void size. The effect of theloading rate is characterized by the constant, _p: It is worth noting that for thesame pressure and density, a smaller void radius is also the result of an increasedloading rate.Potential nucleation sites are equivalent if they are characterized by the same

cavitation pressure, pc: These sites all nucleate simultaneously, and their radii haveidentical evolutions given by Eq. (10). Because the void volume for one site is givenby 4

3pa3; the volume generated by dn sites with cavitation pressure pc is

dV ¼ 43pa3 dn ¼ 4

3pNa3gðpcÞdpc. (11)

The lower limit of integration is p0c for the Weibull law, and for convenience ofnumerical analysis with the Gaussian law it is set to an arbitrary value, pG0c; which issmall enough with respect to pc (in the following we take pG0c ¼ pc � 5s). Definingx ¼ hpc � p0ci and X ¼ hp � p0ci for the Weibull distribution and using the sameexpression for the Gaussian distribution with p0c being replaced by pG0c; the totalvoid volume generated per unit initial volume at pressure pXp0c (Weibull) or pXpG0c

(Gaussian) is:

V ¼4

3

8

33

� �3=2 pN

_p3r3=2X 9=2

Z X

0

1�x

X

� �9=2gWðx; 0; a; gÞdx : Weibull,

V ¼4

3

8

33

� �3=2 pN

_p3r3=2X 9=2

Z X

0

1�x

X

� �9=2gGðx; ¯pc;sÞdx : Gauss, (12)

where ¯pc ¼ pc � pG0c: Clearly the number Sp ¼ N= _p3r3=2 is important for scalingresults with different physical properties and loading histories, as will becomeevident in several illustrative examples.

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The porosity f is defined by the current volume fraction of voids:

f ¼V

1þ V, (13)

which is approximated in the following by

f � V (14)

for small values of V, and so Eq. (12) is also an approximate formula for the porositywhen porosity is small. It is important to remember that the interaction betweenvoids has been neglected. The net result is that the early evolution of porosity isgoverned by Eq. (12).In the present model, the concept of a critical porosity, f �; is introduced, such that

for fpf � void interaction is negligible. Beyond f �; voids start to interact and it isassumed that a critical phenomenon occurs which results in rapid material failure. Arough estimate may be made of the value of f � by considering the size of a plasticzone surrounding a growing void. According to Hill (1950) for quasi-static loadingand verified by Wu et al. (2003a) for dynamic loading, the plastic zone extendsapproximately 5 or 6 radii beyond the void itself. Now suppose that neighboringsites, with average radius, a; interact when they are on average 4a–6a apart. Thenf �

� ð43pa3Þ=ð4a26aÞ3 � Oð0:0220:07Þ: The effect of inertia in the radial directionfor each void may well modify the effective interaction distance, but at this stage ofthe analysis we can only speculate. Therefore, we can only estimate what areasonable value for f � might be, but f �

¼ 0:05 is likely to be representative,although Davison et al. (1977) report that significant interaction seems to haveoccurred in their calculation by 1 or 2 percent void volume fraction at the maximumlocation.An interesting point is to analyse the effect of the rate of loading on the statistical

distribution of damage at the critical porosity f �:

5. Discussion of the effects of the loading rate

Simulations have been conducted for values of material parameters typical forsteel, and for the statistical distribution of flaws given in Table 1: The value of N inthe table corresponds to a mean separation of all sites of about b ¼ 10mm; but in anycase the influence of N is shown by parametric variations below. The choice of aparticular value is only meaningful for a specific material, although parametric

Table 1

Material parameters

r ðkg=m�3Þ sY ðMPaÞ pc ðMPaÞ p0c ðMPaÞ s2 ðGPa2Þ N ðm�3Þ

7800 500 5sY 1644 0.2 1015

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Fig. 2. Gaussian and Weibull probability density functions of potential nucleation sites corresponding to

the parameters given in Table 1. Both laws have same mean value and same standard deviation.


variation of the mean separation of sites (through variation of N) should correspondroughly to parametric variation of the average grain size. In this section, we explorehow the evolution of dynamic damage is affected by the rate of loading, by materialproperties, and by the statistical distribution of flaws.Fig. 2 shows the Gaussian and Weibull probability density functions gG and gW

defined in Eq. (3). The mean value and the variance of the Gaussian law are given inTable 1. Parameters of the Weibull law have been chosen so as to have the samemean value and variance as for the Gaussian law. For the Gaussian law, theprobability of having a defect with a critical nucleation pressure, pc; being in therange ½p1; p2� is given by

Prðp1; p2Þ ¼1ffiffiffiffiffiffi2p

ps

Z p2

p1

exp½�ðpc � pcÞ2=2s2�dpc

¼ 12½erfððp2 � pcÞ=

ffiffiffi2

psÞ � erfððp1 � pcÞ=

ffiffiffi2

psÞ�, ð15Þ

where erfðxÞ is the error function. The corresponding number of potential nucleationsites is

n ¼ N Prðp1; p2Þ. (16)

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For the Weibull law, we have

Prðp1; p2Þ ¼

Z p2

p1

gWðpc; p0c; a; gÞdpc ¼ exp �ðp1 � p0cÞ

g

ag

� �� exp �

ðp2 � p0cÞg

ag

� �.

(17)

The effect of the loading rate is analysed in Fig. 3. Two loading rates areconsidered, _pA ¼ 1GPa=ns and _pB ¼ 0:01GPa=ns: Fig. 3a shows, respectively forthe Weibull and the Gaussian laws, the densities of porosity distribution pc !

hWðpc; p�Þ and pc ! hGðpc; p

�Þ; defined by Eqs. (19) and (21). Results are shown atthe values p� (different for the Weibull and the Gaussian laws) of the appliedpressure for which the assumed critical value of the total porosity, f �

¼ 0:05; isreached.According to Eq. (12), the porosity f � is given, for the Weibull law, by:

f �¼

Z p�

p0c

hWðpc; p�Þdpc (18)

with the density of porosity distribution defined as

hWðpc; p�Þ ¼

4

3

8

33

� �3=2 pN

_p3r3=2ðp� � pcÞ

9=2gWðpc; p0c; a; gÞ. (19)

For the Gaussian law we have

f �¼

Z p�

p0cG

hGðpc; p�Þdpc (20)

with

hGðpc; p�Þ ¼

4

3

8

33

� �3=2 pN

_p3r3=2ðp� � pcÞ

9=2gGðpc; pc;sÞ. (21)

Note that hW vanishes for pcpp0c and that both hW and hG can be considered asvanishing for pcXp�: Note also that the areas defined by the curves in Fig. 3a havethe same values for critical porosity, f �

¼ 0:05; however, the corresponding criticalpressures, p�; are different as shown in Table 2.For either distribution law the porosity density curves shift toward higher

pressures with increasing rate of pressurization, but the Gaussian curve appears to bemuch more sensitive to the change. The reason for this appears to be that at low ratesof pressurization only the lower tail of the Gaussian contributes to the result, but themain part of the Weibull distribution is immediately engaged, at least for g ¼ 2: Athigher rates, the two porosity density curves are more nearly equal. Fig. 3b shows thedistribution of the micro-void radii for each cavitation stress at the same loadingrates, and amplifies on the previous remark. It shows clearly that the distribution ofvoid radii is strongly skewed toward the lower end for the Gaussian at the lower rate,but is more uniform at the higher rate for both Gaussian and Weibull distributions.The actual frequency of occurrence of voids of a certain size is weighted by theprobability distribution function, of course.

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Fig. 3. Effect of the loading rate ( _pA: solid lines, _pB: dashed lines) on: (a) porosity densities pc ! hGðpc; p�Þ

(Gauss) and pc ! hWðpc; p�Þ (Weibull) (with p� depending on the loading rate and the probability law, see

Table 2), and (b) void radius versus nucleation pressure. Calculations are made for the value f �¼ 0:05 of

the critical porosity.


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Table 2

Comparison of results for the Gaussian and the Weibull probability laws

Radius adom of

dominant voids

Nucleation

pressure of

dominant voids

Number of

nucleated voids

pressure p� at

porosity f �

Gaussian distribution

_pA ¼ 1GPa=ns 2:34mm 2.34GPa N 7.96GPa

_pB ¼ 0:01GPa=ns 10:4 1.23 0:10N 1.94

Weibull distribution

_pA ¼ 1GPa=ns 2:45mm 2.17GPa N 7.96GPa

_pB ¼ 0:01GPa=ns 6:05 1.76 0:32N 2.25

The number of potential nucleation sites per unit volume is N ¼ 1015 m�3 and the critical porosity has the

value f �¼ 0:05:


From Fig. 3a the maxima of the porosity density curves occur at the cavitationstress for the dominant void for either rate of pressurization and either distribution.Then from Fig. 3b the size of the dominant void for each case may be read off. Inaddition the total number of sites which have been effectively nucleated at the criticalporosity ðf �

¼ 0:05Þ and the pressure p� at which f � is reached may also bedetermined. All these results are summarized in Table 2. Two loading rates areconsidered, _pA ¼ 1GPa=ns and _pB ¼ 0:01GPa=ns: For _pA ¼ 1GPa=ns; the pressureat the critical porosity (f �

¼ 0:05) is p�A ¼ 7:96GPa for the Gaussian law and nearly

the same for the Weibull law. That level of critical pressure is much larger than thecavitation pressure, pc; of all significant void families. Therefore, almost allnucleation sites are activated (Nnucl

A � N). In addition, the dependence of the voidradii with respect to pc decreases for larger pressure rates, _p; as is apparent in Eq. (10)and in Fig. 4, which shows the evolution in terms of the applied pressure, p, of thevoid radii, a, for three values of nucleation stress (pc ¼ 1; 2; 3GPa). From theseconsiderations, it can be deduced that the distribution of void radii is more uniformfor higher values of the loading rate. In contrast, for the lower loading rate _pB ¼

0:01GPa=ns; the pressure at f � has the smaller value p�B ¼ 1:94GPa (Gauss) and

p�B ¼ 2:25GPa (Weibull). For this pressure, just a fraction of the potential nucleationsites is activated, Nnucl

B ¼ 0:10N(Gauss) and NnuclB ¼ 0:32N (Weibull). Because p�

B isclose to the cavitation pressures of the sites that have been nucleated, Eq. (10) andFig. 4 show that the dispersion in the sizes of the radii is large. This discussionfurnishes an explanation of the void distribution shown in Fig. 3b.To summarize, it can be concluded that for both distributions at the assumed

critical porosity, f �¼ 0:05; the dynamic damage has the following characteristics:

�
The distribution of void radii is more uniform at higher rates than at lower rates ofpressurization.
�
The voids having a dominant contribution to the total porosity have a smallerradius at higher rates (this effect is more pronounced for the Gaussian than for theWeibull distribution).

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Fig. 4. Evolution, in terms of the applied pressure, p , of void radii for three sites with nucleation pressures

pc ¼ 1; 2 and 3Gpa, respectively. Two loading rates are considered, _pA ¼ 1GPa=ns and _pB ¼ 0:01GPa=ns:Note that for the smaller loading rate, the void evolution with pressure is much faster.


�
The total number of micro-voids (nucleated sites) is larger at higher rates. � The pressure, p�; is larger at higher rates. � At higher rates the two distributions give more nearly comparable results than atlower rates.
These conclusions regarding certain quantitative aspects of spallation, althoughperhaps suggested by some observations, appear not to have been discussed inprevious reviews of experimental observations of dynamic damage, Meyers andAimone (1983), Davison et al. (1996), and Antoun et al. (2003). However, thecalculations do strongly suggest that a higher rate of tensile loading corresponds tonucleation and early growth of a larger number of micro-voids with smaller sizesmaking an effective contribution to the global porosity. This last observation isperhaps suggested in the preceding references, but does not seem to be explicitlystated. However, in their section on spall in iron and steel Meyers and Aimone (1983)discuss the evidence that when the material is loaded past the phase transition at13GPa, a rarefaction shock can form in the unloading wave preceding spall.The unloading shock can lead to extremely fast tensile pressurization and a‘‘smooth’’ spall surface, which nevertheless has been characterized as ‘‘ductile’’because dimpling on a scale of 5mm or less is still visible in scanning electronmicrographs.

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6. Influence of material parameters

A simple parametric analysis can be performed by using Eq. (12) in which totalporosity has been written as a function of the applied pressure p and the loading rate_p: Dynamical effects are characterized by the terms _p

ffiffiffir

p: Thus, changing the loading

rate by a factor of l ( _p ! l _p) is equivalent to changing the density by a factor of l2

(r ! l2r), and in either case the critical value of the porosity f � will be reached forthe same value of the applied pressure, p�; and for the same void radii, as given byEq. (10). For illustration, the results shown in Fig. 5 are obtained for two values ofthe mass density, r ¼ 7800 and 1000 kg=m3; all other parameters being identical andgiven in Table 1. The loading rate is _p ¼ 1GPa=ns: Results are identical by keepingthe mass density at the value r ¼ 7800 kg=m3 and changing _p ¼ 1GPa=ns into _p ¼ffiffiffiffiffiffiffiffiffiffiffi1=7:8

pGPa=ns: Thus the trends observed in Fig. 3 are retrieved. Increasing the rate

of loading, _p; or increasing the mass density, r; for a given value of the criticalporosity, f �; results in larger pressure, p�; more uniformity of void radii, and anincreasing number of activated voids.The dependence of the critical pressure p� and of the dominant void radius

(corresponding to the cavitation pressure that nucleates voids making the largestcontribution to the global porosity) upon the loading rate _p is shown in a log–logdiagram, Fig. 6. For _p41GPa=ns; the increase of p� with _p is well described byEq. (22) for the Weibull distribution, and for the Gaussian distribution by replacing

Fig. 5. Effect of the mass density r ¼ 7800kg=m3 compared to r ¼ 1000kg=m3 for the Gaussian (thin

lines) porosity density pc ! hGðpc; p�Þ and the Weibull (thick lines) porosity density pc ! hWðpc; p

�Þ: Theloading rate is _p ¼ 1GPa:

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Fig. 6. (a) Effect of the loading rate, _p; on the critical pressure, p�; for the value of the critical porosity,f �

¼ 0:05: The estimate p�est given by Eq. (22) is shown to be relevant for _p41GPa=ns: (b) Effect of _p on

the void radius, adom; corresponding to the family having the dominant contribution to porosity. For largeloading rates, adom tends to the asymptotic value given by expression (25), assuming that all voids have the

same radius. The gap between adom and aunif is an indication of the void radius heterogeneity (decreasing

function of _p).


p0c by pG0c:

p�est � p0c ¼

3

4p

� �2=933

8

� �1=3f �

N

� �2=9

_p2=3r1=3. (22)

This estimate is obtained in the following way. For a large enough pressure, p�; itfollows from Eq. (10) that void radii become identical for all significantlycontributing voids:

a� ¼ aunif �

ffiffiffiffiffi8

33

r1

_pffiffiffir

p ðp� � p0cÞ3=2. (23)

Result (22) follows from the expression of the critical porosity

f �¼ ð4=3ÞpNa3unif . (24)

Another way to obtain Eq. (22) is from direct examination of Eq. (12). For Xbx

throughout the peak region of gðpcÞ (that is, when pc lies within the peak region ofgðpcÞ and p � p0cbpc � p0c) the term 1� ðx=X Þ is nearly equal to one, and therefore

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may be taken outside the integral for purposes of estimation. Then becauseR1

0 gðxÞdx ¼ 1 for any law of distribution, (approximate for the Gaussian, butessentially true in our applications) Eq. (22) follows directly.In relationship (22), the power law dependence p�

est � p0c / _p2=3 provides theconstant slope 2

3shown in Fig. 6a for large values of p� � p0c: Another asymptotic

result for small values of p� � p0c is easily obtained for the Weibull distribution. Inthis case the loading history only penetrates to the lowest part of the probabilitydensity function, as shown in Fig. 2b for example. Then with the Weibulldistribution expressed as gW�

gag hxi

g�1 þ h:o:t:; and with the change of the variableof integration to Z ¼ x=X ; the expression for porosity reduces to

f�4p3

8

33

� �3=2N

_p3r3=2gag

hp � p0cið9þ2gÞ=2xðgÞ þ h.o.t. (25)

where xðgÞ ¼R 1

0 ð1� ZÞ9=2Zg�1 dZ is just a constant. Inversion of Eq. (25) at thecritical porosity and stress leads to the expression

p� � p0c�ðf � _p3r3=2=NÞ2=ð9þ2gÞ, (26)

which should hold for small values of the quantity in parentheses.Fig. 6b shows the dependence of the dominant void radius, adom; on _p: Note that

for large loading rates, adom tends to an asymptotic value given by the expression

aunif ¼3f �

4pN

� �1=3

¼3f �

4p

� �1=3

b, (27)

which is independent of _p and r; as well as the law of distribution. In order to beconsistent with the previous discussion, Eq. (27) is obtained from Eq. (24) byassuming that all voids have the same radius. The gap between adom and aunif isindicative of the heterogeneity in the distribution of void radii which, as discussedbefore, decreases with increasing _p: Because the effect of _p is carried by the term _p

ffiffiffir

p;

results similar to those of Fig. 6 would be obtained by varying the value of r:The effect of the number of potential nucleation sites, N, may be analyzed in a

similar way. As should be clear from Eq. (12), decreasing N1=3 is equivalent toincreasing _p

ffiffiffir

p: In fact, because the mean spacing between potential nucleation sites

is given by b � N�1=3; the porosity will remain invariant for b _pffiffiffir

p¼ const:

Consequently, increasing b (i.e., increasing grain size or decreasing N1=3) is predictedto be exactly equivalent to increasing _p with all other parameters and structuralfeatures held constant in both cases, and should have the same effects of flatteningthe distribution of void sizes and shifting the distribution of differentialcontributions to porosity toward higher cavitation pressures. The physical effectseems to be that because there are fewer potential sites for nucleation, more of thetotal population must be initiated and the pressure must be higher to achieve thesame porosity. Alternatively, the larger the number of potential nucleation sites(everything else being unchanged) the higher the loading rate must be to reach thecritical porosity at the same value of pressure. Furthermore, Eq. (27) leads us toconclude that the average void size will be smaller in this case, as well.

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Fig. 7. Effect of the number of potential nucleation sites N, for the loading rate _p ¼ 0:01GPa=ns and for aGaussian density law: (a) porosity densities pc ! hGðpc; p

�Þ; (b) void radius versus nucleation pressure,

and (c) normalized void radius versus nucleation pressure. Results are shown at the value f �¼ 0:05 of the

overall critical porosity. The effects of decreasing the number, N, of potential nucleation sites, are

qualitatively similar to the dynamical effects associated with the increasing of the loading rate, _p; or of themass density, r; illustrated in Figs. 3 and 5.


For the Gaussian law and for a given value of the critical porosity, f �¼ 0:05;

Fig. 7 shows the effects of N at the loading rate _p ¼ 0:01GPa: The trends are similarfor the Weibull distribution. The distribution of void radii in Fig. 7b indicates thatessentially all potential nucleation sites are activated when the smaller value N ¼

1012 is considered, while only a fraction of these sites is activated for the larger valueN ¼ 1015: Because the total porosity is given, the void radii are smaller on averagefor the larger value of N as shown in Fig. 7b. The distribution of void radii,normalized by the radius of the largest void, is presented in Fig. 7c and shows thatthe distribution is more uniform for a smallerN. Thus, it is confirmed that decreasingthe number of potential nucleation sites is analogous to the dynamical effectsassociated with higher values of the loading rate, _p; or of the mass density, r:The Gaussian distribution (3) of potential nucleation sites is characterized by the

mean cavitation pressure, pc; and the variance, s2: The Weibull distribution ischaracterized by the minimum critical stress, p0c; and the parameter, a; but as givenin Eq. (4), for a given value of g these can be obtained from the Gaussian parameters.Of course, the equivalent Gaussian parameters could also be determined from givenWeibull parameters.We consider, first for the Gaussian distribution, the effect of changing the value of

pc into pc þ Dp0; while s is fixed. From expression (12) for the porosity, f ¼ V ; and

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the definitions X ¼ hp � pG0ci; pG0c ¼ pc � 5s; ¯pc ¼ pc � pG0c ¼ 5s; it is clear that f isinvariant if p is also shifted by an amount Dp0: As a result, we expect that the valueof the applied pressure, p�; for which the critical porosity, f �; is reached in a materialwith mean nucleation pressure, pc; will change into p� þ Dp0 if a material with meannucleation pressure pc þ Dp0 is subjected to the same loading rate, _p: The void radiiwill be left invariant by the change pc ! pc þ Dp0; p� ! p� þ Dp0: An illustration isgiven in Fig. 8 where two values, pA ¼ 2:5GPa and pB ¼ 5GPa; are considered. Thenumber of potential nucleation sites is taken as N ¼ 1014 m�3: The evolution of p� interms of _p is shown in Fig. 8a in a log–log diagram. Note that p�

B � p�A ¼ pB � pA:

The evolution of the void radius for the family having a dominant contribution tothe porosity is reported in Fig. 8b.For the Weibull case, expression (12), f ¼ V ; together with X ¼ p � p0c and

relationship (4) for p0c (when the Weibull parameters are tied to the Gaussianparameters) show that X and f are invariant under the changes pc ! pc þ Dp0 andp ! p þ Dp0: Therefore, the conclusions obtained for the Gaussian law can bereadily extended to the Weibull law and are not shown in Fig. 8. However, as inFig. 6, at the lower rates of pressurization the dominant void sizes for the Weibulldistribution will fall well below those for the equivalent Gaussian distribution.

Fig. 8. Effect of increasing the average nucleation pressure from pA ¼ 2:5GPa to pB ¼ 5GPa: The criticalpressure, p�; at the porosity, f �; is shifted by Dp ¼ pB � pA; but the void radii are left unchanged. The

number of potential nucleation sites per unit volume is taken as N ¼ 1014 m�3: The Gaussian distributionis analysed here. The trends for the Weibull distribution are similar and are not shown here, but adom will

be less for Weibull than for Gauss at low rates, as in Fig. 6b.

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Fig. 9. Gaussian laws are considered with mean value p ¼ 2:5GPac: The effect of the variance, s2; isanalysed by taking two values, s2A ¼ 0:2 ðGPaÞ2 and s2B ¼ 0:002 ðGPaÞ2: (a) probability density functions

of potential nucleation sites pc ! gGðpc; pc; sÞ; (b) porosity densities pc ! hGðpc; p�Þ and (c) distribution of

void radii, for the loading rate _p ¼ 0:01GPa=ns and for the critical value of the total porosity f �¼ 0:05:

By decreasing the standard deviation, s; the distribution of void radii has clearly to be more uniform. In

addition, the pressure, p�; at the critical porosity, f �¼ 0:05; is a decreasing function of s; p�A ¼ 1:94GPa

and p�B ¼ 2:74GPa for respectively s2A and s2B: Note that the total number of activated nucleation sites

decreases as s increases. The dominant void radii are: aAdom ¼ 10:4mm and aB

dom ¼ 2:56mm:


The effect of the variance s2 is shown in Fig. 9 for N ¼ 1015 m�3 and _p ¼

0:01GPa=ns: Two probability density functions for the potential nucleation sites areshown in Fig. 9a for two values of the variance s2 and for a Gaussian law with meanvalue given in Table 1. Clearly, by decreasing s2; the distribution of void radii withrespect to nucleation pressures must be more uniform. This point is illustrated byconsidering the density of porosity distribution and the distribution of void radii, a,for s2A ¼ 0:2 ðGPaÞ2 and s2B ¼ 0:002 ðGPaÞ2 and for the loading rate _p ¼

0:01GPa=ns; see Fig. 9b–c. In Fig. 10, all data are identical to those of Fig. 9,except that the loading rate has been increased to _p ¼ 1GPa=ns: The results shownin Figs. 9 and 10 are summarized in Table 3.To summarize, it can be concluded that for both variances at the assumed critical

porosity, f �¼ 0:05; the dynamic damage has the following characteristics:

�
The distribution of void radii is more uniform at higher rates or smaller variance. � The voids having a dominant contribution to the total porosity have a smallerradius at higher rates or with a smaller variance.

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Fig. 10. The same parameters as those of Fig. 9 are used, except for the higher loading rate, _p ¼ 1GPa=ns:The effect of the standard deviation, s; on the critical pressure, p�; and on the dominant void radius has

become negligible: p�A ¼ 7:95GPa; p�B ¼ 8:02GPa; aA

dom ¼ 2:34mm (pc ¼ 2:34GPa), aBdom ¼ 2:29mm

(pc ¼ 2:50GPa).

Table 3

Effect of loading rate and variance on the dominant void size, number of sites nucleated, and pressure at

f �¼ 0:05 for a Gaussian distribution of potential sites

_p ðGPa=nsÞ s2 ðGPaÞ2 adom ðmmÞ # Voids p� ðGPaÞ

0.01 0.2 10.4 Less 1.94

0.01 0.002 2.56 More 2.74

1.0 0.2 2.34 All 7.95

1.0 0.002 2.29 All 8.02


�
The total number of micro-voids (nucleated sites) is larger at higher rates or withsmaller variance.
�
The pressure, p�; is larger at higher rates or with smaller variance.
The effect of the variance, s2; on the critical pressure, p�; and on the dominant voidradius becomes negligible when _p is large enough. As before, asymptotic evaluationof Eq. (12) is independent of the details of the function itself, provided that p �

p0cbpc � p0c throughout the peak of the distribution function.Results for both Weibull and Gaussian laws are compared in Fig. 11

( _p ¼ 0:01GPa=ns) and Fig. 12 ( _p ¼ 1GPa=ns) for two values of the standard

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Fig. 11. Effects of the standard deviation on porosity densities, respectively hGðpc; p�Þ and hWðpc; p

�Þ for

the Gaussian and Weibull laws which both have the same mean value (2.5Gpa) and standard deviation, s:Parameters are identical to those of Fig. 9. Note that the results become close for small values of the

standard deviation.

Fig. 12. Effects of the standard deviation on the distribution of porosity densities. Parameters are identical

to those of Fig. 11, except for the pressure rate, _p ¼ 1GPa=ns: Results are compared for Gaussian and

Weibull laws which have the same mean value (2.5Gpa) and standard deviation, s:


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Fig. 13. Effect of the loading rate _p on the critical pressure p� and the dominant void radius adom for

Gaussian laws with two values of the variance, s2A ¼ 0:2 ðGPaÞ2 and s2B ¼ 0:002 ðGPaÞ2: As it has to be,

variations produced by _p are more important for the largest value of the variance . Note also that the

results become independent of s for _p41GPa=ns: Indeed, for large loading rates p� and adom were shown

in Fig. 6 to be independent of the statistical distribution of potential nucleation sites. The number of

potential nucleation sites per unit volume is taken as N ¼ 1014 m�3:


deviation. Note that the results of those laws become comparable for small values ofthe standard deviation, as well as for higher rates of pressurization with a fixedstandard deviation.For the Gaussian law the effects of the loading rate, _p; on the critical pressure, p�;

and on the dominant void radius, adom; are illustrated in Fig. 13 for two values of thevariance s2A ¼ 0:2 ðGPaÞ2 and s2B ¼ 0:002 ðGPaÞ2: As expected, variations producedby _p are more important for the larger value of the variance. Note also that theresults become independent of s2 for _p41GPa=ns: Indeed, for large loading rates,the results of Fig. 6 have demonstrated that p� and adom do not depend on thestatistical distribution of potential nucleation sites.

7. Summary and discussion

In this paper we have tried to construct a simple, but physically based theory forthe nucleation and early growth of voids under intense dynamic loading. Such atheory is necessarily only part of a general theory of spall. Our approach has fourprincipal ingredients: (1) the concept of a cavitation pressure at which the local

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solution experiences a bifurcation, (2) a dynamical growth law for a single void thatapplies after the cavitation pressure has been exceeded, (3) a statistical description ofcavitation pressures that characterizes a particular material, and (4) a history ofapplied pressure, which has been assumed to be linear in time for illustrativepurposes in this paper. In the initial stage of the nucleation process, the void volumefraction is generally small enough to neglect void interaction; therefore, the analysisof a single void isolated in an infinite matrix remains consistent until significantinteraction between voids has developed.Our approach differs from previous theories such as those of Seaman et al. (1976),

Davison et al. (1977), Eftis and Nemes (1991), or Addessio and Johnson (1993) all ofwhom adopt some form of a viscous growth law for the basic response describingvoid growth. In contrast, convinced of the relatively unimportant effects ofviscoplasticity by fundamental studies of void growth, we have adopted themechanism of integrated inertia responding to excess pressure, given by Eq. (1), asbeing fundamental at the local level, and have explicitly incorporated the assumedlocal statistics of cavitation stress in the overall dynamics. However, recall that ourtreatment to date attempts to describe only the earliest stages of spall, whereas theothers strive for a complete theory.To illustrate some of the principal consequences of the theory, a similarity solution

for early growth of an isolated void following cavitation was introduced andexpressed in terms of a constant rate of tensile pressurization and an excess pressureabove the local critical stress. This simple dynamical growth law was combined withtwo choices for a statistical description of critical stresses: either a Gaussian or aWeibull probability distribution function. The combining of these two ideas, i.e., adynamical growth law and a suitable probability distribution function, isstraightforward because both are expressed in terms of a local critical stress, whichin turn can be used as a variable of integration in the probabilistic calculations.An immediate result is to identify the importance of a spall parameter, Sp ¼

N= _p3r3=2; which appears in Eq. (12) and linearly scales the calculation for earlyvolume fraction of voids. It contains two physical quantities that are associated withthe characteristics of the material and one that reflects the imposed dynamicalenvironment. The first two are the volumetric density of potential nucleation sitesand the mass density, and the last is the rate of tensile pressurization. The compositeparameter, Sp; summarizes in a simple way two extremely complex aspects of anyspall experiment. Although the mass density of the material is straightforwardenough, the other two quantities are not. The total number of potential sites per unitvolume is not an easily accessible number, although it can be related to the meanseparation between potential nucleation sites, which should be roughly proportionalto grain size (all other metallurgical factors being held constant). Finally, the rate oftensile pressurization at a potential site depends on wave propagation andinteraction of intersecting waves, which in turn depend strongly on experimentaldesign, as well as elastic and thermoviscoplastic constitutive properties of thematerial. In effect, Sp announces that spall strength is not a unique physicalproperty, but is the result of a dynamical process that connects a complex,heterogeneous material with a specific loading history.

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Eq. (12) also shows the importance of the loading history and its interaction withthe statistical nature of the local critical stress. At the highest rates of loading thedetailed nature of the distribution function becomes unimportant, as noted inSection 6. This fact permits the deduction of several asymptotic results connectingcritical porosity, pressure, and void size.At lower rates of pressurization the details of the distribution function are more

important and show up especially in Figs. 3, 6, 8 and 13. The Gaussian distribution,which includes a finite probability for nucleation at all pressures, begins to developporosity well before the applied pressure reaches the mean value of cavitationstresses. As a consequence, at low rates of loading it is possible to reach a criticalporosity when the applied pressure is still in the statistical tail below the meancavitation pressure. Because the lowest value for calculation, pG0c; is actually chosenarbitrarily, voids at the ‘‘weakest’’ points, which are the first to be excited as thetensile pressure increases, grow to unrealistically large size. Although the probabilityof encountering such a site is small, the net effect is to bias the contribution toporosity strongly toward the weaker sites. This may clearly be seen in Fig. 3a,compared with Fig. 2a, when we recall that the mean cavitation pressure is 2.5GPain this example. Fig. 3b shows another aspect of this bias because the weakest sitesgrow to the largest sizes. Fig. 6b shows the same thing; because of the bias towardearlier excitation, the dominant void radius for the Gaussian distribution has alonger time to grow than for the Weibull distribution, and as a consequence the ratioof the two at the lowest rates exceeds a factor of five or so.Of course, parametric calculations for the Weibull distribution were only made

with g ¼ 2: For significantly larger values of g the Weibull distribution would alsoshow a stronger bias toward growth of the weakest critical stresses at low rates,because like the Gaussian, the function would also have a leading shape with anincreasing slope, in particular gW�Oðhpc � p0ci

g�1Þ; as may easily be seen fromEq. (3), and has already been used in obtaining Eqs. (25) and (26). In some sensethen, the two probability laws chosen for illustration in this paper might beconsidered as extreme cases. At lower rates of loading, the choice of probability lawseems to be of greater consequence than at the highest rates where many effectsbecome independent of the choice.The major parametric effects are shown in the tables and the figures. To summarize

very briefly, at lower rates only the weakest sites for nucleation are engaged, and thepressure required to reach a critical void volume ratio remains relatively low. Athigher rates of loading the driving pressure penetrates farther into the distributionfunction before there is time for the void volume fraction to reach a critical level, andat the highest rates of loading most of the potential sites are activated.Figs. 6a, 8a, and 13a show that over a limited range of rates of pressurization, the

critical pressure can be considered to rise nearly linearly in a log–log plot.Consequently the two should appear to have the relationship p� ¼ A _pn; wherenp2=3; and the lower the rates of tensile pressurization, the smaller the value ofQJ;the exponent. It is suggestive to note that Antoun et al. (2003) summarize theexperimental rate dependence of spall strength for several materials by therelationship s� ¼ Að _V=V 0Þ

m; where _V=V0 is the rate of decompression preceding

ARTICLE IN PRESS


spall. The exponents reported are of the order of 0.1–0.2 with considerable scatter inthe data for spall strength, most of which lie in a range of estimated rates of104–107 s�1: In Fig. 6a the average slope over the first three decades of the rate ofpressurization is about 0.1 or a little less, and in Figs. 8a and 13a approximately 0.2or 0.3 for the two curves in both figures. In decompression, followed by tensilepressurization, the volumetric strain rate should be largely elastic, and therefore, _pshould be roughly proportional to _V=V0; which would bring the experimental andtheoretical exponents into rough agreement. Moreover, with a typical elastic bulkmodulus of order 100GPa, the volumetric strain rates that have been experimentallyobserved correspond to rates of tensile pressurization of 10�3–100 GPa=ns: Thus,both the range of data and the exponents given in Antoun et al. (2003) correspond tothe general predictions in this paper.More precise predictions would depend on the details of the probability law, as is

also evident in Fig. 8a where the higher (and more realistic) value for meancavitation pressure corresponds to the lower exponent, or in Fig. 6a where theWeibull law gives a slightly lower exponent than the Gaussian law. Note also,however, that according to the asymptotic expression (26), if the plots were made asp� � p0c vs _p; rather than p� vs _p; the part of the curve at lower rates would be muchsteeper where p� is only slightly greater than p0c: At higher rates the difference woulddisappear in the log–log plot.Of course our suggested model only shows how early void growth might be

predicted. It does not predict void sizes at complete material failure, and so it cannotbe directly compared to the dimpling that appears on many typical spall surfaces. Itis highly suggestive, however, to note that the theory clearly predicts that a higherrate of loading can be expected to result in more numerous but smaller voids atcritical porosity, as seen in Figs. 6b, 8b and 13b. More precise predictions willrequire further development of the model (including full, two-way coupling betweenpressure and porosity, rather than the one-way coupling that we have used), newexperimental techniques to determine the parameters in the distribution function,and full calculations of impact experiments that lead to spall. Only then will it bepossible to compute not only spall strength, but also porosity and void distributionboth on and adjacent to the spall plane with confidence. Nevertheless, the resultsobtained so far seem encouraging.

Acknowledgements

The authors are grateful to Prof. K.T. Ramesh for discussions during thepreparation of this manuscript.

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Documents

A Physical Model for Nucleation and Early Growth of Voids in Ductile Materials Under Dynamic Loading