Numerical study of spalling in an aluminum alloy 7020-T6

International Journal of Impact Engineering 22 (1999) 649}673

Numerical study of spalling in an aluminum alloy 7020-T6

S. Hanim, J.R. Klepaczko*

Institut Supe& rieur de Ge&nie Me& canique et Productique, Laboratoire de Physique et Me& canique des Mate& riaux,URA-CNRS 1215, Universite& de Metz, Ile du Saulcy, 57045 Metz Cedex.01, France

Received 23 May 1997; received in revised form 21 January 1999; accepted 18 April 1999

Abstract

Planar impact experiment is frequently used to investigate dynamic fracture of materials, particularly thespall phenomenon. Spalling is caused by the superposition of rarefaction waves re#ected from free surfacesand the spall zone is found in the interior of the target. Behavior of materials in this kind of experiment isstrongly a!ected by the stress level, time of loading and temperature. The rate and temperature e!ects areclosely related to the thermally activated micromechanical processes [1]. Thus, in a stressed body thecreation of new fracture surfaces frequently occurs with the assistance of thermal activation. For a moredetailed study, it is therefore necessary to take into account the physical aspects of spalling, includingdynamic plasticity and temperature coupling. This paper reports the numerical analysis performed usinga "nite element FE code by implementation of a cumulative fracture criterion proposed in [2] where theapparent energy of activation for spalling depends on stress, temperature and load history. Initially, a seriesof calculations have been run for the purely elastic case to analyze the minimum critical impact velocityneeded to obtain the spall stress and it has been determined to be a function of the critical time of loading.Such analysis is of great value in designing experiments that are relatively expensive. Next, a viscoplasticconstitutive relation together with the cumulative criterion, and the equation of heat conduction have beenimplemented in a FE code. The set of relations takes into account strain hardening, strain rate sensitivity andtemperature. This series of FE calculations have been performed in order to take into account, changes oftemperature due to volume dilatation as well as conversion of plastic work into heat. In addition to spalling,the free surface velocity}time pro"les have been calculated for a number of impact velocities. Speci"cvariations of the free surface velocity indicates the creation of a new fracture surface inside the target plate.The two sets of FE calculations reported in this paper led to some discussion on the in#uence of physicalparameters on spall mechanics. ( 1999 Elsevier Science Ltd. All rights reserved.

*Corresponding author. Tel.: 0033-03-8731-5356; fax: 0033-03-8731-5366.E-mail address: [email protected] (J.R. Klepaczko)

0734-743X/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 2 3 - 8

1. Introduction

Spallation is a particular kind of dynamic fracture leading also to fragmentation. The "rst andmore serious observation of spall has been probably reported in the XIX century. Since the spallbecame the subject of much interest due to a variety of applications, particularly in terminalballistics and detonics, many studies have been reported so far in literature. The propagation of(elastic}plastic) stress waves and their interaction, with thermal instability phenomena (micro shearbands), and the intrinsic strain rate dependence of material response render dynamic fracturea uniquely complex phenomenon that can be, and most often is, quite di!erent from quasistaticfracture. A comprehensive discussion of the physical features associated with dynamic fracture ofsolids can be found in the review articles by Klepaczko [2], Meyers and Aimone [3], and Curranet al. [4].

Spalling can be de"ned as a complete or partial separation of material near the free surfacesresulting from the tension induced by the interaction of two waves, incident and re#ected. Thesewaves can be produced in the laboratory conditions by direct explosive experiment, by impact ofexplosively driven plate on a target, by a gas gun driven projectile impact on the target plate or bydeposition of an intense pulse of energy on the opposite surface of the target, for example laserirradiation.

In this study, a plate impact experiment was used to produce plane waves in order to study thephenomenon of spallation in Al-Alloy 7020-T6. This experimental technique seems to be the bestsuited because the impact velocity of the #yer plate and planarity of impact can be preciselycontrolled. In experiments of this kind, the impact is produced by a thin #yer plate travelling atvelocities up to several hundred meters per second. In a target plate, the compressive and re#ectedtensile stress pulses travel across the plate thickness typically in about from 100 ns to 2 ls,producing local strain rates of the order of 104}107 s~1. It is clear that in such an experiment, theelastic or plastic wave propagation, strain rate and temperature e!ects are all superimposed at thesame time. It was attempted in this study, as a second step after experiments, to analyse by the "niteelement method, the in#uence of these factors on spallation in the aluminum alloy 7020-T6. Anextensive series of numerical simulations were performed to model the spall behavior of thisaluminum alloy, by implementing in a FE code (Abaqus-explicit), the elasto}viscoplastic constitut-ive relations, thermal coupling and the cumulative fracture criterion which includes e!ects oftemperature and the history of loading [2].

2. Remarks on propagation of plane waves

Any impact, at any velocity, results in the propagation of stress waves through a solid. Since thestress waves propagate at "nite speed, which is a material property, gradients in strain or stress willexist in both space and time. If the magnitude of the applied stress pulse in uniaxial strain is abovethe Hugoniot Elastic Limit (HEL), two types of waves will propagate through the medium. First anelastic precursor propagates with the speed C1 which is followed by a family of plastic wavespropagating with di!erent speeds C

p(ep), where e

pis the plastic strain. This is the so-called simple

wave approach which is rate independent [5]. In general, during wave propagation the internalenergy of a solid change causing changes in temperature. If the level of stress is below the HEL,

650 S. Hanim, J.R. Klepaczko / International Journal of Impact Engineering 22 (1999) 649}673

Table 1

¹ (K) E (GPa) l C1

(mm/ls) C2

(mm/ls) pHEL

/p0

100 75 0.310 6.1151 3.2089 1.815300 72 0.320 6.0878 3.1321 1.888500 65 0.345 6.0606 2.9482 2.1129600 60 0.360 6.0227 2.8169 2.2857

only then is the thermoelastic e!ect present and the wave speed changes accordingly. However,when the plastic acceleration waves are present, adiabatic heating occurs and the wave speedchanges. Because the rate of strain hardening decreases as a function of temperature, the wavespeed will be lower. Thus, during spalling the temperature is already slightly increased due tothermal coupling.

We note that the elastic wave speeds, longitudinal C1 and transversal C2 are a function of thetemperature. The wave speed of the elastic precursor for longitudinal one-dimensional strain isgiven by

C1"S

E(¹)(1!l(¹))o(1#l(¹))(1!2l(¹))

(1)

and the propagation velocity for elastic shear waves is

C2"S

E(¹)2o(1#l(¹))

, (2)

where E, l, o, represent respectively Young's modulus, Poisson's ratio and the density.Based on the experimental results of Koster [6] for E(¹) and l(¹), the calculated C

1and

C2

values for aluminum at four di!erent temperatures are given in Table 1. As expected, C1

andC

2slightly decrease as a function of temperature. The temperature gradient caused by the wave

propagation can change the time variations of the free surface velocity, which is important inevaluation of experimental data.

The Hugoniot elastic limit (HEL), is known to be temperature dependent and it is given by thefollowing expression:

pHEL

(¹)"A1!l(¹)1!2l(¹)Bp0

(¹), (3)

where p0

is the yield limit in uniaxial stress conditions.The plastic wave speed in one-dimensional strain for rate-independent simple waves is

C1(e1zz

)"S1o

Lpzz

Le1zz

. (4)

The longitudinal elastic wave speed is de"ned by Eq. (1). Of course, the tangent modulusE5"Lp

zz/Le

zzusually decreases as a function of temperature and so does the plastic wave speed.

S. Hanim, J.R. Klepaczko / International Journal of Impact Engineering 22 (1999) 649}673 651

Fig. 1. Schematic arrangement of planar plate impact experiment.

Fig. 2. Schematic x}t diagram showing interaction of incident and re#ected waves which generate tension zone withinthe material where spalling occurs (if the tension is high enough).

These relations for simple elasto}plastic waves permit a preliminary analysis of the plate-impactexperiments.

It is brie#y discussed below how the elastic or plastic waves interact and generate a tension wavethat may produce spalling in a material. A typical experimental con"guration used at LPMM-Metz is shown in Fig. 1. A #at #yer plate of thickness h and diameter d travelling at velocity<

0strikes a stationary target of thickness larger than h, typically 2h. For symmetric impact, that is

the target and the #yer plates are made of the same material, symmetric compressive waves aregenerated in the target and in the #yer. The phase diagram for the #yer and the target is shown inFig. 2. Re#ection of the compressive incident wave from the free surface of the target producesa tensile stress wave, and at distance h from the free surface a high tensile stress occurs before thearrival of release waves from the edges of the plate. Thus, the central portion of the target is ina con"ned state of one-dimensional strain. The compressive wave in the #yer plate is re#ected by


the free surface as a tensile wave and returns to the impact surface, the time of contact is t0"2h/C

1.

Consequently, duration of the tensile wave generated in the target is t0. If the magnitude and

duration of this tensile stress wave are high enough, spallation occurs. The stress amplitude of theincident wave can be obtained from the relation p

z"oC

1lz, where l

zis the mass velocity in the

z-direction [5]. This is the so-called acoustic approximation used frequently to analyze test datanear HEL.

The phase diagram shown in Fig. 2 is most helpful in understanding the time process ofspalling. The left-hand side represents the impactor (#yer plate), and the right-hand side representsthe target. At the beginning of the impact, the elastic compression waves are emitted intothe projectile and target. If the impact velocity is high enough, the elastic waves are followedby slower plastic waves. The inverse of the slope of the characteristic lines gives the wave speeds(cf. Eqs. (1)}(4)). It can be readily seen that the elastic precursor has a higher velocity than thesimple plastic waves. As the elastic and Riemann plastic waves encounter the free surfacesof the target and projectile, they re#ect back as elastic waves (if the Bauschinger e!ect is neglected).When spalling initiates, the release waves emitted from the newly created free surfaces com-pletly change the pattern of waves inside the target plate. The pull-out velocity measuredafter spalling at the free surface of the target is frequently used to analyze spall dynamics (see forexample [7]).

3. Remarks on experimental techniques

The material considered in the present work is aluminum alloy 7020-T6, with densityo"2.780 g/cm3, Young's modulus E"72.0 GPa. The target thicknesses used in the experimentswere respectively h"10, 8 and 5 mm and the diameter was d"57 mm. The longitudinal elastic-wave speed at room temperature, and under uniaxial strain condition, determined from Eq. (1) isC

1"6.0878 mm/ls with l"0.32.Simple compression tests have been carried out on this aluminum alloy. The quasi-static and

fast compression tests were performed using a fast universal hydraulic machine. The averagestress}strain curves of three tests obtained at two strain rates, e

1+10~3 s~1 and e

2+11.3 s~1 are

shown in Fig. 3. It is found that this alloy is moderately rate sensitive with relatively high strainhardening rate. The elastic wave speed, Eq. (1), as well as the speeds of the simple plastic waves,Eq. (4), calculated on the basis of experimental data of Fig. 3 are shown in Fig. 4.

Symmetric impact experiments using a gas-gun [8], were performed in vacuum to "ndcritical conditions of spall. The gas-gun was completely automated and was equipped withan optical system to measure the impact velocity of the projectile. The results of a series of the"nal tests using as many as seven projectile-target geometries are reported elsewhere [9].The average results for three geometries are reproduced in Fig. 5. The specimens were softlyrecovered with a specially designed catcher to prevent any secondary damage. The thicknessof the impactor was half of the target thickness. This geometry was used in order to assurethe proper interaction of the waves in the interior of the target. Such a situation is shownin the form of the phase diagram in Fig. 2. These experimental results clearly indicate that thecritical time t

#of material separation (a complete spall) is a function of the amplitude of the critical

stress pF.


Fig. 3. True stress}true strain curves for compression tests of aluminum alloy Al 7020-T6, at strain rates e51"10~3 s~1

and e52"11.3 s~1.

4. Constitutive equations

The aluminum alloy 7020-T6 is assumed to obey rate-dependent J2 (Huber}Mises) #ow theoryof plasticity with isotropic hardening. It is further assumed that the total deformation rate can bedecomposed additively into elastic and inelastic part.

e5ij"e5 %

ij#e5 1

ij. (5)

The elastic deformation rate is governed by Hooke's law, Eq. (6), and the inelastic one is governedby the rate equations (7)}(10).

Elasticity:

p5ij"2ke5 %-

ij#jd

ije5 %-kk

. (6)

Plasticity:Yield function:

J32sijsij!p

e(e6 1-)"0, s

ij"p

ij!1

3dijpkk

. (7)

Equivalent plastic strain:

e6 1-"Pt

0

e65 1- dt, e65 1-"J23e5 1-ije5 1-ij. (8)

Plastic yow rule:

e5 1-ij"

32

sij

pe

e65 1-. (9)


Fig. 4. Longitudinal wave speed C1

calculated using data of Fig. 3.

Fig. 5. Critical spall stress pF

versus critical time of loading t#for 7020-T6 aluminum alloy, solid line represents criterion

(23).


Table 2

Const. B0

q ¹.

n0

m0

e50

e0

Value 1352.0 1.118 1877.0 0.289 0.02248 1.88 e`12 0.007Units (MPa) (1) (K) (1) (1) (s~1) (1)

Here e5 1-ij

is the e!ective plastic strain rate, e65 1- is the equivalent plastic strain rate, p%

is the Misese!ective stress and s

ijis the deviatoric part of the stress tensor.

The strain rate dependence of material is assumed as a power law of the form [10]:

p%"p

0(e6 1-, ¹)A1#A

e65 1-e50B

m(T)

B (10)

with

p0(e6 1-, ¹)"B

0A1!q¹

¹.B(e6 1-#e

0)n(T) (11)

and

n(¹)"n0A1!

¹

¹.B, (12)

m(¹)"m0A

¹

¹.B, (13)

where p0(e6 1-, ¹) is the quasi-static #ow stress which depends on plastic strain e1- and the absolute

temperature ¹, B0

is the plasticity modulus, e0

and e50

are, respectively, the reference strain and thereference strain rate, n

0and m

0are the strain hardening exponent and the strain rate sensitivity

(both at ¹"0 K), respectively, q is the temperature sensitivity and ¹.

is the melting temperature.To determine the material constants, which describe strain hardening and strain rate sensitivity

of aluminum alloy Al 7020-T6, the test data shown in Fig. 3 were used. By doing a "t to theexperimental true stress}true strain data at strain rate e5 "11.3 s~1, the values of the constantsB0, q, e

0and n have been determined. In order to "nd the material constants describing the rate

sensitivity, the following procedure has been applied. Eq. (10) may be rewritten as

log e5 1-"A1mB logA

p%

p0

!1B#log e50

. (14)

The values of the constants have been determined by "tting linear relation (14) to the experimentaldata, that is the "tting to the linear regression log e5 1- as a function of log(p

e/p

0!1) for a total strain

e"0.05. The values of the constants in Eqs. (1), (10)}(13) are presented in the Table 2.

5. Application of the heat conduction

Plastic deformation occurring at a fast pace involves increases in temperature due to adiabaticheating. To accurately predict the response of the material, the e!ect of temperature on the #ow


stress must be included in a constitutive model. Thus, in coupled themomechanical problems anadditional "eld equation is added to the usual "eld equations. This additional equation, the heatconduction one, provides a link between mechanical deformation "elds and the additionalunknown temperature "eld, ¹(x, t), and is given by

i+2¹!¹Q "!

bpe5 1-oC

1

#

aoC

1

E(¹)(1!2 l(¹))

¹0tr(e5 %-), (15)

where the dot refers to di!erentiation with respect to time, i is the thermal di!usivity, o is thedensity, C

1is the heat capacity, b is Taylor}Quiney coe$cient, a is the coe$cient of thermal

expansion, E is Young's modulus, l is Poisson's ratio, and ¹0

is the initial temperature. The "rstterm on the right represents heating due to irreversible plastic deformation and the second termrepresents heating due to the reversible thermoelastic e!ect. When the heat equation is introducedinto FE code, it can be shown that the approximation of adiabatic heating transforms the problemto an ordinary di!erential equation which makes it possible to update the temperature in theconstitutive subroutine. Since the heat conduction term is neglected, the heat equation is trans-formed from a partial di!erential equation to an ordinary di!erential equation. Hence, it can beintegrated locally by the same subroutine as the constitutive equations. This approach makes theformulation computationally advantageous. The equation of heat conduction is then given as

¹Q "bp ) e5 1oC

1

!

aoC

1

E(1!2l)

¹0tr(e5 %). (16)

Here, all temperature changes of elastic constants are neglected.This equation has been implemented in the FE code Abaqus together with the constitutive

model and the cumulative criterion of spalling which is discussed below.

6. Discussion of spall criteria

Fracture is the result of a variety of microscopic rate processes, but each of these processes is notaccessible directly for global analysis and modeling. Development of useful engineering models,together with the use of modern numerical methods, are of great importance in predicting failureevents at di!erent loading rates and temperatures. The models of fracture or spalling must beconstructed to describe, as closely as possible, microscopic observations and experimental data.

Post-fracture photo-micrographic observations, [3,4,11,12], have shown that spallation repres-ents the end result of an accumulation of microcracking or microvoids and adiabatic bridgestriggered by plasticity which takes place during the tensile phase of the stress wave loading.

The microstatistical approach to fracturing has been considered by Curran et al. [4]. Theydeveloped a model that involves the nucleation, growth and coalescence of voids in a regionundergoing tensile stresses (the NAG model). This approach involves the following expressions forthe rate of nucleation, NQ , and the rate of growth, RQ , of microvoids:

NQ "NQ0

expC(p!p

30)

p*D, RQ "A

p!p'0

4g BR, (17)


where p530

is the tensile threshold stress, NQ0

is the threshold nucleation rate, p*is the stress sensivity

for nucleation, p'0

is the tensile threshold for void growth, g is the viscosity parameter, and R is thecrack/void radius. This approach requires determination of a number of material constants byexperiments with application of statistical methods. Those constants are not known a priori. Thus,a great amount of e!ort is needed to be put in to characterize each material before numericalanalysis. The NAG model should be preferrably used for soft metals that are prone to microvoidevolution. In harder materials, the main micro-mechanisms are micro-cracks and adiabaticshearing.

Besides, spalling is dependent on both the amplitude and duration of the tensile pulse. Timedependence of dynamic fracture is observed by the plate impact experiment. For a pulse of longduration, the stress to cause fracture is lower than that for a short duration pulse [9,13]. A similardependence has also been observed in a direct explosive experiment with triangular shaped stresspulse as shown by Breed et al. [14]. Skidmore [15], has reinterpreted the explosively driven #yingplate experiments using the stress rate at fracture. Typically, the plate impact data is usuallyrepresented in terms of the square pulse geometry, which is an idealization of an instantaneoussquare loading}unloading (acoustic approximation).

Tuler and Butcher [16], have proposed a cumulative relation between various loading condi-tions and spall fracture, including time dependence of spalling in the form of a generalized criterion.Spalling is related to the overstress in tensile mode and time duration of the pulse at the spall plane.This criterion has the form

K"Ptc

0

[pF(t)!p

F0]d dt, (18)

where K, pF0

and d are the material constants. This approach is based on only one evolutionequation. Another form of this criterion can be written as

t#0"P

t#

0ApF(t)

pF0

!1Bddt with t

#0"

KpdF0

, (19)

where t#0

is the characteristic time, which is di!erent for di!erent materials. The criterion is basedon the overstress concept and in the limit p

FPp

F0, t

#PR. This is not exactly true since the

longest critical times observed experimentally are "nite and relatively short. For real conditions inmetals, it is about several microseconds. For longest intervals of loading near the threshold stresspF0

, only the incipient damage is usually found.To determine the critical conditions in spalling, Cochran and Banner [17] used the peak of the

velocity formed on the free surface of the target when a spall occurs. In such a case, the free surfacevelocity does not return to zero. It was found that the ratio between the free-surface velocities of thespall to the compression pulse peaks provided a good correlation with void densities at the spallplane. The pull-back free-surface velocity was used in [7] to calculate the stress initiation ofspalling.

It is generally accepted that the rate and temperature e!ects are closely related to the thermallyactivated micromechanical processes of plasticity and fracturing. The concept that thermal activa-tion is involved in material separation during fracture processes has been pursued by Zhurkov [1].In other words, the creation of free surfaces in a stressed body occurs with assistance of thermal


activation processes (thermal vibration of the crystalline lattice provides some additional energy tofracture).

The kinetic concept of the mechanism of fracture is understood as a time-dependent process forwhich the rate is determined by stress and temperature. From this standpoint, the investigation oftime and temperature factors become very important for the understanding of the mechanism offracture phenomenon. In this regard, systematic studies of the relationship between the life-timeof solids under load and the magnitude of the tensile stress and temperature have been previouslycarried out [1]. However, those studies were limited to the long-time span. In such a case, therelationship observed between the life time t

#, the critical stress p

F, and the absolute temperature

¹ can be written in the form of the linearized kinetic equation [1]:

t#"t

#0expA

G0!<Hp

Fk¹ B , (20)

where k is the Boltzmann constant, t#0

and G0represent material constants and<H is the activation

volume. The dependence of barrier energy *H"G0!<Hp

Fon the stress results in sharp

acceleration of the fracture process in a stressed body, and in a decrease of its life time under load.More recently, Dremin and Molodets [18], have proposed a modi"cation of the above equation tobe applied in spall mechanics. A more general discussion of thermally activated processes inspalling can be found in [4].

Klepaczko [2,9], has proposed a di!erent approach to the Boltzmann statistics to formulatea fracture criterion for short and very short loading times. A logical approach is to assume that thefrequency of the rate of microcrack evolution is in the form

NQ "NQ0

expA!*G(p

F)

k¹ B, (21)

where NQ and NQ0

are the rate of damage and the fundamental rate respectively, *G(pF) is the

stress-dependent free energy of activation. It is also assumed that the time interval of themicrocrack growth is very short (typically few nanoseconds in hard materials) in comparison tothe critical time to spalling. The energy of activation is taken after Yokobori [19]:

*G(pF)"*G

0lnA

pF0

pFB, (22)

where pF

is the local stress, pF0

is the threshold stress, *G0

is the barrier energy for non stressedbody. Thus, the following cumulative fracture criterion in the integral form was proposed byKlepaczko [2]:

t#0"P

t#

0ApF(t)

pF0B

a(t)dt; t

#)t

#0, p

F*p

F0, (23)

a(¹)"*G

0k¹

, (24)

where pF0

, t#0

and a(¹) are three material constants at constant temperature ¹, t#0

is the longestcritical time, when p

F(t#0)"p

F0, for t

#'t

#0, p

F"p

F0and p

F0"const. The exponent a(¹) is


temperature dependent and is related to the activation energy *G0, with k the Boltzmann constant

and ¹, the absolute temperature. When the process is non-isothermal, as it happens in case ofspalling, the exponent a(¹) is time dependent via changes of temperature during loading orunloading, a(¹)"*G

0/k¹(t). In such a case Eq. (23) must be integrated accordingly, including the

temperature history ¹(t). The correlation between the experiments (aluminum alloy 7020-T6) andthe criterion Eq. (23) is shown in Fig. 5. The material constants determined for this alloy are:pF0"1.05 GPa, a"1.28, t

#0"2.0 ls.

It should be noted that the criterion in the form of Eq. (23) has some physical motivation basedon thermally activated rate processes. On the other hand it predicts well the critical time t

#as

a function of the spall stress pF

for di!erent materials.

7. Numerical simulation and discussion of results

The constitutive Eqs. (5)}(13), the equation of heat conduction (16), and the cumulative fracturecriterion, Eq. (23), have been implemented in the "nite element code Abaqus [20,21] (explicitversion), by writing user subroutines.

In order to analyze the mechanics of wave propagation together with the spall phenomenon,a large series of numerical FE simulation have been performed for aluminum alloy 7020-T6. Allmaterial constants used in the simulations are given in Table 2. A circular target plate d"57.0 mmloaded by a square pulse has been analyzed. Di!erent thicknesses of the #yer plate were assumed,the same as used during the experiments. The thickness of the #yer plate ¸

1"5.0 mm, and the

thickness of the target plate ¸2"10.0 mm. Due to radial symmetry only half of the #yer and the

target has been simulated. The mesh adopted had 725 axisymmetric elements (element typeCAX4R). The time increment was assumed constant, *t"21.6 ns. The majority of the simulationswas carried out up to the total time of 4.6 ls which is su$cient for several elastic wave passagesacross the target thickness, and long enough for complete spallation. Before chosing the initialinput parameters for every simulation, the value of HEL has been evaluated for 7020-T6 Alloy. Theformula which relates the mass velocity l

HELand the p

HELis p

HEL"oC

1lHEL

. Introducing theconstants given in Table 2, the following values are determined: p

HEL"0.72 GPa and

lHEL

"42.5 m/s. Having determined these values, it is possible to "nd that the mass velocity belowlHEL

creates only the thermo-elastic coupling. When the impact velocity exceeds lHEL

the thermo-plastic coupling is also present.

First series of preliminary numerical simulations have been performed with low mass velocitieswhere the thermo-elastic coupling dominates. The initial mass velocity was assumed v

*"40 m/s

and next 80 m/s. By varying the thickness of the projectile (time of loading) and the amplitude, thecontour plots of the longitudinal stress p

zzat di!erent times in the target plate for v

*"80 m/s are

obtained and shown in Fig. 6. The incident elastic wave is clearly visible. The re#ected tensile waveis slightly dispersed. The thermoelastic wave pro"les of temperature for v

*"40, 80 and 250 m/s in

the middle of the target are shown in Fig. 7a. The thermoelastic e!ect due to dilatational response(or volume change) usually neglected in standard calculations is well visible. It is also known thatthermoelastic increments of temperature are reversible. The temperature increment *¹ is positivefor the compressive wave and negative for the tensile wave. The reversible thermoelastic e!ect isclearly noted for all initial velocities (Fig. 7a). This preliminary result con"rms the accuracy of the


Fig. 6. 3D plot of the longitudinal stress pzz

at di!erent times in the target plate for v*"80 m/s.

approach to the thermal coupling. As expected, the thermoelastic temperature increments arerelatively small: *¹+$26 K for v

*"250 m/s (see Fig. 7a). The elasto}plastic approach is

di!erent for v*"250 m/s as is shown in Fig. 7b. The incident elastic}plastic compressive wave heats

up the material and after re#ection the negative thermoelastic part is superimposed on the positiveincrement of temperature due to conversion of plastic work into heat. When the calculation timereaches its maximum value (+4.5 ls), the total increment of temperature is about 50 K.

The "rst series of simulations, performed over a range of relatively low initial velocities permittedto verify the spall criterion, Eq. (23), with the condition t

#"t

., where t

.is the maximum time of the

incident wave. These calculations have given the minimum critical velocity v*and the minimum

critical stress pF

of spall-fracture as a function of the critical time of loading t#. It has been found

that for the con"guration analyzed, the spalling can occur above a velocity v*"80 m/s for

pF"1.25 GPa and t

#"1.64 ls. These calculations are also permitted us to determine numerically

three stages of the material inside the target: no spall, incipient and advanced spall (creation of thefree surface). The "nal results are shown in Fig. 8. The results of these simulations are shown in


Fig. 7. Temperature changes due to dilatation (7a) and due to dissipation of plastic work (7b), at z"5.0 mm for v*"40,

80 and 250 m/s.

Fig. 8a in the form pF(t#). Fig. 8b shows the comparison of the numerically determined state of

the incipient spall with experiments (three average points of "ve tests each for three values of t#).

In the case of instantaneous loading, which is an ideal case without risetime, the critical spalltime t

#is slightly shorter than in the real case, Fig. 8b. In reality, the rise time before spall is not zero


Fig. 8. (a) Minimum critical fracture stress pF

as a function of corresponding critical time of loading, t#; (b) comparison

of the numerically determined state of the incipient spall with experiments (three mean points of "ve tests each for threevalues of t

#).

and an introduction of the rise time is necessary in numerical schemes. In the case of moreadvanced plastic deformation, the critical time is longer than that for the elastic case. This is causedby a delay time in initiation of yielding and then the plastic #ow increases the rise time.

At high incident velocities, the re#ected tensile waves are more distorted in comparison to theincident compressive waves. This is shown in 3D (p

zz, z, t) for v

*"250 m/s in Fig. 9a and b with two


Fig

.9.

3Dpl

ots

(pzz,z

,t)fo

rv *"

250

m/s

with

two

angl

esofob

serv

atio

nw

ithoutsp

allin

g,sh

owin

gdisper

ssiv

eas

pect

ofvi

scopla

stic

wav

e.


angles of observation. In Fig. 9b the wave dispersion of the elastic front of the incident compressivewave is clearly visible. Similar calculations have been performed for several incident velocities butwith constant time of loading t

."1.6 ls. The main reason was to "nd time histories of the free

surface velocity, usually measured during experiments by the laser-Doppler velocimetry [22].Calculations were run twice for the same initial conditions. The "rst time without fracture criterionand the second time with the spall criterion, Eq. (23). The results are shown, respectively, in Fig. 10aand b. Of course, the free surface velocity is non zero when the spalling occurs, Fig. 10b.

In order to "nd numerically the critical conditions for spalling and creation of a new free surfaceinside the target, the following technique has been applied. Upon reaching the critical conditionsde"ned by the criterion Eq. (23), the stress in the particular element is set to zero and the element iseliminated from further calculations. This operation creates a new free surface inside the targetplate giving rise to a new re#ected wave. When the incident velocity increases, the spall peak of thefree surface velocity becomes more and more evident as it is shown in Fig. 11 where the minimaland maximal free surface velocities are shown as a function of the incident velocity. Thesesimulations correctly reproduce the free surface velocity. Results shown in Fig. 10b demonstrateoccurence of the new free surface in the interior of the target material. The maxima are delayed forthe lower incident velocities, which means that spalling occurs with some delay as it is predicted bythe cumulative criterion. If the critical time t

#is delayed until t

#0, then spall does not occur.

A systematic observation of the calculated contour plots of the equivalent plastic strain,equivalent plastic strain rate and temperature at di!erent instants were carried out durringnumerical analyses. For example, the intensity of strain rate in the spall zone is about 106 s~1.Finally, when the time of calculation was su$cient, the contour plots of the free surface createdinside the target disc could be evaluated. For example, the contour plots at di!erent stages ofspalling for two incident velocities: 160 and 250 m/s are shown, respectively, in Figs. 12 and 13. Theevolution of the spall zones as a function of time is demonstrated for three instants, for impactvelocity 160 m/s: t

1"3.515 ls, t

2"3.581 ls and t

3"3.614 ls; and at 250 m/s: t

1"3.256 ls,

t2"3.308 ls and t

3"3.464 ls. In both cases the spall starts near the external diameter where the

tensile release wave superimposes with the plane incident wave. However, the spall is almostinstantaneous, and "nally the free surface is visible as the white areas. The hypothetical inwardfracture velocity is estimated as a5

#"3.5H102 mm/ls. This value substantially exceeds the Rayleigh

wave speed expected in such situation, that is +3 mm/ls. Thus, the spalling occurs instantaneouslywithin the whole internal surface.

The distance of the spall surface from the free surface of the target disc depends on the incidentparticle velocity v

*. Higher values of v

*cause the position of the spall surface is closer to the free

surface. Such behavior, based on the numerical analysis, is shown in Fig. 14.

8. Analysis of the pull-back velocity

It had been suggested some time ago (see for example [7]) that the variations of the free surfacevelocity can be directly related to the incipient spall strength. The characteristic points duringvariations of the free surface velocity are shown schematically in Fig. 15. Unloading starts at pointA and continues until point B where the new compressive wave arrives. This wave is created by thespall inside the target. Finally, the amplitude of spall is signalled by point C. The di!erence of the


Fig. 10. (a) Sequences of calculated free surface velocity without spalling for di!erents mass velocities; (b) sequences ofcalculated free surface velocity for spalling situation for di!erents mass velocities.

free surface velocities *<&4"<

A!<

Bcalled the pull-back velocity, can be directly related to the

incipient spall strength pF*

by the following relation:

pF*"1

2oC

1R*<

&4, (25)


Fig. 11. Evolution of the maximums and minimums of the "rst peak of the pro"les of free surface velocity as a function ofincident velocity.

where C1R

is the speed of the rarefaction wave emitted by the spall surface, C1R+C

1. This speed

can be found from the distance of the spall surface Z#to the free surface and the travel time *t

Rof

the rarefaction wave, C1R"Z

#/*t

R. It is assumed in this calculation that the rarefaction wave is

purely elastic. The critical time of spalling t#

can be found from the formula t#"t

B!t

A.

Another approch has been proposed in [23]. This approch is based on propagation ofelasto}plastic wave with the bilinear material behavior, and consequently with two wave velocities,elastic C

1and plastic C

2. The formula derived in [23] is

pF*"oC

1*<

&4A1#C

1C

1B

~1. (26)

The critical time can be calculated using the elastic and plastic slownesses S1"1/C

1and

Sp"1/C

1,

*t#"Z

#(S

1!S

1), (27)

where *Z#is the distance between the spall surface and the free surface of the target where *<

&4is measured.

A slightly di!erent method of determining the critical conditions of spalling has been proposed in[24,25]. This method is based on the increment of the free surface velocity d<

&4and the formula

used is

pF*"oC

1(*<

&4#d<

&4), (28)


Fig. 12. The contour plots at di!erent stages of spalling for incident velocity: v*"160 m/s.


Fig. 13. The contour plots at di!erent stages of spalling for incident velocity: v*"250 m/s.


Fig. 14. The distance of the spall surface from the free surface of the disc as a function of the incident particle velocity v*.

Fig. 15. The characteristic points during variations of the free surface velocity.


Fig. 16. Comparison of the three methods, based on the measurement of the free surface velocity (pull-back velocity).

where d<&4

is the velocity correction caused by plastic waves. When the acceleration or decelerationof the free surface velocity <Q

&4is determined, the "nal formula for the velocity correction takes the

form

d<&4"*Z

#(S

1!S

1)

D<Q1<Q

2D

D<Q1D#<Q

2

, (29)

where <Q1

and <Q2

are accelerations of mass velocity at the wave front before and after spalling.Since the numerical study has provided relatively precise data on spalling, which are discussed in

the previous part of this paper, a comparison can be made as to which of those three methods,based on the measurement of the free surface velocity, give the best approximation of the criticalparameters (p

F, t

#). The result of this comparison is given in Fig. 16. The solid line is the `ideala

behavior assumed in the numerical calculations. The acoustic approximation, Eq. (25), gives higherstresses p

F, or shorter critical times t

#. The method based on determination of accelerations, Eq.

(29), gives much better estimations. However, this method is a source of a higher scatter, becausethe "rst derivatives of velocities are used in Eq. (29). Quite reliable results, which are close to theassumed behavior, has been obtained with Eq. (26) shown as a black circles in Fig. 16. The criticaltimes t

#are slightly longer, which is probably caused by assumptions of plastic wave speeds at

e"0.05 for Al-alloy 7020-T6. It may be stated that the acoustic approximation, Eq. (25), can betreated as a "rst approach in "nding the critical condition of spalling when the impact velocities arenot so high.


9. Conclusions

The numerical analysis reported herein clearly demonstrates the ability to predict the spallprocess in structural alloys. The cumulative criterion in the form of Eq. (23), proposed byKlepaczko [2], and introduced in the Abaqus FE explicit scheme enables one to analyze numer-ically the spalling over a wide range of impact velocities. The range of impact velocities studied wasbetween 80 and 560 m/s (twice the initial mass velocity v

*) with the pulse duration from 0.4 to 1.7 ls.

Although the calculated temperature increments are not so high, the numerical scheme appliedallows for studies of temperature e!ects on spalling, including di!erent initial temperature, fromlow to elevated.

The numerical simulations performed for di!erent loading conditions reproduce exactly, for thealuminum alloy studied, the dependence of the spall stress p

Fon pulse duration, as revealed in

Fig. 8b. The extent of spall advancement in the target plate can be estimated numerically atdi!erent impact velocities including thermal e!ects. This could help in the design of new experi-ments.

Finally, the numerical FE scheme combined with the cumulative spall criterion, Eq. (23), is ableto reproduce the `pull-backa velocities of the free surface caused by the creation of the internalspalling, Fig. 16. Among the methods analysed the accurate values for p

Fand t

#have been obtained

with Eq. (26) as derived in [23]. The acoustic approximation, Eq. (25) can be used as a "rstapproximation. In summary, this numerical scheme allows for designing spall experiments witha minimum number of tests which are complex and expensive to conduct.

Acknowledgements

Part of the research reported herein has been supported by the Regional Government ofLorraine and part by CNRS-France.

References

[1] Zhurkov SN. Kinetic concept of the strength of solids. Int J Frac 1965;1:311.[2] Klepaczko JR. Dynamic crack initiation, some experimental methods and modelling. In: Klepaczko JR, editor.

Crack dynamics in metallic materials. Vienna: Springer, 1990. p. 255}453.[3] Meyers MA, Aimone CT. Dynamic fracture (spalling) of metals. Prog Mater Sci 1983;28:1}96.[4] Curran DR, Seaman L, Shockey DA. Dynamic failure of solids. Phys Rep 147 1987;5}6:253}388.[5] Kolsky H. Stress waves in solids. New York: Dover Publ., 1963.[6] Nadai A. Theory of #ow and fracture of solids. New York: MacGraw-Hill, 1963.[7] Speight CS, Taylor PF. Dynamic fracture criteria from free surface velocity measurements. In: Murr LE,

StandhammerKP, Meyers MA, editors. Metallurgical applications of shock-wave and high-strain-rate phenomena,vol. 805, 1986.

[8] Chevrier P, Funfrock F. Automatisation et informatisation du fonctionnement d'un canon aH gaz haute perfor-mance, et eH tude de l'endommagement dynamique d'un alliage d'aluminum soumis a une onde plane induite par unimpact plaque sur plaque, PFE, Univ. de Metz, 1994.

[9] Chevrier P, Klepaczko JR. Spalling of aluminum alloy 7020-T6, experimental and theoretical analyses. In:Proceedings of the ECF11, Mechanisms and Mechanics of Damage and Failure, vol. 1, UK: EMAS, 1996. p. 693.


[10] Klepaczko JR. A practical stress}strain}strain rate-temperature constitutive relation of the power form. J MechWorking Technol 1987;15:143.

[11] Letian S, Bai Y, Shida Z. Experimental study of spall damage in an aluminum alloy. In: Proceedings of theInternational Symposium on Intense Dynamic Loading and its E!ects. Beijing, China: Science Press, 1986. p. 753.

[12] Butcher BM, Barker LM, Munson DE, Lundergan CD. In#uence of stress history on time-dependent spall inmetals. AIAA 1964;2:977}90.

[13] Novikov SA, Divnov II, Ivanov AG. Investigation of the structure of compressive shock waves in iron and steel.Sov Phys JETP 1965;20:545.

[14] Breed BR, Mader CL, Venable D. Technique for determination of dynamic-tensile-strength characteristics. J ApplPhys 1967;38:3271.

[15] Skidmore IC. Introduction to shock waves in solids. Appl Mat Res 1965;4:131.[16] Tuler FR, Butcher BM. A criterion for the time dependence of dynamic fracture. Int J Fract Mech 1968;4:431.[17] Cochran S, Banner D. Spall studies in uranium. J Appl Phys 1977;48:2729.[18] Dremin AN, Molodets AM. On the spall strength of metals. In: Proceedings of the International Symposium on

Intense Dynamic Loading and its E!ects, Vol. 13. Beijing, China: Science Press, 1986.[19] Yokobori T. The Cottrell}Bilby theory of yielding of iron. Phys Rev 1952;88:1423.[20] Hibbitt, Karlsson, Sorensen. Abaqus/Explicit, User's Manual, Version 5.5.[21] Hibbitt, Karlsson, Sorensen, Abaqus. Theory Manual, Version 5.5.[22] Barker LM, Hollenbach RE. Laser interferometer for measuring high velocities of any re#ecting surface. J Appl

Phys 1972;43:4669.[23] Romanchenko VI, Stepanov GV. Dependance of the critical stresses on the loading time parameters during spall in

copper, aluminum and steel. Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 1980;4:141.[24] Bushman AV, Kanel GI, Ni AL, Fortov VE. Intense dynamic loading of condensed matter. London: Taylor and

Francis, 1993.[25] Rasorenov SV, Utkin AV, Kanel GI, Fortov VE, Yarunichev AS, Baumung K, Karow HU. Dynamic deformation

and fracture of high-purity titanium. In: Murr LE, Staudhammer KP, Meyers MA (editors), Metallurgical andmaterials applications of shock-wave and high-strain-rate phenomena, 1995, p. 235.


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Numerical study of spalling in an aluminum alloy 7020-T6