On the dynamic cavitation problem in viscoplastic solids

MechanicsRe~amhCommunications,%ol.23, No. 5. pp.461-474, 1996

Cop~Sht O 1996l~ev~r ScienceUd Printedin the USA. Allrighl~~ 0093-6413/96 $12.00 + .00

Pergamon

Pn s0093..6413(96)000464

On the Dynamic Cavitation Problem in Viscoplastic Solids L. B a d e a *M. Predeleanu t

(Received 5 March 1996; accepted for print 3 July 1996)

1

Introduction

The sudden formation of voids in solids (a phenomenon named "cavitation" as in fluid mechanics) has been observed in various kind of materials (rubbers, elastomers, metals, composites) submitted to tension loadings. The paper of Gent and Lindley [1] has drawn special attention because the experimental critical load for the occurrence of a void in a short rubber cylinder pulled in tension agrees with that theoretically deduced one. Analysing the behaviour of a spherical cavities in a infinite elastic (neo-Hookean) body loaded by a hydrostatic tension, they found that there is a critical value of the load at which the void grows rapidly without bound ("cavitation instability"). Consequently, Gent and Lindley explained the cavitation phenomenon by the rapid growth (without bound) of a pre-existing microscopic void. An alternative approach for the cavitation problems was proposed by Ball [2] which used the discontinuous radially symmetric solutions for the equilibrium equations of an elastic solid sphere submitted to traction loadings on the boundary. He shown that for certain class of elastic materials there exists a critical value of the loading at which a non-homogeneous solution describing the formation of a central cavity bifurcates from the trivial homogeneous solution, which becomes iustable. The cavitation solution is energetically favorable. Due to scaling of the two above problems in finite elasticity, the critical load obtained by Gent and Lindley for cavitation instability is that obtained by Ball for the bifurcation of the discontinuous solutions. *Institute of Mathematics of the Romania~ Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania tLabor&toire de [email protected] Technologie /E.N.S. de Cachan / Univ@rsit4Paris 6/ 61, Avenue du Pr4sident Wilson, 94235 Cacha~, France

461

462

L. BADEA and M. PREDELEANU

Equivalent results were reached by Sivaloganathan [3] and Horgan and Abeyaratne [4], who have treated the cavitation problem for the sphere by considering at its center an infinitesimal pre-existing void (the initial radius approaches zero at critical load). The study of the solutions that allows the cavitation for the full three-dimensional problem in nonlinear elasticity is given in [5]. The cavitation phenomenon has been examined also for elastic-plastic solids for symmetric loadings in [6]-[11] and for non-symmetric loadings in [12]. The radially symmetric cavitation problem for rate-dependent materials has been treated in [13]. A comprehensive review of the literature on cavitation may be found in [18]. Very few studies have been concerned with the dynamical problem of cavitation ([14], [15]) and that only in the context of finite elasticity. In this paper the dynamic cavitation problem for visco-plastic materials governed by an overstress model is considered. In the next section we state the formulation of this mechanical problem for a hollow sphere submitted to a symmetric traction loading in the dynamical case. By considering the incompressibility of the material our problem reduces to solving two differential equations. The first one, which proceeds from the equations of motion and which has as unknown the void current radius, is a second order nonlinear differential equation. The second one is concerned with the temperature and proceeds from the heat conduction equation. In Section 3 we transform these equations considering the initial void radius to be infinitesimal, and consequently, we shall take this radius as vanishing. In what follows we find an expression for the critical load and we give some theoretical results concerning the dependence of the solution on this critical load and various initial conditions. In the last section we make some remarks concerning the results of the previous section specialized for a given material.

2

Mechanical problem formulation

Consider a hollow sphere with its center situated at the origin of a spherical coordinate system (R, {9, ~), and denote by A and B its inner and outer radii in the undeformed configuration. The sphere is supposed to have in time only a spherical symmetric motion and therefore, the coordinates in the current deformed configuration (r, O, ~) are of the form r = r(R,t)

0=0

(1)

CAVITATION IN VISCOPLASTIC SOLIDS

463

for the time interval 0 <_ t < oo. The inner and outer radii in the current deformed configuration will be denoted by a(t) and b(t), respectively. The deformation gradient tensor corresponding to (1) is given by ar

r

F = " ~ e R ® eR 4- ~ ( e o ® ee + e~ ® e¢),

(2)

where eR, e® and e¢, are the unit orthonormal vectors for the spherical coordinate system. Taking into account (1), the velocity field is ar

v = - ~ e r = vrer ® er,

(3)

and consequently, the Eulerian strain rate tensor has the form 0vr D = --~r e~ ® er + v-[~ ( e ° r ® e0 + e~o ® e~).

(4)

The Cauchy stress tensor can by written as o" = arrer ® er + Croo(eo ® e0 + e~0 ® e~0).

(5)

We shall suppose that the sphere is composed of an is®tropic and homogeneous material, and that the constitutive law is defined by a viscoplastic model in overstress form which is widely used in dynamics problems. More precisely, we shall consider its generalized form given by Perzyna [16], 7

F

D = ~ < ¢(~

OF

-

1) > ~-~,

(6)

where ~ is a temperature-dependent viscosity function, ¢ is a control function, ay is the material is®tropic hardening-softening temperature-dependent function, and ¢ is the viscoplastic overstress function. The symbol <: x > means the positive part of x, i.e. < x > = x if x > 0 and < x > = 0 if x _< 0. For a von Mises material, F = ~g2(o "d) ~- --~/l°'d : o "d, the temperaturedependent stress-strain relations are the following s i s n ( r ) a < ¢(Irl _ 1) > V.Kr_

~

~ , i ,,£(.L~..[ _ 1~ %

(7)

1 where 7- = -~(aoo - a ~ ) . The equations of motion are reduced in our case to a single one, p~r

-

Or

--T. r

(8)

464

L. B A D E A and M. P R E D E L E A N U

The simplified heat conduction equation may be written as

pc~i" = -divq + Xtr : D,

(9)

where T is the absolute temperature, c the specific heat, q the heat flux and X the Taylor-Quinne coefficient that takes into account the stored elastic energy (X -~ 0.9). In our spherical case the heat flux, using the Fourier's law, is of the form 0T

q = - k gradT = - k 0---Ter ® er.

(10)

The mechanical problem (8) - (10) will be completed with the following boundary conditions O'rr = O, q =

~rr = P ( t ) ( ~ )

0

on r = a(t), t > 0 2, q = 0

o n r = b(t), t > O,

(11)

and the initial conditions,

T = T o , Vr = v R = O

fort=OandA
(12)

Consequently, the interior cavity surface is traction-free and the outer surface of the sphere is subjected to a uniform nominal radial stress. The heat flux is vanishing on the both surfaces. Now, supposing [r] _> Cry in (6), and that ¢-l(x) exists, from (7) we get IT] = ay[1 + ¢-1(2v~¢- Ivr])]. 7 r

(13)

From (7) we also get sign(T) = sign(v~) for ITI > ~ , and in the above expression we used this fact. In what follows we shall transform equations (8) and (9) using equation (10), boundary conditions (11) and relation (13). Substituting the expression to T from (13) in (8), integrating it with respect to r and taking into account the boundary conditions (11), we obtain

b(t)

P fJa(t) i~rdr = p(t)(~-~)2-- 2v~ fb(t)sign(vr)ay[l + ¢ - l ( 2 v ~ - ) ] d - - - ~ .

(14)

~a(t) On the other hand, replacing (4), (5) and (10) in (9) we get

02T pcT = k ~ + 2v~ayX[1 +

¢-1(2v~¢-Iv~l)] Iv~l ,-f r r

(15)

CAVITATION IN VISCOPLASTIC SOLIDS

465

In (6) the viscoplastic overstress function will be considered of the form ¢(x) = x 1/m,

(16)

0 < m <_ 1 being a constant depending on the material. Concerning the temperature-dependent coefficient in the constitutive equation (6), we shall consider the viscosity function 3' and the material function cry of the form

ay = ~3 Y(T)-~ 3"m = ~1 ,-F3-' Y(T)

(17)

¢ being supposed in this paper to have the value 1. In the above formulae we have adopted for Y(T) and ~?(T) the expressions used by Carroll et al [17],

Y(T) =

{ K(1-T/Tm) 0

ifT Tin,

and

= 71m e x p H ( 1 / T - 1/Tin), where K, H and Tim are constants of the material, and Tm is the melting temperature. Also, 0 < n _< 1 is a constant depending on the material and

= f~Ddt, where D = ~ / 2 D : D. Taking into account (4) and (7) we have -- 2 vrl~~ and consequently, = 2j l n ( c ~r ) h

(18)

where ~ > 0 depends on the time evolution of r from 0 until the last sign change of vr; if r is monotonous, then ~ = 1. Using (16) and (17) in equations (14) and (15), we get [9 --/b(t) i;~dr = ga(t) p(t)(b-~t))2 - 2 fb(t) sign(vr)[Y(T) + v~TI(T)(2v/-3[Vri)ml~ndr

Ja(t)

and

r

r

02T peT = k ~ + 2x[Y(T ) + v~7?(T)(2x/-3iVrl)'n]-g nlvrl , r

(19)

(20)

r

respectively. With a view to simplifying the problem (19) - (20), we shall suppose that the material is incompressible and therefore the motion is isochoric, so

L. BADEA and M. PREDELEANU

466

r 2 Or that, det F = 1, F being given in (2), that is, ~-,x~-g = 1. Integrating this last equation we get,

r ( R , t ) = (a3(t) - A 3 + R3) 1/3,

(21)

b(t) = (a3(t) - A 3 A- B3) 1/3,

(22)

and therefore, Also, the radial velocity and its time derivative can be expressed by, v~(r, t) - a(t)2h(t) r2

(23)

~)r(r, t) = 2a(t)a(t) 2r3 -r5a(t)3 +/i(t) a'"_~2)2 '

(24)

and

respectively. Also, from (18) we get, r

~(r, t) = 21 lnc~ (r 3 _ a(t) 3 + A3)U 3 I.

(25)

The time material derivative of T was considered as,

T(r,t) = -~-(r,t) +

(r,t) yr.

(26)

Taking into account (23) - (26), equations (19) - (20) become a4h2 1 1 +a2a)( ~ 1)] p[----~-(~ - ~-~) - (2ah 2 = p(B)2-2sign(h)

2h fab[y(T) + x / 3 7 / ( T ) ( 2 V / - 3 ~ ) m]

r )ndr (21 l n ~ ( r 3 _ a 3 + A3)1/3 ~-

and

OT ~ a2h] = k 02T p c [ - ~ + Or r 2 "~r2+

a2 " r 2X[Y(T) + v / 3 ~ / ( T ) ( 2 v / 3 ~ ) m ] ( 2 [ l n ~ ( r 3 _ a3-~_ A3)1/3 I)n

respectively.

(27)

,

(28)

CAVITATION IN VISCOPLASTIC SOLIDS

3

467

Cavitation, the critical load

M. S. Chou-Wang and C. O. Horgan [15] have studied dynamic cavitation for neo-Hookean materials and have proved that the critical load in nonlinear elastodynamics is the same as for the corresponding static problem. Here, we propose to study the cavitation phenomenon for a viscoplastic material in the dynamic case.

In the problem (27) - (28) we shall consider the initial inner radius of the sphere to be infinitesimal, taking A ~ 0. Also, we shall change the variable r ra_a3 by setting u = ---7v - , and finally, in the place of a we shall introduce x - a In this way, we obtain for our problem the following equations 2x~214(1 x 4 +x3)4/3

x

3

2..

x

px

(l+x3)U 3+~]+x x[l-(l+x3)1/3]=

2x ..... I'~ 3-~szgn(x)[] o

1

2

a

Y(T)(311nu])

pB2( 1+x3)2/3

n du

1 --u +

~o~+-~ v/-~/(T)(2x/~J( 1 - u ) ) m 2( ~ l l n ual ) n I_---Z-~] du , 07'

p c x [ x - ~ + 3~(1 -

u)20T. 9k .02T ~uuj = B-{ [ ~ u 2

4

1

31- u

](1

--

(29) It)8/3-}

2XI~clx[Y(T)+ ~ / - 3 ~ ( T ) ( 2 v ~ - ( 1 - u ) ) m ] ( ~ [ l n - ~ [ ) n ( 1 - u ) ,

-

(30)

with the boundary conditions 0T Ou

0 foru=O,

1 x3+1' andt>O

u

(31)

and the initial conditions, T = To(u), O < u < _

x=0,

k=0

(32)

fort=O.

In what follows we shall also use another form of equation (29) using the following notation 1

P(t) - 2(1 + x3) 2/3

3

Y(T)(

J0

I ln

I)n du 1 - u'

(33)

468

L. BADEA and M. PREDELEANU

Q(t) = ~4~21~1J0['+~ '(T)(2'/5~(1-~))~(~11n ~ I)"1d~_~"

(34)

On letting t -+ 0 + in (33), the critical load is obtained as

2 [1Z(To)(21nl)n P~ - P(O) = -3 J o 3 u

du l-u"

(35)

In this way, equation (29) can be written as

d{x3[1

X (1 -{- x3) 1/3]:~2} zr" q ( t ) =

~d{ ( 1 + x 3 ) U 3 } p ~ ) ( t )

- sign(k)P(t)]

(36)

where P(t) > 0 and Q(t) > 0 for 0 < T < Tin. Since the above equation is obtained by the multiplication of equation (29) by 2xk, it is satisfied by all the solutions of that equation. R e m a r k 1. In the following we shall focus especially on equation (29) or its variant (36). In these equations, the temperature T will considered as a continuous function of x and ~, there is, the solution T of equation (30) depends continuously on the coefficients. By a solution x of equation (29) on the interval [0, oo) we shall consider the classical solutions, that is, ~).

x c C2(0, 00) n C~[0,

The following lemma gives the behaviour of a non zero solution on a time interval. L e m m a 1.I] on a time interval (tl,t2), 0 < tl < t2, a solution x of equation (29) satisfies

x(t) >0,5:(t) ~ 0

on(tl,t2),

(37)

x(tl) = 0 or ~(tl) = 0,

(38)

• (t2) > 0 and ~(t2) # 0,

(39)

and sign(p(t) - s i g n ( x ) P ( t ) ) = sp # O,

c o n s t a n t on (tl, t2),

(40)

CAVITATION IN VISCOPLASTIC SOLIDS

469

then sign(So) = sp

(41)

and consequently, on (tl,t2) we have the following equivalences i)p(t) > P(t) ¢==>~c > 0 ii)p(t) < - P ( t ) ¢=~ ?c < O, the case Ip(t)l < P(t) being impossible. Proof. First, we remark from (37) - (39) that the integral on (tl, t2) of the left side in (36) is positive, and d ( 1 + x3) 1/3 has the sign of k in (tl, t2). Then, using (40), there exists ml > 0 and m~ > 0 such that d (+Z3) 1 mlSp-~{ 1/3} < d

{x3[1

x d (1 + x3)1/3]k2} + Q(t) <_ m2sp~{(1 + z3) 1/3}

(42)

From these last inequalities we get (41). The other conclusions of the statement easily derive from this equality. Concerning the solution of the problem corresponding to applied loads smaller than the critical load Pcr~we have

Proposition 1 . 1 ] the applied load p(t) is continuous and 0 < p(t) < Per on any closed finite time interval, then problem (29) - (32) has a unique solution, that is, it has only the zero solution. Proof. First, we remark that problem (29) - (32) has always as a solution the identical zero function. Suppose that a solution of this problem is not identical zero. In the following we shall verify that all the suppositions of Lemma 1 hold, and its conclusions will be in contradiction with 0 < p(t) < Pc~. Our supposed solution being non zero, there exists a time t o = sup{s > 0 : x(t) = O,t < s}. From equation (30) and boundary conditions (31) we get T(u, t °) = To(u), and consequently, we can suppose that t o = 0. Therefore for any t~ > 0 there exists 0 < to < tE such that x(to)~(to) ~t O. Indeed, if in an interval (0, t~) we have x(t)~(t) = 0, we obtain, integrating from 0 to t, that x(t) = 0 for any 0 < t < t~, which contradicts the choice of t °. Now, x(t) and ~(t) being continuous, let 0 < tl < to < t2 < te such that (37) and (39) hold. Moreover, because x(0) = ~(0) = 0, we can suppose also that tl satisfies (38). Let p0 = sup p(t) and then P0 < PET' Because we can take t~

zc[tl ,t2]

470

L. BADEA and M. PREDELEANU

however small, and supposing that the temperature T depends continuously on x and k, we may suppose that ] P ( t ) - P c r l < Pcr--Po in (tl, t2). In this way we have p(t) <_ Po _ Pcr, and consequently, for p(t) =Pcr we may have a bifurcation of the solution. Let us consider now nonhomogeneous initial values for our problem, that is in the place of the conditions (32) we shall consider the following initial conditions 1 T = To(u), 0 < u < ~ (43)

x = xo, k = yo

fort=O,

where x0 > 0 or Y0 > 0 and xoYo = 0. The following proposition proves that for the above initial conditions, the problem has a solution only if the applied load exceeds the critical load, or, in other words, we can get cavitation only with applied loads greater t h a n the critical load. P r o p o s i t i o n 2. Supposing that the applied load p(t) is a continuous function for t > O, we then have i) if 0 < p(O) < Per then problem (29) - (31) and (43) has no classical solution, ii) if p(O) > Per and x(t) is a solution of problem (29) - (31), (43), then there exists a t2 > 0 such that k(t) > 0 for t E (0, t2). Proof. As in the proof of Proposition 1, we shall verify that the conditions (37) - (40) in Lemma 1 are satisfied, and then the proposition results from the conclusions of the lemma. Suppose that x(t) is a solution of the problem. First, we remark that (38) holds with tl = 0. Because p(0) # Pcr, there exists a to > 0 such that p(t) - P(t) # 0 on [0, to]. Consequently, since P(t) > 0 we get p(t) + P(t) # O, t • [0, to]. (44) If xo > 0, then there exists 0 < t2 _< to such that x(t) # 0 on [0, t2]. Suppose that there is a t3 • (0,t2] such that &(t3) _< 0. Since p(t) > 0 on this interval, taking into account (29), ~(t3) > 0, and consequently, there exists 0 ~_ t4 < t5 <__ t3 such that all the assumptions of Lemma 1 are satisfied on [t4, ts], with ~ < 0. From item ii) of this lemma, this fact is in contradiction with p(t) > 0. Therefore, if our problem has a non-zero solution, then ~ > 0 on [0, t2], and, in this case, p(0) > Per. If Yo > 0, then

CAVITATION IN VISCOPLASTIC SOLIDS there exists a t2 6 (0, to] such that k(t) > 0 on [0, t2], a n d consequently,

x(t) > 0 on (0, t2]. Using again (44) we conclude that the conditions of the lemma are satisfied for tl : 0. R e m a r k 3. The result of item i) in the above proposition is n a t u r a l taking into account that in our model the initial radius of the sphere is vanishing. The following theorem describes the local behaviour of a solution for a time tl > 0. Its proof is similar to the proof of the Propositions 1 and 2. T h e o r e m 1. Let 0 < tl < t2 and x(t), t 6 [0,t2] be a solution o/problem (29) - (31) with initial conditions (32) or (43). Then i) i/ O < p(tl) < P(tl) then there exists a neighbourhood o/ tl on which

either x(t) = &(t) =- 0 or x(t)k(t) # O, ii) i]p(tl) > P(tl) andx(tl) > 0 or&(tl) # 0 then there is at3 such that tl < t3 < t2 such that &(t) > 0 on (tl,t3), or there exists a neighbourhood of tl on which x(t)~(t) # O.

4

Conclusions

We remark that in the above results the connection between the applied load p(t) and P(t) is essential. In what follows we shall make some remarks concerning the behaviour of a nonzero solution and the critical load at which the cavitation take place, plotting in a plane P - x the curves

P ( x , T ) - 2(1 +

X3)2/3 /1+-~ y ( T ) ( 2 3

J0

o

In 1 ) , du u 1---u

in which the temperature T is considered as a parameter. In Fig. 1 we have considered a specific example to point out the shape of these curves. This example corresponds to n = 0.24 and K = 2206 M P a and the initial and melting value of the temperature were To = 293 K and Tm = 1673 K , respectively. First, we remark that P(x, T) is a decreasing function depending on T. Also, the curves are left turning for n < 1.0 (see also Abeyaratne and Hou, [3], for the case of rate-dependent materials), and we shall assume that the temperature is an increasing function during the deformation. Suppose now that the load p(t) > 0 will remain greater t h a n Pcr if it exceeds this value at a certain time (even if it starts from a value p(0) < P c r ) , and that the

471

472

L. BADEA and M. PREDELEANU

problem has homogeneous initial conditions. In this case, we shall prove below that for p(t) >Pcr a non-zero solution is an increasing function in time. Consequently, for such loads, the point (p, x) will be situated in our figures, either on the segment [0, Pcr], if p Pcr. Now, let x(t) be a non-zero solution and let a tl > 0 such that x(tl) > 0 or ~(tl) ~ 0. Taking into account Proposition 1 we get p(tl) > P c r , and using the above remarks concerning the curves P - x we get p(tl) > P ( t l ) . If there is a t3 as in item ii) of Theorem 1, then our assertion holds. Suppose, as in the second part of the same item, that there is a neighbourhood of tl, (to, t2), such that x(t)k(t) ~ 0 for any t C (to, t2). Evidently, we can take to = sup{t < tl : x(t):~(t) = 0}, and then x(to)k(to) = O. Using again the Proposition 1 as above, we get p(t) > P(t) for t E (to, t2). Now, we have exactly the conditions of the L e m m a 1 satisfied with to in the place of tl. From the item i) of this lemma we get &(t) > 0 on (to, t2), and our assertion is proved. On the other hand, if the load p(t) > 0 assumes values less than Pcr more

than once, then the point (p, x) can theoreticaly be situated in the left region of the curve corresponding to the temperature T(t}. In this case, from item i) of T h e o r e m 1 it results that for a time to, the solution will be discontinuous in two situations: either the point is not on the segment [0, Pcr] and ~(to) = O, or the point lies in [0, Pcr] and it(to) ~ O.

1/

\

X

0.8

\ \ \

\ \

0.6

~ \

0.4

\

T = 293 K T = 493 K T = 693 K T = 893 K T =1093 K T =1293 K T--1493K

--------

1

\

0.2

o

0

500

\

1000

P (MPa) 1500 2000 2500 3000 3500 4000 4500 5000

Fig. 1. Post-cavitation for n = 0.24 and different temperatures As for the points on the curves P - x ,

we remark from equation (29) that

CAVITATION IN VISCOPLASTIC SOLIDS there are no static states (in which the sphere remains undeformed while stressed) if p(t) > O. Indeed, if the point lies on a curve P - x, replacing &(to) = 0 in (29), we get ~(to) > O, and consequently, in an time interval (to,tl) the solution will pass into the region to the right of the curves.

Suppose now that, for the problem with homogeneous initial conditions, until the time to we have only the zero-solution and at this time we perturb the solution taking x(to) = E > O. We remark from equation (29) that if p(to) > 0 then ~(t0) > 0. Taking into account Proposition 2, we can conclude that: if p(to) < Per then the solution will go back to zero (having a jump) and if p(to) > Per then x(t) for t > to will be an increasing function. The obtained critical load Per, for our example, calculated from (3.5), was of 4767.6 M P a .

References [1] A. N. Gent and P. B. Lindley, Proc. R. Soc. Lond., A, 249, 195 (1958~). [2] J. M. Ball, Phil. Trans. R. Soc. Lond., A, 306, 557 (1982). [3] J. Sivaloganathan, Arch. Rational Mech. Anal., 96, 97 (1986). [4] C. O. Horgan and R. Abeyaratne, J. Elasticity, 16, 189 (1986). [5] S. Miiller and S. J. Spector, Arch. Rational Mech. Anal., 131, 1 (1995). [6] R. F. Bishop, R. Hill and N. F. Mott, Proc. Phys. Sco. 57, 147 (1945). [7] R. Hill, The mathematical theory of plasticity, Clarendon Press, Oxford (1950). [8] D. Durban and M. Baruch, J. Appl. Mech., 46, 633 (1976). [9] D.-T. Chung, C. O. Horgan and R. Abeyaratne, Int. J. Solids Structures 23, 983 (1987). [10] Y. Huang, J. W. Hutchinson and V. Tvergaard, J. Mech. Phys. Solids, 39, 223 (1991). [11] V. Tvergaard, Y. Huang and J. W. Hutchinson, Eur. J. Mech., A/Solids 11,215 (1992). [12] H.-S. Hou and R. Abeyaratne, J. Mech. Phys. Solids, 40, 571 (1992).

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L. BADEA and M. PREDF,I,F,ANU [13] R. Abeyaratne and H.-S. Hou, J. Appl. Mech., 56, 40, (1989). [14] K. A. Pericak-Spector and S. J. Spector, Arch. Rational Mech. Anal., 101, 293 (1988). [15] M.-S. Chou-Wang and C. O. Horgan, Int. J. Engn Sci., 27, 967 (1989). [16] P. Perzyna, Adv. Appl. Mech., 9, 243 (1966). [17] M. M. Carroll, K. T. Kim and V. F. Nesterenko, J. Appl. Phys., 59(6), 1962 (1986). [18] C. O. Horgan and D. A. Polignone, Applied Mechanics Reviews, 48, 471, (1995). Acknowledgement-One of the authors (L.B.) gratefully acknowledges the French Minist~re de l'Enseignement Supdrieur et de la Recherche for the allotment of a post-doctoral grant.