1-s2.0-002250969390029F-main

J. Mech. Phyx Solids Vol. 41, No. I I, pp. 1723-I 754, 1993. Printed in Great Britain.

0022-5096/93 $6.00+0.00 ,i 1993 Pergamon Press Ltd

APPROXIMATE MODELS FOR DUCTILE METALS CONTAINING NON-SPHERICAL VOIDS-CASE OF

AXISYMMETRIC PROLATE ELLIPSOIDAL CAVITIES

MIHAI GOLOCANU and JEAN-BAPTISTE LEBLOND

Laboratoire de Modtlisation en Mecanique, Universite Paris VI, Tour 66. 4 place Jussieu, 75252 Paris Cedex 05, France

and

JOSETTE DEVAUX

FRAMASOFT+CSI, 10 rue Juliette Recamier. 69398 Lyon Cedex 03, France

(Recrh~I IO March 1993)

ABSTRACT

THE AIM OF THIS PAPER is to extend the classical Gurson analysis of a hollow rigid ideal-plastic sphere loaded axisymmetrically to an ellipsoidal volume containing a confocal ellipsoidal cavity. in order to define approximate models for ductile metals containing non-spherical voids. Only axisymmetric prolate cavities are considered here. The analysis makes an essential use of an expansion velocity field satisfying conditions of homogeneous boundary strain rate on every ellipsoid confocal with the cavity. A two-field estimate of the overall yield criterion is presented and shown to be reducible, with a few approximations, to a Gurson-like criterion depending on the shape parameter of the cavity. The accuracy of this estimate is assessed through comparison with some results derived from a numerical minimization procedure. The two-field approach is also used to derive an approximate evolution equation for the shape parameter ; comparison with some finite element simulations reveals a reasonable qualitative agreement, and suggests a slight modification of the theoretical formula which leads to acceptable quantitative agreement. The application of these results to materials containing axisymmetric prolate ellipsoidal cavities with parallel or random orientations is finally discussed.

1. INTRODUCTION

IN THE MID-SEVENTIES, following the pioneering studies of MCCLINTOCK (1968) and RICE and TRACEY (1969) on the growth of cylindrical and spherical cavities in infinite rigid ideal-plastic media, GURSON (1977) developed an approximate model for ductile metals containing spherical cavities which has been used quite extensively since. (He also considered the case of cylindrical cavities, but this model has received comparatively little attention.) However, voids are often non-spherical in real materials. They can have, for instance, the shape of long, prolate ellipsoids if they are nucleated around segregations previously elongated by a rolling process; at the opposite extreme, they can look like wide, oblate ellipsoids if they happen to grow from cleavage cracks generated in the hard phase of a dual-phase structure (see e.g.

1723

1124 M. ciol OGANCI

Models of ductile metals with non-spherical voids 1725

chosen here represents a crude approximation of such a unit cell, just as Gursons hollow sphere. The choice of a confocal ellipsoid for the external boundary is admit- tedly primarily dictated by considerations of mathematical tractability, but it is not unreasonable. Indeed, for a given porosity, one wishes the shape of the representative volume studied to conform to that of the void to some extent (for an infinite cylindrical void for instance, the only conceivable representative volume is clearly a coaxial cylinder), and this is precisely what happens with a confocal ellipsoid; also, for a given, finite void shape parameter, one wants the representative volume to become a sphere in the limit of vanishing porosities since the problem then becomes equivalent to that of an isolated cavity embedded in an infinite medium, and the choice of a confocal ellipsoidal external boundary again fulfils this requirement.

For such a non-stackable geometry immediately arises the problem of boundary conditions ; the classical alternative is between homogeneous boundary strain rate (v = D *x, where v denotes the velocity, D the overall strain rate and x the current position) and homogeneous boundary stress (a-n = E-n, where rr is the local stress, n the unit normal vector to the boundary and E the overall stress). Neither of these alternatives correctly represents periodic boundary conditions on the unit cell of a periodic medium ; therefore whatever the choice made, it is unfortunately certain that some adjustments will be necessary in order to apply the results derived from the model problem to real materials, analogous for instance to TVERGAARDS (1981) introduction of some parameters q,, q2, q3 into Gursons original criterion [see also the work of PERRIN and LEBLOND (1990), who tried to rationalize this concept by deriving the value of these parameters from a self-consistent analysis]. Thus there is no clear reason, from the point of view of strict realism, to choose one type of boundary conditions rather than the other.

We do have, however, a good technical reason to prefer conditions of homogeneous boundary strain rate. Indeed, since the space of admissible velocity fields is larger for conditions of homogeneous boundary stress than for those of homogeneous boundary strain rate, it is probable that more trial fields will be necessary to reach a good estimate of the true minimum of the plastic dissipation for the former type of boundary conditions. This is illustrated by GURSONS (1977) finding that his simple two-field criterion yielded notably too stiff predictions if additional velocity fields involving rigid blocks (compatible with conditions of homogeneous boundary stress, but vio- lating those of homogeneous boundary strain rate) were admitted in the minimization procedure, and also by HUANGS (1991) and LEE and MEARS (1992) conclusion that a relatively large number of trial fields is necessary in the case of an infinite body (for which no kinematic constraints arise, except of course for the requirement of uniform strain rate at infinity). Therefore, since our search for an approximate analytical criterion for ellipsoidal voids will rely on a Gurson-like approach involving only two fields, it can be hoped to be successful only for conditions of homogeneous boundary strain rate.

The two fields we shall use are a uniform deviatoric strain rate and another velocity field satisfying conditions of homogeneous boundary strain rate on every ellipsoid confocal with the cavity. The latter field basically represents an expansion of the cavity, but it also includes a change of shape. It bears some resemblance with LEE and MEARS (1992) expansion field, as is best revealed by an electrostatic analogy expounded

I726 M. (;OLO(;ANI

Models of ductile metals with non-spherical voids

42

1727

FIG. I. The geometry studied

We shall occasionally use spherical coordinates Y, 0, cp (associated orthonormal basis e,, e,, e,), but most of the time we shall employ both cylindrical ones p, cp, z (associated orthonormal basis e(,, e,, e,) and, just as LEE and MEAR (1992), classical elliptic coordinates A, fi, cp (associated orthonormal basis e,, eg, e,,,) defined by

i

p = L sinh i sin /J

(P=(P (lE[O, +m[,BE[O,71l,(PE[0,271[) (3) , = c cash I cos fi

[see e.g. MLSHCHENKO et al. (1985), Chap. 21. The iso- surfaces are confocal ellipsoids with foci at z = kc and major and minor semi-axes

u=ccoshI.; h = c sinh I, (4)

while the iso-fl surfaces are orthogonal hyperboloids. The volume studied will be supposed to be made of a rigid ideal-plastic material

obeying the Von Mises criterion (with yield stress c,, in simple tension) and the associated flow rule. It will be subjected to axisymmetric macroscopic stresses (C,, = C,., # 0. C,, # 0, other C,,s = 0) via conditions of homogeneous boundary strain rate.

172% M. &LOGANL


FIG. 2. Comparison of the velocities in the Lee-Near expansion field (1) and a homogeneous deformation rate (2).

a multiple of the unit tensor, so that the velocity would be collinear to the position- vector x, in contradiction with the above mentioned property that it is orthogonal to the void surface (see Fig, 2). In other words, the Lee-Mear expansion field looks like a uniform deformation rate of the cavity if one considers onfy the ~~~~~~~ transformation of the latter, but a detaited examination of the transformation of i~~~zli~~~~ points reveals that the latter do not go where they should on the transformed ellipsoid for the cavity strain rate to be homogeneous. The same is of course true on every confocal ellipsoid. This forbids the use of the Lee-Mear expansion field in the present work because of the boundary conditions adopted.?

Tit may be argued that when one speaks of the cavity strain rate, one does not regard the void boundary as a material surface and therefore that one cares only about the global transformation of this boundary, not about that of individual, material points; in this sense Lee and Mears statement is acceptabIe. But this does not change the main conclusion that the Lee-Mear expansion field is incompatible with the boundary conditions adopted here.

I730 M. GOl.O(ANII (I (I/

3.2. At1 r.\-lm.siott firlrl srttisfi~itzg cotditiot~s ofhotttogc~r1colr.r ho~rtdrr~~~ .vtsrrirt txtc

We shall now look for a distribution of charges generating an electric field ( = velocity field) satisfying conditions of homogeneous boundary rate ott cwt:,~ c~//i~t.wir/ confbcul \l.it/t thr curit~~. The existence of such a distribution is not LI priori obvious. but it will turn out that a solution dots exist. The advantage of finding it is that the corresponding, .singk expansion field can then be used for all values of the porosity and the void shape parameter.

The velocity field looked for must be of the form

v = T(i.) * x (5)

on every 2 = constrrtzt ellipsoid, where T(A) is an axisymmetric, second-rank tensor depending on j. ; equivalently. the components of v in the e,,. e,,. e. basis must be of the form

I I,, = R(i)/

l,. = 0 (6)

I, = %(i.)r

where R(i) and Z(i) arc functions to bc determined. In order to be identiliable to an electric field, this velocity field must have a 7cro

curl (to be the gradient of a potential) and a zero divergence (to respect the requirement of absence of charge outside the central segment, equivalent to the incompressibilit> condition). The non-zero components of the velocity gradient in cylindrical coordinates are readily calculated to bc

(Vv) ,,,, = 13 ,,,,, = R(i)+pR(i) ;; = R(i) + rrhR(i) sin/{

11

(Vv),,,, = I = R(L) I

^ . (Vv),, = 1,,_ = Z(R)+rZ(i) ( k = z(i)+ trbZ( 2) cos2 /i

(7: D

9.

(VV),,, = I,,,; = pR(2) 1: = hR(i) sin 11 cos /j

D

ii (Vv).,, = l,,,, = :Z( i) (,) =

(/Z(i) sin /I cos /i

D

where N, h and I? are defined by (4) and

D = u2 sin /I + h cos2 /j ;

in these expressions, account has been taken of the relations

(7)

Models uf ductile metals with ~o~-s~~e~i~aI voids

m? Bl

=+f3(p,) = [

an&m? &A:& 1

c3Pji3p tT@jaz 1 ~ u sin p i"iCOSff = 6 _- hcoSj3 -a sin 8 I Hence the condition that the velocity be curl-free is equivalent to

01,,2 - v,.P = 0 0 h2R(A) -aZ(A) = 0.

On the other hand, the requirement of zero divergence reads

r>[2R(& +Z(d)] + ab [R(A) sin /3 f Z(A) cos* fl] = 0 ;

f73I

(9)

using (&), writing cos/J as 1 -sin* p and ~dent~fy~n~ terms independent of @ and ~ro~ortiona1 to sin 8 (since the above refation must hold for every fi), one gets from there

b[2R@) +z(n)] -i-az(n) = 0

c2[2R(;1)+Z(;1)]+ah[R(h) -Z(A)] = 0,

or equivalently

fbR(1) -a2Z(l) = 0

i R(a) = - b [2R@.) -5 Z(A)].

Now one sees that (10,) is just the same equation as (9) ; this means that the conditions of i]l~ompressib~Iity and uniformity of the boundary strain rate on every confocal ellipsoid automatically enforce the requirement that the veiocity be the gradient of some potential. Because of this circumstance, the problem reduces to two differential equations only for the two unknown functions R(A) and Z(L), and therefore possesses a so1ution.t This solution is easily found by first combining (IO,) and (IO,) to get 2R(EJ f Z(A) :

2R@) + Z(d) 2a b --------= ----= -2cotanhA-tanhA

;?R@) + Z(2) b a

where C is a constant, then using (IO,) to get R(A) :

R(a) = - & I

=a R(l) = C f -;;&

c 2Jq4 + Z(& = -.I. > ----~-

sinh- 2 cash a

where C is another constant, and finally deducing Z(A) from the two above equations :

t In fact. evet~ if (9) were not contained in (IO), there would still exist an incompressible v&city field saizsfying conditions of ~~~~~e~~us boundary strain rate nn every confocaf ellipsoid; the coincidence of (9) and (10,) is just a bonus that aitaws one to interpret it as an electric field,

I732

In these expressions. the terms involving C represent a uniform deviatoric strain rate : such a velocity field will be included in the analysis to follow, but here WC concentrate on the expansion field and therefore take C = 0. Taking C = 2 to obtain the simplest possible expressions, we finally get

cash i &I.) = sinh? i -

1 coshJ+l LI(

2 ln cash A- I = h - tanh (c;n)

(11)

Z(A) = ~ cofh I +ln cash i+ I 2c

= cash i.- I

~ +2 tanh ((,!,u) N

Since the velocity field defined by (6), (I I) corresponds to a homogeneous boundary strain rate on all i, = mmtant ellipsoids, it deforms any such surface into another ellipsoid but the two ellipsoids are readily verified to be neither confocal nor homo- thetic (as for the LeeeMear expansion field). The latter property means that the deformation of the cavity is not a pure expansion, but also involves some change of shape.

Let us finally determine the corresponding distribution of charge. Since the divergence of the electric field (= velocity field) is zero for 0 < i < + 1;. the charges can only be located on the central segment (I. = 0) and at infinity ; but the latter possibility is ruled out by the fact that, as is easily checked, the velocity vanishes for I. + + yc. To get the density of charge q on the central segment, one can evaluate the divergence of the velocity using the theory of distributions; but a simpler solution consists in applying Gauss theorem stating that the total charge enclosed in a given volume is proportional to the flux of the electric field through its boundary. Taking this volume to be a small cylinder of radius p and height /I centered on the segment, and taking into account the fact that I) + 0 *i -0 *(I - cr. sin /I and r - c cos /3. one readily sees that the flux through the top and bottom caps is O(L) In p) and therefore negligible, whereas that through the lateral surface is 2nol1 x R(l.)p - 2rtpl1 x ,1/E. - 2nhc, sin /) ; equating this expression to y/z, one gets

This equation exhibits both the similarities and the differences between LEE and MEARS (1992) expansion field and that defined by (6) and (I I) : both of them correspond to electric fields generated by some distributions of charges over the segment extending between the foci. but the distribution is uniform for the former whereas it is quadratic for the latter.

Another way of comparing these fields is to decompose that defined by (6) and (I 1) on the family of independent fields proposed by LEE and MEAR (1992). It is easy to check that with these authors notations, the only non-zero coefficients of the decomposition are A and Bz2 = A/6; the first coefficient corresponds to Lee and


Mears expansion field and the second one to another field involving a change of shape of the cavity.

4. TWO-FIELD ESTIMATE OF THE YIELD CRITERION

We shall now derive a rigorous upper bound of the yield criterion by using a velocity field of the form

y = A@ + By(B) (13)

where A and B are constants and v() and vCB) the fields defined by (6), (11) and

VW = x ---e,-ye,>+&= --Pe,+x, 2. 2 2

(14)

respectively.

4.1. Macroscopic plastic dissipation

The average macroscopic plastic dissipation is defined by

where fi denotes the domain considered, V = $a&: its volume and deq the equivalent microscopic strain rate. Now, by definition,

dc, = :d,,d,, = A2d:

2R+Z+ 2R+Z D (2hR cc& /+(N sin/Ghcos[i)Z) 1 Z-cos2i(2R+Z)

D II -_ (15)

where j., and i.? denote the values of the elliptic coordinate 2 on the inner and outer surfaces respectively. It is reminded that the quantities U. /J. R and Z in this equation are functions of E. defined by (4) and (1 1). and D a function of both i and /j given

by (8).

Since conditions of homogeneous boundary strain rate are met, the components of the macroscopic strain rate are simply given by

B D,, = D,, = ;: (i2) = AR(i,)- 3 .

I_ D_, = (lz) = ilZ(i,)+B. (16)

z

The macroscopic stresses associated with the estimated yield locus are then given by (see e.g. GURSON, 1077) :

c,, = c,, = fw

c7D,, ; X~y+

;n,, (17)

where care must be taken to consider D,, and D,, as distinct when evaluating the first derivative. Equations (16) and (17) imply that

3c PD,, CD,, =

,?A + c;: iA = 2R(iu,)C,,+Z(i,)C,,

and similarly

defining the function

x(j_) = R(i) Llh

2R(i) +Z(i.) = 2~ R(i)

or equivalently, in terms of the eccentricity (1 = (1~ = I /cash j. :

I r(e) = ~

I --(G

zr 2c tanh c.

one readily puts the expression of (7 W/CA under the form

(19)

(19)


aW 87cc3 p= C3A

31/ &I, CA = 2a& + (l -2a,)C,,

where c(~ = cl(eZ) is the value of a(e) on the outer boundary. Equations (18) and (20) define the yield locus under a parametric form, W being given in terms of A and B by (15). Numerical examples will be given in Section 6.

4.3. Special cases

In the particular case of a cylindrical cavity (S -+ + CXI, e, and e2 + l), the yield criterion defined by (15), (1 S), (20) is identical to the classical Gurson criterion for a cylindrical void, as desired since the latter is exact in this case (for a coaxial cylindrical representative volume). This can be directly checked on (15), (18) and (20), at the expense of a somewhat heavy calculation ; another, simpler justification, which makes reference to the simplified criterion presented below, consists of noting that the latter is strictly equivalent to that defined by (15), (1 S), (20) for a cylindrical cavity because all the approximations made to derive it become exact in that case, and that it itself reduces to Gursons criterion in the cylindrical case (see Section 5).

Before discussing the particular case of a spherical void (S + 0, e, and e, --f 0), we must give some precisions about Gursons criterion for such cavities. Two approximations were involved in the derivation of this criterion. The first one consisted of representing the velocity field as a sum of only two fields, namely a uniform deviatoric strain rate and the l/r* expansion field for a hollow sphere. The second one consisted of performing a first-order expansion of the expression [analogous to (1 52)] of d, with respect to the crossed term proportional to AB; in fact, once integrated to get the expression of W, the contribution of this crossed term was found to be zero, so that this second approximation was equivalent to discarding the crossed term in the expression of deq. The criterion obtained in that way will simply be called the Gurson criterion (without any precision) in the sequel ; it is the familiar one with its charac- teristic hyperbolic cosine. On the other hand, the criterion obtained by retaining the first approximation but rejecting the second one, which does not yield any simple, explicit expression of the yield function, will be referred to as the Gurson criterion with crossed term. [In fact the difference between the two is numerically quite small, as illustrated on Fig. 11 of GURSON (1977), which compares the criteria obtained by performing the expansion with respect to the crossed term up to the first and second orders.]

Let us now come back to the criterion defined by (15), (18) and (20). For a spherical void, the segment extending between the foci becomes a point; it follows that the expansion field corresponding to the distribution of charge defined by (12) then becomes identical to that generated by a point charge, which was precisely that used by Gurson. Thus considering a velocity field of the form (13) is equivalent to Gursons first approximation, which means that the criterion obtained in that way, without any further approximation, is equivalent to Gursons criterion with crossed term. (In contrast, again for a spherical cavity, the simplified criterion derived in the next section will be equivalent to the usual Gurson criterion.)

1736

We shall now derive an approximate. analytic expression of the two-field yield function possessing the nice property of always rrsc~mhfiny the classical Gurson yield functions. and exactly reducing to them for spherical and cylindrical cavities.

In order to simplify the expression (I 5) of PV, we shall use three approximations denoted .c/, , -cd2 and xJ;

this approximation does not introduce any error then. Furthermore, for a spherical void, it is exactly the simplification that must be made to reduce the Gurson criterion with crossed term to the usual Gurson criterion, and the resulting error was shown by Gurson to be quite small. Also, approximation .d, of course becomes exact in the two limiting cases where A + 0 or B + 0, whatever the shape of the cavity. For all these reasons, it can be hoped to be a sound approximation in the general case.

In the spherical case, D = u2 sin /J+ h co? B is independent of (3 since (I = h. and the same is also true of L&,, provided that approximation .d, is made, since L/:$ depends onty on I [because of the spherical sylnmetry of the veiocity field P] and db: = 1 ; therefore, once the change of variable u = cos a has been made, the integrand in the right-hand side of (15,) is independent of II, so that approximation .&? is then exact. In the cylindrical case, it can be checked that the terms involving sin B or co? p in the expression (1%) of dCq are negligible, so that the latter quantity is independent of U; D being a linear function of u, the same is true of the integrand in (15,) ; therefore approximation .tiz is again exact in that case. It is thus probably a good approximation in the general case.

Approximations .d, and ,QY? being made, the expression ( 15) of LV becomes

We now introduce the change of variable defined by

2A C1 2A .Y =

3 ah = 3cosh 3. sinh ;1

similar to that used by Gurson for spherical and cylindrical cavities; it is especially appealing because the new variable x takes on nice values on the inner and outer ellipsoids :


D,, 3 ftrD; 2Ac D

X2=3-== m

[by (11) and (16)]. It is easy to verify that

dJ = _?~4.;~~dx 3 b(3a -c) x

so that (21,) becomes

Now we rewrite (21,) in the form

deq = [F2(x)x2 + B2] Ii

where the function F is given by

(23)

(24)

(25)

F2(X) = ; !!!.g 2R+Z (~R+

1738

variable .Y generally

M. ~k)LOGANL (I t/l.

varies greatly in the interval of integration [fI,lj, !I,,,:,/], because ,f is quite small in practice. This suggests the introduction of the following final

approximation, which again becomes exact in the case of spherical or cylindrical cavities since the eccentricity is then constant (equal to 0 or I) over the interval of integration.

From now on. the calculation is heavy but straightforward. Evaluation of the integral yields

differentiating with respect to A and H [accounting for the relation (23?) between D,,, and A]. one then gets

elimination of FD,,,,!B finally yields. using (I 8) and (20,) :

(27)

where IL,, is given by (2(L) and K by

3 li = -

I; (28)

Equation (27) provides the Gurson-like expression of the approximate yield function.

It remains to give a formula for the coefficient K. or equivalently the mean value i? The precise definition of this mean value must be such that the error made when replacing the function F(e) by the constant Fin (24) and (25) be minimal. Since this cannot be achieved for all values of the parameter B simultaneously, we shall only demand that the error be zero for B = 0; this condition reads

Mod& of ductile metals with non-spherical voids

y$z+ ~~-+-$

or equivalently, using the change of variable

2A c3 2A e3 .& = 3. ;~ = __ _. .-i ) 3 l-e

1739

(29)

The expression of the function F(e) [equation (26))l is unfortunatel~~ too complex for this integral to be calcuiabfe analytically. However, it may be remarked that the function t j(e( I -e)) varies enormousty in the interval 10, I( (it becomes infinite at Y = 0 and c = I), whereas the function (3-e)F(e) varies modestly, from the value 6 for e =1: 0 to the value 2J3 = 3.464., . for e = 1; hence a reasonable approximate value of the integral may be obtained by replacing the expression (3-e2)F(e) by a poly- nomial approximation and then calculating the resulting integral exactly. Since by (26), i;(e) is an even function, it is natural to define the function

G(e) = (3 -e2)F(e) ; (30)

Figure 4 shows the errors made when replacing G(e) by a linear function having the same boundary values and a quadratic Function having in addition the same derivative at the point e2 = t. The first replacement yietds

and the second one,

0.2 0.4 0.6 0.8 ep

3

FIG. 4, The functh C(2) (full Line) and its apprffximat~~~s by linear (dotted he) and parabolic (dashed line) functions.

1740

the corresponding values of the parameter K are

Equations (27) and (3 I) completely define the approximate yield criterion. Numeri- cal examples and comparisons with the exact two-field criterion defined by (1.5), (IX) and (20) will be presented in Section 6.

In the particular case of cylindrical cavities, LJ, = cZ = 1 so that r: = x(P?) = I,2 by (19) and K = %/3 by (31) (whatever the ~~pproxitnation adopted); hence (20,) and (27) take the form

which is Gursons criterion for cylindrical voids. To study the particular case of spherical cavities. one must let both TV, and P? tend

toward zero; the limit of c(: is then easily calculated to be l/3. and since P, and o3 are tied by (2?). the limit of K is 312 (again whatever the approximation adopted); hence the criterion (24) reads

which is Gursons criterion (without crossed term) for spherical voids.

6. NUMERICAL EVALUATIOK OF THE EXACT YIELD CRITERION

We shall now assess the accuracy of both the exact two-held criterion and its analytical approximation through a numerical minimization procedure of the plastic dissipation (for the same typical geometry and boundary conditions as before), using velocity fields of the form


+ C,,P;(cosh A)]P/Jcos /I)

where the Bkms and Ckms are constants and the Pks, Qs, Pis and Q:s the associated Legendre functions of the first and second kinds of order 0 and 1 (see e.g. GRADSHTE~ and RYZHIK, 1980). This is exactly the family of fields considered by LEE and MEAR (1992) except that these authors considered all coefficients C,,,, to be zero except Cz2 (which is the coefficient of the term representing a uniform strain rate)? ; the omission of the terms involving the Ckms, (k, m) # (2,2), was logical for the infinite medium they considered since these terms do not meet the requirement of uniformity of strain rate at infinity, but would be unjustified here since the representative volume studied is finite.

The coefficients Bkms and Ckms in the above expansion are not arbitrary. Indeed the condition of uniform boundary strain rate reads, in elliptic coordinates :

taking the expressions of the trigonometric polynomials P,(cos /3) and Pi (cos /?) into account, one readily sees that these equations are equivalent to

9F2f&) = a&&&, -D.J

3G2(&) = a:L& -b:D,,

,F&) = G,(d,) = 0 (k = 4,6,8,. . .)

where

Fk(A) = +f (B,,Q;(cosh %)+C&,f,(cosh A)) m=O

and

t Also, OUT expression of ui! differs from theirs by a factor of - 1 ; this is probably due to a different definition of the Legendre functions [the definition adopted here is that of GRADSHTEYN and RYZHIK (1980)].

1742 M. CiOLWANL

Models of ductile metals with non-spherical voids

0 Gz ~ ~xx)/Q i

9 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 !

(2G%c+ (1-2a*)C.,)/cTo

FIG. 6. Same as Fig. 5, but for u Jh, = 5

1143

several porosities : (i) the exact two-field criterion [equations (15), (18) and (20)] ; (ii) the approximate two-field criteria corresponding to first- and second-order approximations of the coefficient K [equations (27) and (31,) or (31,)] ; (iii) discrete points belonging to the yield locus determined through numerical minimization of W (k varying between 2 and 10 and m between 0 and 4, just as in the work of Lee and Mear). The close agreement of the (exact) two-field criterion and the numerical points provides evidence of the accuracy of the two-field approach (for the boundary conditions adopted). Also, both formulae (3 1 i) and (3 1 J lead to good approximations of the criterion; if the numerical points are taken as a reference, it is seen that the theoretically more exact formula (31,) does not, in fact, represent any significant improvement over the simpler formula (3 1,). It seems therefore reasonable to adopt the latter in practice.

Figure 7 shows the results obtained in the extreme case of a spherical cavity (u ,/h, = 1). In this case, as mentioned above, the exact two-field criterion is identical to Gursons criterion with crossed term, and both approximate ones to Gursons usual criterion. In addition to these criteria and numerical points, we have also exceptionally plotted here points determined again numerically but for conditions of homogeneous boundary .stress. Most of these points lie notably below the dotted line representing Gursons criterion, in full agreement with what Gurson himself found when using velocity fields involving rigid blocks. On the other hand, the Gurson model can be seen to give an excellent approximation of the numerically determined criterion ,for conditions qf homogeneous boundary strain rate ; this remark does not appear to have been made before, in spite of the attention that has been paid to Gursons model.

There would be no point in showing results for the other extreme case of cylindrical cavities since all criteria then become identical to that of Gurson, which is exact in that case (for a cylindrical representative volume and axisymmetric loadings, as considered here).

1744

12c,, t X,,),3n,,

FIG. 7. Yield loci for a spherical cavity and sewxal porositics. dull line : exact two-lield criterion (Gursons criterion with crossed term) : dotted lint : approximate two-licld criterion (Gursons usual criterion) : triangles : numerical minimization : circles : numerical minimuxtion for conditions of homugcncous

boundary stress.

7. EVOLUTION OF THE SHAPF. PARAMETER

We shall now consider the problem of providing an evolution equation for the shape parameter S. This is a more dificult task than defining an approximate yield criterion ; indeed, since the plastic dissipation M/is minimum and hence stationary for the exact solution, its value for a trial velocity field differing modestly from the latter should be close to the minimum, so that the estimated yield criterion, which immediately derives from I+, should also be close to the exact one ; on the other hand nothing warrants that the corresponding estimate of 3 (which has no direct connection with W) will be accurate.

In spite of this, we shall again use the simple two-field approach to derive a first, crude estimate of s.

Since conditions of homogeneous boundary strain rate are met on the surface of the void as well as on the external boundary, the void strain rate D is given by the same formulae as the overall strain rate D [equations (16)]. except for the replacement of R(A2) and Z(A2) by I?()_,) and Z(i,) :

B D:, = D:,, = AWL,)- z; D;, = AZ(~.,)+B.

It then follows from the definition (I?) of S that


Expressing A and B in terms of D,, and Dzz with the aid of (16), one readily transforms this expression into

3 = D:= - D,r, i- 3 z(n,)-R(I,)-Z(E,*)+R(lz)D

2R(AJ + Z&) m

or equivalently in terms of the eccentricities e, and e2 [using (19)]

s = D,,-D,,y+3 1-3a, -__ f

+3a,- 1 D,,, tl, = a(e,), a2 E a(e2). (32)

Let us now consider some special cases, first that of cylindrical voids. Then CI, = CI> = ~(1) = l/2 so that the preceding equation becomes

Now 3 can be calculated exactly for a cylindrical cavity of radius p,, height h, embedded in a coaxial cylindrical representative volume of radius p *. Indeed, because of the cylindrical symmetry and the incompressibility condition, the velocity field is of the form

i

c Cp up=----

P 2 VP = 0

v, = Cz

where C and C are constants ; therefore the components of the overall strain rate are

I D,, = ;(A) = C, and those of the void strain rate,

c C D;, = D;.y = 2 (p,) = p: - T-

Drz = ; (h) = C.

Eliminating C and C between these equations, one readily gets the same expression of s = Dr - D,, in terms of D,, and Dzz as above. This means that (32) is exact in the case of a cylindrical cavity (embedded in a coaxial cylindrical representative volume).

L,et us now assume the representative volume considered to undergo a pure expansion (D,, = D_v.b, = 0,;). Then (32) is exact for a spherical cavity, since it is based

1746 M. C;OLO(;A~~I PI


a b

FIG. 8. The volume element considered in the numerical simulations : a : overall view : b : the mesh used.

performed using the Large Strain Plasticity option of the SYSWELD code developed by the FRAMASOFT+CSI Company (LEBLOND, 1989).

Numerical values of the shape parameter are deduced from the displacements of those points of the void surface which lie on the axes (this involves a slight approximation, since the void obviously does not remain strictly ellipsoidal). They are plotted versus the overall equivalent strain Ees = ji (:D:, 01,) 'j2 dt, where D denotes the deviator of D (the components of the latter tensor being deduced from the increments of normal displacement imposed on the outer boundary). Theoretical values of S are derived through integration of (32), using D,,- and D,,-values deduced from the numerical simulation, f-values obtained by integration from those of D,,, and D,, plus the incompressibility condition (elasticity being neglected), and c(,- and cc,-values deduced from those of S and .f; this is thought to be a better solution than deducing the D,,- and D,,-values from the approximate yield criterion (27) and the associated flow rule, because what we wish to do here is to assess the validity of the sole evolution equation (32) of St and adopting this other solution would mean testing both (32) anA the approximate criterion proposed.

t The simulations described here are not quite appropriate for testing the criterion (27) : indeed, since the volume element considered approximately represents an elementary cell in a periodic medium, there are some interaction effects between neighbouring voids, while such effects are not accounted for in (27), since it was derived from a typical geometry involving a single cavity. Conversely, the numerical minimization procedure used in Section 6 to assess the accuracy of (27) would not be appropriate for testing (32) because, as remarked earlier, such a procedure determines s much less accurately than the criterion.

174x M. GOLSXANII ci rrl.

Figure 9 presents the results obtained numerically for the triaxialities T = I /3, 2j3 and I [which fully confirm the few pieces of information provided by KOPLII< and NEEDLEMAN (I 988) and HOM and MCMEEKING (I 989)] ; those for T = 2 are not shown because the cavity immediately becomes ablate (5 < 0) for this triaxiality and (32) was derived for prolate ones only. The sudden sharp decrease of S which occurs for T = I (and also probably for T = 2/3. for values of EC,, lying outside the range represented here) is due to coalescence between adjacent voids, which induces a concentration of the strain in the ligament separating them horizontally and therefore a necking of the latter. The figure also presents the theoretical results during the pre- coalescence period (predictions are stopped at the onset of coalescence since the theoretical analysis does not account for this effect). It clearly appears that the theoretical curves do correctly reproduce the qualitative features of the numerical ones. but that the agreement is rather poor from a fully quantitative point of view.

Let us analyse the origin of the discrepancies in more detail. It can be observed that for large strains (for T = l/3 and 2/3), the numerical and theoretical curves become more or less parallel; since S is rather large then, this means that the value of 3 provided by (32) is reasonable for cavities differing notably from a sphere, in agreement with what was foreseen at the end of the preceding section. On the other hand, the slopes of the numerical and theoretical curves are markedly different for small values of Ecq and S, which means that (32) is inaccurate for nearly spherical voids. However, as remarked above, the term proportional to D,,, in (32) vanishes for spherical cavities, so that the inaccuracy cannot cotne from there ; again, this remark is compatible with what was foreseen above. It can thus be concluded that the discrepancies arise from notable inaccuracies in the term proportional to the deviatoric strain rate D,,- D,,, for small values of S or c,. The extent of these inaccuracies is not very surprising if one considers that in the case of a spherical void and a deviatoric overall strain rate, the yield limit (maximum value of the macroscopic Von Mises equivalent stress) predicted by the two-field approach which forms the basis of (32) is o,,(l -,f). which violates the Hashin-Shtrikman upper bound, as extended to the

0:o 0:2 0:4 oh oi E -q

FIG. 9. Comparison of numerical S Ecq curves (full lines) with those deduced from (32) (dashed liws).


non-linear case by PO~TE-CASTANEDA (1991) and TALBOT and WILLIS (1986, 1992) [see also MICHEL and SUQUET (t992), who gave a later but notably simpler proof] ; in itself, this violation is of relatively minor importance since the yield limit under purely deviatoric loading is close to LT~ anyway (f being always small in practice), but it is a clear symptom of the basic inadequacy of the two-field approach for such loadings.

It thus appears that in order to improve the predictions of (32), one must modify it into

S = W,, - D,) + 3 (

I-3a, -- + 3c1* - 1

> D,,z (320

where A is an empirical factor. This factor must depend upon the shape parameter or the void eccentricity e,, since it must be unity for el -+ 1 (cylindrical case) and non- unity for e, + 0 (spherical case). Unfortunately, it cannot be a function of this sole parameter ; indeed, the numerical curves on Fig. 9 have different slopes at the origin, whereas the void eccentricity is then the same (almost 0). The simplest solution consists of making it a function of e, and T. The expression h(e, = 0, T) = 2 - T* is found to well reproduce the slopes of the numerical curves at the origin (even for T = 2, for which it predicts a negatiue slope). The formula

h(e,,T) = 2-T2+(T2-l)e: (34)

constitutes a simple interpolation between this expression and the unity value required for f, = 1 ; Fig. 10 shows that it leads to an acceptable quantitative agreement with the numerical results. Further improvement of this agreement would certainly be possible, but at the expense of the introduction of more empiricism into (32).

It must finally be remarked that the initial porosity is the same (.fO = 0.0104) in all computations reported here. Thus, strictly speaking, it is not certain that (34) remains

FIG. 10. Comparison of numerical S-E, curves (full lines) with those deduced from (32) and (34) (dashed lines).

1750 M, C;OLOGANIJ (I r/l.

a good approximation for other porosities. Additional simulations are obviously needed to settle that question.

8. MODELS FOR MATERIALS CONTAINING AXISYMMETRIV PROLATF ELLIPSOIDAI

CAVITIES WITH PARAI.LCL OR RANDOM ORIF.NTAIIONS

The criterion (27) was derived for a ,sinyl>~ ~Gktlgcometry loaded n.~i.s,~~r~~~~~c~t~ic~rrl/~,~. The only modifications that must imperatively be brought to apply it to materials containing distributions of caritics lixitll prrraNrl wicntotiom and subject to gcncrol /orr&~y.r are (i) the introduction of TVERGAARDS (19X1) (/, parameter, in order to account for interactions between voids; (ii) the replacement of (C,,- IX,,)? by C,,, where IX:,,, -.($C:,C:,) (C 3 deviator of C) denotes the Von Mises equivalent stress (in order to recover the Von Mises criterion for a zero porosity) ; (iii) the replacement of C,, by @,,+C,.,) in the expression of C,, (in order to account for transverse isotropy perpendicularly to the common axis of the voids). The criterion then becomes

cP(C, ,f: S) = ;I +2q,,fcosh 0,

- 1 -qf,/? = 0,

where the quantities K and C,, are given by

and

c,, = %(C,, +~,,)+(I -2x?)&:.

c,. CJ? and a, being defined in terms of the shape parameter S and the porosity f by the equations

The model next includes some expressions of the plastic and elastic strain rates. The former is given by the macroscopic normality rule (see e.g. GUKSON. 1977) :

and the latter by some hypoelastic law of the form

D_l+v E

c- $rZ)l

where E and 1 denote Youngs modulus and Poissons ratio and

z Ez k+c*n-n*z


some co-rotational derivative (i.e. time-derivative in some matter-tied frame) of the stress tensor, for instance that of Jaumann [for It = j (VV - (VV)), V s macroscopic velocity] or that of Green-Naghdi (for R = ti *R- I, R E rotation in the polar decomposition of the gradient of the transformation).

The model is finally completed by evolution equations for the internal parameters. That of the porosity is classically deduced from the condition of plastic incompressibility of the matrix (porosity variations due to elasticity being neglected) :

,f = (1-J)trDP,

while that of the shape parameter reads

where T z Z,/C,, (C, = 4 tr C) is the triaxiality, DP the deviator of DP, and M., E cl(e,) ; this is the same equation as above, D being replaced by DP (this means neglecting variations of the void shape due to elasticity) and 0:: -O$ by $Lf = O& - :(@,+OP,) (in order to account for transverse isotropy). Also, one must keep track of the unit vector e,? collinear to the axis of the voids, since the zz components of the stress and strain rate tensors do not play the same role in the model as the xx and yy components. The simplest hypothesis here consists of assuming that e, rotates with the same velocity as the matter, i.e. that

where Sz is the same antisymmetric tensor as in the hypoelasticity law.

8.2. Cavities with random orientations

It must first be stressed that even if the hypothesis of random orientations of the voids is a good one initially, it cannot remain indefinitely true because large strains tend to orient the axes of the voids toward the same direction; a full theory of materials containing non-spherical voids with (initially) random orientations should account for the progressive development of such a damage texture. What follows is only a crude attempt, based on a semi-heuristic approach, to describe the first stages of the deformation, when the hypothesis of random orientations is still acceptable. We shall only account for the dispersion of the void orie~tfftions and disregard that of the void shapes. The influence of the shape parameter will be described through a single quantity S which in fact represents the average value of the shape parameters of individual voids.

The model described above for cavities with parallel orientations, being basically anisotropic, obviously does not apply to the case considered here. However, it can reasonably be assumed to be valid at the meso scale of elementary cells containing a single cavity (which corresponds to the macro scale in the Gurson-like analysis presented above), and then be used as a basis for a homogenization-based derivation of some macroscopic model. In this derivation, mechanical fields defined at the meso scale, which were previously represented by capital letters, will be denoted by small

I752 M. GOLOGANU PI trl.

letters, capital ones being reserved for quantities defined at the new macroscopic scale (Fig. 1 I).

Let us first consider the problem of the overall yield criterion. At the meso scale, the criterion can be written under the form

ocq = l)u// < CT,) I +qf.f~-2q,fcos1l ;; 1 ( 11 I, 2

where the symbols and /) 11 denote the deviatoric part and the Von Mises norm of a tensor (/(TIl = (;T,, T,,) ). Using the triangular inequality, one then gets at the macroscopic scale

where the symbol ( ) denotes an averaging process (performed over the various elementary cells). Now the square root is a concave function; thus (,,/.Y) ,< v:/(.u>, which implies that

Also, cash x is a convex function ; thus

Calculating (a,!) = (a?(~,, +a,,) + (I -26)(r,,) exactly wo Id require evaluating o

in terms of Z, which is a difficult task. A rough estimate can however be obtained by adopting the Reuss hypothesis 0 r E. Then (a,,) 2 c(~(C,, +C,.,.) + (1 - h,)(C,,), and since the -_ axis varies from cell to cell in a completely random manner.

----I----

0 , /a ny -;- -:\\;o;Q

FIG. 1 I. Schematic basis of the derivation of a macroscopic model for materials containing non-spherical cavities with random orientations.


:(C,,+C,:,.) = (C,,) = f tr E = C,, so that (oh) N C,. The same result can be arrived at by making the weaker assumption that correlations between the meso stress tensor and the direction of the void axis are negligible. Indeed one then has (cJ,,) = (a: (e,@e,)) 1: (a): (eZ@e,) = E: fl = X:, and similarly for :(a,,+cr,,,). Adopting this estimate, we get

which suggests the following form for the macroscopic criterion (equality being supposed to be attainable in this inequality) :

c2 2 +2q, f cash

This is the same criterion as for cavities with parallel orientations but for the replacement of C,, by C,,, which accounts for macroscopic isotropy. It is identical to the Gurson-Tvergaard criterion for spherical cavities with parameter qI [i.e. with f&,/a0 in the hyperbolic cosine replaced by jq2C,M/o,,; see TVERGAARD (1981)] with a value of q2 comprised between 1 (for K = 3/2, spherical cavities) and 2/,,6 = 1.15. . . (for K = &, cylindrical cavities).

Let us now consider the problem of the evolution equation of S. Unfortunately, even if the shape parameters of the various cavities are identical, their rates are different, because the z direction varies from cell to cell ; hence the only possibility is to identify s with the average value of the S-rates of the different voids :

sz i

i[2-t2+(t2-l)ei]dit+3

where t = CT,JCJ~~. Replacing t by T = C,,,/&, (which is a reasonable approximation in view of the fact that the whole expression between square brackets was introduced on purely heuristic grounds) and neglecting correlations between the meso strain rate tensor and the direction of the void axis, we then get

S= 3 ( 1-3c(, P+3az-1 DE. f 1 It is easy to check that this equation predicts negative S-values for positive DE,-values ; in other words, the voids tend to become spherical on average as they expand. This prediction is reasonable at the beginning of the loading process (but not afterwards, because it fails to account for the progressive development of a damage texture).

Other aspects of the model are unchanged with respect to the case of voids with parallel axes.

REFERENCES

BUDIANSKY, B., HUTCHINSON, J. W. and SLUTSKY, S. (1982) Void growth and collapse in viscous solids. Mechanics of Solids (ed. H. G. HOPKINS and M. J. SEWELL), pp. 1345. Pergamon Press, Oxford.

I754 M. GoL.tXiAXrl

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