A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids—I: Limit-analysis of some representative cell

Journal of the Mechanics and Physics of Solids 60 (2012) 1020–1036

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A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids—I: Limit-analysis of some representative cell Komlanvi Madou, Jean-Baptiste Leblond n UPMC Univ Paris 6 and CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France

a r t i c l e in f o

abstract

Article history: Received 9 September 2011 Received in revised form 8 November 2011 Accepted 19 November 2011 Available online 3 December 2011

Gurson (1977)’s famous model of the behavior of porous ductile solids, initially developed for spherical cavities, was extended by Gologanu et al. (1993, 1994, 1997) to spheroidal, both prolate and oblate voids. The aim of this work is to further extend it to general (non-spheroidal) ellipsoidal cavities, through approximate homogenization of some representative elementary porous cell. As a first step, we perform in the present Part I a limit-analysis of such a cell, namely an ellipsoidal volume made of some rigidideal-plastic von Mises material and containing a confocal ellipsoidal void, loaded arbitrarily under conditions of homogeneous boundary strain rate. This analysis provides an estimate of the overall plastic dissipation based on a family of trial incompressible velocity fields recently discovered by Leblond and Gologanu (2008), satisfying conditions of homogeneous strain rate on all ellipsoids confocal with the void and the outer boundary. The asymptotic behavior of the integrand in the expression of the global plastic dissipation is studied both far from and close to the void. The results obtained suggest approximations leading to explicit approximate expressions of the overall dissipation and yield function. These expressions contain parameters the full determination of which will be the object of Part II. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Porous ductile solids Ellipsoidal voids Homogenization Limit-analysis Homogeneous boundary strain rate

1. Introduction The most classical model of the overall behavior of porous ductile solids is due to Gurson (1977). This model was derived from homogenization, using a limit-analysis of a spherical cell made of some rigid-ideal-plastic von Mises material, containing a spherical void and loaded arbitrarily through conditions of homogeneous boundary strain rate (Mandel, 1964; Hill, 1967). As such, it applied to materials containing spherical voids. Voids actually encountered in real materials are often non-spherical, which induced Gologanu et al. (1993, 1994, 1997) to extend Gurson (1977)’s model to spheroidal voids; the extended model is currently known as the GLD model. These authors again combined homogenization and limit-analysis to derive estimates of the yield surface of spheroidal, prolate (Gologanu et al., 1993) and oblate (Gologanu et al., 1994) cells containing a confocal spheroidal void, and subjected to conditions of homogeneous boundary strain rate. The treatment used incompressible axisymmetric trial velocity fields satisfying conditions of homogeneous strain rate on every spheroid confocal with the boundaries of the void and the cell.1 The model was further improved by Gologanu

n

Corresponding author. E-mail address: [email protected] (J.-B. Leblond). 1 A variant using a velocity field orthogonal to each such spheroid was proposed by Garajeu (1995) and Garajeu et al. (2000).

0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2011.11.008

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

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(1997) and Gologanu et al. (1997) by considering more trial velocity fields, and using some general rigorous results of PonteCastaneda (1991), Willis (1991) and Michel and Suquet (1992) on homogenization of nonlinear composites. Variants and extensions of the Gurson and GLD models have been proposed by various authors. Some of these were based on use of more sophisticated trial velocity fields (Garajeu, 1995: exact solution for a hollow elastic sphere loaded through conditions of homogeneous boundary strain rate; Monchiet et al., 2007: exact Eshelby solution for the ellipsoidal inclusion problem in an infinite elastic matrix). Some others considered matrices obeying Hill’s anisotropic yield criterion instead of that of von Mises (Keralavarma and Benzerga, 2008; Monchiet et al., 2008). Benzerga and Leblond (2010)’s recent review paper provides a synthesis of these works. But more general (non-spheroidal) ellipsoidal voids have not been considered within the approach initiated by Gologanu et al. (1993, 1994, 1997), although such voids are common in practice, for instance in laminated plates. The problem of general ellipsoidal voids was however attacked from another angle by Kaisalam and Ponte-Castaneda (1998) and Danas and Ponte-Castaneda (2008a,b). The first authors initially derived a model based on the concept of ‘‘linear comparison material’’. Unfortunately, although this model theoretically applied to porous plastic (and visco-plastic) solids containing arbitrary ellipsoidal voids and subjected to arbitrary loadings, its predictions revealed rather inaccurate for purely hydrostatic loadings. Danas and Ponte-Castaneda (2008a,b) recently proposed an improved model based on results of some ‘‘second-order homogenization method’’ (Ponte-Castaneda, 2002; Idiart and Ponte-Castaneda, 2005); but the accuracy of this new proposal has not yet been assessed. The approach initiated by Gologanu et al. (1993, 1994, 1997) remains an interesting alternative for the derivation of models for porous plastic materials containing general ellipsoidal voids, since it offers the major advantage of yielding explicit, and basically simple approximate expressions of the overall plastic dissipation and yield function. The aim of this work is to develop such a model. In the present Part I, we begin by extending Gologanu et al. (1993, 1994)’s first works on spheroidal voids, by performing a limit-analysis of some general ellipsoidal cell made of some rigid-ideal-plastic von Mises material, containing a confocal ellipsoidal void and loaded arbitrarily through conditions of homogeneous boundary strain rate. The paper is organized as follows:  As a geometric preliminary, Section 2 is devoted to general ellipsoidal coordinates.  Section 3 then recalls the expression of a family of incompressible velocity fields recently discovered by Leblond and Gologanu (2008). These fields satisfy conditions of homogeneous strain rate on an arbitrary family of confocal ellipsoids, and thus appear as generalizations of those used by Gologanu et al. (1993, 1994), which satisfied such conditions on confocal spheroids.  Section 4 presents the principle of the application of this family of trial velocity fields to some limit-analysis of the ellipsoidal cell considered.  Section 5 then expounds a thorough study of the asymptotic behavior of the integrand in the integral expression of the estimated overall plastic dissipation, both far from and near the void.  Finally Section 6 shows how to deduce from this study reasonable approximations allowing for an explicit calculation of this dissipation and the associated, Gurson-like approximate overall yield function. The parameters appearing in the approximate expression of the overall yield function are not fully determined yet at this stage. Part II will be devoted to their determination. 2. Geometric preliminaries We first define general ellipsoidal coordinates, following Morse and Feshbach (1953). To each triplet ða,b,cÞ of numbers such that a 4 b4 c 4 0

ð1Þ

is attached a set of curvilinear coordinates ðl, m, nÞ such that

l 4c2 4 m 4 b2 4 n 4a2 : The relations connecting these coordinates to ordinary Cartesian ones ðx,y,zÞ are 8 !1=2 8 > > ða2 þ lÞða2 þ mÞða2 þ nÞ > x2 y2 z2 > > x ¼ 7 > > > > þ 2 þ 2 ¼ 2 b2 Þða2 c2 Þ > > ða 2 þl > > a c þl > > b þ l > > !1=2 > > > > 2 2 2 2 2 2 < x < y z ðb þ lÞðb þ mÞðb þ nÞ þ 2 þ 2 ¼ y ¼ 7 3 2 þm 2 2 a c þm 2 2 > > b þ m ðb c Þðb a Þ > > > > > > > > !1=2 > > x2 y2 z2 > > > > þ 2 ¼ ðc2 þ lÞðc2 þ mÞðc2 þ nÞ > > : a2 þ n þ 2 > > z ¼ 7 c þn b þn > 2 : ðc2 a2 Þðc2 b Þ

ð2Þ

1 1 1

ð3Þ

1022

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(Eqs. (3)4,5,6 implicitly define l, m, n in terms of x, y, z through polynomial equations of the third degree). Note that Eqs. (3)1,2,3 leave the signs of x, y, z unspecified so that there are in fact eight possible triplets ðx,y,zÞ for each triplet ðl, m, nÞ. Also, (3)4,5,6, combined with inequalities (2), show that surfaces of constant l are confocal ellipsoids, denoted E l in the ffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 sequel, of semi-axes a2 þ l, b þ l, c2 þ l, whereas surfaces of constant m and n are hyperboloids of one and two sheets, respectively. Define the function qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vðrÞ  9a2 þ r9 9b þ r9 9c2 þ r9 ð4Þ This function appears in the expression of the infinitesimal volume element: dO ¼ 

ðlmÞðmnÞðnlÞ dl dm dn 8vðlÞvðmÞvðnÞ

ð5Þ

and in other instances, as will appear below. Note that up to a factor of 4p=3, vðlÞ represents the volume enclosed within the ellipsoidal surface E l . (The geometrical interpretations of vðmÞ and vðnÞ are less straightforward and will not be needed). In the sequel, the short notation v will often be used to represent vðlÞ, without any ambiguity. The expressions of the derivatives @l=@x, @l=@y, @l=@z will be needed; they are readily found by differentiating Eq. (3)4 at constant (y,z), (z,x) and (x,y), respectively: 8 @l 2x > > ¼ , > > > @x ða2 þ lÞT > > > < @l 2y x2 y2 z2 ¼ , T þ 2 þ : ð6Þ 2 2 @y 2 2 2 > ðb þ lÞT ða þ lÞ ðc þ lÞ2 ðb þ lÞ > > > > @l 2z > > > : @z ¼ ðc2 þ lÞT ,

3. Incompressible velocity fields satisfying conditions of homogeneous strain rate on an arbitrary family of confocal ellipsoids The velocity fields mentioned in the title were recently discovered by Leblond and Gologanu (2008), who established the existence, uniqueness and explicit expression of a field of such a type for any possible value of the overall strain rate tensor imposed on the outermost ellipsoid. We concentrate here on their results, postponing a short overview of their treatment to Appendix A for completeness. Leblond and Gologanu (2008)’s conclusions are as follows: the velocity fields considered are, in ellipsoidal coordinates, of the form 8 v ðrÞ  Dxx ðlÞx þ Dxy ðlÞy þDxz ðlÞz > < x  Dyx ðlÞx þ Dyy ðlÞy þ Dyz ðlÞz v y ðrÞ ð7Þ vðrÞ  DðlÞ  r 3 > : v ðrÞ  D ðlÞx þD ðlÞyþ D ðlÞz z zx zy zz r  xex þ yey þzez being the current position vector and DðlÞ a symmetric second-rank tensor depending on l of the form DðlÞ  AD0 ðlÞ þ D

ð8Þ 0

where A is an arbitrary number, D an arbitrary constant traceless symmetric second-rank tensor, and D ðlÞ the symmetric second-rank tensor depending on l given by 8 R þ1 dr > >   > D0xx ðlÞ  > l 2 þ r vðrÞ > 2 a > > > > > R þ1 dr > >   > D0yy ðlÞ  < l 2 2 b þ r vðrÞ ð9Þ > > > 0 R d r > þ1 >   > Dzz ðlÞ  > l > 2 c2 þ r vðrÞ > > > > > : D0xy ðlÞ  D0yz ðlÞ  D0zx ðlÞ  0 The integrals defining the diagonal components of D0 ðlÞ are of elliptic type. However, it is important to note that the trace of this tensor has a simple expression: !   Z þ1 Z þ1 1 1 1 dr d 1 1 0 ¼  dr ¼ ð10Þ þ þ tr D ðlÞ ¼ dr vðrÞ vðlÞ a2 þ r b2 þ r c2 þ r 2vðrÞ l l

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

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Some remarks pertaining to the velocity field v0 ðrÞ  D0 ðlÞ  r

ð11Þ

are in order:

 In the spherical case ða ¼ b ¼ cÞ, this velocity field is just the radial field inversely proportional to the square of the distance to the origin.

 In the circular cylindrical case ða ¼ þ 1,b ¼ cÞ, it is the planar radial field inversely proportional to the distance to the axis of rotational symmetry.

 In the spheroidal, prolate ða 4 b ¼ cÞ or oblate ða ¼ b 4 cÞ cases, it coincides with the trial field used by Gologanu et al. (1993, 1994) to describe the expansion of the void. These elements suggest that the field v0 ðrÞ may plausibly be used to represent the expansion of an arbitrary ellipsoidal void.

4. Limit analysis of an ellipsoidal cell containing a confocal ellipsoidal void 4.1. Presentation of the cell—notations We shall now consider an ellipsoidal cell containing a confocal ellipsoidal void, and loaded arbitrarily through conditions of homogeneous boundary strain rate (Mandel, 1964; Hill, 1967). Although this shape is admittedly somewhat arbitrary, it is commonly considered as an acceptable approximation of the actual shape of some representative elementary cell in a porous material.2 The boundary conditions imposed on it are also commonly accepted as representative, at least prior to the onset of strain localization phenomena which lie outside of the scope of the present work. The semi-axes of the inner ellipsoid (the boundary of the void) and the outer one (the boundary of the cell) are denoted a,b,c ða 4 b 4cÞ and A,B,C ðA 4B 4CÞ, respectively; they are related through the confocality conditions 2 A2 a2 ¼ B2 b ¼ C 2 c2 . The volumes of these ellipsoids are ð4p=3Þo and ð4p=3ÞO where

o  abc, O  ABC

ð12Þ

and the porosity (void volume fraction) is f

o

ð13Þ

O

We shall use the ellipsoidal coordinates ðl, m, nÞ associated to the triplet ða,b,cÞ, as defined in Section 2. The values of l on the inner and outer ellipsoids are l  0 and l  L, respectively, so that these ellipsoids may be identified, with the notation introduced above, to E 0 and E L , on which the values of vðlÞ are vð0Þ ¼ o and vðLÞ ¼ O, respectively. The semi-axes A,B,C of E L are related to those, a,b,c, of E 0 plus the parameter L through the relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A  a2 þ L, B  b þ L, C  c2 þ L ð14Þ It follows from this equation, the definition (13) of the porosity and the expression (12) of o and O, that the parameter L is determined in terms of the semi-axes of the void and the porosity by the following third-degree polynomial equation: 2

ða2 þ LÞðb þ LÞðc2 þ LÞ

2

a2 b c2 f

2

¼0

ð15Þ

The completely flat ellipsoid confocal with E 0 and E L will play a fundamental role in the sequel. The semi-axes a,b,c of this ellipsoid are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a  a2 c2 , b  b c2 , c  0 ð16Þ The family of confocal ellipsoids E l may be characterized by the single dimensionless parameter sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b b c2 k ¼ 2 a a c2

ð17Þ

2 Danas and Ponte-Castaneda (2008a) have argued that for the small porosities of practical interest, elementary cells of any shape are acceptable, provided that they respect the given value of the porosity and the shape of the voids; indeed the influence of the distribution function of the centers of the voids is of second order in the porosity.

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or the related one qffiffiffiffiffiffiffiffiffiffiffiffi 2 k  1k ¼ 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 b 2 2 a c

ð18Þ

0

0

0

In general, k and k are in the interval ð0; 1Þ. The special values ðk ¼ 0,k ¼ 1Þ and ðk ¼ 1,k ¼ 0Þ correspond to spheroidal, 0 prolate ða 4 b ¼ cÞ and oblate (a ¼ b 4 c) voids, respectively. For a spherical void ða ¼ b ¼ cÞ, k and k are indeterminate. The hollow cell will be assumed to be made of some rigid-ideal-plastic material with yield stress s0 in simple tension, obeying von Mises’s criterion and the associated flow rule. 4.2. Limit-analysis The cell is loaded through conditions of homogeneous boundary strain rate: vðrÞ ¼ D  r,

8r 2 E L

ð19Þ

where D is the macroscopic strain rate tensor. There is a unique incompressible velocity field of the type discussed in Section 3 obeying these boundary conditions, that is, the equation AD0 ðLÞ þ D ¼ D

ð20Þ

has a unique solution in ðA, DÞ; indeed taking the trace of both sides, one gets by Eq. (10) plus the fact that D must be traceless: A tr D0 ðLÞ ¼

A A ¼ ¼ tr D vðLÞ O

)

A ¼ O tr D

ð21Þ

and it follows that

D ¼ DAD0 ðLÞ ¼ DO ðtr DÞD0 ðLÞ

ð22Þ

The thus unambiguously defined velocity field may be used in a limit-analysis of the domain considered. This analysis provides an upper estimate P þ ðDÞ of the overall plastic dissipation, identical to the average value of the local plastic dissipation corresponding to the trial velocity field considered over the ellipsoidal domain: ! Z l ¼ L Z m ¼ c2 Z n ¼ b2 X 8 1 ðiÞ P þ ðDÞ  s d ð l , m , n Þ dO ð23Þ 0 eq ð4p=3ÞO l ¼ 0 m ¼ b2 n ¼ a2 i¼1 ðiÞ

In this equation the symbols deq ðl, m, nÞ ði ¼ 1, . . . ,8Þ denote the values of the local von Mises equivalent strain rate deq ðrÞ  ð23 dðrÞ : dðrÞÞ1=2

ð24Þ

(where dðrÞ is the local strain rate tensor) for the eight triplets ðx,y,zÞ corresponding to the triplet ðl, m, nÞ, and the elementary volume element dO is given by Eq. (5). It is unfortunately impossible to provide an exact explicit expression of the integral in Eq. (27), even in the simplest case of a spherical void. We therefore introduce some approximation here. First, integrating over successive confocal ellipsoids E l , we re-express P þ ðDÞ as Z s l¼L P þ ðDÞ ¼ 0 /deq ðrÞSE l dðvðlÞÞ ð25Þ

O

l¼0

In this expression the symbol /f ðrÞSE l denotes the average value of an arbitrary function f ðrÞ over the ellipsoidal surface E l , with a weight equal to the infinitesimal volume element between the surfaces E l and E l þ dl : ! Z m ¼ c2 Z n ¼ b2 X 8 1 ðiÞ f ðl, m, nÞ dO /f ðrÞSE l  4p 2 n ¼ a2 3 dðvðlÞÞ m ¼ b i¼1 ! Z m ¼ c2 Z n ¼ b2 X 8 1 ðlmÞðmnÞðnlÞ ðiÞ dm dn ¼ f ðl, m, nÞ ð26Þ dv 8vðmÞvðnÞ m ¼ b2 n ¼ a2 4p i ¼ 1 vð l Þ ð l Þ 3 dl ðiÞ

where Eq. (5) has been used; the f ðl, m, nÞ here again represent the values of f ðrÞ for the eight triplets ðx,y,zÞ corresponding qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 to the triplet ðl, m, nÞ. It then follows from the classical inequality /f ðrÞSE l r /f ðrÞSE l (applicable whatever the weighting function chosen to define the average value) that Z s l ¼ L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 P þ ðDÞ r P þ þ ðDÞ  0 /deq ðrÞSE l dðvðlÞÞ

O

l¼0

and the following approximation is introduced: A1 : The quantity P þ þ ðDÞ represents an acceptable approximation of P þ ðDÞ.

ð27Þ

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

1025

This is exactly the hypothesis which must be made to derive Gurson’s criterion for spherical voids (see e.g. Leblond, 2003, Chapter 8) and the GLD criterion for spheroidal ones (Gologanu et al., 1993, 1994, 1997). Since it was found to be quite acceptable in these cases, it may be hoped to again yield satisfactory results for general ellipsoidal voids. The calculation of the integral in Eq. (27) is still a difficult task which will require further approximations detailed in the sequel. Once this task is completed, an external estimate of the overall yield surface of the cell may be obtained from the equation

R¼

@P þ þ ðDÞ @D

ð28Þ

where the components of D act as parameters. (This equation defines a 5D surface in the 6D space of overall stresses R since the six components of R depend only on the ratios of five components of D to the last one, the function ð@P þ þ =@DÞðDÞ being positively homogeneous of degree 0).

5. Asymptotic study of the integrand in the expression of the overall plastic dissipation 5.1. Generalities The aim of this section is to study the asymptotic behavior of the integrand in Eq. (27)2 defining P þ þ ðDÞ, both far from and close to the origin. The results found will play a fundamental role in the search for a reasonable approximate expression of this integrand allowing for an explicit calculation of the overall plastic dissipation and the associated yield criterion. We first note that by Eqs. (7) and (8) 0

dðrÞ ¼ Ad ðrÞ þ D

ð29Þ

0

where d ðrÞ is the local strain rate tensor corresponding to the velocity field v0 ðrÞ defined by Eq. (11); the components of this tensor are readily deduced from Eqs. (6) and (9): 8 > 0 > dxx ðrÞ > > > > > > > < 0 dyy ðrÞ > > > > > > > > d0zz ðrÞ > :

D0xx ðlÞ 

¼

x2 ða2 þ lÞ2 TvðlÞ

8 0 > dxy ðrÞ > > > > > > < 0 dyz ðrÞ > > > > > 0 > > : dzx ðrÞ

,

y2

D0yy ðlÞ 

, 2 ðb þ lÞ2 TvðlÞ z2 D0zz ðlÞ  , 2 ðc þ lÞ2 TvðlÞ

¼ ¼

¼



¼



¼

xy 2

ða2 þ lÞðb þ lÞTvðlÞ yz ð30Þ

2

ðb þ lÞðc2 þ lÞTvðlÞ zx  ðc2 þ lÞða2 þ lÞTvðlÞ

It then follows from Eq. (24) that 2

0 2

0

deq ðrÞ ¼ A2 deq ðrÞ þ D2eq þ 43A d ðrÞ : D

ð31Þ

where

Deq  ð23D : DÞ1=2

ð32Þ

so that 2

0 2

0

/deq ðrÞSE l ¼ A2 /deq ðrÞSE l þ D2eq þ 43A/d ðrÞSE l : D

ð33Þ 0

One may immediately note that the off-diagonal components of the average value /d ðrÞSE l here are zero since, by 0 Eqs. (30)4,5,6, the off-diagonal components of d ðrÞ change sign from one of the eight points ðx,y,zÞ corresponding to a given triplet ðl, m, nÞ to another. 0 2 0 The asymptotic study of the scalar /deq ðrÞSE l p and limit the ffiffiffi the tensor /d ðrÞSE l near infinity is easy. Indeed in this 3=2 ellipsoids E l become spheres of large radius r  l (by Eq. (3)4) and large volume ð4p=3ÞvðlÞ, vðlÞ  v  l  r 3 (by Eq. (4)), the tensor D0 ðlÞ becomes identical to ð1=3vÞ1  ð1=3r 3 Þ1 (by Eq. (10)), and the velocity field v0 ðrÞ becomes identical to the radial incompressible field ð1=3r 3 Þr (by Eq. (11)). It follows that the equivalent strain rate deq ðrÞ becomes uniform and equal to 2=3r 3  2=3v on the surface E l so that 0 2

/deq ðrÞSE l 

4 4  9r 6 9v2

for v- þ1

ð34Þ

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K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036 0

Also, the diagonal components of the tensor /d ðrÞSE l , of a priori order Oð1=r 3 Þ ¼ Oð1=vÞ, become asymptotically equal for 0 reasons of isotropy, and their sum is zero since d ðrÞ is traceless; hence they are zero at dominant order, that is, 8     1 1 > 0 > > > /dxx ðrÞSE l ¼ o r 3 ¼ o v > > >     > < 1 1 0 /dyy ðrÞSE l ¼ o 3 ¼ o for v- þ1 ð35Þ v r > >     > > > 1 1 > 0 > > : /dzz ðrÞSE l ¼ o r 3 ¼ o v 0 2

0

The asymptotic study of /deq ðrÞSE l and /d ðrÞSE l in the other limit, that is for surfaces E l close to the completely flat confocal ellipsoid of semi-axes a,b,c ¼ 0, is unfortunately much more involved. It will require distinguishing between the generic case where a 4 b4 c and three special cases where a 4 b ¼ c, a ¼ b 4 c or a ¼ b ¼ c, corresponding to voids of arbitrary ellipsoidal, prolate spheroidal, oblate spheroidal and spherical shapes, respectively.

5.2. Generic case 0 2

0

In the generic case where a 4b 4 c, the asymptotic study of /deq ðrÞSE l and /d ðrÞSE l for surfaces E l close to the completely flat confocal ellipsoid is made easier by considering the ellipsoidal coordinates ðl , m , n Þ associated to the semiaxes a,b,c ¼ 0 of this ellipsoid. These new coordinates are related to the old ones ðl, m, nÞ through the formulae

l  l þc2 , m  m þ c2 , n  n þ c2

ð36Þ

and the Cartesian coordinates ðx,y,zÞ are related to them through formulae analogous to (3), with a,b,c ¼ 0, l , m , n instead of a,b,c, l, m, n. The surface E l is close to the completely flat confocal ellipsoid when l -0; therefore, what we are interested in is the value of the limits 8 0 > Lx  lim /dxx ðrÞSE l , > > > l -0 > > < 0 Ly  lim /dyy ðrÞSE l , L2  lim /d0 2 ðrÞS ð37Þ El eq l -0 > l -0 > > > 0 > > : Lz  lim /dzz ðrÞSE l , l -0

To evaluate these limits, the first task is to derive the asymptotic expressions of the Cartesian coordinates x,y,z, the function vðlÞ and the quantity T for l -0. The expressions (3)1,2,3 of x,y,z and (4) of vðlÞ become in this limit 8 !1=2 > > ða 2 þ m Þða 2 þ n Þ > > , x  7 > 2 > > > a 2 b > > > !1=2 > 2 2 < pffiffiffi ðb þ m Þðb þ n Þ vðlÞ  ab l ð38Þ y  7 , 2 > > > b a 2 > > q ffiffiffiffiffiffiffiffiffi ffi > > > > l mn > > > , :z  7 ab 0

These equations imply that x and y are of order Oðl ¼ 1Þ while z is of order Oðl quantity T becomes    x2 y2 z2 z2 1 ¼O T 4þ 4þ 2  2 l a l l b

1=2

Þ, so that the expression (6)4 of the

ð39Þ

The next task is to derive the asymptotic expressions of the diagonal components of the tensor D0 ðlÞ. The coordinates l , m , n being used instead of l, m, n, the general expressions (9)1;2,3 of D0xx ðlÞ,D0yy ðlÞ,D0zz ðlÞ become in the limit l -0: 8 R þ1 dr > 0 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 0 > Dxx ðlÞ  > 2 2 > > 2 ða þ r Þ3 ðb þ r Þr > > > > > R þ1 dr < 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dyy ðlÞ  0 2 2 > 2 ða þ r Þðb þ r Þ3 r > > > > > R þ1 dr > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D0zz ðlÞ  > 0 > 2 > 2 : 2 ða þ r Þðb þ r Þr 3

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

1027

where Eq. (4) has been used. The first two integrals here are provided by formulae (3.133.18) and (3.133.12) of Gradshteyn and Ryzhik (1980), and the third is obviously divergent at the origin. One thus gets lim D0xx ðlÞ ¼

l -0

K 0 E0 02 3

k a

2

,

lim D0yy ðlÞ ¼

E0 =k K 0

l -0

02 3

k a

lim D0zz ðlÞ ¼ þ1

,

ð40Þ

l -0

where E0 and K 0 denote classical complete elliptic integrals: Z p=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p=2 df 0 02 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0  Eðk Þ  1k sin2 f df, K 0  Kðk Þ  02 0 0 1k sin2 f

ð41Þ

It now becomes possible to evaluate the limits Lx, Ly and Lz defined by Eqs. (37)1,2,3. By Eqs. (38)1,2,4 and (39), the 4

2

quantities x2 =ðða2 þ lÞ2 TvðlÞÞ  x2 =ða 4 TvðlÞÞ and y2 =ððb þ lÞ2 TvðlÞÞ  y2 =ðb TvðlÞÞ are both of order Oðl they can be neglected with respect to 0

dxx ðrÞ  D0xx ðlÞ,

D0xx ðlÞ

and

D0yy ðlÞ

in the general expressions (30)1,2 of

0 dxx ðlÞ

1=2

Þ for l -0, so that

0

and dyy ðlÞ which become

0

dyy ðrÞ  D0yy ðlÞ for l -0

0

ð42Þ

0

The components dxx ðrÞ and dyy ðrÞ thus become asymptotically constant over the ellipsoid E l in the limit l -0. It then follows from Eqs. (37)1,2 (40)1,2 and (42) that Lx ¼ lim D0xx ðlÞ ¼ l -0

K 0 E0 02 3

k a

2

Ly ¼ lim D0yy ðlÞ ¼

,

E0 =k K 0

l -0

ð43Þ

02 3

k a

The limit Lz defined by Eq. (37)3 cannot be evaluated in a similar way because the two terms in the right-hand side of Eq. (30)3 can be checked to be of the same order. The simplest solution here consists in using the incompressibility of the 0

0

0

velocity field v0 ðrÞ, which implies that dzz ðrÞ ¼ dxx ðrÞdyy ðrÞ and therefore, by Eqs. (43), that Lz ¼ Lx Ly ¼ 

E0

ð44Þ

2 3

k a

(Note that Lz aliml -0 D0zz ðlÞ ¼ þ 1). 0 2 To finally evaluate the limit L2 defined by Eq. (37)4 , one must calculate the limits of the average values /dxx ðrÞSE l , 0 2 0 2 0 2 0 2 0 2 0 0 /dyy ðrÞSE l , /dzz ðrÞSE l , /dxy ðrÞSE l , /dyz ðrÞSE l , /dzx ðrÞSE l for l -0. From the fact that in this limit dxx ðrÞ, dyy ðrÞ and 0 0 0 dzz ðrÞ ¼ dxx ðrÞdyy ðrÞ become asymptotically constant and equal to Lx, Ly and Lz ¼ Lx Ly over the surface E l , it follows that 0 2

lim /dxx ðrÞSE l ¼ L2x ,

l -0

0 2

lim /dyy ðrÞSE l ¼ L2y ,

l -0

0 2

lim /dzz ðrÞSE l ¼ L2z

ð45Þ

l -0

The limits of the three remaining average values may be computed by using Eq. (26) to write the integrals explicitly with the ellipsoidal coordinates ðl , m , n Þ, and using various changes of variables and formulae of Gradshteyn and Ryzhik (1980). When 0 2

0 2

0

0

one calculates the limits of /dyz ðrÞSE l and /dzx ðrÞSE l , all complete elliptic integrals E  EðkÞ, K  KðkÞ, E0  Eðk Þ, K 0  Kðk Þ momentarily appear but finally cancel out upon use of Gradshteyn and Ryzhik (1980)’s formula (8.122) connecting them all. The very simple final results are as follows: 0 2

lim /dxy ðrÞSE l ¼ 0,

l -0

0 2

lim /dyz ðrÞSE l ¼

l -0

1 4

k a6

,

0 2

lim /dzx ðrÞSE l ¼

l -0

1 2

k a6

Gathering Eqs. (37)4 and (43)–(46), one gets upon a straightforward calculation:     

4 1 1 1 1 1 1 02 02 0 0 L2 ¼ 1 þ þK  1 þ K þ E E þ 2 4 2 2 4 3a 6 k04 k k k k k

ð46Þ

ð47Þ

The conclusion is that in the generic case where a 4 b4 c, all limits Lx ,Ly ,Lz ,L2 defined by Eqs. (37) are finite. 5.3. Prolate spheroidal case The prolate spheroidal case where a 4b ¼ c is special for both mathematical and physical reasons. From the mathematical viewpoint, the calculations presented above, based on the use of ellipsoidal coordinates, become invalid 2 since the representation of 3D space by such coordinates breaks down, as is clear from the presence of the factor b c2 ¼ 0 in the denominators of the expressions (3)2,3 of y and z. From the physical viewpoint, the values of the limits looked for are intimately tied to the geometric properties of the completely flat confocal ellipsoid; now this ellipsoid, which is a flat elliptic disk in the generic case, becomes a straight segment in the prolate spheroidal case, which is obviously a very different geometric object. It is important to note that although the coordinatespm , n can longer be used, the coordinate l (or l defined by ffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffi pno 2 Eq. (36)1) still makes sense, as a definer of the semi-axes a2 þ l, b þ l ¼ c2 þ l of the spheroid E l . The definitions (37) of the limits Lx ,Ly ,Lz ,L2 also still make sense.

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K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

In the limit l -0, the expression (4) of vðlÞ  v becomes v  al

ð48Þ 2

2

2

0

2

2

2

1

2

Also, Eq. (3)4 becomes x =a þðy þ z Þ=l ¼ 1, which implies that x is of order Oðl ¼ 1Þ, while y þ z is of order Oðl ¼ l Þ. Hence the expression (6)4 of T becomes    x2 y2 þ z2 y2 þ z2 1  ¼O T 4þ ð49Þ 2 2 l a l l With regard to the asymptotic expressions of the components of the tensor D0 ðlÞ, the expression (9)1 of D0xx ðlÞ reads in the case considered Z þ1 dr D0xx ðlÞ ¼ 2 2ða þ r Þ3=2 r l which obviously implies that D0xx ðlÞ  

ln l



2a 3

ln v 2a 3

for v-0

ð50Þ

where Eq. (48) has been used. Also, Eqs. (10) and (50), combined with the rotational symmetry about the direction x, imply that

1 1 1 D0xx ðlÞ  for v-0 ð51Þ D0yy ðlÞ ¼ D0zz ðlÞ ¼ 2 v 2v Hence D0xx ðlÞ is negligible with respect to D0yy ðlÞ ¼ D0zz ðlÞ in the limit v-0. 0 Now the expression (30)1 of dxx ðrÞ becomes in the limit l -0 0

dxx ðrÞ  D0xx ðlÞ

x2 4

a Tv

and it then follows from the fact that x2 ¼ Oð1Þ plus Eqs. (48), (49) and (50) that 0

dxx ðrÞ  D0xx ðlÞ  

ln v

for v-0

2a 3

ð52Þ

0

Thus dxx ðrÞ again becomes asymptotically constant and equal to D0xx ðlÞ over the surface E l in the limit v-0. This implies that 0

/dxx ðrÞSE l  

ln v

for v-0

2a 3

ð53Þ 0

The asymptotic expressions of the other diagonal components of the tensor /d ðrÞSE l are then trivially deduced from the fact that it is traceless, combined with the rotational symmetry: 1 ln v 0 0 0 /dyy ðrÞSE l ¼ /dzz ðrÞSE l ¼  /dxx ðrÞSE l  2 4a 3

for v-0

ð54Þ 0 2

0 2

0 2

It only remains to determine the asymptotic expressions of the average values /dxx ðrÞSE l , /dyy ðrÞSE l ¼ /dzz ðrÞSE l , 0 2 0 2 0 2 0 2 /dxy ðrÞSE l ¼ /dxz ðrÞSE l and /dyz ðrÞSE l to get that of /deq ðrÞSE l . It first follows from Eq. (52) that 0 2

/dxx ðrÞSE l 

ðln vÞ2

for v-0

4a 6

ð55Þ

Also, by Eqs. (30)2,3,5, (48), (49), and (51), 0 2

0 2

0 2

dyy ðrÞ þ dzz ðrÞ þ 2dyz ðrÞ ¼

D0yy ðlÞ

y2 2

!2 þ D0zz ðlÞ

z2

!2

2

þ2

yz

!2

2

l Tv l Tv l Tv " 2  2  2 # 1 1 y2 1 z2 yz 1  2   2 þ þ 2 ¼ 2 y þ z2 2 y2 þ z2 v y2 þz2 2v2

so that 0 2

0 2

0 2

/dyy ðrÞ þdzz ðrÞ þ2dyz ðrÞSE l 

1 2v2

for v-0

Finally, by (30)46, (48) and (49),    2  2 xy xz x2 1 0 2 0 2 dxy ðrÞ þ dxz ðrÞ ¼ þ  ¼ O v a 2 l Tv a 2 l Tv a 6 ðy2 þ z2 Þ

ð56Þ

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

1029

since x2 and y2 þz2 are of order Oð1Þ and O(v), respectively, and it follows that   1 0 2 0 2 for v-0 /dxy ðrÞ þ dxz ðrÞSE l ¼ O v

ð57Þ

Combination of Eqs. (55), (56) and (57) implies that 0 2

/deq ðrÞSE l 

1 3v2

for v-0

ð58Þ

This result coincides with that derived by Gologanu et al. (1993) for the velocity field v0 ðrÞ in question, account being taken of the different notations used. In conclusion, Eqs. (53), (54) and (58) show that in the prolate spheroidal case where a 4b ¼ c, all limits Lx, Ly, Lz, L2 defined by Eqs. (37) are infinite. 5.4. Oblate spheroidal case From a mathematical viewpoint, the oblate spheroidal case where a ¼ b4 c seems just as special as the prolate spheroidal case, the representation of 3D space by ellipsoidal coordinates becoming again invalid because of the presence 2 of the term a2 b ¼ 0 in the denominators of the expressions (3)1,2 of x and y. From a physical viewpoint, however, one may expect that nothing particular will in fact occur in this case. Indeed the completely flat confocal ellipsoid, which is an elliptic disk in the generic case, simply becomes a circular disk in the oblate spheroidal case. Now one may fix one semiaxis of this disk, say a, and vary the other one, b, from values smaller than a to values larger than it, with clearly nothing special occurring when a ¼ b. This means that one may expect the limits Lx, Ly, Lz, L2 provided by formulae (43), (44) and (47) in the generic case to 02 04 simply go to finite values in the oblate spheroidal case, in spite of the presence of the diverging terms 1=k and 1=k in 0 these formulae. (Recall that k ¼ 0 in the oblate spheroidal case). Indeed, this is fully confirmed by a calculation of the 0 limits of Lx, Ly, Lz, L2 for k -0, based on Gradshteyn and Ryzhik (1980)’s asymptotic expressions (8.113.1) and (8.114.1) of 0 0 the elliptic integrals E and K 0 . The values of Lx, Ly, Lz, L2 for k ¼ 0 are thus found to be Lx ¼ Ly ¼

p 4a

3

,

Lz ¼ 

p 2a

3

,

L2 ¼

3p2 þ 32

ð59Þ

12a 6

Again, these results coincide with those found by Gologanu et al. (1994) for the velocity field v0 ðrÞ in question, with the necessary changes of notation. One may thus conclude that in the oblate spheroidal case where a ¼ b4 c, the limits Lx, Ly, Lz, L2 are finite just like in the generic case. 5.5. Spherical case The spherical case where a ¼ b ¼ c is special for both mathematical and physical reasons: the system of ellipsoidal coordinates once again breaks down, and the completely flat confocal ellipsoid becomes a mere point. The results looked for are easily found by a direct reasoning analogous to that pertaining to ellipsoids E l of very large dimensions, presented at the end of Section 5.1; the only difference being that here formulae are exact instead of being just asymptotic. One thus gets 0

0

0

/dxx ðrÞSE l ¼ /dyy ðrÞSE l ¼ /dzz ðrÞSE l ¼ 0,

0 2

/deq ðrÞSE l ¼

4 9v2

ð60Þ

for all values of r or v ¼ r 3 . Thus, in the spherical case where a ¼ b ¼ c, the limits Lx, Ly, Lz are zero whereas L2 is infinite. 5.6. Comments 0 2

Some qualitative comments are in order with regard to the asymptotic behavior of the average value /deq ðrÞSE l when the surface E l gets close to the completely flat confocal ellipsoid. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 With approximation A1 , the quantity /deq ðrÞSE l represents the average value over the ellipsoidal surface E l of the plastic dissipation associated to the trial velocity field v0 ðrÞ used to represent the expansion of a confocal ellipsoidal void contained in E l . Thus its asymptotic behavior for surfaces E l close to the completely flat confocal ellipsoid is connected to the value of the plastic dissipation necessary to open a void coinciding with the interior of this ellipsoid. Now this void is an elliptic crack in the generic case, a needle in the prolate spheroidal case, a circular crack in the oblate spheroidal case, and a mere point in the spherical case. Thus Eqs. (47) and (59)3 imply that opening an elliptic or circular crack only requires a locally finite plastic dissipation, whereas Eqs. (58) and (60)2 imply that opening a needle- or point-shaped void requires a locally infinite dissipation. In other words, these equations mean, quite reasonably, that it is much easier to open an elliptic or circular crack than a needle- or point-shaped void.

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K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

6. Approximate overall plastic dissipation and yield function We shall now use the results of the preceding section to propose a reasonable approximate expression of the average 0 2 value /deq ðrÞSE l , allowing for an analytic calculation of the integral in Eq. (27)2 defining P þ þ ðDÞ and the associated approximate yield function. 0 2

6.1. Simplification of the ‘‘crossed term’’ in the expression of /deq ðrÞSE l 0

0 2

It has already been noted that in the ‘‘crossed term’’ 43 A/d ðrÞSE l : D of the expression (33) of /deq ðrÞSE l , the off0 diagonal components of the tensor /d ðrÞSE l make no contribution. It follows that 0

0

dg /d ðrÞSE l : D ¼ /d ðrÞSdg E l :D

ð61Þ

where, for any symmetric second-rank tensor T, T 0 1 T xx B C Tdg  @ T yy A T zz

dg

3

2 R denotes the vector made from its diagonal components: ð62Þ 0

dg are Now let P  R3 denote the plane consisting of vectors whose components have a zero sum. Both vectors /d ðrÞSdg E l and D 0 in P since both tensors d ðrÞ and D are traceless. Also, let ðU,VÞ denote the orthonormal basis of P consisting of the vectors3 0 1 0 1 0 1 Lz Ly Lx 1 1 1 B C 1 B C B C U  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ Ly A, V  pffiffiffi @ 1 A  U ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ Lx Lz A ð63Þ 2 2 2 3 þL þ L Þ L2x þ L2y þ L2z Lz 3ðL Ly Lx x y z 1

This basis may be used to express the vector Ddg :

Ddg  BU þ CV, B  Ddg  U, C  Ddg  V

ð64Þ

then 0

0

0

dg dg ¼ B/d ðrÞSdg /d ðrÞSdg El  D E l  U þ C/d ðrÞSE l  V

so that, by Eqs. (33) and (61) 2

0 2

0

0

dg 4 /deq ðrÞSE l ¼ A2 /deq ðrÞSE l þ D2eq þ 43 AB/d ðrÞSdg E l  U þ 3AC/d ðrÞSE l  V

ð65Þ 0 2 /deq ðrÞSE l ,

proportional to the components Eq. (65) shows that there are in fact two crossed terms in the expression of B and C of the vector Ddg in the plane P. But the basis ðU,VÞ of the plane P has precisely been defined so as to allow for the following approximation: 0 0 2 A2 : The term 43 AC/d ðrÞSdg E l  V may be discarded in the expression (65) of /deq ðrÞSE l . Some explanatory comments are in order here. An indispensable preliminary remark is that in an arbitrary quadratic form A11 X 21 þ A22 X 22 þ 2A12 X 1 X 2 with positive coefficients A11 and A22, the ‘‘crossed term’’ 2A12 X 1 X 2 is negligible for all values of the arguments X 1 ,X 2 if and only if A212 5 A11 A22 . (The easy proof is left to the reader). Applied to the terms 0 2 proportional to A2 , C2 and AC in the quadratic form (65) defining /deq ðrÞSE l ,4 this property implies that the term 0 dg 4 AC/d ðrÞS :V is negligible if and only if El 3 0

0 2

2 ½/d ðrÞSdg E l :V 5 /deq ðrÞSE l

ð66Þ

One may then note that:

 For surfaces E l of large dimensions, the left-hand side of condition (66) is oð1=v2 Þ by Eqs. (35), whereas the right-hand 

 

side is Oð1=v2 Þ by Eq. (34); hence condition (66) is satisfied. 0 For surfaces E l close to the completely flat confocal ellipsoid, the vector /d ðrÞSdg E l goes to a limit collinear to the vector U, by the very definition (63)1 of the latter vector; since U and V are orthogonal, the left-hand side of condition (66) goes to zero. In contrast, the right-hand side goes to some nonzero finite or infinite limit, depending on the case considered (see Eqs. (37)4, (47), (58), (59)3 and (60)2). Hence condition (66) is again fulfilled. Condition (66) being thus satisfied when the surface E l is either distant from or close to the completely flat confocal ellipsoid, should be approximately satisfied for all positions of this surface. In addition, in the spheroidal, prolate and oblate cases, the left-hand side of (66) is exactly zero. Indeed the 0 direction of the vector /d ðrÞSdg in R3 is independent of v in both cases for symmetry reasons El

3 It can be checked that U and V are well-defined even in the prolate spheroidal case where Lx, Ly, Lz are infinite; this is tied to the fact that U gives the 0 3 direction of the vector /d ðrÞSdg E l in R in the limit v-0, and that in this special case this direction is fixed and independent of v for symmetry reasons 0 0 0 dg dg 1 ð/dyy ðrÞSdg ¼ /d ðrÞS ¼  /d ðrÞS zz xx El El E l Þ. In the spherical case U and V are ill-defined but may harmlessly be considered as zero. 2 4 The term in C2 in this quadratic form arises from the term D2eq .

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036 0

0

0

0

0

1031 0

dg dg dg dg dg 1 1 (/dyy ðrÞSdg E l ¼ /dzz ðrÞSE l ¼  2 /dxx ðrÞSE l in the prolate case, /dxx ðrÞSE l ¼ /dyy ðrÞSE l ¼  2 /dzz ðrÞSE l in the oblate

case); and it follows that this vector is orthogonal to V not only in the limit v-0, but for all values of v. Hence condition (66) should quite generally be met, implying that the error resulting from approximation A2 should be small. With this approximation, Eq. (65) becomes 2

0 2

0

/deq ðrÞSE l CA2 /deq ðrÞSE l þ D2eq þ 43AB/d ðrÞSdg El  U

ð67Þ

0 2

6.2. Simplification of the spatial dependence of /deq ðrÞSE l With respect to Gologanu et al. (1993, 1994)’s treatment of the prolate and oblate spheroidal cases, approximation A2 is 0 new. (There was no need in the spheroidal cases to disregard the term 43 AC/d ðrÞSdg E l  V which was zero anyway). In contrast, we shall essentially follow from now on Gologanu et al. (1994)’s treatment of the oblate spheroidal case; the transition to the general ellipsoidal case will not introduce any further major novelty. Denoting by L the square root of the limit defined by Eq. (37)4, we define the variable w

1 3v=2þ w=L

ð68Þ

where w is a dimensionless constant of order unity. The advantage of introducing such a change of variable, with such an adjustable constant, will appear below. Note that when v varies from 0 to þ 1, w varies from L=w to 0. 0 2 0 We then write the quantities /deq ðrÞSE l and 43 /d ðrÞSdg E l :U in the following form, which defines the functions F(w) and G(w): 0 2

/deq ðrÞSE l  F 2 ðwÞw2 ,

0 dg 4 3/d ðrÞSE l

 U  2FðwÞGðwÞw2

ð69Þ

Also, we note that by Eq. (64)1, since the basis ðU,VÞ is orthonormal,

D2eq ¼ 23 Ddg  Ddg þ 43 ðD2xy þ D2yz þ D2zx Þ ¼ 23 ðB2 þ C2 Þ þ 43ðD2xy þ D2yz þ D2zx Þ It follows that with the notations just introduced, Eq. (67) becomes 2

/deq ðrÞSE l CA2 F 2 ðwÞw2 þ

2B2 2C2 4 2 þ2ABFðwÞGðwÞw2 þ þ ðDxy þ D2yz þ D2zx Þ 3 3 3

which can be rewritten in the form 2

/deq ðrÞSE l C½AFðwÞ þ BGðwÞ 2 w2 þ

2 2 2 2C2 4 2 B H ðwÞ þ þ ðDxy þ D2yz þ D2zx Þ 3 3 3

ð70Þ

where H2 ðwÞ  132G2 ðwÞw2

ð71Þ

(The quantity H2 ðwÞ thus defined is positive since the quadratic form of A, B, C, Dxy , Dyz , Dzx defined by Eq. (70) is obviously positive-definite). We then introduce the following final approximation: A3 : The functions F(w), G(w), H(w) in Eq. (70) may be replaced by suitable average values F , G, H. Explanatory comments are again in order here.  Consider first the function F(w), defined by the expression (69)1 of /d0eq2 ðrÞSEl . Near infinity ðv- þ 1Þ, w  2=3v-0 0 2 by Eq. (68), so that Eq. (69)1 becomes /deq ðrÞSE l  4F 2 ð0Þ=9v2 . Comparison with Eq. (34) then reveals that Fð0Þ ¼ 1

ð72Þ

Consideration of the other limiting case ðv-0Þ now requires to distinguish between the different geometrical cases. J

In the generic and oblate spheroidal cases, close to the completely flat confocal ellipsoid (v-0), w goes to the limit L=w 0 2 so that /deq ðrÞSE l goes to the limit F 2 ðL=wÞL2 =w2 , which must be equal to L2 by Eq. (37)4; it follows that FðL=wÞ ¼ w

J

ð73Þ

Eqs. (72) and (73) show that when w varies within its interval of definition ð0,L=wÞ, F(w) varies from 1 to w. Provided that w is chosen of order unity, this variation is modest, which justifies the replacement of the function F(w) by a constant. In the prolate spheroidal case, L ¼ þ1 so that Eq. (68) becomes w ¼ 2=3v for all values of v. Therefore, close to the 0 2 completely flat confocal ellipsoid ðv-0Þ, w goes to þ 1 and /deq ðrÞSE l  4F 2 ð þ1Þ=9v2 . Comparison with Eq. (58) then

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K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

shows that Fð þ1Þ ¼

J

pffiffi 3 2

ð74Þ pffiffiffi Hence the function F(w) again varies modestly over the interval of definition ð0, þ 1Þ of w, from 1 to 3=2 C0:87, so that it may again be replaced by a constant. In the spherical case, again, w ¼ 2=3v since L ¼ þ1; Eqs. (60)2 and (69)1 then imply that F(w) is rigorously constant, equal to 1 for all values of w.

Note that these nice properties are intimately connected to the definition (68) of the new variable w, which justifies this definition except, momentarily, for the introduction of the adjustable constant w.  Consider now the function G(w), defined by the expression (69)2 of 43 /d0 ðrÞSdg E l  U. First, near infinity ðv-þ 1Þ, note 0 2 that by Eq. (72), 43 /d ðrÞSdg E l  U  2GðwÞw . Comparison with Eqs. (35) then shows that   1 ¼ oðwÞ for w-0 ð75Þ GðwÞw2 ¼ o v 0

0 2

2 2 2 2 2 2 Therefore, in this limit, ½43 /d ðrÞSdg E l  U  ½2GðwÞw ¼ oðw Þ is much smaller than /deq ðrÞSE l  4=9v  w . By the 0 dg 4 remark on quadratic forms made in Section 6.1, this implies that the crossed term proportional to 3 /d ðrÞSE l  U in the 2 expression (67) of /deq ðrÞSE l is negligible near infinity. This property has been established for the true function G(w), but it remains true if it is replaced by some constant G, since again ð2Gw2 Þ2 5w2 in the limit w-0. Hence, near infinity, replacing G(w) by G just means replacing a negligible term by an equally negligible approximation, which is harmless whatever the value chosen for G. Close to the completely flat confocal ellipsoid, it is again necessary to distinguish between the different geometrical cases.

J

In

the

generic 2

and 2

oblate

spheroidal

cases,

in

the

limit

v-0,

Eq.

(69)2

yields

0 dg 4 3 /d ðrÞSE l

 U-

2

2FðL=wÞGðL=wÞL =w ¼ 2GðL=wÞL =w where Eq. (73) has been used. Now by the definitions (37)1,2,3 of Lx, Ly, Lz and (63)1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of U, the value of this limit is 43 L2x þ L2y þL2z , and it follows that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2w ð76Þ GðL=wÞ ¼ 2 L2x þ L2y þL2z 3L

J

Thus replacing the function G(w) by some constant is reasonable near the completely flat confocal ellipsoid provided that the chosen value of this constant is close to that given by Eq. (76). 0 In p the spheroidal case, Eqs. (69)2, (53) and (54) yield, in the limit v-0, 43 /d ðrÞSdg El  ffiffiffi prolate 2 2 U  3GðwÞw ¼ oðln vÞ ¼ oðln wÞ where Eq. (74) has been used, so that GðwÞ ¼ oðln w=w Þ and consequently Gð þ 1Þ ¼ 0

J

ð77Þ

Hence close to the completely flat confocal ellipsoid, the replacement of the function G(w) by G is again reasonable provided that G is taken to be zero or very small.5 In the spherical case, the function G(w) is uniformly zero by Eq. (60)1 so that G ¼ 0 is again a good value.

 Consider finally the function H(w), defined by Eq. (71). Near infinity, it follows from Eq. (75) that G2 ðwÞw2 ¼ ½GðwÞw2 2 w2 ¼ oð1Þ-0

for w-0

so that Hð0Þ ¼ 1

ð78Þ

On the other hand, on the completely flat confocal ellipsoid, Eq. (76) implies that " !  2 #1=2 2 2 2 1=2 3 L 2 Lx þ Ly þ Lz HðL=wÞ ¼ 1 G2 ðL=wÞ ¼ 1 2 3 w L2

ð79Þ

Using Eqs. (43), (44) and (47), one may check that when k varies from 0 (prolate spheroidal case) to 1 (oblate spheroidal pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi case), the value of HðL=wÞ given by Eq. (79) varies from 1 to 32=ð3p2 þ32Þ C 0:72. Hence HðL=wÞ is always rather close to unity. By Eq. (78), it follows that the variation of the function H(w) over the interval of definition of w is modest. Hence it is again safe to replace it by a constant. This concludes the justification of approximation A3 . With this approximation, Eq. (70) becomes 2

/deq ðrÞSE l C ðAF þBGÞ2 w2 þ

5

2 2 2 2C2 4 2 B H þ þ ðDxy þ D2yz þ D2zx Þ 3 3 3

Gologanu et al. (1993) made the first choice, and Gologanu (1997) and Gologanu et al. (1997) the second.

ð80Þ

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

1033

6.3. Approximate yield function Since dv ¼  23 dw=w2 by Eq. (68), the expression (27)2 of P þ þ ðDÞ may be rewritten in the form Z wmax qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2s dw 2 /deq ðrÞSE l 2 P þ þ ðDÞ C 0 3O wmin w

ð81Þ

where wmin  wðvðl ¼ LÞÞ ¼

1 , 3O=2 þ w=L

wmax  wðvðl ¼ 0ÞÞ ¼

1 3o=2 þ w=L

ð82Þ

2

and /deq ðrÞSE l is given by Eq. (80). From there, the derivation of the approximate yield function associated to the estimate P þ þ ðDÞ of the overall plastic dissipation is based on the following result, established in Appendix B: Gurson’s lemma:6 consider the integral Z u2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du Iða, bÞ  a2 þ b2 u2 2 ð83Þ u u1 where u1 and u2 denote positive parameters. Then the derivatives @I=@a and @I=@b are connected through the following relation independent of a and b:  2   @I 2 @I 1 1 þ cosh ð84Þ  2 2 ¼0 @a u1 u2 @b u1 u2 The procedure involves four steps:

2

1. Using Gurson’s lemma with a  ð23 B2 H þ 2C2 =3þ 43 ðD2xy þ D2yz þ D2zx ÞÞ1=2 , b  AF þ BG, Iða, bÞ  ð3O=2s0 ÞP þ þ ðDÞ  ð3O=2s0 ÞP þ þ ða, bÞ, u  w, u1  wmin and u2  wmax , write Eq. (84) in terms of the derivatives @P þ þ =@a, @P þ þ =@b. 2. Relate the derivatives @P þ þ =@A, @P þ þ =@B, @P þ þ =@C, @P þ þ =@Dxy , @P þ þ =@Dyz , @P þ þ =@Dzx to the derivatives @P þ þ =@a, @P þ þ =@b using the expressions of a and b. 3. Use the formulae found to express ð@P þ þ =@aÞ2 and @P þ þ =@b as linear and quadratic forms, respectively, of the derivatives @P þ þ =@A, @P þ þ =@B, @P þ þ =@C, @P þ þ =@Dxy , @P þ þ =@Dyz , @P þ þ =@Dzx , with coefficients independent of A, B, C, Dxy , Dyz , Dzx . Rewrite Eq. (84) in terms of these derivatives. 4. Use Eqs. (20) and (64)1 to express @P þ þ =@A, @P þ þ =@B, @P þ þ =@C, @P þ þ =@Dxy , @P þ þ =@Dyz , @P þ þ =@Dzx in terms of the derivatives @P þ þ =@Dij , then in terms of the macroscopic stress components Sij using Eq. (28). Rewrite Eq. (84) in terms of these components. The calculations involved are tedious but straightforward. The final output is the following Gurson-like approximate overall yield criterion of the representative cell considered:

QðRÞ LðRÞ FðRÞ  2 þ2ð1 þ gÞðf þ gÞ cosh ð85Þ ð1 þ gÞ2 ðf þ gÞ2 ¼ 0

s0

s0

In this expression:

 The parameter g, which plays the role of a kind of ‘‘second porosity’’ (whence the notation), is given by g



2w 3OL

ð86Þ

Note that the constant w appears in this formula; in Part II, advantage will be taken of the relative freedom of choice of this constant to select a value allowing for a simple, appealing geometric interpretation of the second porosity. LðRÞ is a linear form of the diagonal components of R given by 8 H  OD0xx ðLÞ > > < x 3 Hy  OD0yy ðLÞ ð87Þ , Sh  Hx Sxx þ Hy Syy þHz Szz , LðRÞ  kSh , k  > 2F > :H  OD0 ðLÞ z

zz

Note that by Eq. (10), the coefficients Hx, Hy, Hz here satisfy the equation Hx þ Hy þ Hz ¼ 1

6

This conventional denomination overlooks the fact that the use of this result was only implicit in Gurson (1977)’s work.

ð88Þ

1034

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

 QðRÞ is a quadratic form of the components of R given by QðRÞ 

("

3 2

# " # " # )2 G G G U x  D0xx ðLÞ Sxx þ U y  D0yy ðLÞ Syy þ U z  D0zz ðLÞ Szz F F F

2H 3 þ ðV x Sxx þ V y Syy þV z Szz Þ2 þ 3ðS2xy þ S2yz þ S2zx Þ 2

ð89Þ

Quite remarkably, the approximate yield criterion (85) for general ellipsoidal voids is of the same basic form as that in the GLD model for oblate spheroidal cavities (Gologanu et al., 1994, 1997; Gologanu, 1997). Note however that expression (85) is not fully explicit yet since the constants w, F , G, H have not been ascribed precise values. 7. Conclusion This work, which extends upon previous ones of Gurson (1977), Gologanu et al. (1993, 1994, 1997) and Gologanu (1997), represents a first step in the development of a Gurson-type model for plastic porous solids containing arbitrary ellipsoidal cavities. It was devoted to a limit-analysis of some representative elementary cell in such a medium, namely an ellipsoidal volume containing a confocal ellipsoidal void, and made of some rigid-ideal-plastic von Mises material. This cell was assumed to be loaded through conditions of homogeneous boundary strain rate (Mandel, 1964; Hill, 1967). We first recalled the expression of the incompressible velocity fields recently discovered by Leblond and Gologanu (2008), satisfying conditions of homogeneous strain rate on an arbitrary family of confocal ellipsoids. We then explained the principle of a limit-analysis of the cell considered based on such a family of trial velocity fields. The next step consisted in a thorough asymptotic study, both near infinity and near the origin, of the integrand appearing in the integral expression of the overall plastic dissipation, identical to some weighted average value of the local plastic dissipation over an arbitrary ellipsoid of the family considered. One notable finding was that this average value remains finite near the smallest, completely flat ellipsoid of the family when it is an elliptic or circular disk, that is in the generic and oblate spheroidal cases, but diverges to infinity when it becomes a needle or a point, that is in the prolate spheroidal and spherical cases. This is connected to the fact that it is obviously easier to open an elliptic or circular crack than a needle- or point-shaped void. These results were finally used to define a few reasonable approximations leading to some simplified form of the integrand in the expression of the overall plastic dissipation. This allowed us to express this dissipation as an analytically calculable integral and derive from there an approximate, Gurson-like expression of the overall yield criterion of the cell considered. This yield criterion was found to be of the same basic form as that in the GLD model for oblate spheroidal cavities (Gologanu et al., 1994, 1997; Gologanu, 1997). It remains to ascribe precise expressions to all coefficients appearing in the yield criterion. This will be the object of Part II. Appendix A. Derivation of Leblond and Gologanu (2008)’s family of velocity fields This appendix briefly recalls Leblond and Gologanu (2008)’s study of incompressible velocity fields satisfying conditions of homogeneous strain rate on confocal ellipsoids. An arbitrary family of confocal ellipsoids is represented, in ellipsoidal coordinates, by the family of surfaces E l of constant l. It follows that the velocity fields looked for must be of the form (7) for some family of symmetric second-rank tensors DðlÞ. Calculation of the divergence of such a field yields upon use of Eqs. (6): dDyy 2 dDxx x2 y2 dDzz z2 ðlÞ 2 þ ðlÞ 2 þ ðlÞ 2 div vðrÞ ¼ tr DðlÞ þ T dl dl dl a þl c þl b þl     dDxy dDyz 1 1 1 1 ðlÞ 2 þ 2 ðlÞ 2 þ 2 xy þ yz þ dl dl a þl b þl b þl c þl  

dDzx 1 1 ðlÞ 2 þ 2 zx þ dl c þl a þl Writing the incompressibility condition in the form T div vðrÞ ¼ 0, accounting for the expression (6)4 of T and reordering terms, one gets "

# " # " #   2 dDxx tr DðlÞ 2 2 dDyy tr DðlÞ 2 dDzz tr DðlÞ 2 dDxy 1 1 2 ð l Þ þ þ ð l Þ þ þ ð l Þ þ ðlÞ 2 þ 2 xy x y z þ2 2 2 2 2 2 2 2 2 2 dl a þ l dl c þ l dl a þl b þl ða þ lÞ ðc þ lÞ b þ l dl ðb þ lÞ     dDyz 1 1 dDzx 1 1 ðlÞ 2 þ 2 ðlÞ 2 þ 2 ðA:1Þ þ2 yz þ 2 zx ¼ 0 dl dl c þl a þl b þl c þl

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

1035

In this equation the triplet ðx,y,zÞ is tied to l through Eq. (3)4. However, even if it is not, one can find a triplet of the form ðkx,ky,kzÞ satisfying Eq. (3)4; this triplet must verify Eq. (A.1), and the homogeneity of this equation in x, y, z implies that it must also be satisfied by the triplet ðx,y,zÞ. Hence it is in fact verified for arbitrary triplets ðx,y,zÞ, which implies that 8 8 dDxx tr DðlÞ dDxy > > > > 2 ð l Þ þ ¼ 0, > > > dl > dl ðlÞ ¼ 0 a2 þ l > > > > > > < dDyy < dD tr DðlÞ yz ðlÞ þ 2 ¼ 0, 2 ðlÞ ¼ 0 ðA:2Þ d l > > dl þ l b > > > > > > > > dDzz tr DðlÞ > > dDzx ðlÞ ¼ 0 > :2 : ðlÞ þ 2 ¼ 0, > dl dl c þl The solution 8 > < Dxy ðlÞ Dyz ðlÞ > : D ðlÞ zx

of Eqs. (A.2)4,5,6 is obviously

Dxy  Cst Dyz  Cst Dzx  Cst

  

ðA:3Þ

To solve Eqs. (A.2)1,2,3, the first step is to take their sum to get a differential equation on tr DðlÞ which is readily integrated into A A tr DðlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ vð lÞ 2 ða2 þ lÞðb þ lÞðc2 þ lÞ

ðA:4Þ

where A is an arbitrary constant. The second step is to reinsert this expression into Eqs. (A.2)1,2,3 and integrate to get the following expressions of Dxx ðlÞ, Dyy ðlÞ, Dzz ðlÞ: 8 R þ1 dr > > þ Dxx > Dxx ðlÞ ¼ A l > 2ða2 þ rÞvðrÞ > > > > < R þ1 dr þ Dyy Dyy ðlÞ ¼ A l ðA:5Þ 2 > 2ðb þ rÞvðrÞ > > > > R dr > > > Dzz ðlÞ ¼ A lþ 1 þ Dzz : 2ðc2 þ rÞvðrÞ where the constants Dxx , Dyy , Dzz are a priori arbitrary. But the expression of tr DðlÞ deduced from there ! Z þ1 1 1 1 dr þ Dxx þ Dyy þ Dzz þ þ tr DðlÞ ¼ A a2 þ r b2 þ r b2 þ r 2vðrÞ l Z

þ1

¼ A l

  d 1 A dr þ Dxx þ Dyy þ Dzz ¼ þ Dxx þ Dyy þ Dzz dr vðrÞ vðlÞ

must coincide with that given by Eq. (A.4), which implies that Dxx þ Dyy þ Dzz must be zero; hence the symmetric secondrank tensor D of components Dxx , Dyy , Dzz , Dxy , Dyz , Dzx must be traceless. Gathering these elements, one gets the conclusions mentioned in Section 3 of the text. An alternative method of construction of Leblond and Gologanu (2008)’s velocity fields, suggested to the authors by Kondo (2008), consists in adding to the solution for an ellipsoidal stress-free void in an infinite elastic matrix, that generated by a uniform ‘‘free strain’’ imposed in the void and adjusted so as to satisfy conditions of homogeneous strain on the outer boundary. (In essence, this idea was suggested in Chapter 6 of Monchiet (2006)’s thesis, in the special case of spheroidal voids).

Appendix B. Proof of Gurson’s lemma Differentiation of the expression (83) of the integral Iða, bÞ with respect to a and b and calculation of the integrals yields 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > R u2 b2 u21 1 b2 u2 a du 1 > @I > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ > 1þ 2  1þ 22 ¼ > u1 > @ u u a u a a < 1 2 a2 þ b2 u2     > R u2 @I b > 1 bu2 1 bu1 > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du ¼ sinh ¼ sinh > @b u1 > a a > : a2 þ b2 u2

K. Madou, J.-B. Leblond / J. Mech. Phys. Solids 60 (2012) 1020–1036

1036

It follows that 8  2 > > @I > > > > < @a >   > > @I > > > : cosh @b

¼

¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! b2 u21 b2 u22 b2 u 2 b2 u2 1 1 2 þ 2 1þ 2  1þ 2 1 1 þ 2 2 1þ 2 2 u1 u2 a a a a u1 u2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 b u b u b u1 u2 1 þ 2 1 1 þ 2 2 2

a

a

a

and the left-hand side of Eq. (84) is then trivially checked to be zero. References Benzerga, A., Leblond, J.B., 2010. Ductile fracture by void growth to coalescence. Adv. Appl. Mech. 44, 169–305. Danas, K., Ponte-Castaneda, P., 2008a. A finite-strain model for anisotropic viscoplastic porous media: I—theory. Eur. J. Mech. A/Solids 28, 387–401. Danas, K., Ponte-Castaneda, P., 2008b. A finite-strain model for anisotropic viscoplastic porous media: II—applications. Eur. J. Mech. A/Solids 28, 402–416. Garajeu, M., 1995. Contribution a l’e´tude du comportement non-line´aire de milieux poreux avec ou sans renfort. Ph.D. Thesis, Universite´ de la Me´diterrane´e, Marseille (in French). Garajeu, M., Michel, J.C., Suquet, P., 2000. A micromechanical approach of damage in viscoplastic materials by evolution in size, shape and distribution of voids. Comput. Meth. Appl. Mech. Eng. 183, 223–246. Gologanu, M., 1997. Etude de quelques proble mes de rupture ductile des me´taux. Ph.D. Thesis, Universite´ Pierre et Marie Curie (Paris VI) (in French). Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754. Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. ASME J. Eng. Mater. Technol. 116, 290–297. Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1997. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.), Continuum Micromechanics, Springer-Verlag, New York, pp. 61–130. Gradshteyn, I.S., Ryzhik, I.M., 1980. Table of Integrals, Series, and Products. Academic Press, New York. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rules for porous ductile media. ASME J. Eng. Mater. Technol. 99, 2–15. Hill, R., 1967. The essential structure of constitutive laws of metal composites and polycristals. J. Mech. Phys. Solids 15, 79–95. Idiart, M., Ponte-Castaneda, P., 2005. Second order estimates for nonlinear isotropic composites with spherical pores and rigid particles. C. R. Mec. 333, 147–154. Kaisalam, M., Ponte-Castaneda, P., 1998. A general constitutive theory for linear and nonlinear particulate media with microstructure evolution. J. Mech. Phys. Solids 46, 427–465. Keralavarma, S.M., Benzerga, A., 2008. An approximate yield criterion for anisotropic porous media. C. R. Mec. 336, 685–749. Kondo, D., 2008. Private communication. Leblond, J.B., 2003. Me´canique de la Rupture Fragile et Ductile, Herme s, Lavoisier, Paris (in French). Leblond, J.B., Gologanu, M., 2008. External estimate of the yield surface of an arbitrary ellipsoid containing a confocal void. C. R. Mec. 336, 813–819. Mandel, J., 1964. Contribution the´orique a l’e´tude de l’e´crouissage et des lois de l’e´coulement plastique. In: Proceedings of the 11th International Congress on Applied Mechanics, Munich, pp. 502–509 (in French). Michel, J.C., Suquet, P., 1992. The constitutive law of nonlinear viscous and porous materials. J. Mech. Phys. Solids 40, 783–812. Monchiet, V., 2006. Contribution a la mode´lisation microme´canique de l’endommagement et de la fatigue des me´taux ductiles. Ph.D. Thesis, Universite´ de Lille (in French). Monchiet, V., Cazacu, O., Charkaluk, E., Kondo, D., 2008. Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids. Int. J. Plasticity 24, 1158–1189. Monchiet, V., Charkaluk, E., Kondo, D., 2007. An improvement of Gurson-type models of porous materials by using Eshelby-like trial velocity fields. C. R. Mec. 335, 32–41. Morse, P.M., Feshbach, H., 1953. Methods of Theoretical Physics, Part I. McGraw-Hill, New-York. Ponte-Castaneda, P., 1991. The effective mechanical properties of nonlinear isotropic materials. J. Mech. Phys. Solids 39, 45–71. Ponte-Castaneda, P., 2002. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I. Theory. J. Mech. Phys. Solids 50, 737–757. Willis, J., 1991. On methods for bounding the overall properties of nonlinear composites. J. Mech. Phys. Solids 39, 73–86.