Choices for the reference stress tensor

In this subsection, the choice of the reference stress tensor introduced in relation (2.164) is discussed.

The assumption that the reference stress tensor, ˇσ, is proportional to the deviatoric macroscopic stress tensorσ⁰ has already been made in relation (2.166). Nonetheless, the estimate (2.183) for the effective stress potential requires a prescription for the “magnitude” of the reference stress tensor ˇσ, which is the main focus in this subsection.

As already discussed in prior work (Ponte Casta˜neda, 2002a; Idiart and Ponte Casta˜neda, 2005), it has not yet been possible to “optimize” the choice of the reference stress tensor. For this reason, it is necessary to prescribe a choice for the reference stress tensor ˇσ. The choices proposed in the earlier studies of Ponte Casta˜neda (2002b) and Idiart and Ponte Casta˜neda (2005) are summarized here as

ˇ

σ= (σ⁽¹⁾)⁰= σ⁰

1−f, (2.192)

and

ˇ

σ=σ⁰, (2.193)

where the prime denotes the deviatoric part of the macroscopic stress tensor σ. Both of these two prescriptions become zero in the purely hydrostatic loading (i.e., whenσ⁰=0), and thus lead exactly to the estimate delivered by the “variational” estimate of Ponte Casta˜neda (1991), described in the previous section, in the hydrostatic limit. However, it is known that, for isotropic porous media, the

“variational” estimate is too stiff when compared to the analytical estimate obtained by the solution of the composite sphere (CSA) or cylinder (CCA) assemblages of Hashin (Hashin, 1962; Leblond et al., 1994; Gurson, 1977) subjected to pure hydrostatic loading. In this connection, it is necessary to propose a reference stress tensor, which remains non-zero in the hydrostatic limit. This procedure is described in the following paragraphs.

In this connection, it should be recalled that when the CSA or CCA are subjected to hydrostatic loading, the problem of estimating the effective behavior of the composite reduces in computing the effective behavior of an individual spherical or cylindrical hollow shell, with hydrostatic pressure

boundary conditions in the external boundary and zero traction in the internal boundary. On the other hand, there is no analytical closed-form solution for the analogous problem of ellipsoidal par-ticulate microstructures, although in certain special cases, such as for a confocal, spheroidal shell subjected to a specific axisymmetric loading (Gologanu et al., 1993), it is possible to extract analyti-cal solutions. This suggests the possibility of choosing a reference stress tensor that would allow the

“second-order” estimate (2.183) to coincide with the analytical spherical and cylindrical analytical shell result in the hydrostatic limit, while still producing relatively accurate results for more general ellipsoidal microstructures.

From this viewpoint then and based on the analysis performed in subsection2.2.1, we consider a particulate porous material that comprises ellipsoidal voids with aspect ratiosw1 andw2 randomly distributed in an isotropic matrix phase. The orientation of the principal axes of the voids is described by the unit vectorsn⁽ⁱ⁾(i= 1,2,3). In turn, the two-point correlation function of the voids, describing the relative position of the centers of the voids, has the same shape and orientation with the voids^†. In the special case of purely hydrostatic loading, the effective stress potentialUe describing the behavior of the porous medium should not depend on the orientation vectorsn⁽ⁱ⁾ (i= 1,2,3), such that

Ue(σm;f, w1, w2,n⁽ⁱ⁾) =U(σe m;f, w1, w2). (2.194) Based on this intuitive hypothesis, the proposed approximate estimate for Ue should reduce to the analytical results available in the literature, which can be recovered by considering special limiting procedures for the aspect ratiosw1 andw2 defined in subsection2.2.1. Those cases are itemized:

1. if w1 = w2 = 1, Ue should recover the analytical result delivered when a spherical cavity is subjected to pure hydrostatic loading.

2. if w1 = w2 → ∞ or w1 = 1 and w2 → ∞ or w1 → ∞ and w2 = 1, Ue should recover the analytical result obtained when a cylindrical shell is subjected to pure hydrostatic loading.

In order to arrive at a prescription for the reference stress tensor ˇσ that respects those two aforementioned conditions, it is useful first to express the effective stress potential, Ue, in terms of the mean, applied macroscopic stressσm and the effective flow stressσew(the subscriptwis used to emphasize the dependence on the aspect ratiosw1 andw2) of the porous material, such that

U(σ;e f, w1, w2) = ε˙oeσw

1 +n µ3

2

|σm| e σw

¶_1+n

. (2.195)

In this last expression, use has been made of the fact that the effective stress potentialUeis homogenous of degree n+ 1 inσ, whereas the form (2.195) implies that estimation of eσw fully determines the effective behavior of the porous material, when subjected to hydrostatic loading conditions. It should be emphasized that there exist no exact solutions for the effective flow stresseσwfor a porous material consisting of ellipsoidal voids, except for the corresponding estimates obtained by the “variational”

method, which are known to be stiff in general. However, σew can be computed exactly for the

†This analysis can be generalized to particulate porous media with ellipsoidal voids whose shape and orientation is different than the shape and orientation of their distribution function (Ponte Casta˜neda and Willis, 1995).

above-mentioned two microstructural configurations, i.e., for spherical voids and cylindrical voids with circular cross-section embedded in an incompressible isotropic matrix phase. In the next few paragraphs, we spell out these exact shell solutions, while comparing them with the corresponding estimates derived by the “variational” method in this case of hydrostatic loading.

In this connection, when an aggregate of an infinite hierarchy of sizes of spherical or cylindrical shells is subjected to hydrostatic loading conditions (i.e.,|XΣ| → ∞), the corresponding effective flow stress can be computed exactly by solving the isolated shell problem, and is given by (Michel and Suquet, 1992)

where eσw=1 andσew→∞ are the effective flow stresses of the spherical and the cylindrical shell with an incompressible matrix phase, respectively. Note that for a cylindrical shell with an incompressible isotropic matrix phase that is subjected to plane-strain loading conditions in the 3−direction, the in-plane pressure (σ11+σ22)/2 is identical to the three-dimensional pressure (σ11+σ22+σ33)/3. This is a direct consequence of the fact that for the cylindrical shell subjected to plane-strain loading, the out-of-plane component of the stress isσ33= (σ11+σ22)/2. Thus, definition (2.195) for the effective stress potential is valid for both spherical and cylindrical shells.

On the other hand, the corresponding effective flow stresses delivered by the “variational” proce-dure of Ponte Casta˜neda (1991) (V AR), for porous media containing spherical (denoted withw= 1) and cylindrical (denoted withw→ ∞) voids subjected to purely hydrostatic loading, can be shown to reduce to

Clearly, the estimates (2.196) and (2.197) delivered by the shell problem and the “variational” method, respectively, deviate significantly at low porosities. However, it is interesting to note that the effective flow stresses delivered by the shell problem and the “variational” bound satisfy the following non-trivial relation Making use now of relation (2.198) and due to lack of an analytical estimate in the case of general ellipsoidal microstructures, we make the assumption that the effective flow stressσewof an ellipsoidal shell with arbitrary aspect ratiosw1 andw2may be approximated by

e where eσ_w^var is the corresponding effective flow stress delivered by the “variational” procedure when a porous material with ellipsoidal voids of arbitrary aspect ratios w1 and w2 is subjected to pure hydrostatic loading. Thus, by making use of the general result for the effective stress potential in (2.150) and the definition (2.195), it is easily seen that

˙

which implies that

withMcgiven by (2.143), which is not analytical for general ellipsoidal microstructures. Even so, this simple prescription may also be applied in the cases that the shape and orientation of the distribution function of the pores is not identical with the shape and orientation of the voids (Ponte Casta˜neda and Willis, 1995; Kailasam and Ponte Casta˜neda, 1998). This is a direct consequence of the fact that the “variational” method provides an estimate for general particulate microstructures as described by Ponte Casta˜neda and Willis (1995), and therefore the effective flow stresseσ_w^var can be evaluated in any of these general cases.

To summarize briefly here, the main advantage of prescription (2.199) is that the estimate forUe automatically recovers the two conditions described previously. On the other hand, it should be noted that when the porous material tends to a porous laminate (or porous “sandwich”), the “variational”

estimate can be shown to be exact, i.e., the effective flow stress eσ_w^var for such a microstructure is identically zero. This implies that (2.199) is not exact in this case, in the sense that the ratioeσw/eσ_w^var should go to unity. However, it follows from (2.199)2that the absolute value foreσwis zero and hence equal to the exact value. In this regard, the absolute error introduced by (2.199) for the determination ofeσwin this extreme case of a porous laminate is expected to be small.

Given the estimate (2.199) for the effective behavior of a porous material containing ellipsoidal voids subjected to purely hydrostatic loading, we propose the following ad-hoc choice for the reference stress tensor:

ˇ σ=ξ¡

XΣ,S, sa, n¢

σ⁰, (2.202)

where S is given by (2.166), s_a denotes the set of the microstructural variables defined in (2.16), n is the nonlinear exponent of the matrix phase, and

ξ¡

is a suitably chosen interpolation function. The coefficients t and β are prescribed in an ad-hoc manner to ensure the convexity of the effective stress potential and are detailed in Appendix III. The coefficientsαmandαeqare, in general, functions of the microstructural variablessa, the nonlinearity n of the matrix, the stress tensor S, but not of the stress triaxiality X_Σ. Before discussing the estimation of the coefficientsαm andαeq, it is remarked that the choice (2.202) reduces to ˇσ =σ⁰ for XΣ = 0 (see relation (2.193)), while it remains non-zero in the hydrostatic limit |XΣ| → ∞, in contrast with all the earlier choices proposed in the literature (Ponte Casta˜neda, 2002b; Idiart and Ponte Casta˜neda, 2005), such as the ones given by (2.192) and (2.193). It also guarantees that the effective stress potential is a homogeneous function of degree n+ 1 in the average stress σ for all stress triaxialitiesXΣ.

In particular, the coefficientαmis computed such that the estimate for the effective stress potential Uesom, delivered by the second-order method in relation (2.183), coincides with the approximate solution forUein relation (2.195) in the hydrostatic limit. This condition may be written schematically

as

Uesom→Ue as |XΣ| → ∞ ⇒ αm=αm

¡S, sa, n¢

, (2.204)

which yields a nonlinear algebraic equation forαm.

On the other hand, in order to compute the coefficient αeq, an appropriate estimate for the deviatoric part of the strain-rateD⁰ is needed in the limit asXΣ→ ±∞. Note that for the general case of anisotropic microstructures, the sign (denoted as “sgn”) ofXΣmatters, i.e.,D⁰ gets opposite values in the limits as XΣ→+∞ andXΣ→ −∞. In addition, it is important to remark that the factor αeq has to be independent of the magnitude of the macroscopic stress tensor σ. To ensure this, it is safe to make the following analysis with the normalized strain-rateE⁰ (the prime denotes the deviatoric part), defined in (2.33), which is homogeneous of degree zero inσ by definition.

In this regard, we first note that there exists no exact result for E⁰ in the case of ellipsoidal voids, except for the special cases of spherical or cylindrical with circular cross-section voids and porous sandwiches, whereE⁰=0asXΣ→ ±∞. This result is is exact for finite nonlinearities, i.e., 0< m <1, while the special case of ideally-plasticity will be considered in a separate section later.

On the other hand, it is emphasized that the “variational” method satisfies the above exact results, i.e., E_var⁰ =0as XΣ → ±∞. In addition, it is important to note that, for the case of hydrostatic loading, E⁰ will be a function of the two aspect ratiosw1 andw2, as well as the orientation vectors n⁽ⁱ⁾ (i= 1,2,3) of the ellipsoidal void, in contrast with the effective stress potentialUe, which is only a function of the two aspect ratios. This observation complicates significantly the estimation ofE⁰. However, based on the fact that, in the hydrostatic limit,E_var⁰ , given by (2.153), is exact in the three (including the porous sandwich) limiting cases mentioned earlier and due to absence of any other information on the computation of the deviatoric part of the strain-rate in the hydrostatic limit, the following prescription is adopted forE⁰:

E_som⁰ = D⁰

˙

εo(Γ^som_n /σo)ⁿ →E_var⁰ = D_var⁰

˙

εo(Γ^var_n /σo)ⁿ as XΣ→ ±∞ ∀w1, w2,n⁽ⁱ⁾, (2.205) where Γ^som_n and Γ^var_n are the gauge factors corresponding to the “second-order” (see (2.184)) and the “variational” (see (2.151)) estimates, respectively. In addition, the above prescription guaranties that the slope of the gauge curve, as predicted by the “second-order” method will be identical to the slope of the corresponding gauge curve of the “variational” method in the hydrostatic limit.

The condition (2.205) provides an equation for the estimation of the coefficientαeq, such that αeq=αeq

¡S, sa, n¢

. (2.206)

It should be emphasized that condition (2.205) implies thatαeq does not depend on the magnitude of the macroscopic stress tensorσ, since the termsEsom and Evar are homogeneous of degree zero inσ. While, the computation of the coefficientαmin (2.204) needs to be performed numerically, the evaluation ofαeq can be further simplified to the analytical expression

αeq =α⁻¹_m

"

1 +

3

2σˇeq(Σ)ⁿ−σˇeq(Σ)(2λ)⁻¹+d||−dvar

(1−f) ˆσ_||(Σ)

µ 1 2λ− 1

2λt

¶−1#

. (2.207)

Here, use has been made of definition (2.30) for the normalized macroscopic stress tensor Σ = σ/Γn(σ), as well as of the fact that ˇσeq, ˆσ||and ˆσeq are homogeneous functions of degree one in their arguments. It is further emphasized that all the quantities involved in the above relation must be evaluated in the hydrostatic limitXΣ→ ±∞, i.e., for

Σ= sgn(XΣ) Σ^H_m

¯¯

¯somI, (2.208)

where Σ^H_m

¯¯

¯som is the corresponding normalized hydrostatic stress obtained by the “second-order”

gauge function and is detailed in Appendix V.

Then, the terms in (2.207) associated with the “second-order” method are defined as λ=k^Hµ, µ= ˆσeq(Σ)¹⁻ⁿ/3, λt= 1

3nσˇeq(Σ)¹⁻ⁿ, (2.209) and

d_||=3

2sgn(XΣ) Σ^H_m

¯¯

¯som S· fM

¯¯

¯k=k^H,µ=µ I, (2.210)

withMfgiven by (2.171). Furthermore, for the computation ofk^H in (2.210), it is necessary to solve the nonlinear equation for the anisotropy ratiok, defined by (2.179), in the hydrostatic limit.

On the other hand, the term dvar in (2.207), associated with the “variational” method, is given by

dvar = 9

2sgn(XΣ)³ Σ^H_m

¯¯

¯var

´_n Ã

3I · cMI 1−f

!ⁿ⁻¹