Phase average fields

¯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

2

S·McI, (2.211)

with Σ^H_m

¯¯

¯var denoting the corresponding normalized hydrostatic stress obtained by the “variational”

gauge function, whileMcis given by (2.143). The evaluation of Σ^H_m

¯¯

¯var is detailed in Appendix V.

In summary, relation (2.202), together with relations (2.204) and (2.206) (or (2.207)), completely define the reference stress tensor ˇσ, and thus, result (2.183) can be used to estimate the effective behavior of the viscoplastic porous material. It is important to emphasize at this point that the

“second-order” procedure developed in the previous section is general enough to be able to deal with general ellipsoidal microstructures, in contrast with earlier models proposed in the literature (Gologanu et al., 1993; Leblond et al., 1994; Gˇarˇajeu et al., 2000; Flandi and Leblond, 2005) that are constrained to spheroidal microstructures and axisymmetric loading conditions aligned with the pore symmetry axis.

2.6.3 Phase average fields

In this subsection, the focus is on estimating the average stress, strain-rate and spin in each phase of the porous material. This is necessary for the prediction of the evolution of the microstructural variables to be considered in the next section. Because of the presence of the vacuous phase the average stress tensors are trivially given by

(1−f)σ⁽¹⁾=σ, σ⁽²⁾=0. (2.212)

On the other hand, the estimation of the average strain-rate and spin in each phase is non-trivial.

As already discussed in the context of relation (2.170), use has been made of the Willis (1978)

estimates for the determination of the effective behavior of the LCC. One of the main results of this procedure is that the fields in the inclusion phase (vacuous phase in the present study) of the LCC are uniform, which can be written asDL(x) =D⁽²⁾_L for all xin Ω⁽²⁾ in the LCC. Because of this result, the corresponding “second-order” estimate forD⁽²⁾ in the nonlinear porous material also delivers uniform fields in the voids. However, the work of Idiart and Ponte Casta˜neda (2007) has shown that in principleD⁽²⁾6=D⁽²⁾_L for a general reference stress tensor ˇσ. On the other hand, the author is not aware of corresponding results concerning the estimation of the average spin Ω⁽²⁾ in the vacuous phase, apart from the fact that the spin is uniform in the inclusion phase of the LCC, i.e., ΩL(x) = Ω⁽²⁾_L , and the nonlinear composite due to the use of the Willis estimates. Regardless of any prescription for the estimation of the average strain-rate and spin in the vacuous phase, the following two relations for the macroscopic and the phase average quantities must always hold, both in the nonlinear composite and the LCC, such that

D= (1−f)D⁽¹⁾+fD⁽²⁾, DL= (1−f)D⁽¹⁾_L +fD⁽²⁾_L , (2.213) and

Ω= (1−f)Ω⁽¹⁾+fΩ⁽²⁾, ΩL= (1−f)Ω⁽¹⁾_L +fΩ⁽²⁾_L . (2.214) In the following, the discussion will be focused on estimating the average strain-rate and spin in the vacuous phase.

Specifically, making use of the identities (2.213) and the incompressibility of the matrix phase, the hydrostatic part of the macroscopic strain-rateDm=Dii/3 (i= 1,2,3), and the average strain-rate in the voids,D⁽²⁾_m are related through

D⁽²⁾_m = 1

f Dm, and D⁽¹⁾_m = 0. (2.215)

It is easy to verify thatD⁽²⁾_m cannot be equal to (D⁽²⁾_m)LsinceDm6= (Dm)L in relation (2.186). The result (2.215) is exact and no approximations are involved.

On the other hand, the computation of the deviatoric part of the average strain-rate,D⁽²⁾⁰, in the vacuous phase is non-trivial. In the work of Idiart and Ponte Casta˜neda (2007), the idea of computing D⁽²⁾ lies in perturbing the local nonlinear phase stress potential, given by (2.23), with respect to a constant polarization type fieldp, solve the perturbed problem through the homogenization procedure and then consider the derivative with respect to p, while letting it go to zero. Consequently, the expression for the average strain-rate in the vacuous phase can be shown to be of the form

D⁽²⁾=D⁽²⁾_L −1−f f g ∂σˇ

∂p

¯¯

¯¯ p→0

, (2.216)

withg given by relation (2.188) andD⁽²⁾_L by (2.127), such that D⁽²⁾_L =A⁽²⁾DL+a⁽²⁾= 1

f (Mf−M)σ+η= 1

1−fQ⁻¹σ+η, (2.217) whereA⁽²⁾,a⁽²⁾ andDL are given by relations (2.121), (2.122) and (2.187), respectively.

As already discussed previously, the evaluation of the reference stress tensor requires the com-putation of the effective stress potential of the shell problem. However, solving the shell problem

assuming a perturbed nonlinear potential law for the matrix phase is too complicated and probably can be achieved only by numerical calculations for general ellipsoidal microstructures. Hence, we are left with the option to evaluate the deviatoric part ofD⁽²⁾ approximately, while the hydrostatic part of D⁽²⁾ is obtained by relation (2.215) exactly within the approximation intrinsic to the method.

Thus, we set

D⁽²⁾⁰ =D⁽²⁾_L ⁰, (2.218)

and attempt to get an estimate of the resulting error introduced by ignoring the second term in relation (2.216). First of all, it is important to mention that the reference stress tensor ˇσ prescribed in relation (2.202) could depend onp only through the coefficients αm and αeq. This implies that relation (2.218) is exact in the low triaxiality limit XΣ= 0, since, in this case, the reference stress tensor becomes equal to the deviatoric part of the applied macroscopic stress tensor σ⁰ and hence does not depend onpby definition. Under this observation, it is expected that prescription (2.218) is sufficiently accurate for low triaxial loadings.

Conversely, at high triaxiality loadings the error is expected to be maximum. However, estimating this error is possible only in special cases. In particular, in the cases of isotropic and transversely isotropic porous media that are subjected to pure and in-plane hydrostatic loading, respectively, the exact result readsD⁽²⁾⁰ =0. It can be verified that in those cases use of relation (2.218) introduces an error less than 1% in the computation of the deviatoric part ofD⁽²⁾. Moreover, in this high triaxiality regime, it is expected (it will be verified in the results section) that the hydrostatic part ofD, i.e., D⁽²⁾_m, is predominant and controls the effective behavior of the porous material. Therefore, evaluation of the average strain-rate in the inclusion from relation (2.218) is expected to be sufficientlyaccurate andsimple for the purposes of this work.

In summary the average strain-rate tensor in the voids will be taken to be approximated by the expression

D⁽²⁾ = 1

f D_mI+D⁽²⁾_L ⁰= 1

f D_mI+K µ 1

1−fQ⁻¹σ+η

¶

, (2.219)

whereK denotes the fourth-order shear projection tensor defined by (2.140).

Following a similar line of thought, and due to lack of results for the estimation of the phase average spin of nonlinear materials, the assumption will be made here that the average spin in the vacuous phaseΩ⁽²⁾in the nonlinear porous material can be approximated by the average spin in the pores of the LCC,Ω⁽²⁾_L . This approximation is written as

Ω⁽²⁾=Ω⁽²⁾_L =Ω+Π L Q⁻¹σ, (2.220) where Ω⁽²⁾_L has already been defined in (2.130), while Q and Π are given by relation (2.59) and (2.74), respectively. Note that similar to the “variational” method, the limit of incompressibility (i.e., κ → ∞) needs to be considered for the evaluation of the term Π L (see Appendix I). Note further that expression (2.220) is identical to (2.154)4 obtained for the average spin in the vacuous phase in the context of the “variational” method.