High-rank sequential laminates

In this section, we study a special class of nonlinear composites, known as “high-rank sequentially laminates”, to estimate the effective behavior of isotropic viscoplastic porous media. In this context, we recall that the “second-order” and the “variational” estimates, discussed in the previous chapter, involvetwoapproximations: thelinearization of the nonlinear phases and thehomogenization of the LCC (relation (2.170)). For the second approximation, use was made of the Willis (1998) estimates (2.171). These linear estimates are known (Frankfort and Mura, 1986; Milton, 2002) to be exact for composites with a special class of “sequentially laminated” microstructures. For this reason, nonlinear sequential laminates are particularly appropriate to assess the accuracy of the “second-order” method and, in general, of any LCC-based homogenization method (such as the “variational”

method of Ponte Casta˜neda (1991)), on condition that the Willis estimates be used for the LCC. In

this section, exact results for this special class ofnonlinear sequential laminates are provided.

A sequential laminate is an iterative construction obtained by layering laminated materials (which in turn have been obtained from lower-order lamination procedures) with other laminated materials, or directly with the homogeneous phases that make up the composite, in such a way as to produce hierarchical microstructures of increasing complexity (e.g., Milton, 2002). Therank of the laminate refers to the number of layering operations required to reach the final sequential laminate. Of the many possible types of sequential laminates, we restrict attention to porous sequential laminates formed by layering at every step a porous laminate with the matrix phase (denoted as phase 1).

Thus, a rank-1 laminate corresponds to a simple laminate with a given layering directionn⁽¹⁾, with matrix and porous phases in proportions 1−f1 and f1. In turn, a rank-2 laminate is constructed by layering the rank-1 laminate with the matrix phase, in a different layering direction n⁽²⁾, in proportionsf2and 1−f2, respectively. Rank-M laminates are obtained by iterating this procedureM times, layering the rank-(M−1) laminate with the matrix phase in the directionn^(M), in proportions fM and 1−fM, respectively. A key point in this procedure is that the length scale of theembedded laminate is assumed to be much smaller than the length scale of the embedding laminate. This assumption allows to regard the rank-(M−1) laminate in the rank-M laminate as a homogeneous phase, so that available expressions for the effective potential of simple laminates (e.g., deBotton and Ponte Casta˜neda, 1992) can be used at each step of the process to obtain an exact expression for the effective potential of the rank-M sequential laminate (e.g., Ponte Casta˜neda, 1992; deBotton and Hariton, 2002). From this construction process, it follows that the microstructure of these sequential laminates can be regarded asrandomandparticulate, with phase 1 playing the role of the (continuous) matrix phase embedding the (discontinuous) porous phase. A distinctive feature of this very special class of porous materials is that the strain-rate and stress fields in theinclusion phase (in this case, the pores, denoted as phase 2) are uniform.

The effective stress potential of the resulting rank-M porous laminate can be shown to be (de-Botton and Hariton, 2002; Idiart, 2006)

UeM(σ) = min

w_j⁽ⁱ⁾

σ⁽²⁾=0

XM

i=1

(1−fi)



 YM

j=i+1

fj



 U⁽¹⁾³ σ⁽¹⁾_i ´

, (3.48)

where U⁽¹⁾ is the matrix potential given by relation (2.23), and the average stress tensors in the matrix phase,σ⁽¹⁾_i (i= 1, ...M), and the pore phase,σ⁽²⁾, are given by

σ⁽¹⁾_i = σ+fi w⁽ⁱ⁾− XM

j=i+1

(1−fj)w^(j), (3.49)

σ⁽²⁾ = σ− XM

i=1

(1−fi)w⁽ⁱ⁾. (3.50)

In these expressions, thew⁽ⁱ⁾,i= 1, ..., M, are second-order tensors of the form

w⁽ⁱ⁾=w₁⁽ⁱ⁾ m⁽ⁱ⁾₁ ⊗m⁽ⁱ⁾₁ +w₂⁽ⁱ⁾m⁽ⁱ⁾₂ ⊗m⁽ⁱ⁾₂ +w⁽ⁱ⁾₃ m⁽ⁱ⁾₁ ⊗sm⁽ⁱ⁾₂ , (3.51) wherem⁽ⁱ⁾₁ andm⁽ⁱ⁾₂ are two orthogonal vectors lying on the plane with normaln⁽ⁱ⁾, and⊗sdenotes the symmetric part of the outer product. The total porosityf in this rank-M laminate is given in

terms of the partial volume fractionsfiby f =

YM

i=1

f_i. (3.52)

Thus, expression (3.48) requires the solution of a 3M-dimensional convex minimization with respect to the scalar variables wα⁽ⁱ⁾ (i = 1, .., M, α= 1,2,3), which, for a given set of fi and n⁽ⁱ⁾ and macroscopic stress σ, can be solved numerically using standard numerical techniques. This minimization problem is constrained by the fact that the (uniform) stress in the porous phaseσ⁽²⁾, as given by (3.50), must be zero. This constraint (in the variables w_j⁽ⁱ⁾) can be enforced in two different ways. One way is to enforce that the magnitude of the second-order tensorσ⁽²⁾ be zero, in which case there is a singlenon-linear constraint, while a different, equivalent way is to enforce that each component ofσ⁽²⁾ be zero, in which case there are sixlinear constraints (see (3.50)). The latter approach has been found easier to implement and was therefore adopted in this work.

It is important to note that the effective behavior of the sequential laminates considered here, unlike that of typical nonlinear composites, does not depend on all the details of the microstructure, but only on partial information of it in the form of the volume fractionsfiand lamination directions n⁽ⁱ⁾. Of particular interest here are porous materials exhibiting overallisotropicsymmetry. In general, the effective potential (3.48) will be anisotropic, even if the matrix potential is isotropic. However, appropriate lamination sequences, i.e., particular choices offi andn⁽ⁱ⁾, can be found such that the effective potential (3.48) tends to be isotropic as the rankM increases (deBotton and Hariton, 2002).

To that end, the following lamination sequence has been adopted in this work:

fi= 1−_Mⁱ (1−f)

1−ⁱ⁻¹_M (1−f), (3.53)

and

n⁽ⁱ⁾= sinψisinφi e1+ cosψisinφi e2+ cosφi e3, (3.54) wheref is the prescribed porosity in the rank-M laminate, and the anglesψiandφi, which determine thei^thdirection (i= 1, ..., M) of lamination relative to a reference basis{eα}, are given by

φj+kMη = arccoshj, hj = 2 j−1

M_η−1−1, (3.55)

j= 1, ..., Mη, k= 0, ..., η−1, ψj+kMη =



ψj−1+ 3.6 pMη

q 1 1−h²_j



mod 2π, (3.56)

j= 2, ..., Mη−1, ψ1=ψMη= 0.

In these expressions,η andMη are two integers such that the rank of the laminate isM =ηMη. The set of angles (3.55)-(3.56) corresponds toMηlamination directions (3.54), uniformly distributed on the unit sphere (Saff et al., 1977), withη laminations for each direction. It has been verified numerically that, for this specific lamination sequence, the effective potential (3.48) becomes progressively less sensitive to the orientation of the principal axes ofσ as the parametersMη andηincrease, meaning that the effective potential tends to be more isotropic with increasing rank. The results provided in the next chapters correspond toM = 1500 withMη= 50 andη= 30.

Finally, the macroscopic strain-rate is obtained by differentiating (3.48) with respect toσ. Noting that the expression is stationary with respect to the variablesw_j⁽ⁱ⁾, we have that

D= XM

i=1

(1−fi)



 YM

j=i+1

fj



 ∂U⁽¹⁾

∂σ

³ σ⁽¹⁾_i ´

+λ⁽²⁾, (3.57)

where the second-order tensorλ⁽²⁾ is the optimal Lagrange multiplier associated with the traction-free constraint in (3.48). The expressions for the computation of the phase average quantities are not included here because they are too cumbersome to be shown here. However, a complete derivation of these results are given in the thesis of Idiart (2006).

In summary, the estimates obtained by the high-rank sequential laminates are very useful in the sense that they can be compared with the homogenization methods presented in the previous chapter and the Gurson-type models. This comparison is particularly pertinent in view of the fact that it has been recently shown (Idiart, 2007) that power-law, porous, sequential laminates with isotropic microstructures reproduce exactly the hydrostatic behavior of the composite-sphere assemblage, as described by expressions (2.196). Furthermore, as already stated in the beginning of the section, the linear estimates of Willis (1978) are exact for the high-rank sequential laminates. In this regard, comparing the “second-order” and the “variational” methods (which make use of the Willis estimates to solve the linear comparison composite) with the high-rank sequential laminates provides a good estimate for the accuracy of these homogenization theories.