Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates

Estimates for two-phase nonlinear conductors via iterated homogenization

“…In §3, we obtain an exact expression for the effective energy-density function of finite-rank, sequentially laminated composites with nonlinear phases, and remark thatunlike the corresponding energies for their linear counterparts-they actually depend on additional microstructural information beyond the H-measures. However, consistent with earlier observations by deBotton [11] and Idiart [10], it is found that, in the limit of dilute concentrations of the 'matrix' phase, the expression for the energies of the nonlinear laminates depends only on the H-measures of the microstructures. As a consequence, we are able to iterate this result in the volume fraction of the matrix phase-again as proposed by deBotton [11] and Idiart [10]-to obtain an exact expression for the energies of these sequentially iterated composites consisting of inclusions of the dilute sequentially layered materials that are in turn repeatedly coated with infinitesimal layers of the matrix material until the desired final volume fraction of the matrix is reached.…”

“…However, consistent with earlier observations by deBotton [11] and Idiart [10], it is found that, in the limit of dilute concentrations of the 'matrix' phase, the expression for the energies of the nonlinear laminates depends only on the H-measures of the microstructures. As a consequence, we are able to iterate this result in the volume fraction of the matrix phase-again as proposed by deBotton [11] and Idiart [10]-to obtain an exact expression for the energies of these sequentially iterated composites consisting of inclusions of the dilute sequentially layered materials that are in turn repeatedly coated with infinitesimal layers of the matrix material until the desired final volume fraction of the matrix is reached. Note that although this second iteration on the volume fraction is essentially a differential scheme, it is not of the standard type, because it starts with 100 per cent of the inclusion phase and it is the matrix phase that is added incrementally (instead of starting with the matrix phase and incrementing the volume fraction of the inclusion phase, as in the usual differential self-consistent scheme).…”

“…Finally, in §5, we demonstrate explicitly that the effective behaviour of the finite-concentration laminates with the same finite concentrations of the phases and (transversely) isotropic H-measures is not only anisotropic, but in fact depends on the order of the lamination sequence. As a consequence of the anisotropy of the finite-concentration laminates, we are able to demonstrate explicitly that the (dilutely) iterated laminate construction does not provide a bound for the class of nonlinear composite conductors with prescribed H-measures, contradicting a conjecture made by Idiart [10] to this effect. We conclude the paper with some remarks concerning the possible extremal character of the estimates.…”

“…To that end, we follow a lamination sequence used by Idiart [10] in the context of viscoplasticity. The sequence is formed by layering at every step a laminate with the matrix material, here identified with r = 1.…”

“…Making repeated use of the formula (3.1) for simple laminates, it can be shown-see Idiart [10] for a derivation in the context of viscoplasticity-that the effective potentialw M of the rank-M laminate is given bỹ…”