Field statistics in nonlinear composites. I. Theory

Abstract

This work presents a means for extracting the statistics of the local fields in nonlinear composites from the effective potential of suitably perturbed composites. The idea is to introduce a parameter in the local potentials, generally a tensor, such that differentiation of the corresponding effective potential with respect to the parameter yields the volume average of the desired quantity. In particular, this provides a generalization to the nonlinear case of well-known formulas in the context of linear composites, which express phase averages and second moments of the local fields in terms of derivatives of the effective potential. Such expressions are useful since they allow the generation of estimates for the field statistics in nonlinear composites, directly from homogenization estimates for appropriately defined effective potentials. Here, use is made of these expressions in the context of the ‘variational’, ‘tangent second-order’ and ‘second-order’ homogenization methods, to obtain rigorous estimates for the first and second moments of the fields in nonlinear composites. While the variational estimates for these quantities are found to be identical to those proposed in previous works, the tangent second-order and second-order estimates are found be different. In particular, the new estimates for the first moments given in this work are found to be entirely consistent with the corresponding estimates for the macroscopic behaviour. Sample results for two-phase, power-law composites are provided in part II of this work.

Onset of Cavitation in Compressible, Isotropic, Hyperelastic Solids

“…In this connection, it is relevant to recall that field statistics in the problem at hand may be expediently formulated in terms of the total elastic energy of suitably perturbed problems. More specifically, for the case of interest here, it is not difficult to show (see, e.g., Proposition 3.2 in [43] for an analogous proof) that…”

Fiber-reinforced hyperelastic solids: a realizable homogenization constitutive theory

“…For the special case of plane-strain loading conditions, ω N = 0 and expression (43) reduces to an earlier result of deBotton [16].…”

“…An equation for the single value of deformation gradient F (2) in the fiber phase can be obtained following a procedure proposed by Idiart and Ponte Castañeda [43] in the context of small-strain elasticity nonlinear. This procedure exploits the identity…”

The effect of interfaces on the plastic behavior of periodic composites

“…In addition, following arguments given in [24], it can be shown that estimates for the corresponding phase and interface averages of the plastic strain are given by expressions (37)-(38) evaluated at the optimal polarizations.…”

“…In addition to estimating the effective response, it is of interest to estimate the statistics of the plastic strain field. Following arguments given in [24], it can be shown that the nonlinear estimates for the average plastic strain in each phase and on each set of interfaces are given by the corresponding averages in the LCC (see below).…”