Exact results for weakly nonlinear composites and implications for homogenization methods

Joshua Furer,Pedro Ponte Castañeda,Martin Idiart

doi:10.5802/crmeca.66

Abstract

Weakly nonlinear composite conductors are characterized by position-dependent dissipation potentials expressible as an additive composition of a quadratic potential and a nonquadratic potential weighted by a small parameter. This additive form carries over to the effective dissipation potential of the composite when expanded to first order in the small parameter. However, the first-order correction of this asymptotic expansion depends only on the zeroth-order values of the local fields, namely, the local fields within the perfectly linear composite conductor. This asymptotic expansion is exploited to derive the exact effective conductivity of a composite cylinder assemblage exhibiting weak nonlinearity of the power-law type (i.e., power law with exponent m=1+δ, such that |δ|≪1), and found to be identical (to first order in δ) to the corresponding asymptotic result for sequentially laminated composites of infinite rank. These exact results are used to assess the capabilities of more general nonlinear homogenization methods making use of the properties of optimally selected linear comparison composites.

Highlights

The development of multiscale approximations for the mechanical and physical properties of disordered solids greatly benefits from the identification of specific material systems whose properties at different length scales can be linked exactly, as they provide guidance and useful benchmarks for evaluating the relative merits of competing schemes
Nonlinear composite conductors are characterized by position-dependent dissipation potentials expressible as an additive composition of a quadratic potential and a nonquadratic potential weighted by a small parameter
The first-order correction of this asymptotic expansion depends only on the zeroth-order values of the local fields, namely, the local fields within the perfectly linear composite conductor. This asymptotic expansion is exploited to derive the exact effective conductivity of a composite cylinder assemblage exhibiting weak nonlinearity of the power-law type, and found to be identical to the corresponding asymptotic result for sequentially laminated composites of infinite rank. These exact results are used to assess the capabilities of more general nonlinear homogenization methods making use of the properties of optimally selected linear comparison composites

Summary

Introduction

The development of multiscale approximations for the mechanical and physical properties of disordered solids greatly benefits from the identification of specific material systems whose properties at different length scales can be linked exactly, as they provide guidance and useful benchmarks for evaluating the relative merits of competing schemes Several classes of such solvable systems have been identified when the constitutive phases exhibit linear responses. It was found that these bounds and estimates were not able to reproduce the exact estimates of [34] for small heterogeneity contrast For these reasons, Ponte Castañeda [35] proposed an alternative “tangent second-order” (TSO) variational approach making use of more general LCCs, C. It is shown that the exact estimate of [15] for the corresponding class of sequentially laminated nonlinear composite conductors agrees exactly to first order in the weakly nonlinear limit with the CCA result

Effective behavior of weakly nonlinear composites

Composite cylinder assemblages with power-law dissipation

Infinite-rank laminates with power-law dissipation

Results and discussion

Closing remarks