Exact results for weakly nonlinear composites and implications for homogenization methods

J. FURER; M. I. IDIART; P. PONTE CASTAÑEDA

COMPTES RENDUS MECANIQUE

ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER

Lugar: Paris; Año: 2020 vol. 348 p. 893 - 909

1631-0721

Weakly nonlinear composite conductors are characterized by position-dependent dissipation potentials expressible as an additive composition of a quadratic potential and a non-quadratic 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 $m=1+delta$, such that $|delta|