A class of coherent potentials for two-phase creeping solids

Abstract

A class of thermodynamic potentials for two-phase creeping solids accounting for microstructural disorder and constitutive nonlinearity is proposed. The potentials depend explicitly on the convex dissipation potentials of the constituent phases and on the one- and two-point spatial correlations of particle arrangement. Their derivation relies on a known class of solvable microgeometries confected by means of a nonlinear differential scheme and sequential laminations, whose effective potentials solve a nonlinear Hamilton–Jacobi equation with the relevant macroscopic field and the inclusion concentration playing the respective roles of ‘spatial’ and ‘temporal’ variables. A variational representation of these exact potentials is exploited to construct approximate potentials bearing lower algorithmic complexity. These approximate potentials remain coherent in the sense that they simultaneously comply with all pertinent bounds for disordered media, recover exact expansions to second order in the heterogeneity contrast and preserve constitutive convexity across length scales. The potentials provide descriptions not only for the effective properties but also for the full probability distributions of the underlying mechanical fields. Sample results for isotropic mixtures exhibiting power-law creep are reported. Comparisons with available results for material models with spherical microgeometries confirm the expected aptness of the proposed potentials to describe steady creep flow in solid dispersions with rounded inclusions or voids.