Variational linear comparison estimates for elasto-viscoplastic composites with isotropic phases and microstructures

Abstract

Generalizing Hill’s classical formulation for purely elastic and purely viscoplastic composites, Ponte Castañeda (2025) proposed a variational framework for the time-incremental macroscopic response of elasto-viscoplastic (EVP) composites based on Rayleigh’s least dissipation principle. This framework enabled a consistent extension of the Variational Linear Comparison method (Ponte Castañeda, 1991), yielding estimates for the time-dependent macroscopic response of EVP composites in terms of the constitutive response of comparison linear viscoelastic (LVE) composites. In this approach, the viscosities of the LVE phases correspond to the secant viscosities of the EVP phases, evaluated at the instantaneous values of the second moments of the stress or strain-rate fields in the phases of the LVE comparison composite. In the present work, we leverage the estimates of Lahellec et al. (2024) for the macroscopic response and phase second moments in LVE composites — derived via the correspondence principle from the corresponding estimates of Willis (1977) for purely elastic composites — to generate predictions for EVP composites with isotropic phases and particulate microstructures. The new estimates significantly improve upon earlier results for the same class of EVP composites obtained by Ponte Castañeda (2025) using alternative LVE approximations. While the previously used estimates for the LVE composite were exact for the macroscopic response, the corresponding second moment predictions were only approximate, leading to inaccuracies for EVP composites with strong nonlinearities and large heterogeneity contrast. Numerical results for various special cases, including compressible non-well-ordered phases, ideally plastic (rate-independent) behavior and cyclic loadings, illustrate the enhanced capabilities of the new estimates.