Functional Methods and Effective Potentials for Nonlinear Composites

TL;DR

This paper introduces a variational formulation using functional integrals for local plastic potentials in nonlinear composites, enabling new approximations and extending previous results to higher dimensions and different potentials.

Contribution

It develops a functional integral approach for variational principles in nonlinear composites, allowing computation of effective potentials and extending prior work to higher dimensions and viscoplastic cases.

Findings

Computed second-order weak-disorder expansions of effective potentials.

Extended 3D results of Suquet and Ponte-Castañeda to any dimension.

Calculated viscoplastic potential for uniform strain rates.

Abstract

A formulation of variational principles in terms of functional integrals is proposed for any type of local plastic potentials. The minimization problem is reduced to the computation of a path integral. This integral can be used as a starting point for different approximations. As a first application, it is shown how to compute to second-order the weak-disorder perturbative expansion of the effective potentials in random composite. The three-dimensional results of Suquet and Ponte-Casta\~neda (1993) for the plastic dissipation potential with uniform applied tractions are retrieved and extended to any space dimension, taking correlations into account. In addition, the viscoplastic potential is also computed for uniform strain rates.