Functional methods and effective potentials for non-linear composites

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-Castaneda (Suquet, P., Ponte-Castaneda, P., 1993. Small-contrast perturbation expansions for the effective properties of nonlinear composites. C. R. Acad. Sci. (Paris) Ser. II 317, 1515–1522) 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.