A pattern-based method for bounding the effective response of a nonlinear composite

This paper deals with the prediction of the effective properties of nonlinear composites. Rather than bounding the effective energy, this work aims at bounding directly the effective stress-strain response, by extending a method originally introduced by Milton and Serkov (J. Mech. Phys. Solids 48 (2000) 1295) and recently refined by Talbot and Willis (Proc. Roy. Soc. 460 (2004) 2705). In this paper, bounding the effective response is achieved by introducing a linear comparison composite with the same micro-geometry as the given nonlinear composite, as Ponte Castañeda (J. Mech. Phys. Solids 39 (1991) 45) did for the energy. It is found that any lower bound for the energy of the linear comparison composite generates a corresponding bound for the stress-strain response of the nonlinear composite. A selection of examples is presented to illustrate the method and compare the bounds obtained with existing results.