The overall constitutive behaviour of nonlinearly elastic composites

Abstract
This work uses the general framework of Hill for the definition of overall mechanical properties of composite materials undergoing large deformations, together with Ball’s constitutive assumption of polyconvexity, and ideas based on a Hashin–Shtrikman variational structure developed recently by Talbot & Willis for nonlinear problems, to obtain rigorous first-order (depending only on the initial volume fractions of the phases) and second-order (depending also on the overall isotropy of the composite) bounds on the overall properties of composite materials with nonlinearly elastic phases. Although the proposed first-order upper bound, or Voigt bound, agrees with a previous result, the corresponding lower bound, or Reuss bound, is significantly tighter than the previously available result. The proposed second-order bounds are completely new, and are of course tighter than the first-order bounds. Additionally, a ‘self-consistent’ estimate is given for the case when the phases are neo-hookean.

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