A second-order homogenization method in finite elasticity and applications to black-filled elastomers

This work is concerned with the development of an analytical method for estimating the macroscopic behavior of heterogeneous elastic systems subjected to finite deformations. The objective is to generate variational estimates for the effective or homogenized stored-energy function of hyperelastic composites, which will be accomplished by means of a suitable generalization of the “second-order procedure” of Ponte Castañeda (Ponte Castañeda, P., 1996. J. Mech. Phys. Solids 44, 827–862). The key idea in this method is the introduction of an optimally chosen “linear thermoelastic comparison composite,” which can then be used to convert available homogenization estimates for linear systems directly into new estimates for nonlinear composites. To illustrate the use of the method, an application is given for carbon-black filled elastomers and estimates analogous to the well-known Hashin–Shtrikman and self-consistent estimates for linear-elastic composites are generated.