Tangent second-order estimates for the large-strain, macroscopic response of particle-reinforced elastomers

An approximate homogenization method is proposed and used to obtain estimates for the effective constitutive behavior and associated microstructure evolution in hyperelastic composites undergoing finite-strain deformations. The method is a modified version of the "tangent second-order" procedure (Ponte Castañeda and Tiberio in J. Mech. Phys. Solids 48:1389, 2000), and can be used to provide estimates for the nonlinear elastic composites in terms of corresponding estimates for suitably chosen "linear comparison composites". The method makes use of the "tangent" moduli of the phases, evaluated at suitable averages of the deformation gradient, and yields a constitutive relation accounting for the evolution of characteristic features of the underlying microstructure in the composites, when subjected to large deformations. Satisfaction of the exact, macroscopic incompressibility constraint is ensured by means of an energy decoupling approximation splitting the elastic energy into a purely "distortional" component, together with a "dilatational" component. The method is applied to elastomers containing random distributions of aligned, rigid, ellipsoidal inclusions, and explicit analytical estimates are obtained for the special case of spherical inclusions distributed isotropically in an incompressible neo-Hookean matrix. In addition, the method is also applied to two-dimensional composites with random distributions of aligned, elliptical fibers, and the results are compared with corresponding results of earlier homogenization estimates and finite element simulations.