In this talk I will present applications of a recently developed “second-order” homogenization technique to generate estimates for the effective behavior, microstructure evolution and loss of ellipticity in reinforced elastomers at finite strains. Two special cases will be considered for illustrative purposes: fiber-reinforced rubbers and thermoplastic elastomers (TPEs), which are nanostructured block copolymer systems with a hard glassy phase serving to provide reinforcement in a softer rubbery matrix phase. The main concept behind the method is the introduction of an optimally selected “linear comparison composite,” which can then be used to convert standard linear homogenization estimates into new estimates for the nonlinear hyperelastic composite. Explicit results are provided for materials with isotropic and strongly elliptic constituents. It is found that their overall behavior may lose ellipticity at sufficiently large deformations, corresponding to the possible development of shear band-type instabilities. The source of these macroscopic instabilities has been identified with the evolution of the microstructure, which, under appropriate loading conditions, can induce “geometric softening” leading to the overall loss of ellipticity.

In this talk I will present applications of a recently developed “second-order” homogenization technique to generate estimates for the effective behavior, microstructure evolution and loss of ellipticity in reinforced elastomers at finite strains. Two special cases will be considered for illustrative purposes: fiber-reinforced rubbers and thermoplastic elastomers (TPEs), which are nanostructured block copolymer systems with a hard glassy phase serving to provide reinforcement in a softer rubbery matrix phase. The main concept behind the method is the introduction of an optimally selected “linear comparison composite,” which can then be used to convert standard linear homogenization estimates into new estimates for the nonlinear hyperelastic composite. Explicit results are provided for materials with isotropic and strongly elliptic constituents. It is found that their overall behavior may lose ellipticity at sufficiently large deformations, corresponding to the possible development of shear band-type instabilities. The source of these macroscopic instabilities has been identified with the evolution of the microstructure, which, under appropriate loading conditions, can induce “geometric softening” leading to the overall loss of ellipticity.