The Curious Case of 2D Isotropic Incompressible Neo-Hookean Composites

The homogenized behavior of a hyperelastic composite material is characterized by an effective stored-energy function that is functionally very different from the stored-energy functions that describe the underlying hyperelastic constituents. Over the past two decades, several analytical and computational results suggest that the case of isotropic incompressible Neo-Hookean composites in 2D may be the exception. This Note conjectures that the homogenized behavior of an isotropic hyperelastic solid made of incompressible Neo-Hookean materials is itself an incompressible Neo-Hookean material. To support this conjecture, earlier results are summarized, a new Reuss lower bound is derived, and a set of computational results is presented for the physically relevant cases of a Neo-Hookean matrix filled with random isotropic distributions of rigid and liquid circular particles of identical size.