The nonlinear elastic response of suspensions of rigid inclusions in rubber: I—An exact result for dilute suspensions

A solution is constructed for the problem of the overall elastic response of ideal (Gaussian or, equivalently, Neo-Hookean) rubber reinforced by a dilute isotropic distribution of rigid particles under arbitrarily large deformations. The derivation makes use of a novel iterative homogenization technique in finite elasticity that allows to construct exact solutions for the homogenization problem of two-phase nonlinear elastic composites with particulate microstructures. The solution is fully explicit for axisymmetric loading, but is otherwise given in terms of an Eikonal partial differential equation in two variables for general loading conditions. In the limit of small deformations, it reduces to the classical Einstein–Smallwood result for dilute suspensions of rigid spherical particles. The solution is further confronted to 3D finite-element simulations for the large-deformation response of a rubber block containing a single rigid spherical inclusion of infinitesimal size. The two results are found to be in good agreement for all loading conditions. We conclude this work by devising a closed-form approximation to the constructed solution which is remarkably accurate and — as elaborated in Part II — proves particularly amenable as a fundamental building block to generate approximate solutions for suspensions with finite concentration of particles.