Intrinsic modulus and strain coefficients in dilute composites with a Neo-Hookean elastic matrix

A finite element modelling of dilute elastomer composites based on a Neo-Hookean elastic matrix and rigid spherical particles embedded within the matrix was performed. In particular, the deformation field in vicinity of a sphere was simulated and numerical homogenization has been used to obtain the effective modulus of the composite ${\mu }_{\text{eff}}$ for different applied extension and compression ratios. At small deformations the well-known Smallwood result for the composite is reproduced: ${\mu }_{\text{eff}}=(1+\left[\mu \right]\phi ){\mu }_0$ with the intrinsic modulus $\left[\mu \right]=2.500$. Here $\phi$ is the volume fraction of particles and ${\mu }_0$ is the modulus of the matrix solid. However at larger deformations higher values of the intrinsic modulus $\left[\mu \right]$ are obtained, which increase quadratically with the applied true strain. The homogenization procedure allowed to extract the intrinsic strain coefficients which are mirrored around the undeformed state for principle extension and compression axes. Utilizing the simulation results, stress and strain modifications of the Neo-Hookean strain energy function for dilute composites are proposed.