Effective elastic properties of a weakly nonlinear particulate composite

The fundamental problem of finding the effective linear and nonlinear elastic properties of a particulate composite subjected to finite elastic deformations is solved when the matrix and particulate phases are assumed to be weakly nonlinear. Weak nonlinearity is adequate to describe common engineering materials and composites loaded in the elastic regime. A nonlinear analogue of the Eshelby solution for the axisymmetric deformation of spherical particles is derived. Based on this solution, explicit asymptotic expressions for the effective linear and third-order (nonlinear) elastic moduli are derived in the case of a dilute distribution of spherical particles based on a general homogenisation methodology proposed by Hill. It is demonstrated that the current solutions correctly recover well-known relationships for the linear material properties of particulate composites as well as previously derived expressions for the effective nonlinear properties for certain special cases considered previously (e.g. hydrostatic loading, and a neo-Hookean matrix containing voids). The obtained theoretical results also agree with limited experimental data available in the literature.