Exact solutions for the effective nonlinear viscoelastic (or elasto-viscoplastic) behaviour of particulate composites under isotropic loading

We consider a composite sphere which consists of a spherical inclusion embedded in a concentric spherical matrix, the inclusion and matrix phases obeying an isotropic nonlinear viscoelastic behaviour. For different isotropic loadings (macroscopic stress or dilatation, swelling of the inclusion phase), the general solutions are shown to depend on the shear stress distribution in the matrix. This shear stress distribution is solution of a first-order nonlinear integro-differential equation, regardless of the inclusion viscoplastic behaviour. When the viscous strain rate potentiel in the matrix is a power-law function of the von Mises equivalent stress, closed-form solutions are given for some special cases clearly identified. Full-field calculations of representative volume elements of particulate composites are also reported. For a moderate volume fraction of inclusions, the composite sphere model turns out to be in excellent agreement with these full-field calculations.