Replacement relations for a viscoelastic material containing multiple inhomogeneities

Classical replacement relations provide a connection between elastic properties of a porous material and the same material with fluid or solid infill of the porous space. We derive such relations for the case when both skeleton and infill materials are viscoelastic. For this goal, we use formalism of compliance/stiffness contribution tensors that lead to replacement relations for anisotropic elastic materials that, in the case of isotropy, coincide with classical Gassmann equation (Gassmann, 1951). We rewrite these relations using creep and relaxation contribution tensors that describe effect of individual inhomogeneities on the overall viscoelastic properties of a heterogeneous material. Explicit analytical expressions are obtained using elastic-viscoelastic correspondence principle and Laplace transform. It becomes possible when viscoelastic properties are expressed in terms of fraction-exponential operators of Scott Blair-Rabotnov. Results are obtained in closed explicit form.