Mean-field descriptions for the viscoelastic response of thermorheologically complex reinforced solids

Mean-field descriptions for the thermo-viscoelastic response of reinforced solids undergoing small deformations but large temperature changes are presented. The descriptions follow from an approximate homogenization scheme that identifies macroscopic internal variables with low-order statistics of the microscopic internal variable fields as a result of a variational model reduction. Unlike descriptions based on the correspondence principle, these descriptions can account for thermorheologically complex constitutive laws with multiple internal times and/or temperature-dependent stiffnesses, and can provide information not only on the macroscopic response but also on the statistics of the microscopic mechanical fields; it is further demonstrated that the descriptions can even accommodate a special class of hereditary laws often employed for polymeric materials. Simple expressions for material systems with elastically rigid but thermally dilatant reinforcements are provided. By way of example, reduced descriptions are presented for a special class of material systems for which the thermomechanical response can be computed exactly by means of the correspondence principle. In the case of isotropic particle-reinforced solids under hydrostatic loadings, the reduced descriptions can reproduce the exact response identically; in the case of transversely isotropic fiber-reinforced solids subject to monotonic coolings, the reduced and exact descriptions provide indistinguishable macroscopic strains and stresses for the entire range of temperature drops and cooling rates considered.