Constitutive modelling for nonlinearly viscoelastic solids where strain depends on stress and stress-rate

Abstract

This work focuses on the derivation and investigation of constitutive relations depending on the stress, the rate of the stress and the temperature, describing the response of viscoelastic solids. Using a thermodynamical approach, we derive conditions to be satisfied in order for the material response to be stable. Under the assumption that the strain function depends on the stress, the stress-rate and the temperature as separate variables, we further investigate these conditions both for large and small strains for various scenarios including isotropic bodies. For each of these cases, there certainly exists an analogy between the current approach and the classical approach where the stress is given in terms of the strain and its rates. We also study the corresponding boundary value problems of the extension/compression of a cylinder, the biaxial extension of a thin plate, and the triaxial compression of a slab, for both of the cases of large deformations and small strains. We look at the case of transversely isotropic viscoelastic solids, when the body is assumed to be inextensible, as well as the case of a viscoelastic body whose mechanical properties depend on the spherical stress (the pressure). Finally, we discuss the potential applications of these constitutive theories.