A generalisation of the Mooney-Rivlin model to finite linear viscoelasticity

The constraint condition of incompressibility leads to consequences which are of physical interest in view of a thermodynamically consistent material modelling. Some of these are discussed within the concept of finite linear visco elasticity. Two possibilities are presented to generalise the familiar Maxwell-model to finite strains; both tensor-valued differential equations are integrated to yield the present Cauchy stress as a functional of the relative Piola or Green strain history. Both types of Maxwell-models are related to a free energy functional such that the dissipation inequality is satisfied. The stress and energy functionals are generalised to incorporate arbitrary relaxation functions; the only restriction for thermodynamic consistency is that the relaxation functions have a negative slope and a positive curvature. The linear combination of the two types of energy functionals can be understood to be a generalisation of the Mooney-Rivlin model to viscoelasticity. In order to motivate a concrete representation of relaxation functions, a series of Maxwell models in parallel may be considered, which leads to a discrete relaxation spectrum. However, continuous relaxation spectra might be more convenient in view of experimental observations.