Finite deformation viscoelasticity laws

Abstract

It is proved that two different decompositions of strain may be assigned to every linear viscoelastic solid. In particular, this is true for the so-called three-parameter solids. For this case, the two decompositions of deformation are in a natural way associated with the two well known spring–dashpot models, the first one being a spring in parallel with a Maxwell element and the second model consisting of a spring in series with a Kelvin element. Furthermore, it is shown how the two decompositions of deformation may be generalized to finite deformations in the framework of a multiplicative decomposition of the deformation gradient tensor. This enables to assign to each version of the three-parameter solids a corresponding class of finite deformation counterparts. Note that the finite deformation models are derived so, that the second law of thermodynamics is satisfied for every admissible process. To this end, use is made of the so-called Mandel stress tensor. As one may expect, unlike the linear case, the finite deformation models obtained do not predict identical mechanical responses generally. This is illustrated for the loading case of uniaxial tension–compression. Also, an analysis of the model responses for simple shear is given.