Evolution Equations for Non-Simple Viscoelastic Solids

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.