Closed-form stress solutions for incompressible visco-hyperelastic solids in uniaxial extension

This paper is concerned with the theory of incompressible visco-hyperelastic solids. Closed-form stress solutions are derived for five different visco-hyperelastic material models in uniaxial extension. Two loading scenarios are considered: uniaxial extension with constant true strain-rate; uniaxial extension with constant engineering strain-rate. The obtained stress solutions are expressed using only elementary functions, when the applied stretch is linear in the true strain. However, for the constant engineering strain-rate case, all the solutions are expressed using special functions, even for the very simple Hencky's hyperelastic material model. The novel solutions can be used to improve the performance and accuracy of the parameter-fitting procedure of a particular visco-hyperelastic solid. In addition, the closed-form expressions for the stresses may serve reference solution for benchmark problems.