Large deformation finite element analysis of non-linear viscoelastic membranes with reference to thermoforming

Abstract

This paper reports on the development of finite element formulations and computer programs for modelling free and constrained inflation of thin polymeric sheets in the context of thermoforming of plastic articles. In recognition of the generally time-dependent viscoelastic behaviour of polymers, and the large strains encountered in thermoforming applications, the material is modelled as non-linear visoelastic. For this purpose the constitutive relation proposed by Christensen (1)† is adopted, assuming the relaxation function to be exponential. Most of the published work on non-linear viscoelastic membranes deals with simple axisymmetric geometries, while the finite element formulations presented in this work are for both axisymmetric and non-axisymmetric membrane inflations, including contact against constraining surfaces. Both frictionless and slipless idealizations of contact conditions are studied. The finite element solutions of free and constrained inflations of circular membranes serve as illustrative examples for the axisymmetric case, while those for elliptical membranes demonstrate the non-axisymmetric cases. Comparison of the finite element results with the analytical solutions obtained (Appendix 1) for some simple free and constrained inflation problems shows good agreement.