Constitutive Models and Singularity Types for an Elastic Biaxially Loaded Rubber Sheet

Abstract

Experiments on a rubber sheet under equal biaxial in-plane tensile loads show that unequal stable equilibrium stretches are possible. To model these unequal stretches, the sheet strain energy function, when parameterized by the load, must have several bifurcations in the equilibrium set or must have paths of equilibria disjoint from the equal stretch equilibria path. A Liapunov-Schmidt reduction for the equilibria of a class of isotropically symmetric energy functions and elementary catastrophe theory are used to classify the degenerate singularity behavior. The classical empirical constitutive models proposed for rubberlike, isothermal, incompressible nonlinear elastic materials are shown by this analysis to fail to generate enough bifurcations or disjoint equilibria paths to represent the experimental rubber sheet behavior under equal biaxial loads. Based on a full description of the equilibria behavior of any Ogden strain invariant near a singularity, a model that has three degenerate singularities and reproduces the qualitative structure of Treloar's sheet data is constructed from linear combinations of three of Ogden's strain invariants. Errors in making the two in-plane tensions equal are represented by an imperfection parameter in the catastrophe universal unfolding of the energy function.