Constitutive models and singularity types for an elastic biaxially loaded rubber sheet

Experiments on a rubber sheet under equal biaxial in-plane tensile loads sh ow that unequal stable equilibrium stretches are possible. To model these u nequal stretches, the sheet strain energy function, when parameterized by t he load, must have several bifurcations in the equilibrium set or must have paths of equilibria disjoint from the equal stretch equilibria path. A Lia punov-Schmidt reduction for the equilibria of a class of isotropically symm etric energy functions and elementary catastrophe theory are used to classi fy the degenerate singularity behavior. The classical empirical constitutiv e models proposed for rubberlike, isothermal, incompressible nonlinear elas tic materials are shown by this analysis to fail to generate enough bifurca tions or disjoint equilibria paths to represent the experimental rubber she et behavior under equal biaxial loads. Based on a full description of the e quilibria behavior of any Ogden strain invariant near a singularity, a mode l that has three degenerate singularities and reproduces the qualitative st ructure of Treloar's sheet data is constructed from linear combinations of three of Ogden's strain invariants. Errors in making the two in-plane tensi ons equal are represented by an imperfection parameter in the catastrophe u niversal unfolding of the energy function.