Asymmetric Equilibrium Configurations of Symmetrically Loaded Isotropic Square Membranes

The homogeneous deformations provided by the equilibrium problem of nonlinear isotropic hyperelastic symmetrically loaded membranes are analyzed. Besides the universal symmetric solutions, the problem considered, depending on the form of the stored energy function, may admit asymmetric solutions. For general incompressible materials, the mathematical conditions governing the global development of these asymmetric solutions are investigated. Explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a Valanis–Landel and an Ogden material. For this last case, which is frequently used to model rubberlike materials, a broad numerical analysis was performed. The more qualitatively interesting branches of asymmetric equilibria are shown and the influence of the material parameters is discussed. Finally, using the energy criterion a number of considerations are made on stability.