Damaged hyperelastic membranes

• We study the equilibrium paths of thin damaged membranes.
• We assume a two-parameter damage law.
• We study the stability of the equilibrium solution trough the energy criterion.
• Mooney–Rivlin damaged membranes under plane stress are studied in detail.

This paper deals with equilibrium problems in non-linear dissipative inelasticity of damaged membranes. The inelastic constitutive law is obtained by modifying the classical constitutive law for a hyperelastic isotropic material through a proper damage function, which allows to measure the effective stress and the dissipated energy. After making the constitutive modeling, the boundary-value problem is formulated for a damaged membrane subjected to biaxial loadings. The purpose of the analysis is to understand how behaves a membrane that, during the deformation process, experiences a progressively increasing damage. Equilibrium multiple branches of symmetric and asymmetric solutions, together to bifurcation points, are computed and it is shown how damage can alter these equilibrium paths with respect to the virgin undamaged case. In particular, the stress reductions caused by damage can give rise to transitions of the constitutive behavior from hardening type to the softening one. These changes can considerably affect the quality of the equilibrium solutions. Accordingly, the analysis is completed by assessing the stability of the solutions. For this aim, the stability analysis based on the energetic criterion is extended to damaged membranes.