Equilibrium configurations and stability of a damaged body under uniaxial tractions

This paper deals with the equilibrium problem in nonlinear dissipative inelasticity of damaged bodies subject to uniaxial loading. To model the damage effects, a damage function, affecting the stored energy function, is defined. In the framework of the continuum thermodynamics theory, the constitutive law for damaged hyperelastic materials and an inequality for the energy release rate are derived. By means of an energy-based damage criterion, the irreversible evolution law for the damage function is obtained. After formulating the equilibrium boundary value problem, explicit expressions governing the global development of the equilibrium paths are written. Successively, the stability of the equilibrium solutions are assessed through the energy criterion. For a damaged body under uniaxial loading, seven inequalities are derived. These conditions, if fulfilled, ensure the stability of the solutions under each type of small perturbation. Finally, a number of applications for compressible neo-Hookean and Mooney–Rivlin materials are performed.