Irreversibility condition and stability of equilibria in the inverse-deformation approach to fracture

Abstract: We derive the irreversibility condition in fracture for the inverse-deformation approach using the second law of thermodynamics. We consider the problem of brittle failure in an elastic bar previously solved in (Rosakis et al 2021). Despite the presence of a non-zero interfacial/surface energy, the third derivative of the inverse-deformation map is discontinuous at the crack faces. This is due to the presence of the inequality constraint ensuring the inverse strain is nonnegative and the orientation of matter is preserved. A change in the material location of a crack results in negative entropy production, violating the second law. Consequently, such changes are disallowed giving the irreversibility condition. The inequality constraint and the irreversibility condition limit the space of admissible variations. We prove necessary and sufficient conditions for local stability that incorporate these restrictions. Their numerical implementation shows that all broken equilibria found in (Rosakis et al 2021) are locally stable.