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

Abstract

We derive an irreversibility condition for fracture in the inverse-deformation approach using the second law of thermodynamics. We consider the problem of brittle failure in one-dimensional, homogeneous, elastic bars undergoing quasistatic motion. Despite the presence of a non-zero interfacial/surface energy, the third derivative of the inverse deformation is discontinuous at crack faces. This is due to the inequality constraint allowing fracture while ensuring that the orientation of matter is preserved. We show that any change in the location of open cracks in their reference configuration result in negative entropy production. We disallow these changes according to the second law of thermodynamics, fixing the location of cracks in their reference domain. The inequality constraint and the irreversibility condition limit the space of admissible variations. We prove second-order conditions for local stability incorporating these restrictions. Their numerical implementation shows that all broken equilibria of the elastic bar under hard loading are locally stable.