Path-independent integrals in finite elasticity and inelasticity, with body forces, inertia, and arbitrary crack-face conditions

In this paper, certain path-independent integrals, of relevance in the presence of cracks, in elastic and inelastic solids are considered. The hypothesized material constitutive properties include: 1. (i) finite and infinitesimal elasticity 2. (ii) rate-independent incremental flow theory of elasto-plasticity 3. (iii) rate-sensitive behaviour including elasto-viscoplasticity and creep. In each case, finite deformations are considered, along with the effects of body forces, material acceleration, and arbitrary traction/displacement conditions on the crack-face. Also the physical interpretations of each of the integrals either in terms of crack-tip energy release rates or simply energy-rate differences in two comparison cracked-bodies are explored. Several differences between the results in the present work and those currently considered well established in literature are pointed out and discussed.