Energy Criterion for Tearing of Elastic and Elastic-Plastic Materials: Independent Derivations from Compliance Relations and a Thermodynamic Model with Internal Variables

Elastic-plastic fracture criteria based on Irwin’s postulate and later J-integral analysis have faced strong limitations. Two alternate criteria, C1 and C2, are explored, applicable to extended crack growth in elastic and fully plastic regimes. Criterion C1, restricted to materials exhibiting linear elastic unloading, follows from mathematical definitions of the unloading compliance C, energy dissipation rate D and an elastic-plastic adaptation I of Irwin’s crack driving force G. It predicts crack extension whenever loading conditions are such that I=D. Criterion C2 emerges from a thermodynamic formulation with internal variables, predicting fracture whenever I^*=D^*, where I^* and D^* are non-linear generalizations of I and D. Criterion C2 reduces to C1 for elastic-plastic materials with linear elastic unloading, and for linear or non-linear elastic fracture it becomes J = D^*, where J is the J-integral. The measured parameter D^* is identified thermodynamically as the sum of heat dissipated per unit crack extension, plus the rate of change of a residual Helmholtz energy component admitting non-dissipative irreversible energy forms such as residual elastic energy and Griffith surface energy. Under variable transformation C2 essentially yields Irwin’s criterion. Associated instability criteria are postulated, showing qualitative agreement with observed instability behavior under displacement- and load-controlled conditions.