Energy methods in fracture mechanics. Stability, bifurcation and second variations

The quasi-static evolution of a system of interacting linear cracks is considered in brittle fracture. Stability and bifurcation criteria are presented in terms of the second variation of the potential energy and with a formulation of the rate boundary value problem following Hill's method. A symmetric description is proposed for this problem involving as principal unknowns the crack propagation velocity and the displacement velocity defined on the current configuration. As a consequence, an explicit expression for the matrix of the second derivatives of energy with respect to the crack lengths is given in terms of new path-independent integrals. The numerical computation of these path-independent integrals by the f.e.m. is also considered and illustrated by some simple examples.