Non-differentiable energy minimization for cohesive fracture

An energy minimization approach to initially rigid cohesive fracture is proposed, whose key feature is a term for the energy stored in the interfaces that is nondifferentiable at the origin. A consequence of this formulation is that there is no need to define an activation criterion as a separate entity from the traction-displacement relationship itself. Instead, activation happens automatically when the load reaches a critical level because the minimizer of the potential no longer occurs at the 0-displacement level. Thus, the activation computation necessary in previous initially rigid formulations is now replaced by the computation of a minimizer of a nondifferentiable objective function. This immediately makes the method more amenable to implicit time stepping, since the activation criterion no longer interacts with the nonlinear solver for the next time step. A novel extension of the functional to the dynamic case is presented. The optimization problem is solved by a continuation (homotopy) method used in conjunction with an augmented Lagrangian and a trust region minimization algorithm to find the minimal energy configuration. Because the approach eliminates the need for an activation criterion, the algorithm sidesteps the complexities of time-discontinuity and traction-locking previously observed in relation to initially rigid models.