  Cohesive fracture is among the few techniques able to model complex fracture
nucleation and propagation with a sharp (nonsmeared) representation of the
crack. Implicit time-stepping schemes are often favored in mechanics due to
their ability to take larger time steps in quasistatic and moderate dynamic
problems. Furthermore, initially rigid cohesive models are typically preferred
when the location of the crack is not known in advance, since initially elastic
models artificially lower the material stiffness. It is challenging to include
an initially rigid cohesive model in an implicit scheme because the initiation
of fracture corresponds to a nondifferentiability of the underlying potential.
In this work, an interior-point method is proposed for implicit time stepping
of initially rigid cohesive fracture. It uses techniques developed for convex
second-order cone programming for the nonconvex problem at hand. The underlying
cohesive model is taken from Papoulia (2017) and is based on a
nondifferentiable energy function. That previous work proposed an algorithm
based on successive smooth approximations to the nondifferential objective for
solving the resulting optimization problem. It is argued herein that cone
programming can capture the nondifferentiability without smoothing, and the
resulting cone formulation is amenable to interior-point algorithms. A further
benefit of the formulation is that other conic inequality constraints are
straightforward to incorporate. Computational results are provided showing that
certain contact constraints can be easily handled and that the method is
practical.
