Second-order cone interior-point method for quasistatic and moderate dynamic cohesive fracture by S. Vavasis, K. Papoulia, M. R. Hirmand

  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.