  The Lip-field approach is a new way to regularize softening material models.
It has already been tested in 1D quasistatic and 2D quasistatic: this paper
extends it to 1D dynamics, on the challenging problem of dynamic fragmentation.
The Lip-field approach formulates the mechanical problem to be solved as an
optimization problem, where the incremental potential to be minimized is the
non-regularized one. Spurious localization is prevented by imposing a Lipschitz
constraint on the damage field. The displacement and damage field at each time
step are obtained by a staggered algorithm, that is the displacement field is
computed for a fixed damage field, then the damage field is computed for a
fixed displacement field. Indeed, these two problems are convex, which is not
the case of the global problem where the displacement and damage fields are
sought at the same time. The incremental potential is obtained by equivalence
with a cohesive zone model, which makes material parameters calibration simple.
A non-regularized local damage equivalent to a cohesive zone model is also
proposed. It is used as a reference for the Lip-field approach, without the
need to implement displacement jumps.
  These approaches are applied to the brittle fragmentation of a 1D bar with
randomly perturbed material properties to accelerate spatial convergence. Both
explicit and implicit dynamic implementations are compared. Favorable
comparison to several analytical, numerical and experimental references serves
to validate the modeling approach.
