  We extend a phase-field/gradient damage formulation for cohesive fracture to
the dynamic case. The model is characterized by a regularized fracture energy
that is linear in the damage field, as well as non-polynomial degradation
functions. Two categories of degradation functions are examined, and a process
to derive a given degradation function based on a local stress-strain response
in the cohesive zone is presented. The resulting model is characterized by a
linear elastic regime prior to the onset of damage, and controlled
strain-softening thereafter. The governing equations are derived according to
macro- and microforce balance theories, naturally accounting for the
irreversible nature of the fracture process by introducing suitable constraints
for the kinetics of the underlying microstructural changes. The model is
complemented by an efficient staggered solution scheme based on an augmented
Lagrangian method. Numerical examples demonstrate that the proposed model is a
robust and effective method for simulating cohesive crack propagation, with
particular emphasis on dynamic fracture.
