Models of Dynamic Damage and Phase-field Fracture, and their Various Time Discretisations

Abstract

Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.

Coupled time discretization of dynamic damage models at small strains

“…Even the rate-dependent model allows for analytical existence result, cf. [24,42] for p = 2; the existence results presented here are only new for p = 2. Yet, our focus is on numerical approximation scheme and on its stability and convergence.…”

Dynamic perfect plasticity and damage in viscoelastic solids

“…We point out that, since damage is in most applications a much faster phenomenon than external loadings, it is a classical modeling assumption to describe it as a rate‐independent process (with or without damage viscosity) having an arbitrary high speed. We refer to [, Subsection 4.3.2] and to [] and the references therein for a thorough overview of the mathematical literature, as well as a comparison of different damage models.…”

“…Note however that plasticity was not involved in all these references. Some modifications have been addressed in [], see also [] for various other models, and [] for the linearized and cohesive‐fracture settings. Remark A combination of damage/fracture with plasticity is sometimes denoted by the adjective “ductile”, in contrast to “brittle”, if plasticity is not considered. There are various scenarios of combination of plastification processes with damage, that can model several phenomena in fracture mechanics.…”

“…A vast literature concerning damage in viscoelastic materials, both in the quasistatic and the dynamical setting is currently available. We refer, e.g., to [42,43,45,50,57,59] and the references therein for an overview of the main results.…”

“…A vast literature concerning damage in viscoelastic materials, both in the quasistatic and the dynamical setting is currently available. We refer, e.g., to [42,43,45,50,57,59] and the references therein for an overview of the main results.The interplay between plasticity and damage has been already extensively investigated, prominently in the quasistatic framework. The interaction between damage and strain gradient plasticity is addressed in [21] whereas a perfect-plastic model has been proposed in [1], where the one-dimensional response is also studied.…”