On the Inviscid Limit of a Model for Crack Propagation

Abstract

We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove the existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

“…Indeed, in order to satisfy the global stability, energetic solutions may change instantaneously in a very drastic way, jumping into very far-apart energetic configurations (see, for instance, [Mie03,Ex. 6.1], [KMZ08,Ex. 6.3], [MRS09, Ex.…”

“…Starting from the seminal paper [EM06], this technique has by now been thoroughly developed both for abstract rate-independent systems [MRS09, MRS12,MZ13], and in the applications to fracture [TZ09,Cag08,KMZ08,KZM10,LT11], and to plasticity [DDMM08,BFM12,DDS11,DDS12,FS13].…”

Quasistatic delamination problem

“…Here it is convenient, like in [13,Sect.4] or [16], to use the additive split u = u + u D (t), u D (t) being an extension of w D (t) to Ω. In view of (2.4) and of the assumption Γ c ∩ Γ D = ∅, we can assume…”

“…Our model is related to [16,17,26,27], where the evolution of a single crack with prescribed path is studied, in the case of a single crack tip. The present model allows for more general crack sets although, for some specific loadings, it seems possible that it would give the same response as the models mentioned above.…”

Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes

“…The quasistatic evolution of rate independent systems has been often obtained as the limit case of a viscosity driven evolution (see [27], [20], [9], [6], [30], [22], [13], [14], [16], [17], [21], [23]). In this paper we present a case study on the approximation of a quasistatic evolution by dynamic evolutions, in a mechanical problem governed by partial differential equations.…”