A Vanishing Viscosity Approach to Fracture Growth in a Cohesive Zone Model With Prescribed Crack Path

Abstract

The existence of crack evolutions based on critical points of the energy functional is proved, in the case of a cohesive zone model with prescribed crack path. It turns out that evolutions of this type satisfy a maximum stress criterion for the crack initiation. With an explicit example, it is shown that evolutions based on the absolute minimization of the energy functional do not enjoy this property.

Quadratic and rate-independent limits for a large-deviations functional

“…The rigorous proof of Statement 1 consists of the following theorem (11). Consider the sequence of Markov processes {X n } with generator Ω n defined in (12) and with X n (0) converging to x 0 for n → ∞.…”

“…The time-dependent case follows by the same modification as above. (11), with ∇ E uniformly continuous, let α n β n → ω, nβ n → 1/ h, and let μ n t be the law of the process {X n (t)} defined in (12) …”

“…[1,13,14,21,28,40,43]), with a perturbative approach (e.g. [11]), or from a chain model of highly nonlinear viscous springs [38]. The general approach in these results is to choose a microscopic model with a component of gradient-flow type (quadratic dissipation, deemed more 'natural') and then take a limit which induces the vanishing of the quadratic behaviour and the appearance of the rate-independent behaviour.…”

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

“…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

“…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. The approximating (and regularized) problem containing the surface term (1.2) has been considered already in [18,19], and is also related to [4,5,9], where a prescribed crack path is considered for cohesive-zone models describing delamination with partially debonded crack surfaces. Moreover, according to Barenblatt's cohesive-zone model [1], the energy density needed to produce a new crack (or to increase an existing one) depends on the crack opening, namely on [u] Γc .…”