We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
98/2005
Crack initiation in elastic bodies
Antonin Chambolle, Alessandro Giacomini and Marcello Ponsiglione
Abstract
In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith's type energy. We prove that, during a load process through a time dependent boundary datum of the type $t \to t g(x)$ and in absence of strong singularities (this is the case of homogeneous isotropic materials) the crack initiation is brutal, i.e., a big crack appears after a positive time $t_i>0$. On the contrary, in presence of a point $x$ of strong singularity, a crack will depart from $x$ at the initial time of loading and with zero velocity. We prove these facts (largely expected by the experts of material science) for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components.
The main tool we employ to address the problem is a local minimality result for the functional $$\varepsilon (u,\Gamma):=\int_\Omega f(x,\nabla v)\,dx+k\mathcal{H}^1(\Gamma),$$ where $\Omega \subseteq \mathbb{R}^2$, $k>0$ and $f$ is a suitable Carathéodory function. We prove that if the uncracked configuration $u$ of $\Omega$ relative to a boundary displacement $\psi$ has uniformly weak singularities, then configurations $(u_\Gamma \Gamma)$ with $\mathcal{H}^1(\Gamma)$ small enough are such that $\varepsilon (u,\emptyset)<\varepsilon (u_\Gamma,\Gamma)$.
