Crack initiation in elastic bodies

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 $$ \Es(u,Γ):=\int_\Om f(x,\nabla v) dx+k\hu(Γ), $$ where $Ω\subseteq \R^2$, $k>0$ and $f$ is a suitable Carathéodory function. We prove that if the uncracked configuration $u$ of $\Om$ relative to a boundary displacement $ψ$ has uniformly weak singularities, then configurations $(u_Γ,Γ)$ with $\hu(Γ)$ small enough are such that $\Es(u,\emptyset)<\Es(u_Γ,Γ)$.