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Crack initiation in brittle materials. (English) Zbl 1138.74042
Summary: We study the crack initiation in a hyperelastic body governed by a Griffith-type energy. We prove that during a load process through time-dependent boundary data of the type $$t \rightarrow tg (x)$$ and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time $$t_{i} > 0$$. Conversely, in the 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 for admissible cracks belonging to a large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ is a local minimality result for the functional
$\varepsilon(\nu, \Gamma):=\int_{\Omega} f(x,\nabla v)\,dx+ k{\mathcal{H}}^{1} (\Gamma),$
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 at most uniformly weak singularities, then configurations $$(u_\Gamma, \Gamma)$$ with $${\mathcal{H}}^{1} (\Gamma)$$ small enough are such that $$\varepsilon(u,\emptyset) < \varepsilon(u_{\Gamma},\Gamma)$$.

MSC:
 74R10 Brittle fracture 74G65 Energy minimization in equilibrium problems in solid mechanics
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