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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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