Crack initiation in elastic bodies
Antonin Chambolle, Alessandro Giacomini, and Marcello Ponsiglione
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Submission date: 11. Nov. 2005
published in: Archive for rational mechanics and analysis, 188 (2008) 2, p. 309-349
DOI number (of the published article): 10.1007/s00205-007-0080-6
with the following different title: Crack initiation in brittle materials
MSC-Numbers: 35R35, 35J85, 35J25, 74R1
PACS-Numbers: 46, 81., 91.60.Ba
Keywords and phrases: free discontinuity problems, energy minimization, crack initiation, variational models
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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 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 . 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
where , k>0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of relative to a boundary displacement has uniformly weak singularities, then configurations with small enough are such that .