Crack Initiation in Brittle Materials |
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Authors: | Antonin Chambolle Alessandro Giacomini Marcello Ponsiglione |
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Institution: | (1) CMAP, Ecole Polytechnique, CNRS, 91128 Palaiseau, France;(2) Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy;(3) Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany |
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Abstract: | In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during
a load process through a time-dependent boundary datum of the type t → t
g(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 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 Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with small enough are such that . |
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Keywords: | |
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