In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [10] valid for straight fractures.

%V 483 %P 123656 %8 2020/03/15/ %@ 0022-247X %G eng %U https://www.sciencedirect.com/science/article/pii/S0022247X19309242 %N 2 %! Journal of Mathematical Analysis and Applications %0 Journal Article %D 2016 %T Confinement of dislocations inside a crystal with a prescribed external strain %A Ilaria Lucardesi %A Marco Morandotti %A Riccardo Scala %A Davide Zucco %X We study screw dislocations in an isotropic crystal undergoing antiplane shear. In the framework of linear elasticity, by fixing a suitable boundary condition for the strain (prescribed non-vanishing boundary integral), we manage to confine the dislocations inside the material. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The novelty of introducing a Dirichlet boundary condition for the tangential strain is crucial to the confinement: it is well known that, if Neumann boundary conditions are imposed, the dislocations tend to migrate to the boundary. The results are achieved using PDE techniques and Ƭ-convergence theory, in the framework of the so-called core radius approach. %G en %U http://urania.sissa.it/xmlui/handle/1963/35247 %1 35558 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2016-10-20T11:44:46Z No. of bitstreams: 1 LMSZ-preprint.pdf: 547676 bytes, checksum: 5c7add921deefef560aac54da6a584cc (MD5) %0 Report %D 2015 %T The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data %A Gianni Dal Maso %A Ilaria Lucardesi %X Given a bounded open set $\Omega \subset \mathbb R^d$ with Lipschitz boundary and an increasing family $\Gamma_t$, $t\in [0,T]$, of closed subsets of $\Omega$, we analyze the scalar wave equation $\ddot{u} - div (A \nabla u) = f$ in the time varying cracked domains $\Omega\setminus\Gamma_t$. Here we assume that the sets $\Gamma_t$ are contained into a prescribed $(d-1)$-manifold of class $C^2$. Our approach relies on a change of variables: recasting the problem on the reference configuration $\Omega\setminus \Gamma_0$, we are led to consider a hyperbolic problem of the form $\ddot{v} - div (B\nabla v) + a \cdot \nabla v - 2 b \cdot \nabla \dot{v} = g$ in $\Omega \setminus \Gamma_0$. Under suitable assumptions on the regularity of the change of variables that transforms $\Omega\setminus \Gamma_t$ into $\Omega\setminus \Gamma_0$, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks. %G en %U http://urania.sissa.it/xmlui/handle/1963/34629 %1 34832 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by lucardes@sissa.it (lucardes@sissa.it) on 2015-10-12T09:08:37Z No. of bitstreams: 1 DM-Luc-15-SISSA.pdf: 376913 bytes, checksum: bfdcf0975c25c9366acb4195e0f08e79 (MD5)