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.

VL - 483 SN - 0022-247X UR - https://www.sciencedirect.com/science/article/pii/S0022247X19309242 IS - 2 JO - Journal of Mathematical Analysis and Applications ER - TY - JOUR T1 - Confinement of dislocations inside a crystal with a prescribed external strain Y1 - 2016 A1 - Ilaria Lucardesi A1 - Marco Morandotti A1 - Riccardo Scala A1 - Davide Zucco AB - 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. UR - http://urania.sissa.it/xmlui/handle/1963/35247 N1 - Preprint SISSA 20/2016/MATE U1 - 35558 U2 - Mathematics ER - TY - RPRT T1 - The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data Y1 - 2015 A1 - Gianni Dal Maso A1 - Ilaria Lucardesi AB - 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. UR - http://urania.sissa.it/xmlui/handle/1963/34629 U1 - 34832 U2 - Mathematics U4 - 1 U5 - MAT/05 ER -