This note collects some results on the behaviour of screw dislocation in an
elastic medium. By using a semi-discrete model, we are able to investigate two
specific aspects of the dynamics, namely (i) the interaction with free
boundaries and collision events and (ii) the confinement inside the domain when
a suitable Dirichlet-type boundary condition is imposed.
In the first case, we analytically prove that free boundaries attract
dislocations and we provide an expression for the Peach--Koehler force on a
dislocation near the boundary. Moreover, we use this to prove an upper bound on
the collision time of a dislocation with the boundary, provided certain
geometric conditions are satisfied. An upper bound on the collision time for
two dislocations with opposite Burgers vectors hitting each other is also
obtained.
In the second case, we turn to domains whose boundaries are subject to an
external stress. In this situation, we prove that dislocations find it
energetically favourable to stay confined inside the material instead of
getting closer to the boundary. The result first proved for a single
dislocation in the material is extended to a system of many dislocations, for
which the analysis requires the careful treatments of the interaction terms.Comment: 12 pages, submitted for the Proceedings volume of the XXIII
Conference AIMETA (The Italian Association of Theoretical and Applied
Mechanics
