Implementing image stresses in a 3D dislocation simulation

A three-dimensional (3D) dislocation simulation has been developed in the last few years in order to fill the gap between atomistic simulations and the more macroscopic approaches. The specific role of such a simulation is to combine all mechanisms responsible for the hardening (e.g. the forest mechanism) and the multiplication of dislocations which typically occur in three dimensions. Many problems involve the presence of interfaces, which can come from cracks, oxide layers, particles etc. The present work deals with a method to tackle such problems in a 3D simulation. It is firstly restricted to the image stress case. A method of calculating the image stress field of dislocation segments due to the presence of a semi-infinite free surface has been proposed by Gosling and Willis. An alternative method is proposed here based on the Boussinesq problem of point loading in a half-space. It is firstly shown on simple cases that the methods are equivalent. The latter one is used in the present simulation, and the image stress field due to populations of dislocations is investigated. From a practical point of view, we calculate the depths within which all dislocations have to be taken into account, in order to get reasonable accuracy on the image stresses, and we also investigate how this stress field decays in space.