Boundary Behavior and Confinement of Screw Dislocations
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Boundary Behavior and Confinement of Screw Dislocations Marco Morandotti1 1 Fakultät für Mathematik, Technische Universität München, Boltzmannstrasse, 3, 85748 Garching bei München, Germany. ABSTRACT In this note we discuss two aspects of screw dislocations dynamics: their behavior near the boundary and a way to confine them inside the material. In the former case, we obtain analytical results on the estimates of collision times (one dislocation with the boundary and two dislocations with opposite Burgers vectors with each other); numerical evidence is also provided. In the latter, we obtain analytical results stating that, under imposing a certain type of boundary conditions, it is energetically favorable for dislocations to remain confined inside the domain.
INTRODUCTION Introduced in a seminal paper by Volterra in 1907 [12], dislocations have been proposed as the mechanism that is ultimately responsible for plasticity in metals: Orowan [9], Polanyi [10], and Taylor [11] reached this conclusion independently in 1934. Yet, it was not until 1956 that dislocations were observed with the help of an electron microscope [5]. Nowadays, much effort is put into studying their behavior, and especially their dynamics. We build on a model proposed in 1999 by Cermelli & Gurtin [3] for screw dislocations undergoing antiplane shear to study two different situations: the behavior of dislocations near the domain boundary and a situation in which dislocations can be confined inside the domain by imposing a suitable boundary condition. The theoretical framework adopted is that of linearized elasticity in the plane. In both cases, the assumptions on the deformation allow to reduce the problem from a fully 3D one in the cylinder to a 2D one in the cross-section , which we will assume to be an open connected set with boundary that does not touch itself. To study the behavior near the boundary, we provide accurate estimates on the Green’s function for the laplacian in two dimensions, which will translate into estimates on the direction of the Peach-Koehler force acting on a dislocation near the boundary. The first result that we obtain (see Theorem 1) is that, if one dislocation is sufficiently close to the boundary and sufficiently far away from the others in the system, then its Peach-Koehler force is directed along the outward unit normal to the boundary at the closest point. Next, we turn to estimates on the collision times: we prove that in the situation just described the dislocation which is closest to the boundary will collide with it in finite time. We can also provide estimates on the collision time of two dislocations with opposite Burgers vectors, provided that they are sufficiently close to each other and that the remaining dislocations are sufficiently far away. Furthermore, in both cases, if the other dislocations are sufficiently diluted, no other collision events will happen (see Theorems 2 and 3). To study the confinement, we resort to variational methods and we prove that, under suitably chosen boundary condition, an
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