We consider systems of @math parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modelled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations, and prove several convergence results in the limit @math . The main aim of the paper is to study the convergence of the evolution of the empirical measure as @math . We consider rate-independent, quasi-static evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasi-static evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slip-plane-confined. Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large self- or core energy. We prove Gamma-convergence of this renormalized energy, and we construct joint recovery sequences for which both the energies and the modified distances converge. With this augmented Gamma-convergence we prove the convergence of the quasi-static evolutions as @math .
As for the slip-plane-confined motion, the only related work in the mathematical domain that we know of is @cite_22 , where the authors consider screw dislocations that may move along a finite set of directions. Such a system presents different mathematical difficulties, since each dislocation can, in theory, reach each point in the plane, in contrast to the single-slip-plane confinement of this paper.
@cite_22
2105814348 2962835414 1976044055 2963892386
The goal of this paper is the analytical validation of a model of Cermelli and Gurtin [Arch. Ration. Mech. Anal., 148 (1999), pp. 3--52] for an evolution law for systems of screw dislocations under the assumption of antiplane shear. The motion of the dislocations is restricted to a discrete set of glide directions, which are properties of the material. The evolution law is given by a “maximal dissipation criterion,” leading to a system of differential inclusions. Short time existence, uniqueness, cross-slip, and fine cross-slip of solutions are proved. We report on a study of the translational sliding motion and dynamic friction in systems of double-walled carbon nanotubes using molecular dynamics simulations combined with theoretical analysis. The sliding motion is described by a one-dimensional analytical model which includes the van der Waals force between the nanotubes, a dynamic friction force, and a small Langevin-type fluctuating force. The dynamic friction force is shown to be linear in the velocity over a large domain of initial conditions in armchair-armchair, zigzag-armchair, and zigzag-zigzag double-walled nanotubes. Beyond this domain, evidence is obtained for nonlinear effects which increase friction. In armchair-armchair systems, the dynamic friction is observed to be nonlinearly enhanced by the excitation of internal modes. In the linear domain, the coefficient of proportionality between the dynamic friction force and the velocity is shown to be given by Kirkwood's formula in terms of the force autocorrelation function. This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ( > 0 ) representing the lattice spacing of the crystal, we remove a disc of radius ( ) around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ( 0 ) , in terms of Γ-convergence. We focus on the self energy regime of order ( 1 ) ; we show that configurations with logarithmic diverging energy converge, up to a subsequence, to a finite number of multiple dislocations and we compute the corresponding Γ-limit. We propose a polynomial force-motion model for planar sliding. The set of generalized friction loads is the 1-sublevel set of a polynomial whose gradient directions correspond to generalized velocities. Additionally, the polynomial is confined to be convex even-degree homogeneous in order to obey the maximum work inequality, symmetry, shape invariance in scale, and fast invertibility. We present a simple and statistically-efficient model identification procedure using a sum-of-squares convex relaxation. Simulation and robotic experiments validate the accuracy and efficiency of our approach. We also show practical applications of our model including stable pushing of objects and free sliding dynamic simulations.