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 .
The focus on the interaction term of the energy for edge dislocations was already present in the work of Cermelli and Leoni @cite_0 . In @cite_0 the authors define the renormalised energy starting from a quadratic dislocation energy, and focus on the interaction term of the energy. They however consider the case of a finite number of dislocations with different Burgers vectors, and do not phrase their result in terms of @math -convergence, but instead they keep @math fixed, and therefore express their renormalised energy in terms of the positions of dislocations, rather than densities.
@cite_0
2046646196 1893146140 1992225844 2962724035
In this work we discuss, from a variational viewpoint, the equilibrium problem for a finite number of Volterra dislocations in a plane domain. For a given set of singularities at fixed locations, we characterize elastic equilibrium as the limit of the minimizers of a family of energy functionals, obtained by a finite-core regularization of the elastic-energy functional. We give a sharpasymptotic estimate of the minimum energy as the core radius tends to zero, which allows one to eliminate this internal length scale from the problem. The energy content of a set of dislocations is fully characterized by the regular part of the asymptotic expansion, the so-called renormalized energy, which contains all information regarding self- and mutual interactions between the defects. Thus our result may be considered as the analogue for dislocations of the classical result of Bethuel, Brezis and Helein for Ginzburg--Landau vortices. We view the renormalized energy as the basic tool for the study of the discrete-to-con... This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework. We give a detailed description of the geometry of single droplet patterns in a non- local isoperimetric problem. In particular we focus on the sharp interface limit of the Ohta- Kawasaki free energy for diblock copolymers, regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long-range interactions. Exploiting fine properties of the regularity theory for minimal surfaces, we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns. where u is the order parameter of a two-phases system confined in ⊂ R n , and and m are two nonnegative numerical parameters. The two terms in the energy mimic attractive short-range and repulsive long-range energies between the phases. More precisely the first term is local, it favors minimal interface area and drives the system toward a partition into few pure phases while the second term, involving a Coulomb-like kernel G, is non-local and favor a fine mixing of the phases. A detailed description of the energy is given in § 1. The competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns, whose geometry strongly depends on the choice of the parameters and m (e.g. spherical spots, cylinders, gyroids and lamellae). 0.1. The Ohta-Kawasaki functional for diblock copolymers. The model we consider arises as a simplification of a Ginzburg-Landau functional proposed by Ohta and Kawasaki in their pioneering paper (41) as a possible description of a diblock copolymers' (DBC) system. Even though it is questionable whether such an energy actually describes DBCs (see Choksi and Ren (15), Muratov (39) and Niethammer and Oshita (40)), nevertheless it is a first, and mathematically non-trivial, attempt to capture some of the main features of these systems. For such a reason it deserved over the last twenty years great attention from both the mathematical and the physical