Abstract. We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations.We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem.The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the -limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the formwhere β e represents the elastic part of the macroscopic strain, and Curl β e represents the geometrically necessary dislocation density. The plastic energy density ϕ is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.
International audienceThis paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. We introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper allows us to establish also new results
We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing ε tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and vortices in superconductors. To this purpose, we introduce a notion of asymptotic variational equivalence between families of functionals in the framework of Γ-convergence. We then prove that, in several scaling regimes, the complex Ginzburg-Landau, the XY spin system and the screw dislocation energy functionals are variationally equivalent. Exploiting such an equivalence between dislocations and vortices, we can show new results concerning the asymptotic behavior of screw dislocations in the | log ε| 2 energetic regime.
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