1,721,008 research outputs found
Elastic energy stored in a crystal induced by screw dislocations: From discrete to continuous
No full textThis paper deals with the passage from discrete to continuous in modeling the static elastic properties of vertical screw dislocations in a cylindrical crystal, in the setting of antiplanar linear elasticity. We study, in the framework of Gamma- convergence, the asymptotic behavior of the elastic stored energy induced by dislocations as the atomic scale e tends to zero, in the regime of dilute dislocations, i. e., rescaling the energy functionals by 1/epsilon(2)vertical bar log epsilon vertical bar. First we consider a continuum model, where the atomic scale is introduced as an internal scale, usually called core radius. Then we focus on a purely discrete model. In both cases, we prove that the asymptotic elastic energy as epsilon -> 0 is essentially given by the number of dislocations present in the crystal. More precisely the energy per unit volume is proportional to the length of the dislocation lines, so that our result recovers in the limit as epsilon -> 0, a line tension model
Stability of Neumann problems and applications to the variational theory of crack propagation
No full text
Stability of some unilateral free-discontinuity problems in two-dimensional domains
No full textThe purpose of this paper is to study the stability of some unilateral free-discontinuity problems in two-dimensional domains, with the density of the volume part having p-growth, with 1 < p < infinity, under perturbations of the discontinuity sets in the Hausdorff metric
A VARIATIONAL APPROACH TO THE STATIONARY SOLUTIONS OF THE BURGERS EQUATION
No full textWe consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini et al. [Comm. Pure Appl. Math., 64 (2011), pp. 649-696], we analyze a Lyapunov functional for such an equation which gives the large deviations asymptotic of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. In particular, we focus on boundary data corresponding to a standing wave solution to the Burgers equation in the whole line. In this case, the limiting functional has a one-parameter family of minimizers and we compute the sharp asymptotic cost corresponding to a given shift of the stationary solution
Variational Equivalence Between Ginzburg-Landau, XY Spin Systems and Screw Dislocations Energies
No full textWe introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing E 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 Gamma-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 vertical bar log epsilon vertical bar(2) energetic regime
Two Slope Functions Minimizing Fractional Seminorms and Applications to Misfit Dislocations
No full textWe consider periodic piecewise affine functions, defined on the real line, with two given slopes, one positive and one negative, and prescribed length scale of the intervals where the slope is negative. We prove that, in such a class, the minimizers of s-fractional Gagliardo seminorm densities, with 0 < s < 1, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals, where the slope is constant, vanishes. Our results, for s = 12 , have relevant applications to the van der Merwe theory of misfit dislocations at semicoherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semicoherent limit as the ratio between the two lattice spacings tends to one
Phase transitions and minimal hypersurfaces in hyperbolic space
No full textThe purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behavior at infinity. Then, we study the limit as the length scale tends to zero through a Γ-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in [3, 21] using geometric measure theory. © Taylor & Francis Group, LLC
Gradient theory for plasticity via homogenization of discrete dislocations
No full textWe 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 Gamma-limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form E = integral(Omega) (W(beta(e)) + phi(Curl beta(e))) dx, where beta(e) represents the elastic part of the macroscopic strain, and Curl beta(e) represents the geometrically necessary dislocation density. The plastic energy density. 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
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