A variational problem modelling behavior of unorthodox silicon crystals

J. Hannon; M. Marcus; Victor J. Mizel

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Displaying similar documents to “A variational problem modelling behavior of unorthodox silicon crystals”

Anisotropic functions : a genericity result with crystallographic implications

Victor J. Mizel, Alexander J. Zaslavski (2004)

ESAIM: Control, Optimisation and Calculus of Variations

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In the 1950’s and 1960’s surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious...

Resonance in Preisach systems

Pavel Krejčí (2000)

Applications of Mathematics

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This paper deals with the asymptotic behavior as $t \to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w} (t) + u (t) = ψ (t)$ , $w = u + 𝒫 [u]$ , where $𝒫$ is a Preisach hysteresis operator, $ψ \in L^{\infty} (0, \infty)$ is a given function and $t \geq 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $ψ$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples...

Gradient theory for plasticity via homogenization of discrete dislocations

Adriana Garroni, Giovanni Leoni, Marcello Ponsiglione (2010)

Journal of the European Mathematical Society

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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 $Γ$ -limit of this energy (suitably...

The Energy Density of Non Simple Materials Grade Two Thin Films via a Young Measure Approach

Giuliano Gargiulo, Elvira Zappale (2007)

Bollettino dell'Unione Matematica Italiana

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Dimension reduction is used to derive the energy of non simple materials grade two thin films. Relaxation and $\Gamma$ convergence lead to a limit defined on a suitable space of bi-dimensional Young measures. The underlying ``deformation'' in the limit model takes into account the Cosserat theory.

Relations between multidimensional interval-valued variational problems and variational inequalities

Anurag Jayswal, Ayushi Baranwal (2022)

Kybernetika

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In this paper, we introduce a new class of variational inequality with its weak and split forms to obtain an $L U$ -optimal solution to the multi-dimensional interval-valued variational problem, which is a wider class of interval-valued programming problem in operations research. Using the concept of (strict) $L U$ -convexity over the involved interval-valued functionals, we establish equivalence relationships between the solutions of variational inequalities and the (strong) $L U$ -optimal solutions...

Regularity of Lipschitz free boundaries for the thin one-phase problem

Daniela De Silva, Ovidiu Savin (2015)

Journal of the European Mathematical Society

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We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $E (u, Ω) = \int_{Ω} {| \nabla u |}^{2} d X + ℋ^{n} ({u > 0} \cap {x_{n + 1} = 0}), Ω \subset ℝ^{n + 1},$ among all functions $u \geq 0$ which are fixed on $\partial Ω$ .

Analysis of a one-dimensional variational model of the equilibrium shapel of a deformable crystal

Eric Bonnetier, Richard S. Falk, Michael A. Grinfeld (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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The equilibrium configurations of a one-dimensional variational model that combines terms expressing the bulk energy of a deformable crystal and its surface energy are studied. After elimination of the displacement, the problem reduces to the minimization of a nonconvex and nonlocal functional of a single function, the thickness. Depending on a parameter which strengthens one of the terms comprising the energy at the expense of the other, it is shown that this functional may have a...

Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems

Jozef Kačur, Alexander Ženíšek (1986)

Aplikace matematiky

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The authors study problems of existence and uniqueness of solutions of various variational formulations of the coupled problem of dynamical thermoelasticity and of the convergence of approximate solutions of these problems. First, the semidiscrete approximate solutions is defined, which is obtained by time discretization of the original variational problem by Euler’s backward formula. Under certain smoothness assumptions on the date authors prove existence and uniqueness of the solution...