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

Abstract

Abstract. 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 stable absolute minimum or only a minimizing sequence in which the term corresponding to the bulk energy is forced to zero by the production of a crack in the material.

Surface Diffusion of Graphs: Variational Formulation, Error Analysis, and Simulation

“…Following [4], we consider a twodimensional thin solid occupying the domain {(x, y) : −1 ≤ x ≤ 1 , 0 ≤ y ≤ u(x)} and undergoing a plastic deformation due to competition of elastic effects and surface tension with volume constraint 1 −1 u = 2. The solid is to adjust its shape in order to minimize the following energy:…”

The Stress Driven Rearrangement Instabilities in Electronic Materials and in Helium Crystals

“…Computational work is based on numerical analysis of the mathematical problem formulated in the concluding section "The problem of equilibrium shape of deformable crystal" of the monograph (Grinfeld,1991) and analyzed further by Grinfeld (1991Grinfeld ( , 1993, Spencer and Tersoff (1997), Bonnetier et al (1999) and others.…”

A Quantitative Second Order Minimality Criterion for Cavities in Elastic Bodies

“…Starting from the paper [17] where Grinfeld follows the Gibbs variational approach to model the morphology of thin films, it became clear that a second order variational analysis could be successfully used. This approach has been used in the context of epitaxial growth first for a one dimensional model in [6]. Then in [5] and [12] the model introduced in [17], which is a more realistic two-dimensional model, corresponding to threedimensional configurations with planar symmetry, is studied and the problem of finding a proper functional setting is successfully addressed.…”