Existence of Minimizers and Microstructure in Nonlinear Elasticity
نویسنده
چکیده
In this paper, the existence of a solution in the form of a minimizer or microstructure is established for the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of St. Venant-Kirchhoff type materials. Necessary and sufficient conditions for minimizing sequences of the potential energy to converge to a minimizer or to microstructure are given.
منابع مشابه
Analysis of Microstructures and Phase Transition Phenomena in One-Dimensional, Non-Linear Elasticity by Convex Optimization
We propose a general method to determine the theoretical microstructure in one dimensional elastic bars whose internal deformation energy is given by non-convex polynomials. We use non-convex variational principles and Young measure theory to describe the optimal energetic configuration of the body. By using convex analysis and classical characterizations of algebraic moments, we can formulate ...
متن کاملNumerical Methods for Minimizers and Microstructures in Nonlinear Elasticity
A standard finite element method and a finite element truncation method are applied to solve the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of St. Venant-Kirchhoff materials. Finite element solutions are proved to exist and to be in the form of minimizers in appropriate sets of admissible finite element functions for both methods...
متن کاملMinimizers for a Double-well Problem with Affine Boundary Conditions
This paper is concerned with the existence of minimizers for functionals having a double-well integrand with affine boundary conditions. Such functionals are related to the so-called Kohn-Strang functional which arises in optimal shape design problems in electrostatics or elasticity. They are known to be not quasi-convex, and therefore existence of minimizers is, in general, guaranteed only for...
متن کاملFlat minimizers of the Willmore functional: Euler-Lagrange equations
Let S ⊂ R be a bounded C domain and let g denote the flat metric in R. We prove that there exist minimizers of the Willmore functional restricted to a class of isometric immersions of the Riemannian surface (S, g) into R. We derive the Euler-Lagrange equations satisfied by such constrained minimizers. Our main motivation comes from nonlinear elasticity, where this constrained Willmore functiona...
متن کاملExistence of Minimizers in Incremental Elasto-Plasticity with Finite Strains
We consider elasto-plastic deformations of a body which is subjected to a time-dependent loading. The model includes fully nonlinear elasticity as well as the multiplicative split of the deformation gradient into an elastic part and a plastic part. Using the energetic formulation for this rate-independent process we derive a time-incremental problem, which is a minimization problem with respect...
متن کامل