Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity
| Autor: | Daniel Spector, Scott J. Spector |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Mechanical Engineering
010102 general mathematics Mathematical analysis 74B20 35A02 74G30 35J57 42B25 42B37 49S05 Elasticity (physics) 01 natural sciences 010101 applied mathematics Nonlinear system Mathematics - Analysis of PDEs Mathematics (miscellaneous) Rigidity (electromagnetism) Bounded function Hyperelastic material FOS: Mathematics Compressibility Boundary value problem Uniqueness 0101 mathematics Analysis Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Archive for Rational Mechanics and Analysis. 233:409-449 |
| ISSN: | 1432-0673 0003-9527 |
| DOI: | 10.1007/s00205-019-01360-1 |
| Popis: | The uniqueness of equilibrium for a compressible, hyperelastic body subject to dead-load boundary conditions is considered. It is shown, for both the displacement and mixed problems, that there cannot be two solutions of the equilibrium equations of Finite (Nonlinear) Elasticity whose nonlinear strains are uniformly close to each other. This result is analogous to the result of Fritz John (Comm. Pure Appl. Math. 25, 617-634, 1972) who proved that, for the displacement problem, there is a most one equilibrium solution with uniformly small strains. The proof in this manuscript utilizes Geometric Rigidity; a new straightforward extension of the Fefferman-Stein inequality to bounded domains; and, an appropriate adaptation, for Elasticity, of a result from the Calculus of Variations. Specifically, it is herein shown that the uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping is a local minimizer of the energy among deformations whose gradient is sufficiently close, in $BMO\cap\, L^1$, to the gradient of the equilibrium solution. 39 pages |
| Databáze: | OpenAIRE |
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