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Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 409–449 | Cite as

Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity

  • Daniel E. SpectorEmail author
  • Scott J. Spector
Article

Abstract

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 John (Commun 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 \({{\rm BMO}\cap\, L^1}\) , to the gradient of the equilibrium solution.

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Notes

Acknowledgements

The authors thank Mario Milman for interesting discussions regarding BMO and interpolation theory. The authors thank the referee for helpful comments and for providing a short proof of Lemma 4.6. The authors also thank one of the referees of [59] for their suggestion that the results in Kristensen and Taheri [45] might lead to an extension of John’s [38] uniqueness theorem to the mixed problem.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Chiao Tung UniversityHsinchuTaiwan
  2. 2.National Center for Theoretical SciencesNational Taiwan UniversityTaipeiTaiwan
  3. 3.Department of MathematicsSouthern Illinois UniversityCarbondaleUSA

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