TY - JOUR

T1 - Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity

AU - Spector, Daniel E.

AU - Spector, Scott J.

N1 - Funding Information:
The first author (DS) is supported by the Taiwan Ministry of Science and Technology under research Grants 105-2115-M-009-004-MY2, 107-2918-I-009-003 and 107-2115-M-009-002-MY2
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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 BMO∩L1 , to the gradient of the equilibrium solution.

AB - 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 BMO∩L1 , to the gradient of the equilibrium solution.

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U2 - 10.1007/s00205-019-01360-1

DO - 10.1007/s00205-019-01360-1

M3 - Article

AN - SCOPUS:85060991818

VL - 233

SP - 409

EP - 449

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 1

ER -