Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity

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 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 ∩ L 1 , to the gradient of the equilibrium solution.In the absence of body forces and surface tractions, the total energy of a deformation u : Ω → R n of a compressible, hyperelastic body is given bywhere W : Ω × M n×n + → [0, ∞) denotes the stored-energy density and we write M n×n + for the set of n by n matrices with positive determinant. We require that u = d on D, where d is prescribed and D ⊂ ∂Ω is nonempty and relatively open. The pure-displacement problem can then be expressed as the condition D = ∂Ω, while the genuine-mixed problem is the condition D ∂Ω. We here consider both problems. With this notation, we call u e an equilibrium solution if it is a weak solution of the corresponding Euler-Lagrange equations:for all w ∈ W 1,2 (Ω; R n ) that satisfy w = 0 on D, while the uniform positivity of the second variation of E at u e is then the condition that δ 2 E(u e )[w, w] =

BMO and Elasticity: Korn’s Inequality; Local Uniqueness in Tension

“…In this manuscript we extend the results obtained in [30,43]. We note that when Ω has sufficiently smooth boundary (Lipschitz suffices), the space BMO(Ω) is a Banach space that is between L ∞ and all of the other L p -spaces, that is, for all p ∈ [1, ∞), We show, in particular, that the L ∞ -neighborhood in which there is at most one solution can be enlarged to a neighborhood in BMO for both the displacement and the mixed problem provided the equilibrium solution u e has nonnegative principal stresses everywhere.…”

“…707-709]), the idea of considering functions whose mean oscillation is bounded was conceived by Fritz John. His motivation appears to have been the analysis of problems in Nonlinear Elasticity, where John had noticed that mappings with small nonlinear strain (see (5.16)) correspond to deformation gradients the are small in BMO (see [29] or, e.g., [43,Proposition 4.3 and Lemma 5.6]).…”

“…Subsequently, although our understanding of the space BMO and its applicability to both Harmonic Analysis and Partial Differential Equations has advanced significantly (see, e.g., the exposition by R. V. Kohn [35, p. 509-512]), the original goal of making use of BMO in problems of Elasticity has not progressed. Recently, the authors [43] extended John's uniqueness result to include the mixed problem. In particular we showed that, given a smooth equilibrium solution u e at which the second variation of the energy is uniformly positive, there are no other equilibrium solutions v e for which the difference of the right Cauchy-Green strain tensors:…”

A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs

“…Theorem 1.1 points to the possibility that there are distinct but equalenergy global minimizers of the functional D (•). Recall that in the context of energy functionals of the form Ω f (∇u) dx, where f is strictly polyconvex, Ω is a domain homeomorphic to a ball and u| ∂Ω agrees with an affine map, the question of the uniqueness of minimizers, as set out in Problem 8 in [4], was settled by Spadaro in [24]. To relate that work to ours, first note that property (P3) enables us (via (1.5) below) to extend W to a polyconvex function.…”

Some uniqueness theorems of solutions for the problemsof elasticity

“…, 若K中另一任意变形χʹ同样满足边 界条件, 且应变能满足W(χʹ)>W(χ), 则χ在K中是唯一 的. 2018年Sivaloganathan等人 [78,125] 将John Gurtin和Spector [94,96] 结合Hill [15,43] 和Stoppelli [100][101][102][103] 以及John [79] 的工作, 建立有限变形位移边值问题和混 合边值问题恒载条件下解的唯一性定理: 平衡方程的 位移解若属于任何凸的、稳定变形的非空集合, 此时 解满足唯一性. Spector [77,95] 在此基础上, 将结论扩展 到一类简单活载荷力边值问题中.…”