A Note on Uniqueness in Linear Elastostatics

Abstract

We obtain a uniqueness theorem for linear elastostatics, for a homogeneous isotropic body, under the condition of strong ellipticity, in the case of "mixedmixed" boundary conditions. The theorem applies to bodies of restricted local concavity and convexity. Some example domains are illustrated.

Uniqueness theorems in the equilibrium theory of thermoelasticity with microtemperatures for microstretch solids

“…Establishing necessary conditions for the uniqueness of solutions is an open problem in the classical theory of thermoelasticity [Kupradze et al 1979], the theory of thermoelasticity with microtemperatures [Ieşan and Quintanilla 2000], the micropolar theory of thermoelasticity, theories of micromorphic elasticity and thermomicrostretch elastic solid [Eringen 1999], and in the theory of thermoelasticity with microtemperatures for microstretch solids [Ieşan 2001]. The necessary condition for uniqueness of solutions have been established only in the classical theory of elasticity (see [Knops and Payne 1971;Fosdick et al 2007], for details).…”

“…A linear theory of thermoelastic bodies with microstructure and microtemperatures which permits the transmission of heat as thermal waves at finite speed was constructed in [Ieşan and Quintanilla 2009], and existence and uniqueness results in the context of the dynamic theory were established. An extensive review and the basic results in the microcontinuum field theories are given in [Eringen 1999;Ieşan 2004].…”

On the contact problem of classical elasticity

“…In (4) is the union of a finite number of nonintersecting bounded domains with connected boundaries.…”

“…Piccioni and G. Puglisi [4] proved that (1) is also sufficient for the uniqueness of a solution of a mixed boundary value problem in a bounded domain which is a combination of the displacement, contact and its dual problems. Precisely, one decomposes ∂ into three portions {∂ i } i=1,2,3 and assigns the displacement on ∂ 1 , the normal component of the displacement and the tangential component of the traction on ∂ 2 , the tangential component of the displacement and the normal component of the traction on ∂ 3 .…”

“…The main purpose of this paper is just to show that under the hypotheses (and thanks to the results) of [4]: (i) the contact problem and its dual in exterior domains admit existence and strong uniqueness theorems; (ii) a mixed problem in a bounded multiconnected domain, where the displacement is assigned on the outer boundary and the contact condition (or its dual) in the inner boundary, has a unique solution.…”