On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions

Abstract

In this paper we establish some continuous dependence and uniqueness theorems for equilibrium solutions of the equations of general anisotropic linear elasticity in exterior domains. The method we employ is that of the weight function which we introduced in previous papers. However, this is the first example where the method is applied to a static problem. The above theorems are obtained by allowing the strain to be unbounded at large spatial distances. In some cases, no growth condition is assumed. Moreover, the displacement and the elasticities are also possibly allowed to grow.

Saint-Venant's principle on unbounded regions

“…(See also the references cited in [24].) On specialisation to the exterior problem and the whole space problem our results reproduce or otherwise complement many previous uniqueness theorems including for example, those due to Fichera [4], Galdi and Rionero [7,8] [30] and Wilcox [33]. Some further references are contained in [19].…”

“…Thus, we assume that n ; ayM, vanishes on the (closed) surface 9Q 0 so that We next derive a first-order differential inequality for /(r). Thus, we have from An application of Wirtinger's inequality, together with the arithmetic-geometric mean inequality, then leads to In accordance with the discussion of (3.11), it follows that either /(r) is positive for r ^ r 0 and grows polynomiaily like r Y where ( 5 - 8 ) or I(r) is negative for r>r 0 and decays polynomiaily like r~y.…”

“…The equilibrium equations are given by In accordance with the discussion of (3.11), it follows that either /(r) is positive for r ^ r 0 and grows polynomiaily like r Y where 8 ) or I(r) is negative for r>r 0 and decays polynomiaily like r~y.…”

Some uniqueness theorems of solutions for the problemsof elasticity

“…对于线弹性平衡问题的其他方面, 将各向同性弹 性体的边值问题中解的唯一性总结为表10 [60][61][62][63][64][65][66] . 将均 质各向异性弹性体和非均质弹性体的边值问题中解的 唯一性结论总结为表11 [67][68][69][70][71][72][73] . 对于有界域中三维问题, 均质各向异性弹性体, 材 料的弹性张量满足正定性, 则在局部可积的索伯列夫 表 9 线弹性动力学初边值问题中解的唯一性定理 Table 9 The uniqueness theorems of solutions to the initial-boundary-value problems in linear elastodynamics…”

On the traction problem for linear elastostatics in exterior domains

“…[1,2,20] and literature cited therein, and for the exterior problem [4], when the boundary displacement is prescribed. Nevertheless, well-posedness still remains an open question for the exterior problem when the traction is given on the boundary, only uniqueness having been established [3,5,6,7].…”