On Failure of the Complementing Condition and Nonuniqueness in Linear Elastostatics

Abstract

Consider a homogeneous, isotropic body composed of a compressible linearly elastic material and assume that the body is at equilibrium in a state of plane strain. The traction problem for such a body (in the absence of body forces and surface tractions) consists of finding a displacement U=(u1,u2) that satisfies (cf., e.g., Gurtin [4]) (1) U + ( + ) div U = 0 in R. (2) [ (U + UT) + (div U) I] n = 0 on ðR. Here R R2 is a regular region, n the outward unit normal to the boundary, ðR, and , and are the (constant) Lamé moduli. It is well-known that (1) and (2) have a unique solution, modulo an infinitesimal rigid deformation, provided that ≠ 0, + ≠ 0, and 2 + ≠ 0. The purpose of this note is to demonstrate that the above mentioned uniqueness result fails when = - . In fact we show that (1) and (2) have an infinite number of linearly independent solutions (in spite of the ellipticity of the equations). The reason for this unusual (for an elliptic system) behavior is that the boundary conditions fail to satisfy the complementing (or Lopatinsky-Shapiro) conditions.