# Uniqueness Theorems in Linear Elasticity by Robin John Knops, Lawrence Edward Payne (auth.)

By Robin John Knops, Lawrence Edward Payne (auth.)

The classical end result for strong point in elasticity concept is because of Kirchhoff. It states that the traditional combined boundary worth challenge for a homogeneous isotropic linear elastic fabric in equilibrium and occupying a bounded 3-dimensional area of house possesses at so much one resolution within the classical feel, supplied the Lame and shear moduli, A and J1 respectively, obey the inequalities (3 A + 2 J1) > zero and J1>O. In linear elastodynamics the analogous consequence, as a result of Neumann, is that the initial-mixed boundary price challenge possesses at such a lot one resolution supplied the elastic moduli fulfill an identical set of inequalities as in Kirchhoffs theorem. most traditional textbooks at the linear thought of elasticity point out purely those classical standards for forte and forget altogether the considerable literature which has seemed because the unique courses of Kirchhoff. To therapy this deficiency it sort of feels applicable to aim a coherent description ofthe a number of contributions made to the learn of distinctiveness in elasticity conception within the desire that such an exposition will supply a handy entry to the literature whereas even as indicating what growth has been made and what difficulties nonetheless watch for answer. clearly, the continued assertion of recent effects thwarts any try to offer a whole evaluation. except linear elasticity concept itself, there are a number of different parts the place elastic strong point is significant.

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Although none of the restrictions listed under I-VI can be justified from a thermodynamical standpoint, several attempts have been made to show that they lead to reasonable consequences. 5) to a number of physically reasonable properties that it was felt an elastic body should possess. The analysis was extended to finite deformations, and consequences of strong ellipticity C3, 6 See also Truesdell and Noll [1965 §51]. 7 Hill [1962] gave an interpretation of the strong ellipticity conditions equally valid in terms of steady velocity fields or in the context of elastostatics, whereas Zorski [1962], Shield [1965], and Knops and Wilkes [1966] explored the significance of the condition for stability.

8) Xl for some positive constant M. Then there is at most one classical solution of the displacement boundary value problem for an anisotropic nonhomogeneous material in a bounded domain B. To prove this theorem, we let r denote the distance from some origin inside B and we let a denote the radius of a circumscribing sphere with centre at the origin. 9) where as before Vi represents the difference of two solutions to the problem. Here we have used integration by parts and the fact that Vi vanishes on oB.

In proving that a certain condition is necessary for uniqueness, it must be shown that in all problems of the given class uniqueness implies satisfaction of the stipulated condition. Rather than embark upon this somewhat formidable task, we confront ourselves with the equivalent but simpler program of establishing the inverse of the logical implication. Thus, we may alternatively prove that: failure of the condition implies non-uniqueness in at least one example. Usually, the example is selected to be elementary as regards both geometry and material symmetry.