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The problem of the uniqueness of elastostatic solutions to various boundary value problems involving unbounded two-and three-dimensional bodies is considered. The general boundary value problem is first considered for anisotropic bodies on which the elasticity tensor is uniformly positive definite. The displacement problem is then considered for the cases where the body is homogeneous and isotropic and the elasticity tensor is strongly-elliptic. Finally, the traction problem on homogeneous, isotropic bodies is considered with fairly little restriction placed on the Lame moduli. In the first two cases no restriction is placed on the geometry of the body other than the normal assumptions allowing for the use of the divergence theorem. For the traction problem, however, it is assumed that one component of the outward normal to the boundary of the body almost never vanishes. In each case it is shown that the desired uniqueness results hold with fairly mild restrictions placed on the displacement or stress fields (depending on the problem) in a neighborhood of infinity. Where possible, the results are shown to hold even though the solutions may possess the sort of discontinuities and singularities commonly found in problems involving cracks, corners, and bonded interfaces.Related results involving the insolvability of certain non-trivial boundary value problems and the behavior of certain elastic states are, where appropriate, developed.

General properties of solutions to elastostatic boundary value problems in which some or all of the functions involved are periodic are studied with particular attention given to problems on bodies unbounded in one direction only. It is shown that, even though the displacement corresponding to a periodic strain may, in a very nontrivial sense, be nonperiodic, it does satisfy a "semiperiodicity" condition. In addition, a theorem of work and energy is derived for periodic strain states on bodies unbounded in only one direction. This formulation of the theorem of work and energy includes extra terms arising from the possible semiperiodicity of the displacement but only explicitly involves one component of the mean stress. This leads to a discussion of the uniqueness of periodic strain solutions to various boundary value problems. Conditions insuring uniqueness are obtained with the necessity of these conditions demonstrated by counter-examples. The degree to which uniqueness can fail is also studied and is shown to be limited.The next portion of the paper discusses the question of whether periodic boundary value problems must have, in some sense, periodic solutions. This leads naturally to the question of the uniqueness of solutions to boundary value problems which, in themselves, are not necessarily periodic but whose corresponding null boundary value problem is periodic. Positive results to both questions are obtained for several fairly broad classes of problems. Counter-examples are then cited to show the necessity of many of the assumptions used in deriving these results. PreliminariesWhenever convenient, the points in k-dimensional Euclidean space will be identified with the vectors in Nk in the standard manner -i.e., through the agency of a suitably chosen Cartesian Coordinate system. The induced orthonormal frame of vectors will be denoted by {e 1, e 2 ..... ek}. As usual, if v and T are, respectively, a vector and a (second rank) trensor, then vi will denote v • e i and T~ will denote e i • Te ~.The (elastic) body will be denoted by N and its closure by clN. It occupies an open connected subset of k-dimensional Euclidean space such that, if @ is any ball of finite radius, then 0(N C~ 9) consists of a finite union of smooth (k-D-dimensional manifolds with boundaries. Whenever a limit "as Ixl approaches infinity" is referred to, it is to be assumed that the limit is over points in N. Unless otherwise stated, k 0374-3535/81/030293-24500.20/0Journal o [ Elasticity 11 (1981)

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The asymptotic behavior of the doubly periodic strain state on a three-dimensional body is discussed. The components of the elastic state are initially assumed to be bounded by an arbitrary polynomial. It is then shown that the components can be approximated by second degree polynomials whose coefficients can be readily computed from data generally available in such problems. The error in using the approximation at different points in the elastic body is on the order of the reciprocal of any polynomial of the distance to the boundary of the body.

The asymptotic behavior of the general periodic strain elastic state is discussed. The components of the elastic state are initially assumed to be bounded by an arbitrary polynomial. It is then shown that many of the components--for the case of plane strain, all of the components--can be approximated by second degree polynomials whose coefficients can be readily computed from data generally available in such problems. The error in using the approximation at different points in the elastic body is on the order of the reciprocal of any polynomial of the distance to the boundary of the body.Consequences of these results are then discussed with regard to Saint-Venant's principle, the periodicity of solutions to periodic boundary value problems, the uniqueness of the solutions to periodic and "slightly periodic" boundary value problems, and various formulations of the theorem of work and energy.