Periodic and “slightly” periodic boundary value problems in elastostatics on bodies bounded in all but one direction

Abstract

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)

Multiple and maximal periodicity in linear elastostatics

“…In particular, we shall obtain a version of Betti's reciprocal theorem appropriate to strain states which are periodic in an arbitrary number of directions. This result is analogous to a previously obtained formula (Theorem 4.1 in [4]) and, in fact, Theorem 4.1 of [4] could be obtained as a corollary of the theorem reported here. One would be hard pressed, however, to base any derivation of the reciprocal theorem developed in this paper on the results reported in [4].…”

“…And so, after making use of the symmetry of S. If, in Theorem 5.2, both elastic states are the same, then formula (5.1) becomes a theorem of work and energy for m-tuply periodic strain states. This new theorem of work and energy generalizes that given by Theorem 4.1 of [4] as well as a version used in [5] and, along with Theorem 5.1 of this paper, can be used to extend most of the results reported in those papers to problems involving multiple periodicity. Most of these extensions are quite straightforward and will be omitted from further discussion in this paper.…”

“…5°1, the surface on which u is prescribed, is non-trivial. 4. The problem is a traction problem and ~ the mean stress over any period section, vanishes.…”

“…Several papers ( [4]- [8]) have appeared recently discussing various aspects of the general periodic and "slightly periodic" boundary value problem in classical elasticity. With the exception of the relatively special situation considered in [8], however, all this work has only involved periodicity in one direction.…”

“…With the exception of the relatively special situation considered in [8], however, all this work has only involved periodicity in one direction. One can, of course, view problems involving periodicity in several directions as special cases of the problems discussed in [4] through [7], but a more satisfying approach would be to reexamine the "basic" theorems used there and to obtain genuinely more general versions from which the previous versions could possibly be derived as special cases (instead of vice versa). This will be the approach taken here.…”