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

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.