Periodic and slightly periodic boundary value problems in elastostatics on bodies unbounded in several directions

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 a direction other than the direction “of periodicity”. 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. Conditions which assure the periodicity of the displacement corresponding to a periodic strain are developed as are conditions which assure that the solution to a periodic boundary value problem has periodic strain. This leads to a discussion of the uniqueness of the solutions to various boundary value problems which, in themselves, are not necessarily periodic but whose corresponding null boundary value problem is periodic. As a special example a uniqueness theorem for the displacement problem and a uniqueness theorem for the traction problem on a homogeneous (but not necessarily isotropic) half-plane are proven using the arbitrariness of the periodicity. Throughout the paper, counterexamples demonstrate the necessity of many of the conditions assumed.