On the Almansi-Michell solution and its numerical implementation for heterogeneous beams with periodic microstructures subject to periodically-varying loads

Abstract

This paper aims to present a general solution to the Almansi-Michell problem of heterogeneous beam structures with periodic microstructures, which take the most general form of periodic configurations with different constituent materials, subject to periodically varying external loads. By analogy with Ieşan’s rational scheme, the displacement solution is generated by introducing periodic functions in place of warping functions for constant cross-sectional beams. The governing equations for unknown displacements and the linear system of equations for undetermined coefficients are non-trivially derived in detail. By properly handling surface integrals, improved FE equations for the displacement solution and unknown coefficients, which are more efficient in numerical computation, are then formulated. A novel numerical implementation approach is further developed, which adopts general FE software/in-house FE code as a black box and calculates the Almansi-Michell solution with simple vector arithmetic. The proposed numerical implementation approach can effectively handle a wide range of microstructures. Several numerical examples are given to show the correctness of the Almansi-Michell solution and the effectiveness of the numerical implementation approach.