Generalized Fourier Series Solution of the Torsion Problem in Linear Elasticity with Microstructure

摘要

The theory of micropolar elasticity [1] was developed to account for discrepancies between the classical theory and experiments when the effects of material microstructure were known to significantly affect the body's overall deformation. The problem of torsion of micropolar elastic beams has been considered in [2] and [3]. However the results in [2] are confined to the simple case of a beam with circular cross-section while the analysis in [3] overlooks certain differentiability requirements required to establish the rigorous solution of the problem. In neither case is there any attempt to quantify the influence of material microstructure on the beam's deformation. The treatment of the torsion problem in micropolar elasticity requires the rigorous analysis of a Neumann-type boundary value problem in which the governing equations are a set of three second order coupled partial differential equations for three unknown anti-plane displacement and microrotation fields [4]. This is in contrast to the relatively simple torsion problem arising in classical linear elasticity in which a single anti-plane displacement is found from the solution of a Neumann problem for Laplace's equation [5]. This means that in the case of a micropolar beam with a non-circular cross-section it is extremely difficult (if not impossible) to find closed-form analytical solution to the torsion problem.