Integral Representation of the Solution of Torsion of an Elliptic Beam with Microstructure

Abstract

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 a 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 that are essential to establish the rigorous solution of the problem (see, for example, [4]). 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 antiplane displacement and microrotation fields. This is in contrast to the relatively simple torsion problem arising in classical linear elasticity, in which a single antiplane 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 noncircular cross section it is extremely difficult (if not impossible) to find a closed-form analytic solution to the torsion problem. In this paper, we use a simple, yet effective, numerical scheme based on an extension of Kupradze’s method of generalized Fourier series [6] to approximate the solution of the problem of torsion of an elliptic micropolar beam. Our numerical results demonstrate that the material microstructure does indeed have a significant effect on the torsional function and the subsequent warping of a typical cross section.