Solving the torsion problem for an anisotropic prism by the advanced Kantorovich–Vlasov method

Abstract

Introduction. The torsion problem for elastic prismatic bodies has been addressed by engineers and mathematicians for the last 100 years. The unremitting interest in this problem is due to practical needs (many structural elements are designed to resist torsion) and theoretical needs (the deformation of such elements should be studied to reveal the mechanism of their failure). Only two classical special cases of torsion of a rod acted upon by opposite torques at its ends are completely understood: pure torsion described by the rigorous theory developed by Saint Venant [5, 6] and generalized torsion theoretically studied by Voight [16] and Lekhnitskii [3]. In the former case, only two tangential stresses appear nonzero, the prism axis remains straight, and cross-sectional warping is constant throughout the height. Such a behavior is typical of isotropic, orthotropic, and anisotropic materials with the plane of elastic symmetry perpendicular to the prism axis. In the latter case, all the stresses are nonzero yet constant throughout the height, the prism axis bends, and the cross-sectional warping linearly varies throughout the height. This case of torsion is observed in materials with general anisotropy or anisotropic materials with one plane of elastic symmetry nonorthogonal to the prism axis. The classical torsion problem was generalized in different ways such as inclusion of large strains or stress concentration around cracks, analysis of stability and vibrations under various torsional loads, complication of the body shape, material properties (viscosity), and loading and boundary conditions [10–15, 17]. The present paper addresses the torsion problem for a rectangular prism made of anisotropic materials with low order of symmetry and the stress–strain state varying throughout the height. Use will be made of the general problem statement in linear elasticity. The problem-solving method to be used reduces the original three-dimensional problem to three coupled one-dimensional problems, each for one of the variables of the domain. The deformation of the cross-section and axis of a square-based prism will be studied for different types of anisotropy of the material. 1. Problem Statement. Consider a rectangular prism occupying a Cartesian domain VV x y z x a y b { , , :| | , | | , 0 zc }. The z-axis is directed along the height of the prism and the x- and y-axes along the principal axes of inertia of the cross-section. The prism base z 0is fixed, the opposite face zc (end) is free from loads. The prism is twisted about the z-axis by stresses a yz (, ) and b xz (, ) acting on the lateral surfaces along the tangent to the boundary and distributed nonuniformly