Second-order torsion and bending of isotropic elastic cylinders

Abstract

The problem of the second-order torsion of a cylinder of compressible isotropic material is reduced to the solution of a single boundary-value problem involving two complex potential functions. Without solving this boundary-value problem, a formula for the fractional elongation of the cylinder is obtained in agreement with that found previously by Rivlin (1953). When the region occupied by any cross-section of the cylinder is bounded by a single closed curve the equation satisfied by the complex potentials reduces to that obtained by Green & Shield (1951) for the case of an incompressible cylinder, and a general method of solution was given by these authors. The problem of second-order bending of a cylinder of compressible isotropic material by couples over its plane ends is also reduced to the solution of a single boundary-value problem for two complex potential functions, in addition to the boundary-value problem for classical torsion of a cylinder. A general formula is found for the change in length of the line of centroids and one applicaton of the theory is made to right circular cylinders.