An Energy Method for Certain Second-Order Effects With Application to Torsion of Elastic Bars Under Tension

Abstract

When a mechanical system has a potential energy, it is a simple matter to show that if the generalized force corresponding to a coordinate p is known to first order in p for a range of the other coordinates of the system, then the other generalized forces can be found immediately to second order in p, without requiring a second-order analysis of the system. By this method the second-order change in the axial force when a finitely extended elastic cylinder is twisted is found from the first-order value of the twisting moment. Numerical results for a realistic form of the strain-energy function for an incompressible material suggest that the second-order expression for the axial force is very accurate for a wide range of twist for circular cylinders of rubber-like materials extended 100 percent or more.