Determination of hourglass coefficients in the theory of a Cosserat point for nonlinear elastic beams

Abstract

The theory of a Cosserat point has recently been used [Int. J. Solids Struct. 38 (2001) 4395] to formulate the numerical solution of problems of nonlinear elastic beams. In that theory the constitutive equations for inhomogeneous elastic deformations included undetermined constants associated with hourglass modes which can occur due to nonuniform cross-sectional extension and nonuniform torsion. The objective of this paper is to determine these hourglass coefficients by matching exact solutions of pure bending and pure torsion applied in different directions on each of the surfaces of the element. It is shown that the resulting constitutive equations in the Cosserat theory do not exhibit unphysical stiffness increases due to thinness of the beam, mesh refinement or incompressibility that are present in the associated Bubnov–Galerkin formulation. Also, example problems of a bar hanging under its own weight and a bar attached to a spinning rigid hub are analyzed.