Ph.D Thesis
Ph.D Student Brand Meir An Elastic-Plastic Cosserat Point Element for Large Deformations of Impulsively Loaded Beams Department of Mechanical Engineering Professor Miles Rubin

Abstract

This thesis presents a Cosserat Point Element (CPE), based on the theory of a Cosserat point, for the numerical solution of problems of nonlinear elastic-plastic beams. This element has been developed for the solution of transient large motions of beams with rigid cross-sections. For simplicity attention has been confined to two-dimensional motions of the beam which includes axial extension, bending and tangential shear deformations. Constitutive equations have been developed for rate-independent response of a beam made from an elastic-perfectly-plastic material and for rate-dependent response of an elastic-viscoplastic material. Specifically, a yield function has been proposed which couples the inelastic responses of tension and shear. Another yield function has been proposed for bending which depends on a hardening variable that is determined by an evolution equation.  In contrast, with standard finite element approaches the CPE model needs no integration through the element region.  Also, time integration algorithms have been developed for the evolution equations which are solved implicitly without iterations. The model can be used for cyclic bending deformations even though it neglects known Bauchinger-type response and coupling of bending and tension. In the numerical formulation the beam is divided into N CPSs which are characterized by N cross-sections. The kinematics of the I'th CPE are determined by six scalar functions: two displacements of the midpoints of its cross-sections and two angles characterizing their orientations.  The kinetics of the CPE are characterized by the resultant forces and mechanical moments applied to each of these cross-sections.  The response of the CPE beam element was examined by comparing with classical solutions of tension, pure bending and pure shear of a single element.  Also, a few example problems were used to validate the CPE formulation for transient large motions of beams which are impulsively loaded. The results indicate that the CPE produces reasonably accurate response relative results in the literature and full three-dimensional calculations using ABAQUS.