Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation

The paper discusses the issue of discretization of the strain–configuration relationships in the geometrically exact theory of three–dimensional (3D) beams, which has been at the heart of most recent nonlinear finite–element formulations. It is demonstrated that the usual discretization procedures for implementing these strain measures within a finite–element framework violate the important property of objectivity: invariance to rigid body rotations. A method is proposed for overcoming this limitation, which paves the way for an objective finite–element formulation of the geometrically exact 3D beam theory. For a two–noded element, this method involves obtaining the relative rotation matrix that rotates one nodal triad onto the other and then interpolating the resulting rotation vector.