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.