Configurational forces and geometrically exact formulation of sliding beams in non-material domains

Abstract

The paper presents a geometrically exact, ALE finite element formulation for dynamic problems of nonlinear sliding beams. Previous researches on finite element methods for sliding beams have not taken into account the equation of configurational momentum and boundary condition of equilibrium of configuration forces at the sliding interfaces. In this paper, dual quaternions are used to represent the exact kinematics of rigid-section motion of the beam. Hamilton’s principle of variation of action is applied to derive the equations of mechanical and configurational momentum, and also boundary conditions of equilibrium of mechanical and configurational forces. The material and mesh motions of the cross-sections, material and mesh variations of motions, and also variation of the non-material domain due to sliding motion are correctly considered to get the complete set of governing equations and boundary conditions. In the finite element implementation, proper integration by parts is employed to weakly enforce natural boundary conditions of mechanical and configurational forces, and also to avoid terms of higher order spatial derivatives in the inertial forces. The configurational variables of internal nodes are associated with the arclength coordinates at the beam’s tips to avoid redundancy of the configurational momentum equation. These proposed strategies lead to robust ALE elements for geometrically exact beams. Numerical examples of increasing complexities are presented to validate correctness of the proposed approach.