On the compatibility equations in geometrically exact beam finite element

This paper discusses the compatibility equations which relate the velocity field and the strain field in geometrically exact beam theory. The analysis is carried out in the context of intrinsic equations, namely the dynamic equilibrium equations are formulated in terms of velocity and strain only. In addition to the well established objectivity and path-independence requirements of the spatial discretization, these compatibility equations show that a consistent spatial interpolation of the velocity field should depend on the curvature of the beam, including initial curvature and curvature from the deformation, and it is shown that this consistency is connected to the ability of the element to represent rigid body motion velocity. A two node interpolation scheme is studied and it appears that, as the element gets smaller under mesh refinement, the effect of this dependency reduces, leading eventually to the classical linear shape functions.