Rotation manifold SO(3) and its tangential vectors

In this paper, we prove that incremental material rotation vectors belong to different tangent spaces of the rotation manifold SO(3) at a different instant. Moreover, we show that the material tangent space as the tangent space at unity is not a possible definition yielding geometrically inconsistent results, although this kind of definition is widely adopted in applied mechanics community. In addition, we show that the standard Newmark integration scheme for incremental rotations neglects first order terms of rotation vector, not third order terms. Finally, we show that the rotation interpolation of extracted nodal values on the rotation manifold is not an objective interpolation under the observer transformation. This clarifies controversy about the frame-indifference of geometrically exact beam formulations in their finite element implementations.