Multiplicative Updating of the Rotation Tensor in the Finite Element Analysis of Rods and Shells - A Path Independent Approach

Rotation tensors play a pre dominant role in many engineering applications. They exhibit a pronounced multiplicative structure, the various aspects of which must be dealt with carefully in order to arrive at a numerically efficient and physically sound treatment. A method of multiplicative updating of rotations in the frame of finite element analysis of rods was suggested by Simo and Vu-Quoc which proved to be path-dependent, even in purely elastic problems, as observed by Jelenic and Crisfield. In this paper a path-independent treatment of rotations is developed which proves to be numerically efficient, physically sound, and preserves the multiplicative structure of rotations. In addition, a unified treatment of rod and shell theories is established which considers them from the point of view of Cosserat continua with same degrees of freedom. In the shell case, the formulation allows in a natural way for the inclusion of drill rotations.