Derivatives of the Stretch, Rotation and Exponential Tensors in n-Dimensional Vector Spaces

Abstract

We present a solution for the tensor equation TX + XTT =H, where T is a diagonalizable ( in particular, symmetric tensor, which is valid for any arbitrary underlying vector space dimension n. This solution is then used to derive compact expressions for the derivatives of the stretch and rotation tensors, which in turn are used to derive expressions for the material time derivatives of these tensors. Some existing expressions for n = 2 and n = 3 are shown to follow from the presented solution as special cases. An alternative methodology for finding the derivatives of diagonalizable tensor-valued functions that is based on differentiating the spectral decomposition is also discussed. Lastly, we also present a method for finding the derivatives of the exponential of an arbitrary tensor for arbitrary n.