On the derivative of some tensor-valued functions (Q5932252)

The author gives an explicit expression of the derivative of square root of a tensor by using the derivatives of eigenvalues and eigenvectors of a symmetric tensor. Subsequently, the derivatives of right and left stretch tensors \({\mathbf U}\), \({\mathbf V}\) and of rotation \({\mathbf R}\) with respect to deformation gradient \({\mathbf F}\) are calculated. See [\textit{Y.-C. Chen} and \textit{L. Wheeler}, J. Elasticity 32, No. 3, 175-182 (1993; Zbl 0805.73003); \textit{M. Šilhavý}, The mechanics and thermodynamics of continuous media. Text and Monographs in Physics. Berlin: Springer. xiv (1997; Zbl 0870.73004)], where the formulae express them in terms of exponential function. The derivatives \(\dot{\mathbf U}\), \(\dot{\mathbf V}\) and \(\dot{\mathbf R}\) with respect to time are alternative expressions to those in above-mentioned works.