On the derivative of some tensor-valued functions

An explicit expression of the derivative of the square root of a tensor is provided by using the expressions of the derivatives of the eigenvalues and eigenvectors of the simmetric tensor. Starting from this result, the derivatives of the right and left stretch tensor U, V and of the rotation R with respect to the deformation gradient F, are calculated. Expressions for the material time derivatives of U, V and R are also given.