The Derivative of Isotropic Tensor Functions, Elastic Moduli and Stress Rate: I. Eigenvalue Formulation

We derive basis-free formulae for the derivative of general isotropic tensor functions of symmetric tensors, and the formulae for elastic moduli and stress rate for compressible isotropic elastic materials. Part I of this paper concerns the tensor functions with the representation of spectral decomposition. The solution is obtained through the directional derivatives on certain subspaces that span the set of symmetric tensors.