Clarifying the representation of isotropic symmetric tensor-valued functions of two symmetric tensors

We reconsider the question of the representation of an isotropic symmetric second-order tensor-valued function of two symmetric second-order tensors. This question goes back roughly 70 years and represents the attempt to duplicate the stunning result possible when the function depends only on one tensor input. Over the years, there have been numerous attempts to derive a general representation theorem for such functions, and the literature is replete with conflicting statements as to the appropriate form. By focusing on the Cayley–Hamilton theorem for three-by-three matrices, we are able to arrive at a general result applicable to isotropic functions using nine tensor-valued basis functions. With the addition of an argument exploiting the Cayley–Hamilton theorem for two-by-two matrices, we are able to show that the complete function basis reduces to only involve eight tensor-valued basis functions. Our result clarifies which historically advocated representations are complete and non-redundant without need for complex qualifying cases based on the eigen-structure of the input tensors. The arguments are straightforward and only involve basic algebraic considerations with an intentional focus on Cayley–Hamilton-based reductions.