Isotropic tensor-valued polynomial functions of fourth-order tensors

Fourth-order tensor-valued functions appear in numerous fields of study. The formulation of practical models for these complex functions often requires their representation in terms of tensors of order two. In this paper, we develop an appropriate representation formula by assuming that the isotropic fourth–order tensor–valued function is a polynomial function in the components of two symmetric second-order tensors of degree $\le$ 2. We illustrate the utility of the result by applying it to obtain a representation of the fluctuating velocity, pressure-gradient correlations of turbulence.