  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 $\leq$ 2. We
illustrate the utility of the result by applying it to obtain a representation
of the fluctuating velocity, pressure-gradient correlations of turbulence.
