Isotropic tensor-valued polynomial functions of fourth-order tensors

TL;DR

This paper develops a polynomial representation for isotropic fourth-order tensor-valued functions based on second-order tensors, with applications demonstrated in turbulence modeling.

Contribution

It introduces a new representation formula for isotropic fourth-order tensor functions as polynomials in symmetric second-order tensors, simplifying complex tensor functions.

Findings

Provides a practical representation formula for complex tensor functions.

Demonstrates application in turbulence velocity and pressure-gradient correlations.

Facilitates modeling of fourth-order tensor functions in various fields.

Abstract

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.