Polar and Cartesian representations of mechanical quantities in granular materials

Characterizing mechanical quantities in granular materials is essential for understanding the relationship between macroscopic behavior and microstructural features. A key step in this process is the anisotropic decomposition of the spatial distributions of these quantities. However, studies on arbitrary-order anisotropic expansions are still scarce, and the problem remains challenging. Building on the introduction of projection operators, this study presents the expressions of irreducible tensor bases in two- and three-dimensional Cartesian coordinates and establishes the correspondence between Cartesian expansions and Fourier or spherical harmonic expansions in polar coordinates. A general theoretical framework is proposed for describing the distribution of mechanical quantities in both coordinate systems, together with the relations between the anisotropy coefficients in the two frames. Polar expansion coefficients, easy to compute, are converted into Cartesian coefficients with clearer physical meaning, allowing macroscopic properties to be explained by microscopic mechanical distributions. Finally, the theory is validated through discrete element simulations of 2D super-elliptic and 3D super-ellipsoidal systems, where stress and elastic modulus anisotropy are calculated to explain macroscopic properties. The proposed method simplifies anisotropy characterization without order limitations.