Characterization of orientational order for arbitrary-shaped particles and its applications in granular assemblies

Abstract

Within granular assemblies, different arrangement structures of the particles can also affect the macroscopic mechanical properties of the system. Characterizing the particle order parameter is a prerequisite for studying the properties of the system. The order parameter of arbitrarily shaped particle can be characterized by Wigner D-matrices or the super-tensor basis, and the super-tensor basis more clearly explains the physical meaning of the order parameter. However, the expressions for the super-tensor basis are complex, and explicit expressions for orders higher than three are rarely provided in the literature. This paper reveals the relationship between the super-tensor basis and the projection operators, which decomposes symmetric tensors into irreducible tensors. Explicit expression for the projection operator is provided, and the consistency with the existing recursive expression and differential expression is demonstrated. Based on this, the expression for super-tensor basis of any order is derived, and the orthogonality of super-tensor basis is provided. Finally, the sound propagation in a biaxial super-ellipsoid particle system is simulated by the discrete element method, and the sound velocity is also calculated. The results indicate that different arrangement structures can affect the sound velocity. From the perspectives of order parameter and elastic moduli, the differences in sound velocities of different packing structures are explained. The explicit expressions of the projection operator and the super-tensor basis provided in this paper facilitate the convenient calculation of the anisotropy of other physical quantities within the system, which can be utilized to elucidate the macroscopic mechanical properties of the system.