Macro–micro relations in granular mechanics

In granular mechanics, macroscopic approaches treat a granular material as an equivalent continuum at macro-scale, and study its constitutive relationship between macro-quantities, such as stresses and strains. On the other hand, microscopic approaches consider a granular material as an assembly of individual particles interacting with each other at micro-scale (i.e., particle-scale), and the physical quantities under study are forces and displacements. This paper aims at linking up the findings from these two scales and to establish the macro–micro relations in granular mechanics.Three aspects of the macro–micro relations are investigated. They are about the internal structure, the stress tensor and the strain tensor. The internal structure is described with geometrical systems at the particle scale. Micro-structural definitions of the stress and strain tensors are derived, which link the macro-stress tensor with the contact forces, and the macro-strain tensor with the relative displacements at contact. In addition to a brief review of the past research work on these topics, further generalizations are made in this paper. In particular, the two cell systems proposed by Li and Li (2009), namely the solid cell system and the void cell system, are introduced and used for the derivation of the macro-structural expressions. The stress expression is derived based on Newton’s second law of motion. The result is valid for both static and dynamic cases. The strain expression is derived based on the compatibility requirement. And the expression is valid for any tessellation subdividing the granular assembly into polyhedral elements.The homogenization for deriving a macroscopic constitutive relationship from microscopic behaviour is discussed. Attention is placed on the macroscopic quantification of the internal structure in terms of a second rank tensor, known as the fabric tensor. Existing definitions of the fabric tensors have been reviewed. The correlations among different fabric tensors and their relations with the stress–strain behaviours have been investigated.