New formulation of graph-theoretical approach in the mechanics of granular materials

This paper proposes a new expression of discrete-mechanical quantities in a granular assembly and a new formulation of the graph-theoretical approach in the mechanics of granular materials. New mechanical quantities for void, i.e., void force, void displacement, etc., are introduced and relations between discrete-mechanical quantities of particle, contact and void are given in a very simple form using the discretized Schaefer operators which are also newly defined in this paper. The expressions of virtual work principle in a granular assembly are given using the discrete-mechanical quantities. Through these considerations the meaning of void quantities are clarified. The proposed theory is limited to the case of circular particles in a dense packing. However, it is shown that the present theory is easily extended to a more general case in which the shape of particle is arbitrary and the packing is not necessarily dense.