Discussion of the paper “Tensorial form definitions of discrete mechanical quantities for granular assemblies” [M. Satake, Int. J. Solids and Structures 2004, 41(21), pp. 5775–5791]

Abstract

The aim of this Discussion is to clarify a terminological issue in a previous IJSS paper. Introductory remarksSatake (2004) proposes a definition for stress and strain for an assembly of unequal spheres with the help of contact cells, a system of polyhedra carrying the stresses as well as the deformations of the assembly according to the approach of the paper. The assemblies considered by Satake ( Fig. 1) consist of spherical particles that may have contacts with each other. The spheres may intersect, but because of the mechanical background of the problem, these intersections are small compared to the sphere radii.The definition of contact cells is based on a cell complex determined by the power planes of the spheres. (The power plane of two spheres is the set of those points having equal tangent lengths to the two spheres.) This cell complex is named 'Dirichlet tessellation' in my paper Bagi (1995), and in Satake (2004) citing my work.Satake and myself were informed recently that the cell complex we called 'Dirichlet tessellation' for a collection of unequal spheres is known under other names in the mathematical literature. In order to clarify this terminological confusion, an overview will be given here on the history and on the different existing names of this and similar other tessellations. (The figures are in 2D, but most of them are illustrations of 3D systems as indicated in the text.) 0020-7683/$ -see front matter Ó