Tensorial form definitions of discrete-mechanical quantities for granular assemblies

This paper explains the necessity of tensorial form definitions of mechanical quantities in the discrete mechanics of granular assemblies and how to make such definitions. Particles are assumed to be circular (in 2D) or spheres (in 3D). First, we explain the Dirichlet tessellation and some important geometric tools defined from this tessellation, i.e. the contact and dual-contact cells and dual branch vectors, which become necessary for the tensorial form definition. Comparing the fundamental quantities in discrete and continuum mechanics and regarding their mechanical properties, especially of stress and strain, we propose a new tensorial form definition for all of discrete-mechanical quantities. Lastly discussions are made on the properties of these quantities, especially on the internal work and the compatibility condition of strain.