Statistical mechanics of stress transmission in disordered granular arrays
Transmission of stress and statistics of force fluc tuations in static granular arrays are fundamental, but unresolved, problems in physics [1,2]. Despite several theoretical attempts [3,4] and a vast engineering literature [5,6] the connectivity of granular media is still poorly un derstood at a fundamental level. In this Letter we propose a theory of stress transmission in disordered arrays of rigid cohesionless grains with perfect friction. A real granu lar aggregate (e.g., sand or soil) is a very complex object . However, simple models are easier to comprehend, and extra complexities can always be incorporated subse quently. In our case the rigid grain paradigm provides a crucial starting point from which to appreciate the theo retical physics of the problem. We model the granular material as an assembly of discrete rigid particles whose interactions with their neighbors are localized at pointlike contacts. Therefore the description of the network of in tergranular contacts is essential for the understanding of force transmission in granular assemblies. Grain a exerts a force on grain ß at a point 'Raß = Ra + raß. The contact is a point in a plane whose normal is n“^. The vector R“ is defined by
so that Ra is the centroid of contacts, and hence
where z is the number of contacts per grain and Σ/? means summation over the nearest neighbors. Hence Ra , r a^, and naß are geometrical properties of the aggregate under consideration and the other shape specifications do not enter. Friction is assumed to be infinite and the geometry is frozen after the deposition and cannot be changed by applying or removing an external force on the boundaries. In a static array Newton’s equations of intergranular force and torque balance are satisfied. Balance of force around the grain a requires
where і = 1,2,3 are Cartesian indices and g" is the external force acting on grain a . Further on in this Letter g is used also for the external forces at the boundaries.