Stresses in two-dimensional isostatic granular systems: exact solutions

It has been recognized that the concept of isostaticity holds the key to understanding stress transmission in cohesionless granular media. Here the field equations of isostaticity theory in two-dimensions are studied and solutions are derived. The equations are first decoupled into integro-differential equations for the three independent stress components, highlighting the role of a particular position-dependent fabric tensor. In disordered, but statistically isotropic, systems the fluctuations decay with length-scale and the decoupled equations can be expanded to first order and solved. The zero-order solutions are obtained in closed form and give rise to force chains that propagate along straight characteristic lines. At this order solutions do not attenuate and chains cross one another without interference or scattering. The analysis of the first-order correction reveals emergence of weaker secondary force chains that branch off the zero-order chains and into a 'cone of influence' that they span.