The emergence of Airy stress function in two-dimensional disordered packings of particles

The packing of hard-core particles in contact with their neighbors is a statically determinate problem. The probability functional method allows deriving the full system of equations for the stress tensor components analytically. For an isotropic and homogeneous two-dimensional packing, we derive the classical Euler–Cauchy and Navier equations; the latter allows expressing the stress tensor of the system in terms of the Airy stress function.