Calculation of the effective moduli of a random packing of spheres using a perturbation of the uniform strain approximation

We consider a dense, random packing of spheres. To calculate the bulk modulus of the medium, we apply a displacement to the boundary. The displacement of the sphere centres and hence the contact forces are determined using a perturbation of the uniform strain approximation. Only the case of a hydrostatic compression is considered, although the method may be used for other initial states and for additional simplicity it is assumed that the spheres are either infinitely rough or perfectly smooth.