The theory studies idealized granular systems—assemblies of material discs and spheres—in conditions of static equlibrium under external loads. Particles are regarded as deformable solids interacting by means of contact forces. The objective of the study is to develop means of phenomenological description suitable for efficient analysis of stress fields in granular masses. Systems are analyzed using principles of mechanics applied at the particulate level and using developed statistical and statistico-geometrical methods.

The phenomenological stress tensor is introduced as a fluctuating function of volume through volume-additive combinations of a system's micro-parameters, contact forces and orientations of contact normals. It is shown that the introduced tensor has properties of the continuum mechanics stress tensor in the limit of an infinite homogeneous assembly under homogeneous loads on infinity.

The average of the phenomenological stress tensor over an ensemble of macroscopically similar systems under inhomogeneous loads has all the field properties of the continuum mechanics stress tensor which becomes related to average forces on contacts of the same orientation and to the contact orientation distribution. Under minimal assumptions regarding the form of directional variation of forces, the latter are expressed in terms of stress tensor invariants and the influence of anisotropy in contact orientations is investigated. It is shown that the displacement gradient considered in continuum mechanics can be introduced through conditions of particle micro-compatibility in contacts.

Distributions of contact forces for assemblies of bonded particles are developed using principles of the information theory. Constitutive relationships for assemblies of bonded particles with linear contact interactions are derived by minimization of complementary work of internal deformations. For isotropic assemblies of spheres, the tensor of elasticity is isotropic;​ Poisson's ratio is given as a function of the ratio of tangential to normal stiffnesses and is automatically within thermodynamic limits. Poisson's result (ν = 1/4) corresponds to zero tangential contact stiffness. Fluctuations of the phenomenological stress tensor in finite volumes are assessed.

The anisotropic case is investigated for plane assemblies only. Resulting equations of linear elasticity correspond to a non-symmetrical stress tensor, and an assumed lack of moment transfer in contacts restricts the deformational modes of anisotropic systems. Deformations are unrestricted for particles with vanishing tangential stiffnesses. The resulting rela- tionships correspond to transversely isotropic plane materials. Deforma- tion moduli are expressed in terms of parameters defining the shape of contact orientation distributions:​

The statistico-geometrical analysis of plane systems is aimed at determining admissible shapes of contact orientation distributions compat- ible with the introduced condition of local stability expressed in geo- metrical form. The analysis results in a realtionship between the para- meters of anisotropy and the system's density. The types of admissible anisotropies qualitatively differ for loose and dense assemblies.

The condition of stability also admits a relationship between the average coordination number and the system's density. The existence of maximum random and minimum stable densities is established. Compari- sons with experimental data support these findings.

Assemblies of cohesionless particles are analyzed in the plane case. It is conjectured that the deformational process is of a quasi-dynamic nature due to continuous micro-cycles of "hardening" and "softening". It is suggested that the magnitudes of dissipative tangential contact forces are insignificant and can be neglected. It is also assumed that, due to low intensity of dynamic processes, a system of contact forces for any fixed structure corresponds to minimum potential energy related to deformations of grains. It is shown that the assumption enables to formally employ stress-strain characteristics of bonded particles to determine stress fields in assemblies of cohesionless particles.

Determination of stress fields from a point source on a homogeneous anisotropic granular half-plane indicates that stress distributions qualitatively differ for loose and dense assemblies. Comparison with the results of plate loading tests on granular materials shows definite qualitative similarity in the theoretical and experimental stress distributions.

The theory of earth pressure (at rest) relates K₀ to parameters of structural anisotropy, so that stress fields in plane homogeneous granular assemblies are controlled by the coefficient of earth pressure (at rest) and their relative density.