The stress-dilatancy relation for static equilibrium of an assembly of particles in contact

The dilatancy and strength of an assembly of individual particles in contact when subjected to a deviatoric stress system is found to depend on the angle of friction $\phi_\mu$ between the particle surfaces, on the geometrical angle of packing, $\alpha$, and on the degree of energy loss during remoulding. The Mohr-Coulomb criterion of failure which is strictly applicable to a continuum is shown not to have general application to a discontinuous assembly of particles. A theoretical and experimental study of ideal assemblies of rods and uniform spheres establishes expressions for the relation between the rate of dilatancy and the maximum stress ratio for any ideal packing. The solution is extended to the case of a random assembly of irregular particles by investigating the conditions under which the mass dilates such that the rate of internal work absorbed in frictional heat is a minimum. Experiments on random masses of steel, glass, and quartz in which all the physical properties are measured independently show that the minimum energy criterion is closely obeyed by highly dilatant dense over-consolidated and reloaded assemblies throughout deformation to failure. An additional rate of energy has to be supplied to account for losses due to rearranging of loose packings, when the value of $\phi$ to satisfy the theory increases to $\phi_f$ by an amount dependent on the degree of remoulding. The external stresses applied to an assembly are to be integrated over the $\alpha$-plane defined with reference to figure 14(a) as a plane of repetition of pattern over which the particles interlock, and the resulting forces are to be in equilibrium for sliding on particle interfaces at (45-$\frac{1}{2}\phi_f$) to the direction of the major principal stress. For the special case of no volume change these two planes are identical and the solution agrees then with that based on the Mohr-Coulomb theory. The well-known slip plane in drained discontinuous assemblies is proved to be the result of failure and nothing whatsoever to do with the peak strength. The findings are discussed in the light of previous contributions to the subject.