Application of Spencer’s Ideal Soil Model to Granular Materials Flow

In 1964, Spencer proposed a model for plane deformation of soils based upon the concept that the strain at any point may be considered as the resultant of simple shears on the two surface elements where Coulomb’s yield condition is met. Gravitation and acceleration terms were neglected in his full field equations. These terms are included in the present treatment, however, since they play an essential role in granular materials flow problems. It is shown that the field equations remain hyperbolic, and the characteristic equations are derived. In addition, a streamline equation, similar to Bernoulli’s classical equation for fluid flow, is derived and used together with the continuity equation to obtain one-dimensional approximate solutions to some typical hopper and chute problems. A solution is obtained for the nonsteady flow from a wedge-shaped hopper when the gate is suddenly opened, including an equation for the minimum slope necessary to prevent arching. Another solution is obtained for the profile of a hopper which has constant wall pressure. Still another solution is obtained for the relationship between the height of a chute and its exit velocity, and it suggests that the maximum trajectory usually is obtained with a horizontal exit since an upward-sloping exit requires a velocity jump at the minimum point in the chute, similar to a hydraulic jump in fluid flow. All of these solutions are ideal in the sense that they include no wall resistance to the flow, and therefore represent maximum flow rates.