Measuring the coefficient of internal friction for soil in spherically symmetric plastic flow

Abstract

In the theory of the static and dynamic behavior of soft soils, wide use is made of Coulomb's law of dry friction:T .... r'+kon, (1) where r m and qn are the tangential and normal stresses on the area of maximum shear, and k and r' are coefficients characterizing the internal friction and cohesion (tenacity) of the soil.Note that Coulomb's law is equally applicable to media which are plastic or loose before loading (clay, loess, or sand) and to hard rocks in the crushing stage of a blast wave, i.e., In problems concerning the explosion of a concentrated charge in soil, the dimensions and stability of underground cavities formed by blasting, and so forth. The coefficients r' and k must be found experimentally. In this article we suggest a method of determining these quantities in the case of a static spherically symmetric load. The method is illustrated by determining the coefficient of internal friction of dry sand.Mathematical Expressions. Suppose that we have a spherical volume of soil loaded from the outside by a pressure P2 and having a spherical cavity at its center. Let the internal and external radii be R 1 and R 2, respectively.Suppose that at first the boundary of the cavity ls stationary (e. g., suppose that it is prevented from collapsing by a rigid porous sphere covered by an elastic film). Now increase the pressure In the cavity and observe the surface of the sphere. From the elasttcoplastic representation of the behavior of soil (here by elastic and plastic we mean the absence and presence of flow, respectively), we get the following picture. As the internal pressure Pi increases, at first the soil behaves "elastically," Then at some moment a state of limiting equilibrium is attained at the boundary of the cavity, and on further increase of Pl the elasticoplastic boundary moves from the center toward the periphery. Intensive flow starts when the boundary emerges at the surface of the sphere so that the whole volume of soil Is in a limiting state.A knowledge of the pressures Pt and P2 at which flow begins at the exterior surface will enable us to determine the constant coefficients in Coulomb's law. We must emphasize that in all our calculations we are assuming that they are constant; in any case this will be a good approximation in a not too wide range of pressures, and in our case is also confirmed by the experimental results.The equation of equilibrium of a continuous medium with spherical symmetry is d% 2 (% --%) ~)Institute of Physics of the Earth (IFZ), Moscow.