Elastoplastic deformation and the limit equilibrium of granular media

A theory of the deformation of a granular medium beyond the limit of elasticity is proposed, which takes into account both elastic and plastic deformations and enables the transition of the medium from a purely elastic state to a limit state to be traced. A system of functions is introduced which define the components of the stress and strain tensors, and a resolving system of equations of elastoplastic deformation of the medium is obtained for the case of a plane strained state. By taking the limit one can obtain the equations of the theory of limit equilibrium from these equations. The elastoplastic problem of the loading of a tube (a thick-walled circular cylinder) by internal pressure is solved. It is shown that the load corresponding to the transition of the tube to a purely plastic state over the whole cross section is not always identical with the load obtained using the limit equilibrium theory. The passage to the limit for which these two loads are the same is determined. The elastoplastic problem of the loading of a circular layer having a hole (a well) at the centre is solved in the new formulation. The depth of the layer for which a region of plastic deformation appears in the neighbourhood of the well are determined. The limit value of the depth at which the layer collapses is obtained. The residual stresses in the neighbourhood of the well when there is a repeated increase in the pressure to the value of the rock pressure are determined. The classical problem of the stressed state of a plane slope is solved in the elastoplastic formulation. A relation is established between the mechanical characteristics of the medium and the angle of the slope for which the slope always remains in the elastic state over the whole depth. It is shown that three different stress-strain states are possible in elastoplastic deformation. The transition to the limit state is traced and it is established that, unlike existing representations, plastic flow of the slope from a rigid plastic material is impossible for slope angles less than the angle of internal friction. For slope angles greater than the angle of internal friction plastic yield does not occur over the whole slope but only over its upper layer of a certain thickness.