On the Plane Strain Deformations of Critical State Models for Sands

Abstract

The deformation and flow of sands and similar granular materials are characterised by (a) the fact that shearing is frequently accompanied by volume changes - either positive or negative and (b) the fact that unlike metals and other crystalline materials the deformation behaviour is strongly dependent on the ambient hydrostatic pressure. Thus, in constructing plasticity models to describe their behaviour, it is necessary to allow the yield function and flow potential to depend on the current density and mean pressure in addition to the deviatoric components of stress. The critical state models described by Schofield and Wroth [1] are examples of these types of plasticity theories. However, their theory assumes a normal or associated flow rule which works well for normally consolidated clays but is inadequate to explain the observed behaviour of sands and granular materials. We are hence concerned with plasticity models with distinct yield functions f and flow potential g, both of which are functions of the stress tensor σij (compressive normal stresses taken as positive) and the specific volume v, defined as the ratio of the total volume to the volume of the solid grains in a material element. Such materials possess a critical state in which the material behaves incompressibly and the associated deformation is isochoric. The state of stress in such a critical state is obtained by eliminating the specific volume v between the yield condition and the condition of zero volume change arising from the flow rule, l.e. between

$$f(\sigma ij,v)=0and\partial g(\sigma ij,v)/\partial {{\sigma }_{kk}}=0$$

(1)

Any successful phenomenological theory in continuum mechanics must be validated by parallel considerations of conceptual micromechanical models. In the flow of granular materials this amounts to looking for planes or surfaces on which some critical condition necessary for local slip to occur is achieved. In the primitive Coulomb model of a cohesionless material, slip is assumed to occur when the ratio of the shear traction ti too, the normal traction on a plane, reaches a critical value. The effect of dilatancy - either positive or negative- can be incorporated in a crude manner by considering the sliding of serrated rather than smooth blocks over each. Models of this sort have been discussed by Taylor [2], de Josselin de Jong [3] and Atkinson and Bransby [4]. The latter authors show that this model leads to a stress-dilatancy condition of the form:

$$(\tau /\sigma )=(-d{{\varepsilon }_{v}}/dy)+\mu $$

(2)

where dε_v and dγ are the normal and shear strain increments across the slip plane, the former being positive when compressive. The constant p. is the value of the stress ratio in the critical state. A very similar expression has been derived by Matsuoka [5] from observations of the behaviour of aluminium rods in model tests. The difference being the presence of a factor x which can have values between 1.1 and 1.5, in front of the strain-increment ratio term in (2). This relation is the basis of the comprehensive ‘spatially mobilised plane’ theory developed by Matsuoka and collaborators - see reference [6] for a recent review of this theory. A number of more detailed studies of the deformation of assemblies of granular particles using statistical arguments which predict dilatancy relations similar to (2) have been made by Nemat-Nasser and co-workers, e.g. [7]. Since (2) involves strain-increments as well as stress components it will be related to the flow rule of the corresponding phenomenological plasticity theory.