Critical state models

The inelastic constitutive theory provided in Abaqus/Standard for modeling cohesionless materials is based on the critical state plasticity theory developed by Roscoe and his colleagues at Cambridge (Schofield et al., 1968, and Parry, 1972). The specific model implemented is an extension of the modified Cam-clay theory. The discussion is entirely in terms of effective stress: the soil may be saturated with a permeating fluid that carries a pressure stress and is assumed to flow according to Darcy's law. The continuum theory of this two phase material is described in Continuity statement for the wetting liquid phase in a porous medium.

The basic ideas of the Cam-clay model are shown geometrically in Figure 1 to Figure 7. The main features of the model are the use of an elastic model (either linear elasticity or the porous elasticity model, which exhibits an increasing bulk elastic stiffness as the material undergoes compression) and for the inelastic part of the deformation a particular form of yield surface with associated flow and a hardening rule that allows the yield surface to grow or shrink.

A key feature of the model is the hardening/softening concept, which is developed around the introduction of a “critical state” surface: the locus of effective stress states where unrestricted, purely deviatoric, plastic flow of the soil skeleton occurs under constant effective stress. This critical state surface is assumed to be a cone in the space of principal effective stress (Figure 1) whose axis is the equivalent pressure stress, p.

Figure 1. Cam-clay yield and critical state surfaces in principal stress space.

The section of the surface in the $\mathrm{\Pi }$-plane (the plane in principal stress space orthogonal to the equivalent pressure stress axis) is circular in the original form of the critical state model: in Abaqus this has been extended to the more general shape shown in Figure 5. In the section of effective stress space defined by the equivalent pressure stress, p, and a measure of equivalent deviatoric stress, $\tilde{t}$ (the definition of $\tilde{t}$ is given later in this section), the critical state surface appears as a straight line, intersecting the pressure axis at $p_t$ (strength of the material in hydrostatic tension), with slope M (see Figure 2 and Figure 3). The modified Cam-clay yield surface has the same shape in the $\mathrm{\Pi }$-plane as the critical state surface, but in the $p$–$\tilde{t}$ plane it is assumed to be made up of two elliptic arcs: one arc passes through $p_t$ with its tangent at right angles to the pressure stress axis and intersects the critical state line where its tangent is parallel to the pressure stress axis, while the other arc is a smooth continuation of the first arc through the critical state line and intersects the pressure stress axis at some nonzero value of pressure stress ($p_c$), again with its tangent at right angles to that axis (see Figure 4). Plastic flow is assumed to occur normal to this surface.

The hardening/softening assumption controls the size of the yield surface in effective stress space. The hardening/softening is assumed to depend only on the volumetric plastic strain component and is such that, when the volumetric plastic strain is compressive (that is, when the soil skeleton is compacted), the yield surface grows in size, while inelastic increase in the volume of the soil skeleton causes the yield surface to shrink. The choice of elliptical arcs for the yield surface in the ($p,\tilde{t}$) plane, together with the associated flow assumption, thus causes softening of the material for yielding states where $p<a+p_t$ (to the left of the critical state line in Figure 2, the “dry” side of critical state) and hardening of the material for yielding states where $p>a+p_t$ (to the right of the critical state line in Figure 3, the “wet” side of critical state).

Figure 2. Shear test response on the “dry” side of critical state ($p<a+p_t$).

The resulting stress-strain behavior under states of constant effective pressure stress but increasing shear (deviatoric) strain is shown in Figure 2 and Figure 3: following initial yield (which is governed by the initially assumed yield surface size; that is, by the extent of initial overconsolidation) strain softening or strain hardening occurs until the stress state lies on the critical state surface when unrestricted deviatoric plastic flow (perfect plasticity) occurs. The terms “wet” and “dry” come from the idea of working a specimen of soil by hand. On the wet side of critical state the soil skeleton is too loosely compacted to support pressure stress—such stress, if applied (such as by squeezing the soil by hand) passes immediately into the pore water and thus causes this water to bleed out of the specimen and wet the hands. The opposite effect occurs when the soil is on the dry side of critical state.