ELEMENTS OF THE CRITICAL STATE MODEL
The equation for the yield surface is an ellipse given by
We can rewrite Equation
The theoretical basis for the yield surface is presented by Schofi eld and Wroth (1968) and Roscoe and Burland (1968). You can draw the initial yield surface from the initial stresses on the soil if you know the values of M and p9c.
Critical State Parameters:
Failure Line in (p9, q) Space The failure line in 1 pr, q2 space is
where qf is the deviatoric stress at failure, M is a frictional constant, and p9f is the mean effective stress at failure. By default, the subscript f denotes failure and is synonymous with critical state. For compression, M 5 Mc, and for extension, M 5 Me. The critical state line intersects the yield surface at p9c /2.
We can build a convenient relationship between M and f9cs for axisymmetric compression and extension and plane strain conditions as follows.
We know that
Axisymmetric Extension In an axisymmetric (triaxial) extension, the radial stress is the major principal stress. Since in axial symmetry the radial stress is equal to the circumferential stress, we get
An important point to note is that while the friction angle, f9cs, is the same for compression and extension, the slope of the critical state line in (p9, q) space is not the same (Figure ). Therefore, the failure deviatoric stresses in compression and extension are different. Since Me , Mc, the failure deviatoric stress of a soil in extension is lower than that for the same soil in compression.