Critical State Models

There are two critical state models, the Modified Cam Clay model and the Double Yield Surface Model. The Modified Cam Clay model is a time independent model and the Double Yield Surface model is a time dependent and time independent model.

Modified Cam Clay Model
The Modified Cam Clay model is an elasto-plastic model with non-linear elasticity prior to yielding.

Elastic Deformation
Elastic deformation is governed by the bulk modulus and Poisson's ratio. A constant Poisson's ratio is used in the model. The bulk modulus is related to the slope of the recompression line by:

where K = Bulk modulus;
e = current void ratio of the material;
p = effective mean stress =;
and = slope of the recompression line in e versus natural logarithm of p plot.

The slope of the recompression line is related to the swelling index, which is the slope of the recompression curve in e versus logarithm to the base 10 of p plot, by: Cr = 2.303;

Associated flow rule is used for the Modified Cam Clay model. The yield function is given by:

where M = critical state parameter;
pc = current isotropic consolidation stress;
p = effective mean stress;
J2 is the second stress invariants.

Double Yield Surface Model

The double yield surface model can be used for time dependent and time independent analysis. The constitutive model is based on the double-yield surface model proposed by Hsieh. The formulation employed in the model is consistent with Kavazanjian and Mitchell's postulate that the total deformation of cohesive soil can be separated into four interdependent volumetric and deviatoric, time independent and time dependent components.

The model employs the concept of "double-yield criteria," and is defined by the ellipsoid of the Modified Cam-Clay model (MCCM) and the Von Mises cylinder inscribed in the ellipsoid.

where the yield function based on MCCM yield criterion;
g is the yield function based on Von-Mises yield criterion;
are the effective stress components;
p' = effective mean stress;
M is the slope of the critical-state line in the p'-q plane;
is the isotropic preconsolidation pressure;
is the shear yield stress of the Von-Mises yield criterion.

In this model, the total strain-rate tensor is decomposed in the following manner:

where superscripts e and p' denote the time-independent elastic and plastic parts, respectively; superscript t denotes the time-dependent (creep) part; and subscripts f and g refer to the MCCM ellipsoid and Von Mises cylinder yield surfaces, respectively.

The elastic part of the time-independent components is evaluated by applying:

1. the generalized Hooke's Law;
2. the elastic stress-strain-tensor requires at least two independent material properties:
   a) The elastic bulk modulus Ke has the accepted form:
   where e is the current void ratio;
   is the recompression index, in natural logarithm scale.
   
   b) The elastic shear modulus is obtained by assuming the trace of the Modified Cam-Clay yield surface on the q-g plane, (the deviator stress-axial strain plane in the triaxial stress condition) is a hyperbola of the form:
   

   

   where = is the shear strain;
   a, b are the hyperbolic stress-strain parameters; is the failure ratio.

The elastic shear modulus is back calculated from the initial tangent modulus of the hyperbola curve by :

<\p> Kavazanjian and Mitchell considered that the time-dependent strain rate tensor can be divided into distinct but interdependent, volumetric and deviatoric components:
They postulated that these creep strain rate tensors can be determined using the following phenomenological volumetric and deviatoric expressions for creep.

where is the secondary compression coefficient;
e is the void ratio; exp is the exponential; is the volumetric age, relative to an initial reference time ; is the instant volumetric time, usually set to unity; A, ,m are the Singh-Mitchell creep parameters; is the instant deviatoric time, usually set to unity; is the deviatoric age relative to ; is the deviator stress level.

The total creep strain rate is evaluated by employing a non-associated flow rule for both the equivalent volumetric and deviatoric yield surfaces associated with the state parameters p' and q. It is only necessary to determine the size of these potential functions as for the ellipsoid F and for the cylinder G. By forcing the creep strain rate to satisfy the secondary compression law for volumetric creep and the Singh-Mitchell law for the deviatoric creep simultaneously, the creep strain rate tensor can be expressed as:

where I is the second-order identity tensor;
;
denotes the Euclidean norm.

The parameters for the model can be determined from triaxial tests and creep tests.