Model: Modified Cam-Clay

This model may be used to represent a solid material with plastic response to loading according to the Modified Cam-Clay (MCC) critical state model. The MCC yield function is defined as:

f = q^2 + M^2 p(p - p_c) = 0 ,

where p_c is the preconsolidation pressure, and M is the slope of the critical state line (CSL). M can be related to the critical state friction angle \phi_{cs} as

M = \frac{6 \sin \phi_{cs}}{3-\sin \phi_{cs}}.

Here f represents the yield surface, as shown in Figure 6.


Fig. 84 Cam-Clay and Modified Cam-Clay yield surfaces in p-q space (Borja, 2013).

Here we use a hyper-elastic constitutive law using the following elastic rate constitutive equation

\dot{p} = - \frac{p}{c_r} \dot{\epsilon}^e_v,

where c_r > 0 is the elastic compressibility index. The tangential elastic bulk modulus is K=- \frac{p}{c_r} and varies linearly with pressure. We assume a constant shear modulus, and can write stress invariants p and q as

p = p_0 \exp \left( \frac{\epsilon_{v0} - \epsilon_v^e}{c_r}\right) , \quad q = 3 \mu \epsilon_s^e,

where p_0 is the reference pressure and \epsilon_{v0} is the reference volumetric strain. The hardening law is derived from the linear relationship between logarithm of specific volume and logarithm of preconsolidation pressure, as show in Figure 7.


Fig. 85 Bilogarithmic hardening law derived from isotropic compression tests (Borja, 2013).

The hardening law describes evolution of the preconsolidation pressure p_c as

\dot{p_c} = - \frac{tr(\dot{\boldsymbol{\epsilon}}^p)}{c_c-c_r} p_c,

where c_c is the virgin compressibility index and we have 0 < c_r < c_c.


The supported attributes will be documented soon.


  <ModifiedCamClay name="mcc"
                   defaultVirginCompressionIndex="0.003" />