Example 22: Constant volume cylindrical compression with Modified Cam Clay.

In soil mechanics, a constant volume cylindrical compression represents the so-called undrained triaxial test. The calculation was performed using a single -quad4 element under axisymmetric conditions. Only small deformations were considered. The parameters of the Modified Cam Clay model correspond to the Weald clay according to [2]: $M=0.882$, $\lambda=0.088$, $\kappa=0.031$, $G=3000$ kN/m$^2$. Initial (effective) stress was isotropic with the mean stress $p'=207$ kN/m$^2$. The void ratio $e=0.627$ remained constant during the compression due to the imposed constant volume condition. The initial preconsolidation mean pressure was $p_0=207$ kN/m$^2$.

The calculated stress-strain curve ($q$ is the stress deviator and $\varepsilon_a$ is the compressive axial deformation) in the left figure corresponds exactly to the numerically integrated equations of the model. The stress path in the right figure coincides with the analytical solution, see [12]:

$$\begin{eqnarray*} \displaystyle \frac{p_i'}{p'} = \left( \frac{M^2+\eta^2}{M^2+\eta_i^2} \right)^\Lambda \end{eqnarray*}$$

where $\eta=q/p'$ and $\Lambda=(\lambda-\kappa)/\lambda$. The subscript $i$ denotes initial values.