NonlinearCamClay

Modified Cam-Clay Model

The NonlinearCamClay class defines a basic framework using modified Cam-Clay yield surface and associative flow rule. Theories can be seen in Chapter 10 in Computational Methods for Plasticity: Theory and Application, in which there are some minor errors in the derivation of consistent stiffness.

This model resembles the one in ABAQUS but the third stress invariant does not enter yield surface. As a result, the yield surface on the $\pi$-plane is a circle.

Interested readers can also refer to the corresponding section in Constitutive Modelling Cookbook for more details.

The following function is chosen as the yield surface.

$$F(\sigma,a)=\dfrac{(p-p_t+a)^2}{b^2}+\dfrac{q^2}{M^2}-a^2$$

where $p(\sigma)=\dfrac{1}{3}(\sigma_1+\sigma_2+\sigma_3)$ is the hydrostatic pressure, $q^2(\sigma)=\dfrac{3}{2}s: s$ with $s$ denotes the deviatoric stress, $a(\alpha)$ is a hardening function in terms of internal hardening variable $\alpha$ that is defined as volumetric plastic strain, that is $\alpha=\varepsilon_v^p$. $b=1$ when $p-p_t+a\ge0$ and $b=\beta$ when $p-p_t+a<0$. This $\beta$ parameter changes the radius of the second half of this ellipse on the compressive side of the hydrostatic axis. The constant $M$ modifies the radius of the ellipse along the $q$ axis.

yield surface

The same function is used for plasticity potential so that $G=F$ and

$$\dot{\varepsilon_p}=\dot{\gamma}\dfrac{\partial{}G}{\partial\sigma}=\dot{\gamma}(\dfrac{3}{M^2}s+\dfrac{2(p-p_t+a) }{3b^2}I)$$

where $I=[1~1~1~0~0~0]^\mathrm{T}$ is the second order unit tensor.

The NonlinearCamClay class allows $a(\alpha)$ to be user defined, where $\alpha$ is the volumetric plastic strain.

$$\alpha=\int\dot{\varepsilon}_v^p\mathrm{d}t.$$

Please note in practical applications, this value is negative as soil is often in compression.

History Layout

location	paramater
initial_history(0)	accumulated plastic strain