# Original Cam-Clay model

The original Cam-clay model (OCC) was developed by Andrew Schofield in the 1960s as a description of the behaviour of saturated soil and sands. It shows how, depending on water content, soils can fail by spalling or by plasticity and liquefaction.

Consider a cylinder of water-saturated sand in a triaxial testing regime as in the above figure.

The cylinder is subjected to the total axial stress, σ_{a}, and total radial stress, σ_{r}. It is more useful to work in terms of the effective stress which takes pore water pressure, u, into account.

\[\sigma ' = \sigma - u (1)\]

This allows us to define the general mean effective compressive stress as \( p' = \frac{1}{3}(\sigma_{1}' + \sigma_{2}' + \sigma_{3}') \). In this case σ_{a} = σ_{1} and σ_{r} = σ_{2} = σ_{3} and so \[ p' = \frac{1}{3}(\sigma_{a}' + 2\sigma_{r}') (2)\]

A deviator stress, *q*, can also be defined, given by the equation

\[q = {\sigma_{a}'} - {\sigma_{r}'} (3)\]

In triaxial yield testing it is found that sands and soils on the ‘wet’ side of the critical state yield on a ductile-plastic continuum. The plastic deformations arise as a change in the specific volume of the sample and a strain along its length. We can therefore define the following two strains:

Axial strain: \[\delta {\varepsilon _a} = \frac{{\delta l}}{l} (4)\]

Volumetric strain:\[\frac{{\delta v}}{v} = {\varepsilon _v} = \delta {\varepsilon _a} + 2\delta {\varepsilon _r} (5)\]

To find the triaxial shear strain, ε_{s}, we separate the axial and volumetric strains:

\[\begin{aligned} \delta {\varepsilon _s} = \delta {\varepsilon _a} - \frac{1}{3}\delta {\varepsilon _v} \\ = \frac{2}{3}{\delta {\varepsilon _a} - \delta {\varepsilon _r}} \end{aligned} (6)\]

The work done per unit volume by elastic straining is

\[\begin{aligned} \delta W & = p'\delta {\varepsilon _v} + p\delta {\varepsilon _s}\\ & = \frac{1}{3}{\sigma _{a}' + 2\sigma _{r}'} {\delta {\varepsilon _a} + 2\delta {\varepsilon _r}} + \frac{2}{3} {\sigma _{a}' - \sigma _{r}'} {\delta {\varepsilon _a} - \delta {\varepsilon _r}}\\ & = \sigma _{a}'\delta {\varepsilon _a} + \sigma _{r}'2\delta {\varepsilon _r} \end{aligned} (7)\]

For work done by plastic straining at failure for soils at the critical state or wetter than the critical state, the relevant dissipation function is defined by the equation

\[p'\delta {\varepsilon _v} + q\delta {\varepsilon _s} = \delta W = Mp'\delta {\varepsilon _s} (8)\]

where Μ is the general coefficient of friction. The work done against friction per unit volume is defined by this equation. The work done in producing a volume change does not explicitly appear in this model because it is a consequence of the interlocking of particles.

In OCC the associated plastic flow vector is locally orthogonal to the tangent of the yield locus so that

\[dp'\delta {\varepsilon _v} + dq\delta {\varepsilon _s} \ge 0 (9)\]

In words, this equation is a recognition that the scalar product of the plastic flow normal to the yield locus at (*p*', *q*) and the incremental loads (*dp*', *dq*) causing failure at (*p*', *q*) must be positive.

To derive the OCC we combine Equations (8) and (9). First we divide equation (8) by p’δε_{s} :

\[\frac{{\delta {\varepsilon _v}}}{{\delta {\varepsilon _s}}} + \frac{q}{{p'}} = M\]

Rearranging Equation (9) after setting the inequality to zero, we have:

\[\frac{{\delta {\varepsilon _v}}}{{\delta {\varepsilon _s}}} = - \frac{{dq}}{{dp'}}\]

so that eliminating δε_{v}/δε_{s} in these two equations, we produce the equation

\[\frac{q}{{p'}} - \frac{{dq}}{{dp'}} = M (10) \]

This is a differential equation on which we impose limits and introduce the stress ratio, η = q/p’. Differentiating η:

\[\frac{{d\eta }}{{dp}} = \frac{1}{{p'}}\left( {\frac{{dq}}{{dp'}} - \frac{q}{{p'}}} \right) = - \frac{M}{{p'}}\]

\[ ⇒ \frac{d\eta}{M} + \frac{dp'}{p'} = 0 (11) \]

As we are finding the locus for ‘wetter than critical’ states the integral is as follows:

\[ \begin{aligned}

\int_{M}^{\eta} \frac{1}{M}\,d\eta &=-\int_{p'_c}^{p'}\frac{1}{p'}\,dp' \\

\frac{\eta}{M}-1&=-ln\left(\frac{p'}{p'_c}\right) \\

\frac{q}{Mp'} =& 1 - ln\left(\frac{p'}{p'_c}\right)

\end{aligned} \]

Therefore, when *p*' is equal to *p*'_{c}, η= *M*. Also, when *q* = 0, so that the soil cannot withstand any shear stress at failure, *p*' = e* p*'_{c} ,where e is the base of the natural logarithm, 2.71828 … .

In practice *q* = 0 is actually a situation difficult to attain – see A. Schofield, *Disturbed Soil Properties and Geotechnical Design*, Thomas Telford Ltd., London, 2005, p. 106.