# 33.7: Original Cam-Clay Model

- Page ID
- 33028

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

\[ \sigma ' = \sigma - u \]

This allows us to define the general mean effective compressive stress as

\[ p' = \frac{1}{3}(\sigma '_1 + \sigma '_2 + + \sigma '_3) \]

this case σ_{a} = σ_{1} and σ_{r} = σ_{2} = σ_{3} and so

\[ p' = \frac{1}{3}(\sigma '_a + 2 \sigma '_r ) \]

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

\[ q = \sigma '_a - \sigma '_r \]

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:

\[ \delta \varepsilon_a = \frac{\delta l}{l} \]

Volumetric strain:

\[\frac{\delta v}{v}=\varepsilon_{v}=\delta \varepsilon_{a}+2 \delta \varepsilon_{r}\]

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}\]

The work done per unit volume by elastic straining is

\[\begin{aligned}

\delta W &=p^{\prime} \delta \varepsilon_{v}+p \delta \varepsilon_{s} \\

&=\frac{1}{3} \sigma_{a}^{\prime}+2 \sigma_{r}^{\prime} \delta \varepsilon_{a}+2 \delta \varepsilon_{r}+\frac{2}{3} \sigma_{a}^{\prime}-\sigma_{r}^{\prime} \delta \varepsilon_{a}-\delta \varepsilon_{r} \\

&=\sigma_{a}^{\prime} \delta \varepsilon_{a}+\sigma_{r}^{\prime} 2 \delta \varepsilon_{r}

\end{aligned}\]

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

\[p^{\prime} \delta \varepsilon_{v}+q \delta \varepsilon_{s}=\delta W=M p^{\prime} \delta \varepsilon_{s}\]

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

\[d p^{\prime} \delta \varepsilon_{v}+d q \delta \varepsilon_{s} \geq 0\]

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} :

\[\frac{\delta \varepsilon_{v}}{\delta \varepsilon_{s}}+\frac{q}{p^{\prime}}=M\]

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

\[\frac{\delta \varepsilon_{v}}{\delta \varepsilon_{s}}=-\frac{d q}{d p^{\prime}}\]

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

\[\frac{q}{p^{\prime}}-\frac{d q}{d p^{\prime}}=M\]

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

\[\begin{array}{l}

\frac{d \eta}{d p}=\frac{1}{p^{\prime}}\left(\frac{d q}{d p^{\prime}}-\frac{q}{p^{\prime}}\right)=-\frac{M}{p^{\prime}} \\

\Rightarrow \frac{d \eta}{M}+\frac{d p^{\prime}}{p^{\prime}}=0

\end{array}\]

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

\[\begin{array}{c}

\int_{M}^{\eta} \frac{1}{M} d \eta=-\int_{p_{c}^{\prime}}^{p^{\prime}} \frac{1}{p^{\prime}} d p^{\prime} \\

\frac{\eta}{M}-1=-\ln \left(\frac{p^{\prime}}{p_{c}^{\prime}}\right) \\

\frac{q}{M p^{\prime}}=1-\ln \left(\frac{p^{\prime}}{p_{c}^{\prime}}\right)

\end{array}\]

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.