Proof that CD is a Straight Line
Line CD represents the behaviour of soft soil during plastic yielding and flow at the critical state C, and on the wet side of the critical state C, through the Original Cam-Clay model. As shown below, C is defined by a point (Γ, M) in (vλ, η) space, while, as will also be shown below, D is defined by a point (Γ + λ - κ, 0).
If, at a constant value of p', and therefore a constant value of ln p', q is altered for a wet soil in a triaxial stress test, so that as q increases, v decreases, there has to be a form of plastic distortion within the wet soil aggregate at yield so that macroscopically it exhibits what civil engineers and geotechnical engineers term ductile behaviour.
In q − p' space, the yield locus of Original Cam-Clay between C and D takes the form of a curve satisfying the equation
\[ \frac{q}{{Mp'}} = 1 - \ln \frac{{p'}}{{{{p'}_C}}} (1)\]
as shown here. In this equation M is the critical state friction constant, i.e., the frictional coefficient exhibited by this granular assembly, while p'C is the mean normal effective pressure at C for the soil under consideration, normalised with respect to a standard stress of 1 kPa.
In order to represent this curve between C and D in q - p' space in (vλ, η) space, consideration has to be given to what happens to v as a function of ln p' for wet soils in the Original Cam-Clay model.
Experimentally, it is found that, for a given reduction in volume Δv at constant ln p' for soils on the wet side of critical, all points formerly lying on a line
\[ v = {v_1} - \lambda \ln p'\]then move to a line
\[ v = {v_2} - \lambda \ln p' \]i.e., a series of ‘λ-lines’ can be drawn. Hence, on a graph of v against ln p', these lines will be parallel lines with the same slope −λ, but with different intercepts where ln p' = 0, i.e., where ln p' is 1 kPa. Hence, in general, the equations of these λ-lines take the general form
\[ {v_\lambda } = v + \lambda \ln p' \]for different values of vλ.
Therefore, different points on the Original Cam-Clay yield locus on a q - p' plot satisfying the equation
\[ \frac{q}{{Mp'}} = 1 - \ln {\rm{ }}(p'/{p'_C}) \] i.e., \[ \frac{\eta }{M} = 1 - \ln {\rm{ }}(p'/{p'_C}) (2) \]
will lie on λ-lines with different vλ when plotted in v - ln p space. Furthermore, it is also found experimentally that all the points on the Original Cam-Clay yield locus on a q - p' plot also fall on a line in (v, ln p') space with the equation
\[ {v = {v_\kappa } - \kappa \ln p'} (3) \]
for a particular value of κ < λ.
At C, we know that
\[ {v_{\rm{C}}} = {v_\kappa } - \kappa \ln {p'_{\rm{C}}} \]
At D, where η = 0, so that the soil is unable to withstand any shear stress imposed on it, we know that
\[ {v_{\rm{D}}} = {v_\kappa } - \kappa \ln {p'_{\rm{D}}} = {v_\kappa } - \kappa \ln e{p'_{\rm{C}}} = {v_\kappa } - \kappa - \kappa \ln {p'_{\rm{C}}} = {v_{\rm{C}}} - \kappa \]
since, using Equation (2), \( {p'_{\rm{D}}} = e{p'_{\rm{C}}} \) when η= 0. Therefore, at D,
\[ {v_\lambda } = {v_{\rm{D}}} + \lambda \ln {p'_D} = {v_{\rm{C}}} - \kappa + \lambda \ln e{p'_{\rm{C}}} = {v_{\rm{C}}} + \lambda \ln {p'_{\rm{C}}} + \lambda - \kappa = \Gamma + \lambda - \kappa \]
and so in (vλ, η) space, D is defined by a point (Γ + λ - κ, 0).
Suppose we now consider a general point B which falls on the yield locus in q - p' between C and D with specific volume vB and a mean normal effective pressure p'B.
Using Equation (3),
\[ {v_\kappa } = {v_{\rm{C}}} + \kappa \ln {p'_{\rm{C}}} = {v_{\rm{B}}} + \kappa \ln {p'_{\rm{B}}} (4) \]
and using Equation (2),
\[ \frac{{{\eta _{\rm{B}}}}}{M} = 1 - \ln {\rm{ }}({p'_{\rm{B}}}/{p'_C}) (5) \]
In addition,
\[ {v_{\lambda {\rm{,B}}}} = {v_{\rm{B}}} + \lambda \ln {p'_{\rm{B}}} (6) \]
Hence, using Equations (4) and (6),
\[ {v_{\lambda {\rm{,B}}}} = {v_{\rm{B}}} + \lambda \ln {p'_{\rm{B}}} = {v_{\rm{C}}} + \kappa \ln {p'_{\rm{C}}} + (\lambda - \kappa )\ln {p'_{\rm{B}}} (7) \]
Using Equation (5),
\[ \ln {\rm{ }}{p'_{\rm{B}}} = \frac{{{\eta _{\rm{B}}}}}{M} + 1 + \ln {\rm{ }}{p'_C} (8) \]
and so Equation (7) can be rearranged in the form
\[ {v_{\lambda {\rm{,B}}}} = {v_{\rm{C}}} + \kappa \ln {p'_{\rm{C}}} + (\lambda - \kappa ){\rm{ }}\left( {\frac{{{\eta _{\rm{B}}}}}{M} + 1 + \ln {\rm{ }}{{p'}_C}} \right) \] i.e.,
\[{v_{\lambda {\rm{,B}}}} = {v_{\rm{C}}} + \lambda \ln {p'_{\rm{C}}} + (\lambda - \kappa ){\rm{ }}\left( {\frac{{{\eta _{\rm{B}}}}}{M} + 1} \right) = \Gamma + (\lambda - \kappa ){\rm{ }}\left( {\frac{{{\eta _{\rm{B}}}}}{M} + 1} \right) \]
or, equivalently,
\[ (\lambda - \kappa ){\rm{ }}{\eta _{\rm{B}}} = M{v_{\lambda {\rm{,B}}}} - M(\Gamma + \lambda - \kappa ) (9) \]
Hence, removing the subscript ‘B’, it is evident that the equation of the yield locus CBD in (vλ, η) space is
\[ (\lambda - \kappa ){\rm{ }}{\eta _{\rm}} = M{v_{\lambda {\rm}}} - M(\Gamma + \lambda - \kappa ) (10) \]
This is an equation of a straight line in (vλ, η) space satisfied by (Γ, M) and (Γ + λ − κ, 0). Hence, we have shown that the Line CD represents the yield surface in (vλ, η) space for the Original Cam-Clay model.