7.6.1 Formulation of the Modified Cam⁃clay model
In the Modified Cam⁃clay model，a logarithmic relation is assumed between void ratio e and the mean effective stress p′in virgin isotropic compression，which can be formulated as
The parameterλis the Cam⁃clay isotropic compression index，which determines the compressibility of the material in primary loading.When plotting relation Eq.，7.60 in a e⁃ln p′diagram one obtains a straight line.During unloading and reloading，a different line is followed，which can be formulated as
The parameterκis the Cam⁃clay isotropic swelling index，which determines the compressibility of material in unloading and reloading.In fact，an infinite number of unloading and reloading lines exists in p′⁃e⁃plane，each corresponding to a particular value of the preconsolidation stress pp.
The yield function of the Modified Cam－clay model is defined as
The yield surface（f＝0）represents an ellipse in p′⁃q⁃plane as indicated in Figure 7.19.The yield surface is the boundary of the elastic stress states.Stress paths within this boundary only give elastic strain increments，whereas stress paths that tend to cross the boundary generally give both elastic and plastic strain increments.
In p′⁃q′⁃plane，the top of the ellipse intersects a line that we can be written as
This line is called the critical state line（CSL）and gives the relation between p′and q in a state of failure（i.e.the critical state）.The constant M is the tangent of the critical state line and determines the extent to which the ultimate deviatoric stress，q depends on the mean effective stress，p′.Hence，M can be regarded as a friction constant.Moreover，M determines the shape of the yield surface（height of the ellipse）and influences the coefficient of lateral earth pressure K
in a normally consolidated stress state under conditions of one⁃dimensional compression.
The preconsolidation stress，pp determines the size of the ellipse.In fact，an infinite number of ellipses exist，each corresponding to a particular value of pp.
Figure 7.19 Yield surface of the Modified Cam⁃clay model in p′⁃q⁃plane
The left hand side of the yield ellipse（often described as the“dry side”of the critical state line）may be thought of as a failure surface.In this region，plastic yielding is associated with softening，and therefore failure.The values of q can become unrealistically large in this region.
For more detailed information on Cam⁃clay type models，the readers can be referred to Muir Wood（1990）.