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Mapping of Mohr-Coulomb theory onto (q,p') space

Mohr-Coulomb theory is a theory in (τ, σ) space, or, if there is pore water pressure u, (τ,σ') space where σ' = σ – u.

When considering how to map Mohr-Coulomb theory onto (q, p') space, we can first plot the yield surface in (τ, σ') space in the way conventionally plotted in soil mechanics, with compressive stresses positive:

Failure occurs when the plane subjected to a normal stress \({\sigma'_{n}}\) represented on the horizontal axis by the point B is further subjected to a shear stress of magnitude BA. If we represent BA by τ, then

\[ {\tau} = \tau ^* + {\sigma'_{n}}\tan \phi       (1) \]

In this failure criterion \({\sigma'_{1}}\)and \({\sigma'_{3}}\) are the maximum and minimum compressive principal stresses. The third compressive principal stress, \({\sigma'_{2}}\) , is between these two and is not specified. If we choose the third compressive principal stress to be the arithemetic mean of \({\sigma'_{1}}\) and \({\sigma'_{3}}\), i.e.,

\[{\sigma '_2} = \frac{{{{\sigma '}_1} + {{\sigma '}_3}}}{2}\]

then the centre of Mohr’s circle is simply p'. Furthermore, if we identify q with the largest possible deviatoric stress, then q = \({\sigma'_{1}}\) \({-}\) \({\sigma'_{3}}\), i.e., the diameter of the Mohr’s circle.

With these links, simple geometry shows that

\(\frac{{p' - {{\sigma '}_n}}}{{q/2}} = \sin \phi \) and \(\frac{\tau }{{q/2}} = \cos \phi \)

Hence, we can transform equation (1) into the equation

\[\frac{q}{2}\cos \phi = \tau^* + (p' - \frac{q}{2}\sin \phi )\tan \phi \]

which simplifies to the equation

\[q = 2\tau^ *\cos \phi + 2p'\sin \phi \]

i.e., an equation of the form

\[q = {q_0} + mp'\]