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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

$q = {q_0} + mp'$