Stress analysis
==========================

Mohr circle
-------------------------

The mohr circle quantifies the relationship between normal stress, shear stress, and rock failure. 
Here I present the deriviation process.


.. figure:: ./Mohr_deriviation.jpg
   :align: center

   Stress distribution on a slope plane (Zeng et al., 2008, Structural Geology, Third edition)

The stable condition will requires below two conditions: :math:`\sum{F_n}=0` and :math:`\sum{F_\tau}=0`, 
whcih corresponds to the normal and shear stress equilibrium. Therefore,

.. math::
   \sigma A_{\theta} - \sigma_1\cos\theta A_{\theta}\cos\theta - \sigma_2\sin\theta A_{\theta}\sin\theta = 0

.. math::
   \tau A_{\theta} + \sigma_2\cos\theta A_{\theta}\sin\theta - \sigma_1\sin\theta A\theta\cos\theta = 0

Then we have:

.. math::
   \begin{align}
   \sigma &= \sigma_1\cos^2\theta + \sigma_2\sin^2\theta \\
   &=\sigma_1\frac{1+\cos{2\theta}}{2} + \sigma_2\frac{1-\cos{2\theta}}{2}\\
   &=\frac{\sigma_1+\sigma_2}{2}+\frac{\sigma_1-\sigma_2}{2}\cos\theta
   \end{align}

.. math::
   \begin{align}
   \tau &= (\sigma_1-\sigma_2)\cos\theta\sin\theta \\
        &= \frac{\sigma_1 - \sigma_2}{2}\sin{2\theta}
   \end{align}
          
Finally, we then have the relationship between (effective) normal stress and shear stress:

.. math::
   (\sigma - \frac{\sigma_1+\sigma_2}{2})^2+\tau^2 = (\frac{\sigma_1-\sigma_2}{2})^2