I am reading Geotechnical engineering by B.M. Das, 6th edition. In the chapter on shear strength of soil this is stated:
"It can be shown that $2\theta = 90 + \phi' $ " where $\theta$ is the angle of the failure plane in soil with major principal plane and $\phi'$ is the angle of internal friction.
My attempt at proving this:
Edit: I realise that I used $\alpha$ instead of $\theta$
$\mu=tan(\phi')$ where $\mu$ is the friction constant for shear force $F_{shear}$, on the plane parallel to the angle of failure, where $F_T$ is the force normal to that plane and $F_{shear}=\mu \cdot F_N$
On an element of soil with $\sigma_x,\sigma_y,dx,dy$ it can be shown that the force component normal to a plane, $F_N$, with angle $\alpha$ to the x-axis is $$cos(\alpha)\sigma_ydx-sin(\alpha)\sigma_xdy$$ and the force parallel to the plane,$F_{shear}$, is $$sin(\alpha)\sigma_ydx+cos(\alpha)\sigma_xdy$$
For a failure along the plane: $\mu F_N=F_{shear}\Leftrightarrow \mu=F_{shear}/F_N$
If we create a right triangle with sides $\sigma_xdy,\sigma_ydx$ and angle $\gamma$ we can express $F_{shear}/F_N=sin(\alpha+\gamma)/cos(\alpha+\gamma)=tan(\alpha+\gamma)\Leftrightarrow \phi=\alpha+\gamma$
Is this correct and where do I go from here?
