= Solution
Use compression-positive <principal stresses> $\sigma_1\geq\sigma_2\geq\sigma_3$. The two-dimensional <Mohr's circle> shown is the $\sigma_1$–$\sigma_3$ circle: its horizontal axis is compressive <normal stress> $\sigma_n$, its vertical axis signed <shear stress> $\tau$, its centre is $C=(\sigma_1+\sigma_3)/2$, and its radius is $R=(\sigma_1-\sigma_3)/2$. The intercepts $(\sigma_1,0)$ and $(\sigma_3,0)$ represent <principal stress planes>, on which <shear stress> vanishes. The top and bottom points have $|\tau|=R$, the maximum <shear stress>.
In the <rock> sketch, $\theta$ is the counterclockwise angle from the $\sigma_1$ direction to the plane's <unit normal vector> $\mathbf n=(\cos\theta,\sin\theta)$. Choose the clockwise unit tangent $\mathbf t=(\sin\theta,-\cos\theta)$ and define $\tau=\mathbf t\cdot\boldsymbol\sigma^c\mathbf n$, with $\boldsymbol\sigma^c=\operatorname{diag}(\sigma_1,\sigma_3)$ compression positive. Thus a positive $\theta$ produces a positive $2\theta$ rotation on the plotted circle. The actual <traction> on the positive-normal face is $-\boldsymbol\sigma^c\mathbf n=-\sigma_n\mathbf n-\tau\mathbf t$. Reversing the tangent convention reverses the plotted shear sign, not the physical prediction.
\Image[/past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3-q3-mohr-stress.png]
{title=Stressed rock, Mohr circle, and the effect of pore pressure}
{description=Original illustrative construction with principal compressions 80 and 20 MPa and a plane normal 30 degrees from the maximum compression. The circle point is approximately (65,26) MPa at twice that angle. A separate panel translates the circle left by 15 MPa pore pressure at fixed total stress, bringing it to a frictional failure envelope.}
In three dimensions there are circles for each pair of <principal stresses>. The largest is the $\sigma_1$–$\sigma_3$ circle; a general three-dimensional plane requires all three <principal stresses> and its normal, rather than a single angle on this circle.
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