In class, we solved the problem of a shaft $($piece of chalk$)$ under torsion to demonstrate that a brittle material fails at an angle of $45$ degrees from the longitudinal axis on the periphery of the shaft. Consider the same problem but this time with a compressive force in the longitudinal direction as shown. The shaft is solid cylindrical with diameter $D=10mm.$ The polar moment of inerti$]$ of a cylindrical rod is $J=\frac{D^4}{32}.$

$(1)$ Determine the $3$D state of stress at a point of point $B_1$ as shown. Draw the state of stress on the cube. Make sure to specify the coordinate system.

$(2)$ Determine the three principal stresses $)$ and the angle of the triaxial state of stress. Draw the $3$D triaxial state of stress on a properly oriented cube. Show all steps of your work clearly. Make sure to specify the orientation of the cube and the appropriate stress values on each visible face of the cube.

$(3)$ If the piece of chalk was going to fail $($fracture$),$ at what angle would it most likely fail? Make a sketch of the piece of chalk and the crack.

$(4)$ Use the Maximum Normal Stress theory to determine whether or not the piece of chalk will fail under this load assuming that the ultimate strength of chalk is $0\mathrm{.}2$MPa in tension and $0\mathrm{.}8$MPa in compression. Use a factor of safety of $1\mathrm{.}0.$ Specify whether it fails under tension or compression.