$(15$ points$)$ The following figures $($Figures $($a$)$ and $($b$))$ show the stress paths for a triaxial consolidated undrained test for a cylinder sample under axisymmetric loading conditions.

a$)$ Explain why the total stress path $($TSP$)$ has a $3$:$1($vertical:horizontal$)$ slope using three$-$dimensional stress.

b$)$ Explain why the effective stress path $($ESP$)$ has a vertical slope using threedimensional stress.

c$)$ Explain why the total stress path $($TSP$)$ has a $45-$degree slope using twodimensional stress.

Figure $($a$)$ Stress Path Using Three$-$Dimensional Stress

$3.$ Stress paths

$a_1$