$[18$ Points$]$ A clay sample was consolidated in a triaxial test chamber under an all$-$around confining pressure of $200k\frac{N}{m^2}.$ The sample was then loaded to failure in an undrained condition by applying an additional axial stress of $240k\frac{N}{m^2}.$ A pore water pressure sensor recorded an excess pore pressure of $140k\frac{N}{m^2}$ at failure. You may assume that the hydrostatic water pressure is equal to zero

$($i$)$ Is it a CD$,$ CU$,$ UU or UC triaxial test?

$($ii$)$ On the same plot, draw the effective$-$stress Mohr's circles right after consolidation stage and at failure state. Clearly label the axes and all the important points on the Mohr's circles $($e$.$g$.$ major and minor effective principal stresses$).$

$($iii$)$ On the same plot from $($ii$),$ draw the total$-$stress Mohr's circle at failure state. Clearly label all the important points on the Mohr's circle $($e$.$g$.$ major and minor total principal stresses$).$

$($iv$)$ Draw the total$-$stress and effective$-$stress Mohr$-$Coulomb failure envelops on the same plot from $($ii$)$ and $($iii$).$ Determine the total$-$stress friction angle and effective$-$stress friction angle. You may assume that the total$-$stress cohesion and effective$-$stress cohesion are equal to zero.

$($v$)$ What is the shear strength for this clay sample?

$($vi$)$ What is Skempton's pore pressure parameter $\frac{?}{\text{b}\text{a}\text{r}}(A{)}_f$ at failure?

$($vii$)$ Where is the location of the pole for the effective$-$stress Mohr's circle at failure? Also, find the orientation of the failure plane.

$($viii$)$ Suppose that another triaxial test is performed. The clay sample prepared under the same condition is subjected to the same confining pressure during the consolidation stage. However, drainage is allowed when the additional axial stress is applied. Sketch three diagrams for the second stage of this triaxial test: $($a$)$ deviator stress against axial strain; $($b$)$ volumetric strain against axial strain; $($c$)$ excess water pressure against axial strain.