$($a$)$ A sand is tested at its densest state under a drained condition and the mobilized friction angle becomes $38.$ The friction angle of the sand is $27$

$(5$ Marks$)$

$($b$)$ The critical state lines $($CSL$)$ and the normally consolidated lines $($NCL$)$ for a reconstituted clay have been determined to be:

$CSL$: $q_{cs}=M{p}_{cs}^{\text{'}}$

$v_{cs}=2\mathrm{.}01-0\mathrm{.}14ln{p}^{\text{'}}$

$NCL$: $v=2\mathrm{.}1-0\mathrm{.}14ln{p}^{\text{'}}$

where $p^{\text{'}}=\frac{{}_1^{\text{'}}+2{}_3^{\text{'}}}{3},q={}_1-{}_3,$ and $v$ is the specific volume. The soil parameter $=0\mathrm{.}02.$

An isotropically consolidated undrained compression $($CIUC$)$ triaxial test was carried out on the clay under an effective consolidation stress of $320$ kPa $.$ The clay was normally consolidated $($NC$)$ under this consolidation stress.

$($i$)$ Given that $\frac{q_{cs}}{p^{\text{'}}}=0\mathrm{.}85,$ calculate the undrained shear strength, the pore$-$water pressure at failure and the friction angle from the CIUC test.

$($ii$)$ Sketch schematically the likely effective stress path $q-p$ and the specific volume $v$ versus $p^{\text{'}}$ curves. Plot the CSL and NCL with all the values marked clearly on the same graphs.

$($iii$)$ If the above CIUC test was carried out by firstly consolidating the soil specimen under $320$ kPa and then under $100$ kPa before the undrained shearing, what would be the undrained shear strength obtained from the test?

$(20$ Marks$)$