$3.($a$)$ A loose sand sample with a critical state friction angle of $\backslash (28^\{\backslash$circ$\}\backslash )$ is tested in a triaxial cell under drained condition. The sand sample is also densified to a dense state with a friction angle of $\backslash (40^\{\backslash$circ$\}\backslash ).$ Calculate the dilatancy ratio.

$(5$ Marks$)$

$($b$)$ A reconstituted clay sample with $\backslash (\backslash$lambda$=0\mathrm{.}22$ ; $\backslash$Gamma$=2\mathrm{.}9$ ; $\backslash$kappa$=0\mathrm{.}02$ ; $\backslash$mathrm$\{$N$\}=3\mathrm{.}1\backslash )$ and $\backslash (\backslash$mathrm$\{$M$\}=0\mathrm{.}9\backslash )$ is isotropically consolidated to a cell pressure of $300$ kPa in a triaxial cell. The sample is subsequently sheared under drained condition. Determine the effective mean stress, deviatoric stress, void ratio and volumetric strain at failure.

$(6$ Marks$)$

$($c$)$ The same reconstituted sample as in Q$3($b$)$ is sheared under undrained condition after being subjected to the same istotropic consolidation pressure as in Q$3($b$).$ Calculate the effective mean stress, deviatoric stress and change in pore$-$water pressure at failure.

$(6$ Marks$)$

$($d$)$ The same reconstituted sample as in Q$3($b$)$ is first subjected to an isotropic consolidation pressure of $500$ kPa $.$ After the completion of consolidation, the isotropic pressure is subsequently reduced to $300$ kPa before subjecting the sample to undrained shearing.

$($i$)$ Calculate the specific volume before shearing.

$($ii$)$ Determine the undrained shear strength of the sample.

$($iii$)$ Compute the ratio of the undrained shear strength to that determined from the sample in Q$3($c$).$