Problem $1.$ Drained Strength

Given the drained triaxial compression stress$-$strain plots and volume change plots for samples of a subrounded sand $($SP$)$ prepared to relative densities corresponding to $($a$)$ Dense, and $($b$)$ Loose soil states. Note the different vertical axis scales in $($b$).$

a$.$ Plot Mohr's circles at failure and interpret the strength parameters c$\text{'}$ and $\backslash (\backslash$phi$^\{\backslash$prime$\}\backslash )$ for each relative density. The values of $\backslash (\backslash$sigma$\_\{\backslash$mathrm$\{$df$\}\}\backslash )$ are marked in the figures by the arrows to save some time. Use c$\text{'}\backslash (\backslash$mathbf$\{0\}\backslash$mathbf$\{0\}\backslash )$ when fitting the envelopes to the maximum obliquity plane on the Mohr's circles. This is real lab data so you may need to make a best fit by eye. There is no "exact" answer.

b$.$ Do the strength and volume change results make sense with our class discussion, your Lab $7$ observations, and the Lab $7$ Notes regarding drained strength mechanics? Why or why not?