Problem $1$

$($a$)$ Consider a single crystal of some hypothetical metal that has the FCC crystal structure and is oriented such that a tensile stress is applied along a $[\backslash (\backslash$overline$\{102\}\backslash )]$ direction. If slip occurs on a $(111)$ plane and in a $[101]$ direction, compute the stress at which the crystal yields if its critical resolved shear stress is $3\mathrm{.}42$ MPa $.(\backslash (\backslash$mathbf$\{12\}\backslash )$ Marks$)$

$($b$)$ Single crystal of aluminum is oriented for a tensile test such that its slip plane normal makes an angle of $\backslash (28\mathrm{.}1^\{\backslash$circ$\}\backslash )$ with the tensile axis$-$ Three possible slip directions make angles of $\backslash (62\mathrm{.}4^\{\backslash$circ$\}\backslash ),\backslash (72\mathrm{.}0^\{\backslash$circ$\}\backslash ),$ and $\backslash (81\mathrm{.}1^\{\backslash$circ$\}\backslash )$ with the same tensile axis. Which of these three slip directions is most favored? If plastic deformation begins at a tensile stress of I. $95$ MPa $(280$ psi $),$ determine the critical resolved shear stress for aluminum $(\backslash (\backslash$mathbf$\{8\}\backslash )$ Marks$)$