$(1)$ Plastic deformation concepts:

a$)$ Define a slip system.

b$)$ List the principal slip system for FCC and BCC crystal structures, and state the number of independent slip systems.

c$)$ Define qualitatively the principal slip system for HCP crystal structures, and state the number of independent slip systems.

$(2)$ Given: single crystal of FCC metal. Tensile stress is applied in $[012]$ direction. If the critical resolved shear stress is $50$MPa, calculate the magnitude of applied stress

$(3)$ Given: single crystal of BCC metal oriented such that a tensile stress with a magnitude of $60$MPa is applied in the $[$ii$12]$ direction.

a$)$ Compute the maximum resolved shear stress on the $(011)$ plane.

b$)$ If ${}_{CRSS}=30$MPa, will slip occur on this plane?

$(4)$ The strengthening mechanisms discussed in class are dependent on restricting or hindering dislocation motion. Explain how dislocations are involved in the following strengthening mechanisms:

a$)$ strain hardening $($cold working$)$

b$)$ grain size reduction

c$)$ solid solution strengthening

$(5)$ Cite the differences between recovery and recrystallization.

$(6)$ An undeformed specimen of some metal alloy has an average grain size diameter of $1\mathrm{.}0mm.$ Explain in qualitative terms how the average grain size diameter could be reduced to $0\mathrm{.}03mm.($The given numbers are arbitrary $-$ you will not need to solve any equations.$)$

$(7)$ It is necessary to select a metal alloy for an application that requires a yield strength of at least $310$MPa while maintaining a minimum ductility $(\%$EL$)$ of $27\%.$ If the metal may be cold worked, which of the following are candidates: copper, brass, or $1040$ steel?