$($a$)(4$ pts$)$ A structural component is fabricated from an alloy that has a plane$-$strain fracture toughness of $\backslash (62$ M P a $\backslash$sqrt$\{$m$\}\backslash ).$ It has been determined that this component fails at a stress of $250$ MPa when the maximum length of a surface crack is $1\mathrm{.}6$ mm $.$ What is the maximum allowable surface crack length $($in mm $)$ without fracture for this same component exposed to a stress of $250$ MPa and made from another alloy that has a plane strain fracture toughness of $\backslash (51$ M P a $\backslash$sqrt$\{$m$\}\backslash )?($b$)(6$ pts$)$ Steady$-$state creep data taken for an iron at a stress level of $140$ MPa are given here:

If it is known that the value of the stress exponent $\backslash ($ n $\backslash )$ for this alloy is $8\mathrm{.}5,$ compute the steadystate creep rate at $\backslash (1100^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and a stress level of $83$ MPa $.($c$)(6$ pts$)$ A cylindrical $1045$ steel bar is subjected to repeated compressiontension stress cycling along its axis $($see right$).$ If the load amplitude is $23,000$ N $,$ calculate the minimum allowable bar diameter $($in mm $)$ to ensure that fatigue failure will not occur. Assume a factor of safety of $2\mathrm{.}0.$