B$2($a$)$ For a metallic alloy specimen with a uniform cross$-$sectional area of $\backslash (325\backslash$mathrm$\{$~mm$\}^\{2\}\backslash ),$ a gauge length of $115$ mm and a modulus of elasticity of $115$ GPa $,$ if the maximum tensile load at which plastic deformation begins is $\backslash (89,375\backslash$mathrm$\{$~N$\}\backslash ),$ estimate the maximum elastic extension of the specimen.

$(4$ marks$)$

$($b$)$ A cylindrical specimen with a diameter of $10$ mm is stressed elastically in tension under a force of $\backslash (15,000\backslash$mathrm$\{$~N$\}\backslash ).$ Determine the reduction in diameter of the specimen. $($Given: Poisson's ratio and modulus of elasticity for this material are $0\mathrm{.}3$ and $150$ GPa $,$ respectively.$)$

$(4$ marks$)$

$($c$)$ Figure B $2$ below gives the fatigue curve of a cylindrical specimen made of Ti alloy. The cylindrical specimen has a diameter of $4$ mm and undergoes repeated tension$-$compression stress $\backslash ((\backslash$pm $\backslash$sigma$)\backslash )$ cycles along its axis. The maximum tensile and compression loads are $\backslash (+10,052\backslash$mathrm$\{$~N$\}\backslash )$ and $\backslash (5,026\backslash$mathrm$\{$~N$\}\backslash ),$ respectively. Determine the fatigue life of the specimen.

Figure B$2-$ Fatigue Curves of the Cylindrical Specimen made of Ti alloy

$($d$)$ The fatigue strength of a Ti$-$alloy casting is increased when subjected to Hot Isostatic Pressing $($HIP$).$ The strength is further increased when the HIP casting is subjected to shot peening. Briefly explain what could have happened during shot peening to lead to an extra increase in fatigue strength.

$($e$)$ The initial internal flaw $($crack$)$ length is $6$ mm in a structural aluminum having a plane$-$strain critical fracture toughness $(\backslash ($ K$\_\{$I C$\}\backslash ))$ of $\backslash (50\backslash$mathrm$\{$MPa$\}\backslash$sqrt$\{$m$\}\backslash ).$ Determine what $\backslash (\backslash \%\backslash )$ this internal crack can increase before this specimen would undergo a brittle fracture under an applied stress of $300$ MPa $.($Given: the shape factor $\backslash (($Y$)\backslash )$ is $1\mathrm{.}5.)$