An Al$-$Cu alloy $($FCC structure$)$ is strengthened by forming ordered, coherent particles $($GP zones$).$ This Al$-$Cu alloy is typically used at a temperature of $200$C$.$ At this temperature, the average particle radius $()$ of GP zones increases with time $($t$)$ according to $3=03+,$ where $0$ is the initial particle radius and K is a kinetic constant $(=10-28$ m$3/$s$).$ Assume that the particle shearing stress is dominantly the order hardening for low $$ at early$-$stage precipitation as GP zones often form at early$-$stage precipitation. When the particles coarsen to a certain size, the strengthening mechanism transits from shearing to dislocation bowing. Data you may need: G $=25$ GPa; lattice parameter of Al $0=0\mathrm{.}405$ nm; APBE$=50$ mJ$/$m$2$ ; the volume fraction of GP zone particles f $=0\mathrm{.}05$; $0=10.($a$)$ Calculate the critical radius of the particles when the mechanism transits from shearing to bowing. $($b$)$ Make a plot to show how the strength of this alloy varies as a function of time $($from t$=0$ through $1000$ hours$)$ at $200$C$.$