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Microfluidic channels will need to be fabricated on a key micro$-$scale sensor used by aerospace

industries. Before running machining tests and analyzing machined quality, preliminary efforts are needed

to evaluate selected materials and factors affecting machining process ${?}^i.$ Three material candidates have

been selected, including $422$SS $($stainless steel$),$ IN$718($nickel alloy$),$ and Ti$64($titanium alloy$)$ with their

measured tensile properties and equation of true stress$-$true strain relationship used listed below. $T_{\text{r}\text{e}\text{f}}=25C.$

Specifically, three factors will need to be evaluated, including different materials, temperature, and size

effect. Please calculate true stress values for true strain ranging between $0-3$ for each case listed below.

$=(A+B{}^n)(1+$clog$\frac{({}^{})}{{}_o^{}})(1-(\frac{T-T_{\text{r}\text{e}\text{f}}}{T_m-T_{\text{r}\text{e}\text{f}}}{)}^m)$

$(1+(\frac{18a^{2}b{G}^2}{L((A+B{}^n)(1+clog\frac{({}^{})}{(({}^{}){)}_o})(1-(\frac{T-T_{\text{r}\text{e}\text{f}}}{T_m-T_{\text{r}\text{e}\text{f}}}{)}^m){)}^2}{)}^{}{)}^{\frac{1}{2}}$

$($a$)$ At a fixed temperature $(T=400C)$ and a strain rate $,$ typical for micromachining$),$ please plot

and compare the true stress$-$true strain curves for different materials. A $1m$ wide channel will be cut, so

the length parameter, $L=10nm,$ may be assumed.

$($b$)$ At a strain rate $(({}^{})={10}^5{s}^{-1})$ with a fixed length parameter $(L=10nm),$ please plot and compare the true

stress$-$true strain curves of Ti$64$ at different temperatures $(T=25C,400C,100C).$

$($c$)$ At a fixed temperature $(T=400C)$ and a strain rate $({}^{}{?}^{?}={10}^5{s}^{-1}),$ please plot and compare the true stress$-$

true strain curves of Ti$64$ if microfluidic channels with different width need to be made, specifically $L=10$

$nm,100nm,1000nm.$