Problem $1(25$ points$).$ A specimen of medium$-$carbon steel with an initial diameter of $0\mathrm{.}503$ in was tested in tension using a gauge length of $2$ in $.$ The following data were obtained for the elastic and plastic states:

$\backslash$table$[[$Elastic state,Plastic state$],[$Load $($lb$),$Elongation $($in$),$Load $($lb$),$Current area in ${?}^2),],[1000,0\mathrm{.}0004,8800,0\mathrm{.}1984,],[2000,0\mathrm{.}0006,9200,0\mathrm{.}1978,],[3000,0\mathrm{.}0010,9100,0\mathrm{.}1963,],[4000,0\mathrm{.}0013,13200,0\mathrm{.}1924,],[7000,0\mathrm{.}0023,15200,0\mathrm{.}1875,],[8400,0\mathrm{.}0028,17000,0\mathrm{.}1563,],[,,16400,0\mathrm{.}1307,],[,,14800,0\mathrm{.}1077,]]$

$($a$)$ For each of the $14$ data points given above, calculate the corresponding engineering stress and engineering strain. $($Hint: To find engineering strain in the plastic state, use the equation with area instead of that with length, i$.$e$.,$ use $lon=\frac{A_0-A}{A}.)$

$($b$)$ Using the results in part $($a$),$ plot the engineering stress$-$strain diagram using MATLAB or Excel. Take the first point in this diagram as $(0,0),$ since initially there is no stress or strain. In other words, use a total of $15$ data points.

$($c$)$ From the plot in part $($b$)($zoom$-$in$,$ if necessary$),$ find the modulus of elasticity, the $0\mathrm{.}2\%$ offset yield strength, and the ultimate strength.

$($d$)$ Based on the stress$-$strain plot, is the material ductile or brittle?

$($e$)$ Find true stress and true $($logarithmic$)$ strain for each of the given data points. Assume that in the elastic region, true stress is same as engineering stress.

$($f$)$ Using MATLAB or Excel, plot the true stress$-$strain diagram from the results in part $($e$).$ Do this once with both axes in arithmetic scale, and one more time with both axes in logarithmic scale. From the plot in arithmetic scale, does true stress continually increase with true strain? Can we make the same observation from the engineering stress$-$strain diagram obtained in part $($b$)?$