Problem $3(40$ points$)$ A steel rod is loaded in a tensile test. The original gage length of the rod is $100$ mm and the original diameter is $20\mathrm{.}0$ mm $.$ The rod exhibits elasto$-$plastic behavior for $5\mathrm{.}00$ mm of deformation. At this point the loading is removed. The following values are measured from the test: $\backslash (\backslash$mathrm$\{$E$\}=200\backslash$mathrm$\{$GPa$\}\backslash ),$ Poisson's ratio $\backslash (=0\mathrm{.}25\backslash )$ and yield stress $\backslash (=250\backslash$mathrm$\{$MPa$\}\backslash ).$

a$.$ Neatly sketch the stress$-$strain curve for the entire test. Be sure to indicate and label points A through D on the graph.

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A$-$The unloaded rad

B$-$The rod when it storts yielding

C$-$The red when it is stretched by $5$ mm

D$-$The rod ofter the looding is remeved

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b$.$ What was the length of the rod when it started to yield?

c$.$ What was the final strain in the rod after the loading was removed?

d$.$ What was the diameter of the rod when it started to yield?