Problem $1(20$ points.$)$

Please examine the following stress$-$strain curves for a certain material as a function of temperature.

$(1)$ Determine the elastic modulus of this material at $25C.$

$(2)$ Determine the $0\mathrm{.}2\%$ offset yield strength of this material at $200C.$

$(3)$ Please comment on how the elastic modulus, strength, and ductility vary with temperature.

$(4)$ At $300C,$ if we are to unload just before failure, estimate the permanent strain.

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Problem $2(20$ points.$)$

Consider a solid made of atoms connected to each other via springs. Given that Young's modulus $=100$ GPa $,$ equilibrium atomic spacing $=0\mathrm{.}3nm,$ and atom cross$-$section area $=0\mathrm{.}09n{m}^2,$ estimate the spring constant.

Problem $3(10$ points.$)$

A $120$ mm long cylindrical rod with a diameter of $15\mathrm{.}0$ mm is to be deformed using a tensile load of $35,000N.$ It must not experience either plastic deformation or a diameter reduction of more than $1\mathrm{.}2{10}^{-2}mm.$ Of the materials listed below, which are possible candidates? Justify your choice$($s$).$

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Please solve the problems including graphing.