Physical Basis for Linear Elasticity d$.$ The temperature dependence of elastic modulus in ceramics can be described using the following

equation $($equation $2\mathrm{.}44,$ Mechanical Behavior of Engineering Materials" by Roesler, Harders and

Baeker$)$:

$E_M=E_M(0K)*(1-0\mathrm{.}3\frac{T}{T_M})$

where $E_M(0K)$ is the elastic modulus for a specific element at $0$ K $,$ and $T_m$ is the melting

temperature of the material. All temperature values should be considered in Kelvin.

Using this equation and similar equation provided for metals in question $3$c$.$ qualitatively describe

how you would expect the elastic modulus of a ceramic to change relative to a metal if both

materials were heated to a temperature of $T=0\mathrm{.}5T_M.$

In lecture, we developed a model for elastic deformation which considered the bonds between

atoms as springs that could be stretched. Using Figure $2($adapted from "Mechanical Behavior of

Engineering Materials" by Roesler, Harders, and Baeker$)$ which describes the relationship

between elastic modulus and melting temperature:

a$.$ Provide a schematic of potential energy $($U$)$ vs$.$ atomic distance $($r$)$ for molybdenum and sodium.

Indicate which curve is for each material.

b$.$ Provide a schematic plot of force $($F$)$ vs$.$ atomic distance $($r$)$ for molybdenum and sodium.

Indicate which curve is for each material.

Figure $2.$ Elastic modulus versus melting temperature in some metals $($adapted from

"Mechanical Behavior of Engineering Materials" by Roesler, Harders, and Baeker$).$

c$.$ The temperature dependence of elastic modulus in metals can be described using the following

equation $($equation $2\mathrm{.}43,$ Mechanical Behavior of Engineering Materials" by Roesler, Harders and

Baeker$)$:

$E_M=E_M(0K)*(1-0\mathrm{.}5\frac{T}{T_M})$

where $E_M(0K)$ is the elastic modulus for a specific element at $0K,T_m$ is the melting temperature

of the material. All temperature values should be considered in Kelvin.

Using this equation, construct a plot of elastic modulus versus temperature for two metals:

Aluminum where $E_M(0K)$ is $75$ GPa and $T_M$ is $933$ K

Titanium where $E_M(0K)$ is $125$ GPa and $T_M$ is $1944$ K