# Materials

Elastic limit
Maximum force one can apply to the end of an elastic object for it to behave elastically.
Plastic deformation
Object permenantly changes shape

## Elastic objects

• Obey Hooke's law
• Return to their original state ($e=0$)

## Hooke's law

Hooke's law states that force is proportional to the extension of an elastic object : $F\propto e$ Therefore: \begin{align}F=ke\end{align} \begin{align}[k]=[\frac{F}{e}] = Nm^{-1}\end{align} Where K is the Stiffness constant, elasticity constant or spring constant

### Experiments

To measure $k$, one should rearrange the formula into $y=mx+c$ format: $F=ke+0$ One should then measure extension ($e$) in meters, mass in Kg, $F$ in N. The gradient will be $k$, and thus the spring constant

### The behaviour of rubber?

Rubber is strange and annoying, and doesn't follow Hooke's law. As a rubber band is streached, it warm up, releasing thermal energy. As a result, it take extra time to return to their original length. It form a graph like the one below. The area under the loading line is work done by the masses on the band, and thus a gain in potential energy. The area under the unloading line is work done by the band on the surrondings. The remaining energy is heat to the surrondings. The net gain is the difference between the two areas, like on a Lorenz curve.

[insert diagram of rubber band graph]

### The problems with Hooke's law

• The spring constant depends on the dimensions of the object.

## A better way: The Young Modulus

### Tensile stress - $\sigma$

\begin{align}\sigma=\frac{F}{A}\end{align} where F is tensile force, A is cross sectional area \begin{align}[\sigma]=Nm^{-2}=Pa\end{align} (pressure)

### Tensile strain - $\varepsilon$

\begin{align}\varepsilon=\frac{e}{L}\end{align} where e is extension, L is original length \begin{align}[\varepsilon] = dimensionless\end{align}

### Young modulus

\begin{align}E=\frac{\sigma}{\varepsilon}\end{align} \begin{align}[E]=Pa\end{align} One should expect a value around $10^9$ (gigapascals).