Stress, Strain and the Young Modulus

$\text{Tensile stress}$ is force per unit cross-sectional area, $\sigma =\frac{F}{A}$, measured in pascals. It captures how intensely the material is loaded, regardless of how thick the sample is.

$\text{Tensile strain}$ is extension per unit original length, $\epsilon =\frac{\Delta x}{x}$. It is a ratio of two lengths and so has no units.

Normalising in this way means a thick wire and a thin wire of the same material, loaded to the same stress, show the same strain.

Within the limit of proportionality, stress is proportional to strain, and the constant of proportionality is the Young modulus $E=\frac{\sigma }{\epsilon }$, in pascals.

Substituting the definitions gives $E=\frac{Fx}{A\Delta x}$, which is the form used with experimental data.

A high Young modulus means a stiff material. Steel has about $2\times 10^{11}$ Pa, rubber around $10^7$ Pa — so steel needs roughly ten thousand times the stress for the same strain.

The standard practical uses a long, thin wire. Length makes the extension large enough to measure; thinness makes the stress large for a modest load.

Measure the original length with a metre rule, the diameter at several points with a micrometer (averaging to reduce random error), then add masses in steps and record the extension each time using a vernier scale or a marker against a fixed ruler.

Plot stress against strain and take the gradient of the straight section. Using a gradient from many readings is more reliable than a single calculation, and it automatically discards any zero error in the extension readings.

A $\text{ductile}$ material such as copper shows a straight region (Hooke’s law), then a $\text{yield point}$ where strain increases rapidly with little extra stress, a long region of plastic deformation, and finally fracture. The peak stress is its $\text{ultimate tensile strength}$.

A $\text{brittle}$ material such as glass or cast iron stays almost straight right up to fracture, with no plastic region. It breaks suddenly without warning.

A $\text{polymer}$ such as rubber has a curved graph and a large strain, and unloading follows a different path. The area enclosed between loading and unloading curves — the $\text{hysteresis loop}$ — is energy transferred to heat per unit volume, which is why a repeatedly stretched rubber band warms up.

The area under a stress–strain graph is energy stored per unit volume, the material equivalent of $\frac{1}{2}F\Delta x$.