In Physics, Steel Is More Elastic Than a Rubber Band

Stretch a rubber band and it snaps back; press a lump of kneaded atta and it keeps the dent. Every solid resists changes in shape, but some recover completely and some do not.

This part covers elasticity and plasticity, stress and strain, Hooke's law, and the stress-strain curve.

A body is elastic if it regains its original size and shape when the deforming force is removed, and plastic if it stays permanently deformed; real materials are elastic for small deformations and become plastic when stretched too far.

Worked example. A $2.000$ m wire is stretched to $2.004$ m.

- **Released, it returns to $2.000$ m — elastic behaviour
- Stretched further and released, it returns only to $2.001$ m** — a permanent extension of $1$ mm, so it has entered the plastic range

Examples:

- Nearly elastic: steel, quartz fibre, a spring used within its range
- Nearly plastic: putty, wet clay, kneaded dough

An everyday example. Kneading atta for rotis reshapes the dough permanently, while a rubber band on a tiffin box springs back every time.

The substance. In physics, steel is more elastic than rubber — it develops a much larger restoring stress for the same strain, even though rubber stretches more easily.

**Stress is the restoring force per unit area, measured in N/m${}^2$ or pascal, and strain is the fractional change in size or shape, which has no unit; each comes in longitudinal, volumetric and shear forms.

- Longitudinal**: stress $\frac{F}{A}$ along the length; strain $\frac{\Delta L}{L}$ (tensile or compressive)
- Volumetric: hydraulic stress equal to pressure $p$; strain $\frac{\Delta V}{V}$
- Shear: tangential force per area $\frac{F}{A}$; strain $\frac{\Delta x}{L}=\tan \theta \approx \theta$

Worked example 1 — a wire. A steel wire of radius $1$ mm carries $314$ N and stretches $1$ mm over $2$ m.

$$A=\pi (10^{-3}{)}^2\approx 3.14\times 10^{-6}{\text{ m}}^2\text{stress}=\frac{314}{3.14\times 10^{-6}}=1.0\times 10^8\text{ Pa}$$

$$\text{strain}=\frac{0.001}{2}=5\times 10^{-4}$$

Worked example 2 — volume. A $100$ cm${}^3$ rubber ball shrinks by $0.02$ cm${}^3$ under water: volumetric strain $=2\times 10^{-4}$.

Worked example 3 — shear. The top of a $10$ cm cube of jelly shifts $0.5$ cm under a $2$ N sideways force.

$$\text{shear stress}=\frac{2}{0.01}=200\text{ Pa}\text{shear strain}=\frac{0.5}{10}=0.05$$

An everyday example. Pushing the top cover of a thick book sideways makes the pages slide over one another — a shear strain.

The substance. Stress is an internal restoring force per area, while pressure is an external force per area, even though both share the pascal.

**Hooke's law states that for small deformations stress is proportional to strain, so stress $=$ modulus of elasticity $\times$ strain; it holds only up to the proportional limit, which lies just below the elastic limit.

For stretching, the modulus is Young's modulus** $Y=\frac{\text{stress}}{\text{strain}}$. For a spring, the same idea gives $F=kx$.

Worked example 1 — the steel wire. With $Y=2\times 10^{11}$ Pa and a stress of $1.0\times 10^8$ Pa:

$$\text{strain}=\frac{1.0\times 10^8}{2\times 10^{11}}=5\times 10^{-4}\Delta L=5\times 10^{-4}\times 2=1\text{ mm}$$

Worked example 2 — a spring. A spring with $k=200$ N/m stretches by

$$x=\frac{F}{k}=\frac{10}{200}=0.05\text{ m}=5\text{ cm under }10\text{ N}$$

Beyond the limit, doubling the load no longer doubles the extension.

An everyday example. The spring balance at a vegetable shop has evenly spaced markings because its spring obeys Hooke's law over the working range.

The substance. Hooke's law is an experimental rule for small strains, not a law that every material obeys at every load.

On a stress-strain curve for a metal, the straight first part ends at the proportional limit, recovery is complete only up to the elastic limit, strain grows rapidly at the yield point, stress peaks at the ultimate tensile strength, and the wire breaks at the fracture point.

Going along the curve:

- O to A — straight line; Hooke's law holds up to A, the proportional limit
- A to B — slightly curved but still recoverable up to B, the elastic limit
- Beyond B — at the yield point, strain increases quickly with little extra stress; unloading leaves a permanent set
- Highest point — the ultimate tensile strength
- End of curve — the fracture point

Worked example 1 — slope. The straight part ends at a stress of $2.5\times 10^8$ Pa and a strain of $1.25\times 10^{-3}$:

$$Y=\frac{2.5\times 10^8}{1.25\times 10^{-3}}=2\times 10^{11}\text{ Pa}$$

Worked example 2 — permanent set. A $2$ m wire is loaded past its yield point and unloaded, leaving a permanent strain of $0.002$: it stays $2\times 0.002=4$ mm longer.

Ductile materials such as copper have a long region between yield and fracture; brittle materials such as glass break soon after the elastic limit.

An everyday example. Copper can be drawn into thin electrical wires, while a glass bangle snaps instead of bending.

The substance. The proportional limit and the elastic limit are close but different points.