Stress–Strain / Elasticity
Stretch a wire and watch its stress–strain curve rise to the elastic limit.
About this simulation
Choose a material, length and radius, then apply increasing load; the wire extends while a stress–strain graph plots the linear Hookean region — whose slope is the Young's modulus — up to yield. Comparing wires shows why a thicker or shorter wire of the same material stretches less for the same force.
How to use it
- Choose a Material (Steel, Copper or Aluminium) and set Wire length and Wire radius.
- Increase Load and watch the dot climb the green straight line on the stress–strain graph while the ΔL mark beside the wire grows.
- Press 'Right at yield' to put a steel wire exactly at the elastic limit, then add a little more Load and see the dot move onto the red plastic curve.
- Reduce Wire radius, or press 'Thin wire snaps', until the wire turns red and is labelled snapped.
- Press 'Twice as long': the wire is drawn longer and its ΔL mark stretches, but the dot on the graph stays where it was, because the strain does not change.
What to notice
- Below the elastic limit stress is proportional to strain, and the slope of that straight line is the Young's modulus of the material.
- Stress depends on the force per unit area, so halving the radius of the same wire makes the stress four times larger.
- Extension ΔL = FL/AY: a longer wire stretches more, a thicker or stiffer wire stretches less for the same load.
- Past the elastic limit the curve bends over and large strain appears for little extra stress, until the wire breaks.
The formulas
- σ = F ⁄ A
- Stress σ is the load F divided by the cross-sectional area A = πr² of the wire.
- ε = ΔL ⁄ L
- Strain ε is the extension ΔL divided by the original length L; it has no unit.
- Y = σ ⁄ ε = FL ⁄ AΔL
- Young's modulus Y is stress divided by strain in the elastic region; it is a property of the material.
Questions
What is Young's modulus?
Young's modulus is the ratio of tensile stress to tensile strain within the elastic limit, Y = σ/ε. It measures how stiff a material is: a larger Y means less stretch for the same stress. Steel's Y is about 200 GPa and aluminium's about 70 GPa. On the graph it is the slope of the straight green part of the curve.
What is the elastic limit?
The elastic limit is the greatest stress a material can take and still return to its original length when the load is removed. Beyond it the material deforms permanently. In the simulation it is marked on the graph where the green line ends and the red plastic curve begins.
Does Young's modulus depend on the length and thickness of the wire?
No. Young's modulus is a property of the material alone. A longer or thinner wire stretches more under the same load, but its stress and strain change together so their ratio stays the same. In the simulation, change Wire length or Wire radius and the 'Y = 200 GPa' figure above the graph does not move; change Material and it does.
What is Hooke's law?
Hooke's law states that, within the elastic limit, stress is directly proportional to strain. For a wire this means the extension is proportional to the load. On a stress–strain graph Hooke's law is the straight line starting at the origin.
What is the difference between stress and pressure?
Both are force per unit area and both are measured in pascals. Pressure is the push a fluid or surface applies from outside, acting normally on a surface. Stress is the internal restoring force per unit area inside a deformed solid, and it can be tensile, compressive or shearing.