Deformation of solids

Learning objectives

• Use Hooke’s law within its range.
• Calculate stress, strain and Young modulus.
• Find elastic energy from a graph.

The big idea

A spring obeys Hooke’s law where force is proportional to extension. The limit of proportionality and the elastic limit describe different ideas: a graph can stop being linear before permanent deformation begins.

Stress is force divided by cross-sectional area; strain is extension divided by original length. Young modulus compares stress with strain in the linear region and describes a material rather than a particular specimen’s dimensions.

1. Measure extension, not final length

Extension is final length minus original length. For a spring obeying Hooke’s law, force is proportional to extension and k = F/x. A 10 N load extending a spring by 0.050 m corresponds to 200 N/m stiffness. A force–extension graph is linear only within its proportional range; do not extend that straight-line relationship indefinitely to larger loads.

2. Geometry and material are separate

Two wires of the same material can have different stiffness because their lengths or areas differ. Stress F/A removes the effect of cross-sectional area from the applied load; strain ΔL/L removes the original length from extension. Young modulus is stress divided by strain in the linear elastic region. It characterises the material under specified conditions, not simply a wire’s overall size.

3. Stored energy comes from a changing force

As an ideal spring extends, the required force rises from zero to F, so the work is the area of a triangle under its force–extension graph: ½Fx. Using Fx would incorrectly treat the final force as constant throughout loading. For a curved loading graph, estimate the actual area rather than automatically using the triangular expression. The graph axes determine whether its area represents energy.

4. Elastic and proportional are not identical

Elastic deformation disappears on unloading; plastic deformation leaves a permanent change. A material may show non-linear elastic behaviour, so the end of proportionality and the onset of permanent deformation need not be the same point. Real loading and unloading can also dissipate energy. In experiments, use small controlled loads and protect against falling masses or a snapping wire; a mathematical model does not set a safe working load.

A method you can reuse

1. Record original length and calculate extension in metres.
2. Find cross-sectional area carefully, including any squared diameter conversion.
3. Locate the proportional region before finding stiffness or Young modulus.
4. Use the appropriate area under the force–extension graph for energy.
5. Distinguish reversible shape change from permanent deformation.

Equations and relationships

Investigate and explain

Add small loads to a spring over a catch tray, record extension and unload. Stay below the teacher-set safe load.

Connect force–extension behaviour with oscillation and an illustrative interparticle model. An ideal spring and a two-particle potential cannot reproduce all real materials or their failure limits.

Open the interactive investigation

Check your understanding