6.1 Stress and strain

A tensile force pulls along a body and tends to extend it; a compressive force pushes along it and tends to shorten or squash it.

Identify the force direction relative to the body axis, then decide whether the deformation is extension or compression. Real materials can experience both locally.

A hanging cable is under tension, while a column supporting a roof is under compression.

“Tension” is not a synonym for any force in a material; it specifically describes pulling stress along the relevant direction.

Load is the applied force; extension is the increase in length and compression the decrease. The limit of proportionality is where extension ceases to be directly proportional to load.

Read these terms from a force–extension graph and distinguish proportional behaviour from later elastic or plastic behaviour.

If a spring extends 2 mm under 4 N and remains proportional, 8 N predicts 4 mm; beyond the limit that scaling may fail.

The limit of proportionality is not automatically the breaking point or the elastic limit; those are separate boundaries.

For a spring or wire within its proportional region, F=kx: extension x is directly proportional to applied force F.

The relation applies only while the material remains proportional. Use the original length and the extension, not the final length.

If 6 N produces 3 mm extension in the proportional region, 10 N predicts 5 mm for the same spring.

Hooke’s law does not hold indefinitely; crossing the limit of proportionality makes the simple linear relation unreliable.

The spring constant is k=F/x in the proportional region. Its unit is N m⁻¹; a larger k means more force is needed for the same extension.

Convert extension to metres before calculating. Compare like-for-like loading conditions and do not use points beyond proportional behaviour.

A 12 N force producing 0.040 m extension gives k=300 N m⁻¹.

k is not the extension itself, and a stiffer spring has a larger k, not a larger extension under the same force.

Stress σ=F/A and strain ε=x/L are dimensionless ratios of deformation. Young modulus E=σ/ε measures a material’s stiffness in the linear elastic region.

Use cross-sectional area and original length, not the deformed dimensions, and keep stress in pascals.

A wire’s stress doubles if the same force acts on half the area; its strain depends on extension relative to its starting length.

Strain has no unit, while stress and Young modulus are measured in Pa; Young modulus is not the same as spring constant.

For a uniform wire in the linear region, E=(F/A)/(x/L)=FL/(Ax).

Measure original length L, extension x, force F and cross-sectional area A; use diameter carefully because A=πd²/4.

A longer wire extends more under the same load, but Young modulus remains a material property when dimensions are accounted for.

Do not compare raw extension to judge material stiffness without allowing for length, area and force.