6. Deformation of solids

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.

Elastic deformation disappears when the load is removed; plastic deformation remains. The elastic limit is the greatest load or stress before permanent deformation begins.

Separate the elastic limit from the limit of proportionality: a material may stop being linear before it becomes permanently deformed.

A metal wire unloaded within its elastic region returns to its original length; loaded beyond its elastic limit it remains longer.

Elastic does not mean perfectly linear, and plastic does not mean the object has already broken.

Work done by a changing force is the area under the force–extension graph, W=∫F dx; for a linear spring from zero, W=½Fx=½kx².

Use the loading path and the correct graph axes. The area represents energy transferred into elastic or other deformation stores.

A spring reaching 0.20 m under 40 N with a straight-line graph stores 4.0 J of elastic energy.

The area is not simply F×final extension unless force is constant; unloading can enclose a different area when energy is dissipated.

A material deformed within its elastic limit stores elastic potential energy, equal to the work done in deforming it.

The energy is recoverable when the load is removed. Use the loading force–extension relationship rather than assuming all deformation energy is retained.

A stretched spring can return to its original length while releasing its stored elastic energy to a moving mass.

Elastic potential energy is not the same as gravitational potential energy, and plastic deformation can dissipate some input energy.

Within the proportional region, E_P=½Fx=½kx² because the average force during loading is F/2.

Use final force and extension only for a straight-line force–extension graph from zero. For a non-linear graph, find the area under the curve.

A spring with k=200 N m⁻¹ stretched 0.10 m stores 1.0 J.

Do not use Fx for a spring whose force changes during loading; that would overestimate the work by a factor of two in the linear case.