3.2Work, energy, power and deformation

Deformation OF Solids

5 min read 10 parts Lesson 3 of 17 1 quiz Virtual lab

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Lesson summary

3.2 Deformation of Solids — AS PHYSICS 9702 notes: Describe tensile and compressive deformation, state Hooke’s law, define stress, strain and the Young modulus, and distinguish elastic from plastic behaviour.Deformation — forces change the size or shape of a body.Tensile forces —…

Compressive forces
What the symbols mean
Spring constant
Springs in series
Springs in parallel
Limit of proportionality

Section 1 of 10Tensile and compressive forces

Deformation

Deformation forces change the size or shape of a body.

Tensile forces

Tensile forces a pair of equal and opposite forces that stretch a body and cause extension.

Compressive forces

Compressive forces a pair of equal and opposite forces that squash a body and cause compression.

Extension

Extension x = L − L₀, the extended length minus the natural length, in metres.

Section 2 of 10Hooke’s law and the spring constant

Hooke’s law

Hooke’s law the extension of a body is directly proportional to the applied force, provided the limit of proportionality is not exceeded.
F = kx, with F in N, k in N m⁻¹ and x in m.
What the symbols mean: F = force (N) · k = spring constant (N m⁻¹) · x or e = extension (m)

Spring constant

Spring constant k = F/x, the force per unit extension and a measure of stiffness.
It equals the gradient of the linear part of a force–extension graph.

Springs in series

1/k = 1/k₁ + 1/k₂.

Springs in parallel

Springs in parallel k = k₁ + k₂.

Section 3 of 10Stress, strain and the Young modulus

Stress

σ = F/A, force per unit cross-sectional area, in Pa.
What the symbols mean: σ = stress (Pa) · F = tensile force (N) · A = cross-sectional area (m²)

Strain

ε = x/L, extension per unit original length, with no unit.

Young modulus

E = σ/ε = FL/(Ax), in Pa.
What the symbols mean: E = Young modulus (Pa) · σ = stress (Pa) · ε = strain · F = force (N) · L = original length (m) · A = cross-sectional area (m²) · ΔL = extension (m)
It is a property of the material, not of the sample.
A stiffer material has a larger Young modulus; steel is about 2 × 10¹¹ Pa.

Wire area

A = πd²/4 = πr².

Graph

Graph the gradient of the linear part of a stress–strain graph is the Young modulus.

Section 4 of 10Elastic and plastic behaviour

Feature	Elastic	Plastic
Return to original length on unloading	Yes	No — permanent extension remains
Region of graph	Up to the elastic limit	Beyond the elastic limit
Energy on unloading	Stored energy is recovered	Some work done is dissipated as thermal energy
Atomic view	Bonds stretch and atoms return	Layers of atoms slide past each other

Limit of proportionality

Limit of proportionality the point beyond which force is no longer proportional to extension.

Elastic limit

Elastic limit the maximum force beyond which the material will not return to its original length;
- it is slightly beyond the limit of proportionality.

Section 5 of 10Brittle, ductile and rubber

Brittle

Brittle glass and ceramics break with little or no plastic deformation, at the end of the elastic region.

Ductile

Ductile copper has a large plastic region and stretches before breaking.

Ultimate tensile stress

Ultimate tensile stress the maximum stress a material can withstand before it breaks.

Rubber

Rubber a non-linear material with hysteresis, where loading and unloading curves differ.
The area between the curves is energy dissipated as heat per cycle.

Section 6 of 10Elastic potential energy

Work done

Work done the area under a force–extension graph is the work done in stretching.

Hooke’s law region

E_p = ½Fx = ½kx², the area of a triangle.

Non-linear graph

Non-linear graph count squares, with energy per square = force per square × extension per square.

Energy density

Energy density the area under a stress–strain graph is the strain energy per unit volume, in J m⁻³.

Beyond the elastic limit

Beyond the elastic limit the unloading line is roughly parallel to the loading line, so only the area under the unloading line is recovered.

Section 7 of 10Experiment to find the Young modulus of a wire

Clamp a long, thin wire at one end and pass it over a pulley to a mass hanger, with a marker on the wire against a scale.

Measure the original length L with a metre rule.

Measure the diameter with a micrometer at several points and orientations, then take the mean and calculate A = πd²/4.

Add masses in equal steps and record the extension for each load F = mg.

Record readings while unloading to check the elastic limit was not exceeded.

Plot force against extension and find the gradient of the linear region.

Calculate E = gradient × L/A.

Precautions
- a long wire gives a larger extension, and the diameter has the largest percentage uncertainty because it is squared.

Section 8 of 10More from the specification

1Hooke's law

Hooke's law extension is proportional to force up to the limit of proportionality, and a force-extension graph is straight in that region

2Young modulus

Young modulus stress divided by strain, a property of the material independent of the object's shape

3Stress and strain

Stress and strain stress is force per unit cross-sectional area and strain is extension divided by original length

4Stress-strain graph

Stress-strain graph the gradient gives the Young modulus and the area under it gives energy stored per unit volume;
- for a metal wire measure diameter and length and the extension for several loads to determine it

5Hooke's law

Hooke's law extension is proportional to force up to the limit of proportionality, and a force-extension graph is straight in that region

6Young modulus

Young modulus stress divided by strain, a property of the material independent of the object's shape

7Stress and strain

Stress and strain stress is force per unit cross-sectional area and strain is extension divided by original length

8Stress-strain graph

Stress-strain graph the gradient gives the Young modulus and the area under it gives energy stored per unit volume;
- for a metal wire measure diameter and length and the extension for several loads to determine it

Section 9 of 10Key terms

Term	Meaning
Hooke’s law	Extension is directly proportional to force up to the limit of proportionality.
Spring constant	Force per unit extension, k = F/x, in N m⁻¹.
Stress	Force per unit cross-sectional area, σ = F/A.
Strain	Extension per unit original length, ε = x/L.
Young modulus	Ratio of stress to strain for a material, E = σ/ε.
Elastic limit	Maximum force beyond which a material does not return to its original length.
Plastic deformation	Permanent deformation after the load is removed.
Elastic potential energy	Energy stored in a deformed body, equal to the area under the force–extension graph.

Section 10 of 10Exam technique

Using the total length instead of the extension in F = kx — use x = L − L₀.

Confusing the elastic limit with the limit of proportionality — they are different points on the graph.

Thinking the Young modulus depends on the wire’s dimensions — it is a property of the material.

Using Ep = ½kx² for a non-linear graph — then the area must be found directly.

Try it in the virtual labHooke’s law \n