Mechanical Properties of Solids: theory

CONCEPT

Elastic Behaviour, Stress & Strain

In one line
A deforming force produces stress and strain; within the elastic limit the body returns to its original shape and stress ∝ strain.

SOLIDS

9.1.1 States of matter and solids

Solid: definite shape and volume. Liquid: definite volume only. Gas: neither.
Crystalline solids have a regular pattern and a sharp melting point; amorphous solids (glass) have neither.

ELASTICITY

9.1.2 Elasticity and plasticity

Elasticity: a body tends to regain its original shape and size when the deforming force is removed.
Plasticity: it does not regain them.
Nearly perfectly elastic: quartz fibre, phosphor bronze.
Nearly perfectly plastic: paraffin wax, wet clay.

Molecules behave as if joined by springs

Reason: deforming a solid changes the distance between molecules, and intermolecular forces pull them back.
Elastic limit: the largest deforming force up to which a body fully recovers.
Elastic fatigue: temporary loss of elastic strength under repeated loading. Bridges become unsafe; spring balances read wrongly.
Elastic after-effect: the delay in regaining the original state.

COMPARING

9.1.3 Which is more elastic?

Same force, less deformation: more elastic.
Steel is more elastic than rubber; water is more elastic than air.
Solids > liquids > gases in bulk modulus.

FACTORS

9.1.4 What changes elasticity

Hammering and rolling: elasticity increases.
Annealing: elasticity decreases.
Rise in temperature: elasticity decreases (invar steel hardly changes).
Impurities can raise or lower it.

STRESS

9.1.5 Stress

Stress = FA

Restoring force per unit area. It balances the applied force at equilibrium.
Unit N m⁻² (pascal). Dimensions [ML⁻¹T⁻²].

The three kinds of stress

Longitudinal: force along the length, on the cross-section. Tensile if length grows, compressive if it falls.
Bulk: normal force at all points; equal to the change in pressure. Occurs in solids, liquids and gases.
Shear: tangential force on a face; only shape changes.

Pressure	Stress
always normal to the area	normal or tangential
always compressive	compressive or tensile

STRAIN

9.1.6 Strain

Longitudinal strain = ΔlL

Volumetric strain = ΔVV

Shearing strain φ = xL

Change in configuration divided by the original: no unit, no dimensions.
A bent beam has both extension and compression.

Outer layer stretched, inner layer compressed

THE CURVE

9.1.7 Stress-strain curve

Load on a metal wire

O to P: straight line, Hooke’s law holds. Slope = Y = tan θ.
P to E: not straight, but the wire still recovers. E is the elastic limit or yield point.
E to A: permanent set; the wire does not recover.
A to B: small increase in stress gives a large strain. B is the breaking strength.
B to C: the wire flows and breaks at C.

Curve shape tells the type of material

Type	Behaviour	Example
brittle	breaks just after the elastic limit	glass, cast iron
ductile	long plastic range, drawn into wires	mild steel
elastomer	large strain, no plastic range	rubber

HOOKE

9.1.8 Hooke’s law

Stress ∝ strain, E = stressstrain

Holds within the limit of proportionality.
E depends on the material and the type of deformation, and falls with temperature.
E does not depend on the dimensions of the body.
E has the unit and dimensions of stress.

Modulus	Stress type	Exists in
Young’s Y	longitudinal	solids
Bulk K	volume	solids, liquids, gases
Rigidity η	shear	solids

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Interatomic force

Force between two atoms: zero at r₀

Electrical in origin; acts at distances of about 10⁻¹⁰ m.
r > r₀: attraction. r < r₀: repulsion, rising very fast.

F = −dUdr

Potential energy curve

U is least at r₀: stable equilibrium

U = Ar¹² − Br⁶

The first term is repulsion, the second attraction.

Intermolecular force

F = −ar⁷ + br⁹

Van der Waals forces: weaker than interatomic forces by 50 to 100 times.
They depend on the distance and also on the orientation of the molecules.

Interatomic force constant

k = Y r₀

Unit N m⁻¹. Atoms in length l of a wire: N = l ÷ r₀.

Elastic hysteresis

Area of the loop = energy lost per unit volume

Strain lags behind stress on loading and unloading.
Car tyres: small loop (B), less heating. Shock absorbers: large loop (A).

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CONCEPT

Young’s Modulus & Elongation

In one line
Young’s modulus is longitudinal stress ÷ longitudinal strain, and it gives the elongation Δl = FL ÷ AY of any wire.

DEFINITION

9.2.1 Young’s modulus

Wire stretched by a hanging load

Y = F ÷ AΔl ÷ L = FLAΔl

Y = MgLπr²Δl

Property of the material only: changing L or r does not change Y.
If the length is doubled, strain = 1, so Y = stress. This is the stress that doubles the length.
Y decreases with a rise in temperature.
Largest elongation within the limit: Δl = (elastic limit ÷ Y) × L.

Material	Y (×10¹⁰ N m⁻²)
Aluminium	6.9
Brass	9.0
Copper	11
Iron	19
Steel	20
Tungsten	36
Diamond	83

ELONGATION

9.2.2 Elongation of a wire

Δl = FLAY

What is fixed	Result
F, Y	Δl ∝ Lr²
F, Y, volume	Δl ∝ L² ∝ 1r⁴
Y, volume, Δl	F ∝ A²
F, L	Δl ∝ 1AY

Same material and same force: greater L ÷ r² means greater elongation.
Same volume: A = V ÷ L, so a thinner wire is also longer.

COMPARING

9.2.3 Different materials

same F and Δl: Y ∝ LA

same A and L: F ∝ Y

same F, A, Δl: L ∝ Y

GRAPHS

9.2.4 Reading graphs for Y

Stress on y-axis, strain on x-axis: steeper line, larger Y.
Strain on y-axis, stress on x-axis: steeper line, smaller Y.
Copper of fixed volume drawn to length l: Δl versus l² is a straight line.

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Very stiff and very soft

Rigid body: no deformation, so Y is infinite. Diamond and carborundum come nearest.
Rubber stretches far but has a smaller Y than steel, so steel is more elastic.

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CONCEPT

Wire Combinations & Special Cases

In one line
For any joined or hanging wire, find the tension in each part first, then use Δl = FL ÷ AY part by part.

JOINED WIRES

9.3.1 Wires joined end to end

Each wire carries everything hanging below it

Light wires in a line with one load: same force in both.
Blocks in between: find the tension part by part.
Total elongation is the sum of the parts.

Δl = FA (L₁Y₁ + L₂Y₂)

same L and A: Y = 2Y₁Y₂Y₁ + Y₂

OWN WEIGHT

9.3.2 Elongation under its own weight

Weight acts at the centre of gravity

Δl = MgL2AY = ρgL²2Y

The weight acts at the centre of gravity, so only L ÷ 2 counts.
This is half of what the same weight would do if hung at the end.

PULLEYS

9.3.3 Wire over a pulley; pull at both ends

Two people pull with F each: tension is F, not 2F.
W hung on each end over a pulley: tension is W, so the elongation is the same as with one W on a fixed wire.

unequal blocks: T = 2m₁m₂gm₁ + m₂

NATURAL LENGTH

9.3.4 Length from two tensions

Lengths l₁ and l₂ under tensions T₁ and T₂: extension ∝ tension.

L = l₁T₂ − l₂T₁T₂ − T₁

Masses 1 kg and 2 kg give L₁ and L₂: L = 2L₁ − L₂.

WIRE AS A SPRING

9.3.5 Force constant

k = FΔl = YAL

f = 12π √YAmL

k depends on the material and on the dimensions.
A spring cut into pieces: k ∝ 1length.

EXPERIMENT

9.3.6 Searle’s experiment and errors

A load on the test wire gives an extension read on a vernier against a reference wire.
Plot extension against load. The graph gives Y.

Y = LA × loadextension

ΔYY = ΔMM + ΔLL + 2Δrr + Δll

The smallest measured quantities give the largest error: radius and extension.
Load fully inside a liquid: Δl falls by the factor (1 − ρ_liquidρ_load).

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Wire sagging under a central load

A wire of length L between two supports sags by a small angle θ under a load Mg at its middle.
Strain = θ² ÷ 2 for small θ.

2T sin θ = Mg, T = YA θ²2

Tapering (conical) wire

Δl = FLπY r₁r₂

r₁ and r₂ are the end radii.

Heavy wire with a load

Tension at distance x from the top = load + weight of the part below x.
So stress is largest at the top.

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CONCEPT

Thermal Stress & Breaking Stress

In one line
A rod not allowed to expand develops thermal stress YαΔθ; a wire breaks when its stress reaches the breaking stress, whatever its length.

THERMAL STRESS

9.4.1 Thermal stress

Rod held between rigid walls

strain = αΔθ · stress = YαΔθ

force = YAαΔθ

Heated: compressive. Cooled: tensile.
Stress does not depend on the length or the area.
Tension in a wire held taut is proportional to the fall in temperature.
Volume coefficient: γ = 3α.

BREAKING

9.4.2 Breaking stress

breaking force = P × A

P is the breaking stress, also called tensile strength.
P depends only on the material, not on the length or thickness.
Double the radius: four times the breaking force.
Cut a wire into parts: each part holds the same load.
Working stress is kept below breaking stress:

safety factor = breaking stressworking stress

ROPES AND LIFTS

9.4.3 Finding the maximum load

Largest tension when accelerating upward

T = M(g + a) ≤ PA

d = [4M(g + a)πS]^½

Two wires in a line: test each wire with what it carries. The smaller limit decides.
Load capacity ∝ area of the rope.

whirling in a circle: mv²r ≤ PA

OWN WEIGHT

9.4.4 Breaking under its own weight

L = Pρg

Largest length that can hang without breaking; it does not depend on the area.

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Two rods between rigid walls

F = (L₁α₁ + L₂α₂)ΔθL₁ ÷ AY₁ + L₂ ÷ AY₂

Volume form and mountain height

thermal stress = KγΔθ · h_max = Kρg

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CONCEPT

Elastic Potential Energy

In one line
Stretching a wire stores ½ × force × extension as elastic energy; per unit volume it is ½ × stress × strain.

WORK DONE

9.5.1 Work done in stretching

Energy = area under the force-extension line

U = 12 Fl = YAl²2L

The force grows from zero to F, so use the average force.
For a fixed wire, U ∝ l².
For the same l: U ∝ r²L.
A hanging load loses Mgl of gravitational energy, but the wire stores only half of it; the rest is lost as heat.

ENERGY DENSITY

9.5.2 Energy per unit volume

u = 12 × stress × strain

u = 12 Y (strain)²

u = (stress)²2Y

Total energy U	Per unit volume u
12 F × l	12 stress × strain
12 Y (strain)² × V	12 Y (strain)²
(stress)²2Y × V	(stress)²2Y

Area under a stress-strain line gives the energy per unit volume.
Same load and material: u ∝ 1d⁴.

CATAPULT

9.5.3 Energy into motion

12 YAL x² = 12 mv²

The energy of a stretched cord becomes the kinetic energy of the stone.

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Force raised from F₁ to F₂

U = F₁ + F₂2 × l

Thermal energy density

u = 12 Y α² (Δθ)²

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CONCEPT

Bulk Modulus & Compressibility

In one line
Bulk modulus is pressure ÷ volumetric strain; its reciprocal is compressibility.

DEFINITION

9.6.1 Bulk modulus

Uniform pressure: shape same, volume less

K = −pΔV ÷ V = −pVΔV

C = 1K

The negative sign shows that volume falls when pressure rises.
Defined for solids, liquids and gases.
Unit of C: m² N⁻¹.
Same p: ΔVV ∝ 1K, so the smallest K compresses most.

Substance	K (×10¹⁰ N m⁻²)
Steel	16
Water	0.22
Air	1.01 × 10⁻⁵

DEPTH

9.6.2 Compression at a depth

p = ρgh

ΔVV = ρghK = Cρgh

DENSITY

9.6.3 Density under pressure

ρ′ρ = 11 − p ÷ K

Δρ = ρpK

Mass is unchanged, so density rises as volume falls.

SPHERE AND CUBE

9.6.4 Change in size

Piston load raises the pressure everywhere

ΔVV = 3 Δrr

Δrr = p3K, with p = mga

A cube follows the same rule: fractional side change = one third of the volume change.

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Bulk modulus of a gas

isothermal: K = P · adiabatic: K = γP

Adiabatic elasticity is larger by the factor γ.

Cork in a bottle

p = B × 2Δaa, side area = 2πab

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CONCEPT

Shear Modulus & Poisson’s Ratio

In one line
Shear modulus connects tangential stress and shape change; Poisson’s ratio connects lateral and longitudinal strain.

SHEAR

9.7.1 Modulus of rigidity

Top face moved by x while the bottom is fixed

η = F ÷ Aφ = FLAx

Shape changes while volume does not.
Only solids have η.
Tangential stress = force ÷ area of the face on which it acts.

cube of side L: x = FηL

slab, force on narrow face: x = Fηd

d is the thickness of the slab.
Same material and force: a bigger cube shifts less.

punching a hole: F = stress × πd × t

Sheared area is the side of the hole: circumference × thickness.
Stretching a coil spring is decided by η, because its wire twists.

POISSON

9.7.2 Poisson’s ratio

Stretched bar: longer and thinner

σ = lateral strainlongitudinal strain = −Δr ÷ rΔL ÷ L

No unit, no dimensions. It is an elastic constant, not a modulus.
Theoretical range: −1 to +½.
Practical range: 0 to +½.
σ = ½: volume stays constant (K is infinite).
σ = 0: no lateral change (cork).

ΔVV = ΔLL (1 − 2σ)

ΔVV = ΔLL − 2 Δrr

RELATIONS

9.7.3 Relations between constants

Y = 3K(1 − 2σ)

Y = 2η(1 + σ)

9Y = 1K + 3η

σ = 3K − 2η6K + 2η

Metal	Y	K	η
Aluminium	6.9	7.0	2.6
Copper	11	13	4.5
Steel	20	16	8.4

Values in 10¹⁰ N m⁻². For aluminium and copper: η < Y < K.

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Torsion of a cylinder

Twist θ at the free end

rθ = φl

torque per unit twist = πηr⁴2l

W = 12 Cθ²

C is the torque per unit twist; a hollow shaft of the same mass is stronger than a solid one.

Bending of a beam

Depression δ under a central load

δ = Wl³4Ybd³

Rectangular beam of breadth b and depth d: more depth reduces δ most, hence the I-shape.
Circular beam: δ = Wl³12πYr⁴.

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EXAM-DAY CARD

Just see this before the exam

E = stressstrain

Hooke

Y = FLAΔl

Young

K = pVΔV

Bulk

η = FLAx

Shear

u = (stress)²2Y

energy density

YαΔθ

thermal stress

FORMULA SHEET

Whole chapter on one page

Stress & strain

stress = FA

strain = ΔlL, ΔVV, φ = xL

unit N m⁻²; strain: none

Young’s modulus

Δl = FLAY

own weight: MgL2AY

k = YAL

L = l₁T₂ − l₂T₁T₂ − T₁

Thermal & breaking

stress = YαΔθ

force = YAαΔθ

breaking force = P × A

L = Pρg

Energy

U = 12 Fl = YAl²2L

u = 12 stress × strain

u = 12 Y (strain)²

Bulk

ΔVV = pK, C = 1K

depth: p = ρgh

Δρ = ρpK

Δrr = p3K

Shear & Poisson

x = FLAη

ΔVV = ΔLL(1 − 2σ)

Y = 3K(1 − 2σ) = 2η(1 + σ)

9Y = 1K + 3η

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Young’s modulus changes with L and r

Pull of F at both ends means tension 2F

Energy stored = Fl

Using d for r in Δl (factor of 4)

CORRECT ✓

Y depends on the material only

Tension = F

Energy stored = 12 Fl

A = πd²4