1. STRENGTH OF
MATERIALS
John Parkinson

2. THE GOLDEN ORB SPIDER’S SILK IS TWICE AS STRONG AS MILD STEEL AND
CAN STRETCH BY UP TO 30% 0F ITS ORIGINAL LENGTH

3. He suspended various masses from springs, and measured the extension.
STIFFNESS
first investigated by
ROBERT HOOKE, 1648 m m
He suspended various masses from springs, and measured the extension.

4. Hooke's Law m e.g. for a spring or wire e F = mg
“Up to a certain level of force, the extension (e) produced is proportional to the applied force (F).”

5. Hooke's Law k is called the spring constant
EXTENSION
Force
e/m
F/N
0.00
0.12 1
0.24 2
0.36 3
0.48 4
0.60 5
0.72 6
0.84 7
0.96 8
1.08 9
1.20 10
k is called the spring constant
k is equal to the gradient of the graph = 8.3 Nm-1
k is a measure of the spring’s stiffness

6. WORK DONE IN STRETCHING / ENERGY STORED IN A STRETCHED SPRING OR WIRE
1. Special case: when F is proportional to e
force
extension e F
Work Done =
Average Force x Extension
W = ½ Fe
which is the area under the graph
W = ½ ke2
or, as F = k e

7. dw = f de 2. The general case de
WORK DONE IN STRETCHING / ENERGY STORED IN A STRETCHED SPRING OR WIRE
2. The general case
force
extension f
dw = f de e de
which again is the area under the graph

8. STRETCHING OF A POLYMER e.g. RUBBER
force
extension F
Initially it is easy to merely straighten the polymer chains. F
It then becomes much harder to separate the atoms of the carbon chain F C F

9. work done in stretching
STRETCHING OF RUBBER
force
extension
loading
work done in stretching
HYSTERESIS
LOOP
unloading
energy recovered
energy LOST as internal energy
The word HYSTERESIS is used to refer to the energy lost in a cyclic process.

10. Extension depends upon: (i) Material Properties
Steel is approximately 1½ STIFFER than copper.
Which is graph below shows the stretching of a copper bar and which a steel bar under loading? A
force
extension B
ANSWER
WE CANNOT TELL!
Extension depends upon:
(i) Material Properties
(ii) Geometry of Bar (i.e. L and A)

11. s ε l STRESS and STRAIN = A = F e
TENSILE STRESS and TENSILE STRAIN are used to compare the stiffness of different materials.
TENSILE STRESS, σ = force applied per unit cross sectional area A F = s A F
TENSILE STRAIN, ε = fractional change in length under load
= extension ÷ original length l e = ε
Original length l
Extension,e

12. Strain can also be expressed as a percentage.
STRESS and STRAIN
UNITS A F = s
N m-2 = Pa
STRESS l e = ε
NO UNITS
STRAIN
Strain can also be expressed as a percentage.

13. WHICH IS COPPER, A or B? ANSWER B must be COPPER
Steel is approximately 1½ STIFFER than copper.
stress
strain A B
WHICH IS COPPER, A or B?
ANSWER
B must be COPPER

14. YOUNG’s MODULUS OF ELASTICITY - E
E measures stiffness
stress
strain
E = gradient of the stress – strain graph
e A
F l = E
UNITS of E are Pa (GPa)

15. YOUNG’s MODULUS OF ELASTICITY
TYPICAL VALUES
MATERIAL
E ( MPa )
Carbon Fibre
270
Steel
210
Copper
130
Kevlar
124
Bone 28
Rubber
0.02

16. Copper – a Ductile Material
STRESS – STRAIN GRAPHS
Copper – a Ductile Material
stress L Y F U
L = Limit of Proportionality
Y = Yield Point
U = Ultimate Tensile Strength (UTS)
F = Fracture
strain

17. (permanent deformation)
STRESS – STRAIN GRAPHS
Copper – a Ductile Material
stress
strain L Y F U
Strain hardening
Necking
UTS
Yield Strength
PLASTIC REGION
(permanent deformation)
ELASTIC REGION

18. If the load is removed a Permanent Set remains in the sample.
STRESS – STRAIN GRAPHS
stress
strain
Some of a specimen’s elasticity may be lost before the yield point is reached. Y
Permanent
Set
If the load is removed a Permanent Set remains in the sample.

19. AREA UNDER STRESS – STRAIN GRAPHSl A
Energy stored per unit volume
Energy stored per unit volume = ½ STRESS X STRAIN

20. STEEL STRESS – STRAIN GRAPHS stress Y = (UPPER) YIELD POINT
LOWER YIELD POINT is the point at which the main plastic deformation begins.
Copper for comparison
strain

21. FRACTURE AT THEIR ELASTIC LIMIT
STRESS – STRAIN GRAPHS
BRITTLE SUBSTANCES
stress
e.g. Cast Iron or Glass X
BRITTLE SUBSTANCES
FRACTURE AT THEIR ELASTIC LIMIT
strain

22. YOUNG’S MODULUS APPARATUS
Two identical wires are suspended from the same support.
This counteracts any sagging of the beam and any expansion due to change in temperature.
Extensions, e, are measured with the vernier scale.
Diameter of the wire, D, is measured with a micrometer.
BEAM l
support wire
test wire
main scale
venier scale
fixed load keeps system taut
F = mg
Plot F vs. e
variable load F e