Young's modulus

concretecakeUrban and Civil

Nov 29, 2013 (3 years and 6 months ago)

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1

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

Tensile means
relating to tension
or stretching.

When something is being pulled or stretched, it is in tension.

There will be two forces involved:


An applied force which puts the substance under tension


A tension within the substance, which acts on the object
putting it under tension

© TPS 2008

2

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

Tensile means
relating to tension
or stretching.

In this example, the force acting on the spring is
the weight of the mass on the end of the spring.

The tension within the spring is a force which acts
on the weight in the opposite direction.

Are these two forces Newton’s third law pairs?

No, they act on the same body and are of
different types.

3

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

Tensile means
relating to tension
or stretching.

The tension, pulling the spring back to its original
length is often called a restoring force.

4

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

Compressive
means relating to
squashing.

When something is being compressed, it will resist this
compression with a restoring force.

There will be two forces involved:


An applied force which puts the substance under compression


A restoring force within the substance which acts on the object
putting it under compression

5

The pillars holding this bridge up are

made from concrete.

Concrete can withstand the large compressive
forces that the bridge and the traffic on the bridge
produce.

Engineers need to understand about the ways in
which tensile and compressive forces behave.

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

Compressive
means relating to
squashing.

6

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

This cartoon
character is clearly
under stress!

If you put a person under stress, it produces a strain.

The same is true in Physics too.

We can apply both tensile (stretching) and compressive stresses to
objects by putting them under stress.

We can define stress as the force per unit area.

7

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

You will sometimes see the symbol
σ

(lower case sigma) used to
represent stress.

Its units are …

See later!

The area is the area perpendicular to the force vector.

area
force
Stress

8

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

If the units are Nm
-
2
, this
is, of course, pascal (Pa)
and you should always
quote stresses in Pa.

area
force
Stress

9

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

As suggested earlier,
stress produces
strain.

Strain is defined as the change in dimension (length) divided by the
original dimension.

This makes it the change in length per unit length.

What are the units of strain?

It has no units. The two units cancel out. It is said to be
dimensionless.

10

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

As suggested earlier,
stress produces
strain.

The symbol for strain is
ε
.





0
l
l
l


0
0
l
l



11

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

If we apply too much stress, the substance it is applied to will break.

The stress that causes the substance to break is called the …

This is also known as the ultimate tensile stress (for a stretching
stress) and has the symbol
σ
u
.

12

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

The measure of how strong a substance is, is its …

The greater the strength of a substance, the higher its breaking
stress.

13

Young’s modulus

Before we investigate Young’s modulus, we shall first define some
terms:

Some substances need huge stresses to produce an appreciable
strain. This is called a substance’s …

Diamond is very stiff; rubber is not at all stiff.

14

Young’s modulus

Young’s modulus is defined by the equation:





i.e.







E
Strain
Stress
E

15

Young’s modulus

You should be able to work out what the units of Young’s
modulus are.


They will be the units of stress divided by the units of strain.


Units of stress


Pa


Units of strain


it hasn’t got any


Units of Young’s modulus are Pa


Strain
Stress
E

16

Young’s modulus

We can also write the equation for Young’s modulus as follows:




Where
F

is the tensile force

A

is the cross
-
sectional area

l
o

is the original length

Δ
l

is the increase in length

0
l
l
A
F
E





17

Young’s modulus

The Young’s modulus for copper is
120 GPa and the Young’s Modulus
for aluminium is 71 GPa.

What does this mean?

Firstly, 1 GPa is 10
9

Pa so the
numbers revert to:


1.2 x 10
11

Pa for copper


7.1 x 10
10

Pa for aluminium

18

Young’s modulus

The Young’s modulus for copper is
120 GPa and the Young’s Modulus for
aluminium is 71 GPa.

What else does this mean?

It means that if we apply the same
force to identically sized pieces of
aluminium and copper, there will be a
greater expansion in aluminium than
copper.

(Density of copper is 8.9 gcm
-
3

and
the density of aluminium is 2.7 gcm
-
3
)

E = stress / strain

So E
x

strain = stress

If the stress is the same for
aluminium (A
l
) and
copper (Cu) then we get:

E
A
l

x

strain
A
l

= stress

E
Cu

x

strain
Cu

= stress

If the Young’s modulus for
copper is the biggest, the
strain for copper must be
the smallest.

So for the same stress,
aluminium stretches more
than copper.

19

Young’s modulus

The Young’s modulus for copper is
120 GPa and the Young’s Modulus for
aluminium is 71 GPa.

What else does this mean?

This means that the stiffness of copper
is greater than the stiffness of
aluminium.

Remember that stiffness is
related to the stress
required to produce a
certain strain.

For a given applied stress,
a stiff material will stretch
less than one which is less
stiff.

20

Young’s modulus

Words like strength, stiffness, stress and strain are words that we
hear used in every
-
day English. However, they have specific
meanings in Physics and you should use them appropriately.

21

Young’s modulus

If Young’s modulus for mild steel is
210 GPa and a 0.7 m long mild steel
girder of cross
-
sectional 10 cm x
10cm supports a 2 tonne car, how
much will the girder extend by?
(Assume the gravitational field
strength to be 10 Nkg
-
1
)

210 GPa is equivalent to

2.1 x 10
11

Pa


10 cm
2

is equivalent to

0.1 x 0.1 m
-
2

i.e. 10
-
2
m
-
2


2 tonne is equivalent to

2 x 10
3

kg


2 x 10
3

kg is equivalent to

2 x 10
4

N

0
l
l
A
F
E


1
.
1
10
10
2
10
1
.
2
2
4
11
l
x
x



m
x
l
5
10
05
.
1



22

Young’s modulus

How would we measure Young’s modulus?

Table

G clamp

Wire

There are various ways of doing this and the simplest is to use the
apparatus that we used to investigate the effect of stretching a
nylon wire.

We use a long wire to maximise the expansion which makes it
easier to measure more accurately.

We use a wire with a thin cross
-
sectional area for the same
reason.

23

Young’s modulus

How would we measure Young’s modulus?

Table

G clamp

Wire

However, making a wire thin will mean that we could introduce
inaccuracies in both measuring the cross
-
sectional area and
because the wire might not have a uniform cross
-
sectional area.

For this reason you should:


Measure the diameter using vernier callipers (some people spell
this as calipers). They are very accurate. You can then use the
formula for the area of a circle (A =

r
2
) to calculate the cross
-
sectional area (diameter = 2 x radius)

24

Young’s modulus

How would we measure Young’s modulus?

Table

G clamp

Wire

However, making a wire thin will mean that we could introduce
inaccuracies in both measuring the cross
-
sectional area and
because the wire might not have a uniform cross
-
sectional area.

For this reason you should:


Make sure that you can read a pair of vernier callipers!

25

Young’s modulus

How would we measure Young’s modulus?

Table

G clamp

Wire

An additional problem for really accurate work is the possibility
of the wire expanding due to the room getting warmer.

A temperature rise of 10
°
C will make a 2m long steel wire expand
by over 2mm.

Air conditioning might sound like a possible solution but would
be expensive and the temperature would still vary, albeit over a
smaller range.

26

Young’s modulus

An alternative arrangement is
to use two identical wires: one
under test, the other next to it
and taut but not heavily loaded.
The expansion is measured
relative to the lightly loaded
wire. If there is any increase in
temperature, both wires will get
longer by the same amount. If
we just measure the relative
expansion, we will have
excluded the thermal
expansion.

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling*

*
The brace must be attached to a strong part of the building.

27

Young’s modulus

The wires should be as long
as possible and their lengths
measured.

Each time a weight is added
to the device, that wire will
get longer.

The spirit level will go out of
balance.

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling*

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling

The expansion is exaggerated in the picture.

28

Young’s modulus

The wires should be as long
as possible and their lengths
measured.

Each time a weight is added
to the device, that wire will
get longer.

The spirit level will go out of
balance.

The knurled dial is then
rotated to level the spirit
level again.

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling

The expansion is exaggerated in the picture

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling

The expansion is exaggerated in the picture.

29

Young’s modulus

The expansion is measured
by noting the number of
turns on the knurled dial.

This is calibrated so that, for
example, one turn is
equivalent to 0.5 mm.

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling

The expansion is exaggerated in the picture

Knurled
dial

Spirit level

Weights

Reference
wire

Wire under
test

Brace attached to the wall or ceiling

The expansion is exaggerated in the picture.

30

Young’s modulus

However, in all of these experiments, it is important to measure
the expansion, as the wire is being loaded and as it is being
unloaded.

If the “paths” are not the same in each case, then either the elastic
limit has been exceeded or the limit of proportionality.

Either of these would make the values obtained invalid.

If the wire appears to show a permanent deformation, this could
be due to a change in temperature.