Deformation of Solids
Forces can cause objects to change shape. The way in which an object deforms depends on material,
size of the force and direction of the force.
Hooke’s Law
Measure how a spring stretches as you apply an increasing force to it and you get:
F
This shows that :
fe
In words, force is proportional to extension.
This is Hooke’s Law. Any object behaving like this is said to be
obeying Hooke’s law. In fact, many objects and materials obey
e
Hooke’s Law for part or all of their deformation, including glass and wire.
The gradient of this graph is constant. Let’s call it ‘k’. The value of the constant can be found from:
In words, ‘k’ is the force per unit extension.
Rearrange to give:
F = ke
This is Hooke’s Law.
‘k’ is called the spring constant (or the spring stiffness).
F
Units for k: Nm-1
A stiffer spring has a greater value of spring constant.
If you continue to stretch a spring it eventually comes
to a point where it stops obeying Hooke’s Law.
Stops obeying
Hooke’s Law here
– Elastic limit, E
e
The point on the graph where it stops obeying Hooke's Law is often called the 'limit of proportionality'
because it is the last point at which the deformation of the material is proportional to the force acting on
the material. At about the same moment as it stops obeying Hooke’s law, you will notice that if you
unload the spring it won’t return to its original shape. It has been permanently deformed. We call this
point the elastic limit – the limit of elastic behaviour.
If a material returns to its original size and shape when you remove the forces stretching or deforming
it, we say that the material is demonstrating elastic behaviour.
Permanent deformation is a sign of plastic behaviour.
Energy in Deformations
F/N
To calculate the energy stored in a deformed object,
find the area under the force – extension graph.
10
In this example:
Work = ½ force x extension = ½ x 10 x 0.02 = 0.1 J
e/m
0.02
Some common examples to learn
Note: + = breaking point. So glass breaks as soon as it stops obeying Hooke’s Law.
Note: the rubber takes more energy to load up (area under the loading line) than it gives back when it
unloads (area under the unloading line). The difference between these two (the area of the gap) is given
out as heat. The rubber gets hot. This is known as a hysteresis curve.
Stress and Strain
The problem with force – extension graphs is that they only give information about the exact object and
material that you are examining.
Stress and strain are measurements that allow us to compare behaviour of materials and objects no
matter what size or shape they are because the force and extension are multiplied up or down to find out
what the force would be if it was spread over 1 m2 or what the extension would be per metre of the
original material.
Stress and Strain, Definitions
Stress is defined as the force per unit area of a material.
Stress = force / area .
Units: Nm-2 or Pa.
Strain is defined as extension per unit length.
Strain = extension / original length.
Strain has no units.
For the description of the elastic properties of linear objects like wires, rods, columns that are either
stretched or compressed, a convenient parameter is the ratio of the stress to the strain, a parameter
called the Young's modulus of the material. Young's modulus can be used to predict the elongation or
compression of an object as long as the stress is less than the yield strength of the material.
Elastic Properties of Selected Engineering Materials
Material
Density Young's Modulus Ultimate Strength Su Yield Strength Sy
(kg/m3)
109 N/m2
106 N/m2
106 N/m2
Steel
7860
200
400
250
Aluminium
2710
70
110
95
Glass
2190
65
50
...
Concrete
2320
30
40
...
Wood
525
13
50
...
Bone
1900
9
170
...
Polystyrene
1050
3
48
...
a Structural steel (ASTM-A36), b In compression, c High strength, d Douglas fir
Data from Table 13-1, Halliday, Resnick, Walker, 5th Ed. Extended
Stress - strain graphs
Instead of force – extension graphs
we can draw stress – strain graphs.
In all the cases that you come across
the shape of the graph is exactly the
same as that for a force – extension graph.
So here are some examples:
Note: the gradient of a stress – strain graph = stress / strain.
For the straight line (proportional) part of the graph while
Hooke’s Law is obeyed, the gradient is constant.
So stress / strain = a constant = E
Or
E = Fl
We call this constant the Young Modulus and
give it the symbol ‘E’.
eA
Units for Young Modulus: Nm-2 or Pa.
The value of Young Modulus is always the same for a particular material, no matter what the size of the
sample being tested. That’s one of the reasons why stress – strain graphs are more useful than force –
extension graphs.
‘E’ gives a measure of the stiffness of a sample. A very big value of E suggests a very stiff material.
(Note that E usually has massive values as stress (a big number) is divided by strain (a small number) to
produce a huge result.)
Energy in stress – strain graphs
Note that the area under a stress – strain graph gives the energy stored per unit volume (how many
joules are stored in 1m3 of the material) not just the energy stored.
but
Al = Area x Length = Volume
so the equation becomes:
Area = energy stored per unit
volume
which is the work done per unit volume
the work done
Deformation of solids
f = ke - Hooke’s law
Equations
Stress/ Strain
- stress equation
Work = ½ force x extension
- strain equation
- Young’s modulus
- Energy per m3
Symbols
Deformation of solids
Stress/ Strain
F – Force, N
E - Young Modulus,
Nm-2 or Pa
k - spring constant (or the spring stiffness). Nm-1
F - Force, N
e – extension, m
A - Area, m2
s - stress, Nm-2 or Pa
e - strain, no units
e - extension, m
l – original length of material, m
Glossary
Tensile / tension forces: forces stretching something.
Compressive / compression forces: forces squashing something.
Elastic deformation: Non-permanent, the object returns to its original shape when the forces are
removed.
Plastic deformation: permanent deformation.
Brittle: a material that can’t deform plastically without breaking.
Ductile: a material that can undergo extensive plastic deformation without breaking.
Hard: very difficult to scratch or mark
Strong: will not break easily under tension or compression.
Ultimate tensile strength: the maximum tensile force that an object / material can stand.
Useful Web sites:
Hooke’s Law applet:
http://webphysics.davidson.edu/Applets/animator4/demo_hook.html
Practical analysis of Hooke’s law:
http://www.phys.utk.edu/labs/pl232hl.pdf
Simplified investigation:
http://www.frontiernet.net/~jlkeefer/hookes.html
Values in Periodic Table:
http://www.webelements.com/webelements/properties/text/definitions/youngs-modulus.html#def
Notes on Young’s Modulus:
http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html
Experimental investigation:
http://schools.matter.org.uk/Content/YoungModulus/experiment_1.html
http://pergatory.mit.edu/2.007/handouts/bending/bending.pdf
Spider’s web example:
http://www.tiem.utk.edu/~gross/bioed/bealsmodules/spider.html