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:
This shows: f µ 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 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).
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.
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
To calculate the energy stored in a deformed object,
find the area under the force – extension graph.
In this example:
Work = ½ force x extension = ½ x 10 x 0.02 = 0.1 J
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.
A useful tip: In calculations stress is usually a very large number and strain is usually a very small number. If it comes out much different than that, you’ve done it wrong!
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
We call this constant the Young Modulus and give it the symbol ‘E’.
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:
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
Loads of revision:
http://www.jcphysics.com/toolbox_indiv.php?sub_id=1