In order to use a material safely an engineer must be able to describe how it behaves under different conditions. In this section you can find out how engineers describe and specify the properties of a material and how the materials are tested and the properties measured.
If you are going to look at the mechanical properties of an object a good starting point is Hooke’s Law. Hooke was a contemporary of Isaac Newton who's famous line, “If I have seen a little further it has been by standing on the shoulders of Giants” may have been a snub against Hooke who had a hunched back.
Hooke’s Law states that for many objects, when a force is applied to them the extension is proportional to the applied force. So if a 2 N force produces an extension of 3 cm then a 6 N force should produce an extension of 9 cm (so long as the elastic limit is not exceeded).
The mathematical description is:
Where F is the applied force, Δx is the extension and k is a constant that depends on the object being tested.
This is fine for a pure science investigation but is not of much use to applied subjects such as the many branches mechanical engineering.
Engineering can be confusing. Different shapes, different materials and different forces; every situation seems to be... different. The easiest way to approach the problem is to look at the material that a component is made from rather than the component itself. This allows each new component’s behaviour to be predicted before it is ever made. The surprising thing is that it took over one and a quarter centuries before the idea occurred to anybody.
The principal engineering figure is Young’s Modulus, introduced by Thomas Young in 1807, it is a measure of the stiffness (or resistance to stretching under tension) of the material. In effect it takes Hooke’s Law and restates it in terms of the material the object is made from.
Hooke’s Law |
Young’s Modulus |
|||
Quantity (symbol) |
Unit |
Quantity (symbol) |
Calculated |
Unit |
Force (F) |
N |
Stress (s) |
|
N m-2 (Pa) |
Extension (Δx) |
m |
Strain (e) |
|
Dimensionless (or expressed as a percentage) |
Constant (k) |
N m-1 |
Young’s Modulus (E) |
|
N m-2 (Pa) |
Now instead of each and every component having a different ‘spring constant’ each material has a value of Young’s Modulus which can be checked up in a reference book or on line. In practice most mechanical designs are now produced with the aid of a computer and the engineer need only specify the material and precise shape of the component for its behaviour to a range of forces to be calculated.
The range of values for Young’s Modulus is very large and because a large stress is generally needed to produce a small strain the absolute numbers tend to be very large. As a result Young’s Modulus is usually quoted in GPa (109 Pa).
Metals |
Non-Metals |
||
Material |
E (GPa) |
Material |
E (GPa) |
Tin |
50 |
Rubber |
0.01 – 0.1 |
Aluminium |
70 |
Spruce (across grain) |
0.003 |
Gold |
78 |
“ (along grain) |
13 |
Copper |
110 |
Bone |
21 |
Steel |
(typically) 190 - 210 |
Carbon Nanotubes |
1,000 + |
Tungsten |
400 |
Diamond |
1,200 |
How much does each sample stretch by?
Calculate the stress and the % strain in each sample.
Which material would you chose to suspend a 10 kg by in practice and why?
Every school boy knows that you can store energy in a stretched spring (or rubber band) and then release that energy (to fire a pellet across the classroom). But how much energy can be stored? The answer lies in the force – extension and stress – strain diagrams:
Force – Extension Diagram for a Steel Wire
In the force – extension diagram the work that has been done stretching the wire is given by the area under the curve since work done is equal to the applied force multiplied by the movement (or extension) in the direction of the force. So long as the force required to stretch the wire increases proportionally to the extension, the final work done is equal to:
As useful as tables of data are, most people find it considerably easier to look at a picture. As well as giving a quick impression of the data a diagram can sum up a lot of detail as well.
This very simple diagram shows very quickly the relative stiffness of tungsten, steel and aluminium; the steeper the line the greater the stiffness of the material (i.e. a steep line means a high Young’s Modulus). The dashed lines show that the stress in the materials would continue to rise linearly if the strain were increased. But how long would it continue to rise for?
Many metals are ductile and would look far more like the following image if they were tested to destruction.
Starting at point A the material could be strained uniformly up to point B (the limit of proportionality). This is in the elastic region and if the stress were removed the sample would return to its original length (zero strain).
A little beyond point B the material reaches its elastic limit. If strained beyond this point it will not return to its original length when the stress is removed. It is now permanently strained and the material has entered the plastic region.
Slightly beyond the elastic limit the material reaches its yield stress at point C’. If the stress is removed the material will follow the dashed line, parallel to the original line, down to point A’. Beyond the yield stress a small increase in stress will produce a large permanent strain (e.g. C” down to A”).
If the stress is increased further there will be a rapidly increasing strain up to point D, the ultimate tensile strength. This is the highest point on the curve and so the maximum stress that the material can be subjected to. At this point the stress in the sample will suddenly decrease as the sample rapidly stretches and fail at point E. In fact what has happened is that the sample has ‘necked’, a small section has stretched and narrowed, this increases the stress in the small volume of the neck which in turn stretches further. Inside the neck, small gaps open up which rapidly combine into a single large void. The stress is now concentrated in a ring of material around the void which quickly tears open, failing at point F.
Necking process with photograph of cup and cone fracture
