Stress and strain: the Young modulus

Introduction

The Young modulus

As we mentioned in our notes on Hooke's Law, when two identical springs are joined in series, we saw that the spring extended twice as much as a single spring on its own

  • The extension depends on its original length

Similarly, if we join two spring in parallel, then the springs extend half as much as a single spring on its own

  • The extension depends on the area over which the force is acting

In each case, the property of the object depends on the dimensions of the object itself

If we want to compare materials fairly, rather than compare two different objects, then it is better to have a measurement that doesn't depend on the shape or size of the object

  • The comparison is made by using stress or strain instead of force and extension

Tensile stress

Tensile stress is a measurement of the force applied over the cross-sectional area of sample of material

The Young modulus

Tensile strain

Tensile strain is the ratio of the extension and original length of the sample

The Young modulus

The Young Modulus

We can now use these quantities to calculate a measure of the stiffness of an elastic material that is independent of the shape of the sample of the material

This is called the Young Modulus, E, of the material

The Young modulus

Young modulus is measured in Pa (or Nm-2)

The Young modulus

This graph shows a stress-strain graph for copper

  • It looks like the force-extension graph shown earlier, but this graph will be valid for ANY sample of copper

O-P on the graph represents the range of tensile stree for which the copper obeys Hooke's law

  • The gradient of this section is the Young modulus for copper
  • This value will be the same no matter what the size or shape of the sample of copper being used
  • Point P represents the limit of proportionality for the material

Point E on the graph represents the elastic limit

  • Up to point E, if the stress is removed, the sample of copper will return to its original length
  • Beyond this point, copper behaves plastically; it does not return to its original length

The yield point of the material is given by Y

  • This is the value of stress beyond which the strain increases rapidly for small increases in stress

The ultimate tensile stress (UTS) of copper is sometimes called the 'maximum strength' or 'strength of the wire'

  • Just before this point the copper becomes narrower at the weakest point, known as necking
  • The UTS is the stress at which the material breaks

Some materials will also show a phenomemon known as creep

  • This is when the sample continues to extend over a period of time, even though the stress applied is not increased

Example

The Young modulus

Experiment

A simple experimental set-up for copper can be used to measure its Young modulus

But, if you want to be accurate, you can use an experimental set-up like this:

  • This is particularly useful for materials such as steel, which generally give smaller values of strain

The Young modulus

Here, the left-hand wire is a reference wire that holds the main scale of the vernier calipers, usually calibrated in milimeters

The reference wire has a mass hung from it to keep it taut

The sample wire (made from the same material) is hung close to the reference wire and holds the smaller scale of the vernier calipers

As weights are added to the sample wire it extends, and the scale moves relative to the main scale - this allows the extension of the wire to be measured

Interpreting stress-strain graphs

Stress-strain graphs allow us to describe the properties of materials, and also to predict the stresses at which changes in these properties might occur

The Young modulus

This graph compares the stress-strain graphs for four different materials: ceramic, steel, glass and copper

  • Ceramics are extremely strong and have very high UTS values. However, they show very little plastic behaviour before they fracture, so they are also very brittle
  • Glasses have lower UTS values than ceramics and so are less strong, but they are also brittle, generally showing no plastic behaviour before they break

Steel is made by adding different elements to iron to form an alloy

  • Common elements used in steel-making are carbon, manganese, and chromium
  • Types of steel differ in the percentage composition of the various elements added to iron to create them
  • This affects the properties of the steels and they are generally much stiffer than ductile metals such as copper
  • This can be seen on the graph from the shallower gradient of copper giving a lower value of Young modulus

The high-carbon steel shown in the graph is a strong but brittle material

  • It shows elastic behaviour at higher values of stress but fractures with little plastic behaviour
  • This type of steel is often used in cutting tools and drill bits because it has a higher UTS value
  • Other types of steel may show plastic behaviour but have a lower UTS value

Copper has a plastic region because it is a ductile material, and this makes it ideal for forming into wires for use in electrical circuits

Strain energy density

The strain energy density is the strain energy per unit volume of a sample

  • Again, this does not depend on the dimensions of the material being tested (the result will be the same for all materials, regardless)

From earlier we saw that: Strain energy = 1/2 F▵l

If l is the original length of the wire, and A is its cross section, then the volume of the wire is Al. Therefore:

The Young modulus

Example

The Young modulus

Extra

Also see our notes on: