Topic 6 Elastic Properties of Solid Materials

We will study how materials behave under compression (squashing) forces and tension (stretching) forces.  Scientists need to know how materials behave so that they can assess how suitable a particular material is to a particular job.

Density

Density is mass per unit volume.  Density, mass, and volume are linked by a simple relationship:

SI Units for density are kg/m³.  In some texts, you will find some densities given in g/cm³.  It is important that you use the SI units otherwise formulae will not work.  To convert you will need the following conversion:

1 g/cm³ = 1000 kg/m³

Click HERE to see some important definitions.  You will see these in the text books.

Hooke’s Law

If we load a spring, we find that the extension (code e) or stretch is proportional to the force (code F).  If we double the force, we double the stretch.

   F µ e

Þ F = ke

The constant of proportionality is called the spring constant and is measured in newtons per metre (N/m). 

Bear trap:  Remember to convert the extension into metres and the load into Newtons before working out the spring constant.

We can plot this as a graph:

We can see that the graph is a straight line and that the gradient gives us the spring constant.  That is why we have the extension on the horizontal axis.

The same is true if we apply a squashing force.

Stress and Strain

If we stretch a wire, the amount it stretches by depends on:

• its length
• its diameter
• the material it’s made of.

If we have two of the same material and length, it is clear that the thicker wire will stretch less for a given load.  We make this a fair test by using the term tensile stress which is defined as the tension per unit area normal to that area.  The term normal means at 90^o to the area. 

We can also talk of the compression force per unit area, i.e. the pressure. 

Stress = Load (N) = F                                             

           area (m2)   A

In some text books you may see stress given the code s (sigma, a Greek letter 's').

You will have met the expression F/A before.  It is, of course, pressure, which implies a squashing force.  A stretching force gives an expression of the same kind.  Units are newtons per square metre (N/m2) or Pascals (Pa).

                                     1 Pa = 1 N/m²

You must always convert areas to square metres for this equation to work.  Remember that radii will often be given in mm or cm.  This is a common bear trap.

1 mm² = 1 x 10^-6 m²

Therefore if you get an area of 10^-2 you probably have forgotten to do the conversion to square metres.

If we have a wire of the same material and the same diameter, it doesn’t take a genius to see that the wire will stretch more for a given load if it is longer.  To take this into account, we express the extension as a ratio of the original length.  We call this the tensile strain which we define as the extension per unit length.

                        Strain = extension (m)               = e

                                   original length (m)            l

There are no units for strain; it’s just a number.  It can sometimes be expressed as a percentage. 

You will find that the same is true for when we compress a material.

Elastic Strain Energy

When we stretch a wire, we have to do a job of work on the wire.  We are stretching the bonds between the atoms.  If we release the wire, we can recover that energy, which is called the elastic strain energy.  Ideally we recover all of it but in reality a certain amount is lost as heat.  This lost energy is called hysteresis. 

The energy is the area under the force-extension graph.

So we can use this result to say:

This result can be obtained using the  process of integration, which is part of the branch of mathematics called calculus.  You are not expected to know this for AS, although you will be if you study Physics at University.  All we need to say here is that the energy is the area of the triangle.

Stress-Strain Curves

Stress-strain graphs are really a development of force-extension graphs, simply taking into account the factors needed to ensure a fair test.  A typical stress-strain graph looks like this:

We can describe the details of the graph as:

• P is the limit of proportionality, where the linear relationship between stress and strain finishes.

• E is the elastic limit.  Below the elastic limit, the wire will return to its original shape.

• Y is the yield point, where plastic deformation begins.  A large increase in strain is seen for a small increase in stress.

• UTS is the ultimate tensile stress, the maximum stress that is applied to a wire without its snapping.  It is sometimes called the breaking stress.  Notice that beyond the UTS, the force required to snap the wire is less.

• S is the point where the wire snaps.

We can draw stress-strain graphs of materials that show other properties.

• Curve A shows a brittle material.  This material is also strong because there is little strain for a high stress.  The fracture of a brittle material is sudden and catastrophic, with little or no plastic deformation.  Brittle materials crack under tension and the stress increases around the cracks.  Cracks propagate less under compression. 

• Curve B is a strong material which is not ductile.  Steel wires stretch very little, and break suddenly.  There can be a lot of elastic strain energy in a steel wire under tension and it will “whiplash” if it breaks.  The ends are razor sharp and such a failure is very dangerous indeed.

• Curve C is a ductile material

• Curve D is a plastic material.  Notice a very large strain for a small stress.  The material will not go back to its original length.

  Rubber has a strange force-extension graph.

Stress strain graphs are used extensively by material scientists.  Machines like this are used:

This will prevent failures like this.

The Young Modulus

The Young Modulus is defined as the ratio of the tensile stress and the tensile strain.  

  So we can write:

            Young modulus = tensile stress

                                      tensile strain

  We know that

            tensile stress = force =  F

                                  area      A

  and that

            tensile strain =  ___extension __   = e

                                   original length        l

So we can write:

Units for the Young Modulus are Pascals (Pa) or newtons per square metre (Nm^-2).

The Young Modulus describes pulling forces. 

We can link the Young Modulus to a stress strain graph.

The Young Modulus is the gradient of the stress-strain graph for the region that obeys Hooke’s Law.  This is why we have the stress on the vertical axis when we would expect the stress to be on the horizontal axis.

The area under the stress strain graph is the strain energy per unit volume (joules per metre³).

Strain energy per unit volume = 1/2 stress x strain.

The units arise because stress is in Nm^-2 and strain is mm^-1 (NOTE: This unit here is not "millimetres to the minus one", but metres per metre which mean no units).

        Nm^-2 x mm^-1 = Nm m^-3.  Nm is joules, hence Jm^-3