There is far more to a material than just tensile stiffness.
Young’s Modulus is a vitally important number that describes how a material behaves under tension. It is probably the single most widely used engineering statistic and it is certainly the most widely taught. It doesn’t however tell the whole story; real engineering materials are not just stretched, they are also compressed, bent and twisted. A further set of numbers have to be used to describe the behaviour of materials to these forces. Most of these new numbers are defined in the same way as Young’s Modulus, that is:
The main difference is that the type of stress and strain will not be tensile.
for TimeThe second commonest type of force an engineering component is subjected to is compression. Typical compression situations include the towers of a suspension bridge, the foundations of a building and a nail being struck by a hammer. The modulus of compression is given by:
This is very similar to the Young’s Modulus. Compressive stress is the change in length divided by the original length and compressive strain is the applied force divided by the cross sectional area that it acts over. This should not be too much of a surprise when you find out that for many materials the compressive modulus is exactly the same as the Young’s Modulus, so long as the applied force is not too great, exceptions are composite materials such as concrete.
If the force is increased then the material will fail. If the material is ductile it will flow into a new, fixed shape (a blacksmith would use this to reduce the thickness and increase the width of a steel bar when making a horse shoe). If however the material is brittle then it is crushed (think about a workman using a pneumatic drill to make a hole in concrete). When this happens the compressive strength of the material has been exceeded.
EnjoymentMany objects are exposed to what are called ‘shearing forces’, two equal and opposite forces as with a compressive or tensile force, however the two forces do not pass through the same point. The result is that the profile of the object becomes distorted. A good example of a shear force is if you try to push the edge of a pencil rubber over a rough surface.
The equation for the Shear Modulus is given by:
Notice that in the shear stress the two distances are not in the same direction the displacement Δx and the height h are perpendicular to each other.
A bicycle brake block is made from rubber with dimensions as shown on the diagram. The brake levers apply a force of 200 Newtons to push the brake surface against the wheel rim. As the bike slows down the brake surface is dragged back by 5 mm.
By how much is the brake block compressed by the 200 N force?
How large is the force that the brake block applies parallel to the wheel rim?
What is the coefficient of friction between the rubber brake block and the steel wheel rim?
Young’s Modulus for rubber is 0.05 GPa and the Shear Modulus is 0.0006 GPa.
Shear forces also occur whenever something is twisted such as a key in a lock a screw being driven into wood or the drive shaft of a car carrying the driving power from the engine to the wheels. You can think of a torsional stress as one that is trying to slide a series of cylinders past each other. As they cannot slide, the individual mini cylinders have to be distorted. The shear force is greatest at the surface and reduces to zero at the centre and this means that torsional cracks tend to start at the surface and then spread inwards. Because a rough surface can be the focus for a crack to start at it is common for components that are subjected to very high torsional stresses to be polished.
As well as a Shear Modulus there is also a shear strength, the maximum shear stress beyond which a material will fail. When a component does fail in torsion it tends to produce a helical crack. So long as the material is ‘isotropic’ (the same structure in all directions) then this happens regardless of whether the material is a steel drive shaft, a human shin bone or a cucumber. Why not try this at home? Shin bones take a long time to heal and drive shafts are very strong so use a cucumber. Take hold of each end and twist it (don’t bend). You should manage to produce the characteristic helical fracture.
We tend to think of solids and liquids as being incompressible this is not strictly true though; all matter, given a large enough pressure, is compressible. The Bulk Modulus tells us just how compressible materials are.
The normal way of expressing a stress is as a force divided by the area that it acts over but this is just a pressure! In the case of the bulk modulus the pressure acts uniformly over the entire surface of the sample and so it is easier to think of pressures rather than forces. The pressure acting on the sample is increased from p to p + Δp and there is a corresponding change of volume from V to V + ΔV (ΔV will be negative if Δp positive and so we add a negative sign to the equation so as to keep the Bulk Modulus as a positive number).
Density at atmospheric pressure: Sea water, 1.03 g cm-3, aluminium, 2.70 g cm-3.
When a sample, such as a wire, is stretched then it gets a little thinner. This is easiest to see with a rubber band. The Poisson Ratio puts a number to this effect.
The axial direction is the direction of the applied force, be that tensional or compressive. The (two) transverse directions are at right angles to this. If the sample is stretched then (usually) it will get thinner and so the Poisson Ratio is defined as:
Each material has its own Poisson Ratio. The value has to be between -1.0 and +0.5. Almost all material are between 0 and +0.5. A value of 0 means that a sample would squash straight down without spreading out to the sides at all (cork is very close to this) and +0.5 is a material that has a volume that does not change when it is stretched or compressed in one direction (rubber comes very close to this). Some materials have a negative Poisson Ratio; they are called auxetics and have the peculiar property that if you try to stretch them they actually get wider!
There are two other, less commonly used moduli; the P Wave Modulus M is used to determine the ratio of tensile or compressive stress to strain assuming that there is no distortion in the other directions and Lamé’s first parameter l has no direct physical meaning but is used to simplify other calculations of elasticity.
One odd point is that even though you have read here about six different measures of elasticity… you only ever need two of them! So long as the material is homogenous (the same at all points in the material) and isotropic (the same in all directions) then given any two of the numbers (and it doesn’t matter which) you have enough information to calculate the other four. The full set of figures are, whilst strictly speaking largely redundant, are used to make life easier for particular sets of calculations. The moduli are also useful for finding the speed of waves in matter as set out below (ρ is the density of the material).
Modulus |
Wave Type |
Speed |
Comments |
Young’s Modulus (E) |
Longitudinal in solids |
|
This is only valid if the sample is a rod or bar that is much thinner than the wavelength of the sound wave. A thin bar allows the material to spread out at right angles to the wave. |
Bulk Modulus (K) |
Longitudinal in fluids |
|
This governs the speed of sound in both water (for whales and synchronised swimmers) and in air (for the rest of us). |
Shear Modulus (G) |
Transverse in solids |
|
Fluids cannot support shear waves. The classical transverse wave is the earthquake S-wave. |
P-wave Modulus (M) |
Longitudinal in solids |
|
This equation has to be used if the object that the wave is traveling through has a breadth at least comparable to the wave length. This is because in this situation the material cannot move sideways as the wave passes. As a result a P-wave is always faster than a simple Young’s Modulus wave unless the Poisson Ratio is zero (in which case the two waves have exactly the same speed. The classical example of this is the earthquake P-wave. |
There are another set of figures that are important in some areas of engineering, these are the ‘Specific’ values. Think of it this way: the heat capacity of an object is the amount of energy required to raise its temperature by one degree Celsius, the specific heat capacity is the energy required to raise the temperature of a unit mass (usually 1 kg) of the material by one degree Celsius. The Specific Young’s Modulus of a material will therefore be equal to Young’s Modulus divided by the density and the Specific Ultimate Tensile Strength will be the Ultimate Tensile Strength divided by the density.
The Ultimate Tensile Strength and the Young’s Modulus of a material will tell you how thick your components need to be to achieve a particular strength and stiffness. The Specific Ultimate Tensile Strength and the Young’s Modulus will tell you how much mass will be needed to achieve the same thing! Knowing your mass budget is critically important in both aircraft and spacecraft. For an aircraft every kilogram that can be saved is a kilogram that does not have to be lifted to an altitude of ten kilometres or more. Saving weight means that less fuel is burned and so flights can cost less and, even more importantly, they do less environmental damage, by emitting less CO2. For space applications weight saving could not be more important as every kilogram carried to low earth orbit costs in excess of $20,000.
What other factors should the two engineers consider when selecting their materials?
A block of material is to be used to support a piece of equipment and to reduce the effects of vibration. The material is to be cut into a square block 90 mm on a side and 20 mm thick which will then be set into a square housing in the floor that is the same depth as the block thickness and 100 mm on a side.
