Bones

Without a framework of bones, we would be quivering immobile blobs of jelly (actually you can see that outside any fast-food outlet in any UK town on a Saturday night).  Bones are hollow tubular structures that have great strength for a relatively low mass.  They can absorb massive loads without breaking.  If they do break, they generally heal.

Bones follow the rules of materials that we have seen before.  The table shows some definitions you have come across before:

Stress and Strain

Stress is defined as force per unit area.  In Physics code it is written:

s = F

       A

The physics codes are:

• s - Stress (the curios letter is s, sigma, a Greek letter 's')

• F - Force (N);

• A - Area (m²).

The units for s are Newtons per square metre (N m-2) or Pascals (Pa).  Pascals are quite small units so we often use kilopascals (kPa) or even megapascals (MPa).  You should know how to convert.

The same formula applies whether the sample is stretched (tensile) or squashed (compressive).

The ultimate tensile (or compressive) stress is the stress needed to break a material.  It's also known as the breaking stress and is a measure of the strength of the material.

Strain is defined as extension (or compression) per unit length.  We know that Hooke's Law tells us that the extension is proportional to the force.  So we can write:

Dx µ s

The extension depends on:

• the length of the sample;
• the area of the sample.

If we keep the areas of two different samples the same, we need only worry about the length.  But suppose we have different lengths.  The longer sample will stretch more.  However we can express the stretch in terms of the original length.  This ratio is the strain.

strain = extension ÷ original length

In Physics code:

e = Dx

       l

The code e is epsilon, a Greek letter 'e'.

Strain has no units, as it's metres per metre.  Often it's quoted as a percentage.

Young's Modulus

When we stretch different wires, we need to ensure a fair test.  If we do a force-extension graph, we see that the gradient is the spring constant.

However it might not be possible to get wires of the same length or thickness.  We want a quantity that doesn't need to take that into account.  The Young's Modulus fits that bill, because it's a property of the material, and is not affected by length or area.

Young's Modulus is given by:

Young's modulus = stress ÷ strain

In Physics code:

E = s

       e

or

E = F ×l

       A  Dx

Units for Young's modulus are Pascals (Pa).  We can draw a stress-strain graph:

In the argument above, we have considered the tensile stress, but the same is true of compressive stress.

Elastic Energy

If we stretch or squash a sample, we do a job of work in deforming it.  Provided that there is no plastic deformation, we can say that when we unload it, we can get work out of the sample.  This is elastic energy, which is the area under the force-extension graph.

The green triangle is the energy so:

DEel = 1/2FDx

Or we can write:

DEel = 1/2k(Dx)²

If we have a material that deforms plastically, we need to resort to counting the squares under the graph.

We can find the energy per unit volume as the area under the stress-strain graph.

The energy is:

Area = 1/2se

We know that the extension is:

e = Dx/l

We know that the force is:

s = F/A

So we can combine these three to say that:

Area = 1/2 F/A × Dx/l

Now the area x length is the volume, so we can say that the area is the energy per unit volume, or energy density, given the Physics code U.  So we can rewrite the equation to give us:

Since FDx is the elastic strain energy (force × distance moved), we can write:

We can also bring in the Young's Modulus to say:

The Inside Story

We have seen how X-rays shining through a crystal form a diffraction pattern, similar to a laser light shining through a fine gauze.

We see a pattern of dots on the film and we can interpret the pattern to see the crystal structure.  It's a laborious process that is nowadays done by a computer.  The theory is beyond these notes.

A similar pattern can be observed with electron diffraction:

The Belgian physicist  Louis de Broglie (1892 - 1987) reasoned that if waves have a particulate properties, it was reasonable to suppose that particles had wave properties.    It is the logical extension of the particulate nature of electromagnetic wave phenomena.

  He combined the following equations:

• Energy of photons:                    E = hf

• Einstein’s mass equivalence:      E = mc²

Therefore:

 hf = mc²

Now

f = c/l

So

mc = h/l

The term mc is mass ´ velocity, which is momentum.  We give momentum the code p.

We can rewrite the equation as  

                                    l = h/p                        or                     l = h/mv

Therefore every particle with a momentum has an associated de Broglie wavelength.

We can accelerate electrons with a high voltage to a high speed, which gives the a high momentum, hence a short de Broglie wavelength, shorter than X rays.  This increases the resolution, so that we can more complex molecules.

Rubber is a natural polymer that consists of complex chains.  If we look at rubber when there is no load, we see that the chains are all jumbled up, like spaghetti. 

When a load is applied, the chains become parallel like so:

There are bonds between the chains, which are broken when the material is loaded.  This requires work to be done, so more energy has to put in than is got out.  This leads to energy lost due to hysteresis.

Many polymers are like this, but high density high molecular weight polythene has a more crystalline structure with neat loops stacked together like plates.  This is a lamella formation.  When loaded, the loops unwind to form fibrils.  Under increasing load, they form parallel chains.

Physicists and chemists are trying hard to produce materials that can be used for medical use that have the right chemical and mechanical properties to make lasting prostheses.