energy science

Alan Arnold Griffith

Everest of fracture surface [By Kaspar Kallip (CC BY-SA 4.0), via Wikimedia Commons]

Some of you maybe aware that I hold the AA Griffith Chair of Structural Materials and Mechanics at the University of Liverpool.  I feel that some comment on this blog about Griffith’s seminal work is long overdue and so I am correcting that this week.  I wrote this piece for a step in week 4 of a five-week MOOC on Understanding Super Structures which will start on May 22nd, 2017.

Alan Arnold Griffith was a pioneer in fracture mechanics who studied mechanical engineering at the University of Liverpool at the beginning of the last century.  He earned a Bachelor’s degree, a Master’s degree and a PhD before moving to work for the Royal Aircraft Establishment, Farnborough in 1915.

He is famous for his study of failure in materials.  He observed that there were microscopic cracks or flaws in materials that concentrated the stress.  And he postulated that these cracks were the source of failure in a material.  He used strain energy concepts to analyse the circumstances in which a crack or flaw would propagate and cause failure of a component.  In order to break open a material, we need to separate adjacent atoms from one another, and break the bonds between them.  This requires a steady supply of energy to do the work required to separate one pair of atoms after another and break their bonds.  It’s a bit like unpicking a seam to let out your trousers when you’ve put on some weight.  You have to unpick each stitch and if you stop working the seam stays half undone.  In a material with a stress raiser or concentration, then the concentration is quite good at delivering stress and strain to the local area to separate atoms and break bonds.  This is fine when external work is being applied to the material so that there is a constant supply of new energy that can be used to break bonds.  But what about, if the supply of external energy dries up, then can the crack continue to grow?  Griffith concluded that in certain circumstances it could continue to grow.

He arrived at this conclusion by postulating that the energy required to propagate the crack was the work of fracture per unit length of crack, that’s the work needed to separate two atoms and break their bond.  Since atoms are usually distributed uniformly in a material, this energy requirement increases linearly with the length of the crack.  However, as the crack grows the material in its wake can no longer sustain any load because the free surface formed by the crack cannot react against a load to satisfy Newton’s Law.  The material in the wake of the crack relaxes, and gives up strain energy [see my post entitled ‘Slow down time to think (about strain energy)‘ on March 8th, 2017], which can be used to break more bonds at the crack tip.  Griffith postulated that the material in the wake of the crack tip would look like the wake from a ship, in other words it would be triangular, and so the strain energy released would proportional to area of the wake, which in turn would be related to the crack length squared.

So, for a short crack, the energy requirement to extend the crack exceeds the strain energy released in its wake and the crack will be stable and stationary; but there is a critical crack length, at which the energy release is greater than the energy requirements, and the crack will grow spontaneously and rapidly leading to very sudden failure.

While I have followed James Gordon’s lucid explanation of Griffith’s theory and used a two-dimensional approach, Griffith actually did it in three-dimensions, using some challenging mathematics, and arrived at an expression for the critical length of crack. However, the conclusion is the same, that the critical length is related to the ratio of the work required for new surfaces and the stored strain energy released as the crack advances.  Griffith demonstrated his theory for glass and then others quickly demonstrated that it could be applied to a range of materials.

For instance, rubber can absorb a lot of strain energy and has a low work of fracture, so the critical crack length for spontaneous failure is very low, which is why balloons go pop when you stick a pin in them.  Nowadays, tyre blowouts are relatively rare because the rubber in a tyre is reinforced with steel cords that increase the work required to create new surfaces – it’s harder to separate the rubber because it’s held together by the cords.

By the way, James Gordon’s explanation of Griffith’s theory of fracture, which I mentioned, can be found in his seminal book: ‘Structures, or Why Things Don’t Fall Down’ published by Penguin Books Ltd in 1978.  The original work was published in the Proceedings of the Royal Society as ‘The Phenomena of Rupture and Flow in Solids’ by AA Griffith, February 26, 1920.

Slowing down time to think [about strain energy]

161-6167_imgLet me take you bungee jumping.  I should declare that I am not qualified to do so, unless you count an instructor’s certificate for rock-climbing and abseiling, obtained about forty years ago.  For our imaginary jump, pick a bridge with a good view and a big drop to the water below and I’ll meet you there with the ropes and safety gear.

It’s a clear early morning and the air is crisp and fresh – ideal for throwing yourself off a bridge attached to a rope.  The rope is the star of this event.  It’s brand new, which is reassuring, and arrived coiled over my shoulder.  A few days ago, I asked you how much you weigh – that’s your real weight fully clothed, at least I hope that’s the number you gave me otherwise my calculations will be wrong and you’ll get wet this morning!  I have calculated how much the rope will stretch when it arrests your free-fall from the bridge parapet; so, now I am measuring out enough rope to give you an exciting fall but to stop you short of the water.  I’m a professor of structural materials and mechanics so I feel confident of getting this bit right; but it’s a long time since I worked as an abseiling instructor so I suggest you check those knots and that harness that we’ve just tightened around you.

You’ve swung yourself over the parapet and you’re standing on the ledge that the civil engineers conveniently left for bridge jumpers.  The rope is loosely coiled ready with its end secured to a solid chunk of parapet.  As you alternate between soaking up the beautiful view and contemplating the chasm at your feet, you wonder why you agreed to come with me.  At this moment, you have a lot of potential energy due to your height above the sparkling water [potential energy is your mass multiplied by your height and gravitational acceleration], but no kinetic energy because you are standing motionless.  The rope is relaxed or undeformed and has zero strain energy.

Finally, you jump and time seems to stand still for you as the fall appears to be happening in slow motion.  The air begins to rush past your ears in a whoosh as you build up speed and gain kinetic energy [equal to one half your mass multiplied by your velocity squared].  The bridge disappeared quickly but the water below seems only to be approaching slowly as you lose height and potential energy.  In reality, your brain is playing tricks on you because you are being accelerated towards the water by gravity [at about 10 metres per second squared] but your total energy is constant [potential plus kinetic energy unchanged].  Suddenly, your speed becomes very apparent.  The water seems very close and you cry out in surprise.  But the rope is beginning to stretch converting your kinetic energy into strain energy stored by stretching its fibres [at a molecular level work is being done to move molecules apart and away from their equilibrium position].  Suddenly, you stop moving downwards and just before you hit the water surface, the rope hurls you upwards – your potential energy reached a minimum and you ran out of kinetic energy to give the rope; so now it’s giving you back that stored strain energy [and the molecules are relaxing to their equilibrium position].  You are gaining height and speed so both your kinetic and potential energy are rising with that squeal that just escaped from you.

Now, you’ve noticed that the rope has gone slack and you’re passing a loop of it as you continue upwards but more slowly.  The rope ran out of strain energy and you’re converting kinetic energy into potential energy.  Just as you work out that’s happening, you run out of kinetic energy and you start to free-fall again.

Time no longer appears to stationary and your brain is working more normally.  You begin to wonder how many times you’ll bounce [quite a lot because the energy losses due to frictional heating in the rope and drag on your body are relatively small] and why you didn’t ask me what happens at the end.  You probably didn’t ask because you were more worried about jumping and were confident that I knew what I was doing, which was foolish because, didn’t I tell you, I’ve never been bungee jumping and I have no idea how to get you back up onto the bridge.  How good were you at rope-climbing in the gym at school?

When eventually you stop oscillating, the rope will still be stretched due to the force on it generated by your weight.  However, we can show mathematically that the strain energy and deformation under this static load will be half the values experienced under the dynamic loading caused by your fall from the bridge parapet.  That means you’ll have a little less distance to climb to the parapet!

Today’s post is a preview for my new MOOC on ‘Understanding Super Structures’, which is scheduled to start on May 22nd, 2017.  This is the script for a step in week 2 of the five-week course, unless the director decides it’s too dangerous.  By the way, don’t try this home or on a bridge anywhere.

More violent storms

I made a mistake last week by initially publishing two posts.  My apologies for confusing you or tantalising you with the prospect of going bungee jumping and then postponing the trip.  We’ll go bungee jumping next week.  I postponed it because it’s a preview of the new MOOC on ‘Understanding Super Structures‘ that I am writing and there was a delay in publishing the registration page for the MOOC.

When I posted my comment about postponing the bungee jump due to rain, I didn’t realize that, the following day Liverpool would be battered by Storm Doris, with 90 miles per hour winds that closed the Port of Liverpool.  As I sat writing week 4 of the new MOOC, the wind was swirling around our house causing the windows to rattle; and, on the top storey of our narrow but tall house, you could feel the house moving in the gusts of wind.  Across the street, people visiting Liverpool Cathedral were hanging onto the railings as they made their way to the entrance, and the trees were being bent over to an angle that made you think there would be a loud cracking and splintering of wood at any moment.  Fortunately, the storm was short-lived in Liverpool and moved on to wreak havoc inland.  Bungee jumping would have been very hazardous!

The number of violent storms appears to be increasing and the graphic shows the number of storms in the Atlantic basin since 1850.  Although there is a lot of scatter in the data, there is a clear concentration in the last couple of decades of years with fifteen of more named storms, which suggests there has been more energy in the weather systems in recent years.  The primary source of this energy is the temperature of the oceans and atmosphere.  There is a good account of the development of storms cells in Manuel Delanda’s book ‘Philosophy and Simulation: The Emergence of Synthetic Reason‘, see chapter 1 – The Storm in the Computer, which is available via Google Preview.

The increased frequency of high-energy storm systems is a very apparent manifestation of climate change that is having an impact on many people.  Yet, some governments refuse to even consider the possibility that our climate is changing and that they need to lead our society in discussing and planning strategies to mitigate the impacts.  It reminds me of the saying, attributed to Henri Poincare: ‘To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.’

More on white dwarfs and existentialism

Image by Sarah

Image by Sarah

When I was writing about cosmic heat death a couple of weeks ago [see ‘Will it all be over soon?’ posted on November 2nd, 2016], I implied that our sun would expire on a shorter timescale of about 4 to 5 billion years but without mentioning what we expect to happen.  The gravitational field associated with every piece of matter is proportional to the mass of the piece of matter and inversely proportional to distance from its centre.  The size of the sun implies it should collapse under its own gravitational forces, except that the fusion of hydrogen in its core causes an outwards heat transfer, which prevents this from happening. The sun remains a sphere of hot gases with diameter of about 864,000 miles by ‘burning’ hydrogen.  When the hydrogen runs out, the gravitational field will take over and the sun is expected to collapse to a 30,000 mile diameter ball of atoms and free electrons, or a white dwarf.

These are all spontaneous processes and so the total entropy must increase although there are some local reductions.  The heat dissipated following the fusion of two hydrogen nuclei generates more entropy in the surroundings than the local reduction caused by the fusion.  The collapse to white dwarf would appear to represent a substantial reduction of entropy of the sun because the atomic particles are crushed together. However, this is countered by the release of photons to the surroundings which ensures that the entropy of the surroundings increases sufficiently to satisfy the second law of thermodynamics.


Isaac Asimov, The roving mind: a panoramic view of fringe science, technology, and the society of the future, London: Oxford University Press, 1987.

An extract is available in John Carey (editor), The Faber Book of Science, London: Faber & Faber, 2005.

Will it all be over soon?

milkywayNASAAs you may have gathered from last week’s post [Man, the Rubbish-Maker on October 26th, 2016], I have been reading Italo Calvino’s Complete Cosmicomics.  In one story, ‘World Memory’ the director of a project to document the entire world memory in the ‘expectation of the imminent disappearance of life on Earth’ is explaining to his successor that ‘we have all been aware for some time that the Sun is halfway through its lifespan: however well things went, in four or five billion years everything would be over’.  The latter is one of the scientific conclusions around which Calvino weaves these short stories and this one put into perspective the concerns expressed by some of my students on both my undergraduate course and MOOC in thermodynamics the prospect of a cosmic heat death resulting from the inevitable consequences of the second law of thermodynamics [see my post ‘Cosmic Heat Death‘ on February 18th, 2015].  The second law requires ‘entropy of the universe to increase in all spontaneous processes’.   Entropy was defined by Rudolf Clausius about 160 years ago as the heat dissipated in a process divided by the temperature of the process.  The dissipated heat flows into random motion of molecules from which it is never recovered.  So, as William Thomson observed, this must eventually create a universe of uniform temperature – an equilibrium state corresponding to maximum entropy where nothing happens and life cannot exist.   Entropy has been increasing since the Big Bang about 13.5 billion years ago.  And as Calvino writes, the sun is about halfway through its life – it is expected to collapse into a white dwarf in 4 to 5 billion years when its supply of hydrogen runs out.  These are enormous timescales: the first human cultures appeared about 70,000 years ago [see my post ‘And then we discovered thermodynamics‘ on February 3rd, 2016]  and history would suggest that our civilization will disappear long before the sun expires or cosmic heat death occurs.  A more immediate existential threat is that our local production of entropy on Earth destroys the delicate balance of conditions that allows us to thrive on Earth.  See my post on Free Riders on April 6th, 2016 for thoughts on avoiding this threat.


Italo Calvino, The Complete Cosmicomics, London: Penguin Books, 2002.