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.