fracture

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.

Advertisements

Engineering archaeology

Last week I spent a relaxing day painting the old railings in front of our house. Since I am not a painter and decorator by trade the end result is not perfect but they look much better in shiny black than two-tone rust and matt black.   One of the fleurs de lis on our railings had been knocked off when either we moved in or the previous occupiers moved out.  It’s a way of life being an engineer, so the shape of the failure surface on the broken railing was bugging me while I was painting the rest.  You would expect wrought iron railings to be ductile, i.e. to deform significantly prior to fracture, and to have a high tensile strength.  Wrought iron’s properties are derived from its very low carbon content (less than 0.25%) and the presence of fibrous slag impurities (typically about 2%), which almost make it a composite material.  It was historically used for railings and gates.  However, my broken railing had exhibited almost no deformation prior to fracture, i.e. it was a brittle failure, and the fleur de lis had broken in half on impact with the stone flags.  So on one of the rainy days last week, when I couldn’t paint outside, I did a little bit of historical research and discovered that in the late 1790s and early 1800s, which is when our house was built, cast iron started to be used for railings.  Cast iron has a high carbon content, typically 2 to 4%, and also contains silicon at between 1 and 3% by weight.  Cast iron is brittle, i.e. it shows almost no deformation prior to fracture, so the failure surface tends be to flat and smooth just like in my fleur de lis.

This seems like a nice interdisciplinary, if not everyday, engineering example.  It would be vandalism to go around breaking iron railings in front of old buildings.  So, if you want Everyday Engineering Examples of ductile and brittle behaviour, then visit a junk shop and buy an old china dinner plate and a set of cutlery.  The ceramic of the china plate is brittle and will fracture without deformation – have some fun and break one!  The stainless steel of the fork and spoon is ductile and can be easily bent, i.e. it is easy to introduce large deformation, in this case permanent or plastic deformation, prior to failure.  In fact you will probably have to bend the fork back and forth repeatedly before it will snap with each bending action introducing additional damage.

The more curious will be wondering why some materials are ductile and others brittle.  The answer is associated with their microstructures, which in turn is dependent on their constituents, as hinted above.  However, I am not going to venture into material science to explain the details.  I have probably already given materials scientists enough to complain about because my Everyday Engineering Examples are not directly analogous at the microstructural level to wrought iron and cast iron but they are more fun.

Sources: http://www3.westminster.gov.uk/spgs/publications/Railings.pdf