cracks

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.

Forensic engineering

Picture1The picture above shows the fracture surface of a thin bar of aluminium alloy that had a circular hole through the middle, like the peep-hole in a front door. The photograph was taken in a Scanning Electron Microscope (SEM) at x160 magnification. There is a scale bar in the bottom right corner showing a length of 100 microns. We are looking approximately in the longitudinal direction, which was the direction of loading, and across the photograph from left to right corresponds to the direction you would look through the hole. The lower one third of the picture shows the machined surface of the hole cut or machined by the drill. The top two-thirds shows the surface created by the fatigue crack as it extended incrementally with each cycle of load. The crack started from edge of the machined surface approximately on the vertical centre-line of the picture. I can tell this because all of the features in the texture of the fracture surface point towards this point because the failure radiated out from this location. The picture below shows the crack initiation area at x1000 magnification. It is a small area at interface with hole above the letters ‘SS40’ in the top photograph; this should be enough to let you identify the common features but the interpretation of these images requires significant skill.

Fractography is the forensic study of failure surfaces such as this to establish the cause of failure. In this example, the hole in aluminium bar ensured that it will always fail with cyclic loading through the growth of a crack from somewhere around the hole. The textured form of the fracture surface occurs because the material is not homogeneous at this scale but made up of small grains. The failure of each grain is influenced by its orientation to the loading which results in the multi-faceted surface in the photographs.

I made the photographs the focus of this post because I thought they are interesting, but may be that’s because I’m an engineer, and because they are a tiny part in a fundamental research programme on which I have been spending a significant portion of my time. A goal of programme is to understand how to use these materials to build more energy-efficient structures that are cheaper and last longer without failing by, for example, fatigue.

More details:

The bar was 1.6mm thick and 38mm wide in the transverse direction and made from 2024-T3 Aluminium alloy. The hole diameter was 6.36mm. A tension load was repeatedly applied and removed in the longitudinal direction which caused the initiation and growth of a fatigue crack from the hole that after many cycles of loading led to the bar breaking in half along a plane perpendicular to the load direction. The pictures were taken at the University of Plymouth by Khurram Amjad with the assistance of Peter Bond and Roy Moate using a JEOL JSM-6610LV.

x1000