A.A. Griffith and the birth of fracture mechanics

Before continuing with the science in progress series, I thought it would be good to provide some theoretical background to help understand my work, so I’m going to do a couple of posts on the development of my field of science: fracture mechanics.

As soon as people started using tools and building shelters they would have been confronted with the question: ‘when will this thing break?’ For millennia craftsmen, engineers, and architects relied on experience and the ‘carpenter’s eye’ to judge how strong their creations were. Undoubtedly though the more observant would have noticed that the strength of an object depends on both its shape and the material it’s made from: make two identical rods out of different materials, and the maximum load they can bear will be different. Keep the material the same, but double the cross-sectional area, and the maximum load doubles as well.

With the dawn of the enlightenment and the mechanistic approach to ‘natural philosophy’, this observation lead to the development of the concept of stress. What mattered, scientists found, was not amount of force exerted on an object, but the amount of force divided by the cross-sectional area carrying that force. This they dubbed stress. It soon turned out that the stress at which a material fails is always the same. This is a very powerful discovery; it means that you can test a small piece of material in your lab, and then use those results to calculate whether the bridge, sky scraper, or steam engine you are designing will be strong enough.

This concept worked well enough for pristine structures, but it broke down when it came to structures containing some kind of flaw: a hole, crack, or even just a scratch. Cracked structures, it appeared, failed well below the maximum stress that pristine lab specimens could withstand.

A first step towards solving this riddle was the realisation that holes and cracks in the structure will concentrate stress. Near a hole or crack the stress will be much higher than in the pristine part of the structure. You might already expect the stress to be higher there, as a crack or hole means there is less material, and thus the cross-sectional area will be smaller. However the stress concentrating effect of cracks and hole is much larger than that. The stress near such a notch in the structure will be several times higher than the nominal stress, far more than you would expect if it was simply proportional to the reduction in surface area.

In 1913 the British engineer Charles Inglis published a paper showing how to calculate the concentration of the stresses in the presence of an elliptical hole in a plate. Yet even with these values cracked materials were failing at a lower stress than expected. Either Inglis’ calculations were off by a factor of 2 to 3, or something more was going on.

This state of affairs brings us to the title character of this blog post: Alan Arnold Griffith, a young English engineer working at the Royal Aircraft Establishment. With a rather elegant experiment on scratched iron wires he showed that Inglis’ calculations were in fact correct. Therefore it was the theory of failure at a maximum stress that needed to be revised, and Griffith set about doing just that.

What Griffith realised was that creating a crack requires energy. The atoms or molecules in a material ‘like’ to surround themselves with molecules of the same kind. This means that an atom that is surrounded on all sides by the same kind of atom has less potential energy than an atom that exposed to a different material. It is a common principle in nature that objects ‘want’ to minimise their potential energy (e.g. that’s why things roll downhill, not uphill). In other words, the most energetically favourable shape for a bunch of atoms is the shape that has the least surface area. You can see this in videos of water in space. The water clumps into spheres, because those have the least surface area for a given amount of volume.

Surface tension causing a water drop to become a sphere. Photo credit: Flickr user Steve Begin, photo used under CC BY- NC-SA 2.0 license
Surface tension causing a water drop to become a sphere.
Photo credit: Flickr user Steve Begin, photo used under CC BY- NC-SA 2.0 license

If you want to create or grow a crack you are separating atoms that were previously joined together, and creating a new surface area in the process. Since this new configuration is less energetically favourable, you need to provide an energy input to create it.

Griffith realised that when a crack grows, not only is energy consumed by that crack growth, but energy is also released. Think of an elastic band: when you pull on it and hold it under tension, you are storing energy (in a form called strain energy) in the elastic band. When you let go (or cut through the elastic), that energy is released. In the same way, if a crack grows, this releases strain energy from the surrounding material.

Griffith showed that for brittle materials, such as glass, you can figure out how much energy will be consumed to create crack surfaces by measuring a property called the surface tension. At the same time you can also work out how much energy will be released by the growth of a crack held under a certain tension. While an uncracked object will break when the stress exceeds the maximum stress (which is a property of the material), a cracked object will break when the amount of energy released by crack growth exceeds the amount of energy such growth would consume. In that case a crack can keep growing, without requiring an external input of energy; the strain energy already present in the material will suffice to power the crack growth. When this happens will depend on the material itself (the surface tension is a material property), but also on the load you are applying and the length of the crack or flaw, as those determine how much energy will be released.

Griffith presented his ideas in paper entitled ‘The Phenomena of Rupture and Flow in Solids’ which was published in the Philosophical Transactions of the Royal Society of London in 1921. Although the basic principle proposed by Griffith is sound, there are some complications. For example, you probably noticed that I mentioned that Griffith proposed his theories for brittle materials. If a material is ductile (more on that difference in a future blog post), things become more complicated. Also, Griffith looked at cracks growing in a material subject to an almost constant load (‘quasi-static’), where the entire material suddenly cracks. In real life many structures are subjected to variable or cyclical loads, and suffer from cracks that grow little by little, every time a new load cycle is applied. Thus 95 years after Griffith’s first paper, fracture mechanics is still a thriving field of research, in which there is still much left to discover (and hopefully I will be doing some of the discovering).

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