From failure of specimens to growth of cracks – the strain energy release rate

Note for new readers, this is part of a series of posts on the history of fracture mechanics, you can find the first post in this series here.

When Griffith started on his research, he was concerned with finding a theory for predicting the failure of entire specimens. Such a theory was already available for uncracked specimens, and as we have seen Griffith was successful in creating a theory that could also predict failure in cracked specimens.

Once Griffith’s framework was in place, people soon started thinking about not just the failure of an entire specimen, but also about what was going on with the crack itself. This is an interesting topic, because cracks can grow without immediately causing failure.

For an example of this, take a piece of paper and make a small tear in the middle. If you now carefully pull on both ends of the paper, you should be able to get the tear in the paper to grow further, without tearing the entire sheet in two. This is exactly what can happen with cracks in other types of materials as well.

The challenge of predicting crack growth was taken up by G.R. Irwin, who we met last time, and his colleague J.A. Kies. They started their research with Griffith’s concept of looking at energy. Since all processes in nature obey the laws of thermodynamics, Irwin and Kies reasoned, it should be possible to write down the energy balance for crack growth.

Sometimes physics is just like bookkeeping! Photo credit: Andreas Praefcke / Public Domain via Wikimedia Commons: http://commons.wikimedia.org/wiki/File:Hauptbuch_Hochstetter_vor_1828.jpg
Sometimes physics is just a matter of good bookkeeping.
Photo credit: Andreas Praefcke / Public Domain via Wikimedia Commons:

Writing an energy balance is just like bookkeeping, only with energy instead of money. On the one side you put the amount of energy going in to some process, and on the other side you put the amount going out. The first law of thermodynamics states that both sides of this equation should be equal. If they’re not, you’ve failed to account for something.

In the case of crack growth, Irwin and Kies identified two terms on the ‘in’ (or ‘energy available’) side: the release of strain energy from material surrounding the crack (because the material is no longer held together) and the amount of work performed by whatever mechanism is loading the specimen. In physics, ‘work’ means exerting a force over a certain distance. For example when a lift moves, it is exerting a force (the tension in the cables) over a certain distance (the amount the cage travels up and down). When the lift is stationary, there is still tension in the cables, but there is no movement, and thus there is also no work.

On the ‘out’ (or ‘energy required’) side, there were also two terms: the amount of energy required to generate the crack surfaces (including the energy required to plastically deform the material around the crack tip, as discussed last time) and the kinetic energy imparted to any parts of the specimen.

As long as no pieces go flying off, you can reasonably assume that the kinetic energy can be ignored. You are then left with the input work and the strain energy release on the ‘available’ side, and the energy consumed by crack growth on the ‘required’ side. This can be simplified even further by realising that the input work and the strain energy release are connected: if you put in work, it can either be stored as strain energy, or it can be used to grow the crack. Similarly, if strain energy is released it can be used for crack growth or to cause the loading mechanism to ‘spring back’, in effect performing negative work. Thus the amount of energy that is available is the difference between the amount of work done, and the amount of released strain energy.

Furthermore, Irwin and Kies argued, what matters is not the absolute amount of energy, but how much is available and required per unit of crack growth. For the difference between input work per unit of crack growth and strain energy release per unit of crack growth, they formulated the term ‘strain energy release rate’, which they gave the symbol G, in honour of Griffith.

The crux of Irwin and Kies’ argument was then as follows: if the amount of energy made available per increment of crack growth (i.e. G) is larger than the amount of energy required for an increment of crack growth, then crack growth will occur. In other words, crack growth will occur if G is larger than some critical value, called the critical strain energy release rate, Gc.

In addition, this crack growth will be self-sustaining: unless growth of the crack makes G decrease again (or somehow increases how much energy you need) you don’t need to put in any more energy. All the energy the crack needs is already available, locked up as strain energy. To put it another way: once G becomes larger than Gc, the crack powers itself: the strain energy released by one increment of crack growth provides the ‘fuel’ needed for the next increment.

Although devised some 60 years ago, this argument still is the basis of how we think about, and predict, crack growth today.

Next post we’ll discuss what G tells us about the stress state near the crack tip, and then we’ll be ready to start working towards my research topic: the growth of cracks due to repeated load cycles, aka fatigue. In the meantime, feel free to ask any questions in the comments below!

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