History of fracture mechanics episode III – Revenge of the plasticity

Last post I discussed how Griffith used the concept of plastic deformation (deformation that remains even when you remove the force that caused the deformation) to prove that Ingliss’ calculation of stress at a crack tip was correct. Unfortunately for Griffith however, plastic deformation would soon cause some serious trouble for his own theories.

Griffith suggested that if an object was cracked, what determined when it would break was not whether the tensile strength of the material was exceeded, but rather the amount of energy being supplied to the crack tip. If this energy is greater than the amount of energy required to create new crack surfaces, then the crack will start growing, and the object will break. How much energy do you need? According to Griffith, the amount is equal to the amount of new surface area you are creating (remembering that you are creating both an upper and a lower surface!) times the surface tension of the material. By pulling apart a series carefully pre-cracked of glass tubes Griffith showed his theory worked, for glass.

Glass is a brittle material. That means it does not deform plastically. Many other materials on the other hand are ductile, which means they are capable of plastic deformation if the stress in the material exceeds the so-called yield stress. One class of ductile materials you are probably familiar with is that of metals. Considering all the things we make out of metal, e.g. bridges, cars, aeroplanes, obviously it would be nice if we could use Griffith’s ideas to calculate if they will fail due to the presence of a crack. However, remember that cracks cause concentrations of stress? That means that the stress near to the tip of a crack will be much higher than in the rest of the material. This can mean that near to the tip of a crack the yield stress is exceeded, causing the material there to deform plastically, even if the rest of the material is only deforming elastically. This fact was how Griffith had been able to confirm Ingliss’ equations. Unfortunately, plastic deformation is a process that dissipates energy. This means that if you are loading a ductile material, not only do you need to supply energy to grow the crack; you also need to provide energy to the plastic deformation that is happening close to the crack tip. As a result, the amount of energy you actually need to grow a crack will not equal the surface area times the surface tension for a ductile material.

So does this mean Griffith’s theories won’t work for ductile materials? Initially it may have seemed that way, but 30 years after Griffith’s paper, a solution was proposed independently by Egon Orowan, a Hungarian/British physicist working at the Cavendish Laboratory at the University of Cambridge, and George Rankine Irwin, an American materials scientist working at the US Navy Research Laboratory.

Their solution? Since, they argued, plastic deformation at the crack tip is unavoidable, the easiest solution is simply summation. Just add the energy needed for plastic deformation to the energy needed to create the crack surfaces. Instead of the crack growing when the supplied energy exceeds the crack surface area times the surface tension, the crack will grow when the supplied energy exceeds this value, plus the amount of energy dissipated by plastic deformation. As long as the plastic deformation is confined to a relatively small area (compared to the entire crack) this approach works and allows us to use Griffith’s theories to predict failure, even in materials that are not perfectly brittle.

Next episode we’ll explore how Irwin, together with his colleague J.A. Kies, formalised Griffith’s ideas to think about crack growth, rather than just the failure of an object. In the meantime, if you have any questions or thoughts about these blog posts, please feel free to post a comment. I’d love to hear from you!

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