Consultant David Jesson explores the J-integral, a method that allows us to calculate the toughness of materials, determining their structural integrity.
Within the AtoZ blogging challenge, there are always some letters that cause consternation for the writer. The exact letters may differ between challenges, usually because a chosen theme offers opportunities in one place, and challenges in another. There is of course always some way of meeting a specific alphabet, although readers may groan at a bad pun or in some other way cry foul. You might be thinking that I’ve taken that route for today’s letter, ‘J’. In fact, the J-integral is very real, and played an important part in my doctoral research.
Fracture mechanics is the study of the way in which cracks affect the performance of a structure. A crack is not necessarily catastrophic – you’ve probably seen a crack in wall which is still standing, or a small crack in a windscreen or bumper. These can be repaired, if caught in time. If they are allowed to grow, then eventually they will reach a critical length and the component will fail. It is understanding when the length of a crack will become critical that is the challenge…
As is often the case with such models, there is a relatively simple equation that governs the overall principles, and some intimidating modifications that can be applied to account for specimen geometry, the location of the crack, and the location of the applied load. If we know the fracture toughness of the material, then we can determine a relationship between the applied load and the crack length – the longer the crack, the lower the applied stress that the structure can survive. Hence, there are all sorts of structures that survive with surprisingly large cracks in them, because the applied loads are small. Until, that is, the applied load increases, perhaps due to flooding or a heavy fall of snow, or simply extremely cold or hot weather causing contraction or expansion of the structure.
So where does the J-integral come in? Mostly we assume that a material behaves elastically when a crack grows through it. By this we mean that all energy used to deform the material is released when the material is unloaded, and there is no permanent deformation of the atoms. That’s not always the case though: whilst most materials display a linear relationship between load applied and extension, up to the point of yielding, some show elastic-plastic behaviour. The J-integral was developed to account for this, and is a method for determining the plastic contribution during crack growth, allowing us to calculate the toughness of such materials. In this way, even challenging materials can be modelled, and the ongoing integrity of an asset determined.