Testing rupture models

How do materials break ? This question is the starting point of my present post-doc in France. People have good ideas on how metals and crystals break, because their structures are known for a century or two. But when you turn your attention to more messy materials, like composites, concrete, paper, gels, etc. the answers are far less convincing.

Popular (because simple) models of fractures are called fibre bundle models. Imagine a cotton yarn, or a rope made of many interlocked fibres. If you pull on it hard enough and for long enough, a fibre will break. The pulling strength will be shared by all the other fibres, until an other one breaks. The more broken fibres, the less remaining fibres, the more often a new fibre breaks. Sounds like a random action movie scene, isn't it?

A first simple model is to consider that the fibres are independent from each other and each of them has a slightly different breaking threshold. The only way fibres interact with each other is by sharing evenly the load. This model is good at describing how paper breaks. It predicts that over a given load threshold, the material will always break. The stronger you pull, the faster it breaks, conversely if you pull just a minute amount above the threshold it will take a very long time to break. For most of the process, you may even not notice that you are damaging your rope. That's why in the movies it's always when the last guy crossing the rope bridge who falls.

A mode refined version of the model says that when a fibre breaks, the load it bear is split among its neighbouring fibres. It makes sense when you consider an actual rope, made of smaller twines, made of even smaller threads, etc. until you get to the individual fibres. What this new model predicts is the absence of threshold load. If you pull on your material long enough, it will break, whatever your strength. Of course, the weaker you are, the longer you'll have to wait. An other interesting prediction is the appearance of fractures, or regions where all the fibres are broken. Can this explain the fractures and cracks observed in many materials? That's on this question that I started my own research.

I selected a system that indeed looks like a network of fibres:

This is a gel obtained by slow acidification of a solution of milk protein (sodium caseinate), or in plain English: a yoghurt. The picture above was made by electron microscopy, and the white fibres you see are one hundred time thinner that a human hair. The black voids are occupied my water that flows throughout the network.

Unstirred yoghurt do behave very much like a hard solid, except that it is much weaker. You don't need a powerful machine to fracture it, a spoon is enough.

And once it is broken, the cracks won't heal. It is very different from a liquid or from pastes that you can stir to erase the memory of the system. If you stir a yoghurt you just destroy it further.

Now, I don't trust myself to apply a constant force with a spoon, especially if the experiment is set to run for a week. So I make my yoghurt between two concentric cylinders of a machine called a rheometer, which is able to rotate the inner cylinder with a constant force until the next power cut. Step by step: I pour the protein solution in the rheometer at rest, the gel forms during 17 hours as the solution acidifies, and only then I apply the force.

As you see on the picture above, after some time fractures appear in the yoghurt and grow vertically. By the way, the picture was taken by a standard webcam. This is not rocket science. The picture above shows only the bottom half of the yoghurt, there are fracture also growing from the top. When bottom and top fracture meet, the gel breaks completely.

I repeated this experiment many times, varying the applied load and thus the time needed to break the yoghurt completely. At high load, I am limited by the recording frequency of the rheometer, so my minimal breaking time is slightly less that a second. On low load, patience is the limiting factor, as well as evaporation and mold growth. Still I have an experiment that lasted 11 days. That is a factor 10 million between the longest and shortest experiments, enough to prove that there is no meaningful load threshold. That is a first win for the model.

Edit March 29th, 2014

A video of the rupture. Nothing seems to append and then ...