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3. Everything / Maths, Science & Technology / Physics

Created: 2nd January 2009
An Introduction to the Fracture Mechanics of Spaghetti
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Spaghetti.

There is intriguing and instructive science in the simplest of systems. Imagine holding each end of a strand of uncooked spaghetti between forefinger and thumb and flexing it until it snaps. How many pieces does the spaghetti break into?

There's a good scientific reason to predict an answer of two. The stress in the spaghetti will increase as we bend it further. At some point, somewhere along the length, this stress will reach a level sufficient to break the material. As soon as a break occurs though, the stress will be relieved, and so there will be no further failures.

Sure enough, if you try it out and break a single strand of spaghetti in this way, the outcome might well be two pieces. Over many repetitions, however, a different statistical pattern emerges. The most frequent number of pieces is indeed two, but not by much. Three pieces is nearly as common. Infrequent instances of four pieces will also be observed if the trial is carried out enough times.

The reasoning that led us to expect two pieces every time was obviously flawed. Before considering why, it's useful to investigate another variant of the experiment. This time the spaghetti is cooked, and it's stretched (ie put in tension along its length) rather than bent. No matter how many repetitions there are, the result is now invariably two pieces, and thus a single point of failure.

Cooked spaghetti is string-like, and strings in tension fail at a single point in everyone's experience, whether it's of threads of cotton or of towropes. On the other hand, if we were to accidentally drop a packet of uncooked spaghetti onto the kitchen floor, we would expect to find many small fragments in the bottom of the packet after opening. Uncooked spaghetti is prone to shattering, in a way that string isn't.

Ductility and Brittleness

We all probably remember Hooke's Law from our schooldays, and the experiments with scale pan and ruler that demonstrate elasticity. The amount by which something stretches is proportional to the force applied to it. The behaviour is reversible, moreover. If we take the force away, the elastic body resumes its original shape.

There is a regime beyond this behaviour, though. Most solid materials, and arguably all the useful ones, follow a progression from elastic behaviour at low strain to plastic behaviour (in which a permanent deformation takes place under further loading) and finally to failure. Some materials exhibit a very long plastic range, and can be drawn into lengths many times greater than their original dimension. Others fail almost as soon as the elastic limit is reached. Some materials even switch from one property to the other depending on other conditions. Glass is highly formable when hot, for example, but it fractures rather than yields if strained when cold.

A material that exhibits plastic deformation is said to be ductile. A material that fails with little or no plastic yielding is said to be brittle. Uncooked spaghetti is a brittle material. Cooked spaghetti is relatively ductile.

Energy and Fracture

There is a significant difference between the way that brittle and ductile materials fail. Broadly speaking, materials store a lot of energy when elastically deformed. Plastic yielding dissipates this energy in a controlled way, and so the more ductile a material is, the less violent will be its failure. To envisage this, think about the difference between a taut elastic band and the cheese strings attached to a slice of pizza. The purely elastically deformed band packs enough energy to sting flesh, but the elastic-plastic cheese stretches and yields over a prodigious length without ever storing up enough energy to develop a snap.

It follows that a brittle material (one that undergoes little or no plastic yielding when stressed beyond its elastic limit) will tend to release stored energy suddenly. This behaviour is important in what happens to our spaghetti.

A further important characteristic of the material under consideration here is that its structure is not homogeneous. It's full of flaws, some of them specks of foreign material in the starch matrix, and some due to discontinuities in the chains of molecules that compose it. When the strand of spaghetti is bent, cracks are initiated at some of these flaws. There will be very many of them. Once a crack exists, the likelihood of failure local to it increases significantly, and the stress level needed to induce failure there will be lower than in the adjacent undamaged material.

The earlier analysis of the bending of uncooked spaghetti was right as far as it went. The bent spaghetti breaks at a single point, usually close to the mid-point where the stress is highest, but this is only the start of an extended process of collateral failure. The elastic energy is released rapidly, and the free ends whip straight. In fact they overshoot, and flex the other way. The cracks that were induced on the original inside of the bend, where the spaghetti was formerly in compression, are now in play. Even though there is no steady force applied to the free ends, these damage sites briefly experience high bending stresses due to the inertia of the fast-moving outboard mass. These stresses might be sufficient to induce a further failure on either side of the original break. It is possible that a secondary failure occurs on both sides, giving rise to two further breaks.

Five?

It should now be clear why the spaghetti may break into two, three or (less often) four pieces. Very occasionally, spaghetti-snappers report higher numbers. Assuming they are being truthful, we now have to postulate a progressive damage mechanism, where new potential failure sites are created by whiplash or shockwave within the recoiling structure. Harmonics may be involved too, whereby certain residual lengths of spaghetti will tend to whip with especially high amplitude and energy.

The author of this Entry has broken quite a lot of spaghetti in the course of its creation, however, and hasn't had a fiver yet.

Beyond Pasta

Though apparently trivial, the behaviour of the spaghetti has an important lesson for engineers. Wherever structures and mechanisms store large amounts of energy while in duty, there is a significant risk of collateral secondary damage after a primary failure. An example of this from our common experience is punctures while driving. Low speed punctures are an irritation, but high speed ones can be catastrophic. Engineers have a responsibility to recognise these secondary self-destructive failure modes, and to design them out, or at least contain them as far as possible.

There is a lot more to fracture mechanics and damage theory than spaghetti-bending teaches, of course. If you'd like to take the next step, you could try bending a plastic ruler until it shatters. Now you have a different aspect ratio and a two-dimensional system. Think about the shape of the fragments, and the characteristic angles of the fractures. If you actually want to try it, eye protection is advisable.

Back with the spaghetti, you probably had a hunch about what would happen in the two tests before you were told the results. Most people guess that the dry spaghetti will shatter when bent, while being fairly sure that the cooked spaghetti, when pulled, will break in two. Intuition based on experience is in play, even before we think through the physics. For hundreds of years, indeed, engineers built things with no more than an intuitive understanding of mechanical failure. That's not good enough today, though. Knowingly or otherwise, we all trust our lives daily to other people's confidence that whatever they designed won't break.



Edited Entry of the Month
January 2009
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ENTRY DATA
Written and Researched by:

Pinniped

Edited by:

The H2G2 Editors

Referenced Entries:

Numbers
Pizza - Not Just Italian
Elastic Band Warfare
Physics and the Knowledge of Ignorance
Glass Blowing
Fractions
Growing Spaghetti
How to Fire an Elastic Band
Divisibility
Unity - the Number One



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