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.

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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.

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A.A. Griffith and the birth of fracture mechanics

Before continuing with the science in progress series, I thought it would be good to provide some theoretical background to help understand my work, so I’m going to do a couple of posts on the development of my field of science: fracture mechanics.

As soon as people started using tools and building shelters they would have been confronted with the question: ‘when will this thing break?’ For millennia craftsmen, engineers, and architects relied on experience and the ‘carpenter’s eye’ to judge how strong their creations were. Undoubtedly though the more observant would have noticed that the strength of an object depends on both its shape and the material it’s made from: make two identical rods out of different materials, and the maximum load they can bear will be different. Keep the material the same, but double the cross-sectional area, and the maximum load doubles as well.

With the dawn of the enlightenment and the mechanistic approach to ‘natural philosophy’, this observation lead to the development of the concept of stress. What mattered, scientists found, was not amount of force exerted on an object, but the amount of force divided by the cross-sectional area carrying that force. This they dubbed stress. It soon turned out that the stress at which a material fails is always the same. This is a very powerful discovery; it means that you can test a small piece of material in your lab, and then use those results to calculate whether the bridge, sky scraper, or steam engine you are designing will be strong enough.

This concept worked well enough for pristine structures, but it broke down when it came to structures containing some kind of flaw: a hole, crack, or even just a scratch. Cracked structures, it appeared, failed well below the maximum stress that pristine lab specimens could withstand.

A first step towards solving this riddle was the realisation that holes and cracks in the structure will concentrate stress. Near a hole or crack the stress will be much higher than in the pristine part of the structure. You might already expect the stress to be higher there, as a crack or hole means there is less material, and thus the cross-sectional area will be smaller. However the stress concentrating effect of cracks and hole is much larger than that. The stress near such a notch in the structure will be several times higher than the nominal stress, far more than you would expect if it was simply proportional to the reduction in surface area.

In 1913 the British engineer Charles Inglis published a paper showing how to calculate the concentration of the stresses in the presence of an elliptical hole in a plate. Yet even with these values cracked materials were failing at a lower stress than expected. Either Inglis’ calculations were off by a factor of 2 to 3, or something more was going on.

This state of affairs brings us to the title character of this blog post: Alan Arnold Griffith, a young English engineer working at the Royal Aircraft Establishment. With a rather elegant experiment on scratched iron wires he showed that Inglis’ calculations were in fact correct. Therefore it was the theory of failure at a maximum stress that needed to be revised, and Griffith set about doing just that.

What Griffith realised was that creating a crack requires energy. The atoms or molecules in a material ‘like’ to surround themselves with molecules of the same kind. This means that an atom that is surrounded on all sides by the same kind of atom has less potential energy than an atom that exposed to a different material. It is a common principle in nature that objects ‘want’ to minimise their potential energy (e.g. that’s why things roll downhill, not uphill). In other words, the most energetically favourable shape for a bunch of atoms is the shape that has the least surface area. You can see this in videos of water in space. The water clumps into spheres, because those have the least surface area for a given amount of volume.

Surface tension causing a water drop to become a sphere. Photo credit: Flickr user Steve Begin, photo used under CC BY- NC-SA 2.0 license
Surface tension causing a water drop to become a sphere.
Photo credit: Flickr user Steve Begin, photo used under CC BY- NC-SA 2.0 license

If you want to create or grow a crack you are separating atoms that were previously joined together, and creating a new surface area in the process. Since this new configuration is less energetically favourable, you need to provide an energy input to create it.

Griffith realised that when a crack grows, not only is energy consumed by that crack growth, but energy is also released. Think of an elastic band: when you pull on it and hold it under tension, you are storing energy (in a form called strain energy) in the elastic band. When you let go (or cut through the elastic), that energy is released. In the same way, if a crack grows, this releases strain energy from the surrounding material.

Griffith showed that for brittle materials, such as glass, you can figure out how much energy will be consumed to create crack surfaces by measuring a property called the surface tension. At the same time you can also work out how much energy will be released by the growth of a crack held under a certain tension. While an uncracked object will break when the stress exceeds the maximum stress (which is a property of the material), a cracked object will break when the amount of energy released by crack growth exceeds the amount of energy such growth would consume. In that case a crack can keep growing, without requiring an external input of energy; the strain energy already present in the material will suffice to power the crack growth. When this happens will depend on the material itself (the surface tension is a material property), but also on the load you are applying and the length of the crack or flaw, as those determine how much energy will be released.

Griffith presented his ideas in paper entitled ‘The Phenomena of Rupture and Flow in Solids’ which was published in the Philosophical Transactions of the Royal Society of London in 1921. Although the basic principle proposed by Griffith is sound, there are some complications. For example, you probably noticed that I mentioned that Griffith proposed his theories for brittle materials. If a material is ductile (more on that difference in a future blog post), things become more complicated. Also, Griffith looked at cracks growing in a material subject to an almost constant load (‘quasi-static’), where the entire material suddenly cracks. In real life many structures are subjected to variable or cyclical loads, and suffer from cracks that grow little by little, every time a new load cycle is applied. Thus 95 years after Griffith’s first paper, fracture mechanics is still a thriving field of research, in which there is still much left to discover (and hopefully I will be doing some of the discovering).

Mars One – The real ethical questions

NASA Mars base concept.

If you’re at all interested in space travel you will probably have heard about Mars One by now; the non-profit foundation that aims to land four people on Mars by the middle of the next decade. To reduce technical complexity, as well as cost, these colonists won’t be coming back.

NASA Mars base concept.
NASA Mars base concept.
Image credit: NASA, via Wikimedia Commons

Much has been made of how Mars One’s astronauts will ‘die on Mars’. ‘Suicide mission’ and ‘one-way trip’ are phrases that appear a lot in articles on Mars One’s plans. But is it really so problematic to send consenting adults to live on Mars permanently? First of all, Mars One is not a suicide mission. Barring accidents, the colonists sent to Mars will live out the remainder of their natural lives on the Red Planet. Is this so different from choosing to permanently emigrate? How many of the millions of people that migrated to the US in the 19th and 20th centuries had any thoughts of going back? How many ever did go back, even if they technically could? Mors certa, hora incerta, applies to us all. Although the place of their death will become certain for the Mars colonists, its hour will be no more or less uncertain than for us here on Earth.

`Ah, but going to Mars is a risky endeavour,’ the detractors will say, ‘can you really give informed consent when a decision is so life altering?’ This argument sounds rather like Catch-22: If you want to go to Mars you must be crazy. Therefore wanting to go to Mars automatically makes you incapable of consent. This argument ignores the fact that people can differ in the level of risk they are willing to accept. Furthermore, it could just as easily apply to any number of decisions we allow people to make every day. Can you really understand in advance the implications of consenting to a risky surgical procedure? To going on an Arctic expedition? To signing a 30-year mortgage? To having a child? If it’s possible to make an informed choice on such matters, then why not on a permanent emigration to Mars? Mars One’s colonists will receive years of training simulating the conditions they will have to live under. Unlike, say, prospective parents.

However, all of this is avoiding the really hard ethical questions. Judging the behaviour and choices of others is easy, potentially an enjoyable pass-time even (see your favourite social media or celebrity news site). But buried in Mars One’s timeline is a scenario that will force us to not judge others, but to look into a mirror and ask ourselves: In cold hard cash, what is a human life worth to me?

Imagine the following hypothetical scenario: It’s 2025. The world looked on in awe as the first humans set foot on Mars. Now, six months later, the world has spent half a year watching the daily routine of 4 high-tech subsistence farmers. Careful psychological assessment has ensured that conflict among the colonists is rare, and if it does occur gets handled in a calm and mature manner. In other words, the reality TV show that Mars One depends on for its funding is unutterably boring, and the ratings plummet. As one cable company after another declares it won’t be buying the rights to a new season, Mars One are forced to announce that they do not have the funds to send new resupply missions with spare parts to Mars. Spare parts the colonists depend on to maintain their critical systems.

How long will the Mars One reality show be interesting?
How long will the Mars One reality show be interesting?
Image credit: Flickr/mwanasimba, CC BY-SA 2.0 licence

At that point the world will have to answer the question: will we sit and watch in HD as the colonists freeze, asphyxiate, dehydrate, or starve to death, or will we mount a rescue mission? If we choose the second option, who should foot the bill? Should it be the country or countries of whom the colonists were previously citizens? What if they can’t afford to pay? Should it be the countries that have the necessary space infrastructure, even if they’re not involved in the project?

An MIT feasibility study (Ho et al, 2014) concluded that 4 colonists would need 13.5 tons of spares every two years. This corresponds to 6 rocket launches at $300 million each. In other words the cost of keeping Mars One’s colonists alive would be $225 million / astronaut / year.

That is roughly 9,000 times more than what my travel insurance would have paid out had I died on my last holiday trip. It is just over 4,750 times more than the limit for the UK national health service to consider a medical treatment cost-effective. Would we consider the Mars One colonists to be worth that much more than other humans, just because we can watch their lives in HD?

Let’s assume for the sake of argument that the EU agrees to pay. Then it would cost the EU’s citizens $1.80 (or €1.47) per person, per year, to keep the colonists supplied. Stated like that it’s not a huge amount. I’m sure most of us would be willing to forgo one cup of coffee or small glass of beer once a year if it meant saving a life. But this poses a new question. If the EU does manage to raise $900 million a year, should that money really be spent on keeping alive 4 astronauts, or should it be spent on Rotavirus vaccines instead, potentially saving hundreds of thousands of lives (Rheingans et al, 2009)?

These are the ethical questions that we will have to answer should Mars One succeed at its goal, but then turn out not to be financially sustainable. Questions that I believe are far more interesting, but also more confronting, than whether we should allow people the choice of dying on Mars. Yet that is precisely the question most coverage of Mars One seems to focus on. Is it really true that none of the reporters covering Mars One have considered the long term financial sustainability of this endeavour, or do we just find the attending ethical questions too uncomfortable to contemplate?

If Mars One cannot meet the financial obligation of maintaining its colonists, we will be forced to explicitly decide how much is the worth of a human life. Can you really not attach a value to a person’s existence, or will we collectively decide that 3 Falcon Heavy launches a year for the remainder of the colonists’ life is just too much?

Perhaps we will never be called upon to make that choice, but does that mean we shouldn’t consider it now? Perhaps we simply prefer the comfort of dismissing Mars One’s candidates as insane; of denying the mental fitness of those who would live out their lives having forever slipped the surly bonds of Earth.

Science in progress part II: Making the computer do the hard work

In my last post I described my test set-up and how my current experiments are performed. At the end of the post we were left with 200 to 300 pictures like this:

Example picture used to measure crack length.
Initial picture of the specimen.

The goal is to use these pictures to measure how long the crack was at each of the measurement points during the test. The simple but time-consuming way of doing this would be to open each picture one-by-one in Photoshop and then measure how many pixels across the crack was. With the graph paper included in the picture you can then convert the number of pixels into a length in millimetres.

While this process is fine if you only have a dozen or so pictures, doing all this by hand for several hundred pictures per specimen sounds way too much like hard work. Fortunately I have a computer to help me out! Using a program called Matlab (known, loved, and possibly hated by engineers and scientists everywhere) I’ve written a simple image processing script that can automatically determine how long the crack was.

I’m not a super-star programmer, so the first step is to crop the image so that (as much as possible) the crack is the only horizontal hard edge left in the image. This helps the program to not get confused. You can see what the cropped image looks like below. Clicking on the image will show you a larger version.

Cropped image
Picture after cropping to only show the crack.

To a computer a picture is just a collection of numbers. In my case each pixel has a number ranging from 0 to 255, representing how black a pixel is. 255 is white, 0 is black, and in between are 254 shades of grey. Whether you realise it or not, doing image manipulation on a computer means doing math with those numbers. Thanks in part to Instagra,m a very common manipulation is applying a filter. A filter is nothing more than an instruction to the computer to change the number associated with a certain pixel based on the numbers of the surrounding pixels (e.g. taking the average of all of them to smooth and blur an image). In my case I apply a Sobel filter ( which has the effect of highlighting edges, as you can see below.

Image after applying a Sobel filter
Image after applying a Sobel filter.

Once the edges are highlighted I convert the image into a binary image. Instead of having 256 possible shades, each pixel is assigned a value of either pure white or pure black. If you want to sound fancy you can call this a ‘thresholding operation’.

Image after thresholding
Image after thresholding.

After thresholding I apply another filter that dilates all the white areas, which means small areas will be joined together if they are separated by only one or two black pixels. This ensures that one or two darker pixels won’t cause the crack to be prematurely terminated.

Image after dilation
Image after dilation.

The final step is to throw out any white areas in the picture below a certain size. If all went well this leaves me with a completely black picture, with one white area representing the crack.

Final image, containing only the crack.
Final image, containing only the crack.

Then it’s just a case of measuring how far to the right the right-most white pixel is and you know how long the crack is!

Repeat this operation for each picture, match up the picture with the number of cycles the test had been running at the moment it was taken, and you can draw a pretty graph of how the crack grew during the test.

Graph of crack length during the test
Graph of the crack growth during the test. Number of fatigue cycles along the horizontal axis and crack length in mm along the vertical axis.

Next time we’ll continue with the hard part: doing data analysis to actually learn something useful from this data.

Science in progress part I: My experimental set up

Overview of my test set-up showing the test machine, a camera, and a lamp.

Recently I made a post about the objectives of my research. In other words, talking about what I do in the abstract sense.  All well and good you might say, but what is it that I actually do all day? Today I thought I’d give a more concrete example of my work, by discussing the experiments I’m currently running.

At the moment I’m performing a set of so-called double cantilever beam (DCB) tests. A regular cantilever beam is just a beam that is completely fixed on one end, but free to move on the other end; like the one in the picture below.

Schematic example of a cantilever beam.
Schematic example of a cantilever beam.
Image credit: Wikimedia Commons / Vladsinger.
Licensed under Creative Commons Attribution-Share Alike 3.0 Unported.

To make a double cantilever beam specimen, you stick together two pieces of material. Then you pull them apart, pulling perpendicular to the interface between the two pieces, creating two ‘arms’. In my case I’m using two aluminium alloy beams stuck together with an epoxy resin. As you can see in the picture I attach a hinge block to each side of the specimen. Then I attach a grip plate to each hinge block. This grip plate fits into the grips of a testing machine, allowing me to apply force to the specimen.

A close-up of a DCB specimen in the test machine
A close-up of a DCB specimen in the test machine

To help matters along a bit I didn’t glue the entire length of the beams, but I inserted some Teflon tape in order to make sure a short length of the beams wasn’t stuck together. In effect you then have two beams that are free to move at one end (the end you are pulling on), and are fixed at the other (the end where they are stuck together). In other words, you have two cantilever beams. Hence the name: double cantilever beam.

Overview of my test set-up showing the test machine, a camera, and a lamp.
Overview of my test set-up showing the test machine, a camera, and a lamp.

Once the specimen is installed in the testing machine it’s time to start running the experiment! I’m doing what are called displacement controlled fatigue crack growth tests. That means that I’m commanding the machine to move (displace) the two beams apart a set distance, then bring them not-quite back together and then pull them apart again. This is repeated five times a second, until I have enough data. Generally that is after about 200,000-300,000 cycles, which is equivalent to one to two and a half days of continuous running.

Remember that we had a portion of the specimen where the two arms are not bonded together? That forms a kind of initial ‘pre-crack’. All the pulling and pushing (well, pulling mainly) of the fatigue machine will cause an actual crack to slowly start growing in the adhesive; beginning at the pre-crack. Each time there is a new fatigue cycle, the crack will grow a little further. What I’m trying to measure in these experiments is how fast the crack will grow.

Fortunately I don’t have to sit next to my specimen with a pen, a notebook, and a ruler all day. Instead I have a camera set up next to the specimen (you can see it in the picture). Every 1,000 cycles the test machine will go to the maximum displacement I’ve set for the test. It will hold the specimen there for a short while. Then the camera will take a picture like the one below. After that the load cycles will continue again.

Example picture used to measure crack length.
Example picture used to measure crack length.

Those of you who are good at mental arithmetic will have realised I have about 200 to 300 of these pictures per specimen. Fortunately I don’t have to go through those by hand. Curious what I do do? Stay tuned for part II of this series…

(hint: it involves something Instagram is famous for)