#28 - Composites in Plain English Pt. 4
There’s a chance that reading this could save your life -- not a very good chance, but you never know. Picture this: one day, you find yourself in the lair of an evil genius called Mr. Evil. In front of you is a pond. It looks calm enough, but when Mr. Evil tosses a fish in, the pond erupts with alligators. Above the pond are two ropes, which Mr. Evil tells you are made with carbon fibers. One is a standard rope of dry carbon fibers. The other is a composite, with carbon fibers glued together with epoxy. The glued fibers seem more like a rod than a rope to you, but you’re not about to argue the point. Mr. Evil tells you that both ropes are the same weight. He is a lot of things -- evil, genius, and a lover of riddles to name a few -- but he is not a liar. Mr. Evil tells you that you will be suspended from one of the ropes, but generously allows you to choose which one. One of the ropes will stretch out and put you within reach of the alligators, the other is stiff enough to keep you safe. Which rope do you choose?
I’ll give you some space to think about which rope you would choose and why. You always learn more if you have an opinion, even if it’s wrong.
Thinking space
If you chose the unglued rope, congratulations! You’re safe. If you chose the glued composite rope, you’re in range of the alligators’ snapping jaws. Hopefully, for your sake, they’re pescatarians.
To understand the reason why, we’ll need to think about the role of glue in composites. We started to explore that question in part 3, but only for a rope/strand being compressed. To recap, we found that the glue binds the fibers together, forcing them to work as one unit. That change in geometry -- from the thousand individual fibers to a single column -- makes for a much more stable configuration. Stability is usually the limiting factor under compression, so the added glue makes the composite a lot stiffer in compression.
This situation is different; we’re now pulling on the composite (the glued rope). It might not seem like a big deal -- it is the same material after all -- but things react very differently to being pushed and being pulled. Under tension, stability don’t matter nearly as much. Think of a hammock; the thick ropes at the ends branch into thinner ropes. But the one thick rope and many thin ropes do an equally good job of supporting your weight. As long as there’s enough material (the rope is thick enough), the way the material is arranged doesn’t really matter.
How quickly things change. Under compression, the geometry was so important that it overshadowed the material’s stiffness. Under tension, the geometry hardly matters at all. That must mean that the material’s stiffness takes on greater importance.
To explore that, I’m going to rely on a mental model that I introduced in a previous post to explain how load chooses where to go in a composite. I’d recommend reading that post, but basically we’re going to think of the glued rope as a bouquet of springs. Each spring represents either a fiber or a column of glue, which is how I think of the glue between the fibers. When the rope is pulled, all of those springs extend in unison.
Thankfully, we can look up how to deal with springs in parallel (parallel meaning that the springs are side-by-side rather than end-to-end). It turns out that we can find the stiffness of this group of springs by adding all their individual stiffnesses together. That makes sense -- it’s like stretching one rubber band versus stretching a whole bunch of rubber bands together. So for our glued rope, which is made up of fiber springs and glue springs, we’d get something like this:
Of course, it’s a pain to count the fibers, which number in the thousands, and impossible to count the glue columns, which are really just part of a mental model. On top of that, if we had a different, thicker rope, you’d have to count the fibers all over again. It would be much nicer if we could describe the rope’s material more generically. Think of it like this: you’ve just eaten an incredible plate of red beans and rice and you want to know how to make it. This spring-counting method is like counting how many beans and grains of rice are in the plate. Literal bean-counting; not too convenient. It would be much easier to have a recipe which gives the amounts of beans and rice in a ratio of volumes. So let’s convert our spring-counting recipe to a ratio-style recipe.
Instead of counting the number of fibers and glue columns, we’ll find the ratio of fiber to glue. (This is a volume ratio, which can be found using weights and densities.) Instead of finding the stiffness/springiness of an individual fiber or glue column, we look up the stiffness of the material. (This is Young’s modulus.) This ratio-style recipe is called the rule of mixtures. Here’s how that looks, assuming the composite is a reasonable 60% fiber and 40% glue:
Textbooks and Wikipedia make the rule of mixtures sound complicated, but conceptually it’s pretty simple. The more of something you have in a composite, the more it’ll influence the overall stiffness.
The only remaining question is about the stiffness of the glue versus the stiffness of the fiber. In part 1 of this series, I talked about how thinness gives fibers fantastic strength and stiffness, so it’s no surprise that the fiber is stiffer than the glue. The chart below gives a great visual of those relative stiffnesses. The higher on the chart, the stiffer the material. The carbon fiber is high up, the resin (the glue) is low down, and the composite is in the middle.
So adding glue actually makes the composite less stiff in tension. That’s not too surprising if you believe, like I do, that there’s no such thing as a free lunch. Composites are an attempt to combine two materials to capture the best of both. It would be far too easy if we could just add the good bits without giving anything up. Combining properties makes composites better for a wide range of uses, but in some special cases, like space elevators where the tether is purely under tension, fibers are the way to go.
For “fun”, let’s do a little calculation of how much more the glued composite rope would stretch. Let’s say the length, L, and the thickness, A, of the ropes are the same. Your weight, P, is also the same. From the chart above, the stiffness, E, of the composite is around 100 GPa and the stiffness of the fiber is 400GPa. 𝛅 is the change in the rope’s length.
So the composite rope stretches four times as much as the fiber rope. For instance, if the fiber rope stretched 3 inches when you dangled from it, the composite rope would stretch a full foot. Of course, it would have to be a very thin and long rope to stretch carbon fiber that much, but finding a reasonable size for the rope is left as an exercise for the reader.
Corrections? Questions? Comments? I’d love to have your input. Leave a comment, email me at surjan@substack.com, or find me on LinkedIn.
Drawing exercise #17. If you missed it, here’s why I’m learning to draw.