#18 - How Does Stiffness Attract Load?
A Visualization of Load Sharing in Composites
I can hear you shouting at the screen, “WHERE IS COMPOSITES IN PLAIN ENGLISH PT. 2?!”. I did leave you hanging at the end of the last post, but I decided I had to take a quick detour. When I was reading up on how the fibers and glue interact in composites, I consistently came across authors talking about the glue transferring load (i.e., force) to the fibers. But how does the load decide where to go? Why doesn’t it just stay in the glue? We engineers have been taught that stiffness attracts load and that the fibers will take the load because they’re stiffer than the glue. But why is that? It’s not like the load is sentient. (As an aside, I should be careful about how deep I go with the “why”s. This might end up as a philosophy self-study.)
So let’s talk about stiffness in solid materials for a second. The diagram below - from the always great J.E. Gordon’s Structures: Or Why Things Don’t Fall Down - shows the mechanism for stiffness. We can think of each atom in a material as connected to neighboring atoms with springs. When you push or pull on a material, those springs compress or extend while resisting the movement. In stiffer materials, these springs (or interatomic bonds) themselves are stiffer. This is true even of materials like steel. That means when you push on steel with your finger, it reacts the same way as rubber. It squishes a little bit and rebounds when you pull back. Rubber is not that stiff, so there is enough movement to be seen. Steel is much stiffer, so the movement is not visible to the naked eye. So stiffness just describes the level of springiness of a material. (For completeness, I’ll note that I’m just talking about the elastic region of materials. I won’t go into it, but you can read about elasticity here.)
Now, imagine yourself sitting on a rope swing, which is hanging from the branch of an old oak tree. The swing has the classic wood plank as the seat, but instead of ropes going up to the tree branch, it has something a little odd. On your left, there is a spring and on the right there is a metal chain. Really picture yourself sitting on that swing. What’s your reaction?
You probably pictured the left end of the swing dropping down and trying to dump you off. If you didn’t resign yourself to falling off, you may have tried to grab on to the chain and scooch yourself closer to the chain on the right. It’s very instinctual and you certainly weren’t thinking about stiffnesses. But what you did is redistribute the load so that more of it goes into the stiffer chain. Let me explain that a bit.
You may remember the equation for springs, which is F=kx. F is the force applied to the spring, k is a measure of a spring’s stiffness, and x is the distance that the spring lengthened or compressed. In the swing example, x is how far the swing sags down. If you want to stay seated on the swing, you need both the spring on the left and the chain on the right to sag the same amount. You don’t really care how much they sag as long as they sag the same amount. That way the seat stays level. So to put that in math terms, we rearrange the equation to get x = F/k and then set both “x”s equal to each other. Which give us:
Let’s just say the chain is 10 times stiffer than the spring and simplify the result.
So in order for the seat to stay level, there must be ten times as much load in the chain as there is in the spring. This is what you were instinctively trying to accomplish by shifting your weight towards the chain. (You could figure out far you’d need to scooch by using your knowledge of statics.)
But what does this have to do with composites? The chain represents the stiffer fiber and the spring the less-stiff glue. More of the load will be redistributed into the fiber. But as I said earlier, the load is not sentient. (Sentience here has been reduced to possession of a butt that can be scooched.) If we replaced you on the swing with a rock, it would certainly slide off and nothing would be redistributed. So our model is a bit of an oversimplification. Let’s lengthen the swing and add one more spring and one more chain for a total of two springs and two chains.
With this set up, no derrieres have to actively shift to maintain the seat in a flat, horizontal position. You can imagine that if we hang a weight off of that seat, it will remain flat. x = F/k still applies and x is the same for all four chains/springs. So, we can reason that the ratio of load in the chain to load in the spring is still the same. Which mean that in our model, the stiffness of a material is directly proportional to the load it carries. Stiffness attracting load. This is a little better model of a composite. Multiple fibers with multiple channels of glue. But what does the seat represent? What makes the glue and fibers expand and contract the same distance? Weirdly, it’s the glue itself. The glue sticking to the fiber prevents it from expanding and contracting much differently than the fiber.
Drawing exercise #8. If you missed it, here’s why I’m learning to draw.
