About volume and foams

If you had the opportunity to learn the basics of rational mechanics in high school or impact dynamics during your degree, you may be familiar with one specific condition which was specially useful in order to solve problems: the conservation of volume. But, what if I told you that there are certain cases where that particular assumption can be totally wrong?

Let’s start with the so-called “conservation of volume condition”. This condition assumes that when we have a component (e.g. a beam or a bar) and it goes from one state to another (e.g. it is impacted by another body), no mass will be lost. In other terms, it considers that the component will not break into pieces. However, this doesn’t mean that your particular component cannot change its shape. Thus, it is usually taken for granted that even if the shape changes, the volume should remain constant due to the fact that the density (mass over volume) of a certain material should remain the same. Or at least, that’s what we are usually taught…

With regards to the change in shape, it can be explained thanks to a concept known as the “Poisson’s ratio”. Basically, what this coefficient does is relate the deformation in one direction with the corresponding one in the transverse direction. Hence, normally, if we have a component under compression, its length will be reduced whereas its width will increase. Therefore, the volume will be the same before and after the compression event. Figure 1 illustrates this concept, where, assuming that the depth is the same, a x b x depth should be equal to c x d x depth.


Figure 1 Block under compression: before (left) and after (right)

This is what we are usually taught in our early days as scientists or engineers. Now a question arises: what do you think will happen to a material with such a small Poisson’s ratio that it could be considered to be equal to zero? Just think about the same example again: if now the ratio is zero, that means that there won’t be any transverse deformation. Hence, the length will be the only dimension subjected to modification. Therefore, the cross section will remain the same and the volume will change. Looking at Figure 2, it is easy to see that a x b x depth is not going to be equal to a x c x depth. But… this would mean that the density of the material can vary! Is that possible in the real world? Well, the answer is yes!


Figure 2 Block under compression (Poisson’s ratio=0): before (left) and after (right)

A few weeks ago, I published a post about materials in surfboards, where I wrote about some types of foams. Why am I saying this now? Let’s say that some foams are a very good example for materials which can modify their density under certain circumstances. Amazing right? So the question now is: how is it possible? Without going into deep details, I can tell you that foams have a cellular structure which can present quite high porosity. This means that there is a considerably big amount of voids which are filled in with air particles. Therefore, when a block of a material like this is compressed, the air particles are removed from the structure and the voids “close”, resulting in a compacted block with the same mass but a smaller volume. That is the reason why the stress-strain response of crushable foams presents an increase of stress at the very end of the curve, phenomenon known as densification (Figure 3). Just so you know, this kind of materials are widely used in the aerospace and automotive industries due to its high energy absorption capability (which means they are good for absorbing energy during impact events) and its low weight.


Figure 3 Typical stress-strain curve of a foam under compression [1]

Hopefully, you’ll have learned something new with this post. When it comes to  science, we have to be careful before doing the same assumptions over and over again without really thinking about them since, as I have tried to show, they can be wrong for some cases!

[1] Li, Q. M., Mines, R. A. W. and Birch, R. S. (2000) ‘The crush behaviour of Rohacell-51WF structural foam’, International Journal of Solids and Structures, 37, pp. 6321-6341.

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