# Understanding Poisson’s Ratio

Hello everyone, and welcome to another video on the Efficient Engineer channel! In this video we’re going to take a look at a really interesting material property, Poisson’s ratio. Poisson’s ratio is such an important material property because it gives us key information about how different materials deform under loading. Let’s get right into it! We can start with a quick example, the rubber band. If you take a rubber band and stretch it along its length, its length will obviously increase.

But it is also quite intuitive that at the same time the band will get thinner. Poisson’s ratio can tell us how much thinner the band is going to get. What about for a generic lump of material? Take this little orange cuboid – if we apply a compressive force to it, it will expand in the directions which are perpendicular to the direction in which we are applying the load. We call the direction along which the load is applied the longitudinal direction, and we call the perpendicular directions the lateral directions. If we reverse the direction of the force to apply a tensile load, the cuboid will obviously extend in the longitudinal direction, and it will contract in the lateral directions. The key concept here is that when you apply a load to a material in one direction, the material will also deform in the lateral directions. Poisson’s ratio is the parameter that tells us how much the material will deform in the lateral directions. To define Poisson’s ratio more precisely, let’s take a closer look at the case where a tensile force is applied. First let’s give names to the various parameters involved.

Lx, Ly and Lz are their original dimensions of our cuboid along the X, Y and A axes. Delta Lx, Delta Ly and Delta Lz are the changes in these dimensions after the load has been applied. They are shown as being divided by two because the change in dimension occurs on both sides of the object. Next we can write out the strains in all three directions. As a quick reminder strain is a measure of deformation, and it is defined as the change in length divided by the original length. It is denoted using the Greek letter epsilon and is usually expressed as a percentage. When we apply our longitudinal load, it turns out that the resulting strains in the lateral directions are equal. And here’s the interesting part – they are proportional to the strain in the longitudinal direction.

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The ratio between the longitudinal strain and the lateral strain is actually a material constant. You might have guessed it by now, but we denote this material constant with the Greek letter nu, and we call it Poisson’s ratio. By the way, Poisson’s ratio is named after this guy, Siméon Denis Poisson, the famous French mathematician who formally defined the ratio in a note published in 1827. You might be wondering why a minus sign slipped into the equation. Remember that by convention tensile strains are positive and compressive strains are negative.

The minus sign is just there so that for the typical case where the lateral strains are opposite in sign to the longitudinal strains, we get a positive Poisson’s ratio. Now is probably a good time to mention that this equation and the concepts described in this video only apply for isotropic materials, which are materials which have the same properties in all directions. We are also assuming that the materials are deforming within the elastic region. Things get a bit more complicated when plastic deformation is involved.

Anyway to summarise, Poisson’s ratio is a dimensionless material property which tells us how much a given material will contract in the lateral directions when we pull on it in a longitudinal direction. Now that we know what Poisson’s ratio is, we can look at some typical values for different materials. Without getting too deep into the math behind it, it’s useful to know that the theoretical range of possible values for Poisson’s ratio goes from -1 to 0.5. In practice, most real materials have a Poisson’s ratio of somewhere between 0 and 0.5. Most metals have a Poisson ratio of around 0.3. Here are some typical values for a few selected materials. Now let’s look at how different values of Poisson’s ratio affect how a material deforms under loading. We’ll keep things simple and consider a two-dimensional case. Most materials have Poisson’s ratios between zero and 0.5. When a tensile force is applied, these materials contract in the lateral directions as we have already seen with our orange cuboid.

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Materials with a Poisson’s ratio of zero are interesting. When a longitudinal tensile force is applied there is no deformation in the lateral directions. One material that behaves in this way is cork, which has Poisson’s ratio close to zero. This property makes it a very useful material for certain applications. A great example is the cork in a bottle of wine. Because it doesn’t expand laterally when compressed, as you can see here, it can easily be inserted into the neck of a bottle. A material with a larger Poisson’s ratio would be much more difficult to insert, as it would expand in the lateral directions when compressed. Finally we have materials with negative Poisson’s ratios. These are known as auxetic materials. They expand laterally when pulled, and contract laterally when compressed, which seems counter-intuitive. These are mostly engineered materials, like special foams, rather than materials occurring naturally in nature. This animation should give you an idea of how it is possible to get negative Poisson’s ratios.

You can see that as the material is compressed, both its longitudinal and lateral dimensions are reduced. For a typical material with a Poisson’s ratio larger than zero, you would expect the lateral dimensions to increase when the material is compressed. Because it tells us how a material deforms, Poisson’s ratio is a very important parameter in continuum mechanics for determining how a body responds to applied stresses. Let’s look at a simple case of uniaxial stress, the tensile test. It is easy enough to determine the strains acting on a small piece of the specimen under test. The applied stress is a normal stress in the X direction, sigma X. The strain in the longitudinal X direction is simply given by Hooke’s law, as sigma X divided by the Young’s modulus E. Even though there are no stresses acting in the lateral Y and Z directions, there will be strains in these directions, as the material is contracting laterally. The strains in the lateral directions are obtained by multiplying the longitudinal strain by Poisson’s ratio.

This is just using the definition of Poisson’s ratio which we covered earlier. But what if we look at a more complex case where we have tri-axial stress, with different stresses in all three directions? In this case we can’t just use Hooke’s law to determine the strain in the X direction, because it will also be affected by the strains in the Y and Z directions. The simple version of Hookes law for uniaxial stress no longer applies because the strains in one direction will depend on the stresses applied in all three directions. We can use Hooke’s law in combination with the equation for Poisson’s ratio, and the principle of superposition, to obtain the equation for strain in the X direction. We can re-arrange this equation into a more practical form, like so, and then apply the same process to obtain equations for strains in the Y and Z directions.

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These equations form what is known as the Generalized Hooke’s Law, and can be used to determine deformations for a case of tri-axial stress. There is one last case I would like to mention, which is materials with a Poisson ratio of 0.5, the maximum value in the theoretical range we discussed earlier in the video. Let’s return to our cuboid to introduce the concept of volumetric strain. Volumetric strain is a measure of the change in volume of an object under load. I’m sure it will come as no surprise that it is closely linked to Poisso’ns ratio.

We can calculate the volumetric strain by summing the strains in all three directions. Let’s use the equations from the Generalized Hooke’s Law we just covered to expand the equation for volumetric strain. We can then re-arrange the terms to end up with the following equation. You might notice that something interesting happens to the volumetric strain when Poisson’s ratio is equal to 0.5. We end up with a volumetric strain equal to zero. This means that for materials with a Poisson’s ratio of 0.5 the volume of the material remains constant as it deforms. These are known as incompressible materials. Rubber is an example of an incompressible material. And that’s it! We’ve reached the end of our review of Poisson’s ratio. If you enjoyed watching, please remember to like and subscribe for more engineering videos!