What is Poisson’s Ratio: Poisson’s Ratio Formula [PDF]

poisson's ratio

Did you notice that when you stretch a rubber band, the width gets noticeably thinner? This is called Poisson’s effect. Passion’s ratio is the measure of that change in width due to the change in length. Let us learn more about Poisson’s Ratio in this article and we will also learn about Poisson’s Ratio Formula.

Poisson’s Ratio

Poisson’s Ratio is the negative of the ratio of lateral strain ( Transverse) to the linear strain ( Axial). As with the example of a rubber band, when we stretch the length ( Axial), the cross-sectional width gets reduced ( Transverse) and vice versa. The ratio of this change is called Poisson’s ratio. This is named after French engineer Siméon Denis Poisson

Since Poisson’s ratio is a measure of one strain divided by another, there is no unit for Poisson’s Ratio. For flexible materials and other soft materials, Poisson’s ratio is close to 0.5 which means that the cross-sectional width gets reduced by up to 50%. In the case of brittle or tough materials, the Poisson’s ratio is near to 0 which means that there is no change in width even though there is a change in length.

Poisson’s Ratio varies from 0.00 to 0.50

What is strain?

We all know that we apply load on a body, it either expands or contracts. The ratio of the amount of change to the original size is called strain. For example, if the length of a body is “L” and because of load it elongated by “ΔL”, the strain is (ΔL/L)

Strain = (ΔL/L)

What is Transverse strain or lateral strain?

The transverse strain is the ratio of change in diameter to the actual diameter of a circular object. For example, if the actual diameter is D and the amount of change in diameter is ΔD, then the transverse strain is (ΔD/D)

Transverse stain= (ΔD/D)

What is linear strain or axial strain?

The axial strain is the ratio of change in length to the actual length. For example if the actual length is L and the change in length is ΔL, then the axial strain is (ΔL/L)

Axial Strain = (ΔL/L)

Poisson’s Ratio Formula

Let us look into the image shown below. The green body is the unconstrained object, means no load is applied to it. When a tensile load is applied to it ( As shown by arrow), the original body deforms and take the shape of the orange body. Due to tensile load, the body is elongated by ΔL, and the cross-section is contracted by ΔL‘. So the Poisson’s ratio of this body is {-(ΔL‘/ΔL)}

poisson's ratio
Credit : Wikipedia

Poison’s Ration={-(ΔL’/ΔL)}

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Negative Poisson’s Ratio

As we said that the Poisson’s ratio values vary from 0 to .5. But for some Auxetics materials, the Poisson’s ratio is negative. What does that mean is that, for a change in length, the cross-sectional width gets increased. Since there is no decrease in width, the Poisson’s ratio becomes negative. This is something not usual but happens only with Auxetics materials.

This happens due to uniquely oriented, hinged molecular bonds. In order for this bond to stretch in the longitudinal direction, those bond need to open up in the transverse direction. That is why we see the increase in cross-sectional width and negative Poisson’s ratio

For some type of woods, the Poisson’s ratio is negative as well. When you stretch the wood, initially it will show positive Poisson’s Ratio but gradually it shows negative Poisson’s ratio when you stretch it more.

Few of the negative Poisson’s ratio materials are , Li, Na, K, Cu, Rb, Ag, Fe, Ni, Co, Cs, Au, Be, Ca, Zn, Sr,

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Poisson’s ratio of Steel and other materials

MaterialPoisson’s Ratio
Rubber.50
Steel.27-.30
Stainless Steel.30-.31
Gold.42-.44
Glass.18-.30
Copper.33
Cast iron.21-.26
Concreate.10-.20
Titanium.26-.34
Lead.43
Brass.35
Aluminium.33
Bronze.34
Marble.20-.30
Zinc.33

Application of Poisson’s effect

Poisson’s effect is considered in many applications where the material tends to compress or elongate due to different loading condition. One of the examples is piping. When the fluid is flowing at very high pressure, it will create stress on pipe walls. Due to this stress, the pipe will try to increase its diameter and decrease its length.

Imagine if there are multiple pipes joined together having this decrease in length. This will make those joints weak and it may break. So Poisson’s ratio is really important here to analyze how much the pipe can expand or contract due to the fluid flowing through it.

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Conclusion

Poisson’s Ratio is one the key factor to understand how a material will behave in different loading condition. An engineer should take Poisson’s ratio into account before designing a part which may be subjected to stress. Without proper consideration, there is a high chance that the part may fail when it is exposed to real-life scenarios.

This is all about Poisson’s ratio. I hope you got a basic idea about Poisson’s ratio. When you practically work and apply Poisson’s ratio, you may need to consider different other aspect and discussing all those is out of the scope of this article. Still, if you face any issue or if you have any question, please do write in the comment section and I will be happy to reply.

Frequently Asked Questions ( FAQ)

What is Poisson’s Ratio?

Poisson’s Ratio is the negative of the ratio of lateral strain ( Transverse) to the liner strain ( Axial)

What is the Poisson’s Ratio range?

0-.50. But there are materials which has negative Poisson’s ratio

What is the Poisson’s ratio of steel

.27-.30

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