The Fundamentals Of Poisson’s Ratio Derivation

Poisson’s ratio derivation tells us about how materials will change their shape (or deform) when a force is applied to them in a particular direction.

For instance, when you stretch a broad elastic band, you see that it stretches along the length. Simultaneously, the breadth of the band also shrinks and becomes thinner. Poisson’s ratio tells us how much the band’s breadth is going to reduce. 

What occurs when a force is applied in one direction?

When we apply a load to any material in one direction, it is obvious that the material will get deformed in that direction. Intuitively, we also know that the material also gets deformed in all the lateral directions.

Consider a cuboid or a cylindrical object. If we were to pull along the length of this object, it would increase in length. At the same time, it would also contract along the width and height (in the case of a cuboid) or along the diameter (in the case of a cylinder). These are known as the lateral or normal or transverse directions to the direction in which we are pulling (i.e., the direction of application of the force). 

The change in shape or deformation on the application of stress (force) is termed as ‘strain.’

Poisson’s ratio derivation

To understand Poisson’s ratio derivation, let us consider a cylinder. Let’s term the original length as L and the original diameter as D. Let △L be the change in length and △D be referred to as the change in diameter. 

Now, when we apply a tensile force along the cylinder’s length (i.e. pull the cylinder), then 

Longitudinal (Linear) Strain= △L / L , and 

Lateral (Transverse) Strain= △D / D 

Poisson’s ratio (ν, Greek letter pronounced as nu) is the measure of the change in lateral (transverse or normal) strain over the change in linear (axial or longitudinal) strain, i.e.,  

Poisson’s ratio (ν)= -Lateral Strain / Longitudinal Strain = – (△D / D) (△L / L) 

Note a few features of Poisson’s ratio

• Since contraction means a ‘negative’ value, thus the lateral strain will typically be a negative value for regular materials. Hence, we add a ‘minus’ sign in the above equation, so that the resultant value for Poisson’s ratio is positive.

• Poisson’s ratio does not have any dimension because it is a ratio of two strains or forces. Therefore, it has no units.

• Under the effect of the longitudinal strain (say along x-axis), all the axes which are perpendicular to the force (i.e. y-axis and z-axis) will undergo an equal lateral strain. This means that in case of a cylinder, the diameter will reduce uniformly in all directions perpendicular to the force and in the case of cuboid, the height and width will be reduced by equal values.

Poisson’s ratio was formally defined by the French mathematician, Siméon Denis Poisson, in his work published in 1827. 

Is Poisson’s ratio universally applicable to all materials?

The above equation and the concepts described are applicable only for isotropic materials. These are materials which display the same properties in all directions. Also, this holds true only in situations where the deformations occur within the elastic range.

Does Poisson’s ratio vary within a limited range?

Theoretically, Poisson’s ratio for materials ranges from -1 to 0.5.

In reality, however, for most of the natural materials that we come across, the value ranges between 0 and 0.5.

A high Poisson’s ratio means that the material will deform to a great extent, even when exposed to a small force. On the other hand, a material having Poisson’s ratio close to zero, will not show elastic deformation even when a large force is applied. 

Practical applications of Poisson’s ratio

Cork has a Poisson’s Ratio of close to 0. This means that it does not deform at all when it is pulled or compressed along the length. This makes it an ideal material for using as a bottle stopper.

Most metals have a value in the range of 0.2 to 0.3. The Poisson’s ratio values for some commonly used materials are given below for reference: 

Magnesium: ν = 0.35

Gold: ν = 0.44

Lead: ν = 0.431

Copper: ν = 0.355

Clay: ν = 0.41

Steel: ν = 0.30

Concrete: ν = 0.20

Zinc: ν = 0.331 

Materials with a value of 0.5 are known as incompressible materials, since their volume remains constant as they deform. A good example will be rubber whose value is very close to 0.5.

Materials used in pipes for handling flow of fluids need to have a low Poisson’s ratio to prevent deformation of the pipes, which may result in leakages. 

Conclusion

In this lesson, we learnt about strain and what happens to materials when they are acted upon by compressive or tensile loads. We learnt about Poisson’s ratio derivation and its practical applications.