Poisson's Ratio

Poisson’s Ratio

When a piece of rubber is in its original shape which is a cuboid and is pulled along its sides what happens? The rubber will get compressed from the middle. The original length and breadth of the rubber, which is L and B respectively when pulled longitudinally, gets compressed laterally. The length of the rubber increases by the amount of dL and the breadth increases by dB.

\[\epsilon t\] =  - \[\frac{dB}{B}\]

\[\epsilon l \] = \[\frac{dL}{L}\]

The formula for Poisson’s ratio is,

\[\mu\] = −\[\frac{\epsilon t}{\epsilon l} \]

\[\epsilon t\] is the Lateral or Transverse Strain.

\[\epsilon l \] is the Longitudinal or Axial Strain.

\[\mu\] is the Poisson’s Ratio

Strain: Strain is the change in the dimension of an object or shape in terms of length, breadth etc divided by its original dimension.

Poisson’s Effect: The material when stretched in one particular dimension will compress in the direction perpendicular to the force applied and vice versa. The measurement of this is given with the Poisson's ratio. Poisson's ratio is used to explain this phenomenon. For example, Thinning of a rubber band when stretched.

Poisson’s Ratio Values for Different Materials

Poisson’s ratio value is different for each material in existence, and it depends on how a material will react under the force of compression and pulling. The range of the values of the poisson’ s ratio lies between -1.0 to +0.5, but for most of the materials the value of poisson’s ratio is between 0 and 0.5. Value of poisson’s ratio for some of the different materials are listed in the table below - 

Some Points to be Noted for the Value of Poisson’s Ratio

• The value of poisson’s ratio is a scalar quantity and it is also a dimensionless  unit less quantity.

• Its value is positive for tensile deformation, while for the compressive deformation its value is in negatives.

• When the Poisson’s Ratio approaches to zero (0), there will be no change in the diameter or the width of an object

Definition of Poisson’s Ratio for a Material:

The ratio of the transverse contraction of a material to the longitudinal extension strain in the direction of the stretching force is the Poisson's Ratio for a material. The stress or stain can be generated by applying the force on the material by the body. The Poisson's ratio is negative for the compressive deformation whereas for the tensile deformation the Poisson's Ratio is Positive. The negative Poisson's ratio suggests that the positive strain is in the transverse direction. The Poisson's Ratio for most of the materials is in the range of 0 to 0.5.

The Poisson's Ratio is between the range of 0 to 0.5 for plastics. When the Poisson's Ratio is 0 there is no reduction in the diameter or one can even say there is no laterally contraction happening when you are elongating the material but the density would reduce. The value of 0.5 indicates that the volume of the material or object will remain the same or constant during the elongation process or when the diameter decreases of material when the material is elastomeric.

Following is the different Poisson’s Ratio for different materials.

Rubber = 0.49

Aluminium = 0.32

Concrete = 0.2

Cork = 0

Usually, Poisson's Ratio is positive because most of the common materials when stretched become narrower in the opposite or cross direction. Most of the materials resist the change in volume which is determined by the bulk modulus K or also called B more than they resist change in shape which is determined by the shear modulus G. The interatomic bonds also realign with the shape deformation.

Poisson’s Ratio in Bending

When you bend a bar or plate the Poisson's Ratio will govern the curvature in the direction perpendicular to the bending. One can easily observe the anticlastic curvature in the case of bending a rubber.

Poisson's Ratio and Anisotropy

When it comes to anisotropic solids such as honeycombs, single crystal, and some fibrous composites, the physical properties of this material including the Poisson’s ratio and Elastic Moduli depends on the direction in which they are stretched or bent. The Poisson’s ratio can be positive or negative for the large magnitude of this kind of anisotropic materials.

Poisson's Ratio in Viscoelastic Materials

The context of the transient test such as creep and stress relaxation etc have an effect on the Poisson's Ratio of the viscoelastic material. The Poisson's Ratio also depends on the frequency and the phase angle if the deformation executed is sinusoidal in nature. In most of the cases, the transverse strain is out of phase and has the longitudinal strain when it comes to viscoelastic solid.

Poisson's Ratio and Phase Transformations

The phase transformation can have a considerable impact on the Poisson's Ratio of a material. The bulk modulus most softens near a phase transformation but the shear modulus does not have much impact. The Poisson's Ratio decreases along with the vicinity of the phase transformation and can even go to negative values. Therefore, it is very important to study the effect of phase transformation on the Poisson's ratio of a material.

Poisson's Ratio, Waves, and Deformation

The speed of propagation and reflection of the stress waves are affected by the Poisson's ratio of the various materials. The ratio of the compression to shear wave is very important from the geographical application point of view as it helps to infer the nature of a rock situated deep in the earth. The wave speed ratio depends upon the Poisson's ratio as well. The Poisson's Ratio affects the distribution of stress around the cracks as well as the decay of the stress.

Sums: 

Show that if E is assumed to be correctly determined, an error of 1% in the determination of G will involve an error of about 6% in the calculation of Poisson’s ratio (v) when it’s correct value is 0.20.

Solution: Since we have 

E = 2G(1 + v)       …(1)

1% error in G will result in G' = 1.01 G

And the value of Poisson’s ratio will become v’.

Then,  E = 2.G’ (1 + v’)      … (2)

Since E is determine correctly; from equations (1) and (2),

2G(1 + v) = 2G’(1 + v’)

2G(1 + v) = 2 × 1.01 G(1 + v’)

1 + v = 1.01 + 1.01 v’

Subtracting both sides from v’.

v’ – v = –20.01 – 0.01 v’ = –0.01 (1 + v’)                       … (3)

The percentage error in v is; using equation (3)

\[\frac{v’ – v}{v} \times 100 =\text –0.01 \frac{(1 + v’)}{v}\times100\]

= –0.01 (1 + 0.20) / 0.20 × 100

= –6%

Poisson's Ratio is used to measure the Poisson effect. The Poisson effect is the phenomenon wherein material tends to expand in the direction perpendicular to the compression. The Poisson effect is different for isotropic material, orthotropic material etc. The Poisson’s Ratio is considerably applied in the pressurised pipe flow, structural geology etc.

Poisson Ratio helps one to judge the properties of a material. The material which does not contract are brittle and those who contract are ductile. The Poisson ratio also gives us the knowledge that the material having a high Poisson ratio can be pulled easily as opposed to those having a low Poisson Ratio, for example, plastic (0.5) and cork (0). The material, which does not have any elongation, breaks on pulling. Poisson’s Ratio also helps one to understand which materials should be used for a particular task. For example, we cannot have elastic rubber as a stopper for Champagne as the rubber may cause leakage of the champagne and spoil the party. Thus, Poisson Ratio plays an important role in deciding the application of the material.

The Expression for Poisson’s Ratio

Poisson’s Ratio varies from object to object and depends upon factors like temperature, density, young’s modulus, etc. Still, we have a unified expression for determining the ratio which is the negative ratio between strain in the direction of load and strain at a right angle to the load. It can be represented as

Poisson's Ratio= \[-\frac{{Transverse}/{\text{Lateral strain}}}{\text{Axial strain}}\]

As both the quantities used to find the Ratio are of the same dimension, Poisson’s Ratio is a dimensionless quantity. The value of Poisson’s Ratio can range from -1.0 to 0.5. 

Poisson’s Ratio Formula

Mathematically, Poisson’s Ratio is denoted by the Greek letter, (Nu). The answer to what is Poisson Ratio formula will be: 

Poisson's Ratio (𝜈) = \[-\frac{{Transverse}/{\text{Lateral strain}}}{\text{Axial strain}}\]

As observed from the formula of Poisson Ratio, the Poisson’s Ratio of an object is directly proportional to lateral strain and inversely proportional to axial strain. 

Based on the Poisson's Ratio equation, further calculations are carried out.

Poisson’s Ratio Values for Different Material

Knowing the Poisson’s Ratio for different materials helps us determine the ideal purpose for that material. The deformations in Poisson’s effect are of 2 types namely Tensile Deformations (Poisson’s Ratio is positive) and Compressive Deformation (Poisson’s Ratio is negative). The negative Poisson’s Ratio suggests that the material will exhibit a positive strain in the transverse direction. 

Poisson’s Ratio Examples

Poisson’s Ratio plays an important role in engineering from designing a house to building rockets. The ability to determine the behaviour of materials under stress beforehand allows us to select an appropriate material for the job. 

For example, Poisson’s Ratio is frequently used to determine building materials like flat plates that will be subjected to bending stress. The higher the value of Poisson’s Ratio, the more rigid the plate will be and the more it will be able to endure more stress.  Poisson’s Ratio of some of the common materials like concrete (0.1-0.2), steel (0.27-0.30), rubber (0.4999), and foam (0.10-0.50) are known very accurately due to their vast applications. 

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