Poisson’s Ratio – Definition and Formula

Poisson’s Ratio consists of strain and stress that we use in the direction of the stretching force. Furthermore. It relates to the tensile strength of an object.

Definition of Poisson’s Ratio

It refers to the transverse shrinkage stress to longitudinal extension stress in the direction of the stretching force. Furthermore, we consider the tensile deformation positive and compressive deformation negative.

In addition, the Poisson’s Ratio contains a negative sign (minus sign) so that the normal materials have a positive ratio.  Besides, we also refer it to as Poisson Ratio, Poisson coefficient or coefficient de Poisson. In addition, it is usually represented by the lower case Greek letter nu, ν.

The formula of Poisson’s Ratio is

ν = – ε_trans / ε_longitudinal

In this, the strain or stress ε is defined in elementary form. Also, the original length divides the change in length.

ε = ΔL/L

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Why the Poisson’s Ratio is positive?

All common materials virtually become narrower in cross-section when we stretch them. Also, if you stretch a piece of rubber you see this phenomenon.

Besides, the reason is that in the continuum view, most of the materials resist a change in their volume as determined by bulk absolute value K (also called B at some places).

Moreover, they resist a change in their shape as determined by shear absolute value G. If we view it structurally then the reason for this is the usual positive Poisson’s Ratio that inter-atomic bonds that realign with deformation.

Besides, if you stretch a honeycomb structure by a vertical force then it will show this effect.  Furthermore, the negative Poisson’s Ratio in some anisotropic materials and some designed materials is well known.

Relation of Poisson’s Ratio with elastic moduli in isotropic solids

Generally, the Poisson’s Ratio relates to the elastic moduli K (also referred as B), the bulk absolute value G as a shear absolute value; and E, Young’s Absolute value, by the following (for isotropic solids, for which the properties are independent of direction).

Besides, the elastic moduli are a measure of stiffness.  Furthermore, they are the ratios of strain to stress.  Moreover, with the direction of both the force and the area specified, the stress is the force per unit of that area.

ν = (3K – 2G)/(6K + 2G)

E = 2G( 1 + ν)

E = 3K(1 – 2ν)

Furthermore, the interrelations of the elastic constants of isotropic solids are as follows. Here B is a Bulk Absolute value.

B = E/(3 (1-2ν)), B = GE/(3 (3G-E))

ν = E/(2G-1), ν = (3B -2G)/(6B+2G), ν = 1/2 – E/6B

E = 2G (1 + ν)

E = 3B (1 – 2 ν) = 9GB/3B + G

C₁₁₁₁ = B + 4/3 G

C₁₁₁₁ = E (1- ν)/((1+ ν)(1 -2ν))

Most noteworthy, the theory of isotropic linear elasticity allows Poisson’s Ratios in the range from -1 to 1/2 for an object, which has a free surface, and no constraints.

Moreover, the materials should be stable and the stiffness must be positive. Also, the shear and bulk stiffness are interrelated by a formula that incorporates Poisson’s Ratio.

Poisson’s Ratio of various materials depends on their structure and the space between their particles. If an object has a high molecular space then it will have high elasticity or Poisson Ratio.

On the contrary, an object, which has dense molecular space, has lower elasticity. Besides, Platinum has a Poisson Ratio of 0.380 and rubber has ~0.550.

Solved Question on Poisson’s Ratio

Question. Poisson Ratio relates to which one of the following.

A. Strength

B. Molecular space

C. Elasticity

D. Atomic Structure

Answer. The correct answer is option C.