Why Normally Poisson Ratio Can't Exceed 0.5?

Poisson's Ratio and its significance:

Poisson's ratio, named after Simeon Poisson, is the ratio of transverse strain to longitudinal strain. It measures the tendency how deformation of one dimension affects another. Suppose if we stretch rubber band the length of the rubber band increases but the thickness/width decreases. This effect of one dimension deformation on another dimension is showed by Poisson's ratio.

Credit: ntu.edu

So mathematically Poisson's ratio can be written as:

Negative sign signifies here inverse relation of lateral and longitudinal deformation. If the length of either dimension increases the length of another dimension decreases.*

Generally we find materials having Poisson's ratio between 0 to 0.5. However some anisotropic materials have Poisson's ratio greater than 0.5 in some directions. Even negative Poisson's ratio values exist in the case of auxetic materials. But we are not going to consider those cases. Here we do mathematical and physical analysis of the general materials we see in our daily life.

Why general isotropic engineering materials can not have Poisson's ratio greater than 0.5?

During any mechanical operations work is done on the material. The work done on the material is converted into heat and internal energy in accordance to First law of thermodynamics. This finally increases temperature of the material.

This increase in temperature results increase in volume of that material as shown by equation below:

Here all the RHS terms are positive so dV is greater than zero.

Volume of any physical object is three dimensional quantity. Let's suppose here the object is having certain cross-sectional area(A) and length(l). 

We write here cross-section area as second-degree function of cross-section linear dimension(taking the case like that of rod).

Here we have denoted a as linear dimension of cross-sectional area.

So,

Differentiating equation (I),

Hence for the general engineering materials which have positive coefficient of expansion and are isotropic in nature the Poisson's ratio is always less than 0.5.

*Some materials may be such that if length of one dimension decreases another dimension length also decreases due to negative coefficient of expansion. An example for such material is Cubic Zirconium Tungstate. Such materials are not common and they are discarded in our analysis. So we are not considering materials that violate our equation (A).