Modulus of rigidity and Poisson's Ratio

Modulus of rigidity (n): It is defined as the ratio of tangential stress to shear strain. It is also called shear modulus and is denoted by η i.e.

      η = Tangential stress / Shear strain

If A is area of body, F is force acting on it tangentially and θ is shear angle, then modulus of rigidity is
                   

                    η = F / Αθ

SI unit of η is N / m² or Pascal (Pa). 

CGS unit of η is dyne/cm² and
dimensional formula η = [ML-1 T-2].

Poisson's Ratio (σ) : When we apply deforming force on a wire of length l and radius R, this results in the increase in length of wire but decrease in radius of wire, hence two strains are produced by a single force. I.e.

Poisson's Ratio (σ) =  Lateral strain /  Longitudinal strain   ...................(i)

Here Longitudinal strain = Δl/l and
Lateral strain = ΔR/R
Put in (i) equation , then we get

Poisson's Ratio (σ) = - ΔR/R / Δl/l  or
Poisson's Ratio (σ) =  - ΔR.l  /  R.Δl

Here -ve sign indicates that if length of wire increases on applying deforming force, then radius of wire will decrease. 

Note : Poisson ratio is a dimensionless and unitless quantity.