Poisson’ s Ratio When a deforming force is applied at the free end of a suspended wire of length 1 and radius R, then its length increases by dl but its radius decreases by dR. Now two types of strains are produced by a single force.
( i) Longitudinal strain = & DElta; U l
( ii) Lateral strain = – R / R ∴ Poisson’ s Ratio( σ) = Lateral strain / Longitudinal strain = – R / R / ΔU l
The theoretical value of Poisson’ s ratio lies between – 1 and 0.5. Its practical value lies between 0 and 0.5.
Relation Between Y, K, η and σ( i) Y = 3K( 1 – 2σ)
( ii) Y = 2 η( 1 + σ)( iii) σ = 3K – 2η / 2η + 6K( iv) 9 / Y = 1 / K + 3 / η or Y = 9K η / η + 3K
Important Points
Coefficient of elasticity depends upon the material, its temperature and purity but not on stress or strain.
For the same material, the three coefficients of elasticity γ, η and K have different magnitudes.
gas.
Isothermal elasticity of a gas ET = ρ where, ρ = pressure of the
Adiabatic elasticity of a gas Es = γρ
where, γ = Cp / Cv ratio of specific heats at constant pressure and at constant volume.
Ratio between isothermal elasticity and adiabatic elasticity Es /
ET = γ = Cp / Cv