Modulus of Elasticity

As there are three kinds of strain, three modulli of elasticity are there corresponding to these strains. These are Young’s modulus, Bulk Modulus and Modulus of rigidity corresponding to linear strain, volume strain and shearing strain, respectively.

Young’s Modulus

The ratio of the longitudinal stress to the longitudinal strain is called Young’s modulus for the material of the body. Suppose that when a wire of length L and area of cross-section A is stretched by a force of magnitude F, the change in its length is equal to ∆L. Then

Longitudinal stress = F/A

Longitudinal strain = ∆L/L

Young’s modulus Y = (F x L)/(A x ∆L)

If the wire of radius r is suspended vertically with a rigid support and a mass M hangs at its lower end, then A = πr² and F = Mg.

Y = (MgL)/(πr²∆L)

The SI unit of Y in is N m^–2.

Bulk Modulus

The ratio of normal stress to the volume strain is called bulk modulus of the material of the body. If due to increase in pressure P, volume V of the body decreases by ∆V without change in shape, then

Normal stress = ∆P

Volume strain = ∆V/V

Bulk modulus B = V ∆P/∆V

The reciprocal of bulk modulus of a substance is called compressibility.

k = 1/B

Gases being most compressible are least elastic while solids are most elastic or least compressible i.e. B_solid > B_liquid > B_gas

Modulus of Rigidity or Shear Modulus

The ratio of the shearing stress to shearing strain is called modulus of rigidity of the material of the body. If a tangential force F acts on an area A and θ is the shearing strain, the modulus of rigidity

η = Shearing stress / Shearing strain

η = F/Aθ

Both solid and fluids have bulk modulus. However, fluids do not have Young’s modulus and shear modulus because a liquid can not sustain a tensile or shearing stress.