Explain the different types of modulus of elasticity.

Explanation

There are three types of elastic modulus.
(a) Young's modulus
(b) Rigidity modulus (or Shear modulus)
(c ) Bulk modulus
Young's modulus:
When a wire is stretched or compressed, then the ration between tensile stress (or compressive stress) and tensile strain (or compressive strain) is defined as Young's modulus.
Young's modulus of a material
`= ("Tensile stress or compressive stress")/("Tensile strain or compressive strain")`
`Y = (sigma_(1))/(epsilon_(1))` or `Y = (sigma_(c ))/(epsilon_(c ))`
The unit for Young modulus has the same unit of stress because strain has no unit. So, S.I. unit of Young's modulus is `N m^(-2)` or pascal.
Bulk modulus:
Bulk modulus is defined as the ratio of volume stress to the volume strain.
Bulk modulus,
K `= ("Normal (perpendicular) stress or pressure")/("Volume strain")`
The normal stress or pressure is
`sigma_(n) = (F_(n))/(Delta A) = Delta P`
The volume strain is
`epsilon_(v) = (Delta V)/(V)`
Therefore, Bulk modulus is
`K = (sigma_(n))/(epsilon_(v)) = - (Delta P)/((Delta V)/(V))`
The negative sign in the equation means that when pressure is applied on the body, its volume decreases. Future , the equation implies that a material can be easily compressed if it has a small value of bulk modulus.
The rigidity modulus or shear modulus:
The rigidity modulus is defined as the ratio of the shearing stress to shearing strain.
`sigma_(s) = ("Tangential force")/("Area over of shear or shearing strain")`
The shearing stress is
`sigma_(s) = ("Tangential force")/("Area over which it is applied") = (F_(t))/(Delta A)`
The angle of shear or shearing strain
`epsilon_(s) = (x)/(h) = theta`
Therefore, Rigidity modulus is
`eta_(R ) = (sigma_(s))/(epsilon_(s)) = ((F_(t))/(Delta A))/((x)/(h)) = (F_(t))/((Delta A)/(theta))`
Further, the above equation implies that a material can be easily twisted if it has small value of rigidity modulus. For example, consider a wire, when it is twisted through an angle `theta`, a restoring torque is developed, that is
`tau alpha theta`
This means that for a larger torque, wire will twist by a larger amount (angle of shear `theta` is large). Since, rigidity modulus is inversely proportional to angle of shear, the modulus of rigidity is small.