From Hooke.s law, the stress in a body is proportional to the corresponding strain, provided the deformation is very small. Here we shall define the elastic modulus of a given material. There are three types of elastic modulus.
(a) Young.s modulus (b) Rigidity modulus (or Shear modulus) (e) Bulk modulus
Young.s Modulus: When a wire is stretched or compressed, then the ratio between tensile stress (or compressive stress) and tensile strain (or compressive strain) is defined as Young.s modules.
`"Young modulus of a material"= ("Tensile stress or compressive stress")/("Tensile strain or compressive strain")`
`Y= (sigma_(t))/(epsilon_(t))`
S.I. unit of Young 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 `stigma_(n)= (F_(N))/(DeltaA)= DeltaP`
The volume strain is `epsilon_(v)= (DeltaV)/(V)`
Therefore, Bulk modulus is `K= -(sigma_(n))/(epsilon_(v))= (DeltaP)/( (DeltaV)/(V))`
The negative sign indicates when pressure is applied on the body, its volume decreases. Further, the equation implies that a material can be easily compressed if it has a small value of bulk modulus. In other words, bulk modulus measures the resistance of solids to change in their volume. For an example, we know that gases can be easily compressed than solids, which means, gas has a small value of bulk modulus compared to solids. The S.I. unit of K is the same as that of pressure i.e., `N m^(-2)` or Pa (pascal).
The rigidity modulus or shear modulus: The rigidity modulus is defined as the ratio of the shearing stress to shearing strain, `eta_(R)= ("shearing stress")/(( "angle of shear or shearing strain")`
The shearing stress is `sigma_(s)= ("Tangential force")/("area over whichit is applied")= (F_(t))/(DeltaA)`
The angle of shear or shearing strain `epsilon_(s)= (x)/(h)= theta`
Therefore, Rigidity modulus is `eta_(R)= (sigma_(s))/(epsilon_(s))= ((F_(t))/(DeltaA))/((x)/(h))= ((F_(t))/(DeltaA))/(theta)`
Further, the above 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 prop 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. The S.I. unit of `eta_(R)` is the same as that of pressure i.e., `N m^(-2)` or pascal.
