Young's modulus (E), Bulk (or) Volume modulus (k), Shearing (or) Rigidity modulus 'n' ~ MECHTECH GURU

# Three modulii of elasticity

There are three types of modulii depending upon the three kinds of strain.

## (a) Young's modulus (E)

It is defined as the ratio of linear stress to linear strain.

Let a wire of initial length L and cross-sectional area ' A', undergo an extension l, when a stretching force 'F', is applied in the direction of its length.

The modulus of elasticity, in this case, is called Young's modulus and is given by

### i.e., Young's modulus ( E ) =Linear stress/ Linear strain

Then, longitudinal or linear stress = F/A

and longitudinal strain = l/L

E =Linear stress/ Linear strain

= FL/Al

The unit for Young’s modulus is newton per square metre with symbol N m-2. The single term unit which is widely used for Young's modulus is 'pascal' with symbol 'Pa'.

## (b) Bulk (or) Volume modulus (k)

It is defined as the ratio of bulk stress to bulk strain.

When a body is subjected to a uniform compressive force, its volume decreases and the strain produced is a bulk or volume strain.

If 'v' is the change in volume and V is the original volume, then

If F is the total compressive force acting on a total area A, then bulk stress = F/A = P

If P is the stress applied i.e. (force/unit area) then,

bulk strain =v/ V

### bulk modulus k = bulk stress/ bulk strain

The unit for bulk modulus is 'newton per square metre with symbol N m-2. The single term unit which is widely used for bulk modulus is 'pascal' with symbol 'Pa'.

## (c) Shearing (or) Rigidity modulus 'n'

The ratio of the shearing stress applied to the body to the shearing strain produced is called the rigidity modulus and denoted by the letter 'n'.

If T is the tangential force/unit area and if θ is the angle of shear measured in radian, then

### rigidity modulus n = T/ θ

The unit for rigidity modulus is 'newton per square metre radian-1' with symbol N m-2 rad-1. The other unit which is widely used for rigidity modulus is 'pascal radian-1' with symbol 'Pa rad-1'.

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