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# Three modulii of elasticity

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

### (a) Youngโs modulus (E)

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

### (c) Shearing (or) Rigidity modulus โnโ

## (a) Youngโs 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 โ Aโ, undergo an extension l, when a stretchingย force โFโ, is applied in the direction of its length.

The modulus of elasticity, in this case, is called Youngโs modulus and is given by

### i.e., Youngโs modulus ( E ) =Linear stress/ย Linear strain

Then, longitudinal or linear stress =ย F/A

and longitudinal strain =ย l/L

E =Linear stress/ย Linear strain

ย ย = FL/Al

The unit for Youngโs modulus is newton per square metre with symbol N m-2. The single term unit which isย widely used for Youngโs modulus is โpascalโ with symbol โPaโ.

## (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 strainย produced is a bulk or volume strain.

If โvโ 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/ย V

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

The unit for bulk modulus is โnewton per square metre with symbol N m-2. The single term unit which isย widely used for bulk modulus is โpascalโ with symbol โPaโ.

## (c) Shearing (or) Rigidity modulus โnโ

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

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

### rigidity modulus n = T/ย ฮธ

The unit for rigidity modulus is โnewton per square metre radian-1โ with symbol N m-2 rad-1. The other unitย which is widely used for rigidity modulus is โpascal radian-1โ with symbol โPa rad-1โ.