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# Three modulii of elasticity

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

## (a) Youngโ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 โ Aโ, undergo an extension l, when a stretchingย force โFโ, is applied in the direction of its length.

The modulus of elasticity, in this case, is called Youngโs modulus and is given by

### i.e., Youngโs 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 Youngโs modulus is newton per square metre with symbol N m-2. The single term unit which isย widely used for Youngโs modulus is โpascalโ with symbol โPaโ.

## (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 โvโ 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 โnewton per square metre with symbol N m-2. The single term unit which isย widely used for bulk modulus is โpascalโ with symbol โPaโ.

## (c) Shearing (or) Rigidity modulus โnโ

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 โnโ.

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 โnewton per square metre radian-1โ with symbol N m-2 rad-1. The other unitย which is widely used for rigidity modulus is โpascal radian-1โ with symbol โPa rad-1โ.