The term “*Modulus of elasticity*” was first used by an English scientist, Thomas Young. According to Hooke’s law, Within elastic limit, stress is directly proportional to strain. \[\text{Stress} ∝\text{Strain}\] \[\text{Stress }= \text{Constant} × \text{Strain}\] \[\frac{\text{Stress}}{\text{Strain}}=\text{Constant}\] This constant is known as **modulus of elasticity** or **coefficient of elasticity** of the given material. It has the unit of stress.

There are three modulus of elasticity; Young’s Modulus, Bulk Modulus and Modulus of Rigidity.

## Young’s Modulus of Elasticity

It is defined as the ratio of normal stress to the longitudinal strain. It is denoted by $Y$. \[Y=\frac{\text{Normal Stress}}{\text{Longitudinal Strain}}\] Suppose a force $F$ is applied to the free end of a wire or rod of length $L$ and cross section area $A$ by which its length increases by $x$. Then, \[\text{Normal stress}=\frac{F}{A}\] \[\text{Longitudinal Strain}=\frac{x}{L}\] \[Y=\frac{F/A}{x/L}\] \[Y=\frac{FL}{xA}\] If the rod has circular cross section area, then, \[Y=\frac{FL}{πr^2x}\]

## Bulk Modulus of Elasticity

It is defined as the ratio of normal stress to the volumetric strain. It is denoted by $K$. \[K=\frac{\text{Normal stress}}{\text{Volumetric strain}}\] Suppose a force $F$ is acting on the whole surface area of a sphere of volume $V$ and surface area $A$ by which its volume decreases by $v$. \[\text{Normal stress}=\frac{F}{A}\] \[\text{Volumetric Strain}=-\frac{v}{V}\] \[K=\frac{F/A}{-v/V}\] \[K=-\frac{FV}{Av}\] The negative sign indicates that when stress increases, volume decreases. Here, the pressure applied over the sphere, \[P=\frac{F}{A}\] \[K=-\frac{PV}{v}\] If pressure is applied on outward direction so that its volume increases, then, \[K=\frac{PV}{v}\]

## Modulus of Rigidity

It is defined as the ratio of tangential stress to the shear strain. It is denoted by $η$. \[η=\frac{\text{Tangential stress}}{\text{shear strain}}\]

Consider a rectangular block whose lower face $ABCD$ is fixed and upper face $EFGH$ is subjected to tangential force $F$. Let $A$ be the area of each face and $AH=L$ be the perpendicular distance between the faces. The tangential stress will shear the rectangular block by displacing the upper face through a distance $HH’=x$.

Suppose $HAH’=\theta$, then, $\theta$ is the angle of shear. \[\text{Tangential stress}=\frac{F}{A}\] \[\text{Shear Strain}=\text{Angle of shear}=θ\] \[η=\frac{F/A}{θ}\] \[η=\frac{F}{Aθ}\]

In practice, for solids, the angle of shear is very small so, In $∆HAH’$, \[θ≈tanθ=\frac{HH’}{AH}=\frac{x}{L}\] The distance $x$ through which the upper face has been displaced is called lateral displacement.

Therefore, the shear strain may also be defined as the ratio of lateral displacement of a layer to its distance from the fixed layer.

- A solid develops restoring forces whenever it is deformed in length or volume or shape, so it has all three types of Modulus of elasticity.
- A fluid develops restoring forces only when its volume is changed, so the fluid has only Bulk Modulus of elasticity.

**More on Elasticity**