Elasticity

# Modulus of Elasticity

​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