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

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

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

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