Elasticity and Modulus of Elasticity

What is Elasticity?

Elasticity is the property of the materials to return to its original size and shape after the deforming force is removed.  When an adequate amount of force is applied to the solid object it will deform. If the solid object is elastic it will come back to the original shape when the external force is removed.

An object is said to be perfectly elastic if it completely regains its original form when the deforming force is removed. In other words, the material which shows perfect elasticity is called perfectly elastic material. Since no material can completely regain its original form so the concept of a perfectly elastic body is only an ideal concept. The quartz fiber is an example of a material that acts almost like a perfectly elastic material.

Stress and Strain

Stress is defined as the restoring force per unit area of the body. The S.I unit of stress is N/m2.

Stress = externally applied force or deforming force/Area of the body 

The strain is the change in the change in the configuration like length, shape or volume to the original configuration of the body. It does not have a unit.

Strain = Change in configuration/original configuration

Modulus of Elasticity

According to Hooke’s law stress is directly proportional to the strain. 

i.e., Stress ∝ Strain

Stress = Constant x Strain

Stress/Strain = constant

This proportionality constant is called the modulus of elasticity.  Thus, modulus of elasticity can be defined as the ratio of stress to the strain. The S.I unit of modulus of elasticity is given by N/m2.

Types of modulus of elasticity

The modulus of elasticity is of three types since the strain is of three types (longitudinal strain,  volume strain and shear strain). The different modulus of elasticity is 

1.Young’s modulus of elasticity

  1. The bulk modulus of elasticity 
  2. Modulus of rigidity or Shear modulus

Young’s modulus of elasticity

Within the elastic limit, the ratio of longitudinal stress to longitudinal strain is called Young’s modulus of elasticity. Let us consider a wire of length L, area A (πr2) and radius r. A force F = mg is applied along its length which causes the wire to expand by the length ΔL. 

Longitudinal stress = F/A = mg/πr2

Longitudinal strain = ΔL/L 

Young’s modulus (Y) = (mg/πr2)/(ΔL/L)

The SI unit of Young’s modulus is N/m2 or Pascal

Bulk Modulus of elasticity (K)

The ratio of bulk stress to the volume strain is called the bulk modulus. Bulk stress is also called compressive stress. Bulk stress is defined as the restoring force that acts perpendicular to the cross-section of the body. Therefore the bulk stress will be equal to the pressure applied (P). The volume strain is given as -Δv/v. The negative sign in the volume strain expression shows that the volume decreases with an increase in pressure.

Bulk modulus (K) = P/(-Δv/v)

The SI unit of Young’s modulus is N/m2 or Pascal

Shear modulus or Modulus of rigidity

Shear modulus is the ratio of shearing stress to shearing strain. The shearing stress or tangential stress is defined as the ratio of the force acting tangential to the surface to the area of the surface. Let us consider cube whose lower surface is fixed and a tangential force F acts on the upper surface of area A. The vertical sides of the cubed shifts through an angle θ.

Shear Modulus (G) = (F/A)/θ

The SI unit of Young’s modulus is N/m2 or Pascal