The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed is known as elasticity.
Hooke’s law
Within elastic limit, the strain is proportional to stress. That is Hooke’s law.
Stress ∝ Strain
Stress = k × Strain
Where k is constant, called the Modulus of the elasticity.
Elastic energy per unit volume is given by,
ε = ½ × Y × σ2
Where,
Y is young’s modulus of elasticity.
σ is the strain.
Modulus of elasticity
According to Hooke’s law
Stress ∝ Strain
or Stress = k × Strain
$$ k = \frac{Stress}{Strain}$$
Where k is a constant, called the modulus of elasticity or coefficient of elasticity.
Types of Modulus of elasticity
1. Young’s modulus of elasticity or Young’s modulus
Young’s modulus of elasticity or Young’s modulus is given by,
$$ Y = \frac{\sigma}{\epsilon}$$
Where $$ \sigma$$ is normal stress and $$ \epsilon$$ is longitudinal strain.
We know that
$$ \sigma = \frac{F}{A}$$ and $$ \epsilon =\frac{\Delta L}{L}$$
So $$ Y = \frac{F L}{A \Delta L}$$
2. Bulk modulus
Bulk modulus is given by,
$$ B = \frac{\sigma_v}{\epsilon_v}$$
Where $$ \sigma_v$$ is normal stress and $$ \epsilon_v$$ is volumetric strain.
We know that
$$ \sigma_v = \frac{F}{A}$$ and $$ \epsilon_v = \frac{\Delta V}{V}$$
So $$ B = \frac{F V}{A \Delta V}$$
Modulus of rigidity
Modulus of rigidity is given by
$$ G = \frac{\sigma_s}{\theta}$$
Where $$ \sigma_s$$ is tangetial stress and $$ \theta$$ is shearing strain.
Poisson’s ratio
Poisson’s ratio is given by,
$$ p.r. =-\frac{d/D}{l/L}$$
where, l/L is the longitudinal strain and d/D is the lateral strain.