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 × σ^{2}

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.