Metal rod either expands or contract on heating or cooling. A stress is produced in rod if it is restricted from expanding or contracting. This stress is directly proportional to the thermal strain.

Thermal strain = Change in length / original length
Thermal strain = Δ L / L

Thermal strain = α L ΔT / L —— Linear thermal expansion (Δ L = α L ΔT).
Thermal strain = α ΔT

Young’s modulus of material = Stress / Strain
Y = Stress / α ΔT
So Stress = Y α ΔT

Therefore, Force = Y  A α ΔT

Y = Young’s modulus of elasticity
α = coefficient of linear expansion.
∆T = increase in temperature
A = Area

Ratio of lateral contraction to linear elongation is called Poisson’s ratio. It can also be defined as the ratio of unit transverse strain to unit longitudinal strain.

The formula for Poisson’s ratio ‘σ’ is given as,

Poisson’s ratio = σ =Lateral contraction / Linear elongation.

or

Poisson’s ratio = σ = – ε_t / ε_{l     ——— where} ε_t = transverse strain and ε_l = longitudinal strain

The decrease in thickness in lateral direction is lateral contraction and extension in length is known as linear elongation.

The ratio of tangential force per unit area to angular deformation occurred in body is known as Shearing (or) Rigidity modulus ‘n’ . It can also be defined as the ratio of shearing stress to shearing strain within elastic limit is Modulus of Rigidity ‘n’.
Rigidity modulus = T / θ   —- where T = the tangential force/unit area and θ is the angle of shear measured in radian,

Rigidity modulus = Force /Area  θ

It can also be  Rigidity modulus = Shear Stress / Shear Strain

Shear Stress = Force (F) /Area (A)

Shear Strain = Δ x / h  —— where Δ x is displacement and h is the separation between the top and bottom surfaces.

using the above formula we get,

Rigidity modulus = F h /A Δx

The unit for rigidity modulus is ‘newton metre-2 radian-1’ with symbol N m-2 rad-1.
The other unit which is widely used for rigidity modulus is pascal radian-1(Pa rad-1).

Bulk or Volume modulus (k) is defined as the ratio of volume stress to volume strain within elastic limit. In other words if the Hooke’s Law is followed,  a small change in pressure produces a proportional change in volume. Bulk modulus of material indicates its resistance to change in volume. It corresponds to volumetric strain.

A body subjected to a uniform compressive force has its volume decreased and the resultant strain is a bulk or volume strain. Consider Δv as the change in volume and V is the original volume, then If F is the total compressive force acting on a total area A, then

Bulk stress = Force / Area = Pressure (P)

Bulk modulus (k) = Bulk Stress/ Bulk strain

If Pressure (P)  is the stress applied i.e. (force/unit area) then,

Bulk strain = Δv / V

The formula for Bulk or Volume modulus (k) is given as,

The unit for bulk modulus is ‘newton metre-2 (N m-2). The single term unit which is widely used for bulk modulus is ‘pascal’ with symbol ‘Pa’.

Tensile Modulus , Modulus of Elasticity, Young modulus is defined as the stress required to cause a unit strain in the material. It is the ratio of linear stress to linear strain. In other words Young modulus is used to measure the elasticity of an elastic material. It corresponds to linear or tensile strain.

Let a wire of initial length L and cross-sectional area ‘ A’, undergo an extension l, when a stretching force ‘F’, is applied in the direction of its length.

Young’s modulus (modulus of elasticity) is given by-

Formula of Young’s modulus  = Linear Stress / Linear Strain.
Longitudinal or linear stress = Force / Area. = F/A
Longitudinal or linear strain = Change in length / Original length = l / L

Using the above formula we get,

The unit for Young’s modulus is newton metre-2 with symbol N m-2.
The single term unit normally used for Young’s modulus is ‘pascal’ with symbol ‘Pa’.