Bulk Modulus: Definition, Formula, Unit, Relation, Calculation

In this article, you will learn a complete overview of the bulk modulus or bulk modulus of elasticity such as its definition, formula, unit, derivation, relation, calculation, and many more. 

The bulk modulus of elasticity or bulk modulus is one of the types of elastic constant.

Elastic constants are those constants that determine the deformation produced by a given stress system acting on any material.

The elastic constant measures the elasticity of the material.

It is used to derive the relationship between stress and strain.

For a homogeneous and isotropic material, there are four types of elastic constants:
  1. Modulus of Elasticity  or Young’s modulus
  2. Modulus of rigidity or Shear modulus
  3. Bulk modulus
  4. Poisson’s Ratio

Here we only discussed the bulk modulus or bulk modulus of elasticity.

Modulus of elasticity, poisson's ratio and modulus of rigidity is already discussed in our previous article.

So without wasting time let's start.

Bulk Modulus

When a uniform element is subjected to equal stresses in three mutually perpendicular directions then, the ratio of direct stress to volumetric strain is called Bulk modulus.

It is also called a bulk modulus of elasticity.

Generally, the bulk modulus is denoted by K or B.

Mathematically,

K = σ/εᵥ

Where,

σ = Direct Stress

εᵥ = Volumetric Strain

As we know,

εᵥ = (ΔV/V)

So,

K = σ/(ΔV/V)

It can also be written in the form of applied pressure or force then,

K = - ΔP/(ΔV/V)

bulk modulus

A negative sign is shown here because when pressure increases in three mutually perpendicular directions, the volume decreases so that the value of V will be negative and when the pressure decreases, the volume increases so that the value of P will be negative so it always gives a negative sign.

Where,

K = Bulk modulus

ΔP =  Change in pressure or force applied per unit area on the material

V = Initial volume of the material 

ΔV = Change in the volume of the material

Units of Bulk Modulus

SI Unit

As we know the unit of stress in the SI system is N/m² and the unit of volumetric strain is unit less.

So,

Bulk Modulus 

= Direct stress/Volumetric Strain
                                    
 =  N/m²

Hence in the  SI system, the unit of the bulk modulus will be N/m² or Pascal.

FPS Unit

As we know the unit of stress in the FPS system is lb/ft² and the unit of volumetric strain is unit less.

So,

Bulk Modulus 

= Direct stress/Volumetric Strain
                                      
 =  lb/ft²

Hence in the  SI system, the unit of the bulk modulus will be lb/ft².

Bulk Modulus Dimensional Formula

As we know,

K = σ/εᵥ

Since,

Stress = Force/Area

As we know,

Force = mass (m) × acceleration (a)

So,

Stress = {mass (m) × acceleration (a)}/A

As we know the unit of mass is kg, the unit of acceleration is m/s² and the unit of area is m².

So,

σ = kg × m s ⁻²/m²
 
σ = kg × s ⁻²/m

σ = ML⁻¹T⁻²

Since the volumetric strain is unit less quantity so the bulk modulus or bulk modulus of elasticity dimensional formula will be:

K = ML⁻¹T⁻²

Relation Between Bulk Modulus and Compressibility

Compressibility is the reciprocal of the Bulk modulus which is defined as the ratio of compressive stress to volumetric strain.

As we know,

Bulk Modulus, 

K = - ΔP/(ΔV/V)

So, Compressibility will be

β = 1/K

β = 1/- ΔP/(ΔV/V)

β = - (ΔV/V)/ΔP

So, if the bulk modulus of a material is high, the compressibility of that material will be low.

For example, if the bulk modulus of steel is 160 GPA and the bulk modulus of water is 2 GPA then water will be more compressible than steel.

Relation Between Bulk Modulus and Modulus of Elasticity

Lets,

σ = Stress on the faces

I = length of the cube

E = Young's modulus for the Material

μ = Poisson Ratios

bulk modulus

As we know,

Bulk Modulus, K = σ/εᵥ

Young's Modulus, E = σ/ε

Volume of cube = l³

So,

Change in Volume,

ΔV = 3l².δl

Both sides dividing by V

ΔV/V = (3l².δl)/V

ΔV/V = (3l².δl)/l³

ΔV/V = 3 × δl/l

ΔV/V = 3 × ε

As we know,

Linear strain,

ε = σ/E(1 - 2/m)

Now, putting these values then,

ΔV/V = 3 × σ/E(1 - 2/m)

Now, bulk modulus

K = σ/(ΔV/V)

K = σ/(3 × σ/E(1 - 2/m)

K = E/3(1 - 2/m)

E = 3K(1 - 2/m)

As we know,

Poisson's Ratio, μ = 1/m

So,

E = 3K(1 - 2μ)

Relation Between Bulk Modulus, Modulus of Elasticity, and Shear Modulus

As we know from the relation between modulus of elasticity and shear modulus,

E = 2G(1 + μ)

So,

μ = (E/2G) - 1

Now, we know

E = 3K(1 − 2μ)

After putting Poisson's Ratio value then,

E = 3K{(1 −  2(E/2G) - 1)}

After calculating these values we will get,

E = 9KG/(3K+G)

Calculation of Bulk Modulus

Question

For certain materials, the modulus of elasticity is 200 N / mm². If Poisson's ratio is 0.35, Calculate Bulk modulus. 

Solution

Given Data,

 E = 200 N / mm²

 μ = 0.35

As we know,

E = 3K(1 − 2μ)

After putting values

200 = 3K(1 - 2 × 0.35)

200 = 0.9K

K = 200/0.9

K = 222.22 N / mm²

So here you have to know all aspects related to the bulk modulus of elasticity or bulk modulus. If you have any doubt then you are free to ask me by mail or on the contact us page.

Thank You.

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