## Relation between Young's modulus, Bulk modulus, modulus of rigidity, Poisson ratio

### Relation between Young's modulus, Bulk modulus, modulus of rigidity, Poisson ratio

Saturday, 26 December 2020

In this post we will define and derive the relation between Young's modulus, Bulk modulus, modulus of rigidity and poison ration. together known as elastic constant. basically we can say that we will find the relation between three moduli of elasticity. But before we discuss the relation between elastic constant. We have to know the Young's modulus, Bulk modulus and modulus of rigidity in more detail for better understanding.

Young's modulus, Bulk modulus, Poisson ratio these three terms are extracted from mechanical properties of solid. As we study in our class, Solid has different types of properties like physical, chemical, mechanical etc. But here we study a small part of the properties of solid. i.e, Mechanical properties. So lets start with Young's Modulus.

Young's modulus, Bulk modulus, modulus of rigidity and Poisson's ratio all are known as elastic constant. So we can say that Relation between Young's modulus, Bulk modulus, Modulus of rigidity and Poisson ratio In simple word relation between Elastic constant.

## Young's Modulus

Young's modulus is simply define as the ration of longitudinal stress and longitudinal strain having the same elastic limit is called Young's modulus.

young's Modulus = Longitudinal stress / Longitudinal strain

Unit of Young's modulus (Y) is N/m2 or Pascal (Pa). Where,
•  Longitudinal stress means restoring force act per unit area of cross sections.
• Longitudinal strain means the deforming force that undergoes to change the shape and size of the body per unit length.
I know many of have even now confusion about stress and strain. So lets understand it by the following given simple example.

As we discuss Young's modulus is the simple ratio of stress and strain. To elaborate this definition take a look.

Suppose we have a rubber band of length 'L' and Area of cross section 'A' in Normal position. Now after expanding it to ΔL, their area of cross section must be decreases due to increasing in length. as shown in fig.

After expanding to  ΔL a restoring force 'Fr' is stored in the rubber band to come back in normal shape and size. That restoring force per unit area called Stress. Now as we know the rubber band expands to  ΔL to its normal size L. So that the deformation (or come back to normal) force per unit length is called strain. therefore, the ration of stress and strain is called Young's Modulus.

Now we will learn the types of stress and strain. There is not necessary to understand the types but Perhaps it would be better to understand it for deep knowledge.

### Types of stress and strain

• Longitudinal stress:- It is the restoring force per unit cross sectional area of a body. When the length of the body increases in the direction of force.
• Compressional stress:- It is the restoring force per unit area when its length decreasing under deforming force.
• Hydrostatic stress:- If a body is in uniform force from all direction then corresponding stress called Hydrostatic stress.
• Tengential or Shearing stress:- When a deforming for acts tangentially to the surface of the body. the shape and size of the body changes. That tangentially force per unit area called Tengential or Shearing stress.
• Longitudinal strain:- It is defined as the increase in length per unit original length.
• Volumetric Strain:- It is defined as the change in volume per unit original volume.
• Shear strain:- it is the relative displacement between 2 parallel planes per unit distance between parallel plane.
After a long discussion about Young's modulus. we should shift to Bulk modulus. Remember to understand the Bulk modulus. We should understood the volumetric strain as mention above. Hope you all understand it well.

## Bulk Modulus

Having the elasticity limit, the ratio of normal stress to the volumetric strain is called Bulk modulus.

Bulk modulus = Normal stress / Volumetric strain

Lets understand bulk modulus by taking an example. Consider a body of volume 'V' and area 'A'. Suppose a force 'F' acts on a body from all its side so that the volume of a body decreases due to applied uniform force over the whole surface. let 'ΔV' be the volume decreases due to force as shown in fig. Then bulk modulus is given by,

K = Normal stress / volumetric strain

Unit of Bulk modulus (k) is N/m2 or Pascal (Pa). please note that the bulk modulus of solids is 50 times larger than that of liquid. Thus solid are less compressible than liquid or gas. Gases are about a million times more compressible than that of solid. Because the atoms of solid are least apart but the molecules of liquid or gases is further apart from each other.

## Difference and Relation between Young's modulus and Bulk Modulus

After discussing Young's modulus and Bulk modulus. we have an idea of the mechanical properties of solid. i.e, the value of bulk modulus indicates that how hard to compress the solid to the compare of liquid or gas. Apart from the definition of Young's modulus and bulk modulus. We also notice that there is a slightly small difference between Young's modulus and bulk modulus. The difference is, Young's modulus is the ratio of longitudinal stress to the longitudinal strain whereas the bulk modulus is the ratio of volumetric stress to the volumetric strain. The relationship is both the moduli has same S.I unit that is Pascal (pa).

## Modulus of Rigidity or Shear Modulus

Having the elastic limit, the ratio of tangential stress to the tangential strain known as Modulus of rigidity. It is denoted by η. Hence the modulus of rigidity is given by,

η = Tangential stress / tangential strain

S.I unit of modulus of rigidity is N/m2 or Pascal (Pa).

Let take an example to understand the modulus of rigidity in detail. Consider a rectangular block whose lower face is fixed. A tangential force F is applied over its upper face having area A. Due to force applied on upper face an equal and opposite force also acts on its lower face. Due to the two equal and opposite force on its faces a couple forms which exert a torque. as we know the lower face of the block is fixed, the couple shears the block into parallelogram shape by displacing its upper face through distance AA' = ΔL.

Let AB = DC = L and angle  ABA' = θ.

Many of even now confusion about modulus of rigidity. So here is one more simple example to explain the shear modulus. If you don't understand the shear modulus. there will be a difficulty to understand the relation between Young's modulus, Bulk modulus, modulus of rigidity and Poisson ratio. So following below picture will help to understand shear modulus.

In the above picture a man push the upper face of a rectangular block tangentially with force F. Consider that the lower face of the block is fixed. Due to the force applied on its upper face. The lower fixed part of the block exert equal and opposite force in the backward direction to the force F. The two equal and opposite forces form a couple which exert a torque. Due to the fixed lower face of the block, the couple shears the block into a parallelogram shape by displacing its face.

Now I hope you understood the above first example to this 3D picture. Now we will discuss a little about Poisson ratio. After Poisson ratio we will take a look on relationship between y, k, η, and  σ. we will also derive these relation.

## Poisson ratio

Having the elastic limit, It is the ratio of lateral strain to the longitudinal strain called Poisson ratio. It is denoted by σ. When a rubber band is expanding due to force applied. Its length increases but its diameter decreases. The strain produced in the direction of applied force is called longitudinal strain and that produced in the perpendicular direction is called lateral strain.

Suppose the length of the rubber band is L. After force applied its length increases to ΔL. And its diameter decreases from D to D - ΔD.

Longitudinal strain = ΔL / L

Lateral strain =  − ΔD / D

Poisson ratio (σ) = Lateral strain / Longitudinal strain

## Relation between Young's modulus, Bulk modulus, Modulus of rigidity and Poisson ratio (elastic constant)

The relation between all these elastic constant Young's modulus, Bulk modulus, Modulus of rigidity, and Poisson ratio. Out of the many relations, the main relation between Young's modulus, Bulk modulus, Modulus of rigidity and Poisson ratio are all these elastic constant has same S.I unit that is Pascal (Pa) except Poisson ratio.

The mathematical relations between elastic constant y, k, η, and  σ are:-
• Y = 3k (1 − 2σ )
• Y = 2η (1 + σ )
• σ = (3k −  2η) / (6k + 2η)

## Derive the relation between Young's modulus, Bulk modulus and Modulus of rigidity

As we know that the relation between Young's modulus, Bulk modulus and Modulus of rigidity is given by,

Bulk Modulus = Young's modulus / 3(1 − 2 Poison ration)

K = Y / 3(1 − 2σ)  = 1 / 3(α  − β) -------------- (1)

Where,
α = Longitudinal strain per unit stress.
β = Longitudinal stress per unit strain.

Now,

Y = 1 / α --------------- (2)

η (Modulus of rigidity) = Young's Modulus (Y) / 2(1 + σ) = 1 / 2(α  + β) --------- (3)

From equ. (1) we get,

α  − 2β = 1 / 3k --------------- (4)

From equ. (2) we get,

2α  + 2β = 1 / η  ------------- (5)

By adding equ. 4 and 5 we get,

3α = 1 / 3k + 1 / η

In other way,

3 / Y = 1 / 3k + 1 / η

Hence, the relation between Young's modulus, Bulk modulus and Modulus of rigidity is 3 / Y = 1 / 3k + 1 / η Proved.

In word we can write above derived relation as 3 divided by Young's modulus is equal to the 1 divided by three times Bulk modulus added with 1 divided by Modulus of rigidity.