**Chapter – 12,**** ****Elasticity**

**Introduction : **

In our daily life, there are some objects whose shapes, volume and length may be changed permanently or temporarily by applying external force. After removal of the applied force, the objects may recover completely or partially its original shape, volume and length.

As for example, if we press the spring, the length of the spring become shorter than previous. Again, if we remove the applied pressure then the spring almost recovers its original length. Similarly, if we stretch the rubber band its length increases. This property of the material is termed as **Elasticity**.

**Definition of Elasticity : **The property of a body by virtue of which the body tries to regain its original shape after the removal of deforming force, is called as Elasticity.

**Deforming Force : **When a force is applied to a body and there is change in the shape of the body. The force is known as Deforming force.

**Elastic Body :** The body having the property of regaining of its original shape after removing deforming force, is called as Elastic body. Wire, spring, Rubber band are the examples of elastic body. In human body, the bones are connected by cartilage tissue which has elastic property.

**Restoring Force :** The force developed with in the elastic body on account of relative molecular displacement is called as restoring force. Due to this force, the elastic body regains its original shape after the removal of deforming force.

**Perfectly Rigid body :** The body which is not deformed under the action of applied force, is called as Perfectly Rigid Body. The distance between the two points of rigid body remains constant under the action of external force. Practically, there is no rigid body. It is an ideal case.

**Perfectly Elastic Body :** If the body regains its original shape completely after the removal of deforming force irrespective of its magnitude then the body is said to be Perfectly elastic body. In reality, there is no perfectly elastic body but a body may be considered as perfectly elastic body with in its elastic limit.

**Partly Elastic Body :** When a deformed body only partially regains its original shape and volume after the removal of external or deforming force, is called as Partly Elastic Body. Most of the bodies may be treated as Partly Elastic Body.

There are some bodies except elastic body which can not regain its original shape after the removal of deforming force. Like heterogeneous mixture of water and flour, metals etc such type of materials. These materials are known as plastic body.

**Plasticity :** The property of remaining deformed after removing the deforming force or external force is termed as Plasticity of the material.

**Plastic body :** The body which remains with the deformed shape after the removal of deforming force, is known as Plastic Body.

**Perfectly Plastic Body / Inelastic Body :** The body which do not recover its original shape completely after the removal of deforming force, is called as Perfectly Plastic Body. It is also known as Inelastic body.

**Elastic Limit :** It is the upper limit of deforming force up to which, if the deforming force is removed, the body regains its original shape completely. If the deforming force is increased beyond its elastic limit, the body loses its elasticity.

**Stress**

The internal force of restitution per unit area of deformed body is called as stress. According to Newton’s third law of motion it is equal in magnitude and opposite in direction to the deforming force.

Unit of Stress :

1 ) In SI – system its unit is N / m^{2}

2 ) In CGS – system its unit is dyne / cm^{2}

**Types Of Stress :**

- Normal Stress
- Tangential stress
- Hydrostatic Or Hydraulic Stress

**1 ) Normal Stress :** If the deforming force acts normally over an area of a body, the stress is said to be normal stress. This stress is of two types. One is tensile stress and other is compressive stress.

**Tensile Stress :** It is the restoring force developed per unit cross sectional area of a body when the length of the body increases in the direction of deforming force.

**Compressive Stress :** It is the restoring force developed per unit cross sectional area of a body when the length of the body decrease under action of deforming force.

**2 ) Tangential Stress Or Shearing Stress :** When the deforming force acting tangentially to the surface of a body changes the shape of the body, then the stress developed in the body is called as Tangentially stress or Shearing stress.

**3 ) Hydrostatic Or Hydraulic Stress :** The stress developed in a body when it is compressed uniformly from all sides is called as Hydrostatic stress.

**Strain**

It is the ratio of change in configuration to the original configuration of the deformed body. It has neither unit nor dimension.

**Types Of Strain :**

- Tensile strain
- Volumetric Strain
- Shear strain

**1 ) Tensile Strain :** It is the ratio of change in length to the original length of the deformed body, is called as Tensile Strain. It is also known as longitudinal strain or linear strain.

**2 ) Volumetric Strain :** It is the ratio of change in volume to the original volume of the deformed body. It is also called as Bulk Strain.

**3 ) Shear Strain :** When change take place in the shape of the body due to the applied deforming force, the strain is termed as Shear Strain.

Shear strain = ∅ = sin∅ / cos∅ = tan∅

**Angle Of Shear :** It is defined as the angle ( in radian ) through which a line originally perpendicular to the fixed face gets turned on applying tangential deforming force.

**Pure Shear :** When force is applied parallel to the surface of a solid body. The body remains in equilibrium as there is no net force or net torque acting the body. Such shear is called as Pure shear.

**Hook’s Law**

**Statement of Hook’s law :** ” With in elastic limit, the extension of an elastic body is directly proportional to the force that is producing it.”

OR

” With in the elastic limit the stress is directly proportional to the strain”.

**Mathematical form of Hook’s Law :**

with in the elastic limit

Stress ∝ Strain

Stress = K * Strain

where K is proportionality constant and is called as modulus of elasticity.

**Modulus of elasticity :** From Hook’s law, the ratio of stress to the strain with in the elastic limit, is called as Modulus of elasticity. Its value depends on the nature of material of the deformed. It is also called as the coefficient of elasticity of the material. The unit and dimension of stress are the unit and dimension of the Modulus of elasticity. It is called as coefficient of elasticity.

**Types of modulus of elasticity :**

- Young’s Modulus
- Bulk Modulus
- Modulus of rigidity

**1 ) Young’s Modulus ( Y ) :** It is the ratio of longitudinal stress to the longitudinal strain.

**Expression Of Young’s Modulus :**

Let L = Original length of the elastic body

ΔL = change in length

F = deforming force

A = Area of cross section ( Shaded region in the above figure ) = πr^{2}

Longitudinal Stress = F / πr^{2}

Longitudinal Strain = ΔL / L

Young’s Modulus = Longitudinal Stress / Longitudinal Strain = ( F / πr^{2} ) / ( C / L ) = ( FL ) / ( ΔLπr^{2})

**Y = ( FL ) / ( ΔLπr ^{2})**

If L = 1 unit

ΔL = 1 unit

πr^{2} = 1 unit

then Y = F

That is young’s modulus is defined as the force required to extent an elastic body of unit length and unit cross sectional area through unity. It means that the Young’s modulus is the longitudinal stress required to double the length of the elastic body.

**2 ) Bulk Modulus ( K ) :** It is the ratio of normal stress to the volumetric strain.

Let

V = Original volume

ΔV = – Change in volume

V’ = Final Volume

ΔP = Normal Stress

volumetric strain = – ΔV / V

Bulk Modulus ( K ) = ΔP / ( – ΔV / V ) = – ( VΔP ) / ΔV

**K = – ( VΔP ) / ΔV**

**Here, the negative sign indicates the decrease in volume.**

Compressibility : It is the reciprocal of Bulk Modulus of elasticity

Compressibility = 1 / Bulk modulus = – ΔV / ( VΔP )

**3 ) Modulus of Rigidity :** It is the ratio of shearing stress to the shearing strain.

**Elastic Energy **

When the the external force is applied to a elastic body, the restoring force is produced inside the body to maintain the original shape of the body. According to Newton’s third law of motion, restoring force and external deforming force are equal in magnitude and opposite in direction. To maintain the original shape of the body, the internal restoring force does the work against the external deforming force. This amount of work done is stored in the body in the form of potential energy. This energy is called as **Elastic energy** or **Elastic Potential Energy**.

**Poisson’s Ratio :** With in the elastic limit, the ratio between lateral strain and linear strain is a constant. This constant is called as Poisson’s ratio. It is represented by **σ.**

**σ = Lateral strain / linear strain**

It has neither unit nor dimension as it is pure number. It is not elastic modulus because it is not a ratio of stress to the strain. It depends only on the nature of the material of the body. It is applicable only to the solid. It is meaningless in case of the gas and liquid.

**Questions – Answer Zone **

**1 ) Define Elasticity.**

Ans : ** **The property of a body by virtue of which the body tries to regain its original shape after the removal of deforming force, is called as Elasticity.

**2 ) What do you mean by plasticity?**

Ans : The property of remaining deformed after removing the deforming force or external force is termed as Plasticity of the material.

**3 ) What is the difference between the plasticity and elasticity ?**

Ans : Elasticity is the property of regaining the original shape after the removal of the deforming force whereas the plasticity is the property of remaining in the deformed shape after the removal of the deforming force.

**4 ) Distinguish between the elastic body and plastic body.**

Ans :

**5 ) What is meant by elastic limit ? **

Ans : It is the upper limit of deforming force up to which, if the deforming force is removed, the body regains its original shape completely. If the deforming force is increased beyond its elastic limit, the body loses its elasticity.

**6 ) State Hook’s law.**

Ans : With in elastic limit, the extension of an elastic body is directly proportional to the force that is producing it.

**7 ) What do you mean by modulus of elasticity ? **

Ans : From Hook’s law, the ratio of stress to the strain with in the elastic limit, is called as Modulus of elasticity. Its value depends on the nature of material of the deformed. It is also called as the coefficient of elasticity of the material. The unit and dimension of stress are the unit and dimension of the Modulus of elasticity. It is called as coefficient of elasticity.

**8 ) Define Young’s modulus. **

Ans :

**Y = ( FL ) / ( ΔLπr ^{2})**

If L = 1 unit

ΔL = 1 unit

πr^{2} = 1 unit

then Y = F

That is young’s modulus is defined as the force required to extent an elastic body of unit length and unit cross sectional area through unity. It means that the Young’s modulus is the longitudinal stress required to double the length of the elastic body.

**9 ) Define Bulk Modulus.**

Ans : It is the ratio of normal stress to the bulk strain.

**10 ) What do you mean by stress ? **

Ans : The internal force of restitution per unit area of deformed body is called as stress. According to Newton’s third law of motion it is equal in magnitude and opposite in direction to the deforming force.

**11 ) Define Strain.**

Ans : It is the ratio of change in configuration to the original configuration of the deformed body. It has neither unit nor dimension.

**12 ) Define tensile stress.**

Ans : It is the restoring force developed per unit cross sectional area of a body when the length of the body increases in the direction of deforming force.

**13 ) Define bulk stress.**

Ans : It is the ratio of change in volume to the original volume of the deformed body.

**14 ) Define shear stress.**

Ans : When the deforming force acting tangentially to the surface of a body changes the shape of the body, then the stress developed in the body is called as Tangentially stress or Shearing stress.

**15 ) Define linear strain.**

Ans : It is the ratio of change in length to the original length of the deformed body, is called as Tensile Strain. It is also known as longitudinal strain or linear strain.

**16 ) Define volumetric strain.**

Ans : It is the ratio of change in volume to the original volume of the deformed body, is called as volumetric Strain.

**17 ) Define shear strain.**

Ans : When change take place in the shape of the body due to the applied deforming force, the strain is termed as Shear Strain.

**18 ) Define angle of shear.**

Ans : It is defined as the angle ( in radian ) through which a line originally perpendicular to the fixed face gets turned on applying tangential deforming force.

**19 ) Define pure shear of a body. **

Ans : When force is applied parallel to the surface of a solid body. The body remains in equilibrium as there is no net force or net torque acting the body. Such shear is called as Pure shear.

**20 ) Define modulus of rigidity.**

Ans : It is the ratio of shearing stress to the shearing strain.

**21 ) Write down the expression for the young’s modulus.**

Ans : **Y = ( FL ) / ( ΔLA)**

F = deforming force

L = Original length

ΔL = change in length

A = area of cross section

**22 ) Define Poisson’s ratio. **

Ans : With in the elastic limit, the ratio between lateral strain and linear strain is a constant. This constant is called as Poisson’s ratio. It is represented by **σ.**

**23 ) Why Poisson’s ratio has not dimension and unit?**

Ans : As it is a pure number, Poisson’s ratio has not dimension and unit.

**24 ) What is the limiting value of Poisson’s ratio ? **

Ans : Theoretically, the limiting values of Poisson’s ratio are -1 and 0.5 but practically, it lies between 0.2 to 0.4 for most of the material.

**25 ) What is the meaning of Poisson’s ratio in case of liquid or gas ?**

Ans : It is meaningless in case of liquid or gas.

**26 ) Define breaking strength.**

Ans : The stress required to cause actual fracture of a material is called as the breaking strength or ultimate strength.

**27 ) Define tensile strength or breaking stress.**

Ans : A maximum stress to which a wire can be subjected just before it breaks is called as breaking stress or tensile strength. Its value depends on the nature of the material.

**28 ) Define breaking force. **

Ans : Braking stress of a material over a cross sectional area is termed as Breaking force. It is the product of Breaking Stress and cross sectional area.

Breaking force = Breaking stress * cross sectional area.

**29 ) Define elastic relaxation time.**

Ans : The time delay in regaining the original shape after the removal of deforming force is called as elastic relaxation time.

**30 ) In what condition, the tangential and normal stress both are developed in a elastic body ? **

Ans : If the deforming force is inclined to the surface at an angle Ø such that Ø is not equal to zero and Ø is equal to the right angle, then both the tangential and normal stress are developed in a elastic body.

**31 ) What should be done to increase the elasticity of the material ? **

Ans : On hammering or rolling, elasticity of the material may be increased.

**32 ) Why Quartz is considered as perfectly elastic body ? **

Ans : As Quartz has very low value of elastic relaxation time, it is considered as perfectly elastic body.

**33 ) Which material has high value of elastic relaxation time ? **

Ans : Glass has high value of elastic relaxation time.

**34 ) Which material has minimum value of elastic relaxation time ? **

Ans : Quartz has minimum value of elastic relaxation time.

**35 ) Why steel is more elastic than the rubber ? **

Ans : Value of Young’s modulus of steel = 2 *10^{11} GPa

Value of Young’s modulus of rubber = 0.05 Pa

The value of Young’s modulus of steel is much larger than that of rubber. Therefore, steel is more elastic than the rubber.

**36 ) What do you mean by compressibility of a material ? **

Ans : The reciprocal of bulk modulus of a body is termed as Compressiblity of the material. It is the ratio of volumetric strain to the normal stress.

**37 ) What is the relation between the compessibility and bulk modulus ? **

Ans : Compressibility and bulk modulus are reciprocal to each other.

**38 ) Explain the Stress versus Strain relationship.**

**39 ) What is the value of Young’s modulus for a perfectly rigid body ? **

Ans : The value of Young’s modulus for a perfectly rigid body is infinity.