**ELASTICITY**

**Elasticity :-** The property by virtue of which a body undergoes a change in size, shape or both due to application of external forces, tend to region its original size and shape when the external forces are withdrawn, is called elasticity.

**Perfectly elastic body :-** Bodies which can regain completely their original conditions or withdrawal of the deforming forces, are said to be perfectly elastic body.

**Perfectly plastic body :-** Bodies which do not show any tendency to regain their original conditions on withdrawal of the deforming forces, are said to be perfectly plastic body.

N.B.- There are no bodies which are perfectly elastic or plastic.

**Stress and Strain**

**Stress –**

When a system of forces, act over a body in equilibrium, the body undergoes a deformation. But an internal resisting forces acts on the body to resist the relative motion of the particles. Thus the body regains its original condition when the external deforming forces are withdrawn.

*Definition :* The internal resisting force that comes to play within the body during its deformation is called stress.

*Measurement* : Stress is measured by the internal force per unit area surrounding the point, i.e.

**Unit –**

**Dimension –**

**Strain-**

The change in dimension ( length, volume or shape ) of a body due to application of a system of forces in equilibrium, is called strain.

*Measurement :*

*Unit :* unitless quantity

*Dimension :* Dimensionless quantity.

### Different types of stresses and strains

There are three types of stresses –

**1) Longitudinal stress,**

** 2) volume stress, **

**3) Shearing stress.**

and three types of strains –

**1) Longitudinal strain.**

**2) Volume strain**

**3) Shearing strain.**

#### I) Longitudinal stress and strain :

If a body is deformed longitudinally by external force then Internal force per unit area is called longitudinal stress and change in length per unit length is called longitudinal strain.

*Calculation-*

If a body of length l, and cross section is deformed longitudinally by an amount dl due to external force dF.

Then

longitudinal stress = dF/dA

and longitudinal strain = dl/l

#### II) Volume stress and strain :

If a surface force is pressure acting on a body produce change in volume without changing shape, the surface force per unit area i.e. pressure is called volume stress. The change in volume per unit volume is called volume strain i.e.

*Calculation* – If a body of volume V is deformed of by volume dv due to application of pressure dp,

then

volume stress = dp

volume strain = dv/v

#### III) Shearing stress and strain :

When a tangential force acts on a body such that change in shape of the body is produced i.e. a relative displacement of different parallel layers of the body is produced.

Then tangential force per unit area is called shearing stress.

And

which is equal to shearing angle.

*Calculation* –

Shearing stress = F/A

where F = tangential force , A = area

Shearing strain = AA’/AD = tanθ ≈ θ = shearing angle (as θ is small)

* Hooke’s law* :- Within elastic limit stress is proportional to strain i.e.

stress **∝** strain

or

This constant is called **‘ modulus of elasticity‘**.

Different elastic constants :

**1. Young’s modulus of elasticity (Y)**

**2. Bulk modules of elasticity (k)**

**3. Rigidity modulus (n)**

**4. Axial Modulus (x)**

##### Young’s modulus of elasticity (Y) –

The ratio of the longitudinal stress to longitudinal strain within elastic limit is called Young’s modulus of elasticity i.e. Young’s modulus

If a body of length l and cross- section A is deformed longitudinally by an amount dl due to external force

then

##### Bulk modulus of elasticity :

The ratio of the volume stress to volume strain is called Bulk modulus of elasticity, i.e.

Bulk modulus (k)

Mathematically,

**Rigidity modulus**

Within elastic limit the ratio of the shearing stress to the shearing strain is called rigidity modulus i.e.

rigidity modulus (n)

When a tangential force F is applied to area A and angle of deformation is θ (say) such that P is displaced

to P’ and Q is displaced to Q’. Then

as shearing strain = PP’/PS =tanθ ≈ θ

**Axial modulus**

The ratio of the longitudinal stress to longitudinal strain without any lateral strain is called axial modulus.

i.e. Axial modulus (x)

**Poisson’s ratio**

A longitudinal extension produces a lateral contraction then the ratio of the lateral strain to longitudinal strain is called Poisson’s ratio.

i.e. Poisson’s ratio (σ)

– dimensionless quantity.

N.B. – σ is not elastic constant as it is the ratio of two strains.

### Strain energy of a rod due to longitudinal strain

Let a rod of length l and cross section A elongated by along the direction of force F. Then Young’s modulus of the material,

For a further extension dx, elementary work done,

Hence total work done for elongation of length l ,

Also,

Strain energy = 1/2 x Stress x Strain x Volume.

So, Strain energy per unit volume = 1/2 x Stress x Strain

### Strain energy due to volume strain :

Let a body of volume V is subjected to an external pressure P and corresponding volume change is v.

Then Bulk modulus

For a further volume change dv, elementary work done,

Total work done,

Hence Strain energy per unit volume

N.B.- Strain energy per unit volume

### Strain energy due to Shearing strain

Let a tangential force F applied on area α corresponding angle of shear is θ.

then rigidity modulus

For further shearing dθ, elementary work done,

Strain energy

where h – linear dimension of the body perpendicular to the applied force.

Strain energy / volume = 1/2 x Stress x Strain

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