 # Elasticity

## What is Elasticity?

When an external force is applied to a rigid body there is a change in its length, volume (or) shape. When external forces are removed the body tends to regain its original shape and size. Such a property of a body by virtue of which a body tends to regain its original shape (or) size when external forces are removed is called elasticity.

## Stress and Strain

### What is Stress?

Elastic bodies regain their original shape due to internal restoring forces. This internal restoring force, acting per unit area of the deformed body is called stress.

$Stress=\frac{Restoring\,Force}{Unit\,Area}$

### Types of Stress

There are three types of stress

• Longitudinal stress
• Volume stress
• Tangential stress (or) shear stress

#### Longitudinal Stress

When the stress is normal to the surface area of the body it is known as longitudinal stress.

Again it is classified into two types

• Tensile stress
• Compressive stress.

Tensile stress: When longitudinal stress produced due to an increase in the length of the object is known as tensile stress.

Compressive stress: Longitudinal stress produced due to the decrease in length of the object is known as compressive stress.

#### Volume Stress

If equal normal forces are applied every one surface of a body then change in volume produced. The force opposing this change in volume per unit area is called volume stress.

#### Tangential Stress

When the stress is tangential (or) parallel to the surface of the body is known as Tangential (or) Shear stress. Due to this shape of body changes (or) gets twisted.

### What is Strain?

The ratio of charge of any dimension to its original dimension is called strain.

$Strain=\frac{Change\,in\,dimension}{initial\,dimension}$

Strain also classified into three types

• Longitudinal strain
• Volume strain
• Shearing strain

#### Longitudinal Strain

$Longitudinal\,strain=\frac{Change\,in\,length\,of\,the\,body}{initial\,length\,of\,the\,body}$

$=\frac{\Delta L}{L}$

#### Volume Strain

$Volume\,strain=\frac{Change\,in\,volume\,of\,the\,body}{\,Original\,volume\,of\,the\,body}$

$=\frac{\Delta V}{V}$

#### Shearing Strain

When a deforming force is applied to a body parallel to its surface its shape (not size) changes this is known as shearing strain. The angle of shear $\phi$

$\tan \phi =\frac{\ell }{L}=\frac{displaceemtn\,of\,upper\,face}{distance\,between\,two\,faces}$

### Stress Strain Graph

1. Proportion limit: The limit in which Hook’s law is valid and stress is directly proportional to strain.
2. Elastic limit: That maximum stress which on removing the deforming force makes the body to recover completely its original state.
3. Yield point: The point beyond the elastic limit at which the length of the wire starts increasing with increasing stress. Is defined as the yield point.
4. Breaking point: The point when the strain becomes so large that the wire breaks down, at last, is called the breaking point.

#### Elastic hysteresis

The strain persists even when the stress is removed. This lagging behind of strain is called elastic hysteresis. This is why the values of strain for the same stress are different while increasing the load and while decreasing the load.

## Hooke’s Law

If deformation is small, the stress in a body is proportional to the corresponding strain this fact is known as Hooke’s law.

Within elastic limit, Stress & strain $\Rightarrow \frac{Stress}{Strain}=Constant$

This constant is known as modulus of elasticity (or) coefficient of elasticity. It only depends on the type of material used. It is independent of stress and strain they are

• Young’s modulus of elasticity “y”
• The bulk modulus of elasticity “B”
• Modulus of rigidity
• Poisson’s ratio

### Young’s modulus of elasticity “y”

Within the elastic limit, the ratio of longitudinal stress and longitudinal strain is called Young’s modulus of elasticity (y).

$y=\frac{Longitudinal\,stress}{Longitudinal\,strain}=\frac{\frac{F}{A}}{\frac{\ell }{L}}=\frac{FL}{A\ell }$

Within the elastic limit, the force acting upon a unit area of a wire by which the length of wire becomes double is equivalent to Young’s modulus of elasticity of the material of the wire. If L is the length of wire, r – radius and is the increase in the length of wire by suspending a weight (mg) at its one end then young’s modulus of elasticity of the wire becomes,

$y=\frac{F/A}{\ell /L}=\frac{FL}{A\ell }=\frac{mgL}{\pi {{r}^{2}}\ell }$

(a) The increment of the length of an object by its own weight:

Let a rope of mass M and length (L) is hanged vertically. As the tension of different point on the rope is different, similarly stress as well as the strain will be different at different points.

• Maximum stress at hanging point
• Minimum stress at lower point

Consider a dx element of rope at x distance from lower end then tension.

$T=\left( \frac{M}{L} \right)\times g$

So stress $=\frac{T}{A}=\left( \frac{M}{L} \right)\frac{xg}{A}$

Let the increase in length of element dx is dy then

$Strain=\frac{Change\,in\,length}{Original\,length}=\frac{\Delta y}{\Delta x}=\frac{dy}{dx}$

Now we got stress and strain then young’s modulus of elasticity “y”

$y=\frac{Stress}{Strain}=\frac{\left( \frac{M}{L} \right)\frac{xg}{A}}{\frac{dy}{dx}}\Rightarrow \left( \frac{M}{L} \right)\frac{xg}{A}dx=\frac{1}{dy}$

The total change in length of the wire is

$\frac{Mg}{LA}\int\limits_{o}^{L}{x\,dx}=y\int\limits_{o}^{\Delta e}{dy}$

$\frac{Mg}{LA}\frac{{{L}^{2}}}{2}=y\Delta \ell$

$\frac{MgL}{2Ay}=\Delta \ell$

(b) Work done in stretching a wire

If we need to stretch a wire, we have to do work against its inter atomic forces, which is stored in the form of elastic potential energy.

For a wire of length (L0) stretched by a distance $\left( x \right)$ the restoring elastic force is

$I=(Stress)(Area)=y\left[ \frac{x}{{{L}_{0}}} \right]A$

Work required for increasing an element length

$dW=F-dx=\frac{{{y}_{A}}}{{{L}_{0}}}x\,dx$

Total work required in stretching the wire is

$W=\int\limits_{0}^{\Delta \ell }{F-dx=\frac{{{y}_{A}}}{{{L}_{0}}}}\int\limits_{0}^{\Delta \ell }{x\,dx}$

$=\frac{{{y}_{A}}{{\left( \Delta \ell \right)}^{2}}}{2{{L}_{0}}}$

$=\frac{{{y}_{A}}}{{{L}_{0}}}\left[ \frac{{{x}^{2}}}{2} \right]_{0}^{\Delta \ell }$

$W=\frac{1}{2}{{(Strain)}^{2}}(Original\,volume)$

$W=\frac{1}{2}(Stress)(Strain)(Volume)$

(c) Analogy of rod as a spring

From definition of young’s modulus

$y=\frac{Stress}{Strain}=\frac{FL}{A\,\Delta L}$

$F=\frac{{{y}_{A}}\Delta L}{L}$ … (1)

This expression is analogy of spring force

$F=kx$

$k=\frac{yA}{L}=$constant

$\frac{yA}{L}=$ constant which only function its material property.

### Bulk Modulus (B)

Within elastic limit the ratio of the volume stress and the volume strain is called bulk modulus of elasticity.

$B=\frac{Volume\,stress}{Volume\,strain}=\frac{\frac{F}{A}}{-\frac{\Delta V}{V}}=\frac{\Delta P}{\frac{-\Delta V}{V}}$

### Rigidity Modulus

Within elastic limit, the ratio of shearing stress and shearing strain is called modulus of rigidity.

$\eta =\frac{Shearing\,stress}{Shearing\,strain}=\frac{\frac{{{F}_{tangential}}}{A}}{\phi }=\frac{{{F}_{tangential}}}{\phi }.$

$\phi -$angle of shear.

### Poisson’s Ratio

Within elastic limit, the ratio of lateral strain and longitudinal strain is called poison’s ratio.

$Poisson’s\,ratio(\sigma )=\frac{lateral\,strain}{longitudinal\,strain}=\frac{\beta }{\alpha }$