Elasticity

# Stress And Strain

## Stress

​When external forces applied to a body change the shape or size of the body, an internal force which is developed to counter the deforming forces is known as restoring force. Stress is the resistance of the body to deformation due to application of external forces.

Stress is defined as the restoring forces per unit area set up in the body when deformed by external forces.

$\text{Stress}=\frac{\text{Restoring forces}}{\text{Area}}$

Within elastic limit, the restoring forces are equal and opposit to the external deforming forces.

$\therefore\text{Stress}=\frac{\text{External forces applied}}{\text{Area}}=\frac{F}{A}$

Stress is a scalar quantity. Its S.I. unit is $\text{Nm}^{-2}$ and its dimensional formula is $[ML^{-1}T^{-2}]$.

## Types of Stress

The stress developed in a body depends upon how the external forces are applied over it. Following are the various types of stress:

### Normal Stress

The stress in which the deforming forces are acting normal to the surface area of the body is known as normal stress. Such stress is developed when there is change in the length of the body. For example, when a wire is pulled by a force, with the force acting perpendicularly is normal stress.

Again the normal stress can be further divided into tensile stress and compression stress. If there is increase in the length of the body in the direction of applied force, then the stress is also known as tensile stress. If there is compression of the body, then the stress is known as compression stress.

### Tangential Stress (Shear Stress)

The stress in which the deforming forces are acting parallel to the surface area of the body is known as tangential stress or shear stress. The shape of the body changes but volume remains same.

### Volumetric Stress (Bulk Stress)

If the body is subjected to the same force on all the faces as shown in the figure given below, then the stress is called volume stress. There is change of pressure due to change of volume, so body is said to be subjected to hydrostatic pressure. Volume stress is also known as hydrostatic stress or bulk stress.

$\therefore\text{Volume stress}=\text{Change of Pressure}$

## Elastic Limit, Yield Point and Breaking Stress

The extension produced in an elastic body is directly proportional to the applied force within certain limit of deformation. This limit is known as elastic limit. When the applied force crosses the elastic limit of the body, permanent change in the shape or size of the body is observed. So, the elastic limit of the body may also be defined as the value of stress at which the permanent deformation begins to occur.

When stress is applied beyond elastic limit, a point occurs at which the deformation increases without any increase in the stress. This point is known as yield point. Then, the body can ultimately snap or break. The value of maximum stress that a material can bear before rupture or breaking is known as breaking stress or ultimate strength or tensile strength.

## Strain

​When deforming forces act on a body, the body is deformed i.e. it undergoes change in its dimension. Strain is defined as the ratio of change in dimensions to the original dimension. $\text{Strain}=\frac{\text{Change in dimension}}{\text{Original dimension}}$

Strain has no unit because it is a simple ratio of two quantities having same unit.

## Types of Strain

Since a body can have three types of deformations, i.e. in length, in volume or in shape, likewise there are three type os strains namely longitudinal strain, volumetric strain and shear strain.

### Longitudinal Strain

If the applied deforming force changes the length of the body only, then the relative change in length of the body is known as longitudinal strain. Mathematically, the change in length per unit original length is longitudinal strain. It is also called linear strain or tensile strain.

Let $L$ be the original length and $x$ be the change in length of the body. Then $\text{Longitudinal Strain}=\frac{\text{Change in length}}{\text{Original length}}$$=\frac{x}{L}$

If $x$ be an increase in length i.e. stress is tensile, then $\frac{x}{L}$ measures tensile strain and if $x$ be a decrease in length i.e. stress is compressive, then $\frac{x}{L}$ measures compressive strain.

### Volumetric Strain

If the deforming force changes the volume of the body, then the relative change in volume of the body is called volumetric strain or bulk strain. Mathematically, the change in volume per unit original volume is known as volumetric strain. If a body is immersed in a fluid, the fluid exerts a force on it so that its volume decreases.

Let $V$ be the original volume of the body and $v$ be the change in volume of the body due to the deforming force. Then $\text{Volumetric Strain}=\frac{\text{Change in Volume}}{\text{Original Volume}}$$=-\frac{v}{V}$

The negative sign is due to the decrease in volume.

### Shearing Strain

If the deforming force changes the shape of the body only, then the ratio of lateral displacement of the top surface of the body to the perpendicular distance of this top from the fixed plane is known as shearing strain. Simply, the strain under the effect of tangential stress is called shear strain. It is measured as an angle $\theta$ (in radian) through which a line originally perpendicular to the fixed face turns due to the application of a tangential force.

The angle $\theta$ is called angle of shear. Let $x$ and $L$ be the lateral displacement of the top surface and its perpendicular distance from the fixed surface respectively. Then $\tan\theta=\frac{x}{L}$

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