# Longitudinal Strain

## Strain Meaning

Stress and strain are two sides of a coin.

Strain is the force inclined to pull or stretch something to an extreme or damaging degree.

When an external force per unit area (stress) is applied to an object, and there is a deformity in its shape.

The inter-atomic particles inside the body try to regain its original position. While the body exceeds the elastic limit.

Such a condition where the internal restoration force fails to bring the body back to its original shape and this condition is called the strain.

The property of strain is that the elastic strain is irreversible.

### SI Unit of Strain

In Mechanics, strain is often said to be “dimensionless” regardless of what system we use as it has no units.

Strain  = Change in length / original length

Δ L/ L  =    [$L^{1}$]/ $[L^{1}]$ =    $[L^{0}]$ = 1

If we use meters/ meters. It will always come as 1.

So,

 SI unit of strain  =  one (no unit)

### Unit of Strain

The unit of strain in non-SI units,

Other units of strain = cm/cm or cubits/ cubits will always give 1.

 Unit of strain = 1 (no unit)

Generally, we identify a number as strain.

We generally use the strain after the number such as 0.012 strain.

The measurement of strain is usually given as the numeric units in (με).

Since the changes in length are usually very small and a typical strain measurement in the English system is given as microinches per inch (i.e., in the order of  $10^{-6}$ ).

Therefore, the numeric value, which is unitless, will remain the same in any system.

### Longitudinal Strain Definition

The longitudinal strain is defined as the ratio of change in length of the material due to the applied force to original length.

### Longitudinal Strain

When stress (external force per unit area) is applied on the body such that this deforming force causes change in the length alone, and the body exceeds its elastic limit.

This condition or strain so produced in the body is called the longitudinal strain.

As the name longitudinal strain suggests that we are talking about the length and strain is causing deformity in its shape by elongating its length.

For example,

In Fig.1, a rod of length ‘Lo’ is stretched along the X-axis with enough external force that its length is extended to ‘Δ L’.

Now, the new length of the rod is Lf, which equals Lo + Δ L.

Δ L is the extended length of the rod.

This has happened due to the strain in the rod.

Longitudinal strain is denoted by a Greek symbol epsilon, ε

So,  formula for the longitudinal strain is given by,

 ε =   Change in length/ original length  =  = Lf- Lo/ Lo  = Δ L/ L

The unit of longitudinal strain is one.

The dimensional formula = $[L^{0}]$

### Young’s Modulus of Elasticity ($\gamma$)

For a given material there can be different types of modulus of elasticity depending upon the type of stress applied, and the resulting strain produced.

One of them is young’s modulus of elasticity.

Young’s modulus of elasticity corresponds to the ratio of longitudinal stress to the longitudinal strain within the elastic limit.

Consider a wire PQ (in  Fig.2) of length L of radius of cross-section, ‘r’ and uniform cross-sectional area ‘A’ is suspended from a rigid support P. When stretched by a suspended load of ‘mg’ from the other end Q.

Therefore, a force (perpendicular force) F is applied at its free end. Such that there is an elongation in the body by Δ L.

Young’s modulus or $\gamma$  = Longitudinal stress/ longitudinal strain…(1)

Where,

Longitudinal stress is the deforming force when applied to the body, the stress is produced in the body causing elongation in its length.

Its formula is the same as the stress which is equal to F/A (Force per unit area)

Longitudinal stress = F/ A = F/ $\pi r^{2}$ …(2)

and, longitudinal strain  =  Δ L/ L..(3)

So,  putting values of (2) and (3)  in (1)

=  F/ $\pi r^{2}$/  Δ L/ L

## longitudinal Strain

 $\gamma$    =      F. L /  $\pi r^{2}$. Δ L

Within the elastic limit, this ratio always remains constant.

The unit of  $\gamma$  in SI is N/ $m^{2}$ or Pascal (Pa).

In CGS system = dyne/ $cm^{2}$.

The dimensional formula for $\gamma$ is $[M^{1}$ $L^{-1}$ $T^{-2}$]

### Key Points

• If the length increases from its natural length, the longitudinal strain is called the tensile strain and if the length decreases from its natural or original length, then it is the compression strain.

### Do You Know?

• In a suspension bridge, there is a stretch in the ropes by the load because of which length of rope varies. Hence young’s modulus of elasticity is involved in real life.

1. A steel wire of length 5 mm and r is 2 cm when stretched by 6 kg-wt. Find the  Δ L if the young’s modulus of steel is 2.4 x 1012 dyne.cm-2

Ans: Here, L = 5 mm = 500 cm , r = 2 cm

F = mg  = 6000 x 980 dyne, 𝛄= 2.4 x 1012 dyne /cm2

𝛄 = F .L/ πr2. Δ L

Δ L = F.L/ πr2 𝛄 =  (6000 x 980) x 500/2.4 x 1012 x 3.14 x 22

On solving we get,

 Δ L = 0.00009753184 cm

2. What is the difference between longitudinal and lateral?

Ans: Force applied parallel the axis - Longitudinal

Force applied perpendicularly to the axis- Lateral

3. What is the ratio of a lateral strain to a longitudinal strain known as?

Ans: The ratio of lateral strain to a longitudinal strain is known as Poisson’s ratio. It is a mechanical property that is used to determine the geometric behaviour of the object under loading.

4. What is a lateral strain?

Ans: Lateral strain is the change in length to the original length when subjected to an axial force in the perpendicular direction.