Dimensional Formula of Strain

Dimensional Formula of Strain

In normal conditions, when an external force is applied to the body, it either sets the body in motion or comes to rest according to Newton’s law. However, when it comes to a rigid body showing elastic properties, applying an external force brings a shift in its dimensions. For example, when you stretch out a rubber band, you apply equal forces on both ends, due to which the band elongates in terms of length. Similarly, when a force acts along the tangential surface of a block object, it causes the uppermost layer of molecules to move along the same direction. This phenomenon is defined and well-explained by the concept of strain and the dimensional formula of strain.

What is strain, and how is it expressed in terms of physics?

To understand the concept of the dimensional formula of strain, it is essential to know more about elastic and brittle bodies. When we apply compressive or stretching forces, the bodies don’t suffer from deformation. Instead, they either resist the forces or break into several pieces. These bodies are brittle as they do not undergo any dimensional change.

On the other hand, certain bodies’ dimensions may increase or decrease based on the type of external force being applied. These are said to be elastic and, thereby, help analyse strain.

Definition of strain

Let’s consider a rope having an original length of l1. When external stress is applied towards the gravitational field, it will stretch out, and therefore, the new length will be more than the original one. Let l2 define the new length of the rope, which is facilitated by the load hanging down from its end.

Now, the increase in length can be defined as (l2 – l1). Strain is defined as the ratio of the change in dimensions to the original dimension of the body. Here, the original length of the rope is l1, while the dimensional change that has occurred due to the linear stress is given by (l2 – l1).

Therefore, the formula or physical expression for the strain can be defined as:

Strain = (l2 – l1) / l1

Or, Strain = ∂l / l1

Examples of strain

Let’s consider a rope having an original length of 30 centimetres. When a load is hung from its end, its length increases further to 45 centimetres. In this scenario, we have to calculate the strain.

So, initial length (l1) = 30 centimeters

  New length (l2) = 45 centimeters

Therefore, the change in linear dimension is (l2 – l1), which is equivalent to 15 centimetres.

According to the physical expression mentioned above, strain will be given as:

Strain = (l2 – l1) / l1

Or, Strain = 15/30

Or, Strain = 0.5

How is the dimensional formula of a strain derived?

Considering strain, it’s a ratio, and its mathematical expression is given as:

Strain = ∂l / l1

Length is considered an independent unit because it does not depend on anything else. It is expressed as [Ln], where n signifies its index. Therefore, if the length is not present in a unit, we can write it as [L0]. So, according to the dimensional formula of strain notes,

Strain = [L1] / [L1]

Or, Strain = [L1-1]

Or, Strain = [L0]

In any dimensional formula, if one dimension has 0 indexes and there is no other finite indexed dimension, the unit is considered dimensionless. Hence, strain is also a dimensionless unit since the dimensional formula of strain is expressed as [L0].

Types of strain and their applications

Strain can be divided into three based on their applications and dimensions. As it is considered directly proportional to stress during the early stage of deformation, the types of strains can be classified based on the stress being applied.

Linear strain

Linear strain is defined as the ratio between the change in length of an object to its original length. This kind of strain is observed when linear stress is applied.

StrainL = ∂l / l

Where ∂l is the change in the length and l is the original length.

Also, as linear stress is directly proportional to strain, the dimensional formula of strain can be written as:

σL ∝ StrainL

Or, σL=Y × StrainL

Here, Y is known as Young’s modulus.

Surface strain

Surface strain is defined as the ratio of change in the surface area to the original surface area of the object on the application of shear stress.

StrainS = ∂A / A

Where ∂A is the change in the area, and A is the original area.

Also, as shear stress is directly proportional to strain, the expression can be written as:

σA ∝ StrainA

Or, σA = G × StrainA

Here, G is known as the Shear’s modulus.

Volume strain

Volume or bulk strain is defined as the ratio of change in the object’s volume to the original volume when bulk stress is applied to the object.

StrainV = ∂V / V

Where ∂V is the change in the volume and V is the original volume.

Also, as volume stress is directly proportional to strain, the expression can be written as:

σV ∝ StrainV

Or, σV = B × StrainV

Here, B is known as the bulk modulus.

Conclusion

Dimensions are very crucial since it helps in determining the parameters on which a physical quantity depends. For example, area depends on length, and hence, the area becomes the dependent unit while length becomes the independent unit. Dimensional formulas are used to establish a proper relationship between these unit types. Apart from this, the knowledge of dimensions also helps understand the behaviour of a dependent unit based on the independent one. If we consider the dimensional formula of strain, we can understand why no SI unit is used for defining this unit, as it’s a dimensional attribute. Also, this particular fact explains why all the three elasticity moduli (Youngs, Bulk, and Shear) have the exact dimensions of the stress being applied.