Dimensional Formula of Rigidity Modulus

Considering an ideal situation, all solid bodies can be considered rigid due to their definite volume and shape. However, that isn’t the same in reality because all these bodies can be compressed, stretched and deformed. A particular amount of force needs to be applied to the concerned body to do so. For example, you need to pull both ends to stretch a rubber band. Similarly, if you want to change the shape of a sponge, you need to squeeze it. Therefore, we can say that these bodies are elastic, and their structures can be changed by applying external forces. To define this phenomenon, the rigidity modulus is considered. 

Elasticity and Deformation 

To understand rigidity modulus, first, it’s crucial to understand elasticity and deformation. Whenever external pull forces are applied to a body, it is stretched and as a result, the dimensions increase. Once the forces are removed, the body will return to its original shape. This property is termed elasticity. 

However, when the forces are applied in huge quantities, the concerned body loses its elasticity. As a result, it is deformed permanently. Once the body is deformed, it won’t return to its original shape. 

Stress and strain

Whenever elasticity is concerned, two different units are considered-stress and strain. Stress is the force acting per unit area of the body that brings changes in its dimensions. It can be categorised into two forms-shear stress and linear stress, based on the application of the force. 

Strain is defined as the ratio of change in the dimensions of the body to its original dimension on the application of a force. 

Modulus of elasticity

As per the concepts of elasticity, if the stress is more or the force applied is more on a small area, then the change in the dimension will also be more. Similarly, the dimensional changes will be more minor when the stress is less. Therefore, we can infer that stress is directly proportional to strain. 

Stress ∝ Strain 

Or, Stress = X * Strain

Here, X is termed as the modulus of elasticity. There are 3 modulus units:

• Linear or Young modulus

• Volume or Bulk modulus 

• Area or Shear modulus 

What is rigidity modulus and how is it derived? 

The modulus of rigidity is defined as the ratio of stress acting tangentially to the shear strain, provided the values are considered within the elastic limit. Tangential stress is considered the force acting per unit area on the body’s surface and along the horizontal direction. Shear strain is defined as the ratio of change in the length of the body to its original length. 

Tangential stress = Force / Area

Or, σ = f /a 

Shear strain = L0 / L

Or, ε = L0 / L

Therefore, tangential stress is directly proportional to the shear strain and it can be expressed in the below format:

σ ∝ ε

Or, σ = ηε

Here, σ is the tangential stress, ε is the shear strain and η is the modulus of rigidity or shear modulus. 

SI unit of rigidity modulus 

The SI unit of the rigidity modulus is Pascal since strain does not have any unit and Stress has the SI unit of Pascal. If considered in the CGS system, rigidity modulus can be defined as dyne/cm2. 

Dimensional formula derivation of rigidity modulus 

For deriving the dimensional formula of the rigidity modulus, first, we need to derive its physical formula. 

σ = F/A

ε = tanθ

Or, ε = Δx/x ; where x is the change in dimension

Therefore, η = σ/ε

Or, η = (F/A) / (Δx/x)

Or, η = Fx/AΔx

Now, considering the dimensions of all these units, we can say: 

Length = [L]

Mass = [M]

Time = [T] 

Force = [M1L1T-1]

Therefore, the dimensional formula of the rigidity modulus can be described as: 

η = ([M1L1T-2][L1]) / ([L2][L1])

Or, η = [M1L2T-2] / [L3]

Or, η = [M1L-1T-2]

From this formula, the following inferences can be derived: 

1. The rigidity modulus is directly proportional to the mass. Therefore, for heavier modulus, the modulus is more. 

2. The shear modulus is indirectly proportional to length. So, if the length of the body is less, the value of the modulus will be more. 

3. Even though time doesn’t play any direct role in the shear modulus, the quantities are indirectly proportional. 

Conclusion

From this above discussion, we can understand that when a force is applied along the body’s surface in a tangential direction, it causes an inclined change. Here, the strain is mainly defined as the distance between the old position of the surface and its new position. Considering the shear modulus, it can be possible to determine how much lateral shift a body will suffer when subjected to a tangential force per unit area. In real-life scenarios, it defines the deformation extent when a body is twisted or bent along the lateral axis, the effect of friction force on the unit area of the surface, and more.