Dimensional Formula of Modulus of Rigidity

Modulus of rigidity is a material’s elastic shear stiffness, usually indicated by the letter G, or sometimes by S. 

Shear modulus is the same as modulus of rigidity. Shear stress is a component of stress that is parallel to the cross-section of material. It is caused by the shear force. 

On the other hand, the strain in a body occurs when particles are moved relative to a reference length. The strains belong to either the normal or shear category. Normal strains are perpendicular to the element’s face and shear strains are parallel. 

Modulus of rigidity 

• Shear stress can be defined as any force that causes material deformation by slippage along a plane. It can also be described as the internal resistance force induced in a body in response to any deformation caused by an external force. Because of this internal resistance, the body tries to regain its original shape. 
• On the other hand, shear strain is defined as the ratio of relative displacement of any layer to its perpendicular distance from a fixed layer.
• According to the principle of modulus of rigidity, all bodies are not perfectly rigid, and when an external force is applied to them, they can bend, compress, and stretch. However, when a body gets deformed due to external force, its internal resisting force tries to regain its original form. This internal resisting force induced to regain the original shape per unit area of the body is called stress. 
• The SI unit of the shear modulus is N/m2 or Pascal (Pa), and the shear modulus is denoted by the letter G.  It is usually expressed in gigapascals(GPa).

Formula for modulus of rigidity 

G represents the modulus of rigidity or shear modulus. 

Shear modulus formula: G = shearing stress shearing strain

Dimensional formula

The physical quantity’s dimensions are the powers to which the basic quantities are elevated to represent that amount. The dimensional formula of any physical quantity is an equation that explains how and which of the base quantities are contained in that amount. 

It is written by enclosing the symbols representing base amounts in square brackets with the corresponding power, i.e. ().

E.g.: the dimension formula of displacement is: (L)

A dimensional equation is obtained by equating a physical quantity with its dimensional formula.

Dimensional formula of modulus of rigidity

Shear modulus formula: G = shearing stress shearing strain

G= Txy/ Yxy= (F/A) / (Δx/l)= Fl/AΔx

Here, Txy= F/A is the shear stress;

An object experiences force F;

An area where a force is exerted is A;

Shear strain is Yxy= Δx/l;

Transverse displacement is represented by Δx.

The initial length of material is l

It is a particular form of Hooke’s law of elasticity.

G = Fl AΔx

We know that dimension of force = [M1L1 T-2]

dimension of length = [L1]

dimension of area = [L2]

dimension of Δx = [L]

Now put the values in equation

G = Fl AΔx   = [M1L1 T-2][L1][L2] [L]  =  [M1 L-1 T-2]

So dimensional formula of modulus of rigidity =  [M1 L-1 T-2]

Dimensional analysis

In a physical relation, the dimensions are examined through dimensional analysis. These analyses can be used in conversion, correction, and derivation.

Applications of dimensional analysis

It determines the dimensional consistency, homogeneity, and accuracy of the mathematical expressions. 

Limitations of the dimensional equation

• The principle of homogeneity of dimensions cannot be used for trigonometric and exponential expressions. The derivation is more complex and complicated.
• The comparing terms or factors are less.
• The correctness of the physical expressions depends only on dimensional equality.
• It is majorly used in the case of dimensional constants. We are not able to find the value of the dimensionless constant. 

Conclusion

Generally, rigidity refers to a solid’s ability to change its shape. It follows that even if a force is applied externally on a solid material, it will not change its shape. There is a strong attraction between the particles, as seen by the close packing of the particles. The rigidity modulus is important because it gives us an indication of the extent of deformation that will occur based on how much stress is applied.