Short Answer:

Modulus of rigidity, also known as shear modulus, is the ratio of shear stress to shear strain within the elastic limit of a material. It measures the material’s ability to resist deformation under tangential or shear forces.

In simple terms, when a body is subjected to a shear force, it changes its shape without changing its volume. The modulus of rigidity determines how stiff or flexible a material is in resisting such shape changes. It is denoted by the letter G or sometimes C, and its SI unit is N/m² (Pascal).

Detailed Explanation:

Modulus of Rigidity

The modulus of rigidity (G) is one of the most important elastic constants that describes how a material behaves under shear stress. It defines the relationship between the tangential force applied and the resulting angular deformation (shear strain) produced in the material.

When a tangential or shear force acts on a body, it causes the shape of the body to distort slightly. The amount of resistance a material offers against this tangential deformation is measured by its modulus of rigidity. Materials with a high modulus of rigidity are stiffer and deform less under shear forces, while materials with a low modulus of rigidity are more flexible and deform easily.

Definition

The modulus of rigidity (G) is defined as the ratio of shear stress to shear strain within the elastic limit of a material.

Mathematically,

Where,

•  = Modulus of rigidity (N/m² or Pascal)
•  = Shear stress (N/m²)
•  = Shear strain (radians, dimensionless)

This relationship follows from Hooke’s Law for shear, which states that within the elastic limit, the shear stress is directly proportional to the shear strain.

Explanation of Modulus of Rigidity

When a body such as a rectangular block is fixed at one face and subjected to a tangential force F on the opposite face, the body gets slightly deformed. The top layer slides over the bottom layer, forming a small angular distortion.

• The force per unit area acting tangentially on the surface is the shear stress (τ).
• The angular deformation or angle of twist (φ) between the vertical and deformed sides is the shear strain.

Within the elastic range, these two quantities are linearly related, and the constant of proportionality between them is the modulus of rigidity (G).

Thus,

This means that for a given material, when the value of G is large, a higher stress is required to produce a certain amount of angular deformation. Hence, the material is stiffer.

Units and Dimensions

1. SI Unit:
   • Newton per square meter (N/m²) or Pascal (Pa)
   • For practical purposes, it is often expressed in Giga Pascal (GPa)
2. CGS Unit:
   • dyne/cm²
3. Dimensional Formula:

Physical Meaning

The modulus of rigidity represents the stiffness of a material under shear forces.

• A high modulus of rigidity means that the material is rigid and does not deform easily under tangential loads.
• A low modulus of rigidity means that the material is flexible and deforms easily.

For example:

• Steel has a high modulus of rigidity (around 80 GPa), meaning it is very stiff.
• Rubber has a very low modulus of rigidity (around 0.0003 GPa), meaning it can deform easily under small shear forces.

Typical Values of Modulus of Rigidity

Material	Modulus of Rigidity (GPa)
Steel	80 – 85
Copper	42
Aluminum	28
Brass	36
Cast Iron	45
Rubber	0.0003

These values show that metals are more rigid, while materials like rubber are flexible and easily deformable.

Relation Between Modulus of Rigidity and Other Elastic Constants

The modulus of rigidity (G) is related to other important elastic constants such as Young’s modulus (E) and Poisson’s ratio (ν) through the following relation:

Where,

•  = Young’s modulus
•  = Modulus of rigidity
•  = Poisson’s ratio

From this relationship, if any two of these constants are known, the third can be calculated.

This relation shows that the modulus of rigidity is directly dependent on the stiffness and the Poisson effect of the material.

Derivation of the Relation Between Shear Stress and Shear Strain

Let us consider a cube of side  fixed on one face and subjected to a tangential force  on the opposite face.

• The shear stress acting on the surface:

where  is the area of the face.

• The shear strain produced:

where  is the tangential displacement of the top surface.

From Hooke’s law for shear:

Substituting the above expressions:

This equation shows that the tangential force required to produce a given shear deformation depends on the modulus of rigidity (G).

Importance of Modulus of Rigidity

1. Material Selection:
   It helps engineers choose suitable materials for parts subjected to torsion, shear, or twisting loads, such as shafts, bolts, and beams.
2. Torsional Analysis:
   Used in calculating the torque and angle of twist in circular shafts.
3. Structural Design:
   Helps in analyzing shear stress and deflection in beams, columns, and other structural components.
4. Elastic Behavior:
   Helps determine how much deformation occurs before the material reaches its elastic limit.
5. Safety and Reliability:
   Ensures components can withstand tangential stresses without excessive deformation or failure.

Example Problem

A steel shaft of diameter 50 mm and length 2 m is subjected to a torque of 150 Nm. Calculate the shear stress if the modulus of rigidity of steel is .

Solution:

Given,

The maximum shear stress in a circular shaft is given by,

where

Substituting the values:

Hence, the shear stress in the shaft is 12.22 MPa.

Conclusion

The modulus of rigidity (G) is the ratio of shear stress to shear strain and represents the material’s stiffness under shear forces. It helps determine how resistant a material is to shape changes caused by tangential loads. A higher modulus of rigidity means the material is more rigid and deforms less under shear stress. It is one of the fundamental elastic constants and is widely used in mechanical and structural engineering for analyzing torsion, beams, and other components subjected to shear loads.