Short Answer:

The modulus of rigidity (G), also known as the shear modulus, is a property of a material that measures its ability to resist deformation under shear stress. It is defined as the ratio of shear stress to shear strain within the elastic limit of the material.

Mathematically,

A high modulus of rigidity indicates that the material is very stiff and can resist shape change easily under shear load, while a low value means the material deforms easily when subjected to tangential forces.

Detailed Explanation :

Modulus of Rigidity (G)

The modulus of rigidity (G), also called the shear modulus, is one of the fundamental mechanical properties that describe the elastic behavior of materials when subjected to shear stress. It represents the relationship between the applied tangential (shear) force and the resulting angular deformation in the material.

When a force acts tangentially on the surface of a body, it causes the material layers to slide relative to each other. The internal resistance developed against this deformation is known as shear stress, and the resulting change in shape is known as shear strain. The ratio of these two quantities within the elastic limit is called the modulus of rigidity.

Mathematically, it is expressed as:

Where,

= Modulus of rigidity (N/m² or Pascal)
= Tangential force applied (N)
= Area of the surface on which force acts (m²)
= Angular deformation or shear strain (radians)

The unit of modulus of rigidity is the Pascal (Pa), but in practical engineering applications, it is often expressed in Giga Pascals (GPa) because its value is quite large for most materials.

Physical Meaning of Modulus of Rigidity

The modulus of rigidity represents how strongly a material resists deformation when shear forces act on it. A material with a high modulus of rigidity is more rigid and less likely to deform, while a low modulus of rigidity material will deform more easily under the same shear stress.

For example:

Steel has a high modulus of rigidity (around 80 GPa), which means it is very stiff.
Rubber, on the other hand, has a low modulus of rigidity (about 0.0009 GPa), meaning it can easily change shape under a small shear force.

This property is crucial for selecting materials for mechanical parts such as shafts, springs, beams, and fasteners, which experience twisting and shear forces during operation.

Relation with Other Elastic Constants

The modulus of rigidity (G) is related to the modulus of elasticity (E) and Poisson’s ratio (μ) by the formula:

Where,

= Young’s modulus or modulus of elasticity
= Modulus of rigidity
= Poisson’s ratio

This equation shows that the three elastic constants are not independent; if any two are known, the third can be calculated. For isotropic materials (having the same properties in all directions), this relationship helps determine unknown material properties easily.

Shear Stress and Shear Strain Relationship

When a tangential force acts on a material, it causes a deformation such that one layer of the material slides over another.

Shear stress (τ) is the force per unit area acting tangentially to the surface, given by

Shear strain (φ) is the angular distortion produced due to this force. It is measured in radians and is given by

As long as the material remains in the elastic limit, the relationship between shear stress and shear strain is linear, following Hooke’s Law for shear, which states:

Beyond the elastic limit, the relationship becomes nonlinear, and permanent deformation occurs.

Practical Significance in Engineering

The modulus of rigidity plays a vital role in the design of many mechanical and structural components that experience shear or torsional loads. Some of the important applications include:

Design of Shafts:
In mechanical systems such as engines and turbines, shafts transmit torque. The modulus of rigidity is used to calculate the angle of twist in shafts to ensure safe operation.
Design of Springs:
Helical springs experience shear stresses when compressed or extended. Their stiffness and energy storage capacity depend on the modulus of rigidity of the spring material.
Beams and Torsion Members:
When beams are subjected to torsion, G helps determine the shear strain and angular deflection, ensuring the component does not twist excessively.
Earthquake-Resistant Structures:
In civil engineering, G helps estimate how building materials will respond to lateral forces caused by wind or seismic loads.
Material Selection:
Modulus of rigidity aids engineers in selecting materials that can withstand shear and torsional stresses without failure.

Example Calculation

Suppose a steel rod of circular cross-section with a diameter of 20 mm and length of 1 m is fixed at one end and subjected to a torque of 10 N·m. The modulus of rigidity of steel is 80 GPa.

To find the angle of twist (θ):

Where,

This means the rod will twist by approximately 0.9°, showing how stiffness is influenced by G.

Typical Values of Modulus of Rigidity for Common Materials

Steel: 80 GPa
Copper: 44 GPa
Aluminum: 26 GPa
Brass: 39 GPa
Rubber: 0.0009 GPa

These values help engineers choose the right material depending on how rigid or flexible a component needs to be.

Conclusion:

The modulus of rigidity (G) is a measure of a material’s resistance to deformation under shear stress. It plays a key role in determining how much a material twists or changes shape when tangential forces are applied. The modulus of rigidity is essential in the design of shafts, springs, and various structural elements where shear forces act. A higher value of G indicates greater stiffness, which is important for materials used in mechanical and structural applications.