Short Answer:

Strain energy due to shear is the energy stored in a body when it is subjected to shear stress within the elastic limit. When an external shear force acts on a material, it causes angular deformation, and the work done by the applied force is stored as strain energy.

In simple terms, when a material is twisted or subjected to a tangential force, it experiences shear deformation. The energy absorbed by the material during this deformation and recoverable upon unloading is called strain energy due to shear. This property is important in designing shafts, beams, and couplings subjected to torsional or shear stresses.

Detailed Explanation :

Strain Energy Due to Shear

When a body is subjected to an external force that causes shear stress, internal resisting forces develop within the material. As the load is applied gradually, these internal forces do work on the material, causing it to deform through a small angular displacement. The energy stored within the body because of this deformation is known as the strain energy due to shear.

This energy remains stored in the body as long as the stress is within the elastic limit, and it is completely recoverable when the load is removed. If the load exceeds the elastic limit, some part of the energy will be lost as heat or plastic deformation.

Understanding the concept of strain energy due to shear is essential for analyzing and designing shafts under torsion, beams subjected to transverse loads, and machine parts that transmit torque or shear force.

Definition

The strain energy due to shear can be defined as:

“The energy stored in a material per unit volume due to the action of shear stress within the elastic limit is called strain energy due to shear.”

It is also referred to as shear strain energy density when expressed per unit volume.

The formula for strain energy due to shear depends on the relationship between shear stress and shear strain as defined by Hooke’s law for shear, which states that within the elastic limit,

where,

•  = shear stress,
•  = shear strain (angular deformation),
•  = modulus of rigidity or shear modulus.

Derivation of Expression

Let us derive the formula for strain energy due to shear.

Consider a block of material subjected to a shear stress  on its top face, while the bottom face is fixed. The applied shear stress causes the upper surface to move through a small angle .

If the block has:

• Area of top surface = ,
• Height of block = ,
• Shear strain = , where  is the horizontal displacement.

Now,
Work done by the applied shear force = Average force × displacement

Since the load is applied gradually, the average shear force = .
Final shear force = .

Therefore,
Work done (W) = .

Substitute :

The strain energy per unit volume (u) is obtained by dividing the total energy by the volume of the block :

From Hooke’s law for shear:

Substitute this value of  into the expression for :

 

Thus,

and the total strain energy stored in a body of volume  is:

Meaning of Terms

•  = total strain energy due to shear (Joules)
•  = strain energy per unit volume (J/m³)
•  = shear stress (N/m²)
•  = modulus of rigidity (N/m²)
•  = volume of the material (m³)

Units

• In SI system: Joules per cubic meter (J/m³)
• In CGS system: Erg/cm³
• Since it is a form of work or energy, the units remain the same as those of mechanical energy.

Physical Significance

Strain energy due to shear represents the recoverable energy stored in a material due to shear deformation.

• It helps engineers estimate the energy capacity of materials before failure under shear loading.
• In torsional members like shafts, this energy is distributed throughout the volume and can be used to calculate twist, strength, and resilience.
• It ensures that components such as couplings, bolts, and beams operate safely within their elastic limit.

Application in Shafts under Torsion

When a circular shaft is subjected to torque , it experiences shear stress throughout its cross-section. The strain energy stored due to this shear stress can be determined as:

where,

•  = polar moment of inertia of the cross-section,
•  = length of the shaft.

This expression is important in designing shafts, axles, and spindles, ensuring they can store and transmit required energy safely under twisting moments.

Example Calculation

A steel shaft of diameter 50 mm and length 2 m is subjected to a maximum shear stress of 80 MPa.
Take .

 

Thus, the total strain energy stored in the shaft is 157 Joules.

Importance of Strain Energy Due to Shear

1. Used in Design of Shafts:
   Helps determine safe torque levels to prevent failure under torsion.
2. Energy Absorption in Couplings and Bolts:
   Ensures safe transmission of power and resistance to shear failure.
3. Elastic Strength Estimation:
   Provides insight into the capacity of materials to recover elastically from shear deformation.
4. Resilience in Materials:
   Used to calculate the shear modulus of resilience — the energy absorbed per unit volume before yielding in shear.
5. Safety in Engineering Applications:
   Helps prevent sudden shear failures in beams, rivets, and fasteners.

Conclusion

The strain energy due to shear is the energy stored in a material when subjected to shear stress within its elastic limit. It is given by the formula , where  is the shear stress and  is the modulus of rigidity. This concept is essential for designing components that experience torsion or shear forces, such as shafts, bolts, and couplings. Understanding strain energy due to shear helps engineers ensure that materials can absorb energy safely and operate within the elastic range, maintaining both strength and flexibility in mechanical systems.