Short Answer:

Strain energy is the energy stored in a material or a structural member when it is subjected to external loads within its elastic limit. When the load is applied, the material deforms, and the work done by the external forces gets stored in the body as strain energy.

In simple terms, strain energy is the elastic potential energy possessed by a material due to deformation under stress. When the load is removed, this energy helps the body return to its original shape. It plays an important role in studying elasticity, resilience, and failure of materials.

Detailed Explanation :

Strain Energy

When a body is subjected to an external load, internal stresses and strains are developed within it. During this process, the external forces perform work on the body. This work done does not disappear but is stored within the body in the form of strain energy as long as the deformation remains within the elastic limit of the material.

If the load is removed, this energy is released, allowing the body to regain its original shape. However, if the load exceeds the elastic limit, permanent deformation occurs, and part of the energy is lost as heat or plastic work. Thus, strain energy plays a key role in understanding the elastic behavior and energy absorption capacity of materials.

Definition

The strain energy can be defined as:

“The energy stored in a body due to deformation under the action of external loads, within the elastic limit, is known as strain energy.”

It is measured in units of work or energy, such as joules (J) or newton-meters (N·m).

Mathematical Expression

Consider a bar of length , cross-sectional area , subjected to an axial load .
The bar elongates by a small amount .

The work done by the external load, which is stored as strain energy, is given by:

Using Hooke’s law ( and ), we can express strain energy as:

or

Hence, the strain energy per unit volume is:

where,

•  = total strain energy,
•  = strain energy per unit volume,
•  = stress,
•  = modulus of elasticity,
•  = cross-sectional area,
•  = length of the body.

Types of Strain Energy

1. Strain Energy Due to Axial Load
   When a member is subjected to a direct tension or compression, strain energy is stored as:

1. Strain Energy Due to Shear Stress
   For a body subjected to shear stress :

where  is the modulus of rigidity.

1. Strain Energy Due to Bending
   For a beam subjected to bending moment :

where  is the moment of inertia and  is the modulus of elasticity.

1. Strain Energy Due to Torsion
   For a shaft under twisting moment :

where  is the polar moment of inertia.

These expressions show that strain energy depends on the type of loading and the elastic properties of the material.

Strain Energy in Terms of Stress and Strain

The relationship between stress, strain, and strain energy per unit volume can also be expressed as:

where,

•  = stress,
•  = strain.

This indicates that the energy stored is equal to half the product of stress and strain within the elastic region.

Importance of Strain Energy

1. Elastic Behavior Analysis:
   It helps to understand how materials deform elastically under applied loads.
2. Resilience and Toughness:
   The ability of materials to absorb energy without failure is evaluated using strain energy concepts.
3. Design of Springs and Beams:
   The theory of strain energy is used in designing springs, beams, and shafts to ensure they store and release energy effectively.
4. Failure Analysis:
   Helps predict yielding and fracture in materials under combined stresses.
5. Finite Element Analysis (FEA):
   Modern computational methods use strain energy to calculate stresses and deformations in structures.

Resilience

Resilience is directly related to strain energy. It is defined as the capacity of a material to absorb energy within the elastic limit and release it upon unloading.

• The strain energy per unit volume up to the elastic limit is called the modulus of resilience.

where  is the yield stress of the material.

Materials with high resilience, such as spring steel, are used where energy absorption is important.

Units of Strain Energy

The units of strain energy are the same as those of work or energy:

• In SI system: Joule (J) = N·m
• In CGS system: Erg
  Since 1 Joule = 1 N·m = 1 (Pa·m³)

Example

Consider a steel rod of length 2 m and cross-sectional area 1000 mm² subjected to a tensile load of 50 kN.
If :

 

Hence, the total strain energy stored in the rod is 12.5 Joules.

Conclusion

The strain energy is the energy stored in a material due to deformation under applied loads within the elastic limit. It represents the ability of the material to resist deformation and return to its original shape when unloaded. Strain energy plays a vital role in elasticity, strength, and design of structural components like beams, springs, and shafts. Understanding strain energy helps engineers ensure that materials and structures perform safely without exceeding their elastic capacity.