Short Answer:

Strain energy is the energy stored in a body due to deformation under the action of applied loads. When an external force is applied to an elastic material, it changes shape slightly, and this deformation causes the material to store energy internally.

When the load is removed, the material tries to regain its original shape, releasing the stored energy. This stored energy is known as strain energy. It depends on the type of material, the magnitude of stress, and the type of loading such as tension, compression, bending, or torsion.

Detailed Explanation :

Strain Energy

When an external force acts on a body, it produces internal resistance within the material. This resistance causes a small deformation, and the work done by the external force is stored inside the body as potential energy. This energy is called strain energy. It exists only as long as the body remains deformed. Once the force is removed, the body tends to return to its original position, and the strain energy is released.

In simple words, strain energy is the elastic potential energy stored in a material due to the application of load. It is a fundamental concept in the study of strength of materials and is used to analyze the elastic behavior of structures under various types of loads.

Expression for Strain Energy

Consider a bar subjected to an axial load . The load causes an extension  in the bar. The work done by the force during this extension is stored as strain energy.

If the load is applied gradually, the average force during loading is , and the work done (strain energy) is:

Now, using Hooke’s law,

Substituting this in the above equation gives:

 

Here,

•  = Strain energy stored in the bar
•  = Applied load
•  = Length of the bar
•  = Cross-sectional area
•  = Young’s modulus of elasticity

Alternatively, strain energy per unit volume (known as strain energy density) is given by:

where  is the stress.

Types of Strain Energy

1. Strain Energy due to Axial Loading:
   When a member is subjected to direct tension or compression, the strain energy stored per unit volume is

This type of strain energy occurs in rods, columns, and tie bars.

1. Strain Energy due to Shear Stress:
   When the body is subjected to shear force, it deforms in shape. The strain energy per unit volume in this case is

where  = shear stress and  = modulus of rigidity.

1. Strain Energy due to Bending:
   For a beam under bending, strain energy is stored due to bending stresses. It can be expressed as

where  = bending moment,  = modulus of elasticity, and  = moment of inertia.

1. Strain Energy due to Torsion:
   When a shaft is twisted by torque , the strain energy stored is

where  = polar moment of inertia and  = modulus of rigidity.

Importance of Strain Energy in Engineering

• Design of Structures: Helps in predicting how much energy a structure can absorb before failure.
• Elastic Stability: Used to study the behavior of beams, columns, and springs under load.
• Resilience: The strain energy stored within the elastic limit represents the resilience of the material.
• Energy Method Applications: In advanced structural analysis, strain energy is used to determine deflections and vibrations.
• Safety Analysis: Prevents overloading by ensuring that the energy stored remains within safe limits.

Energy Storage and Release

When the applied stress is within the elastic limit, all the strain energy can be recovered after the load is removed. However, if the stress exceeds the elastic limit, some of the energy is lost in plastic deformation, and the body does not return to its original shape. The recoverable portion of energy is known as elastic strain energy, while the non-recoverable part is known as plastic energy.

Example

Let’s take a mild steel rod of 1 m length and 100 mm² cross-section. If it is subjected to a tensile stress of 100 MPa, and , the strain energy per unit volume is:

This means that every cubic meter of material stores 25,000 J of strain energy before it yields.

Conclusion:

Strain energy is the elastic potential energy stored in a material when it is deformed by external forces. It plays a crucial role in understanding how materials and structures behave under various loads. The energy stored depends on the type of stress, the elastic constants of the material, and the magnitude of deformation. Knowing strain energy helps engineers design safer and more efficient components by ensuring materials work within their elastic limits.