Define strain energy.

Short Answer:

Strain energy is the energy stored in a body due to deformation when external forces act on it within its elastic limit. When the load is removed, this stored energy is released as the body returns to its original shape.

In simple words, when a material is stretched, compressed, twisted, or bent, it stores some amount of energy due to the internal resistance developed within it. This energy is called strain energy. The strain energy is proportional to the applied stress and strain, and its SI unit is Joule (J).

Detailed Explanation:

Strain Energy

When an external force acts on a body, it produces deformation in the material. As the material resists this deformation, work is done by the external force, and this work gets stored in the body as strain energy. The strain energy is a form of potential energy stored due to internal stresses and strains developed inside the body.

If the applied load is removed, and the material returns to its original shape (that is, within the elastic limit), the stored strain energy is completely recovered. However, if the material goes beyond its elastic limit, some energy is lost due to permanent deformation.

Hence, strain energy is an important concept in mechanics of materials and structural analysis, as it helps engineers understand how much energy a material can absorb before failure or yielding.

Definition of Strain Energy

Strain energy may be defined as:

“The energy stored in a body due to deformation under the action of externally applied loads within the elastic limit.”

It is also defined as the work done by external forces in deforming a body, which is stored internally as energy in the material.

Mathematically,

or,

where,

  •  = Strain energy (Joule)
  •  = Stress in the material (N/m²)
  •  = Strain in the material (dimensionless)
  •  = Volume of the material (m³)

Derivation of Strain Energy Formula for a Bar Under Axial Load

Consider a prismatic bar of:

  • Length ,
  • Cross-sectional area ,
  • Subjected to an axial load .

The bar experiences an extension .
From the definition of work, the work done by the load (and stored as strain energy) is:

Since the load increases gradually from 0 to , the average load is .
Thus,

From Hooke’s Law,

Substitute this value into the equation for :

This is the total strain energy stored in the bar.

The strain energy per unit volume (u), also known as strain energy density, is given by:

Since stress ,

Hence,

This formula gives the strain energy stored per unit volume in a body subjected to direct stress.

Strain Energy for Different Types of Stresses

  1. For Direct Stress:
  1. For Shear Stress:

where  = Modulus of rigidity.

  1. For Bulk or Volumetric Stress:

where  = Bulk modulus,  = Hydrostatic pressure.

Units and Dimensions of Strain Energy

  1. SI Unit:
    • Joule (J)
    • Since it is a form of energy,
  1. Dimensional Formula:

Significance of Strain Energy in Engineering

  1. Design Safety:
    • Helps engineers ensure that structures can absorb stress energy without failure.
    • Prevents sudden fracture by checking how much energy a material can safely store.
  2. Elastic Behavior:
    • Explains how materials return to their original shape when loads are removed.
  3. Resilience:
    • The resilience of a material is defined as the capacity of a material to absorb strain energy within its elastic limit.
  4. Impact Resistance:
    • Materials with high strain energy can absorb sudden impact loads without breaking.
  5. Material Selection:
    • Used for choosing materials for springs, beams, and shafts that must store and release energy repeatedly.
  6. Structural Analysis:
    • Strain energy concepts are used in determining deflections and stresses in beams, trusses, and other members.

Examples of Strain Energy

  1. Spring Mechanisms:
    • When a spring is compressed or stretched, it stores strain energy which is released when the spring returns to its original position.
  2. Bridge and Building Structures:
    • Beams and trusses store strain energy when subjected to loads; this helps prevent sudden collapse.
  3. Flywheels and Shafts:
    • Shafts under torsion store strain energy which helps in energy transmission systems.
  4. Automotive Suspension Systems:
    • Springs and shock absorbers in vehicles rely on strain energy to absorb shocks and vibrations.
  5. Elastic Materials:
    • Rubber bands, steel wires, and other elastic materials store strain energy when stretched.

Relation Between Strain Energy and Stress-Strain Curve

The area under the stress-strain curve up to the elastic limit represents the strain energy per unit volume stored in the material.

  • The total area up to the yield point = Strain energy per unit volume.
  • The area up to the elastic limit = Resilience of the material.

Therefore, the greater the area under the elastic portion of the curve, the more energy the material can store without permanent deformation.

Applications in Mechanical Engineering

  1. Design of Springs:
    To calculate energy storage and release capacity.
  2. Analysis of Shafts:
    To estimate strain energy under torsional loading.
  3. Impact Analysis:
    To ensure materials can absorb energy safely during collision or impact.
  4. Resilience Testing:
    To determine energy absorption in materials before yielding.
  5. Vibration and Energy Storage Systems:
    Used in flywheels, suspension, and dynamic systems.
Conclusion

In conclusion, strain energy is the energy stored in a material when it undergoes deformation under applied loads within the elastic limit. It is a form of potential energy due to internal stresses developed inside the material. The strain energy depends on the applied stress, strain, and modulus of elasticity of the material. It plays a vital role in mechanical and structural design by ensuring that materials can store and release energy safely without permanent deformation or failure.