# Elastic Strain Energy for Normal Stresses

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Mechanical Engineering
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JoVE Core Mechanical Engineering
Elastic Strain Energy for Normal Stresses

### Vidéo suivante27.4: Elastic Strain Energy for Shearing Stresses

When a material is subjected to an axial loading, it experiences normal stress, leading to strain energy generation. This equation is valid for uniformly distributed stress, and the strain energy density is constant throughout the material.

For a material having non-uniform stress distribution, the strain energy density is defined for the small volume of the material.

Here, strain energy density is expressed as the product of the applied stress and the produced strain.

Integrating strain energy density over the entire volume of the material gives the total strain energy stored in the material.

The obtained equation for strain energy only applies to elastic deformation and is also termed elastic strain energy.

When the material is subjected to centric axial loading, the normal stresses are assumed to be uniform over any transverse section. Here, the normal stress is written as the ratio of the internal forces to the cross-sectional area under consideration.

The strain energy stored in such cases is written in terms of internal force and the modulus of elasticity.

## Elastic Strain Energy for Normal Stresses

Strain energy quantifies the energy stored within a material due to deformation under loading conditions, a fundamental concept in materials science and engineering. The strain energy can be modeled when a material is subjected to axial loading with uniformly distributed stress. In this scenario, the stress experienced by the material is the internal force divided by the cross-sectional area, and the strain induced is directly proportional to this stress through the modulus of elasticity.

If the stress distribution is uniform, the strain energy density, defined as the product of stress and strain, can be integrated over the entire volume of the material to yield the total strain energy stored.

However, calculating strain energy becomes more complex for materials with non-uniform stress distributions. In such cases, the strain energy density must be defined for the small volumes to account for local variations in stress and strain. The total strain energy is the sum of these densities across the material's volume.

This consideration assumes elastic behavior, where the deformation is reversible, and the material returns to the original shape when the load is removed. Understanding and calculating strain energy is vital for designing materials and components that can withstand operational stresses without failure.