Define strain energy & dervie the equation fot the same

Strain energy is the energy stored in a body due to deformation. When an external force is applied to a body, it causes a change in the shape of the body. The body resists this change and in doing so, stores energy. This stored energy is known as strain energy.

The strain energy (U) in a deformed body is given by the integral of the product of the stress (σ) and strain (ε) over the volume (V) of the body. Mathematically, this is represented as:

U = ∫V σε dV

However, stress and strain are related by Hooke's Law, which states that stress is proportional to strain. The constant of proportionality is the modulus of elasticity (E). Therefore, we can write stress as:

σ = Eε

Substituting this into the equation for strain energy gives:

U = ∫V Eε^2 dV

This is the general equation for strain energy in terms of strain. If the strain is not uniform throughout the body, the integral must be evaluated to find the total strain energy. If the strain is uniform, the integral can be replaced by multiplication by the volume, giving:

U = EVε^2

This is the equation for strain energy in a body with uniform strain.