# Strain Energy

3 min read## Table of Contents

## Strain Energy

Strain Energy Within the elastic limit, the work done by the external forces on a material is stored as deformation or strain that is recoverable.

On removal of load, the deformation or strain disappears and the stored energy is released. This recoverable energy stored in the material in the form of strain is called elastic strain energy.

Total Work Done = \int_{0}^{\delta} Pd\delta

This equation is valid for both elastic and nonelastic materials.

If the material is linearly elastic, then the load-displacement diagram will become linear.

The elastic strain energy stored in the material is determined from the area of triangle OAB.

U = \frac{1}{2} P_1 \delta_1 And \delta_1 = \frac{P_1L}{AE} And U = \frac{1}{2} K\delta_1^2

Since Strain Energy is a nonlinear function of load (quadratic), it does not follow the principle of superposition.

Strain energy stored per unit volume of the material is referred to as strain energy density.

Strain energy density= \frac{1}{2} \sigma_1 \varepsilon_1 = \frac{1}{2} * \frac{\sigma_1^2}{E}

When the stress in the material reaches the yield stress the strain density attains its maximum value and is called the modulus of resilience.

Modulus of resilience = \frac{1}{2} * \frac{\sigma_y^2}{E}

Modulus of resilience is a measure of energy that can be absorbed by the material due to impact loading without undergoing any plastic deformation

If the material exceeds the elastic limit during loading, all the work done is not stored in the material as strain energy. This is because part of the energy is spent on deforming the material permanently and that energy is dissipated out as heat. The area under the entire stress-strain diagram is called the modulus of toughness, which is a measure of energy that can be absorbed by the material due to impact loading before it fractures.

Hence, materials with higher modulus of toughness are used to make components and structures that will be exposed to sudden and impact loads.

Strain Energy for Bar hanging under its weight U = \frac{\gamma^2 AL^2}{6E}

Strain Energy of Bar hanging under Axial load P U = \frac{1P^2L}{2E}

Strain Energy of Bar hanging under Axial load and self-weight U = \frac{\gamma^2 AL^2}{6E} + \frac{1P^2L}{2E} + \frac{\gamma P^2L^2}{2E}

The elastic deformation of a material is linked to the strain energy. A substance stores energy internally when it deforms elastically due to external stresses. When the forces are relaxed, the potential energy that has been stored can be released, allowing the material to revert to its original shape. The applied forces, the material’s characteristics (such as elasticity and stiffness), and the degree of deformation all affect how much strain energy is stored in a material.

**Example-1 A 25 kN load is applied gradually on a steel rod ABC as shown in the figure. Taking E=200 GPa, determine the strain energy stored in the entire rod and the strain energy density in parts AB and BC.**

**Solution:- **

Strain energy density in part AB, u_{AB} = \frac{\sigma^2_{AB}}{2E}

u_{AB}= \frac{1}{2 \times 200 \times 10^9}[\frac{25 \times 10^3}{\frac{\pi}{4}(0.024)^2}]^2= 7.63 kJ/m^3

Strain energy density in part BC, u_{BC} = \frac{\sigma^2_{BC}}{2E}

u_{AB}= \frac{1}{2 \times 200 \times 10^9}[\frac{25 \times 10^3}{\frac{\pi}{4}(0.016)^2}]^2=38.65 kJ/m^3

Strain energy in the entire rod,

U = u_{AB}V_{AB} + u_{BC}V_{BC}

= 7.63 \times 10^3 \times [\frac{\Pi}{4}(0.016)^2 \times 0.8]

## Saint Venants Principle (1855)

Saint Venants Principle If the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally but has a negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed.

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