Strain Energy

When a wire is stretched by an external force, work is done on the wire and energy is stored in it. This energy remains stored in the wire as long as it is within its elastic limit and is known as strain energy. Strain energy is an important concept in understanding the mechanical properties of solids.

The elastic potential energy gained by a wire during elongation by a stretching force is called as strain energy.

W = \[\frac {1}{2}\]Fl

Where:

• W = Work done (Strain energy)
• F = Stretching force applied
• l = Extension/elongation produced
• Y = Young's modulus
• Stress = Force per unit area = \[\frac {F}{A}\]
• Strain = Change in length per unit length = \[\frac {l}{L}\]

• Elastic property: Strain energy is stored only when deformation is within the elastic limit​
• Potential energy: It is a form of elastic potential energy stored in the deformed body​
• Recoverable: The stored energy can be recovered when the deforming force is removed (within the elastic limit)​
• Proportional to deformation: The Greater the extension, the higher the strain energy stored​
• Depends on material properties: Strain energy depends on Young's modulus of the material​
• Volume dependent: Total strain energy depends on the volume of the material

Setup:

Consider a wire of original length L and cross-sectional area A stretched by a force F acting along its length. The wire gets stretched, and elongation l is produced in it. The stress and the strain increase proportionately.​

1. Basic relationships

Longitudinal stress = \[\frac {F}{A}\]

Longitudinal Strain = \[\frac {l}{L}\]

Young's modulus = \[\frac{\text{longitudinal stress}}{\text{longitudinal strain}}\]

Y = \[\frac{(F/A)}{(l/L)}=\frac{FL}{Al}\]

\[\therefore\]F = \[\frac{YAl}{L}\]

2. Work done during stretching

The magnitude of the stretching force increases from zero to F during the elongation of the wire. At a certain stage, let 'f' be the force applied and 'x' be the corresponding extension. The force at this stage is given by:​

f = \[\frac{YAx}{L}\]

For further extension dx in the wire, the work done is given by:

Work = (force) . (displacement)

dW = f dx

∴ dW = \[\frac{YAx}{L}dx\]

3. Total work done

When the wire gets stretched from x = 0 to $=l$, the total work done is given as:​

\[W=\int_0^ldW\]

\[\therefore W=\int_0^l\frac{YAx}{L}dx\]

\[\therefore W=\frac{YA}{L}\int_0^lxdx\]

\[\therefore W=\frac{YA}{L}\left[\frac{x^2}{2}\right]_0^l\]

\[\therefore W=\frac{YA}{L}\left[\frac{l^2}{2}-\frac{0^2}{2}\right]\]

\[W=\frac{YAl^2}{2L}\]

\[W=\frac{1}{2}\frac{YAl}{L}\cdot l\]

\[W=\frac{1}{2}Fl\]

Therefore:

Work done = \[\frac {1}{2}\] × load × extension

The work done by the stretching force is equal to the energy gained by the wire. This energy is strain energy.​​

Strain energy = \[\frac {1}{2}\] × load × extension

4. Strain energy per unit volume

Strain energy per unit volume can be obtained by using the above formula and various formulas of stress, strain, and Young's modulus.​

Work done per unit volume = \[\frac{\text{work done in stretching wire}}{\text{volume of wire}}\]

\[=\frac{\frac{1}{2}F\cdot l}{A\cdot L}\]

= \[\frac{1}{2}\left(\frac{F}{A}\right)\left(\frac{l}{L}\right)\]

Work done per unit volume = \[\frac {1}{2}\] × stress × strain

Strain energy per unit volume = \[\frac {1}{2}\] × stress × strain

• Energy storage: Helps understand how much energy a material can store when deformed within elastic limits​
• Design of structures: Essential for designing springs, cables, and structural components that undergo stretching​
• Material selection: Enables comparison of different materials based on their energy storage capacity​
• Failure prediction: Helps predict when a material might fail by exceeding its elastic limit​
• Work-energy calculations: Useful in calculating work done in deforming elastic bodies​
• Stress analysis: Important in analyzing stress distribution in loaded structural members​
• Safety engineering: Critical for ensuring safety margins in engineering applications involving elastic deformation​