Question 7: Explain strain energy in a deformed wire by drawing its graph.

## Definition of strain energy

When a wire is stretched within its elastic limits, work is done on it. This work stores in the wire as the elastic strain energy of the wire.

When a wire is stretched within its elastic limits, work is done on it. This work stores in the wire as the elastic strain energy of the wire.

Explanation

Consider a wire of length ‘l’ suspended from one end and to the other end, hang some weight as shown in the fig.

An extension occurs in the length of the wire, which increases as we increase the hung weight.

Let’s calculate the work done on the wire during an extension segment of e.

Let the force F varies from 0 at the start to F at the end for this extension. Then the average force during this interval is = ½ (0+F) = ½ F. As the extension is ‘e’, therefore, the work done is

This work is stored in the wire as elastic strain energy in the wire. Remember, this equation is applied within the elastic limits of the wire, i-e, when the Hook’s law is obeyed.

## Strain energy per unit volume

Sometimes, when we deal with engineering objectives, it is desired to find the strain energy in the unit volume of the wire.

Let ‘A’ is the cross-sectional area of the wire in our above discussion and ‘l’ is its length. Then the volume of the wire is ‘Al’. Total strain energy is given by the above equation (1). Therefore, energy per unit volume can be determined by dividing the energy on the volume of the wire,

Now F/A = stress and e/l = strain, therefore,

Stress energy per unit volume may also be called strain energy density, therefore, the above relation gives the energy density in the extended wire.