Question

Long answer type question.

Derive an expression for strain energy per unit volume of the material of a wire.

Answer 1

The expression for strain energy per unit volume:

1. 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.
2. If the wire is perfectly elastic then,
   Longitudinal stress =`"F"/"A"`
   Longitudinal strain = `l/"L"`
   `"Young’s modulus (Y)" = "longitudinal stress"/"longitudinal strain"`
   Y = `("F"//"A")/(l//"L") = "F"/"A" xx "L"/l`
   ∴ F = `"YAl"/"L"`      ....(1)
3. The magnitude of stretching force increases from zero to F during the elongation of wire. Let ‘f’ be the restoring force and ‘x’ be its corresponding extension at certain instant during the process of extension.
   ∴ f = `"YAx"/"L"`     .....(2)
4. Let 'dW’ be the work done for the further small extension ‘dx’.
   Work = force × displacement
   ∴ dW = fdx
   ∴ dW = `"YAx"/"L"`dx      .....(3) [From (2)]
5. The total amount of work done in stretching the wire from x = 0 to x = l can be found out by integrating equation (3).
   W = \[\int\limits_{0}^{l} dW = \int\limits_{0}^{l}\frac{YAx}{L} dx = \frac{YA}{L} \int\limits_{0}^{l} x dx\]
   ∴ W = `"YA"/"L" ["x"^2/2]_0^l`
   ∴ W = `"YA"/"L" [l^2/2 - 0^2/2]`
   ∴ W = `("YA"l)/"L" xx l/2`
   But, `("YA"l)/"L" = "F"`       .....[From (1)]
   W = `1/2 xx "F" xx l`
   ∴ Work done in stretching a wire,
   W = `1/2 xx "load" xx "extension"`
6. Work done by stretching force is equal to strain energy gained by the wire.
   ∴ Strain energy = `1/2 xx "load" xx "extension"`
7. `"Work done per unit volume" = "work done instretching wire"/"volume of wire"`
   `= 1/2 xx ("F" xx l)/"V"`
   `= 1/2 xx ("F" xx l)/("A" xx "L")`
   `= 1/2 xx "F"/"A" xx l/"L"`
   `= 1/2 xx "stress" xx "strain"`
   ∴ `"Strain energy per unit volume" = 1/2 xx "stress" xx "strain"`
8. Other forms:
   Since, Y = `"stress"/"strain"`

• Strain energy per unit volume
  `= 1/2 xx "stress" xx "stress"/"Y" = 1/2 xx ("stress")^2/"Y"`
• Strain energy per unit volume
  `= 1/2 xx "Y" xx "strain" xx "strain" = 1/2 xx "Y" xx ("strain")^2`