Question

Define strain energy and derive the equation for the same.

Answer 1

Step 1: Given

Strain energy:

1. Strain energy is the potential energy stored in a body when it is under strain( deformation).

Step 2: Formula used

$W=F.dX$ Where $W$ is work done, and $dX$ is the displacement.

Step 3: Derivation of the expression

1. Consider deformation of the body under force, the Youngs modulus (Y) is given by
   $$\begin{gathered} Y=\frac{stress}{strain} \\ Y=\frac{(F/A)}{X/L} \\ \Rightarrow F=YAx/L \end{gathered}$$

Where$F$ is force,$L$is length, $A$ is the area, and $X$ is the change in length.

3. Work done for elongation is given by
$$\begin{gathered} W=\int dW=\int dF.dx \\ W=\int (YAx/L).dx \\ \Rightarrow W=YA/L\int Xdx \\ =YA/L(X^2/2) \\ =\frac{YA{X}^2}{2L} \end{gathered}$$

Step 4: Stain energy

4. Thus strain energy is $U=\frac{YA{X}^2}{2L}$

$=\frac{1}{2}\times (Y\times \frac{X}{L})\times AL\times \frac{X}{L}$ ( Multiplying and dividing by L)

$=\frac{1}{2}\times (Y\times \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n})\times AL\times \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}$

l$=\frac{1}{2}\times \mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\times \mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\times \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}(\mathrm{because}\mathrm{Y}\times \mathrm{strain}=\mathrm{stress})$

$orstrainenergy,U=\frac{1}{2}{\left(stress\right)}^2\times volumeofthematerial$