Explanation:
Initial length of the rod,
\({L_t} = {L_0}(1 - \alpha \Delta \theta )\)
\( \Rightarrow \quad \Delta L = {L_t} - {L_0} = {L_0}\alpha \Delta \theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
If the same rod of length \(L_{0}\) is subjected to stress along its length, then extension in length
can be calculated by Hooke's law.
\(Y = \frac{{{\rm{ stress }}}}{{{\rm{ strain }}}} = \frac{{{\rm{ stress }}}}{{\Delta L/{L_0}}} = \frac{{{L_0} \times {\rm{ stress }}}}{{\Delta L}}\)
\(\therefore \Delta L = \frac{{{L_0} \times {\rm{ stress }}}}{Y}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
If the rod is prevented from expanding, we have
\(L_{0} \alpha \Delta \theta=\dfrac{L_{0} \times \text { stress }}{Y}\)
\(\therefore\) Stress \(=Y \alpha \Delta \theta\) (independent of \(L_{0}\) )