A metal rod, fixed at both ends, has a length l,area of cross section A and youngs modulus Y and coefficient of linear expansion α is heated through t0 C. The work done in heating the rod is
12YAlα2t2
The work done is nothing but the potential energy stored.
Also, the thermal stress is given by TA=Yαt. We have,
W=12 (stress) (strain) (volume)
W=12×Yl×x2l×(Al) ..... (1)
Since x=TlYA and T=YAαt, we have on substituting in (1),
W=12YAlα2t2