$5$: A long solid cylinder of radius $a$ is rapidly heated around its curved surface producing an

instantaneous temperature distribution

$T=T_0+T_1\frac{r^2}{a^2}$

If $T_l$ is large enough for some yielding to occur, find the radius of the elastic region and the

instantaneous stress distribution in the cylinder. Assume that the material obeys Tresca's yield

criterion and that the tensile yield stress SY is independent of temperature.