In the present paper, the problem of dynamic thermal stresses in an infinite elastic cylinder of radius a, with its axis along the z-axis, subject to certain boundary conditions, is studied. This type of situation can arise due to melting at constant rate of an insulating material at zero temperature deposited on the positive half of the infinite cylinder. A solution and numerical results are obtained for the stress components, displacement components, and temperature. Numerical results are given and illustrated graphically for each case considered.