Thermally induced elastic fracture around a localized energy source on a pressurized cylindrical vessel is studied in this work. Analytical solutions are obtained via the method of dual integral transform, with emphasis on the identification of the dominating parameters in thermal cracking. Directions for crack extension from the heat source are examined by the stress and strain-energy-based criteria, including the effects of internal pressure and axial stress. Special features include the intrinsic transition between the thermally driven and the mechanically driven fracture patterns. The physical parameters governing such transition are determined.