Approximate relationships for thermoelastic stresses in thick-walled vessels under an arbitrary thermal shock

Approximate solutions were derived for the transient thermal stress fields developed in thick-walled vessels subjected to a monotonic thermal-shock. In order to accomplish this, Duhamel's integral was first used to relate the arbitrary thermal loading to a previously derived unit kernel for tubular geometries. Approximate rules for direct and inverse Laplace transformations were then used to modify the resulting Volterra equation to an algebraically solvable and relatively simple form. The desired thermoelastic stress distributions were then easily determined using the calculated transient thermal distributions and elasticity theory. Good agreement was seen between the derived temperature and stress relationships and a finite-element analysis of a cylinder subjected to an asymptotic exponential heating on the internal surface with convection to the outer environment. The use of a smoothed polynomial demonstrated the feasibility of employing the method with empirical data that may not be easily represented by standard functions. Furthermore, the derived relationships can be used with polynomials or other suitable functions to solve the more difficult inverse (deconvolution) thermal problem when the temperature history on one of the surfaces is know. For any of the cases explored, the resulting relationships can be used to verify and calibrate finite-element or other numerical calculations.