Approximate Direct and Inverse Relationships for Thermal and Stress States in Thick-Walled Vessels Under Thermal Shock

Approximate solutions were derived for the transient through steady-state thermal-stress fields developed in thick-walled vessels subjected to a potentially arbitrary 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 determined using the calculated thermal states and elasticity theory. Good agreement was seen between the derived temperature and stress relationships and earlier analytical and finite-element studies of a cylinder subjected to an asymptotic exponential heating on the internal surface with convection to the outer environment. It was also demonstrated that the derived relationships can be used to approximate the more difficult inverse (deconvolution) thermal problem for both exponential (monotonic) and triangular (non-monotonic) load histories. The use of polynomial of powers $tn∕2$ demonstrated the feasibility of employing the method with empirical data that may not be easily represented by standard functions. For any of the direct and inverse cases explored, the resulting relationships can be used to verify, calibrate, and/or determine a starting point for finite-element or other numerical methods.