Thermoelastic Stresses in an Axisymmetric Thick-Walled Tube Under an Arbitrary Internal Transient

A closed-form axisymmetric solution was derived for the transient thermal-stress fields developed in thick-walled tubes subjected to an arbitrary thermal loading on the internal surface with convection to the surrounding external environment. Generalization of the temperature excitation was achieved by using a versatile polynomial composed of integral-and half-order terms. In order to avoid the difficult and potentially error prone evaluation of functions with complex arguments, Laplace transformation and a ten-term Gaver-Stehfest inversion formula were used to solve the resulting Volterra integral equation. The ensuing series representation of the temperature distribution as a function of time and radial position was then used to derive new relationships for the transient thermoelastic stress-states. Excellent agreement was seen between the derived temperature and stress relationships, existing series solutions, and a finite-element analysis when the thermophysical and thermoelastic properties were assumed to be independent of temperature. The use of a smoothed polynomial in the derived relationships allows the incorporation of empirical data not easily represented by standard functions. This in turn permits an easy analysis of the significance of the exponential boundary condition and convective coefficient in determining the magnitudes and distribution of the resulting stress states over time. Moreover, the resulting relationships are easily programmed and can be used to verify and calibrate numerical calculations.