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RESEARCH PAPERS

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

[+] Author and Article Information
A. E. Segall

Engineering Science and Mechanics, The Pennsylvania State University, 212 EES Building, University Park, PA 16803aesegall@psu.edu

R. Akarapu

Engineering Science and Mechanics, The Pennsylvania State University, 212 EES Building, University Park, PA 16803

J. Pressure Vessel Technol 129(1), 52-57 (Feb 28, 2006) (6 pages) doi:10.1115/1.2389002 History: Received July 25, 2005; Revised February 28, 2006

## Abstract

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.

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## Figures

Figure 1

Thick-walled geometry showing the surfaces at r=a and r=b and their switching identity depending on the whether a direct or inverse approach is used during the analysis

Figure 2

Comparison of numerical approximations for the recurring integral required for the evaluation of the half-order polynomial terms

Figure 3

Transient temperature distributions at various radial locations of a cylinder subjected to a unit step temperature change of the form ΔT(t)=1 on the internal surface with convection allowed on the outer surface

Figure 4

Transient temperature distributions at various radial locations of a cylinder subjected to exponential heating on the internal surface of the form Δ(t)=1(1−e−0.5t) with convection allowed on the outer surface

Figure 5

Transient hoop stress distributions at various radial locations of a cylinder subjected to exponential heating on the internal surface of the form Δ(t)=1(1−e−0.5t) with convection allowed on the outer surface

Figure 6

Transient axial stress distributions at various radial locations of a cylinder subjected to exponential heating on the internal surface of the form Δ(t)=1(1−e−0.5t) with convection allowed on the outer surface

Figure 7

Transient radial stress distributions at various radial locations of a cylinder subjected to exponential heating on the internal surface of the form Δ(t)=1(1−e−0.5t) with convection allowed on the outer surface

Figure 8

Approximate inverse predictions at r*=1 of the underlying Δ(t)=1(1−e−0.5t) excitation as a function of radial distance from applied surface loading

Figure 9

Comparison of approximate and least-squares inverse predictions at r*=1 for a triangular excitation on the tube’s inner surface with external convection

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