Heat conduction in plates and shells with emphasis on a conical shell

Abstract

This paper is concerned with analyzing heat conduction in rigid shell-like bodies. The thermal equations of the theory of a Cosserat surface are used to calculate the average (through-the-thickness) temperature and temperature gradient directly, without resorting to integration of three-dimensional results. Specific attention is focused on a conical shell. The conical shell is particularly interesting because it has a converging geometry, so that the shell near its tip is “thick” even though the shell near its base may be “thin”. Generalized constitutive equations are developed here in a consistent manner which include certain geometrical features of shells. These equations are tested by considering a number of problems of plates, circular cylindrical shells and spherical shells, and comparing the results with exact solutions. In all cases, satisfactory results are predicted even in the thick-shell limit. Finally, a problem of transient heat conduction in a conical shell is solved. It is shown that the thermal bending moment produced by the average temperature gradient is quite severe near the tip, and it attains its maximum value in a relatively short time