Restrictions on linear constitutive equations for a rigid heat conducting Cosserat shell

Although the three-dimensional field equation for linear heat conduction is simple, it is still challenging to obtain solutions of boundary value problems for shells with general geometry. The formulation of such problems can be simplified by using specialized equations which model heat conduction in rigid shells in terms of two temperature fields: one for the average temperature and the other for the average temperature gradient through the shell's thickness. The resulting equations are simpler because the field quantities are independent of the coordinate through the shell's thickness. However, constitutive equations for the heat fluxes in the shell theory are complicated because they depend on both the heat conduction coefficient of the material being considered and on the shell's geometry. The objective of this paper is to develop restrictions on the constitutive equations in the linear Cosserat theory of rigid heat conducting shells which ensure that the Cosserat equations produce exact steady-state solutions for Fourier conduction with an arbitrary constant temperature gradient for all shell geometries including variable thickness. Constitutive equations are proposed which satisfy these restrictions and example problems of a plate and of circular cylindrical and spherical shells are solved which examine the accuracy of the Cosserat theory. The results of these examples show that the Cosserat theory is accurate for moderately thick shells and moderately strong variation of the temperature field through the shell's thickness. In particular, the Cosserat solution converges smoothly to the exact solution as the shell becomes thin. In contrast, the two other theories considered are shown to predict incorrect slopes at the thin shell limit. (C) 2004 Elsevier Ltd. All rights reserved.