Boundary element methods for axisymmetric heat conduction and thermoelastic deformations in nonhomogeneous solids

Abstract

This thesis is concerned with the development of boundary element techniques for the numerical solution of several important classes of axisymmetric heat conduction and thermoelastic problems involving nonhomogeneous solids with material properties that vary continuously in space. The problems under consideration have applications in the analyses of functionally graded materials which play an important role in engineering.

The classes of axisymmetric problems considered in this thesis may be categorized as follows: (a) nonsteady heat conduction in a nonhomogeneous solid with temperature dependent material properties, (b) nonclassical heat conduction, based on the dual-phase-lag heat conduction model, in a nonhomogeneous solid, and (c) thermoelastostatic and thermoelastodynamic deformations in nonhomogeneous solids.

The problems are formulated in terms of boundary-domain integral equations. The dual-reciprocity method is applied to transform approximately the domain integrals in the boundary-domain integral equations into boundary integrals. New axisymmetric interpolating functions, which are bounded in the axisymmetric coordinate plane but are expressed in terms of relatively simple elementary functions, are proposed for use in the dual-reciprocity method.

In the last part of the thesis, an alternative boundary element approach, based on the theory of complex variables, is proposed for solving an axisymmetric problem involving steady heat conduction in a nonhomogeneous solid. With the aid of the axisymmetric interpolating functions, the problem is recast into one requiring the construction of a complex function which is analytic in the axisymmetric solution domain and which is such that the relevant boundary conditions are satisfied. Cauchy integral formulae are discretized to obtain a boundary element procedure for constructing the required complex function numerically. Unlike the real boundary element approach for axisymmetric problems, the complex variable approach does not require the evaluation of complicated fundamental solutions involving elliptic integrals.

To check the validity of all the boundary element procedures, they are applied to solve specific problems with known analytical solutions. The numerical results obtained agree well with the analytical solutions. New results are also obtained for some problems which may be of some practical interest, such as axisymmetric laser heating of particular functionally graded solids.