Study On The Two-dimensional Novel General Solutions And Green’s Functions For Isotropic Thermoelastic Materials

The thermoelastic problems widely exist in aerospace, mechanical, civil, electronic engineering and so on. The study on the general solutions of governing equations and Green’s function are the foundation to handle these thermoelastic problems. The general solution is the starting point of analytical method; hence the concise and practical general solution is a powerful tool to solve these problems. The Green’s function, which is the solution under a concentrated load, is a foundation of a lot of further works. It can be used to construct analytical solutions of practical problems under arbitrary load by the principle of superposition, and it can also provide necessary precondition to many high precision numerical methods, such as boundary element method (BEM). Furthermore, they are essential in the study of cracks, defects and inclusions. In this dissertation, the two-dimensional general solution and Green’s function for thermoelastic isotropic material are investigated.For the two-dimensional steady-state general solution of thermoelastic materials, based on the constitutive equations, equilibrium equations and steady-state heat conduction equation of thermoelastic isotropic materials, three groups of general solutions are expressed in functions which satisfies a six-order partial differential equation by the differential operator theory. By virtue of the Almansi’s theorem, the above solutions can be further transferred to three groups of general solutions expressed by two harmonic functions. Finally, the more complete general solutions are obtained by superposing these above solutions. As two checking examples, the Green’s functions for both a line heat source in the interior of infinite plane and a line heat source on the surface of semi-infinite plane are presented by virtue of the obtained general solutions.For the Green’s function for a line heat source applied in the interior of a semi-infinite plane, three harmonic functions with nine undetermined constants are constructed. Substituting these harmonic functions into the steady-state general solutions, all the components of the thermoelastic field in the semi-infinite plane can be obtained. And these nine undetermined constants can be determined by the continuous conditions of displacement and stresses, boundary conditions on free surface and mechanical-thermal equilibrium conditions. The thermal resistance effects of thermoelastic material in different surface conditions are analyzed by the numerical example.For the Green’s function for a line heat source applied in the two-phase infinite plane, six harmonic functions with fifteen undetermined constants are constructed. Substituting these harmonic functions into the steady-state general solutions, all the components of the thermoelastic field in each semi-infinite plane can be obtained. And these fifteen undetermined constants can be determined by the continuous conditions of displacement and stresses in the plane containing the line heat source, the mechanical-thermal equilibrium conditions and the interface continuous conditions of the two-phase infinite plane. The interface effects of all components in thermoelastic fields under different interface conditions are investigated by numerical results.For the Green’s function for a line heat source applied in the fluid and thermoelastic two-phase infinite plane, four harmonic functions with seven undetermined constants are constructed. Substituting these harmonic functions into the steady-state general solutions, all the components of thermoelastic field in both the fluid and solid semi-infinite planes can be obtained. These seven undetermined constants can be determined by the continuous conditions of displacement and stresses in the plane containing the line heat source, the thermoelastic equilibrium conditions, the boundary condition of solid semi-infinite plane surface, and the continuous condition of the temperature and heat flux at the interface between the fluid and solid two-phase infinite plane. The thermal resistance effects in different interface conditions are analyzed by numerical results.For the Green’s function of the coated semi-infinite plane, a series of corresponding harmonic functions with undetermined constants are constructed for a line heat source applied on the surface of coating layer, in the interior of coating layer and in the interior of substrate, respectively. Substituting these harmonic functions into the steady-state general solutions, all the components of thermoelastic field in the coating layer and substrate can be obtained. And these undetermined constants can be determined by the continuous conditions of displacement and stresses in the plane containing the line hear source, the mechanical-thermal equilibrium conditions, the boundary condition on the coating surface, and the continuous condition at the interface of coating and substrate. Through the numerical results, the complex interface effect, the shear and stretch debonding region, and the tensile failure in the coating layer are investigated in detail. All these provide some valuable suggestions and an efficient analysis tool to the accurate design of thermal resistance coating layer.For the two-dimensional quasi-static general solution of isotropic thermoelastic materials, based on the constitutive equations, equilibrium equation and quasi-static heat conduction equation, a general solution expressed in two functions are obtained by differential operator theory, Almansi’s theorem, and proper transition. One function satisfies the harmonic equation and the other satisfies quasi-static heat conduction equation. As a checking example, the Green’s function for a pulse line heat source in the infinite plane is presented by virtue of the obtained quasi-static general solutions. This general solution can be used to get Green’s functions of more complex structures under dynamic thermal loads and solve various engineering problems analytically.