Three-dimensional Green’s Functions For Isotropic Thermoelastic Materials

Green’s functions or fundamental solutions play an important role in both applied andtheorical studies on the physics of solid. They are the foundations for lots of furtherworks.The research on analytical method of thermoelastic structures not only can providean exact analytical solution for engineering applications， but also can be used as areference criterion for various approximate theories and calculation methods. In the case ofthermoelastic materials，the introduction of thermal coupling equation, which breaks manyexcellent properties of the original equations. And it increases the difficulty to solve theseproblems. However, the temperature change in pratical working conditions always happens,and hence the thermalelastic coupled effect can not be avoided. To solve these problems,the General solutions of thermalelastic field for three-dimensional steady and quasi-staticissues are deduced in this contest. And the Green’s functions for infinite solid, semi-infinitesolid and two-phase infinite solid, on which the steady or quasi-static point heat sourceapplied, are given by virtue of the general solutions obtained in this paper.For the completeness, the consititutive equations and equilibrium equations ofisotropic thermoelastic materials are introduced firstly. The general solutions are expressedin two displacement potential functions with the use of differential operator theory. Byvirtue of Almansi’s theorem, they can be expressed in terms of three harmonic functions.Finally, a more completed general solution is obtained by the superposing method. Basedon the obtained general solutions, the Green’s functions of infinite and semi-infinitethermal-elastic materials with point heat source acted on are investigated. Three harmonicfunctions with undetermined constants are introduced and these constants can bedetermined with the consideration of corresponding continuous boundary conditions andthermal-elastic equilibrium conditions. At last, the Green’s functions are obtained.For the two-phase isotropic thermalelastic materials, the Green’s functions ofsolid-solid and solid-liquid infinite body are studied, in which the point heat source applied.Six harmonic functions are introduced for the case of solid-solid two-phase infinite bodyand four harmonic functions for the other case. Considering the boundary conditions,continuity conditions and equilibrium conditions, we can get the undermined constants ofharmonic functions. And then the corresponding Green functions of two-phase infinitebodies are obtained.The three-dimensional quasi-static issues of isotropic thermal-elastic materials arealso investigated here, and the same method is employed. The Green’s functions of infinite body under the load of instantaneous point heat and sudden steady point heat source areobtained.The numerial examples are implemented at last, and the contours of all stresscomponents and temperature increment correspongding to the cases discussed above aregiven. And then some valuable conclusions for engineering are presened. It is worth topoint out that, since the general solutions and harmonic functions constructed in this paperis in compact form and all the components in the field are present explicitly, and it isconvenient to further use in both engineering and theorical studies. The combination of theGreen’s functions obtained here and the numerical method, such as tht bondary elementmethod, can provide more exact solution to many practical problems.