Tkachenko I. The solution of the basic and mixed boundary thermoelasticity problems for the multilayer foundations

The thesis contains the well-known method of exact solution of the basic boundary elasticity problems for the elastic multilayer foundations - the compliance function technique. It is extended to the new class of objects - thermoelastic multilayer foundations. The one-dimentional and the two-dimentional Fourier trans-formation (for the planar deformation and the polydi-mensional deformation respectively) and Hankel trans-formation (for the axisymmetrical deformation) are used. An integral Laplase transformation is used to ob-tain the solutions of the differential equilibrium equa-tions in the space of transforms. An auxiliary functions for each layer are introduced. These functions are asso-ciated with the transforms of the temperature, flow, stresses and displacements on the upper boundary of the layer. The compliance functions for thermoelastic foundations are introduced. These functions allows to express some auxiliary functions in terms of the other ones. The properties of the compliance functionsare studied. The method of their determination is proposed. It is shown, that in the case, when the temperature ef-fects are not taken into consideration, the obtained re-sults coincide with the results for the elastic founda-tions. The algorithms of the solution of the basic boundary thermoelastic problems in the cases of the planar and polydimensional deformations are presented in the thesis. The great attention is paid to the numerical implementation of the obtained solutions. The explicit expressions for the compliance functions are obtained with the help of the theory of matrices. The modified compliance functions are introduced and applied. The results of the solution of the concrete problems are given. The calculations were carried out with the help of the programs in Maple. The integral equations of the planar and axisymmetrical mixed problems are obtained in the thesis. The temperature on the one part of the upper boundary of the multilayer foundation is given, and the another part is thermal insulated. The forces, applied to the boundary are also given. The kernels of the integral equations contain the modified compliance functions, introduced before. The conditions of the existence of the solution in the form of the integral equalities are obtained. It is impossible to obtain the exact solution, taking into consideration the complexity of the kernels. The technique of an approximate solution of the obtained integral equations and systems is proposed. This technique is the symbiosis of the method of the orthogonal polynomial and the method of the finite sum. In the case, when the foundation is the uniform thermoelastic semispace, we obtain the integral Karleman equation. It has an exact solution. This result coincides with the result, obtained by N.M. Borodachev. The contact problems, connected with the deformation of the multilayer thermoelastic foundation by the heated stamp with the plane bottom, are considered. The systems of two integral equations for the planar and the axisymmetrical deformation are obtained. The first equation expresses the condition of deformation compatibility of the surface of the stamp and the upper boundary of the foundation. Since this equation and the corresponding equation for the mixed problems have the same structure, one can use the same mathematical apparatus. The second equation represents the condition of equilibrium of the stamp. The numerical results shows the stability of the method. These results are in agreement with the physical meaning.