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The work is devoted to the proof of the existence theorem and the development of analytical numerical methods for finding solutions of geometrically nonlinear boundary value problems of the three-dimensional elasticity theory. Currently, the solvability of nonlinear spatial boundary-value problems for isotropic homogeneous and piecewisehomogeneous elastic bodies is most fully studied. Therefore, the development of mathematical methods to investigate the solvability and prove theorems on the existence of solutions of spatial nonlinear problems for anisotropic inhomogeneous elastic bodies is very important. In this paper, we study the solvability of nonlinear boundary value problems of the threedimensional elasticity theory for an isotropic inhomogeneous ellipsoid under kinematic boundary conditions. In the case of three-dimensional problems, the proposed research is based on integral approximations for displacements and on the fundamental solutions of Laplace, with which equilibrium equations are reduced to a system of three-dimensional singular integral equations in the volume occupied by an elastic body. The solvability of the system of integral equations is established using topological methods. Representations for displacements constructed in this way allow us to reduce the original system of equilibrium equations to a single nonlinear operator equation, the solvability of which is investigated using the principle of contracted mappings.
When creating computer programs that allow solving complex problems of calculating elastic structures, the problem of adequacy of the model chosen to real processes comes to the fore. The solution to this problem is based on a rigorous mathematical study of the solvability of boundary value problems. Availability of existence theorems makes it easy to prove the convergence of numerical methods to the exact real solution. Therefore, a rigorous study of the solvability of boundary value problems and the proof of existence theorems are a very topical problem in the mathematical theory of elasticity. In this paper, the method is used based on application of integral approximations for the components of displacements to study the solvability of nonlinear boundary problems for an isotropic inhomogeneous elastic ellipsoid. The problem is reduced to a system of singular spherical integral equations, the solvability of which is established with the involvement of a singular operator.
An existence theorem is proved and an analytical method is developed for finding solutions of geometrically nonlinear spatial boundary problems for an elastic isotropic in homogeneous ellipsoid under kinematic boundary conditions.
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