539.3

багатошаровий циліндр
власні функції
метод найменших квадратів
напруження
переміщення
eigenfunctions
multilayer elastic cylindern
method of least squares
displacement
stresses

Розглянуто тривимірний напружено-деформований стан скінченного багатошарового циліндра, дослідження якого, шляхом відокремлення кутової змінної, зведено до розв’язання одновимірних крайових задач. Уперше побудовано власні функції для багатошарового циліндра з ненавантаженими торцями. Компоненти вектора переміщень і тензора напружень подано у вигляді рядів, які визначаються власними функціями. З використанням методу найменших квадратів, розв’язання крайових задач зведено до мінімізації узагальненої квадратичної форми. Вперше теоретично встановлені числові критерії збіжності методу і показано, що точність задоволення крайових умов оцінюється одним числом – мінімумом квадратичної форми.
Multilayer elastic cylinders are widely used in construction and engineering designs. In the calculation of the stress-strain state (SSS ), the simplified two-dimensional model of cylindrical shells are used or axisymmetrical boundary value problems under static load, which depend only on one spatial variable are considered. However, publications that describe a technique for solving boundary value problems in three- dimensional setting are not available. Investigation of the three-dimensional boundary-value problems of elasticity theory for multilayer cylindrical bodies are significantly complicated by the need to satisfy many ( more than ten ) conditions of perfect contact of adjacent layers and the boundary conditions. The problem of justification the completeness of solution as the series, research on the convergence of the method, theorem proving the existence and uniqueness of the solution, installation error estimates the meet all conditions has not studied. These issues are discussed in the article. To solve the three-dimensional boundary value problems the representation of general solution of equations of elasticity theory in a cylindrical coordinate system in three harmonic functions has been used. Countable set of eigenfunctions with nonzero and zero eigenvalues has been built. Components of displacement vector and the stress tensor are presented as series that defined eigenfunctions. The resulting strain is identically zero satisfying the boundary conditions at the ends of the layers. An approximation SSS of each cylinder layer is approximated by the finite sum of eigenfunctions. Basing on the method of least squares the algorithm for numerical solution of boundary value problems for multilayer cylinder has been developed. Satisfaction of all conditions is reduced to minimizytion of the generalized quadratic form. The method of analytical calculation of the coefficients of the quadratic form that allows to find the value of unknown coefficients of the high accuracy and to improve the performance of the method has been proposed. For the first time the numerical convergence criterion method was established theoretically. It is shown that the accuracy of satisfaction of the boundary conditions and the conditions of perfect contact is measured by a single number - the minimum quadratic form.