ГЕОМЕТРИЧНО НЕЛІНІЙНИЙ ЗГИН ФУНКЦІОНАЛЬНО-ГРАДІЄНТНИХ ПОЛОГИХ ОБОЛОНОК НА ПРУЖНІЙ ОСНОВІ

The paper considers the problem of geometrically nonlinear bending of shallow elements of structures made of functional gradient materials (FGM) under the influence of various transverse loads. The shallow shells under consideration can have arbitrary plan shapes and be in contact with an elastic base of the Winkler-Pasternak type. The mathematical formulation is made within the framework of classical geometric-nonlinear theory. To linearize the nonlinear system of differential equations of equilibrium, the sequential load method in combination with Newton's method is used. The effective mechanical properties of FGM vary along the thickness and are calculated according to the power law. The use of the theory of R-functions made it possible to build the necessary systems of coordinate functions for shells with arbitrary geometry and support conditions. The proposed approach is implemented in software, tested, and applied to solve the problems of bending shallow shells of complex plan shapes with holes. The influence of the elasticity coefficients of the base, the gradient index in the distribution of metal and ceramic particles, as well as other parameters on the deflections of structural elements, was studied.