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Документ Analysis of free vibration of porous power-law and sigmoid functionally graded sandwich plates by the R-functions method(Shahid Chamran University of Ahvaz, 2023) Kurpa, Lidiya; Shmatko, Tetyana; Awrejcewicz, Jan; Timchenko, Galina; Morachkovska, IrynaInvestigation of free vibration of porous power and sigmoid-law sandwich functionally graded (FG) plates with different boundary conditions is presented in this paper. The FG sandwich plate includes three layers. The face layers are fabricated of functionally graded material (FGM) and middle layer (core) is isotropic (ceramic). Imperfect sigmoid FG sandwich plates with even and linear-uneven porosities and nonporous core layer are studied. Developed approach has been realized in the framework of a refined theory of the first-order shear deformation theory (FSDT) using variational methods and the R-functions theory. The analytical expressions are obtained for calculating the elastic characteristics with the assumption that the values of Poisson's ratio are the same for constituent FGM materials. For rectangular plates, the obtained results are compared with known results and a good agreement is obtained. Vibration analysis of a complex-shaped porous sandwich plate made of FGM has been performed. The effect of various parameters on the dynamic behavior of the plate, such as the type and values of porosity coefficients, power index, lay-up scheme, types of FGM, has been studied.Документ Динамічний аналіз функціонально-градієнтних пористих сигмовидних сендвич пластин(Національний технічний університет "Харківський політехнічний інститут", 2023) Курпа, Лідія Василівна; Шматко, Тетяна Валентинівна; Лінник, Ганна Борисівна; Морачковська, Ірина Олегівна; Тимченко, Галина МиколаївнаВ роботі розглянуто проблему дослідження вільних коливань функціонально-градієнтних (ФГ) пористих сигмовидних пластин типу сендвіч, які можуть мати складну геометричну форму та різні типи закріплення. Для розв'язання поставленої задачі використано варіаційно-структурний метод (RFM), який поєднує теорію R-функцій та варіаційний метод Релея-Рітца. Математичну постановку задачі виконано в рамках деформаційної теорії пластин першого порядку(FSDT. Розглянуто пластини, зовнішні шари яких вироблено із функціонально-градієнтних матеріалів (ФГМ), а заповнювач є ізотропним. Для різних моделей розподілення пор (сигмовидне рівномірне та нерівномірне) виведені формули для обчислення ефективних властивостей ФГМ. Числові результати для прямокутних пластин порівняно з відомими результатами, отриманими за допомогою інших методів. Досліджено власні коливання пластин зі складною формою плану. Отримані результати представлені у вигляді таблиць та графіків. Проаналізовано вплив об’ємної долі кераміки, різних видів ФГМ та коефіцієнту пористості на власні частоти коливань пластини.Документ Stability of steady states with regular or chaotic behaviour in time(Wydawnictwo Politechniki Łódzkiej, 2019) Mikhlin, Yuri V.; Goloskubova, Natalyia S.; Shmatko, Tatyana V.Документ Дослідження вільних коливань функціонально-градієнтних пологих оболонок методом r-функцій(Дніпровський національний університет імені Олеся Гончара, 2019) Шматко, Тетяна ВалентинівнаFree vibration analysis of laminated functionally graded shallow shells is fulfilled for subjects of the complex plan form by the R-functions method. The different combinations of the material for face sheet layers and core are considered. The effective properties of FGM are calculated according to power law. Numerical results for shallow shells of the complex form are presented in the plots and tables.Документ Free vibration analysis of FGM shell with complex planform in thermal environments(Wydawnictwo Politechniki Łódzkiej, 2019) Awrejcewicz, Jan; Kurpa, Lidiya; Shmatko, TetyanaSummary. In the present study free vibrations of FGM shallow shells of an arbitrary planform in thermal environment are investigated via R-functions method (RFM). First-order shear deformation theory of shallow shells is employed. Material properties are assumed to be temperature-dependent and expressed as nonlinear functions of temperature. The generic material properties are not only functions of temperature, but also functions of thickness direction. It is supposed that material properties vary through thickness according to a power-law distribution of the constituent’s volume fraction. The developed method is based on the combined applications of the R-functions theory, variational Ritz’s method. A comparison of the obtained results with available ones is carried out for rectangular plates and shallow shells. Vibration of shell panels with complex planform and different boundary conditions including mixed ones are studied. Solution structures and related admissible functions for shells with complex planform have been constructed by the R-functions theory. The effect of the temperature rise, geometry of the shell, material properties and constituent volume fraction index is examined.Документ Вільні коливання багатошарових циліндричних панелей з функціонально-градієнтними шарами(Інститут прикладних проблем механіки і математики імені Я. С. Підстригача НАН України, 2019) Курпа, Лідія Василівна; Шматко, Тетяна ВалентинівнаThe R-functions theory and variational Ritz’s method is employed to research free vibrations of the laminated shallow shells with functionally graded layers. Mathematical formulation has used classical and Timoshenko’s theories. Created software is applied to investigate laminated FGM cylindrical shallow shells of the complex plan form and different boundary conditions. Effects of different geometrical and mechanical parameters on natural frequencies have been investigateДокумент Vibration analysis of laminated functionally graded shallow shells with clamped cutout of the complex form by the Ritz method and the R-functions theory(Brazilian Association of Computational Mechanics, 2019) Kurpa, Lidiya; Shmatko, Tetyana; Awrejcewicz, JanThe R-functions theory and Ritz approach are applied for analysis of free vibrations of laminated functionally graded shallow shells with different types of curvatures and complex planforms. Shallow shells are considered as sandwich shells of different types: a) face sheets of the shallow shells are made of a functionally graded material (FGM) and their cores are made of an isotropic material; b) face sheets of the shallow shells are isotropic, but the core is made of FGM. It is assumed that FGM layers are made of a mixture of metal and ceramics and effective material properties of layers are varied accordingly to Voigt’s rule. Formulation of the problem is carried out using the first-order (Timoshenko’s type) refined theory of shallow shells. Different types of boundary conditions, including clamped, simply supported, free edge and their combinations, are studied. The proposed method and the created computer code have been examined on test problems for shallow shells with rectangular planforms. In order to demonstrate the possibility of the developed approach, novel results for laminated FGM shallow shells with cut of the complex form are presented. Effects of different material distributions, mechanical properties of the constituent materials, lamination scheme, boundary conditions and geometrical parameters on natural frequencies are shown and analyzed.Документ Nonlinear vibrations of functionally graded shallow shells of a complex planform in thermal environments(CongressLine Ltd., Hungary, 2017) Awrejcewicz, Jan; Kurpa, Lidiya; Shmatko, TatianaGeometrically nonlinear vibrations of FGM shallow shells of an arbitrary planform subjected to thermal environment are investigated with the use of R-functions theory and variational methods. Nonlinear firstorder shear deformation shallow shells are employed. Material properties are assumed to be temperaturedependent and varying along the thickness direction accordingly to Voigt’s law. The developed method is based on the combined applications of R-functions theory, variational Ritz’s method, procedure by Bubnov-Galerkin, and Runge-Kutta’s approach. The effect of the temperature rise, geometry of the shell, and constituent volume fraction index is examined. A comparison of the obtained results with available ones is also carried out for rectangular plates and shallow shells.Документ Дослідження стійкості багатошарових пластин з отворами складної форми за допомогою теорії R-функцій та варіаційних методів(Інститут прикладних проблем механіки і математики імені Я. С. Підстригача НАН України, 2018) Курпа, Лідія Василівна; Ткаченко, Вікторія Валеріївна; Шматко, Тетяна ВалентинівнаThe laminated plates with cuts of a complex form are studied with meshless approach, based on combined application of the R-functions theory and variational methods. The proposed method is developed for thin plates of an antisymeric form along thickness. Mathematical formulation is presented within the framework of classical nonlinear theory of plates using Kirgoff-Love’s hypothesis. In order to investigate the laminated plates with a complex cut and different boundary conditions, the corresponding solution structures and admissible functions were constructed. The software was developed and tested on many problems. In particular, the obtained results were compared with available ones for a cross three-layered plate with free rectangular cut. For plates with cuts of a complex form effect of different geometrical and physical parameters was stydied. Various types of fastening, geometry of the plate and different materials properties are considered. The nondimensional buckling load, instability regions and response curves are presented for plates with complex form of cut.Документ Застосування теорії R-функцій для дослідження нелінійних коливань функціонально градієнтних пологих оболонок з урахуванням температурного середовища(Інститут прикладних проблем механіки і математики імені Я. С. Підстригача, 2018) Курпа, Лідія Василівна; Шматко, Тетяна ВалентинівнаGeometrically nonlinear vibrations of FGM shallow shells of an arbitrary plan-form subjected to thermal environment are investigated with the use of R-functions theory and variational methods. Nonlinear first-order shear deformation of shallow shells is employed. Material properties are assumed to be temperature dependent and varying along the thickness direction according to Voigt’s law. The effect of the temperature rise, shell geometry, and constituent volume fraction index is examined. A comparison of the obtained results with the available ones is carried out for rectangular plates and shallow shells.