Yu. V. Yurin (Bauman Moscow State Technical University) :
Articles:
539.3 Asymptotic theory of thermocreep for multilayer thin plates
Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2014-4-1836
The suggested thermocreep theory for thin multilayer plates is based on analysis of general three dimensional nonlinear theory of thermalcreep by constructing asymptotic expansions in terms of a small parameter being the ratio of a plate thickness and a characteristic length. Here we do not introduce any hypotheses on a distribution character for displacements and stresses through the thickness. Local problems were formulated for finding stresses in all structural elements of a plate. It was shown that the global (averaged by the certain rules) equations of the plate theory were similar to equations of the Kirchhoff–Love plate theory, but they differed by a presence of the three-order derivatives of longitudinal displacements. The method developed allows to calculate all six components of the stress tensor including transverse normal stresses and stresses of interlayer shear. For this purposes one needs to solve global equations of thermal creep theory for plates, and the rest calculations are reduced to analytical formulae use.
Dimitrienko Y., Gubareva E., Yurin Y. Asymptotic theory of thermocreep for multilayer thin plates. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 18-36
Dimitrienko Y. I. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Zakharov A. A. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2021-3-4261
A coupled problem of high-speed aerothermodynamics and internal heat and mass transfer in heat-shielding structures of reentry spacecraft made of ablative polymer composite materials is considered. To determine the heat fluxes in the shock layer of the reentry vehicle, the chemical composition of the atmosphere is taken into account. The mathematical formulation of the conjugate problem is formulated and an algorithm for the numerical solution is proposed. An example of the numerical solution of the problem for the reentry spacecraft Stardust is presented. It is shown that taking into account chemical reactions in the gas flow around the surface of the reentry vehicle is essential for the correct determination of the gas temperature in the boundary layer. It is also shown that the developed numerical method for solving the problem makes it possible to determine the parameters of phase transformations in a heat-shielding structure depending on the heating time, in particular, it allows calculating the pore pressure field of gaseous products of thermal decomposition of a polymer composite, which, under certain conditions, can lead to material destruction.
мДимитриенко Ю.И., Коряков М.Н., Юрин Ю.В., Захаров А.А., Сборщиков С.В., Богданов И.О. Сопряженное моделирование высокоскоростной аэротермодинамики и внутреннего тепломассопереноса в композитных аэрокосмических конструкциях. Математическое моделирование и численные методы, 2021, № 3, с. 42–61.
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2017-3-4970
The study centers around a technique developed for modeling damageability of laminated composite structures with defects of the delamination type under cyclic loading. The procedure consists of 3 stages iteratively repeated in time: finite element simulation of a macroscopic stress-strain state of the structure with defects; simulation of the microscopic stress-strain state near the defects; modeling the damage accumulation in the matrix, which connects the layers of reinforcing fibers near the defect. The model takes into account the curvilinear anisotropy of the composite material in the structure of complex geometric shapes. The study gives an example of a numerical calculation of a fragment of a composite structure of a helicopter carrier blade, taking into consideration the defect of the delamination type. The results suggest that there is a possibility of using the developed technique for damageability modeling in complex composite structures. The finite-element solution of the macroscopic problem is found by means of the SMCM software platform developed at the Scientific and Educational Center for Supercomputer Engineering Modeling and Software Development (SEC “SIMPLEX”) of Bauman Moscow State Technical University.
Dimitrienko Yu.I. ,Yurin Yu.V. Finite element modeling of damageability and durability of composite structures with local delaminations .Маthematical Modeling and Computational Methods, 2017, №3 (15), pp. 49–70
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University), Yakhnovskiy A. D. (Bauman Moscow State Technical University), Baymurzin R. R. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2020-1-327
The problem of multilevel model development for calculating of an elastic property of polymer composite materials with a complex reinforcement structure at high temperatures is considered. It is assumed that thermal destruction processes take place in the matrix and fibers at high temperatures. In order to take into account the change in the elastic properties of the composite depending on the temperature and heating time, a 3-level structural model of the composite is proposed. At the lower level mono-fibers and a matrix consisting of 4 phases, the ratio between which changes when heated are considered. At this level, the analytical relations proposed earlier in the works of Yu.I. Dimitrienko. At the next level of the model, a unidirectional composite is considered, consisting of bundles of monofilaments and a matrix. To calculate elastic properties at this level, the method of asymptotic averaging is used, and a finite element algorithm for solving local problems of the theory of thermoelasticity arising in this method. At the 3rd structural level of the model, composites with complex reinforcement structures, in particular, fabric composites, are considered. The method of asymptotic averaging is also used to calculate the elastic properties of the composite at this level. For the numerical calculation of the elastic characteristics of polymer composites at high temperatures, specialized software has been developed that operates under the control of the SMCM software package created at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Systems of the Bauman Moscow State Technical University. The article provides examples of the application of the developed multilevel model and software for textile composites based on an epoxy matrix and glass fibers. The values of all components of the tensor of the elastic moduli of the composite are calculated, which vary depending on the heating program of the composite. The microstress fields in the composite are obtained. A comparison is made of the fields of microstresses and effective elastic constants at normal temperatures, with similar values obtained using the ANSYS software package, which has been modified to enable the calculation of effective elastic constants in accordance with the proposed model. A very good agreement was obtained between the calculation results, both of the effective constants and of the microstresses fields, which allows us to speak of the high accuracy of the developed software.
Димитриенко Ю.И., Юрин Ю.В., Сборщиков С.В., Богданов И.О., Яхновский А.Д., Баймурзин Р.Р. Конечно-элементное моделирование упругих свойств тканевых полимерных композитов при высоких температурах. Математическое моделирование и численные методы. 2020. № 1. с. 3–27
539.36 Finite element modeling of natural vibrations of shell structures
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2023-3-317
A numerical algorithm for solving the problem of natural vibrations for thin-walled shell structures based on the finite element method is proposed. A software module has been developed as part of the SMCM software package, which implements the proposed numerical algorithm. A test problem was solved for natural vibrations of a cylindrical shell structural element. A comparative analysis of eigenfrequencies and eigenmodes was carried out with similar results obtained using a two-dimensional shell solution in the ANSYS software package, as well as with the results of solving a three-dimensional problem for natural vibrations in the ANSYS software package.
Димитриенко Ю.И., Юрин Ю.В., Богданов И.О., Маремшаова А.А. Конечно-элементное моделирование собственных колебаний оболочечных конструкций. Математическое моделирование и численные методы, 2023, № 3, с. 3–17.
539.3 Finite element modeling of non-stationary thermal buckling of composite structures
Dimitrienko Y. I. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University), Anokhin D. S. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2024-1-3854
The problem of modeling for buckling analysis of the composite structures due to nonstationary thermal effects on them, taking into account the temperature dependence of the properties of the composite components, is considered. Systems of equations are formulated for calculating the basic and varied states of the structure. A classification of buckling analysis problems is proposed. The application of the finite element method to determine the critical temperature and the corresponding buckling mode of a structure is described. A local generalized eigenvalue problem was formulated and the proposed model was verified using the SMCM software package developed at the Simplex Research Center of Bauman Moscow State Technical University, as well as using ANSYS. It is shown that the results of calculating the eigenforms and eigenvalues in the test problem coincide quite well.
Димитриенко Ю.И., Богданов И.О., Юрин Ю.В., Маремшаова А.А., Анохин Д. Конечно-элементное моделирование нестационарной термоустойчивости композитных конструкций. Математическое моделирование и численные методы, 2024, № 1, с. 38–54.
539.3 Finite element modeling of temperature fields in thin-walled multilayer anisotropic shells
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2023-1-4363
The problem of developing a model for calculating temperature fields in thin-walled multilayer curvilinear-anisotropic thin shells of arbitrary geometric shape, including composite ones, is considered. As a rule, to solve this problem, a specific coordinate notation of the equations of the theory of heat conduction is used, which creates certain difficulties for calculating complex composite shells. In this paper, it is proposed to use an invariant record of the variational formulation of problems in the theory of heat conduction, followed by the application of the finite element algorithm procedure. As a result, a matrix differential equation is derived for determining the temperature field at the nodes of a finite element mesh. A software module has been developed for the finite element solution of the problem of non-stationary thermal conductivity of shells. The module functions as part of the SMCM software package, created at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Systems, Bauman Moscow State Technical University (REC SIMPLEX). An example of solving the problem of calculating a non-stationary temperature field in a cylindrical shell with longitudinal-transverse reinforcement is given. Comparison of numerical simulation with similar calculations in the ANSYS software was carried out, which showed the high accuracy of the proposed method: the relative deviation of the results does not exceed 0,5%
Димитриенко Ю.И., Юрин Ю.В., Коряков М.Н., Маремшаова А.В. Конечно-элементное моделирование температурных полей в тонкостенных многослойных оболочечных элементах конструкций. Математическое моделирование и численные методы, 2023, No 1, с. 43–63
Dimitrienko Y. I. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Zakharov A. A. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2019-2-1534
The coupled task of aero-thermo-mechanics of heat-loaded structures from thermally destructive polymer composite materials under the influence of an intense aerodynamic flow is considered. The mathematical formulation of the conjugate problem is formulated and algorithms for the numerical solution of this problem are proposed. The algorithms are based on an iterative solution of three types of problems: aerodynamics, internal heat and mass transfer, and thermomechanics of the modeling aircraft structure. An example of a numerical solution to the problem for an aircraft structural element in the form of a blunt cone is presented. It is shown that the effect of high temperatures of aerodynamic heating of the structure leads to thermal degradation of the polymer composite material, which results in the generation of a large amount of gases in the pores and thermo-chemical shrinkage, which under certain conditions can lead to premature destruction of the heat-loaded composite structure.
Димитриенко Ю.И., Коряков М.Н., Юрин Ю.В., Захаров А.А. Конечно-элементное моделирование термонапряжений в композитных термодеструктирующих конструкциях при аэродинамическом нагреве. Математическое моделирование и численные методы, 2019, № 2, с. 15–34.
539.3 Finite element simulation of the rock stress-strain state under creep
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2015-3-101118
A model for calculation of a rock stress-strain state considering creep is suggested. The algorithm for finite element solving the three-dimensional creep problem using finite-difference scheme of Euler's method with respect to time is presented. The specialized software is developed allowing the computer to build 3D-models of rock areas based on the initial series of 2D images, obtained with the seismic data, and to perform finite element calculation of variations in rock strain-stress state with time. Numerical simulation of rock stress-strain state was conducted on the example of a zone of the Astrakhan oil and gas field. It was found that there occurs rock mass rising in some points, and in the other points it can slope down with time. The creep rate of different layers is not the same — the highest values of the creep rate are realized in the layers of clay and sand, filled with fluid, which have the most notable creep properties. The developed algorithm and software for numerical simulation proved to be quite effective and can be applied to the study of rock stress-strain state.
Dimitrienko Y., Yurin Y. Finite element simulation of the rock stress-strain state under creep. Маthematical Modeling and Coтputational Methods, 2015, №3 (7), pp. 101-118
Dimitrienko Y. I. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2021-4-1744
A model of constitutive relations for elastic-plastic composites with cubic symmetry of properties is proposed. This class includes a significant number of composite materials: dispersed-reinforced composites, which have an ordered rather than a chaotic reinforcement system, as well as some types of spatially reinforced composites. To build a model of nonlinear constitutive relations, a tensor-symmetry approach was used, based on the spectral expansions of stress and strain tensors, as well as the spectral representation of nonlinear tensor relations between these tensors. The deformation theory of plasticity is considered, for which the tensor-symmetric approach is used, and specific models are proposed for plasticity functions that depend on the spectral invariants of the strain tensor. To determine the model constants, a method is proposed in which these constants are calculated based on the approximation of deformation curves obtained by direct numerical solution of three-dimensional problems on the periodicity cell of elastic-plastic composites. These problems arise in the method of asymptotic averaging of periodic media. To solve problems on a periodicity cell, a finite element method and special software was used that implements solutions to problems on periodicity cells, developed at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Packages of Bauman Moscow State Technical University. An example of calculating the constants of a composite model using the proposed method for a dispersed-reinforced composite based on a metal matrix is considered. Also, the verification of the proposed model for various ways of multiaxial loading of the composite was carried out with direct numerical simulation. It is shown that the proposed microstructural model and the algorithm for determining its constants provide a sufficiently high accuracy in predicting the elastic-plastic deformation of the composite.
Димитриенко Ю.И., Сборщиков С.В., Димитриенко А.Ю., Юрин Ю.В. Микроструктурная модель деформационной теории пластичности квази-изотропных композиционных материалов. Математическое моделирование и численные методы, 2021, № 4, с. 17–44.
Dimitrienko Y. I. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2022-1-1541
Within the framework of the deformation theory of plasticity under active loading, a model of constitutive relations for elastic-plastic composites belonging to the class of transversally isotropic materials is proposed. The theory of spectral expansions of stress and strain tensors and the spectral representation of nonlinear tensor functions for transversely isotropic media are used to develop a nonlinear constitutive relations. Specific models of plasticity functions are proposed, depending on the spectral invariants of the strain tensor. To determine the model constants, a method is proposed in which these constants are calculated based on the approximation of deformation curves obtained by direct numerical solution of three-dimensional problems on the periodicity cell of elastic-plastic composites. Problems on the periodicity cell are formulated using the method of asymptotic averaging of periodic media. The numerical solution of problems on the periodicity cell is carried out using the finite element method within the framework of software developed at the Scientific and Educational Center "Supercomputer Engineering Modeling and Development of Software Systems" of Bauman Moscow State Technical University. An example of numerical calculation of the constants of a composite model using the proposed method for a unidirectionally reinforced composite based on carbon fibers and an aluminum alloy matrix is given. Examples of verification of the proposed model for different loading trajectories of the composite in a 6-dimensional stress space are given. It is shown that the proposed microstructural model and the algorithm for determining its constants provide a sufficiently high accuracy in predicting the elastic-plastic deformation of transversely isotropic composites
Димитриенко Ю.И., Сборщиков С.В., Димитриенко А.Ю., Юрин Ю.В. Микроструктурная модель деформационной теории пластичности трансверсально-изотропных композитов. Математическое моделирование и численные методы, 2022, № 1, с. 15–41.
539.3 Modeling of effective elastic–plastic properties of composites under cyclic loading
Dimitrienko Y. I. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2020-4-326
A method is proposed for calculating the effective elastic–plastic properties of composites under cyclic loading. The technique is based on the application of the method of asymptotic averaging of periodic structures for the case of materials with elastic-plastic properties under cyclic loading. A model of the deformation theory of plasticity by A.A. Il’yushin – V.V. Moskvitin under cyclic loading using the Masing model for changing the plasticity function under cyclic deformation. Local problems of the theory of plasticity for the periodicity cell of a composite material, as well as averaged problems of the theory of anisotropic plasticity under cyclic loading are formulated. A software module has been developed for the finite element solution of local problems on the periodicity cell. The software of the SMCM complex developed at the Scientific and Educational Center "Supercomputer Engineering Modeling and Development of Software Systems" of the Bauman Moscow State Technical University was used. The SMCM complex is designed for finite element modeling of the properties of composite materials. Numerical calculations of the elastic-plastic properties of dispersed-reinforced composites based on an aluminum alloy and SiC ceramic particles have been carried out. Calculations have shown that the developed technique can be used to predict cyclic deformation diagrams of elastic-plastic composites in a wide range of loading conditions, as well as to design new composite materials with specified properties.
Димитриенко Ю.И., Сборщиков С.В., Юрин Ю.В. Моделирование эффектив-ных упруго–пластических свойств композитов при циклическом нагружении. Ма-тематическое моделирование и численные методы, 2020, № 4, с. 3–26.
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Yakhnovskiy A. D. (Bauman Moscow State Technical University), Baymurzin R. R. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2020-3-2246
The problem of calculating the integral characteristics of the viscoelasticity of composite materials is considered, based on information on similar characteristics of the composite components and its microstructure. An algorithm is proposed for predicting the effective relaxation and creep kernels of composites with an arbitrary reinforcement microstructure. The algorithm is based on the Fourier transform application and the inverse Fourier transform, as well as the method of asymptotic averaging for composites under steady-state polyharmonic vibrations. The algorithm uses exponential relaxation and creep kernels for the initial components of the composite. The basis of the computational procedure of the proposed algorithm is the finite element solution of local viscoelasticity problems over the composite periodicity cell. The result of the algorithm application is the determination of the exponential relaxation and creep kernels parameters for composite materials, which makes it possible to obtain a problem solution in a completely closed form. As an example, a numerical simulation of the viscoelastic-tic characteristics of unidirectionally reinforced carbon /epoxy composites has been carried out. It is shown that the developed algorithm allows one to obtain effective relaxation and creep kernels of the composite with high accuracy, without oscillations, which, as a rule, ac-company the methods of inverting Fourier transforms.
Димитриенко Ю.И., Юрин Ю.В., Сборщиков С.В., Яхновский А.Д., Баймурзин Р.Р. Моделирование эффективных ядер релаксации и ползучести вязко-упругих композитов методом асимптотического осреднения. Математическое моделирование и численные методы, 2020, № 3, с. 22–46.
539.3 Multiscale modeling of thin multilayer composite plates with solitary defects
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2016-4-4766
We developed a multiscale model of deformation of thin multilayer composite plates with solitary defects. The model is based on the asymptotic analysis of general threedimensional equations of deformable solid mechanics. The general solution of threedimensional equations is reduced to the solution of two classes of problems: problems for thin plates without defects and local three-dimensional problems in the vicinity of the defect with the condition of damping solution at the distance from the defect. A solution of local problems is used for averaged problems of the multilayer plates theory, which enables us to find an explicit solution for all six components of the stress tensor in the field without the defect, based on the solution of the averaged two-dimensional problem of the plate theory. In the defect area the general solution is a superposition of the two solutions: the one obtained on the basis of the plates theory and local three-dimensional mechanics problems. The paper gives an example of a numerical finite element solution of the local mechanics problem for the three-layer composite plate with a solitary defect in the middle layer. Moreover, findings of the research show that the defect impact is localized in its immediate vicinity and the maximum transverse stress concentration is achieved in the vicinity of the defect peak.
Dimitrienko Y., Yurin Y. Multiscale modeling of thin multilayer composite plates with solitary defects. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 47-66
539.3 Numerical modeling of deformation and strength of sandwich composite structures with defects
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Fedonyuk N. N. (Krylov State Research Centre, St. Petersburg)
doi: 10.18698/2309-3684-2016-3-323
The purpose of this research was to develop a multilevel model for multiscale deformation of three-layer (sandwich) structures made of polymeric composite materials such as plates with a foam based filler. We took into account the micromechanical processes of deformation and damageability in the matrix and reinforcing filler and foam, as well as macroscopic defects such as non-impregnation of the composite skins. First, we did a finite element modeling of stress-strain state, damageability and destruction of the sandwich plates with skins made of hybrid carbon fiber composites, with different types of defect such as non-impregnation, under the flexural uniform pressure. Then we found the characteristic features of the deformation and damageability process in this type of composite structures. Finally, we developed a method which can be used to calculate the deformation, damageability and destruction of sandwich plates made of polymer composite materials applied in various industries: shipbuilding, aviation, rocketry.
Dimitrienko Y., Yurin Y., Fedonyuk N. Numerical modeling of deformation and strength of sandwich composite structures with defects. Маthematical Modeling and Coтputational Methods, 2016, №3 (11), pp. 3-23
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Yegoleva E. S. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2017-4-4859
The article proposes a numerical method for solving the inverse three-dimensional problems of recovering the fields of loads acting upon composite structural elements based on the results of the experimental diagnostics of structural displacements on a certain surface. The problems of this type arise when creating the systems of the built-in diagnostics of structural movements and intelligent composite structures. The restored field of loads acting upon the parts of the outer surface of the composite structure is used to calculate the stress-strain state and forecast the structural life. The proposed method uses an alternating algorithm for solving the inverse problems of restoring loads in the problem of elasticity theory, in combination with the finite element method for solving the direct problems in the theory of elasticity. We consider an example of solving the inverse problem of restoring loads acting on the structural elements made from layered fibrous composite materials.
Dimitrienko Yu.I., Yurin Yu.V., Egoleva E.S. Numerical solution of inverse three-dimensional problems of recovering the loads acting upon composite structural elements. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 48-59
539.3 Timoshenko-type asymptotic theory for thin multi-layered plates shells
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2018-1-1640
The paper presents a new modification of asymptotic theory describing thin multi-layered shells with finite shear rigidity. It is based on asymptotic analysis of general threedimensional equations from the elasticity theory for multi-layered bodies. This modification allows us to derive averaged equations from a Timoshenko-type plate theory. We identified the small geometrical parameter and used it to carry out our asymptotic analysis. We stated local elasticity theory problems which may be solved analytically. We show that when only the dominant terms of asymptotic expansions are taken into account, an asymptotic theory will result in the averaged plate equations of the Kirchhoff — Love type. When taking into account those terms that follow the dominant ones in asymptotic series in a self-similar way as compared to the previous approximation, an asymptotic theory will lead to Timoshenko-type averaged equations. At the same time, theoretical accuracy of the resulting truncated asymptotic solution is as high as that of the solution according to a Kirchhoff — Love type theory. The asymptotic theory modification that we developed makes it possible to use explicit analytical expressions to compute all six stress tensor components for a multi-layered plate with a high degree of accuracy. We used our method to perform a numerical simulation of stresses and displacements in a multi-layered plate subjected to uniform pressure that causes the plate to bend. Numerical computations show that our Timoshenko-type asymptotic theory provides a similarly high accuracy of computing flexural, shear and lateral stresses as compared to a three-dimensional finite element solution over a very fine mesh and a Kirchhoff — Love-type asymptotic theory. A Timoshenko-type theory will provide a better result for computing buckling than a Kirchhoff — Love-type theory, especially for relatively short plates. When the displacement is longitudinal, a Timoshenko-type theory will only provide a good result for elongated plates.
Димитриенко Ю.И., Юрин Ю.В. Асимптотическая теория типа Тимошенко для тонких многослойных пластин. Математическое моделирование и численные методы, 2018, № 1, с. 16-40