and Computational Methods

doi: 10.18698/2309-3684-2016-1-105122

The article considers the concept of applying the multidimensional continuum model to one of the main problems emerging in the theory of large scale data array treatment i.e. forecasting the dynamics of data cluster change. The concept is based on the model of multidimensional continua in spaces of high dimensionality (more than three) earlier developed by the authors. The model includes the integral conservation laws, which are reformulated for informational data clusters, as well as the model of motion kinematics and cluster deformation. The model of deformable multidimensional cluster is developed. The movement of the cluster in multidimensional data space includes translational and rotational motion and uniform tension-compression strain. The system of differential tensor equations describing the dynamics of the deformable multivariate cluster motion over time is formulated. A numerical algorithm for solving the system of differential equations for the ellipsoidal model of multidimensional cluster is worked out. An example of the developed model application for predicting the dynamics of economic data (data on goods purchases in a large supermarket) is considered. The results of forecasting the data on purchases of different consumer groups are shown.

Dimitrienko Y., Dimitrienko O. A model of multidimensional deformable continuum for forecasting the dynamics of large scale array of individual data. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 105-122

doi: 10.18698/2309-3684-2015-4-7591

This article deals with the finite-element RKDG method (Runge-Kutta Discontinuous Galerkin) and its application for numerical integration of three-dimensional system of equations of ideal gas on unstructured grids. By means of the described algorithm we solved two test tasks. For each task we conducted the analysis and compared the task solution with well-known analytical solutions or with tabular data. We also give error assessment in the solution.

Dimitrienko Y., Koryakov M., Zakharov A. Application of RKDG method for computational solution of three-dimensional gas-dynamic equations with non-structured grids. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 75-91

doi: 10.18698/2309-3684-2014-1-3656

The theory of thin constructive-orthotropic plates with a two-periodic structure was suggested. Examples of such structures are honeycomb sandwich panels and backed plates. The theory is based on equations of a three-dimensional elasticity theory with the help of asymptotic expansions in terms of a small parameter being the ratio of a plate thickness and a characteristic length without introducing 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 are similar to equations of the

Dimitrienko Y., Gubareva E., Sborschikov S. Asymptotic theory of constructive-orthotropic plates with two-periodic structures. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 36-56

doi: 10.18698/2309-3684-2022-2-2862

The problem of development a theory for calculating the stress-strain state of thin multilayer elastic plates, for which linearized slip conditions are specified at the interface between the layers, is considered. The solution of this problem is constructed using an asymptotic analysis of the general equations of the 3-dimensional theory of elasticity with the conditions of non-ideal contact of the layers. The asymptotic analysis is carried out with respect to a small geometric parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local quasi-one-dimensional problems of the theory of elasticity with slip are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged equations of elastic equilibrium of multilayer plates is presented, taking into account the slippage of the layers. It is shown that due to the effect of slippage of layers, the system of averaged equations of the theory of multilayer plates has an increased - 5th order of derivatives, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff-Love plates. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 6 components of the stress tensor in the layers of the plate. As a special case, the problem of calculating the stress-strain state of a 4-layer plate under uniform pressure bending with one slip coefficient is considered. A complete analytical solution of this problem is obtained, including explicit expressions for all non-zero components of the stress tensor. A numerical analysis of the solution of the averaged problem for a composite plate is carried out, in which the layers are unidirectional reinforced fibrous materials oriented at different angles. A comparative analysis of the influence of the fiber reinforcement angles and the slip coefficient of the layers on the displacement of the plate and the distribution of stresses in the layers was carried out. It is shown that the problem of bending a plate with slip admits the existence of a spectrum of critical values of the slip coefficient, when passing through which the displacements and stresses in the layers of the plate change significantly, and these critical values depend on the angle of reinforcement of the composite layers.

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория многослойных тонких упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы, 2022, № 2, с. 30–64

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

doi: 10.18698/2309-3684-2023-2-3366

The problem of development of a theory for calculating the stress-strain state of thin multilayer elastic plates in the moment (micropolar) theory, is considered. The solution of this problem is built using an asymptotic analysis of the general equations for a 3-dimensional quasi-static problem of the moment theory of elasticity. The asymptotic analysis is carried out with respect to a small parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local problems of the moment theory of elasticity are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged system of equations for multilayer plates is presented. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 9 components of the stress tensor and the moment stress tensor (in general) in the plate. As a special case, the problem of calculating the stress-strain state of a centrally symmetrical hingedly fixed plate when bending under the action of a uniformly distributed pressure. A complete analytical solution of this problem for all non-zero components of the stress tensor and the moment stress tensor is obtained. A numerical analysis of the solution of the problem for a single layer plate for the stress tensor is carried out, basing on the obtained expressions. A comparative analysis of the obtained results with similar calculations for the classical theory of elasticity is carried out, with revealing of similarities and differences for all components of the stress tensor.

Димитриенко Ю.И., Бойко С.В. Асимптотическая теория многослойных тонких микрополярных упругих пластин. Математическое моделирование и численные методы, 2023, № 2, с. 33–66.

doi: 10.18698/2309-3684-2019-1-326

The problem of deformation of thin two-layer plates, for which a slip condition is speci-fied at the interface between the layers, instead of the classical case of ideal contact, is considered. The method of asymptotic analysis of the general equations of the 3-dimensional theory of elasticity is used to solve this problem under the influence of transverse pressure, longitudinal and shear forces on the end surfaces. Asymptotic analysis is performed using a small geometric parameter representing the ratio of thickness to the characteristic length of the plate. Recurrent formulations of local quasi-one-dimensional problems of elasticity theory with slippage are obtained. For these problems, explicit analytical solutions are obtained. The averaged equations of elastic equilibrium of a two-layer plate with slippage of layers are derived. It is shown that, due to slippage, the order of the averaged equations of the theory of plates increases to 5 orders of magnitude, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff – Love plates. Additional boundary conditions to this 5th order system are formulated and its analytical solution is obtained for the case of a rectangular plate under the influence of uniform pressure. A numerical analysis of the solution of the averaged problem is carried out. It is shown that the presence of layer slippage significantly increases the deflection of the plate in comparison with the conditions of ideal contact of the layers.

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория тонких двухслой-ных упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы. 2019. № 1. с. 3–26.

doi: 10.18698/2309-3684-2015-1-6782

The article presents a suggested method of numerical finite-element solving the ‘hole ovalization’ problem. This method can be applied for experimental development of advanced aviation materials with the aim of determining structure element resistance against deforming with stress concentrators, mainly, connectors. The method is based on three-dimensional finite element solution of the problem of lastoplastic deformation of plates with a hole under crushing. It is appropriate for reduction of xperimental studies and replacing them by the numerical experiments. The Ilyushin model of small lastoplastic deformations has been used. The results of numerical simulation of a threedimensional stress-strain state of elastoplastic plates under crushing are presented as well as results of experimental nvestigations of deforming plates of Al-alloy 163. It is shown that the results of numerical and experimental modeling for deforming plates under crushing agree quite well.

Dimitrienko Y., Gubareva E., Sborschikov S., Erasov V., Yakovlev N. Computational modeling and experimental investigation of elastic-plastic plates deforming under crushing. Маthematical Modeling and Coтputational Methods, 2015, №1 (5), pp. 67-82

doi: 10.18698/2309-3684-2014-3-324

In the article we propose an algorithm for the numerical simulation of conjugate gasdynamic and thermomechanical processes in composite structures of high-speed aircraft. The algorithm allows calculating all parameters of the three-dimensional gasdynamic flow near the surface of the aircraft, heat exchange on the surface, heat and mass transfer processes in the internal structure of thermodestructive polymer composite, as well as processes of composite construction thermodeformation, including the effects of changes in the elastic characteristics of the composite, variable thermal deformation, shrinkage caused by thermal degradation, building up interstitial gas pressure in the composite. An example of numerical simulation of conjugated processes in a model composite construction of high-speed aircraft illustrates the possibilities of the proposed algorithm.

Dimitrienko Y., Koryakov M., Zakharov A., Stroganov A. Computational modeling of conjugated gasdynamic and thermomechanical processes in composite structures of high speed aircraft. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 3-24

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.

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

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

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.

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.

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

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.

doi: 10.18698/2309-3684-2014-2-2848

We propose a method for calculating effective viscoelastic properties of composite materials under steady-state cyclical vibrations. The method is based on asymptotic averaging of periodic structures and finite-element solution of local problems of viscoelasticity in periodicity cells of composite materials. We provide examples of numerical simulation of viscoelastic properties for composites with unidirectional reinforcement, and of calculations of complex tensors of stress concentration in a periodicity cell. The paper presents a comparative analysis of dependencies of loss tangent of complex composite elasticity

modulus on vibration frequencies obtained through FEA calculations and rough mixed formulae. We show that rough mixed formulae, often used for calculating dissipative properties of composite materials, can yield appreciable calculation errors.

Dimitrienko Y., Gubareva E., Sborschikov S. Finite element modulation of effective viscoelastic properties of unilateral composite materials. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 28-48

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

doi: 10.18698/2309-3684-2017-1-3254

The article considers the modeling results of incompressible layered composites with finite strains deformation according to the individual layers characteristics. The article proposes an asymptotic averaging method version for layered nonlinearly elastic incompressible composites with finite deformations and periodic structure. We are using a universal representation of the defining relations for incompressible composite layers, proposed by Yu.I. Dimitrienko, which allows us to simulate simultaneously for a complex of various nonlinear elastic media models characterizedby the choice of a pair of energy tensors. We proved that if all composite layers are incompressible, the composite as a whole is also an incompressible, but anisotropic, medium. The article considers the problem of laminated plate uniaxial stretching from incompressible layers with finite deformations. Using the developed method, we calculated the effective deformation diagrams connecting the averaged Piola — Kirchhoff stress tensors components and the strain gradient, as well as the stress distribution in the composite layers.

The developed method for calculating effective deformation diagrams and stresses in composite layers can be used in the design of elastomeric composites with specified properties.

Dimitrienko Y., Gubareva E., Kolzhanova D., Karimov S. Incompressible layered composites with finite deformations on the basis of the asymptotic averaging method. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 32-54

doi: 10.18698/2309-3684-2018-2-7095

The finite element method is used to simulate the nonisothermal flow of non-Newtonian viscous fluids in complex geometries. The Carreau-Yasuda model of a non-Newtonian fluid is considered, in which the dependence of the viscosity coefficient on the second invariant of the strain rate tensor has a power form. A variational formulation of the problem of the motion of a non-Newtonian fluid for a plane case is obtained. The iteration algorithm of Newton-Raphson is used to solve the Navier-Stokes equations system, and the Picard iteration algorithm is used to solve the energy equation. The problem of the movement of a polymer mass in a mold of complex variable cross section in the presence of an uneven temperature field is considered. With the help of finite element modeling, a numerical analysis of the effect of various parameters on the movement of a liquid and the heat transfer of a polymer material at different values of external pressure was carried out. It is shown that the nature of the motion of a non-Newtonian fluid essentially depends on the rheological properties of the fluid and the characteristics of the geometric shape, which must be taken into account in technological processes of plastics processing.

Димитриенко Ю.И., Шугуан Ли Конечно-элементное моделирование неизотермического стационарного течения неньютоновской жидкости в сложных областях. Математическое моделирование и численные методы, 2018, № 2, с. 70–95.

doi: 10.18698/2309-3684-2022-3-4770

A microstructural model of layered elastic-plastic composites based on the anisotropic flow theory is proposed. The model represents the effective constitutive relations of the transversally isotropic theory of plastic flow, in which the model constants are determined not experimentally, but on the basis of approximations of the deformation curves of composites obtained by direct numerical solution of problems on the periodicity cell for basic loading trajectories, which arise in the method of asymptotic averaging. The problem of identifying the constants of this composite model is formulated; for the numerical solution of this problem, methods of optimizing the error functional are used. The results of numerical simulation by the proposed method for layered elastic-plastic composites are presented, which showed good accuracy of approximation of numerical strain diagrams.

Димитриенко Ю.И., Черкасова М.С., Димитриенко А.Ю. Микроструктурная модель анизотропной теории течения для упруго-пластических слоистых композитов. Математическое моделирование и численные методы, 2022, № 3, с. 47–70.

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.

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.

doi: 10.18698/2309-3684-2020-2-2645

The paper is devoted to the development of a method for calculating the nonlinear dielectric properties of composites with a periodic structure. Methods for predicting of the nonlinear dielectric properties of composites play an important role in the design of dielectric materials with specified properties, in particular for heterogeneous ferroelectrics, which are widely used to create various devices and electrical devices, for example, to create memory storage devices for computers. A quasi-static problem of the distribution of an electric charge in an inhomogeneous polarizable medium with a periodic structure and nonlinear dielectric properties is considered. To solve this nonlinear problem, the asymptotic homogenization method proposed by N.S. Bakhvalov, E. Sanchez-Palencia, B.E. Pobedria. As a result, local nonlinear problems of electrostatics on the periodicity cell are formulated, an algorithm for calculating effective nonlinear constitutive relations for dielectric properties, and an averaged problem for a composite with effective properties are proposed. For the case of a composite with a layered structure, the solution of local problems is obtained, and effective defining relations for the nonlinear dielectric properties of the composite are constructed. It is shown that a laminated composite is a transversely isotropic nonlinear dielectric material if it is isotropic materials. A numerical example of calculating the nonlinear properties of a 2-layer composite based on barium titanate and ferroelectric ceramic varicond VK4 is considered. A model is proposed that describes the nonlinear dependence of the dielectric constant of these materials on the vector of the electric field strength. It is shown that the nonlinear dependence of the dielectric constant tensor of the composite on the strength vector differs significantly for the direction of the field in the plane of the layers and in the transverse direction. It is shown that the developed technique can serve as a basis for designing nonlinear dielectric composite materials with anisotropic properties.

Димитриенко Ю.И., Губарева Е.А., Зубарев К.М. Моделирование нелинейных диэлектрических свойств композитов на основе метода асимптотической гомогенизации. Математическое моделирование и численные методы. 2020. № 2. с. 26–45

doi: 10.18698/2309-3684-2019-3-1938

The paper describes investigating a mathematical model of the process of a non-Newtonian liquid multiscale filtration in three-dimensional periodic porous media by asymptotic homogenization. The so-called local problem of filtration in a single pore is formulated as well as the local non-Newtonian-viscous defining relationship. An iterative finite element method is developed for solving a local problem in 1/8 periodicity cell, based on the physical symmetry of the structure. The distribution of the components of the filtration rate, pressure micro-fields and non-Newtonian viscosity in a single pore is calculated. On the basis of Darcy's law the nonlinear filtration law is analyzed, the effect of liquid rheological properties on permeability is shown.

Димитриенко Ю.И., Шугуан Ли. Моделирование проницаемости неньютоновских жидкостей в трехмерных композитных структурах на основе метода асимптотической гомогенизации. Математическое моделирование и численные методы. 2019. № 3.c.19–38.

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.

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.

doi: 10.18698/2309-3684-2023-4-4763

A mathematical model of phase transformations in steel alloys during resistance spot welding is proposed, taking into account all stages of the process: from heating and partial melting of the metal, which cause irreversible physical and chemical transformations of the steel microstructure, to the cooling stage, during which solidification and “return” formation of alloy phases occurs . The model describes changes in the 3D microstructure of a steel alloy during heating and subsequent cooling with the formation of ferritic and austenitic structures. An algorithm for calculating model constants using a special procedure for solving the inverse problem is proposed, as well as an algorithm for numerically solving the problem of predicting changes in the elastic properties of steel during the welding process, which includes finite element 3D modeling using the SMCM software package, developed at the Department of Computational Mathematics and Mathematics physics" of Bauman Moscow State Technical University. An example of numerical simulation using the proposed model and algorithm for a steel alloy is given.

Димитриенко Ю.И., Сальникова А.А., Орешникова Е.А. Моделирование изменения микроструктуры и упругих свойств сплавов в процессе контактной точечной сварки. Математическое моделирование и численные методы, 2023, № 4, с. 47–63

doi: 10.18698/2309-3684-2021-1-330

The problem of calculating the stress–strain state of a composite laminated panel during cylindrical bending under conditions of finite deformations is considered. To solve the problem, the method of asymptotic averaging of periodic nonlinear elastic structures with finite deformations was applied, which was developed in detail earlier in the previous works of the authors. A feature of this problem is the use of universal models of constitutive relations for isotropic components of the composite, as well as for the composite as a whole, which is a transversely isotropic nonlinear elastic medium. Universal models make it possible to obtain solution of problems within the framework of a single solution algorithm simultaneously for several classes of models of nonlinear elastic media corresponding to different conjugate pairs of stress tensors–deformation. An analytical solution is obtained for the problem of cylindrical bending of a composite panel. A numerical analysis of the solution is carried out using the example of a composite, the periodicity cell of which consists of two layers: polyurethane and rubber. It is shown that for thin panels the stresses, both averaged and true, practically do not depend on the class of the model of the constitutive relations. At the same time, for thicker panels, the stresses differ significantly for different classes of models of composite layers.

Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Моделирование напряжений в композитной нелинейно упругой панели при цилиндрическом изгибе. Математическое моделирование и численные методы, 2021, № 1, с. 3–30.

doi: 10.18698/2309-3684-2018-4-324

The paper considers the problem of thermal convection in the melt zone during unidirectional crystallization of a metal axisymmetric sample with a free surface boundary (liquid bridge) under microgravity. The mathematical problem includes a system of Navier-Stokes equations in the Boussinesq approximation with an equation for the mass transfer of impurity particles in a liquid, as well as equations for the motion of the liquid free surface.

A numerical algorithm for solving the problem based on the vortex and current function method, linearization of the problem, and finite-difference approximation using the variable direction method to solve the difference system of linear equations is developed. The physical parameters of thermal convection processes in the melt zone are calculated. It is shown that taking into account the motion of the free boundary near the crystallizing liquid phase leads to a change in the distribution of impurities near the curing surface, which in turn causes a change in the characteristics of the cured material.

Димитриенко Ю.И., Леонтьева С.В. Моделирование термоконвективных процессов при однонаправленной кристаллизации сплавов с учетом движения свободных границ. Математическое моделирование и численные методы. 2018. № 4. с. 3–24.

doi: 10.18698/2309-3684-2022-4-330

The general asymptotic theory of thin multilayer shells developed by the authors earlier in Part 1 of this study is applied to cylindrical anisotropic thermoelastic shells. It is shown that for cylindrical shells the general theory is substantially simplified: general two-dimensional averaged thermoelasticity equations for multilayer shells are obtained. These equations are similar to the classical equations of cylindrical shells in the Kirchhoff-Love theory, but they are obtained in a completely different way: on the basis of only an asymptotic analysis of the general three-dimensional equations of the theory of thermoelasticity. No hypotheses regarding the distribution of displacements or stresses over the thickness are used in this theory, which makes it logically consistent. In addition, the developed theory makes it possible to obtain explicit analytical expressions for all 6 components of the stress tensor in cylindrical anisotropic shells. Explicit expressions are obtained for all tensor constants included in these stress formulas. An example of calculating thermal stresses in a cylindrical composite shell with axisymmetric bending due to the combined action of external pressure and one-sided non-stationary heating is given. An example of a layered-fiber 4-layer shell with different angles of helical winding of reinforcing fibers is considered. It is shown that the developed one allows one to study in detail such complex effects as the formation of significant transverse thermal stresses during heating, which significantly exceed the level of interlayer shear stresses, which are traditionally considered the most dangerous for layered composites.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е., Белькова К.В., Борин Д.М. Моделирование термонапряжений в композитных оболочках на основе асимптотической теории. Часть 2. Расчет цилиндрических оболочек. Математическое моделирование и численные методы, 2022, № 3, с. 3–30

doi: 10.18698/2309-3684-2018-3-114132

The previously developed general asymptotic theory of thin multilayer shells is used for the case of cylindrical shells. The ratios are presented in explicit analytical form for all six components of the stress tensor in a thin multilayer elastic cylindrical shell, depending on the deformations, curvatures of the middle surface of the shell, as well as their derivatives along the longitudinal coordinates. The obtained formulas make it possible to calculate all the distributions of the components of the stress tensor over the thickness in a cylindrical shell after finding solutions to the two-dimensional problem of the theory of KirchhoffLyav shells. An example is given of the calculation of stresses in a cylindrical composite shell underaxisymmetric bending by pressure. To calculate stresses by these formulas, only a differentiation of displacements is required - a deflection and two displacements of the middle surface of the shell, for which an analytical solution is obtained.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование напряжений в тонких композитных цилиндрических оболочках на основе асимптотической теории. Математическое моделирование и численные методы, 2018, № 3, с. 114–132.

doi: 10.18698/2309-3684-2021-2-1537

The aim of this work is to find the constitutive relations for a layered elastoplastic composite according to the flow theory using the method of asymptotic averaging. This goal is achieved by developing an algorithm for solving the problem of the theory of plastic flow for a layered composite material, taking into account various characteristics and properties of these layers of the material, followed by visualizing the result in the form of effective plasticity diagrams connecting the components of averaged stress tensors and components of averaged strain tensors.

Димитриенко Ю.И., Губарева Е.А., Черкасова М.С. Моделирование деформирования слоистых периодических композитов на основе теории пластического течения. Математическое моделирование и численные методы, 2021, № 2, с. 15–37.

doi: 10.18698/2309-3684-2018-4-7292

The paper considers a model of effective constitutive relations for a transversal-isotropic incompressible composite with finite strains. The model belongs to the so-called class of universal models that connect several pairs of energy stress and strain tensors simultaneously. A method is proposed for separating coupled problems of micro- and macroscopic deforming composites with finite strains that arise when the method of asymptotic homogenization (AH) of periodic structures is used. The method is based on the application of the effective constitutive relation model as an approximation dependence of the results of numerical simulation of the composite deformation curves obtained using the exact AH method. To find the elastic constants of the transversely isotropic composite model the method of minimizing the deviation of the approximation of deformation diagrams from the AH diagrams is used for a series of problems of standard deforming at finite strains. To solve minimization problems, the Nelder—Mead method is used. The results of numerical simulation by the proposed method for nonlinear elastic layered composites are presented, which showed good approximation accuracy, achieved due to application of the proposed method for the separation of coupled problems of micro- and macroscopic deforming.

Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Модели-рование эффективных характеристик трансверсально-изотропных несжимаемых композитов с конечными деформациями. Математическое моделирование и чис-ленные методы, 2018, № 4, с. 72–92.

doi: 10.18698/2309-3684-2015-2-322

The article presents a model of microstructure of two-phase monocrystalline intermetallic alloys in the form of a periodic structure of the hexagonal type, as well as a mathematical model of elastic-plastic deformation of monocrystalline alloy, based on the method of asymptotic smoothing periodic structures. Deformation plasticity theory under loading is used for the phases with due regard for the effect of their damage level during loading. For numerical calculations of the developed model the heat-resistant monocrystalline alloy of the type VKNA-1V was used. Finite element calculations of deformation and fracture micromechanical processes in monocrystalline alloy of the type VKNA-1V were carried out. It was found that under tension maximum values of phase damagability, which determine the beginning of the alloy micro-fracture zone, are achieved in the areas adjacent to the phase interface and in areas of maximum curvature of the geometric shape of the phases. Calculations of heat-resistant alloy strain diagrams in plastic range are proved to be consistent with experimental data.

Dimitrienko Y., Gubareva E., Sborschikov S., Bazyleva O., Lutsenko A., Oreshko E. Modeling the elastic-plastic characteristics of monocrystalline intermetallic alloys based on microstructural numerical analysis. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 3-22

doi: 10.18698/2309-3684-2016-2-323

The purpose of this article is to propose a model of deformation of elastic-plastic composite materials with periodic structures with an allowance for fault probability of the composite phases. The model is based on a variant of the deformation theory of plasticity with the active loading. To simulate the effective characteristics of elastic-plastic composites, we applied the method of asymptotic homogenization of periodic structures. For numerical solution of linearized problems on the periodicity cell we offered the finite elements method using SMCM software medium developed at the Scientific-Educational Center of Supercomputer Engineering Modeling and Program Software Development of the Bauman Moscow State Technical University. We provide the research with the examples of numerical computations for dispersion-reinforced metal composites (aluminum matrix filled with SiC particles). Finally, we present the results of numerical modeling of deformation processes, damage accumulation and metal-composite destruction.

Dimitrienko Y., Gubareva E., Sborschikov S. Multiscale modeling of elastic-plastic composites with an allowance for fault probability. Маthematical Modeling and Coтputational Methods, 2016, №2 (10), pp. 3-23

doi: 10.18698/2309-3684-2017-2-327

A mathematical model for the multiscale process of filtration of weakly compressible liquids and gases in periodic porous media is proposed with reference to the process of composite material production based on the RTM method. Using the method of asymptotic averaging made it possible to formulate the so-called local filtration problems for a single pore and the global problem of unsteady filtration of weakly compressible liquids. Two models of a weakly compressible fluid are considered: classical and generalized. The classical model is based on the Musket’s equation of the state, which requires initial constant values for fluid pressure and density to be preset. The generalized model is based on the same equation, but requires presetting only the initial fluid density, using the unknown hydrostatic pressure instead of the initial constant liquid pressure. The results of simulation of the impregnation process of a of filler material sample by a binder are presented using the two models of a weakly compressible liquid.

Dimitrienko Yu. I.Bogdanov I.O. Multiscale modeling of liquid binder filtration processes in composite structures manufactured by RTM Маthematical Modeling and Coтputational Methods, 2017, №2 (14), pp. 3-27

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

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

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

doi: 10.18698/2309-3684-2019-4-100116

The problems of identifying individual customers are formulated based on the analysis of large amounts of data on cash receipts in a large supermarket. Models of behavior of various categories of individual customers in the supermarket are developed. A computational algorithm is proposed for solving the problems of identifying individual customers using panel data from cash receipts. The algorithm is universal, since it does not use any personal data about the buyer, but is based on an analysis of only buying activity, calculated on the basis of panel data on cash receipts. The algorithm allows you to identify groups of customers, as well as with a certain probability, an individual customer. As an example of the application of the developed models and computational algorithms, commodity checks from the X5Retail Group company supermarket chain for a certain period of time were used.

Димитриенко Ю.И., Котельникова А.В. Задачи идентификации индивидуальных покупателей на основе анализа больших объемов панельных данных о кассовых чеках. Математическое моделирование и численные методы, 2019, № 4, с. 100–116.

doi: 10.18698/2309-3684-2020-4-84110

An asymptotic theory of thermoelasticity of multilayer composite shells is proposed, the derivation of the basic equations of which is based on the asymptotic expansion in terms of a small geometric parameter of three-dimensional thermoelasticity equations. This method was previously developed by the authors for thin composite plates, and in this article it is applied to thin-walled shells of an arbitrary frame. According to the developed method, the original three-dimensional problem of thermoelasticity decomposes into a recurrent successor of one-dimensional local problems of thermoelasticity and an averaged two-dimensional problem of thin shells. For local problems of thermoelasticity, analytical solutions are obtained, which make it possible to close the averaged formulation of the problem of the theory of shells with respect to 5 unknown functions: longitudinal displacements, deflection, and two shear forces. It is shown that the averaged problem for multilayer shells coincides with the classical system of equations for Kirchhoff–Love shells, however, it is more substantiated, since the asymptotic theory does not contain any assumptions regarding the pattern of the distribution of permutations and stresses over thickness. In addition, the asymptotic theory makes it possible to calculate all the stresses in the shell, without solving any additional problems, but only by differentiating the averaged displacements.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование термона-пряжений в композитных оболочках на основе асимптотической теории. Часть 1. Общая теория оболочек. Математическое моделирование и численные методы, 2020, № 4, с. 84–110.

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

doi: 10.18698/2309-3684-2014-1-None

Aleksandrov A., Dimitrienko Y. Математическое и компьютерное моделирование — основа современных инженерных наук. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 3-4