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