Problem Of Thermal Conductivity Research Articles

Two-dimensional temperature oscillations and elastic thermocyclic stresses in sphere and space with spherical cavity under spatially inhomogeneous heat transfer conditions

The problem of steady-state temperature oscillations and thermocyclic stresses in regions bounded from inside or from outside by spherical surface is considered. To find them, the uncoupled quasistatic cyclic thermoelasticity problem under axisymmetric heat transfer conditions has been solved. The given definition corresponds to a solution that depends on two spatial variables. The thermal conductivity problem is solved by method of separation of variables for three types of boundary condition. Solution of the elastic displacement problem is obtained by using thermoelastic potential of displacements and Papkovich-Neuber solution for isothermal deformation.

Условия теплового взрыва в пластине при конвективно-радиационном теплообмене

The processes of obtaining and storing energy-saturated substances are characterized by energy release in their volume. The intensity of this energy release increases with increasing temperature. The stability of the stationary temperature state of a solid with a temperature-dependent intensity of volumetric energy release is directly related to the conditions of heat transfer of this body with the environment. If the heat energy released in the volume of the body can no longer be diverted into the environment, the steady temperature state of the body becomes impossible. The paper studies the conditions for a thermal explosion in a solid in the form of a plate with a temperature-dependent coefficient of thermal conductivity and convective-radiation heat transfer on its surfaces. The statement of the nonlinear problem of steady-state thermal conductivity in the plate is represented by a system of integral relations. The limits of integration of the integrals included in these relations are the desired functions and parameters which determine the temperature state of the plate. A quantitative analysis of these relationships makes it possible to establish the influence of the parameters which determine the intensity of heat transfer and the dependence of the thermal conductivity of the plate material on the conditions for a thermal explosion with an arbitrary law of variation with temperature of the volumetric power of the energy release in the plate. The results of such an analysis are presented in the framework of a one-parameter model of the stationary theory of thermal explosion

Modified method of direct in problems of thermal conductivity of annular plates

In this paper, to solve the initial boundary value problem of thermal conductivity using a numerical-analytical method - a modified method of lines is proposed. The initial equations of thermal conductivity defined in the cylindrical coordinate system are considered in the spatial formulation, which greatly complicates them. As an object on which they are defined, an annular plate is considered, the overall dimensions of which are commensurate. In the problems of calculating of thermal effects in load-bearing elements the first step is to determine the temperature fields, especially if the overall dimensions of the structures are proportional. Such elements include non-thin annular plates. The boundary conditions are considered in a general form too - these are the conditions for convective heat transfer, which using the passage to the limit, turn into boundary conditions of the first and second types. The application of the modified method of lines to reduce the dimensionality of the initial system of equations of nonstationary thermal conductivity used to determine the temperature fields of the load-bearing elements is shown in this paper.
 The application of the modified method of lines involves solving these initial boundary value problems in two stages. At the first stage, the dimensionality of the initial equations with respect to variable z is reduced. The Bubnov-Galerkin-Petrov projection method is used to reduce the dimensionality. The so-called functions-"caps" are accepted as basic functions, which are related to the lines plotted on the definition domain of the problem. The projection method is also used to reduce the dimension of the initial and boundary conditions, that allows to formulate a reduced initial-limit problem, which is convenient to solve using the numerical finite-difference method, using explicit or implicit difference schemes. The most successful form of writing the original equations was found, which ensures ease of application of dimensionality reduction of the initial system of equations using a modified method of lines. The calculation took into account the impact of the environment. Reduced equations, boundary and initial conditions are obtained. As a result, the reduced problem has a form convenient to its solution by modern numerical methods.

Simulation and experimental study of induction heat treatment of titanium disks

During induction heat treatment (IHT) of titanium disks, the influence of the main electrotechnological parameters, in particular the inductor current, on the temperature of their surfaces was established at an exposure within t = 5 min. In the course of numerical simulation, i.e. when solving the self-consistent boundary value problem of electrodynamics and thermal conductivity for the "inductor – titanium disk" system, the influence of the inductor current in the range 2.7–4.5 kA on the average exposure temperature of the titanium disk was determined, which varied from 800–850°C to the melting temperature (about 1670°C) and above. In order to confirm numerical calculations, experimental studies on IHT in the same temperature range were performed. In addition to the inductor current IIND varying in the range from 2.7 to 4.5 kA, the influence of the corresponding value of the electric power PE in the range from 0.21 to 0.95 kW on the exposure temperature during IHT was considered. Changes in the surface morphology of the titanium disks were studied over a wide temperature range. The initial oxide coating was retained on the surfaces of the samples processed at a minimum exposure temperature of T = 800–850°C. A thick scale of titanium dioxide was also preserved on the surfaces of semi-molten titanium disks. In the IHT exposure range at temperatures from 1000–1050°C to 1500–1550°C, the scale was spontaneously separated from the surfaces of titanium disks; however, a high-strength oxide sublayer was formed under it.

Effect of coupling phase change materials and heat pipe on performance enhancement of Li‐ion battery thermal management system

SummaryA thermal management system (TMS) including heat pipes, heat‐conducting glue, phase change materials (PCM), and micro‐channel plates is built in this research. The heat pipes and heat‐conducting glue can solve the problem of poor thermal conductivity and large internal temperature difference (Td) of TMS including PCM and liquid cooling. The micro‐channel plates can solve the problem that TMS combined with the PCM and the heat pipes is difficult to transfer heat. The designed TMS can cool and heat li‐ion battery. Through the li‐ion battery discharge test experiment by 1C, 2C, and 3C, the li‐ion battery heating power is obtained. The temperature change curve of li‐ion batteries at 1C, 2C, 3C discharge and when ambient temperature (Ta) is −20°C, −10°C, 0°C, and 20°C are studied. When the Ta is lower than 0°C, the designed TMS can heat the li‐ion battery to 20°C, and make the Td inside the li‐ion batteries approach 0°C within a short period.

Математичні моделі визначення температурних режимів у елементах літій-іонних акумуляторних батарей

Удосконалено раніше розроблені та наведено нові математичні моделі визначення та аналізу температурних режимів в окремих елементах літій-іонних акумуляторних батарей, які геометрично описано ізотропними півпростором і простором із внутрішнім джерелом тепла циліндричної форми. Також розглянуто випадки для півпростору, коли тепловиділяючий циліндр є тонким, а для простору, коли він є термочутливим. Для цього з використанням теорії узагальнених функцій у зручній формі записано вихідні диференціальні рівняння теплопровідності з крайовими умовами. Для розв'язування отриманих крайових задач теплопровідності використано інтегральне перетворення Ганкеля і внаслідок отримано аналітичні розв'язки в зображеннях. До цих розв'язків застосовано обернене інтегральне перетворення Ганкеля, яке дало змогу отримати остаточні аналітичні розв'язки вихідних задач. Отримані аналітичні розв'язки подано у вигляді невласних збіжних інтегралів. Для визначення числових значень температури в наведених конструкціях, а також аналізу теплообміну в елементах літій-іонних батарей, зумовленого різними температурними режимами завдяки нагріванню внутрішніми джерелами тепла, зосередженими в об'ємі циліндра, розроблено обчислювальні програми. Із використанням цих програм наведено графіки, які відображають поведінку кривих, побудованих із використанням числових значень розподілу температури залежно від просторових радіальної та аксіальної координат. Отримані числові значення температури свідчать про відповідність наведених математичних моделей визначення розподілу температури реальному фізичному процесу. Програмні засоби також дають змогу аналізувати середовища із внутрішнім нагріванням, зосередженим у просторових фігурах правильної геометричної форми, щодо їх термостійкості. Як наслідок, стає можливим її підвищити, визначити допустимі температури нормальної роботи літій-іонних батарей, захистити їх від перегрівання, яке може спричинити руйнування не тільки окремих елементів, а й всієї конструкції.

Contactless non-destructive thermal control of materials

Physical and mathematical models of the heat transfer process in the reviewed control object are considered. On the basis of the formulated boundary value problem of thermal conductivity (using the Laplace and Hankel transformations) the calculated dependences for calculating the desired values of thermal conductivity and thermal diffusivity are obtained. An algorithm for the functioning of the information-measuring and control system for implementing the developed method for measuring the thermophysical properties of materials is presented.

Influence of Thermoelastic Phenomena on the Energy Conservation in Non-Contacting Face Seals

The purpose of this study was to develop a mathematical model for non-contacting face seals to analyze how their performance is affected by thermoelastic phenomena. The model was used to solve thermal conductivity and thermoelasticity problems. The primary goal was to calculate the values of thermal deformations of the sealing rings in a non-contacting face seal with a flexibly mounted rotor (FMR) for a turbomachine. The model assumes the conversion of mechanical energy into heat in the fluid film. The heat flux generated in the fluid film is transferred first to the sealing rings and then to the fluid surrounding them. Asymmetric distribution of temperature within the sealing rings leads to the occurrence of thermal stresses and, consequently, a change in the geometry of the rings. The model is solved analytically. The distributions of temperature fields for the sealing rings in the cross-sections are calculated using the Fourier-Bessel series as a superficial function of two variables (r,z). The thermoelasticity problems described by the Navier equations are solved by applying the Boussinesq harmonic functions and Goodier’s thermoelastic displacement potential function. The proposed method involves solving various theoretical and practical problems of thermoelasticity in FMR-type non-contacting face seals. The solution of the mathematical model was made use of analytical methods, and the most important obtained results are presented in graphical form, such as the temperature distributions and axial thermal distortions in cross-sections of the rings. The calculated thermal deformations of the sealing rings are used to determine the most important seal performance parameters such as the leakage rate and power loss. The article also presents a multi-criteria analysis of seal rings materials and geometry, which makes it easier to choose the type of materials used for the sliding rings.

IMPACT ASSESSMENT OF MODERNIZATION OF STEAM TURBINE END SEALS ON THE RESIDUAL RESOURCE

Modernization of sealing units with a change of design is carried out in order to improve the technical and economic parameters of steam turbines. It is known from experience of operation of the turbine equipment that, ring cracks in rotors arise in sealing areas. This is due to both seals design features and the accumulation of thermocyclic fatigue. The research is devoted to the study of resource parameters of the high-pressure rotor of the T-250/300-240 turbine equipped with seals of labyrinth, honeycomb and direct-flow type. Numerical experiments were performed on the basis of three-dimensional geometric models of interflow end seals of the high pressure rotor. The finite element method is used to discretize the computational area. The thermal state of the turbine is calculated for the starting mode from the cold state of the metal by solving the boundary value problem of thermal conductivity in a non-stationary setting. The calculation of the stress-strain state of the turbine takes into account the obtained data on temperature stresses and forces from the non-uniformity of temperature fields, as well as centrifugal forces and steam pressure. Differences in the stress-strain state of the rotor for different seal designs are established. It is indicated that after grooving of labyrinth interflow seals to the honeycomb structure, the stress level in the rotor decreased by 8 %, and when grooving to the direct-flow structure — by 21 %. The accumulated damage in the main metal of the turbine was determined using experimental curves of long-term strength of steel 25Cr1Mo1V. The calculation of resource indicators showed that in the transition from the labyrinth to the honeycomb design of the seals, the individual resource of the high-pressure rotor increases by 6.1 %, and in the transition to direct-flow — by 14.4 %. Ref. 10, Fig. 5, Tab. 1.

Thermoelasticity of elastomeric constructions with initial stresses

The article presents an algorithm for solving linked problems of thermoelasticity of elastomeric structural elements on the basis of a moment scheme of finite elements. To model the processes of thermoelastic deformation of structures with initial stresses the incremental theory of a deformed solid is used. At each step of deformation, the stiffness matrix is adjusted using an incremental geometric stiffness matrix. The use of triple approximation of displacements, deformations and volume change function allows to consider the weak compressibility of elastomers. The components of the stress tensor are calculated according to the Duhamel-Neumann law. To solve the problem of thermal conductivity, a thermal conductivity matrix considering the boundary conditions on the surface of a finite element is constructed. A sequential approximation algorithm is used to solve the thermoelasticity problem. At each stage of the solution, the characteristics of the thermal stress state are calculated. Based on the obtained components of stress and strain tensors, the intensity of internal heat sources is calculated as the dissipative energy averaged over the load cycle. To calculate the dissipative characteristics of the viscoelastic elastomer the parameters of the Rabotnov’s relaxation nucleus are used. Solving the problem of thermal conductivity considering the function of internal heat sources allows you to specify the heating temperature of the body. At each cycle of the algorithm, the values of the physical and mechanical characteristics of the thermosensitive material are refined. This approach to solving thermoelastic problems is implemented in the computing complex "MIRELA+". Based on the considered approach, the solutions of a number of problems are obtained. The results obtained satisfactorily coincide with the solutions of other authors. Considering the effect of preload and the dependence of physical and mechanical properties of the material on temperature leads to significant adjustments to the calculated values.

Thermostressed State of a Nozzle Vane from Max Phase Ceramics

Numerical simulation results for the unsteady thermal and stressed states of a nozzle vane under unsteady operation conditions of the gas turbine are presented. The vane and nozzle block duct 3D computer model was constructed. The finite element mesh form, its sizes, and calculation time steps were optimized. The vane and duct contained 1.7 and 3.6 mln elements and 2.3 and 0.8 mln nodes, respectively. The mesh smoothing with corresponding nodal displacement near the transition zones and in the vicinity of curvilinear surfaces was used. The simulation consisted in successive numerical solutions of nonstationary flow problems for a moving medium and heat, transfer, thermal conductivity, and thermoelasticity problems for the vane. The parameters of unsteady thermal loading conditions were defined. The thermal properties of the moving medium and vane material were assumed to be time-dependent. For turbine power improvement conditions, time variations of duct flow parameters, temperatures, thermal stresses in different vane zones, their gradients and rates were analyzed. The essential nonuniformity of the thermostressed vane state and zones of thermal stresses, almost reaching the ultimate strength of the material, were noted. The effect of general heat flow components on the disturbance of thermal stresses was shown. Emphasis was placed upon the appearance of tensile stresses on a heated trailing edge of the vane (most critical zone). The conclusion was drawn regarding a potential application of a MAX phase Ti2ALC us a structural material for the vanes of short-life gas turbines.

Software for motion analysis of devices installed on polymer shock absorbers

A description of the method and program for solving the problem of analyzing the motion of the device installed on the polymer shock absorber is given. The equation of dynamics using a small parameter is used to describe the motion of the device. The values of displacement and rotation angle of the device is obtained by modeling the motion by solving integro-differential equations. The procedures of the Green’s functions building as well as grid methods are used. The influence of temperature on the properties of shock absorbers is taken into account by way of solving the three-dimensional problem of non-stationary thermal conductivity for a shock absorber in the form of a parallelepiped. The Fourier method of dividing variables is used as a method of solving this problem. Temperature distributions by shock absorber volume are set and analyzed. Than temperature values in nonlinear expression are used in the kernel structure in Boltzmann - Volterra equations. The critical mode of motion of the device, when external excitation has a fast growth mode, and temperature range exceeds one hundred Celsius degrees, is analyzed and the dependence of the motion of the center of mass of the device on time and temperature is obtained. The description of the software in which the presented methods of calculation for problems of nonstationary thermal conductivity and device dynamics are realized, is given. The program works in console mode, a window interface is developed for displaying graphs of functions. The examples of the program’s operation both the nonstationary heat conduction problem and for shock absorber’s dynamics are presented.

Analytical study of joint heat transfer between a gasdynamic boundary layer and an anisotropic strip

When solving the problems of coupled heat transfer between viscous flows and streamlined bodies under the conditions of aerodynamic heating of aircraft, it is necessary to overcome significant difficulties. They associated primarily with determining the boundary conditions. The paper investigates the joint (coupled) heat transfer between a heat and gas dynamic boundary layer and an anisotropic strip under conditions of aerodynamic heating based on the obtained analytical solution of the second initial boundary value problem of thermal conductivity in an anisotropic strip with arbitrary boundary conditions. Since the system of equations of the gasdynamic boundary layer is essentially nonlinear, mainly numerical methods are used to solve it. For an incompressible boundary layer near the critical point of a blunt wedge, an analytical solution is obtained to determine the components of the velocity vector, density, temperature, and heat fluxes. The closed-form solution to the conjugate problem was received in the form of a Fredholm integral equation of second kind. The results of numerical experiments are obtained and analyzed.

Coupled heat transfer between a viscous shock gasdynamic layer and a transversely streamlined anisotropic half-space

The purpose of the article is to analytically solve the conjugate problem of heat transfer in a viscous shock layer on a blunt object and thermal conductivity in an anisotropic half-space. A feature of the flow around such bodies near the critical point is that in this case, the boundary layer equations are not satisfied and it is necessary to solve the Navier-Stokes equations together with the equations of continuity, state, and energy, however, in a stationary formulation of an incompressible flow. The problem of coupled heat transfer between a viscous shock gasdynamic layer and anisotropic half-space with the assumption of incompressibility of the gasdynamic flow behind the normal part of the shock wave is posed. An analytical solution to this problem is obtained with a boundary condition at the gas-solid object interface in the form of a parameter – temperature, as well as an analytical solution to the thermal conductivity problem in an anisotropic half-space with the same boundary condition.

Experimental study of the solar water heater integrated with shell and finned tube latent heat storage system

Abstract In this work, a new design of shell and finned tube latent heat storage system was designed, constructed and tested when it was integrated with the flat plate solar water heater. This design not only provides large heat transfer area between the storage material and the heat transfer fluid, but also it overcomes the problem of low thermal conductivity of paraffin wax (PW) by dividing it into tens of thin slices. Three configurations of storage techniques were proposed; where water, PW as phase change material (PCM) and both water and PW were used as thermal storage materials. The proposed systems were tested at different mass flow rates of heat transfer fluid (0.08–0.16 kg/s). The experimental results showed that the highest daily efficiency of 65% was achieved when the configuration that combines the PCM and water storage tank was used. Moreover, this configuration was capable of providing hot water of temperature ranging from 50 to 60.4 °C through 24 h/day. The results of this configuration showed also that the released energy from PW constitutes 52% of the total stored energy; while the remaining energy of 48% was considered backup stored energy that may help in any sudden thermal loads during night.

Determination of thermal diffusivity of the material by numerical-analytical model of a semi-bounded body

A mathematical description of the material thermal diffusivity aт in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical by solving the inverse problem of thermal conductivity. To solve the problem, it is necessary to obtain the temperature values of the unbounded plate as a result of a thermophysical experiment. A plate can be conditionally considered as a semi-bounded body as long as the Fourier number Fo ≤ Foк (Foк ≈ 0.04–0.06). It is assumed that the temperature distribution over cross-section of the heated layer of the plate R is sufficiently described by a power function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating aт in the time interval ∆τ from the dynamics of temperature change T(Rп , τ) of a plate surface with thickness Rп heated under boundary conditions of the second kind. Temperature of the second surface of the plate T(0, τ) is used only to determine the time of the end of experiment τк. The moment of time τк, in which the temperature perturbation reaches the adiabatic surface x = 0, can be set by the condition T(Rп , τк) – T(0, τ = 0) = 0,1 K. The method of approximate calculation of dynamics of changes in depth of the heated layer R by the values of Rп , τк , and τ is proposed. Calculation of a т for the time interval ∆τ is reduced to an iterative solution of a system of three algebraic equations by matching the Fourier number, for example, using a standard Microsoft Excel procedure. Estimation of the accuracy of a т calculation was made by the test (initial) temperature field of the refractory plate with the thickness Rп = 0.05 m, calculated by the finite difference method under the initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rп) at radiation-convective heating. The heating time was 260 s. Calculation of aт, i was performed for 10 time moments τi + 1 = τi + Δτ, τ = 26 s. Average mass temperature of the heated layer for the whole time was T = 302 K. The arithmetic-mean absolute deviation of aт(T = 302 K) from the initial value at the same temperature was 2.8 %. Application of the method will simplify the conduct and processing of experiments to determine the thermal diffusivity of materials.

NUMERICAL IMPLEMENTATION OF THE MODIFIED METHOD OF LINES

An important step in modern combined methods is the application of numerical methods to solve reduced problems. This was the disadvantage of the classical method of lines. The numerical method was used to reduce the measurability (reduction) of the original equations, as a result of which the reduced equations had a complex appearance. This prevented the use of modern numerical methods to solve them. Reducing the measurability of the initial boundary and initial boundary value problems for the equations of the theory of elasticity and thermoelasticity using the projection method [1] retains the form of classical boundary and initial boundary value problems in mathematical physics and requires little adaptation to modern numerical methods [2-6]. This work is devoted to these issues. The application of the modified method of lines can be extended to static problems of elasticity theory and stationary problems of thermal conductivity [7], to problems of steady oscillations of elastic structures, to problems of finding dynamic characteristics (frequencies and forms of natural oscillations), problems of nonstationary thermal conductivity [7] and nonstationary elastic oscillations. objects. Consider the adaptation of modern numerical methods to solve the corresponding reduced problems. It is important in what form it is necessary to present the reduced equations depending on their structure and features of the corresponding numerical method.

Determination of Thermal Diffusivity of Material by the Numerical-Analytical Model of a Semi-Bounded Body

A mathematical description of a material’s thermal diffusivity ae in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical calculation by solving the inverse problem of thermal conductivity. To solve the problem, it is necessary to obtain the temperature values of the unbounded plate as a result of a thermophysical experiment. A plate can be conditionally considered as a semi-bounded body as long as Fourier number Fo ≤ Foe (Foe = 0.04–0.06). It is assumed that the temperature distribution over a cross-section of the heated layer of the plate with thickness R is sufficiently described by a power-like function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating ahc in time interval Δτ from the dynamics of temperature change T(Rp, τ) of a plate surface with thickness Rp heated at boundary conditions of the second kind. Temperature T(0, τ) of the second surface of the plate is used only to determine end time τe of the experiment. Moment of time τe, at which the temperature perturbation reaches adiabatic surface x = 0, can be set by the condition T(Rp, τe) – T(0, τ = 0) = 0.1 K. An approximate method of calculating the dynamics of changes in depth of heated layer R by values of Rp, τe, and τ is proposed. The calculation of ahc for time interval Δτ is reduced to an iterative solution of a system of three algebraic equations by matching the Fourier number, for example, using a standard Microsoft Excel procedure. Estimation of the accuracy of calculation of ahc at radiation-convective heating was performed using the initial temperature field of the refractory plate with thickness Rp = 0.05 m, calculated by the finite difference method under initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rp). The heating time was 260 s. Calculation of ahc, i has been performed for ten time moments τi + 1 = τi + Δτ, Δτ = 26 s. Average mass temperature of the heated layer for the entire time was τe T = 302 K. The arithmetic-mean absolute deviation of ae (T = 302 K) from the initial value at the same temperature was 2.8%. Application of the method will simplify conducting and processing experiments to determine thermal diffusivity of materials.

Different Techniques of Solar Rooftop Combo-PV/T System Implementation: Materials and Installations

Sun radiation been utilized to produce clean and environmental-friendly solar energy. Beside has been used as heat energy, it also has been used as electricity. Solar photovoltaic and thermal (PV/T) collector is a combo system that utilize solar PV technology and solar heater technology all together in one simply module. At the time PV system is irradiated with solar energy, the cell temperature increases significantly. As the heat that extracted is passing through a heat extracting fluid in such way under the module, then this heat is transferred become hot air and water. This paper aims to reveal several materials that been used to design solar combo-PV/T collector modules and their gains and drawbacks, and the installation onto households’ rooftops. Water-based and nanofluids-based PV/T modules are two technologies that being considered in this paper. All the comparisons that have been made turned out to match the assumption that the greater the difference of temperature between ambient temperature and temperature of the cells of the module, the less efficient the electrical efficiency and electrical output of the PV module and also to overcome the problem of low thermal conductivity. The considered water-nanofluids PV/T solar collector modules then put into account for better performance to improve the raised problem of thermal conductivity.

Thermal Force Loading of a Physically Nonlinear Three-Layer Circular Plate

The article considers the effect of a heat flux of constant intensity on a circular, three-layer, nonsymmetrically thick plate subjected to the action of an axisymmetric local loading. The temperature field in the plate was calculated by the formula obtained in solving the problem of thermal conductivity by averaging thermophysical parameters over the thickness of the packet. The materials of the layers deform physically nonlinearly. For describing the kinematics of the packet, the hypotheses of a broken normal are adopted. In thin carrying layers, the Kirchhoff hypotheses are valid. In the normally uncompressed relatively thick filler, the hypothesis on rectilinearity and incompressibility of the deformed normal holds. The work of the filler in the tangential direction is taken into account. The formulation of the corresponding boundary-value problem is given. Equilibrium equations are obtained by the variational Lagrange method. Boundary conditions on the plate contour are formulated. The solution of the boundary-value problem is reduced to finding three unknown functions: of deflection, shear, and of radial displacement. For these functions, an inhomogeneous system of ordinary nonlinear differential equations is written out. A method of rigid Il′yushin solutions is applied for its solution. Iteration analytical solutions are obtained in Bessel functions at circular and annular loadings. Their parametric analyses are made at different local loadings and for the hinged support of the plate contour. The influence of temperature and nonlinearity of the materials of the layers on the displacements in the plate is investigated numerically.