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The paper formulates the nonlinear problem of steady-state heat conduction at the constant electric potential difference on the surfaces of a plane dielectric layer with the temperature-dependent heat conduction coefficient and electrical resistivity. A fixed temperature value is set on one of the layer surfaces, and the convective heat exchange with the ambient medium occurs on the opposite surface. The formulation of the problem is transformed into integral ratios, which allows the calculation of the temperature distribution over the layer thickness, governed both by the monotonic and nonmonotonic function. The quantitative assay of the temperature state of a layer of a polymer dielectric made of amorphous polycarbonate is given as an example, as well as the analysis of nonuniformity of the absolute value of electric field intensity over the thickness of this layer.

For the reliable operation of a high-voltage DC cable with high-temperature superconducting current-carrying conductors with a sufficiently high difference in electrical potentials, it is necessary to maintain a fixed temperature state not only of the conductors but also of other cable elements, including the electrical insulation layer. In this layer, despite the high electrical resistivity of its material, which can be polymer dielectrics, Joule heat is released. The purpose of this study was to build a mathematical model that describes the temperature state of an electrical insulation layer made in the form of a long hollow circular cylinder, on the surfaces of which a constant potential difference of the electric field is set. Within the study, we consider an alternative design of a cable with central and external annular channels for cooling liquid nitrogen. Using a mathematical model, we obtained integral relations that connect the parameters of the temperature state of this layer, the conditions of heat transfer on its surfaces, and the temperature-dependent coefficient of thermal conductivity and electrical resistivity of an electrical insulating material with a given difference in electrical potentials. A quantitative analysis of integral relations is carried out as applied to the layer of electrical insulation of the superconducting cable. The results of the analysis make it possible to assess the possibilities of using specific electrical insulating materials in cooled high-voltage DC cables under design, including superconducting cables cooled with liquid nitrogen

The paper presents a mathematical model describing the distribution of stresses in a hollow cylinder caused by a one-dimensional steady temperature field occurring at an assigned fixed difference of electrical potentials on the surfaces of a cylinder made of polymer dielectric. A quantitative analysis of the model makes it possible to identify the effects of temperature dependences of the specific electrical resistivity, thermal conductivity coefficient, and mechanical characteristics of the dielectric material on the thermo-stressed state of the cylinder. The model reflects the conditions of work of the dielectric layer of a single fiber high DC-voltage cable. Estimates are given of the conditions for the occurrence of a thermal breakdown of a dielectric cylindrical layer and of the cylinder’s boundary state of stress which occurs in the case of planned or emergency nullification of electric potential difference. The results of such analysis can be useful for identifying possible areas of application of polymer dielectrics in various electrical devices.

The nonstationary problem of thermal conductivity in a two-layer thermal protection coating is under consideration. The solution is given for a metal hemispherical shell with axisymmetrical distribution of the heat flux density over the outer surface of the coating.

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

On the basis of the presented differential form of the mathematical model describing the steady-state thermal conductivity in the layer of electrical insulation of a single-core high-voltage dc cable, the variation form of this model is elaborated. The analysis of stationary points of the functional obtained is carried out and a critical combination of the main parameters preceding the thermal breakdown of this layer has been determined. To find the best trial functions allowable for the functional, which approximate the stationary temperature distribution in the polymer layer of electrical insulation, residuals are compared that arise when substituting these functions into an original differential equation.

Based on a constructed mathematical model that describes, at alternating voltage, the steady thermal state of a plane homogeneous layer of a polymer dielectric with a nonmonotonous dependence of dielectric properties on temperature, an integral relation has been obtained that accounts for nonuniform distribution of electric field strength over the layer thickness and for possible change of the thermal conductivity of a dielectric with temperature. A quantitative analysis has been made of the integral relation, intervals have been established that correspond to the existence of stable temperature distribution in the dielectric layer, and a condition was determined under which a state of thermal breakdown of the dielectric occurs.

A mathematical model describing the thermoelastic characteristics of a composite reinforced by anisotropic ellipsoidal inclusions is proposed. The model is applied to estimating the thermoelastic properties in the case of strengthening the material with single-walled carbon nanotubes. The estimates obtained by the self-consistent method and dual variational formulation of a thermoelasticity problem for an inhomogeneous solid body are compared.

A complex mathematical model of the thermo-deformed state of a circular cylindrical shell reinforced by stringers for a vertically installed cryogenic tank during its filling was constructed. The model describes the local deviation of the shell generatrix due to the nonuniform temperature distribution and the action of an axial compressive load. Relations describing the temperature distribution along the shell and its radial displacement were obtained. The quantitative analysis of these relations was carried out. Estimations of the shell generating line deviation from the original rectilinear form, which can decrease the level of the critical axial loads and cause the shell buckling, are presented.

Single-walled carbon nanotubes (SWCNT) are a seamless cylindrical structure formed by folding a single-walled graphene sheet. Nanotubes are of great interest due to their unique thermo-mechanical characteristics. Such objects can be applied in the energy, rocket and aerospace engineering as a filler of advanced structural composite materials.One of the most important SWCNT characteristics is a complex of its elastic properties. The most reliable way to determine the elastic characteristics of a nanotube is to conduct an experiment. However, due to high cost and lengthy time of the experiment completion, it is more preferable to use mathematical modeling of these properties of SWCNT with which the paper deals.The paper proposes a mathematical model that is constructed by methods of generalized continuum mechanics. To use these methods, the nanotube, on the assumption of its transverse isotropy, is presented simultaneously as a cylindrical shell and a continuous circular rod.As a result of consideration of various stress-strain states of the shell and rod, are obtained relations to establish a quantitative relation between the elements of the matrix of compliance coefficients of a rod that simulates a single-walled carbon nanotube, and the elastic characteristics of graphene taken as its basis in the isotropy plane. Also, the constructed mathematical model allows us to define the influence of the nanotube configuration (chiral indices) on its elastic characteristics.The paper presents the application results of the model proposed, using as an example, a SWCNT with chirality indices (7, 0), for which were constructed mutually inverse matrices of compliance and elasticity coefficients that completely describe the elastic properties of the nanotube.Using the proposed mathematical model significantly reduces the time and cost for estimation of the elastic characteristics of the SWCNT, that is especially important at the design stage of new materials based on them.

A mathematical model is constructed to describe a steady one-dimensional temperature distribution in a hollow circular cylinder made of a polymer dielectric with a constant difference in the electric field potentials on the cylinder surfaces. Based on systematized data on the dependence of the thermal conductivity and electrical conductivity of polymer materials used as dielectrics in engineering on temperature, a qualitative analysis of the model is performed for a prescribed density of the heat flux supplied to the inner surface of the cylinder and intensity of convective heat transfer on the outer surface The results obtained in the study allow one to determine the area of applicability of polymer dielectrics used in various electrical engineering applications, including electrical insulation of DC high-voltage cables, and to formulate the conditions for the thermal breakdown of the cylindrical layer of the dielectric.

For spherical layer of insulation surface, we give a solution of the linear time-dependent heat transfer problem with axially symmetric distribution of the heat flux density over the layer exterior surface. The insulation material has anisotropic heat conduction properties characterized by the ratio of the thermal conductivities along and across the layer. The solution obtained is used for qualitative estimate of the effect of this ratio on the leveling of temperature distributions for non-uniform heating specific for conditions of an axially symmetric hypersonic flow past a hemispherical blunting.

The purpose of the study was to build a nonlinear mathematical model governing the steady-state one-dimensional temperature distribution in the electrical insulation layer made in the form of a long hollow circular cylinder whose surfaces are given a constant electric field potential difference. By means of this model, integral ratios, which connecting the parameters of the temperature state of this layer, the heat transfer conditions on its surfaces and the temperature-dependent heat conduction coefficient and the electrical resistivity of the electrical insulating material with a given electrical potential difference, were built. A quantitative assay of the integral ratios is carried out with regard to the electrical insulation layer of a superconducting cable cooled by liquid nitrogen.

We have obtained estimates of equivalent heat conductivity coefficients of nanostructured elements on the basis of the constructed mathematical model of the thermal energy transfer in a single-layer and a multilayer nanotube surrounded by a homogeneous material in the composite matrix. The values of such coefficients permit conditional replacement of nanotubes by solid anisotropic fibers, which makes it possible, in predicting the effective heat conductivity coefficient of a composite modified by carbon nanotubes, to use the known methods developed in detail to fit fibrous composites.

On the base of the simplified analytical model for the bimetallic shell of the combustion chamber of the reusable liquid-propellant rocket engine, the analysis of the loading of connectors between the shell walls and the accumulation of inelastic deformations in the inner wall material was carried out. The results obtained can be used for evaluation of the maximal number of engine operation cycles.

A two-level mathematical model is constructed to describe the thermomechanical interaction between structural elements of a composite (nanoclusters formed by randomly distributed anisotropic single-walled carbon nanotubes and matrix particles) and an isotropic medium possessing the desired thermoelastic characteristics. This model was first employed to obtain the thermoelastic properties of a nanocluster by the self-consistency method and then the same technique was used to describe the thermomechanical interaction of nanoclusters with an isotropic matrix of the composite. A comparative analysis of the calculated dependences for the elastic moduli of the composite and its thermal coefficient of linear expansion was carried out with two-sided estimates of these characteristics based on the dual variational formulation of the thermoelasticity problem. For comparison, the results of a numerical experiment are also used. The presented relationships make it possible to predict the thermoelastic properties of promising composites reinforced by nanoclusters.

A mathematical model is constructed describing the thermomechanical action of the elements of a composite structure (platelike inclusion and matrix particles) and isotropic elastic medium with the required thermomechanical characteristics. The model is used at the first stage to obtain the matrix relations by the self-consistent method to find the elastic modulus of the composite. At the second stage, it is used to determine the temperature coefficient of linear expansion. Using the variation approach for the composite considered, the two-way estimates of the volumetric elasticity modulus, shearing modulus, and temperature coefficient of linear expansion are determined. The estimated dependences presented allow forecasting the thermoelastic characteristics of the composite, which is reinforced with the anisotropic platelike inclusions (including in the form of nanostructural elements).

The variational form of a mathematical model of a thermal explosion has been developed based on a variational formulation of a nonlinear problem of stationary thermal conductivity in a homogeneous solid body. The model takes the temperature dependence of the thermal conductivity of a solid body into account. The presented example of quantitative analysis of the model demonstrates a method for finding the combination of parameters for determining a thermal explosion in a plate with an exponential temperature dependence of the thermal conductivity. At the same time, the analysis allows one to identify the number of steadystate temperature distributions inside a body whose energy release intensifies with a temperature increase.

A mathematical heat transfer model in variational form is constructed for steady-state heat transfer through an inhomogeneous anisotropic solid of arbitrary configuration. This model is used to obtain two-sided thermal heat transfer resistance estimates for such a solid. These estimates permit calculating the maximum possible error when approximating the thermal resistance by half the sum of these estimates.

A composite with chaotically oriented fibers with different elongations and different anisotropy of elastic characteristics is considered. A mathematical model of interaction of such fibers and matrix particles with an isotropic elastic medium whose elastic moduli have to be found as required characteristics of the composite is constructed. The relations derived by the self-consistency method determine the moduli of the composite as functions of the volume concentration, elongations, and elastic properties of each type of fibers, and also of the elastic characteristics of the isotropic matrix. A quantitative analysis of the mathematical model is carried out, and boundaries of the domains of determining parameters within which the effect of fiber elongation is considerable are found. The relations presented allow one to estimate the elastic characteristics of a composite reinforced with various types of short fibers (in particular, high-strength and high-modulus needle-shaped and thread-like crystals, and nanostructural elements).

A modification has been made of the previously constructed mathematical model of a fragment of a flat thermostating coating including a thermoelectric module based on the variation formulation of the stationary problem of heat conduction in an inhomogeneous solid body. With the use of the Fourier finite integral transform the dependences have been obtained for calculating the temperature distribution in the heat insulating layer in the vicinity of the thermoelectric element and commutating conductors. This enabled us to refine one of the diagnostic variables of the model — the total heat resistance of the heat insulator between commutating plates and conductors of the thermoelectric module influencing the energy characteristics of the thermostating coating under investigation.

The article constructs the dual variational form of the model based on the presented differential form of a mathematical model describing the steady-state process of thermal conductivity in a plane or circular cylindrical layer of a solid dielectric at an alternating voltage. This form includes the main and alternative functionals, whose stationary points analysis makes it possible to establish a combination of the determining parameters corresponding to the occurrence of thermal breakdown of the dielectric layer. An example of the analysis of stationary points for two variants of test functions that are admissible for these functionals and approximating the steady temperature distribution in the dielectric layer is given. The estimation of the approximation error, which makes it possible to choose the function closest to the limiting temperature distribution in the layer preceding the thermal breakdown of the dielectric, is presented.

Spheroplastics are composite materials composed of a polymer or organosilicate binder and hollow spherical inclusions (mostly, of glass, but there are also of carbon, phenol, and epoxy), which are called microspheres and have a diameter within a millimeter with the wall thickness of several micrometers. To reduce the material density in watercraft constructions sometimes are used so called macrospheres of up to 40 mm in diameter and shell thickness of 0,5--1,5 mm from spheroplastic with microspheres. Microspheres may contain inert gases such as nitrogen. Many countries have commercialised quartz microspheres. The USA, in particular, produces Q-Gel microspheres with density of 300 kg / m3, the bulk density - 100 kg / m3 and the average diameter of 75 microns, characterized by a high mechanical strength and low cost. Carbon microspheres having low mechanical properties can absorb radio waves in certain frequency ranges. Spheroplastic with silicone microspheres combine relatively high mechanical and dielectric properties. In virtue of low thermal conductivity spheroplastics are used in various heat-insulating structures. As the thermal insulation coatings, the spheroplastic covers the outer surface of the pipes, in particular oil and gas pipelines in the permafrost zones, regions of swampy ground, and underwater. The effective heat conductivity factor, primarily, determines the specific application of spheroplastic as a thermal insulation material. To quantify the value of this factor is necessary to have a mathematical model describing heat ransfer in spheroplastic. The paper presents a four-phase mathematical model of the heat transfer in a representative element of a spheroplastic structure placed in an unlimited array of homogeneous material, the thermal conductivity of which is to be determined as desired characteristics of spheroplastic. This model in combination with a dual variational formulation of stationary heat conduction problem in the inhomogeneous solid first is used to define the guaranteed two-sided boundaries of the parameter space in which there are the true values of effective thermal conductivity of spheroplastic, and then to calculate the dependences of this factor on the bulk concentration of microspheres. The paper conducts a quantitative analysis of the calculated dependences and determines the values of their greatest possible accuracy, which allow us to measure a reliability degree of the predicted effective thermal conductivity of the spheroplastic.

Heat transfer through a thermal insulation layer in radiative-convective heat transfer on its nonconcave surface is considered. Combinations of determining parameters at which the heat flux passing through this layer reaches an extremal absolute value (a maximum corresponds to a critical thickness of thermal insulation, and, a minimum, to an optimal). The qualitative analysis of the dependence of the heat flux on the determining parameters makes it possible to choose the thermal characteristics of the insulation and its thickness, providing the desired reduction in the intensity of heat transfer through the insulation with a nonconcave surface of double curvature in conditions of radiative-convective heat transfer.

The temperature state of a solid body, in addition to the conditions of its heat exchange with the environment, can greatly depend on the heat release (or heat absorption) processes within the body volume. Among the possible causes of these processes should be noted such as a power release in the fuel elements of nuclear reactors, exothermic or endothermic chemical reactions in the solid body material, which respectively involve heat release or absorbtion, heat transfer of a part of the electric power in the current-carrying conductors (so-called Joule’s heat) or the energy radiation penetrating into the body of a semitransparent material, etc. The volume power release characterizes an intensity of these processes. The extensive list of references to the theory of heat conductivity of solids offers solutions to problems to determine a stationary (steady over time) and non-stationary temperature state of the solids (as a rule, of the canonical form), which act as the sources of volume power release. Thus, in general case, a possibility for changing power release according to the body volume and in solving the nonstationary problems also a possible dependence of this value on the time are taken into consideration. However, in real conditions the volume power release often also depends on the local temperature, and such dependence can be nonlinear. For example, with chemical reactions the intensity of heat release or absorption is in proportion to their rate, which, in turn, is sensitive to the temperature value, and a dependence on the temperature is exponential. A further factor that in such cases makes the analysis of the solid temperature state complicated, is dependence on the temperature and the thermal conductivity of this body material, especially when temperature distribution therein is significantly non-uniform. Taking into account the influence of these factors requires the mathematical modeling methods, which allow us to build an adequate nonlinear mathematical model of the heat conductivity process in the volume power release body. Quantitative analysis of these models requires using the numerical methods, as a rule. At the same time, such a simple body, which is an unlimited plate of the constant thickness allows us, under certain assumptions, to solve analatically a nonlinear heat conductivity problem taking into account the thermal conductivity of the plate material and the power release intensity versus temperature. This solution enables us to reveal a number of significant effects that have impact on the thermal state of the plate, including those related to conditions of available steady temperature distribution, and it can be used to test the results obtained by numerical methods.

The self-consistency method is used to deduce a relation between the linear thermal expansion coefficient of a composite reinforced with anisotropic spherical inclusions and the thermoelastic characteristics of the matrix and inclusions and their volume concentration. This method considers the thermomechanical interaction of a separate inclusion with a homogeneous isotropic medium having the required value of the coefficient mentioned. On the basis of the dual variation formulation of a problem of thermoelasticity for a solid heterogeneous body, bilateral limits of possible values of this coefficient allowing one to check the reliability of the dependences obtained and to estimate the limit value of the error of calculation results are constructed. The relations presented can be used to predict the linear thermal expansion coefficient of a composite with disperse anisotropic inclusions, nanostructural ones among them.

We construct a mathematical model describing thermomechanical interaction between composite structure elements (isotropic particles of the matrix and anisotropic short fibers) and the macroscopically isotropic elastic medium with desired thermoelastic characteristics. At the first stage of this model, the self-consistency method is used to obtain relations determining the elasticity moduli of the composite, and at the second stage, the model permits determining its linear thermal expansion coefficient. The dual variational statement of the linear thermoelasticity problem in an inhomogeneous solid permits obtaining two-sided estimates for the bulk elasticity modulus, shear modulus, and linear thermal expansion coefficient of the composite under study. The calculated dependencies presented in the paper permit predicting the thermoelastic characteristics of a composite reinforced by anisotropic short fibers (including those in the form of nanostructure elements).

This article presents the formulation and solution of a linear non-stationary heat conduction problem with an axisymmetric hypersonic flow around a hemispherical blunting of the aircraft head. The material of the heat-shielding layer of blunting has anisotropy of thermal conductivity, characterized by the ratio of thermal conductivity coefficients along and across the layer. The temperature over the thickness of the power metal shell of the head part is assumed uniform. A quantitative analysis of the temperature state of the head part in the laminar flow regime and during the transition to the turbulent regime was carried out.This article presents the formulation and solution of a linear non-stationary heat conduction problem with an axisymmetric hypersonic flow around a hemispherical blunting of the aircraft head. The material of the heat-shielding layer of blunting has anisotropy of thermal conductivity, characterized by the ratio of thermal conductivity coefficients along and across the layer. The temperature over the thickness of the power metal shell of the head part is assumed uniform. A quantitative analysis of the temperature state of the head part in the laminar flow regime and during the transition to the turbulent regime was carried out.

Results of mathematical simulation of the elastic properties of an isotropic matrix composite reinforced with spherical inclusions are presented. As the spherical inclusions, ball nanoclusters composed of randomly oriented single-wall carbon nanotubes (SWCNT) are taken. Two approaches are used: numerical simulation (using the ANSYS software suite together with a newly developed software module) and analytical treatment using the self-consistent method and the dual formulation of the elasticity problem in a non-heterogeneous solid body. The numerical simulation is carried out in the context of a composite with a reinforcement pattern similar to the cubic lattice. In this formulation, two periodic lattice alternatives are considered: a cube with 1/8 of a sphere in each corner, and a cube with a sphere in its center. Comparison is made between the results of numerical simulation and those obtained analytically. The results make it possible to predict the elastic properties of advanced materials such as a composite reinforced with spherical inclusions.

Thermal insulation coatings are applied to the heat transfer areas of structures exposed to intensive ambient heating to ensure the operability of the structures. If the heat transfer rate is non-uniform, the effectiveness of thermal insulation can be enhanced by using an anisotropic material that has a higher conductivity in the longitudinal direction parallel to the heat transfer surface compared to the conductivity in the direction of the normal to it. We present a mathematical model of the transient heating of a layer of such a coating. A quantitative analysis of the model allows estimating the influence of the anisotropy ratio on the leveling of temperature in the longitudinal direction and evaluating the possibility of reducing the ratio for the area where the heat transfer is maximum.Thermal insulation coatings are applied to the heat transfer areas of structures exposed to intensive ambient heating to ensure the operability of the structures. If the heat transfer rate is non-uniform, the effectiveness of thermal insulation can be enhanced by using an anisotropic material that has a higher conductivity in the longitudinal direction parallel to the heat transfer surface compared to the conductivity in the direction of the normal to it. We present a mathematical model of the transient heating of a layer of such a coating. A quantitative analysis of the model allows estimating the influence of the anisotropy ratio on the leveling of temperature in the longitudinal direction and evaluating the possibility of reducing the ratio for the area where the heat transfer is maximum.

For a plane layer of a polar dielectric, which has a nonmonotonic dependence of the dielectric losses on temperature, a mathematical model describing the temperature state of this layer before its thermal breakdown is constructed. On the basis of this model, the connection between the breakdown voltage and the parameters determining the properties of the dielectric and the temperature distribution in the layer preceding the onset of thermal breakdown is established in an integral form. A quantitative assay of the model is carried out for the two types of polymer materials used as dielectrics.

Composites comprising a matrix with multishaped inclusions are widely used as the engineering structural, building, and functional materials in a variety of instrumentation devices. Among the composites a majority of materials used in engineering are heterogeneous solids. Among these materials there is a large group of composites that have inclusions in the form of fine particles. Those should also include a variety of nanostructured particles with the outlook for their using to provide a capability to enhance mechanical properties of composites.To improve processability of the binder polymer as a component part of the composite are used fine fillers in the form of powder, flakes or fibrous particles. This makes it possible to receive the exotherm during the curing of the binder, to reduce its shrinkage, and improve the mechanical, thermal, electromagnetic and other operational characteristics of the produced composite. However, such fillers available in the binder can cause porosity that impaires the composite properties. The pore emergence is also caused by a long binder shrinkage during its solidification reaching several percent. Particle sizes can vary from a fraction of micrometer to several tens of micrometers. One would expect a similar range of variation of pore sizes.One of the composite characteristics, which is sensitive to inclusions available therein as pores and fine particles, is an effective thermal conductivity that is significant in terms of defining the scope of use for such composites. Calculation formulas for evaluating this factor obtained in the prior art, as a rule, either as a result of experimental data processing in relation to specific materials, or by setting a priori distribution of temperature and heat flow in models of heterogeneous body structure. However, building the mathematical models of a heat transfer process in the composite allows us to define a quantitative effect of the concentration of pore volume and fine particles on this coefficient.This study compares the quantitatively estimated results of the effective thermal conductivity of the spherically-shaped disperse inclusions composite obtained using various approaches to describing the process of heat transfer in such a composite.

On the basis of a variational formulation of the problem of stationary heat conduction in a heterogeneous solid, a mathematical model of a fragment of a flat heat-insulating layer containing a thermoelectric module has been constructed. This model has been used to establish conditions under which, when fulfilled, the heat-insulating layer can serve as a thermostating coating for an object with a given fixed temperature under convective-radiative heat exchange on the outer surface of the fragment under consideration. The results of the qualitative analysis of the proposed model are presented.

A mathematical model describing joint radiative-conductive heat transfer in a spherical cavity, the shape of which can be considered as a statistical average in relation to the forms of closed pores in solids, is developed. The model determines an equivalent thermal conductivity of an arbitrary diathermic medium in a cavity that allows a material with a porous structure to be considered as a continuous inhomogeneous solid. The effect of the temperature field gradient in the vicinity of the cavity and the thermal conductivity of the diathermic medium on the equivalent coefficient of thermal conductivity is determined. In the case of a thermally nonconducting medium, the calculated dependence of this coefficient is compared to equations, similar in structure, derived on the basis of different approaches to account for heat transfer by radiation in pores.

The rational use of composites as structural materials, while perceiving the thermal and mechanical loads, to a large extent determined by their thermoelastic properties. From the presented review of works devoted to the analysis of thermoelastic characteristics of composites, it follows that the problem of estimating these characteristics is important. Among the thermoelastic properties of composites occupies an important place its temperature coefficient of linear expansion.Along with fiber composites are widely used in the technique of dispersion hardening composites, in which the role of inclusions carry particles of high-strength and high-modulus materials, including nanostructured elements. Typically, the dispersed particles have similar dimensions in all directions, which allows the shape of the particles in the first approximation the ball.In an article for the composite with isotropic spherical inclusions of a plurality of different materials by the self-produced design formulas relating the temperature coefficient of linear expansion with volume concentration of inclusions and their thermoelastic characteristics, as well as the thermoelastic properties of the matrix of the composite. Feature of the method is the self-accountability thermomechanical interaction of a single inclusion or matrix particles with a homogeneous isotropic medium having the desired temperature coefficient of linear expansion. Averaging over the volume of the composite arising from such interaction perturbation strain and stress in the inclusions and the matrix particles and makes it possible to obtain such calculation formulas.For the validation of the results of calculations of the temperature coefficient of linear expansion of the composite of this type used two-sided estimates that are based on the dual variational formulation of linear thermoelasticity problem in an inhomogeneous solid containing two alternative functional (such as Lagrange and Castigliano). These functionals on the true distribution of strains and stresses in an inhomogeneous body matching meaningfully reach extremes (minimum and maximum respectively). On the convergence of the distribution of the minimized functional application gives an upper estimate of the desired values of the temperature coefficient of linear expansion of the composite, and maximized use - their lower bound. The difference between these estimates allows to predict the maximum possible error in calculations obtained dependences.

The spherical shell made with high accuracy is one of the possible geometric shapes for passive orbital signal repeaters and calibration-adjustment spacecrafts used to determine the energy potential of the radar channel of the ground-based complex needed to control the space object motion. Under the solar radiation, an uneven temperature distribution over the surface of such a shell causing a deviation of its shape from the spherical one occurs. To determine these deviations, the steady-state temperature distribution for the shell with a fixed orientation relative to the Sun obtained by solving the Fredholm integral equation of the second kind is used. A quantitative analysis of the possible alignment of a quasi-stationary distribution of the shell temperature in case of the shell rotation with a constant angular velocity around the axis perpendicular to the direction to the Sun is carried out.

The steady-state process of heat conduction in the rotating disk of a d.c. unipolar generator has been considered. A mathematical model has been constructed and on the basis of its analysis the features of the radial temperature distribution in the disk have been elucidated and the influence on this distribution of the temperature dependence of the electrical resistivity and the heat conductivity coefficient of the disk material have been determined. We have established the limiting value of the temperature coefficient of electrical resistivity at which the existence of a stable temperature distribution in a disk of hyperbolic profile under certain boundary conditions becomes impossible. A solution and a numerical example for a linear problem as applied to a disk with a power dependence of its thickness on the radial coordinate are given.

Composites are widely used as engineering structural materials in different technical areas, given their higher mechanical properties combined with their smaller weight as compared with the “pure” materials. An important characteristic of a composite is a set of its elastic properties. To determine these properties, both experimental methods and mathematical simulation are used. Due to large material costs and time expenditures inherent in each experiment (fabrication of the material and the test sample, performance of the experiment and processing of the results), on the material design stage, as well as when determining the material application area, preference is given to the simulation methods.This paper is focused on the comparative analysis of the available mathematical simulation methods in terms of performance study of the unidirectional fiber-reinforced composite in the plane normal to the fiber orientation. For illustrative purposes, composites with two different kinds of matrix, polymeric and metal, are considered.

A mathematical model of heat transfer is constructed for a composite inhomogeneous body of arbitrary shape, which includes elements of anisotropic materials. Thermal contact between these elements can generally be imperfect. A dual formulation of the variational problem of stationary heat transfer is used in obtaining two-sided estimates for the thermal resistance of the body.

The composites, dispersion-reinforced with inclusions from high-strength and high-modulus materials are widely used in technology. Nanostructure elements can perform the role of such inclusions as well. Possible applications of such composites in heat-stressed structures under heavy mechanical and thermal influences significantly depend on a complex of thermo-mechanical characteristics including the values of the moduli of elasticity and coefficient of linear thermal expansion. There are different approaches to construction of mathematical models that allow calculating dependences to estimate elastic characteristics of composites. Relation between thermoelastic properties of matrix and inclusions of the composite with its temperature coefficient of linear expansion is studied in less detail. Thus, attention has been insufficient in estimating a degree of reliability and a possible error of derived dependencies.A dual variation formulation of the problem of thermo-elasticity in a non-uniform solids simulating the properties and structure of the composite with dispersed inclusions, makes it possible to define two-sided limits of possible values of the volume elasticity modulus, shear modulus, and coefficient of linear thermal expansion of such composite. These limits allow us to estimate the maximum possible error, if to take a half-sum of the limit values of these parameters as the thermoelastic characteristics of the composite. Implementing this approach to find possible errors, arising when using one or another calculating dependency, improves reliability of predicted thermo-elastic characteristics as applied to existing and promising composites.

Consideration is given to some thermomechanical models of the behavior of structural materials that are based on the thermodynamics of irreversible processes and on the analysis of physical processes in polycrystalline materials.

In order to justify a reliable application domain for the calculation dependence which allows one to determine the effective heat conductivity coefficient of transversely isotropic unidirectional fiber composite in the plane perpendicular to the fibers; a mathematical model of heat transfer in representative elements of this composite's structure was created. An ordered arrangement of parallel fibers was considered for the case when their centers coincide with the nodes of cross sectional plane grid with square cells or cells in the form of equilateral triangles. Quantitative analysis of developed mathematical model was performed using the finite element method with controlled integrated computational error. Presented results are important for assessing the temperature condition and efficiency of heat-stressed structures made of unidirectional fiber composite.