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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.

The variational formulation of the nonlinear problem of steady heat conduction in an arbitrary configuration domain is applied to investigate the conditions for the existence of a steady-state temperature distribution in a stationary medium with the intensity of volumetric energy release rising with increasing temperature. Based on the relations of the time-independent theory of thermal explosion, a variational model of this phenomenon is constructed, which makes it possible to obtain estimates of the critical values of the parameters that determine the temperature state of the medium preceding the thermal explosion. The examples of a comparative analysis of such values for a solid and a hollow cylinder of finite length are given.

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.

According to the formulation of the nonlinear problem of stationary heat conduction in a homogeneous ellipsoid, when the intensity of volume energy release increases with temperature, a variational form of a mathematical model of a thermal explosion has been developed. This form contains a functional defined on a set of continuous and piecewise differentiable functions. These functions approximate the temperature distribution in the ellipsoid volume and accept a given temperature on its surface. The study of stationary points of the functional makes it possible to estimate a combination of defining parameters in which the temperature state in the ellipsoid precedes the thermal explosion. A quantitative analysis of the variational form of the model has been carried out with the exponential growth of the energy release intensity with temperature increase. The study introduces a relation for estimating an integral error arising when a specific approximating function is used. The comparison of such estimates for various approximating functions makes it possible to choose the function closest to the temperature distribution preceding the thermal explosion in the ellipsoid.

Currently, the design of large space structures attracts engineers due to the need to solve pressing problems in the area of creating promising space systems for longitudinal operation in the outer space. A special class of space systems is formed by transformable structures whose layout can be transformed. Usually, the structures of this class are delivered to the working orbit in the densely-packed position. When the orbit having the required parameters is reached, the structure is deployed. The deployment is done by force actuators, for example, springs, electric motors, and so on. Transformable space structures are multiple-element systems comprised of many interconnected elements. Despite massive progress in the area of designing such structures, the task of ensuring smoothness and reliability of the deployment of structures with guaranteed functioning after deployment is still relevant. The paper proposes using actuators based on shape memory alloys to ensure controlled deployment of large transformable structures. Experimental investigations of a simple actuator based on the Ti-Ni alloy element confirmed the possibility of using it for the deployment of transformable space structures.

The relevance of the study is due to the increase in recent years, both in Russia and abroad, in the number of computer attacks on production automation systems. One of the most dangerous classes of malware is computer worms. In this regard, this article is aimed at investigating attacks using computer worms on automated systems, assessing the damage received from their implementation and choosing ways to protect against them. As an approach to the study of this problem, methods of rationing and averaging the risk of possible damage from a computer attack are used. The article presents statistics of crimes using computer worms in the USA, China, Russia and the world, short-term forecasts are built. The materials of the article are of practical value for further forecasting of this type of attacks, caused by damage, as well as the choice of ways to counteract them in automation systems.

A significant increase in cargo turnover between the Moon and the Earth is expected in case of full implementation of recently announced programmes of exploration and development of the Moon (Luna, Artemis, Chang’e). A feature of the spacecraft trajectory when it returns from the Moon’s orbit is that the spacecraft speed at the entry into the Earth’s atmosphere is close to parabolic. Compared to orbital landing sections, those of lunar interplanetary spacecraft expose to more intensive radiation and heat flows, and due to it and the above circumstance, the problem of the design of thermal shield for these spacecraft becomes much more complicated.This paper considers two approaches, which help decrease heating of the thermal shield. The first one is the use of trajectory with multiple entrance in the atmosphere. This approach allows for reducing the heat flow intensity due to the gradual deceleration during the repeated re-entries. The second one is the use of anisotropic heat-shielding materials referring to the heat conductivity. This approach allows for reducing the surface temperature in the area of the maximum density of heat flow by redistribution of thermal energy in the thermal shield along the tangential direction.As follows from the analysis provided in the paper, the use of two approaches described herein provides the reduction of the temperature of the spacecraft surface either to the level not destroying the thermal shield or to that providing to reduce its destruction significantly. It allows for improving the spacecraft reliability and, in prospect, provides the multiple uses of it.

The formulation of the nonlinear problem corresponding to the process of stationary heat conduction in homogeneous triaxial ellipsoid with increasing temperature and intensity of volumetric energy release was used to build a variational form of a mathematical model of this process. This form includes a functional defined on a set of continuous and piecewise differentiable functions that approximate the temperature distribution in the volume of an ellipsoid and take a given value of temperature on its surface. An analysis of the stationary points of the functional makes it possible to estimate the combination of determining parameters corresponding to the temperature distribution in the ellipsoid before the occurrence of a thermal explosion. Comparison of the integral error caused by the use of various approximating functions allows to choose the function that most accurately describes the temperature state of the ellipsoid preceding the thermal explosion. Estimations of the parameters of the thermal explosion are obtained under the assumption of an exponential increase in the intensity of volumetric energy release in an ellipsoid with increasing temperature.

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.

Porous composite materials are widely used in engineering as structural and heat-insulating materials. Pores available in such materials are due to both their manufacturing technology and the operating conditions.One of the most important factors in the process of designing products from a porous composite is a set of thermo-physical characteristics of the material. This characteristic determines the application area of the material.Among the thermo-physical properties a thermal conductivity coefficient plays a key role. For some porous materials this coefficient can be determined experimentally, however, to reduce time and resources needed, a theoretical study of this characteristic is more relevant.Theoretical investigation of the thermal conductivity coefficient of a porous composite allows us to predict its possible values depending on the composition of the material and its porosity. Such information about the composite is necessary at various processing stages of the material from its preparation to its using for the structure fabrication.There are many papers on the approaches to the theoretical estimate of the thermal conductivity coefficient of a porous material. However, due to a significant spread of its values, a relevant task is to have the guaranteed two-sided estimates of the possible values of this material characteristic.As is well known, there are some difficulties in making lower estimates of the properties of a porous material. To overcome this difficulty, the paper proposes to use a modification of the structural model of the porous body in conjunction with the dual formulation of the stationary thermal conductivity problem in an inhomogeneous solid.In the paper the structural model modification of a porous body is as follows: a solid sphere with an equal external radius replaces a hollow spherical particle. The solid sphere, in turn, is represented by a composite ball consisting of an inner ball of some conventional material and an outer spherical layer of the skeleton material of the porous body. The equivalent thermal conductivity of the material of the inner ball is to be determined.The structural model modification of the porous body proposed in the paper allowed us to obtain the two-sided estimates of the possible value of this coefficient. Also, the obtained estimates were compared with the improvable upper bound for this characteristic.The obtained results will allow us to predict two-sided estimates of the thermal conductivity coefficient of advanced heat-insulating and structural porous materials.

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 thermal model for the aluminized polymer shell of a gauge-alignment spacecraft was developed to calculate the steady-state temperature distribution of this shell at a fixed orientation to the Sun. A modified version of the model was used to analyze the quasistationary distribution of the shell temperature in the case of its rotation with a constant angular velocity about an axis perpendicular the direction to the Sun.

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).

The use of developed prospective dielectric materials in various modern electrotechnical and electrophysical devices requires reliable forecast of attainability of the required level of final characteristics depending on properties of those materials. Such forecast is based (among others) on solution of electrostatic problems in an inhomogeneous anisotropic medium allowing one to estimate the ability to satisfy the qualifying standards for effective characteristics of created materials. A dual variational formulation of an electrostatic problem in an inhomogeneous anisotropic dielectric is used to solve these problems.

The equipment uses heat-shielding and structural materials that, when exposed, absorb radiation both on the surface and in the volume. In a variety of technical devices, absorption processes of penetrating radiation of materials and structural elements are typical for a number of process steps and operating conditions. Absorption of radiation penetrating into material volume may significantly affect the temperature state and runability of construction made of such material.The process of material-absobed penetrating radiation is associated with transition of the electromagnetic wave energy into the excitation energy of this material microparticles that, after all, leads to increasing internal energy and temperature growth. With radiation passing through the layer of material its flow density and hence energy of penetrating radiation decreases exponentially with increasing distance from the exposed layer surface. This law was experimentally established by the French physicist P. Bouguer and bears his name. In general, a certain fraction of this energy is radiated and dissipated in the material volume, and the rest is absorbed. A mathematical model describing these processes is an equation of the radiative energy transfer.In mathematical modeling of thermomechanical processes there is a need to consider the effect of penetrating radiation on the temperature state of materials and construction elements. The P. Bouguer law is used also when the volume radiation and scattering of penetrating radiation in the material can be neglected, but it is necessary to take into account its absorption. In this case, a negative indicator of the exponential function is represented by the product of the distance from the irradiated surface and integral or some average absorption factor that is constant for a given material and spectral distribution of penetrating radiation. However, with increasing power of radiation passing through the material layer there is a dependence of the absorption factor on the local intensity of this radiation. Furthermore, it can be a significant dependence of this factor on the local value of the material temperature, reflecting the above-mentioned relationship between the absorption of electromagnetic wave energy and the excitation of material microparticles. This process can be described by Boltzmann distribution function that comprises the energy to activate microparticles and the local value of temperature.This paper presents a variational formulation of the nonlinear problem of stationary heat conduction in a plate for the case when the radiation reduction factor in relation to the Bouguer law depends on the local temperature. This formulation includes a functional that can have several fixed points corresponding to different steady states of the plate temperature. Analysis of the properties of this functional enabled us to identify the stationary points, which correspond to the realized temperature distribution in the plate.

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.

Heat-insulating porous materials and structural ones having also a porous structure, which are produced by pressing and powder metallurgy methods, are widely used in engineering. One of the important thermophysical characteristics of such materials is a coefficient of thermal conductivity, which affects the choice of specific areas of their application. Along with the experimentally determined coefficient of thermal conductivity of porous materials, there are various approaches to estimate this coefficient. Most of these approaches have an empirical character and are based on various models of the structure of porous solid skelton, which enable us to approximately estimate contribution of this skeleton to the value of effective thermal conductivity of the entire porous body.A reliable estimate of the thermal conductivity of a porous solid skelton can be based on a modification of its structural model through conditional replacement of pores with their surrounding shells of the material by solid particles with an equivalent coefficient of the thermal conductivity.Such a replacement allows us to extend constructibility of computational dependencies, primarily, to obtain the guaranteed two-sided estimates of the effective thermal conductivity of a porous solid, including using the dual variational formulation of the problem of a steady-state heat conductivity in an inhomogeneous solid. The peculiarity of this formulation is that it includes two alternative functionals (minimized and maximized) that reach equal extremal values at the true temperature distribution in an inhomogeneous body. This property of alternative functionals makes it possible, according to their values, calculated at the approximate temperature distributions in this body, to obtain, respectively, the upper and lower bounds of its effective thermal conductivity.However, the use of the initial structural model of the porous solid skelton, provided that there is no thermal energy transfer through the pores, ensures the preservation of the physical sense only for the upper estimate of the effective thermal conductivity of this skeleton, and the lower estimate is either incorrect or corresponds to the zero value. The introduction of an equivalent thermal conductivity coefficient for a conditional solid particle replacing a pore surrounded by a layer of the solid skelton material allows using the variational approach to find correct two-sided estimates of the effective thermal conductivity and simultaneously to estimate the possible greatest error that can arise when using the computational dependencies based on different structural models and assumptions.

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.

Based on the dual variational formulation of an electrostatic problem in an inhomogeneous anisotropic dielectric, we construct two-side bounds for possible values of second rank tensor components for the dielectric permittivity of a unidirectional fibrous composite taking into account mutual positions of anisotropic reinforcing fibers in the plane perpendicular to their axes. We consider variants of arrangement of cross sections of fibers such that the composite is transversal-isotropic or orthotropic relative to the property of dielectric permittivity. Some estimates of the greatest error appearing in the choice of true values of each component of the dielectric permittivity tensor of the composite in the form of a half-sum of its boundary values are obtained. A successive improvement of two-side bounds for domains where the components of this tensor should lie is carried out. A quantitative analysis of calculation dependencies is presented.

A temperature state of the solid body may depend both on the conditions of heat exchange with external environment surrounding its surface and on the energy release within the body volume, caused, for example, by the processes in nuclear reactor elements or exothermic chemical reactions, absorption of penetrating radiation energy or transformation of a part of the electrical power into heat with flowing electric current (so-called Joule heat).If with growing temperature the intensity of bulk power density increases, a limited steady temperature state can emerge at which heat extracted to the body surface and released within its volume reaches maximum. Thus, small increments of temperature lead to an increase of heat release, which can not be extracted to the body surface by conduction without further temperature increase. As a result, the steady temperature distribution in the body becomes impossible that determines the state of the thermal explosion, so named due to the fact that in this case the appropriate mathematical model predicts an unlimited temperature increase.A lot of published papers and monographs concerning the study of the combustion and explosion processes in a stationary medium analyse the thermal explosion state. The most famous papers consider a mathematical model to describe a temperature distribution in the case when heat release is because of exothermic chemical reactions the rate of which increases with temperature growth. The dependence of the chemical reaction rate on temperature is usually described by the exponential Arrhenius law, which makes it necessary to consider an essentially nonlinear mathematical model containing differential equation, which includes the term, nonlinearly rising with increasing temperature. Even with simplifying assumptions, this model allows an exact closed form solution only in the case of one-dimensional temperature distributions in the two areas of the canonical form: in the plate, infinite in its plane, and in the circular cylinder unlimited in length.An approximate numerical solution of the differential equation that is included in a nonlinear mathematical model of the thermal explosion enables us to obtain quantitative estimates of combination of determining parameters at which the limit state occurs in areas of not only canonical form. A capability to study of the thermal explosion state can be extended in the context of development of mathematical modeling methods, including methods of model analysis to describe the thermal state of solids.To analyse a mathematical model of the thermal explosion in a homogeneous solid the paper uses a variational approach based on the dual variational formulation of the appropriate nonlinear stationary problem of heat conduction in such a body. This formulation contains two alternative functional reaching the matching values in their stationary points corresponding to the true temperature distribution. This functional feature allows you to not only get an approximate quantitative estimate of the combination of parameters that determine the thermal explosion state, but also to find the greatest possible error in such estimation.

Various technical devices quite widely use bimetallic shells as the structural elements. A chamber combustion design of the liquid rocket engine (LRE) is a typical use of the bimetallic shells. In LRE operation a combustion chamber shell is subject to intense thermal and mechanical effects, which necessitates cooling. A cooling shell path is formed by a gap between its inner and outer walls connected to each other by milled or grooved spacer ribs. The outer wall of the shell serves as a load-bearing element, the inner wall is in direct contact with high-temperature combustion products and exposed to intense heat. The difference in functions of shell walls calls for their manufacturing from different materials with different thermophysical and mechanical properties. Interaction between the shell walls of different materials in heating and cooling leads to emerging thermal strains of various values in the walls. In terms of mechanical properties the inner wall material, usually ranks below the outer wall material strength, which uses the high strength stainless steel 12Х21Н5Т. The inner wall is typically made from copper-based highly heat-conductive alloys. (eg.: chromium bronze). Therefore, the result of the difference in temperature deformations, arising in the walls, is inelastic nonisothermal strain of the inner wall material with (usually) elastic behavior of the outer wall material. For reusable LRE, a cyclic sequence of the loading steps of the inner wall can lead to accumulating damages in its material because of the low-cycle fatigue and cause destruction of the wall or the loss of the cooling tract tightness. The main parameter that determines the level of low-cycle fatigue, is an absolute value of the accumulated inelastic strain (both plastic and evolving over time creep deformation). Quantitative evaluation of this parameter involves analysis of the inner wall loading with multiple starts and shutdowns of LRE. The paper represents an approximate analysis of such loading under certain simplifying assumptions using the approaches typical for mathematical modeling of structures under thermal stress.

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.