Zarubin

pp. 155-170

pp. 182-195

pp. 27-45

pp. 44-60

pp. 49-63

pp. 88-104

pp, 64-82

pp. 318-335

pp. 51-63

pp. 51-67

pp. 59–72

pp. 6-16

pp. 37-49

pp. 36-48

pp 197-215

pp. 708-723

pp.237-255

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.

The self-consistency method was used to estimate the components of a thermal conductivity tensor of unidirectional fiber composite, which is transversally isotropic with respect to an axis arranged parallel to the fibers. In this method it is possible to estimate effective values of parameters for the composite as a whole by averaging the perturbed distributions of the parameters over the elements of the composite structure and equating averaged results to zero. A perturbed temperature field and heat flux density distribution over the fibers and composite matrix were calculated on the basis of the developed mathematical models of heat interaction between these elements of the structure and a transversely isotropic homogeneous material which has required coefficients of thermal conductivity. For validation of the final calculated dependences, bilateral estimates based on dual variation formulation of the stationary problem of heat conduction in an inhomogeneous solid body were used. The calculated dependencies can be used for forecasting effective thermal conductivity of unidirectional fiber composites.

The method of self-consistency was applied to obtaining an estimate of the effective value of thermal conductivity of a composite with ball inclusions. This method was based on averaging of parameters of a perturbed temperature field in elements of a composite structure. The perturbed temperature field in inclusions and ball particles of the composite matrix was found by solving the steady-state heat conduction problem formulated on the basis of a constructed mathematical model of thermal interaction of these elements with a homogeneous material with an unknown coefficient of thermal conductivity. Quantitative analysis of the derived calculated relation was carried out to a sufficiently wide range of the defining parameters to verify reliability of the obtained estimate. The calculation results with respect to this relation are located within the interval between the lower and upper estimates which could be obtained from both a singular approximation of the theory of random functions and the dual variation principle of Hashin - Shtrikman.

On the basis of a developed mathematical model of heat transfer in composite with anisotropic inclusions of ellipsoidal shape, a procedure for calculating components of an effective heat-conduction tensor of the textured composite was proposed. A variant of inclusions in the form of ellipsoids of rotation was considered. The main axes of the heat-conduction tensor of such inclusions coincide with symmetric axes of an ellipsoid. In case of conic texture of a composite its possible dispersion was taken into account. The obtained results could be used for estimating effective thermal conductivity of composites modified with nanostructured elements (including carbon nanotubes). Due to electrothermal analogy these results are applicable when consideringcharacteristics of electrical conductivity and dielectric capacitance of textured composites.

The authors consider different mathematical models of thermal interaction of ball inclusions and the matrix of a composite, used for estimating the effective heat conduction coefficient of this kind of composite. Application of a dual formulation of the variation problem of stationary thermal conductivity in a nonhomogeneous solid body allowed to get double-sided estimations of possible values of the coefficient. It was determined that transition from ball inclusions to cubic ones influenced slightly the effective heat conduction coefficient of a composite. The authors derived an evaluation formula that allowed to obtain a reliable estimation of this coefficient within the whole range of possible variations of the volume concentration of inclusions. Due to the electrothermal analogy it was possible to apply the obtained results in order to estimate characteristics of electrical conductivity and of dielectric capacitance of composites modified with ball inclusions (considering nanostructured elements).

The authors carried out a comparative quantitative analysis of mathematical models for estimating effective thermal conductivity coefficient of polycrystalline material which consists of chaotically oriented anisotropic crystal grains with identical crystal grid. Along with well-known methods for generating mathematical models, approaches which allowed to obtain new calculating dependences were also considered. The upper and lower bounds for accepted values of this coefficient were obtained with the use of a dual variational formulation of a stationary thermal conductivity problem for a non-homogeneous solid body. Formulas for calculating best-attested estimations were recommended. These formulas could be used for estimating electric conductivity of polycrystalline materials due to electro thermal analogy.

A mathematical model of heat transfer in a composite material with identically oriented anisotropic ellipsoidal inclusions was created. Formulae for calculating effective thermal conductivity of composite materials, which are anisotropic with respect to the property of thermal conductivity, were obtained on the basis of a developed mathematical model. To estimate the possible error of the results, a dual formulation of the variational problem of stationary heat conduction in a heterogeneous solid body was applied. These results can be used to predict effective thermal conductivity of composite materials with ellipsoidal inclusions, which are among other nanostructured elements (including carbon nanotubes). The obtained formulas can be used for estimating electric conductivity of composite materials with identically oriented anisotropic ellipsoidal inclusions due to electro thermal analogy.