Savel'eva

pp. 49-63

pp. 88-104

pp, 64-82

pp 197-215

pp. 708-723

The article offers a mathematical model of heat transfer in a composite with spherical shape inclusions. The model takes into account that filler particles and pores are available in the binder. The effective thermal conductivity of such composite has been estimated, including the use of dual variation to formulate a problem of stationary thermal conductivity in the non-uniform body. The presented results can be used to predict the thermal conductivity of the composite affected both by fine-dispersed fillers introduced in the polymeric binding structure and by arising porosity when curing the binder.

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.

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).