A new model for predicting the thermal conductivities of a composite with spherical microballoons is proposed and consists of two consecutive procedures, the computation of the thermal conductivity of the microballoon and the composite. The microballoon is first replaced by the equivalent filler of a known thermal conductivity, so the composite is treated as the matrix containing the equivalent fillers and its thermal conductivity is derived by using Eshelby model modified with Mori-Tanaka's mean field approach. The present model is validated comparing the predicted, experimental, and numerical results from the literature. Parametric studies in terms of the microballoon volume fraction, its relative wall thickness, and the thermal conductivity ratio of the shell to the matrix have been made and their results are discussed.

SUMMARYIn this study, we present a new highly accurate composite laminated hybrid-mixed curved beam element. The present element, which is based on the Hellinger-Reissner variational principle and the ÿrst-order shear deformation lamination theory, employs consistent stress parameters corresponding to cubic displacement polynomials with additional nodeless degrees of freedom in order to resolve the numerical di culties due to the spurious constraints. The stress parameters are eliminated and the nodeless degrees are condensed out to obtain the (6 × 6) element sti ness matrix. Several numerical examples conÿrm the superior behaviour of the present hybrid-mixed laminated curved beam element.

Effective thermal conductivity of three-phase composites, consisting of matrix and two kinds of spherical inclusions, has been derived as an explicit form by extending modified Eshelby model (MEM) for two-phase composites. The present results are compared with those by differential effective medium model (DEMM), which are also compared with the experimental results of two-and three-phase composites in the literatures to be validated. For two-phase composites, the results by MEM are better than those by DEMM for the inclusion volume fraction smaller than 0.5. Comparisons between the results by two models and experimental results have been made for three-phase composite, resulting in that MEM predicts better than DEMM for smaller volume fraction of the inclusion having larger inclusion-to-matrix thermal conductivity ratio, but DEMM predicts better as its volume fraction increases. It has been observed through parametric study that its volume fraction is the critical factor affecting the deviation of predictions by the two models. The results by them show a good agreement with the three-phase composite proposed by Molina et al..

An examination of the concept of a microgeometry proposed by Benveniste reveals that the thermal conductivity of the concentric sphere adopted by generalized self-consistent model (GSCM) is equal to that of the composite. It is also noted that the thermal conductivities of the composite with spherical fillers predicted by GSCM and modified Eshelby model (MEM) are the same. These equivalencies enable to propose a simple and alternative approach for determining the thermal conductivity of the composite with multiply coated spherical fillers by applying MEM repeatedly. The present result is compared and shows the exact agreement with the results from literatures.