The potential of the fractional derivative technique is demonstrated on the example of derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimensionality of a temporal fractal ensemble (in a sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of a microstructure of inhomogeneous media exhibiting nonexponential dielectric relaxation is built by singling out groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level, the relaxation should be of a classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times.

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A fractal model of structure [V. V. Novikov et al., Phys. Rev. E 63, 036120 (2001)] is generalized and applied to study the viscoelastic properties of a two-component, strongly inhomogeneous medium of chaotic structure with one component exhibiting a negative shear modulus. It is shown that similar to the results obtained recently in frames of the Hashin -Strikman model [R. S. Lakes, Phys. Rev. Lett. 86, 2897 (2001)], the present fractal model predicts the existence of composites having effective shear and damping coefficients much higher than those of the component phases. The viscoelastic properties of the strongly inhomogeneous medium are, however, qualitatively different from the properties of the Hashin-Strikman medium.

The elastic properties of an inhomogeneous medium with chaotic structure were derived within the framework of a fractal model using the iterative averaging approach. The predicted values of a critical index for the bulk elastic modulus and of the Poisson ratio in the vicinity of a percolation threshold were in fair agreement with the available experimental data for inhomogeneous composites.

Basic physical concepts and limitations of current approaches to the theoretical description of the composition dependence of heat conductivity of microheterogeneous polymer materials (MHM) are reviewed. All "pragmatic" approaches (i.e., those assuming the existence of a infinitely thin, "mathematieal" interface between the components) fail to account explicitly for salient structural features of MHM such as the onset of an "infinite" duster of a disperse component at the percolation threshold, and the transition of a portion of a continuous component into a structurally different "boundary interphase" (BI). Among the "physical" approaches, it is apparently the Step-by-Step-Averaging (SSA) model which accounts simultaneously for both cited structural features of MHM. The SSA model was shown to provide a quantitative description of the experimental data available by an appropriate choice of relevant BI parameters (i.e., thickness and "partial" heat conductivity): at the present stage, however, the numerical values of the latter should be considered as fitting variables, rather than true material properties of BI.