Conductivity Of Two-dimensional Composites Research Articles

Conductivity of two‐dimensional composites with randomly distributed elliptical inclusions

AbstractAnalytical approximate formulae for the effective conductivity tensor of the two‐dimensional composites with elliptical inclusions are derived in the form of polynomial approximations in concentration. New formulae explain the seeming contradiction between various formulae derived in the framework of self consistent methods. Random composites with high conducting inclusions of two different shapes (elliptical and circle) of the same area are compared. It is established that greater relative concentration of ellipses increases the effective conductivity.

Conductivity of Composites Containing Anisotropic Graded Fibers

The conductivity of two-dimensional composites with fibers of graded anisotropic materials, whose conductivity profile in the radial direction is different from that in the tangential direction, is investigated in this paper. A formalism for treating the conducting properties of graded composites with anisotropic inclusions is proposed. The three model graded fibers, whose conductivities have the power-law, linear, and exponential profiles, respectively, are studied in detail. In these cases, the potentials in the host and fiber regions are both derived analytically. The effective conductivity for these kinds of graded composites is calculated based on the analytical solutions for graded fibers.

Quasiconformal mappings as a tool to study certain two-dimensional G-closure problems

A recent theorem due to Astala establishes the best exponent for the area distortion of planar K-quasiconformal mappings. We use a refinement of Astala's theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions. For n= 2 we prove the optimality of the bounds under certain additional assumptions on the G-closure parameters.

Bounds on the effective conductivity of two-dimensional composites made of n ≧ 3 isotropic phases in prescribed volume fraction: the weighted translation method

We establish lower and upper bounds which are valid for the overall conductivity of twodimensional composites. They are based on a method which modifies the so-called translation method in a way which makes it effectively much more flexible. When specialised to composites of n > 2 isotropic phases, the new bounds are often strictly better than all the previously known ones. From the mathematical point of view, the improvement is due mainly to a new regularity result in p.d.e.s [2]. From the physical point of view the latter can be interpreted as a result bounding in a suitable sense the fluctuations of the ‘electric field’.