Isotropic conductivity of two-dimensional three-component symmetric composites (original) (raw)
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Estimating effective conductivity of unidirectional transversely isotropic composites
Vietnam Journal of Mechanics, 2013
Three-point correlation bounds are constructed on effective conductivity of unidirectional composites, which are isotropic in the transverse plane. The bounds contain, in addition to the properties and volume proportions of the component materials, three-point correlation parameters describing the micro-geometry of a composite, and are tighter those obtained in [1]. The bounds, applied to some disordered and periodic composites, keep inside the numerical homogenization results obtained by Fast Fourier method.
Physical Review B, 2005
Exact linear relations are found among different elements of the macroscopic conductivity tensor of a three-dimensional, two-constituent composite medium with a columnar microstructure, without any further assumptions about the forms of the constituent conductivities: Those can be arbitrary nonscalar, nonsymmetric, and nonreal ͑i.e., complex valued͒ tensors. These relations enable all the elements of the macroscopic conductivity tensor of such a system to be obtained, from a knowledge of the macroscopic conductivity tensor components only in the plane perpendicular to the columnar axis. Exact linear relations are also found among different elements of the macroscopic resistivity tensor of such systems. Again, these relations enable all the elements of the macroscopic resistivity tensor of such a system to be obtained, from a knowledge of the macroscopic resistivity tensor components only in the plane perpendicular to the columnar axis. We also present simple exact expressions for all elements of the macroscopic conductivity tensor of a three-dimensional composite medium with a parallel slabs or laminar microstructure and an arbitrary number of constituents, again without making any assumptions about the forms of the constituent conductivities, which can be arbitrary nonscalar, nonsymmetric, and nonreal tensors. The latter results were obtained previously, but their great generality and extreme simplicity were not realized by most physicists.
Bounds for effective properties of multimaterial two-dimensional conducting composites
Mechanics of Materials, 2009
The paper suggests exact bounds for the effective conductivity of an isotropic multimaterial composite, which depend only on isotropic conductivities of the mixed materials and their volume fractions. These bounds refine Hashin-Shtrikman and Nesi bounds in the region of parameters where they are loose. The bounds by polyconvex envelope are modifies by taking into account the range of fields in optimal structures. The bounds are a solution of a formulated finite-dimensional constrained optimization problem. For threematerial composites, bounds for effective conductivity are found in an explicit form. Three-material isotropic microstructures of extremal conductivity are found. It is shown that they realize the bounds for all values of conductivities and volume fractions. Optimal structures are laminates of a finite rank. They vary with the volume fractions and experience two topological transitions: For large values of m 1 , the domain of material with minimal conductivity is connected, for intermediate values of m 1 , no material forms a connected domain, and for small values of m 1 , the domain intermediate material is connected.
Polarization versus Mori-Tanaka approximations for effective conductivity of isotropic composites
Vietnam Journal of Mechanics, 2018
Our polarization approximations for the effective conductivity of isotropic multicomponent materials, constructed recently as approximate solutions to the minimum energy principles, are compared with the widely used Mori-Tanaka approximation, derived as an approximate solution of the field equations. The similarities and differences, advantages and disadvantages of both approaches are analysed with illustrating numerical examples.
On bounding the effective conductivity of isotropic composite materials
Zeitschrift für angewandte Mathematik und Physik, 1991
In this paper inequalities for the effective conductivity of isotropic composite materials are derived. These inequalities depend on several coefficients characterizing the microstructure of composites. The obtained coefficients can be exactly calculated for models of a two-component aggregate of multisized, coated ellipsoidal inclusions, packed to fill all space. As a result, new bounds for effective conductivity, considerably narrower than those of Hashin-Shtrikman, are established for such models of composite materials.
Conductivity of a two-dimensional composite containing elliptical inclusions
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009
We develop a method of functional equations to derive analytical approximate formulae for the effective conductivity tensor of the two-dimensional composites with elliptical inclusions. The sizes, the locations and the orientations of the ellipses can be arbitrary. The analytical formulae contain all the above geometrical parameters in symbolic form.
Journal of Physics A: Mathematical and General, 2004
We present a new numerical method solving exactly Kirchhoff laws to determine the effective ac and dc conductivity and the local field for large scale 3D composites with any number of components. This method is an extension of a previous one restricted to 2D and two-component composites. It is much slower than the Frank and Lobb method for conductivity but calculates in addition exactly the local field for 3D systems with large sizes (which was not done by any previous method). The local field enhancement obtained by this method is in good agreement with recent experimental results and is two orders of magnitude lower than the values obtained by a widely used real space renormalization group method. We further discuss some 3D example calculations of impedance spectra and Cole and Cole diagrams for a threecomponent sample as well as the local field at the surface plasmon resonance.
Modern Physics Letters B, 2001
We study the effect of the local arrangement of the metallic grains in metal-dielectric composite films on the statistical properties of the effective conductivity and the local field at the percolation threshold and for a characteristic frequency (where the conductivities of the two components are of the same magnitude). It is found that the segregation enhances sensitively the effective conductivity of the system as well as the local field. Further discussions on the field localization and other phenomena are provided.