Two-component composites whose effective conductivities are power means of the local conductivities (original) (raw)
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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.
Reciprocal Relations, Bounds, and Size Effects for Composites with Highly Conducting Interface
SIAM Journal on Applied Mathematics, 1997
We provide a reciprocal relation linking the effective conductivity of a composite with highly conducting phase interfaces to that of a composite with the same phase geometry but with an electrical contact resistance at phase interfaces. A field relationship linking the electric field inside a composite with highly conducting phase interfaces to the current in a composite with contact resistance between phases is found. New size effects exhibited by isotropic particulate suspensions with highly conducting interface are obtained. The effective properties of periodic composites are shown to be monotonically increasing as the size of the period cell tends to zero. The role of surface energy for energy minimizing polydisperse suspensions of disks is examined; a necessary condition for isotropic polydisperse suspensions with minimal effective conductivity is found. For monodisperse suspensions of spheres, a critical radius is found for which the electric field is uniform throughout the composite.
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.
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.
Statistical and Scaling Properties of the acacac Conductivity in Thin Metal-Dielectric Composites
1998
We Study in this paper the scaling and statistical properties of the acacac conductivity of thin metal-dielectric films in different regions of the loss in the metallic components and particularly in the limit of vanishing loss. We model the system by a 2D RL−CRL-CRL−C network and calculate the effective conductivity by using a real space renormalization group method. It is found that the real conductivity strongly fluctuates for very small losses. The correlation length, which seems to be equivalent to the localization length, diverges for vanishing losses confirming our previous results for the decay of the real conductivity with the loss. We found also that the distribution of the real conductivity becomes log-normal below a certain critical loss RcR_{c}Rc which is size dependent for finite systems. For infinite systems this critical loss vanishes and corresponds to the phase transition between localized modes for finite losses and the extended ones at zero loss.
Isotropic conductivity of two-dimensional three-component symmetric composites
Journal of Physics A: Mathematical and General, 2000
The effective dc-conductivity problem of isotropic, twodimensional (2D), three-component, symmetric, regular composites is considered. A simple cubic equation with one free parameter for σe(σ1, σ2, σ3) is suggested whose solutions automatically have all the exactly known properties of that function. Numerical calculations on four different symmetric, isotropic, 2D, three-component, regular structures show a non-universal behavior of σe(σ1, σ2, σ3) with an essential dependence on micro-structural details, in contrast with the analogous two-component problem. The applicability of the cubic equation to these structures is discussed. An extension of that equation to the description of other types of 2D three-component structures is suggested, including the case of random structures.
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.
Bounds for the effective conductivity of a composite with an imperfect interface
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2001
The problem of bounding the e¬ective conductivity of a two-phase composite with an imperfect interface is considered. The interface can be either highly conducting or resistive, and both the material properties and geometric arrangement of the phases can be anisotropic. The problem is formulated variationally and by choosing appropriate trial elds, new bounds are obtained in terms of upper and lower bounds on the e¬ective conductivity of a composite with the same microgeometry in which the phases are perfectly bonded. The methodology also applies to composites with a nonlinear interface, and a particular example is described.
Link between the conductivity and elastic moduli of composite materials
Physical Review Letters, 1993
We derive relations linking the conductivity a+ and elastic moduli of any two-dimensional, isotropic composite material. Specifically, upper and lower bounds are derived on the effective bulk modulus x+ in terms of a~and on the effective shear modulus p~in terms of a~. In some cases the bounds are attainable by certain microgeometries and thus optimal. Knowledge of the conductivity can yield sharp estimates of the elastic moduli (and vice versa) even for infinite phase contrast.