Effective Medium Theory for the Elastic Properties of Composite Materials with Various Percolation Thresholds

Abstract

It is discussed that the classical effective medium theory for the elastic properties of random heterogeneous materials is not congruous with the effective medium theory for the electrical conductivity. In particular, when describing the elastic and electro-conductive properties of a strongly inhomogeneous two-phase composite material, the steep rise of effective parameters occurs at different concentrations. To achieve the logical concordance between the cross-property relations, a modification of the effective medium theory of the elastic properties is introduced. It is shown that the qualitative conclusions of the theory do not change, while a possibility of describing a broader class of composite materials with various percolation thresholds arises. It is determined under what conditions there is an elasticity-theory analogue of the Dykhne formula for the effective conductivity. The theoretical results are supported by known experiments and show improvement over the existing approach. The introduction of the theory with the variable percolation threshold paves the way for describing the magneto-elastic properties of magnetorheological elastomers, as it has been achieved for magneto-dielectric and magnetic properties recently.

Theoretical Modeling of Magnetoactive Elastomers on Different Scales: A State-of-the-Art Review

“…Since MAEs can be considered as composite materials, it is natural to employ effective medium theory (EMT) or effective medium approximations to calculate their macroscopic properties from the known physical properties of the constitutive materials. This approach has been developed in the works of Snarskii et al [ 125 , 126 , 127 , 128 ]. The initial MAE microstructure was assumed to be random heterogeneous, in particular, randomly located spherical inclusions of the first phase (carbonyl iron) dispersed in a continuous polymer matrix (second phase).…”

“…The new term is defined in the following way: where is the function of the external magnetic field (15) [ 127 ].…”

Analytical Study of Porous Organosilicate Glass Films Prepared from Mixtures of 1,3,5- and 1,3-Alkoxysilylbenzenes

“…Such a difference in the structure, apparently, leads to different values of the percolation thresholds for conductivity and the threshold for hardness. Good examples comparing stiffness and electrical percolation thresholds have been demonstrated in the references [39,40]. Snarskii et al [40] showed that in the frameworks of effective medium approximation, the electrical percolation thresholds and the stiffness threshold have different values in the three-dimensional case: percolation threshold = 0.3 and stiffness threshold = 0.5.…”

“…Good examples comparing stiffness and electrical percolation thresholds have been demonstrated in the references [39,40]. Snarskii et al [40] showed that in the frameworks of effective medium approximation, the electrical percolation thresholds and the stiffness threshold have different values in the three-dimensional case: percolation threshold = 0.3 and stiffness threshold = 0.5. As another example, we can refer to work [39], in which randomly distributed networks of carbon nanotubes (CNTs) were studied using theoretical analysis as well as numerical modelling.…”

Percolation threshold and effective properties of CNTs-reinforced two-phase composite materials

“…It is well known that for a matrix-composite material its thermal, electrical and mechanical properties, often named as effective properties, are dependent on the properties of the matrix and fillers or inclusions forming the composite, the volume fraction and aspect ratio of the fillers, and the way how the fillers are dispersed and distributed in the matrix [1,2]. It has been widely reported that, the effective property of a composite, such as its electrical conductivity, changes slightly when the volume fraction of fillers is very small, but sharply when the volume fraction of fillers reaches to a certain narrow range [3]. This phenomenon is characterised as the percolation [4], which describes the connectivity of fillers within the matrix and its effect on the macroscale properties of the composite.…”