A New Numerical-Homogenization Method to Predict the Effective Permittivity of Composite Materials
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TECHNICAL ARTICLE
A New Numerical‑Homogenization Method to Predict the Effective Permittivity of Composite Materials Younes Jarmoumi1 · Abdelali Derouiche1 · Fatna Benzouine1 Received: 13 July 2020 / Accepted: 11 November 2020 © The Minerals, Metals & Materials Society 2020
Abstract The effective permittivity of composite materials depends highly on the geometry, the arrangement, and the permittivity of each component. This research proposes a new numerical method that takes into account those dependencies through assemblies of virtual capacitors (electrical circuit). Then, the effective permittivity is calculated from the equivalent capacity of the suggested circuits. The new presented method delimits the effective permittivity of heterogeneous materials through two obtained different summation expressions. This new method was applied to some inclusions to investigate its validity by comparing its results to the existing models and the results of the finite element method. The new results are shown in good agreement with the literature. Moreover, the presented model takes less time in simulation to estimate the effective permittivity compared to the finite element method. The new model can be applied to all kinds of composite materials including the ones that involve multiple phases and complex geometries. Keywords Effective properties · Simulation techniques · Dielectric constant · Circuit-based models · Heterogeneous media
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Introduction 𝜀W,ser = Owing to the important role played by the effective permittivity in physics and other different science fields like geology [1, 2], biology [3–5], hydrology [6] etc., considerable research efforts have been devoted for studying and predicting the effective permittivity of heterogeneous structures as well as other physical properties. Generally, the heterogeneous biphasic structures show an effective permittivity’s value that is between those of the constituents (at least for non lossy mixtures [7]). The first suggested rules known to delimit the effective permittivity using the volume fraction of the constituents were the ones established by Wiener in 1912; in this case, the effective permittivity bounds are given by the following expressions [8]. ) ( 𝜀W,par = 1 − 𝜙inc ⋅ 𝜀mat + 𝜙inc ⋅ 𝜀inc (1)
1 − 𝜙inc 𝜙inc + 𝜀mat 𝜀inc
1
Laboratory of Polymer Physics and Critical Phenomena, Physics Department, Faculty of Sciences Ben M’Sik, University of Hassan II Casablanca, Casablanca, Morocco
(2)
where 𝜙inc , εinc, 𝜙mat , and εmat denote, respectively, the volume fraction of the inclusion, the relative permittivity of the inclusion, the volume fraction of the host matrix, and the relative permittivity of the host matrix. The Eq. (1) corresponds to the exact permittivity of two capacitors mounted in parallel to form one capacitor as shown in the Fig. 1a, and the Eq. (2) gives the exact permittivity of two capacitors mounted in series which corresponds to the Fig. 1b. In 1962, based on a variational treatment of functional energy for mixtures that involve particles distr
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