The Inverse Micromechanics Problem given Dielectric Constants for Isotropic Composites with Spherical Inclusions

In this article, convex optimization is introduced as a promising tool to study Eshelby based inverse micromechanics problems. The focus is on inverse micromechanics using the Eshelby-Mori-Tanaka model given the dielectric constants of the composite material and of all of its components. The model is exactly the same for the conductivity properties as well. This choice of model is made since the model is fairly simple and has a closed form analytical solution for the case of spheroidal inclusions as well. The forward or direct micromechanics problem deals with the determination of effective properties of a composite material given the properties of its components and microstructural information. The focus is on isotropic composites and the distribution of inclusions is assumed to be such that this holds. The inverse micromechanics problem considered in this paper deals with the determination of microstructural information given the properties of the composite material and all of its components. Since in this paper, isotropy of the composite and only spherical inclusions are considered, the goal is to determine just volume fractions of the components of the composite material. The inverse problem is formulated as a Linear Programming problem and is solved. Before this, the inverse problem and certain important variants of it are examined through the lens of convex optimization. Lastly, promising results regarding the relationship between dispersive materials, noise in measurements, and quality of obtained volumetric splits are showcased. The scope of the use of convex optimization in inverse micromechanics is discussed.

💡 Research Summary

The paper introduces a convex‑optimization based framework for solving an inverse micromechanics problem in which the goal is to determine the volume fractions of the constituents of an isotropic composite material that contains spherical inclusions, using only the measured effective dielectric constant of the composite and the known dielectric constants of the individual phases. The authors adopt the Eshelby‑Mori‑Tanaka (EMT) mean‑field homogenization model because it yields a closed‑form expression for the effective dielectric constant when the inclusions are spherical and the overall composite is statistically isotropic. Under these assumptions the EMT reduces to a simple linear‑fractional relation

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