The inverse micromechanics problem given the 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 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 (thermal and electrical) as well. This choice of model is made since the model is fairly simple, has a closed form analytical solution, and is known to perform well 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 the isotropy of the composite and only spherical inclusions are considered, the goal is to determine only the 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 are presented on the relationship between dispersive materials, noise in measurements, and the quality of the obtained volumetric splits. The scope of the use of convex optimization in inverse micromechanics is discussed.