A novel method for depolarization tensor and average form of an arbitrarily shaped inclusion: Extension to different physical fields and their effective transport properties of composites

Depolarization tensor (DT), exclusively controlled by the inclusion shape, is crucial in assessing composite transport behavior. However, the solution of DT for a non-ellipsoidal inclusion exhibits nonuniformity and lacks a closed-form expression. Average DT is then introduced to ensure uniformity and seamlessly integrate the evaluation of effective transport properties of composites. In this work, we devise a novel method to obtain DT and average DT of an arbitrarily shaped inclusion by leveraging mathematical similarity from Greenʼs function to degenerate the Eshelby tensor and average Eshelby tensor in an elastic field. The method circumvents singularity and complexity in direct integration and successfully extracts all tensor components. As natural extensions, we also derive the Eshelby conduction tensor, polarizability tensor, and their averages. The accuracy of the method is validated by comparing the average polarizabilities of Platonic solids and stem cells with existing literature data. Furthermore, we incorporate average DT into a developed homogenization model to predict effective transport properties of two-phase composites comprising randomly distributed congruent irregular inclusions. Comparisons with finite element simulations demonstrate accurate predictions of the effective transport properties like effective permeability, diffusivity, conductivity, and permittivity. The proposed framework offers valuable guidance for tailoring inclusion shapes to design particulate/fibrous/porous/biological composites.