# Enhancing the Electrical and Thermal Conductivities of Polymer Composites via Curvilinear Fibers: An Analytical Study

## TL;DR

This paper develops an analytical framework for optimizing electrical and thermal conductivities in polymer composites with curvilinear fibers, enabling enhanced material performance through precise fiber path design.

## Contribution

It introduces a closed-form solution for electrostatic problems in curvilinear fiber-reinforced materials, facilitating the design of composites with improved conductivity.

## Key findings

- Derived equations for fiber paths optimizing electric conductivity.
- Provided exact electrostatic potential and field expressions validated by finite element analysis.
- Discussed potential for materials with superior conductivity and damage sensing.

## Abstract

The new generation of manufacturing technologies such as e.g. additive manufacturing and automated fiber placement has enabled the development of material systems with desired functional and mechanical properties via particular designs of inhomogeneities and their mesostructural arrangement. Among these systems, particularly interesting are materials exhibiting \textbf{Curvilinear Transverse Isotropy} (CTI) in which the inhomogeneities take the form of continuous fibers following curvilinear paths designed to e.g. optimize the electric and thermal conductivity, and the mechanical performance of the system. In this context, the present work proposes a general framework for the exact, closed-form solution of electrostatic problems in materials featuring curvilinear transverse isotropy. First, the general equations for the fiber paths that optimize the electric conductivity are derived leveraging a proper conformal coordinate system. Then, the continuity equation for the curvilinear, transversely isotropic system is derived in terms of electrostatic potential. A general exact, closed-form expression for the electrostatic potential and electric field is derived and validated by Finite Element Analysis. Finally, potential avenues for the development of materials with superior electric conductivity and damage sensing capabilities are discussed.