  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.
