Modeling complexity in multifunctional materials

Multifunctional materials exhibit field-coupled properties that often manifest as nonlinear macroscale behavior given their complex multi-particle interactions. The rules governing how relatively simple atomic interactions lead to macroscale complexity have drawn significant interest in harnessing this behavior to predict material behavior at lower computational costs. Here we propose relating fractal material structure to non-integer operators to help understand multiscale complexity in multifunctional materials at the continuum scale. We highlight how this has been used to understand nonlinear viscoelasticity in soft dielectric elastomers, heat transport through fractal 3D printed structures, and a fully coupled fractal form of the time dependent electromagnetic Maxwell’s equations. In each case, we highlight the role of (multi)fractal structure and non-integer operators in understanding complex structure-property relationships in multifunctional materials.