Hyper–Micro/NanoStructures and SuperHyper–Micro/NanoStructures: A Unified Framework for Multi-Scale Structural Modeling

Hyperstructures and their hierarchical extensions [1]—SuperHyperStructures—provide a versatile formalism for modeling multi-layered and complex phenomena [2, 3]. A MicroStructure is a measurable mapping that assigns to each material point a single local state, representing attributes such as phase, crystal orientation, chemical composition, and porosity at the microscale [4–8]. A NanoStructure is a physical feature in which one or more dimensions are confined to the nanoscale range (0.1–100 nm), a constraint that significantly influences the material’s properties and functionality [9–12]. Related concepts include Nanofoam [13, 14], Nanomaterials [15–17], Nanohole [18, 19], Nanomesh [20–22], and Nanoparticle [23–25]. In this paper, we define extended concepts of both MicroStructure and NanoStructure using the frameworks of HyperStructure and SuperHyperStructure, and examine their theoretical properties.