Hierarchical n-point polytope functions for quantitative representation of complex heterogeneous materials and microstructural evolution

Effective and accurate characterization and quantification of complex microstructure of a heterogeneous material and its evolution under external stimuli are very challenging, yet crucial to achieving reliable material performance prediction, processing optimization and advanced material design. Here, we address this challenge by developing a set of hierarchical statistical microstructural descriptors, which we call the “n-point polytope functions” P_n, for quantitative characterization, representation and modeling of complex material microstructure and its evolution. These polytope functions successively include higher-order n-point statistics of the features of interest in the microstructure in a concise, expressive, explainable, and universal manner; and can be directly computed from multi-modal imaging data. We develop highly efficient computational tools to directly extract the P_n functions up to n = 8 from multi-modal imaging data. Using simple model microstructures, we show that these statistical descriptors effectively “decompose” the structural features of interest into a set of “polytope basis”, allowing one to easily detect any underlying symmetry or emerging features during the structural evolution. We apply the P_n functions to quantify and model a variety of heterogeneous material systems, including particle-reinforced composites, metal-ceramic composites, concretes, porous materials; as well as the microstructural evolution in an aged lead-tin alloy. Our results indicate that the P_n functions can offer a practical set of basis for quantitative microstructure representation (QMR), for both static 3D complex microstructure and 4D microstructural evolution of a wide spectrum of heterogeneous material systems.