Material Defects Characterized by Trees

Abstract The detailed characterization of defects, including cracks, in materials has posed a significant challenge in the field of solid mechanics. Currently, irregularly shaped defects that cannot be accurately described using Euclidean geometry are often simplified into shapes such as thin strips and ellipses, which limits their comprehensive characterization. In this study, we propose a novel approach utilizing non-Euclidean trees to characterize various defect details by skeletonizing them into tree structures and constructing their heterogeneous tensors. By employing image processing skeleton algorithms, we introduce the concept of "skeletonized trees" as a new way to characterize defects. Furthermore, we establish the "tree heterogeneous tensor," which incorporates parameters such as diameter, aspect ratio, fractal dimension, centrality, and minimum radius of curvature to represent the properties of defects through node and edge representations within the skeletonized trees. We also introduce for the first time a similarity criterion for single defects in 3D printing and develop the rigidity degradation index equation for determining the maximum parameter weightage of the tree heterogeneous tensor- aspect ratio and centrality. The utilization of skeletonized trees allows for closer representation of defect details related to material degradation while providing a theoretical foundation for understanding damage mechanisms and enhancing model generalization capabilities.