Uncertainty Quantification of the Mechanical Properties of 2D Hexagonal Cellular Solid by Analytical and Finite Element Method Approach

The mechanical properties of cellular materials are critical to their performance and must be accurately determined through both analytical and numerical methods. These approaches are essential not only for understanding material behavior but also for evaluating the effects of parametric variations within the unit cell structure. This study focuses on the in-plane comparison of analytical and numerical evaluations of key mechanical properties, including Young's modulus, yield strength, and Poisson's ratio of a 2D hexagonal unit cell subjected to systematic geometric and material variations. Analytically, the mechanical properties were derived based on the geometric configuration of the hexagonal unit cell. Numerically, the finite element method (FEM) simulations employed three different meshing methods: quadrilateral, quad-dominated, and triangular elements, to ensure precision and consistency in the results. The elastic response (Young's modulus) was examined through a parametric sweep involving segmental length variations (4.41 to 4.71 mm) and material modulus (66.5 to 71.5 GPa), revealing percentage differences between analytical and numerical results ranging from -8.28% to 10.87% and -10.58% to 11.95%, respectively. Similarly, yield strength was evaluated with respect to variations in segmental length (4.41 to 4.71 mm) and wall thickness (1.08 to 1.11 mm), showing discrepancies between -2.86% to -5.53% for segmental length and 7.76% to 10.57% for thickness. For Poisson's ratio, variations in the same parameters led to differences ranging from -7.05% to -12.48% and -9.11% to -12.64%, respectively. Additionally, uncertainty was assessed through relative entropy measures-Bhattacharyya, Kullback-Leibler, Hellinger, and Jeffreys-to evaluate the sensitivity of homogenized properties to input variability. These entropy measures quantify the probabilistic distance between core material distributions and their effective counterparts, reinforcing the importance of precise modeling in the design and optimization of cellular structures.