Predicting Crack Propagation in Brittle TPMS Under Uniaxial Loading: A Dual Concept Approach via Geodesic Lines and Cross‐Sectional Analysis

Triply periodic minimal surface (TPMS) structures have been rapidly adopted in engineering applications due to their superior mechanical properties, geometric continuity, and suitability for additive manufacturing. This study investigates the influence of the unique morphologies in four representative TPMS unit cells (Primitive, Gyroid, Neovius, and IWP) on brittle crack propagation. Two geometry‐driven concepts are proposed: (1) the prediction of crack propagation along geodesic lines on the surface and (2) the evaluation of damage localization through the distribution of cross‐sectional area along the loading direction. A mathematical framework is developed to compute analytical geodesics on curved surfaces and to compare them with numerically obtained crack paths. In addition, the areas of irregular cross‐sectional shapes in each TPMS unit cell are analyzed to assess how cross‐sectional variation influences compressive resistance. The base material of the TPMS structures is alumina ceramic modeled in finite element (FE) analysis using the Johnson–Holmquist II constitutive model. The results demonstrate that the numerical crack propagation paths are highly consistent with the analytical geodesics and that damage localization is governed by regions with small cross‐sectional areas. This approach provides a robust geometric basis for predicting fracture in complex lattice topologies.