PSI - Issue 79

Marco Piacentini et al. / Procedia Structural Integrity 79 (2026) 394–403

395

1. Introduction

Fragility fractures are placing an increasingly significant health and economic burden as life expectancy rises Borgstro¨m et al. (2020). A major factor contributing to age-related bone fragility is osteoporosis—a systemic disease characterized by reduced bone mass and deterioration of trabecular microarchitecture. However, a comprehensive understanding of how microscale structural features govern mechanical performance—and in particular fracture initiation and propagation—remains incomplete Buccino et al. (2021). This knowledge gap hinders both early diagnosis of bone embrittlement and the development of treatment strategies. Recent advances combining synchrotron radiation micro-computed tomography with quasi-synchronous mechanical testing have enabled in situ observation of crack nucleation and propagation at sub-micrometer resolution Buccino et al. (2022). However, acquisitions are taken only at limited time intervals and do not give direct access to internal fields such as stresses. Hence, pairing these methods with mechanical simulations can o ff er precious insights into local damage evolution and its dependence on trabecular morphology. Nevertheless, bones are intrinsically hierarchical, with mechanical properties emerging from interactions across multiple structural scales. Furthermore, even characterizing and modeling bone constituent behavior at the trabecular level is well-known to be challenging Pahr and Reisinger (2020); Wu et al. (2018). Therefore, accurate simulations are computationally intensive, and often beyond reach for large statistical studies. Deep learning might help to reduce the computational burden, but still require extensive training datasets and specific adaptations to the task. To address these di ffi culties, it is useful to decompose the problem into simpler subproblems. In this paper, we focus on the influence of the first level of structural hierarchy—the trabecular network—while simplifying other complexities. The simplification is achieved by employing artificial structures that can mimic trabecular architecture. Unlike natural bone, such synthetic structures are not limited by biological variability, material composition, or experimental constraints, and they can be generated and fabricated at low cost with diverse morphologies using additive manufacturing. Importantly, the trabecular network exhibits a non-periodic architecture with a controlled degree of structural disorder. Among the structural families capable of reproducing such architectures, two are most widely reported in the literature: spinodoids Wang et al. (2025) and Voronoi tessellations Chen et al. (2020). In this work, we choose Voronoi tessellations because they allow fine local control over the generated morphology. Methods have been developed to smoothly introduce gradients of density, orientation, and aspect ratio in 3D while filling complex geometries Cao et al. (2024), to perform hierarchical inverse design through topology optimization Padhy et al. (2024), and to modify local morphology through diverse post-processing strategies Zhou et al. (2023); Chen et al. (2021); Kou and Tan (2010). Furthermore, Voronoi-based structures can be tuned continuously from disordered to ordered configurations, enabling systematic investigation of the role of disorder on mechanical behavior Karapiperis and Kochmann (2023); Egmond et al. (2021); Zhu et al. (2001). For this family of structures, deep learning approaches have already been proposed to predict mechanical properties, such as sti ff ness in 2D anisotropic Padhy et al. (2024) and 3D isotropic Zheng et al. (2023a) configurations, or fracture paths in 2D isotropic structures as a function of disorder Karapiperis and Kochmann (2023). However, frameworks capable of simultaneously addressing multiple combinations of tridimensionality, anisotropy, order–disorder transitions, or nonlinear behaviors are still missing. The scope of this study is therefore to take the first steps toward establishing a unifying deep learning framework for predicting the nonlinear properties of Voronoi-based structures, as extracted from physics-based finite element (FE) analysis. Specifically, this work examines the combined e ff ects of anisotropy and nonlinear behavior in a simplified setting: 2D anisotropic beam-based Voronoi tessellations subjected to periodic displacement boundary conditions in uniaxial compression. The use of periodic boundary conditions ensures the elimination of undesired edge e ff ects. Following the present introduction, the paper is structured as follows: Section 2.1 summarizes the methods used for structure generation; Section 2.2 details the FE analysis performed on the generated structures and the postprocessing used to extract the mechanical properties of interest; Section 2.3 introduces the implemented deep learning frameworks; Section 3.1 reports the FE analysis results and relates them to the structural parameters; Section 3.2 illustrates the preliminary results of the deep learning models and discusses limitations and possible improvements in the context of current literature; finally, Section 4 summarizes the findings and limitations of the present study and outlines directions for future work.