PSI - Issue 77

Zihao Liu et al. / Procedia Structural Integrity 77 (2026) 190–197 Z. Liu et al./ Structural Integrity Procedia 00 (2026) 000–000 Z. Liu et al./ Structural Integrity Procedia 00 (2026) 000–000

191

2 2

RVE

Representative volume element

S

Mechanical stimulus Strain energy density Strain energy density

SED t Time ε Strain tensor t ε Strain tensor

Strain energy density Strain energy density

Cauchy true stress tensor Cauchy true stress tensor

© 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers Keywords: Trabecular bone adaptation, finite element analysis, mechanical stimulus. © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers Keywords: Trabecular bone adaptation, finite element analysis, mechanical stimulus. 1. Bone is a dynamic living material that continually adjusts its mass and structure in response to external loading and (BA), introduced by Julius Wolff in 1982 (see translation into English by Sim (1987). This ‘intelligence’ is mediated through bone remodelling, with cells sensing a mechanical stimulus and coordinating biochemical signals to renew or remove parts of the tissue. Bone remodelling plays a crucial role in skeletal maintenance, structural integrity and mineral content of bone. investigate the mechanisms of trabecular BA (TBA) across various length scales without the potential risks and high costs related to experiments. Various theories and computational models were suggested to describe the BA process. as computational power for finite-element analysis (FEA) enable accurate predictions of BA patterns. osteocytes function as sensor cells in the matrix sensing mechanical signals and transmit a decaying signal to surface osteoblasts and osteoclasts. The model of Huiskes et al. (1987) uses the strain energy density (SED) per unit mass as control parameter of BA. The site stimulus in this case is a distance-weighted sum of nearby osteocyte signals. If the site stimulus is above or below respective thresholds, bone forms or resorbs, respectively; otherwise, it keeps its original state. To better represent the insensitivity of trabecular bone (TB) near homeostasis, van Rietbergen et al. (1995) introduced a lazy-zone concept with remodelling only occurring beyond it. Adachi et al. (1999) use stress non uniformity as BA mechanical stimulus, where the stress detected is some neighbourhood contributes to a mechanical stimulus based on weighted von Mises stress. This accounts for each element’s stress multiplied with its weight linked to the distance from the considered point. The model proposed by Prendergast & Taylor (1994) uses the stimulus based on the levels of strain and accumulated damage. Although many studies compared the effects of different mechanical stimuli (Allena, 2024; Smotrova et al., 2022; Su et al., 2019), the comparison of evolution of 3D trabeculae morphology for different types of stimuli remains underexplored. In this research, a 3D FE bone-adaptation model is suggested, using the non-uniformity method with different mechanical stimuli - total SED, deviatoric SED, and von Mises stress - to compare the realisation of TBA processes. 2. © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers 1. Introduction Bone is a dynamic living material that continually adjusts its mass and structure in response to external loading and metabolic demands, a process known as bone adaptation (BA), introduced by Julius Wolff in 1982 (see translation into English by Sim (1987). This ‘intelligence’ is mediated through bone remodelling, with cells sensing a mechanical stimulus and coordinating biochemical signals to renew or remove parts of the tissue. Bone remodelling plays a crucial role in skeletal maintenance, structural integrity and mineral content of bone. In situ analysis of this process is challenging; so, computational models are used, having the advantage to investigate the mechanisms of trabecular BA (TBA) across various length scales without the potential risks and high costs related to experiments. Various theories and computational models were suggested to describe the BA process. The advancement of high-resolution peripheral quantitative computed tomography (HR pQCT) and micro-CT as well as computational power for finite-element analysis (FEA) enable accurate predictions of BA patterns. So far, different mechanical stimuli for bone adaptation were investigated, describing mechanisms in which osteocytes function as sensor cells in the matrix sensing mechanical signals and transmit a decaying signal to surface osteoblasts and osteoclasts. The model of Huiskes et al. (1987) uses the strain energy density (SED) per unit mass as control parameter of BA. The site stimulus in this case is a distance-weighted sum of nearby osteocyte signals. If the site stimulus is above or below respective thresholds, bone forms or resorbs, respectively; otherwise, it keeps its original state. To better represent the insensitivity of trabecular bone (TB) near homeostasis, van Rietbergen et al. (1995) introduced a lazy-zone concept with remodelling only occurring beyond it. Adachi et al. (1999) use stress non uniformity as BA mechanical stimulus, where the stress detected is some neighbourhood contributes to a mechanical stimulus based on weighted von Mises stress. This accounts for each element’s stress multiplied with its weight linked to the distance from the considered point. The model proposed by Prendergast & Taylor (1994) uses the stimulus based on the levels of strain and accumulated damage. Although many studies compared the effects of different mechanical stimuli (Allena, 2024; Smotrova et al., 2022; Su et al., 2019), the comparison of evolution of 3D trabeculae morphology for different types of stimuli remains underexplored. In this research, a 3D FE bone-adaptation model is suggested, using the non-uniformity method with different mechanical stimuli - total SED, deviatoric SED, and von Mises stress - to compare the realisation of TBA processes. 2. Method This research employs the finite-element analysis (FEA) to simulate TBA at the single-trabeculae scale. The developed model was implemented in Abaqus/CAE 2024 software package with a USDFLD material subroutine (written with Intel Fortran 2025). The model systematically compares the effect of different mechanical stimuli on resultant TB morphology employing dimensionless non-uniformity metrics. (written with Intel Fortran 2025). The model systematically compares the effect of different mechanical stimuli on

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