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Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations

Hao Wang, Yongtao Lyu, Jian Jiang, Feihu Zhao Orcid Logo, Sergei Bosiakov, Hanxing Zhu

Frontiers in Bioengineering and Biotechnology, Volume: 13, Start page: 1626104

Swansea University Author: Feihu Zhao Orcid Logo

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DOI (Published version): 10.3389/fbioe.2025.1626104

Abstract

Introduction: Bone scaffolds are widely used for repairing bone defects. As a biomimetic structure for bone scaffolds, the triply periodic minimal surface (TPMS) structure is an ideal choice. To evaluate/characterize the mechanical properties of TPMS structures, multiple methods (e.g., via experimen...

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Published in: Frontiers in Bioengineering and Biotechnology
ISSN: 2296-4185
Published: Frontiers Media S.A. 2026
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Using approximate analytical approach, the mechanical properties of structures can be predicted quickly and efficiently. Therefore, it is necessary to determine the applicable range to ensure that the calculated mechanical properties of TPMS structures with varying porosity and strut orientation are acceptable. Methods: In this paper, approximate analytical prediction of elastic properties of TPMS structures (i.e., Diamond) with varying porosities and strut orientations was investigated, and finite element (FE) method and theory of elasticity were compared with the approximate analytical approach. The ranges for porosity were from 70% to 90%. The ranges for orientation were defined by rotating the scaffold from 0&#xB0; to 90&#xB0; along the [100] and [110] directions, and from -30&#xB0; to 90&#xB0; along the [111] direction. Due to the cubic symmetry of Diamond structure, these angular ranges ensure that the structure is non-repeating and is comprehensively analyzed in all three directions. Additionally, experimental tests were performed to validate the feasibility of the non-experimental methods. Results: It was shown from the experimental validation that the results from non-experimental methods were acceptable at certain porosities and orientations. The FE method, which is commonly used and a reliable approach, was utilized to represent the non-experimental methods and was compared with the experimental results. Therefore, the approximate analytical solutions and the results from elasticity theory were indirectly compared with experimental results. When the porosity of the structure was 85%, the approximate analytical solution showed differences of 17.65% relative to the FE result and 39.13% relative to the elasticity theory result. Therefore, the approximate analytical solution was considered acceptable at a higher porosity. The acceptable ranges of the porosity for applying the approximate analytical approach were higher than 85% in the [001] and [110] directions, and higher than 90% in the [111] direction. At the same structural porosity, in the (100) plane, the predicted results were acceptable when the structural orientation was close to 0&#xB0; or 90&#xB0;. In the (110) plane, the predicted results were acceptable when the structural orientation was close to 0&#xB0;. In the (111) plane, whether the predicted results can be accepted or not was basically independent of the structural orientation but was dependent on the porosity of the structure. The planes of (100), (110) and (111) are defined as the planes perpendicular to the directions of [100], [110] and [111], respectively. 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spelling 2026-01-19T10:45:50.6193580 v2 71269 2026-01-19 Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations 1c6e79b6edd08c88a8d17a241cd78630 0000-0003-0515-6808 Feihu Zhao Feihu Zhao true false 2026-01-19 EAAS Introduction: Bone scaffolds are widely used for repairing bone defects. As a biomimetic structure for bone scaffolds, the triply periodic minimal surface (TPMS) structure is an ideal choice. To evaluate/characterize the mechanical properties of TPMS structures, multiple methods (e.g., via experiment or theoretical analysis) can be used. Each method has its advantages and disadvantages. Using approximate analytical approach, the mechanical properties of structures can be predicted quickly and efficiently. Therefore, it is necessary to determine the applicable range to ensure that the calculated mechanical properties of TPMS structures with varying porosity and strut orientation are acceptable. Methods: In this paper, approximate analytical prediction of elastic properties of TPMS structures (i.e., Diamond) with varying porosities and strut orientations was investigated, and finite element (FE) method and theory of elasticity were compared with the approximate analytical approach. The ranges for porosity were from 70% to 90%. The ranges for orientation were defined by rotating the scaffold from 0° to 90° along the [100] and [110] directions, and from -30° to 90° along the [111] direction. Due to the cubic symmetry of Diamond structure, these angular ranges ensure that the structure is non-repeating and is comprehensively analyzed in all three directions. Additionally, experimental tests were performed to validate the feasibility of the non-experimental methods. Results: It was shown from the experimental validation that the results from non-experimental methods were acceptable at certain porosities and orientations. The FE method, which is commonly used and a reliable approach, was utilized to represent the non-experimental methods and was compared with the experimental results. Therefore, the approximate analytical solutions and the results from elasticity theory were indirectly compared with experimental results. When the porosity of the structure was 85%, the approximate analytical solution showed differences of 17.65% relative to the FE result and 39.13% relative to the elasticity theory result. Therefore, the approximate analytical solution was considered acceptable at a higher porosity. The acceptable ranges of the porosity for applying the approximate analytical approach were higher than 85% in the [001] and [110] directions, and higher than 90% in the [111] direction. At the same structural porosity, in the (100) plane, the predicted results were acceptable when the structural orientation was close to 0° or 90°. In the (110) plane, the predicted results were acceptable when the structural orientation was close to 0°. In the (111) plane, whether the predicted results can be accepted or not was basically independent of the structural orientation but was dependent on the porosity of the structure. The planes of (100), (110) and (111) are defined as the planes perpendicular to the directions of [100], [110] and [111], respectively. Discussion: Data in the present study provide valuable guidance on applying the approximate analytical approach to efficiently predict the mechanical properties of TPMS structures prior to performing formal calculations and experiments. Journal Article Frontiers in Bioengineering and Biotechnology 13 1626104 Frontiers Media S.A. 2296-4185 triply periodic minimal surface, diamond structure, approximate analytical approach, finite element method, theory of elasticity, effective elastic modulus 5 1 2026 2026-01-05 10.3389/fbioe.2025.1626104 COLLEGE NANME Engineering and Applied Sciences School COLLEGE CODE EAAS Swansea University Another institution paid the OA fee This work was supported by the National Natural Science Foundation of China (12572376, W2521087), the National Science Foundation of Liaoning Province (2025-MS-365), the National Key R&D Program of China (2024YFE0213500), the Fundamental Research Funds for the Central Universities (DUT25YG245), and the State Key Laboratory of Structural Analysis, Optimization, and CAE Software for Industrial Equipment (GZ25110). 2026-01-19T10:45:50.6193580 2026-01-19T10:39:21.2545273 Faculty of Science and Engineering School of Engineering and Applied Sciences - Biomedical Engineering Hao Wang 1 Yongtao Lyu 2 Jian Jiang 3 Feihu Zhao 0000-0003-0515-6808 4 Sergei Bosiakov 5 Hanxing Zhu 6 71269__36039__e84334ad588e4cfa9ad13d96bc4b4c48.pdf fbioe-13-1626104.pdf 2026-01-19T10:39:21.2252142 Output 4197840 application/pdf Version of Record true © 2026 Wang, Lyu, Jiang, Zhao, Bosiakov and Zhu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). true eng https://creativecommons.org/licenses/by/4.0/
title Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
spellingShingle Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
Feihu Zhao
title_short Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
title_full Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
title_fullStr Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
title_full_unstemmed Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
title_sort Approximate analytical prediction on elastic properties of Diamond structures with varying porosities and orientations
author_id_str_mv 1c6e79b6edd08c88a8d17a241cd78630
author_id_fullname_str_mv 1c6e79b6edd08c88a8d17a241cd78630_***_Feihu Zhao
author Feihu Zhao
author2 Hao Wang
Yongtao Lyu
Jian Jiang
Feihu Zhao
Sergei Bosiakov
Hanxing Zhu
format Journal article
container_title Frontiers in Bioengineering and Biotechnology
container_volume 13
container_start_page 1626104
publishDate 2026
institution Swansea University
issn 2296-4185
doi_str_mv 10.3389/fbioe.2025.1626104
publisher Frontiers Media S.A.
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Engineering and Applied Sciences - Biomedical Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Engineering and Applied Sciences - Biomedical Engineering
document_store_str 1
active_str 0
description Introduction: Bone scaffolds are widely used for repairing bone defects. As a biomimetic structure for bone scaffolds, the triply periodic minimal surface (TPMS) structure is an ideal choice. To evaluate/characterize the mechanical properties of TPMS structures, multiple methods (e.g., via experiment or theoretical analysis) can be used. Each method has its advantages and disadvantages. Using approximate analytical approach, the mechanical properties of structures can be predicted quickly and efficiently. Therefore, it is necessary to determine the applicable range to ensure that the calculated mechanical properties of TPMS structures with varying porosity and strut orientation are acceptable. Methods: In this paper, approximate analytical prediction of elastic properties of TPMS structures (i.e., Diamond) with varying porosities and strut orientations was investigated, and finite element (FE) method and theory of elasticity were compared with the approximate analytical approach. The ranges for porosity were from 70% to 90%. The ranges for orientation were defined by rotating the scaffold from 0° to 90° along the [100] and [110] directions, and from -30° to 90° along the [111] direction. Due to the cubic symmetry of Diamond structure, these angular ranges ensure that the structure is non-repeating and is comprehensively analyzed in all three directions. Additionally, experimental tests were performed to validate the feasibility of the non-experimental methods. Results: It was shown from the experimental validation that the results from non-experimental methods were acceptable at certain porosities and orientations. The FE method, which is commonly used and a reliable approach, was utilized to represent the non-experimental methods and was compared with the experimental results. Therefore, the approximate analytical solutions and the results from elasticity theory were indirectly compared with experimental results. When the porosity of the structure was 85%, the approximate analytical solution showed differences of 17.65% relative to the FE result and 39.13% relative to the elasticity theory result. Therefore, the approximate analytical solution was considered acceptable at a higher porosity. The acceptable ranges of the porosity for applying the approximate analytical approach were higher than 85% in the [001] and [110] directions, and higher than 90% in the [111] direction. At the same structural porosity, in the (100) plane, the predicted results were acceptable when the structural orientation was close to 0° or 90°. In the (110) plane, the predicted results were acceptable when the structural orientation was close to 0°. In the (111) plane, whether the predicted results can be accepted or not was basically independent of the structural orientation but was dependent on the porosity of the structure. The planes of (100), (110) and (111) are defined as the planes perpendicular to the directions of [100], [110] and [111], respectively. Discussion: Data in the present study provide valuable guidance on applying the approximate analytical approach to efficiently predict the mechanical properties of TPMS structures prior to performing formal calculations and experiments.
published_date 2026-01-05T05:33:32Z
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