×

Analytical relationships for the mechanical properties of additively manufactured porous biomaterials based on octahedral unit cells. (English) Zbl 1443.74053

Summary: Additively manufacturing (AM) techniques make it possible to fabricate open-cell interconnected structures with precisely controllable micro-architectures. It has been shown that the morphology, pore size, and relative density of a porous structure determine its macro-scale homogenized mechanical properties and, thus, its biological performance as a biomaterial. In this study, we used analytical, numerical, and experimental techniques to study the elastic modulus, Poisson’s ratio, and yield stress of AM porous biomaterials made by repeating the same octahedral unit cell in all spatial directions. Analytical relationships were obtained based on both Euler-Bernoulli and Timoshenko beam theories by studying a single unit cell experiencing the loads and boundary conditions sensed in an infinite lattice structure. Both single unit cells and corresponding lattice structures were manufactured using AM and mechanically tested under compression to determine the experimental values of mechanical properties. Finite element models of both single unit cell and lattice structure were also built to estimate their mechanical properties numerically. Differences in the bulk mechanical properties of struts built in different directions were observed experimentally and were taken into account in derivation of the analytical solutions. Although the analytical and numerical results were generally in good agreement, the mechanical properties obtained by the Timoshenko beam theory were closer to numerical results. The maximum difference between analytical and numerical results for elastic modulus and Poisson’s ratio was below 6%, while for yield stress it was about 13%, both occurring at the relative density of 50%. The maximum difference between the analytical and experimental values of the elastic modulus was \(<15\%\) (relative density\(=50\%\)).

MSC:

74-10 Mathematical modeling or simulation for problems pertaining to mechanics of deformable solids
74E30 Composite and mixture properties
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Heinl, P.; Müller, L.; Körner, C.; Singer, R. F.; Müller, F. A., Cellular Ti-6Al-4V structures with interconnected macro porosity for bone implants fabricated by selective electron beam melting, Acta biomaterialia, 4, 5, 1536-1544 (2008)
[2] Murr, L. E.; Gaytan, S. M.; Martinez, E.; Medina, F.; Wicker, R. B., Next generation orthopaedic implants by additive manufacturing using electron beam melting, Int. J. Biomater. (2012), 2012, Article ID 245727
[3] Vaezi, M.; Seitz, H.; Yang, S., A review on 3D micro-additive manufacturing technologies, Int. J. Adv. Manuf. Technol., 67, 5-8, 1721-1754 (2013)
[4] Zadpoor, A. A.; Hedayati, R., Analytical relationships for prediction of the mechanical properties of additively manufactured porous biomaterials, J. Biomed. Mater. Res. Part A (2016)
[5] Van Bael, S.; Chai, Y. C.; Truscello, S.; Moesen, M.; Kerckhofs, G.; Van Oosterwyck, H.; Kruth, J.-P.; Schrooten, J., The effect of pore geometry on the in vitro biological behavior of human periosteum-derived cells seeded on selective laser-melted Ti6Al4V bone scaffolds, Acta Biomaterialia, 8, 7, 2824-2834 (2012)
[6] Truscello, S.; Kerckhofs, G.; Van Bael, S.; Pyka, G.; Schrooten, J.; Van Oosterwyck, H., Prediction of permeability of regular scaffolds for skeletal tissue engineering: a combined computational and experimental study, Acta Biomaterialia, 8, 4, 1648-1658 (2012)
[7] Rumpler, M.; Woesz, A.; Dunlop, J. W.; van Dongen, J. T.; Fratzl, P., The effect of geometry on three-dimensional tissue growth, J. R. Soc. Interface, 5, 27, 1173-1180 (2008)
[8] Bidan, C. M.; Kommareddy, K. P.; Rumpler, M.; Kollmannsberger, P.; Bréchet, Y. J.; Fratzl, P.; Dunlop, J. W., How linear tension converts to curvature: geometric control of bone tissue growth, PloS one, 7, 5, e36336 (2012)
[9] Bidan, C. M.; Kommareddy, K. P.; Rumpler, M.; Kollmannsberger, P.; Fratzl, P.; Dunlop, J. W., Geometry as a factor for tissue growth: towards shape optimization of tissue engineering scaffolds, Adv. Healthc Mater., 2, 1, 186-194 (2013)
[10] Bidan, C. M.; Wang, F. M.; Dunlop, J. W., A three-dimensional model for tissue deposition on complex surfaces, Comput. Methods Biomech. Biomed. Eng., 16, 10, 1056-1070 (2013)
[11] Wauthle, R.; Ahmadi, S. M.; Yavari, S. A.; Mulier, M.; Zadpoor, A. A.; Weinans, H.; Van Humbeeck, J.; Kruth, J.-P.; Schrooten, J., Revival of pure titanium for dynamically loaded porous implants using additive manufacturing, Mater. Sci. Eng.: C, 54, 94-100 (2015)
[12] Sobral, J. M.; Caridade, S. G.; Sousa, R. A.; Mano, J. F.; Reis, R. L., Three-dimensional plotted scaffolds with controlled pore size gradients: effect of scaffold geometry on mechanical performance and cell seeding efficiency, Acta Biomaterialia, 7, 3, 1009-1018 (2011)
[13] Yavari, S. A.; Ahmadi, S.; Wauthle, R.; Pouran, B.; Schrooten, J.; Weinans, H.; Zadpoor, A., Relationship between unit cell type and porosity and the fatigue behavior of selective laser melted meta-biomaterials, J. Mech. Behav. Biomed. Mater., 43, 91-100 (2015)
[14] Li, S.; Xu, Q.; Wang, Z.; Hou, W.; Hao, Y.; Yang, R.; Murr, L., Influence of cell shape on mechanical properties of Ti-6Al-4V meshes fabricated by electron beam melting method, Acta Biomaterialia, 10, 10, 4537-4547 (2014)
[15] Murr, L.; Amato, K.; Li, S.; Tian, Y.; Cheng, X.; Gaytan, S.; Martinez, E.; Shindo, P.; Medina, F.; Wicker, R., Microstructure and mechanical properties of open-cellular biomaterials prototypes for total knee replacement implants fabricated by electron beam melting, J. Mech. Behav. Biomed. Mater., 4, 7, 1396-1411 (2011)
[16] Wang, X.; Xu, S.; Zhou, S.; Xu, W.; Leary, M.; Choong, P.; Qian, M.; Brandt, M.; Xie, Y. M., Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: A review, Biomaterials, 83, 127-141 (2016)
[17] Hollister, S. J., Porous scaffold design for tissue engineering, Nature Mater., 4, 7, 518-524 (2005)
[18] Gibson, L. J.; Ashby, M. F., Cellular solids: structure and properties (1999), Cambridge university press
[19] Shulmeister, V.; Van der Burg, M.; Van der Giessen, E.; Marissen, R., A numerical study of large deformations of low-density elastomeric open-cell foams, Mech. Mater., 30, 2, 125-140 (1998)
[20] Babaee, S.; Jahromi, B. H.; Ajdari, A.; Nayeb-Hashemi, H.; Vaziri, A., Mechanical properties of open-cell rhombic dodecahedron cellular structures, Acta Materialia, 60, 6, 2873-2885 (2012)
[21] Borleffs, M., Finite Element Modeling to Predict Bulk Mechanical Properties of 3D Printed Metal Foams (2012), Delft University of Technology: Delft University of Technology Delft, TU Delft
[22] Zheng, X.; Lee, H.; Weisgraber, T. H.; Shusteff, M.; DeOtte, J.; Duoss, E. B.; Kuntz, J. D.; Biener, M. M.; Ge, Q.; Jackson, J. A., Ultralight, ultrastiff mechanical metamaterials, Science, 344, 6190, 1373-1377 (2014)
[23] Warren, W.; Kraynik, A., Linear elastic behavior of a low-density Kelvin foam with open cells, J. Appl. Mech., 64, 4, 787-794 (1997) · Zbl 0920.73318
[24] Ahmadi, S.; Campoli, G.; Amin Yavari, S.; Sajadi, B.; Wauthlé, R.; Schrooten, J.; Weinans, H.; Zadpoor, A. A., Mechanical behavior of regular open-cell porous biomaterials made of diamond lattice unit cells, J. Mech. Behav. Biomed. Mater., 34, 106-115 (2014)
[25] Smith, M.; Guan, Z.; Cantwell, W., Finite element modelling of the compressive response of lattice structures manufactured using the selective laser melting technique, Int. J. Mech. Sci., 67, 28-41 (2013)
[26] Ptochos, E.; Labeas, G., Elastic modulus and Poisson’s ratio determination of micro-lattice cellular structures by analytical, numerical and homogenisation methods, J. Sandw. Struct. Mater., Article 1099636212444285 pp. (2012)
[27] Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A. A., Mechanics of additively manufactured porous biomaterials based on the rhombicuboctahedron unit cell, J. Mech. Behav. Biomed. Mater., 53, 272-294 (2016)
[28] Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A. A., Mechanical properties of regular porous biomaterials made from truncated cube repeating unit cells: analytical solutions and computational models, Mater. Sci. Eng.: C, 60, 163-183 (2016)
[29] Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A., Mechanical behavior of additively manufactured porous biomaterials made from truncated cuboctahedron unit cells, Int. J. Mech. Sci., 106, 19-38 (2016)
[30] Doyoyo, M.; Hu, J. W., Multi-axial failure of metallic strut-lattice materials composed of short and slender struts, Int. J. Solids Struct., 43, 20, 6115-6139 (2006) · Zbl 1120.74786
[31] Wang, A.-J.; McDowell, D., Yield surfaces of various periodic metal honeycombs at intermediate relative density, Int. J. Plast., 21, 2, 285-320 (2005) · Zbl 1114.74361
[32] Deshpande, V. S.; Fleck, N. A.; Ashby, M. F., Effective properties of the octet-truss lattice material, J. Mech. Phys. Solids, 49, 8, 1747-1769 (2001) · Zbl 1011.74056
[33] Sun, J.; Yang, Y.; Wang, D., Mechanical properties of Ti-6al-4V octahedral porous material unit formed by selective laser melting, Adv. Mech. Eng. (2012), 2012, Article ID 427386
[34] Yang, L., Experimental-assisted design development for an octahedral cellular structure using additive manufacturing, Rapid Prototyp. J., 21, 2, 168-176 (2015)
[35] Hedayati, R.; Sadighi, M.; Mohammadi Aghdam, M.; Zadpoor, A. A., Mechanical properties of additively manufactured thick honeycombs, Materials, 9, 8, 613 (2016)
[36] Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A. A., Effect of mass multiple counting on the elastic properties of open-cell regular porous biomaterials, Mater. Design, 89, 9-20 (2016)
[37] Wauthle, R.; Vrancken, B.; Beynaerts, B.; Jorissen, K.; Schrooten, J.; Kruth, J.-P.; Van Humbeeck, J., Effects of build orientation and heat treatment on the microstructure and mechanical properties of selective laser melted Ti6Al4V lattice structures, Addit. Manuf., 5, 77-84 (2015)
[38] Park, S.-I.; Rosen, D. W.; Choi, S.-k.; Duty, C. E., Effective mechanical properties of lattice material fabricated by material extrusion additive manufacturing, Addit. Manuf., 1, 12-23 (2014)
[39] Sekulovic, M.; Salatic, R., Nonlinear analysis of frames with flexible connections, Comput. Struct., 79, 11, 1097-1107 (2001)
[40] Park, S.-i.; Rosen, D. W., Quantifying mechanical property degradation of cellular material using as-fabricated voxel modeling for the material extrusion process, Annual solid freeform fabrication symposium, Austin (2015)
[41] Rho, J.-Y.; Kuhn-Spearing, L.; Zioupos, P., Mechanical properties and the hierarchical structure of bone, Med. Eng. Phys., 20, 2, 92-102 (1998)
[42] Zadpoor, A. A.; Sinke, J.; Benedictus, R., Elastoplastic deformation of dissimilar-alloy adhesively-bonded tailor-made blanks, Mater. Design, 31, 10, 4611-4620 (2010)
[43] Kidd, T.; Zhuang, S.; Ravichandran, G., In situ mechanical characterization during deformation of PVC polymeric foams using ultrasonics and digital image correlation, Mech. Mater., 55, 82-88 (2012)
[44] Wang, Y.; Cuitiño, A. M., Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation, Int J Solids Struct., 39, 13, 3777-3796 (2002)
[45] Hild, F.; Roux, S., Digital image correlation: from displacement measurement to identification of elastic properties-a review, Strain, 42, 2, 69-80 (2006)
[46] Zadpoor, A. A.; Sinke, J.; Benedictus, R., Global and local mechanical properties and microstructure of friction stir welds with dissimilar materials and/or thicknesses, Metall. Mater. Trans. A, 41, 13, 3365-3378 (2010)
[47] Fleck, N.; Olurin, O.; Chen, C.; Ashby, M., The effect of hole size upon the strength of metallic and polymeric foams, J. Mech. Phys. Solids, 49, 9, 2015-2030 (2001) · Zbl 1093.74500
[48] Olurin, O.; Fleck, N.; Ashby, M., Deformation and fracture of aluminium foams, Mater. Sci. Eng.: A, 291, 1, 136-146 (2000)
[49] Chen, C.; Fleck, N.; Lu, T., The mode I crack growth resistance of metallic foams, J. Mech. Phys. Solids, 49, 2, 231-259 (2001) · Zbl 1048.74036
[50] Amin Yavari, S.; Wauthlé, R.; Böttger, A. J.; Schrooten, J.; Weinans, H.; Zadpoor, A. A., Crystal structure and nanotopographical features on the surface of heat-treated and anodized porous titanium biomaterials produced using selective laser melting, Appl. Surf. Sci., 290, 287-294 (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.