خواص مکانیکی و رفتار سازه‌ای استخوان در سطح نانو با به‌کارگیری المان‌های چسبنده

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه سازه و زلزله، دانشکده مهندسی عمران، دانشگاه صنعتی نوشیروانی بابل، بابل، ایران

2 گروه سازه و زلزله، دانشکده مهندسی عمران، دانشگاه صنعتی نوشیروانی بابل، ایران

3 عضو هیئت علمی و پروفسور بیومکانیک، دانشگاه برن، برن، سوئیس

چکیده

استخوان بافت بیولوژیکی معدنی بوده که اجزاء اصلی آن دارای خواص مکانیکی بسیار متفاوتی می‌باشند. علائم بالینی برخی از بیماری‌ها در سطح ماکرو رخ داده ولی علت آن تغییر ساختار استخوان در سطح نانو می‌باشد، بنابراین انجام مطالعات در سطوح نانو و میکرو ضروری می‌باشد. در تحقیق حاضر خواص الاستیک و رفتار استخوان در سطح نانو با روش اجزاء محدود مدل‌سازی شده و از لایه‌های چسبنده استفاده شده است. پس از صحت‌سنجی مدل، نتایج توزیع تنش و خواص الاستیک آن با نتایج تحلیلی مقایسه شده است. نسبت کرنش مواد معدنی به کل مدل که در مطالعات پیشین ارزیابی نشده، برای مصالح بینابینی مختلف ارائه شده است. تاثیر تغییر نسبت حجمی مواد معدنی و خصوصیات مکانیکی کلاژن بر رفتار مدل نیز بررسی شده است. نتایج مقایسه مدل عددی با سایر مدل‌ها نشان دهندۀ تطابق بسیار خوبی می‌باشد. مصالح بینابینی کلاژن و صفحات معدنی با خواصی ناشناخته مهم‌ترین عامل گسیختگی فیبریل کلاژن معدنی بوده و حضور لایه ضخیم آب با پیوندهای واندوالسی و برشی ویسکوز به عنوان محتمل‌ترین لایۀ چسبنده می‌باشد. خواص غیرخطی کلاژن سبب ایجاد تغییرات چشمگیری در مدل شده و جهت کاهش محاسبات در مدل‌های سه‌بعدی، می‌توان از ساختار تک سلولی با شرایط مرزی پریودیک استفاده نمود.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Mechanical Properties and Structural Behavior of Bone at Nano Scale with Cohesive Element

نویسندگان [English]

  • Elham Alizadeh 1
  • Mehdi Dehestani 2
  • Philippe Zysset 3
1 Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran
2 Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran
3 Professor of Biomechanics, University of Bern, Switzerland
چکیده [English]

Bone is a biological tissue whose main components are different from the mechanical aspect. Some of the bone diseases are due to mutations in the bone structure at the nano scale, while their clinical symptoms appear at the macro scale. Therefore, the evaluation of bone at micro and nano scales is important. In the current study, the finite element modeling is performed to evaluate the mechanical properties and behavior of bone at the nano scale and the cohesive element is applied. After its verification, the stress distribution and elastic properties are compared with the analytical model. Limited studies are available on strain ratio and it is presented for different cohesive elements in the current study. The influence of mineral volume fraction and mechanical properties of collagen is investigated. The comparison between finite element models and the other ones demonstrate an excellent agreement. The collagen- hydroxyapatite interface with unknown mechanical properties is the most important parameter in the model and the thick water layer with Van der Waals interaction and viscous shear is determined as the most probable cohesive layer. The parametric studies indicate the significant effect of nonlinear collagen on the model. To decrease the calculation in models, the proposed unit cell with periodic boundary conditions could be employed.

کلیدواژه‌ها [English]

  • Bone
  • Nano
  • cohesive element
  • finite element
[1] A.L. Mescher, Junqueiras basic histology: text and atlas, Mcgraw-hill, 2013.
[2] A. Miller, Collagen: the organic matrix of bone, Philosophical Transactions of the Royal Society B: Biological Sciences, 304 (1984) 455-477.
[3] P. Fratzl, H. .Gupta, E. Paschalis, P.Roschger, Structure and mechanical quality of the collagen–mineral nano-composite in bone, Journal of materials chemistry, 14(14) (2004) 2115-2123.
[4] S. Eppell, W. .Tong, J. Katz, L. Kuhn, M. .Glimcher, Shape and size of isolated bone mineralites measured using atomic force microscopy, Journal of orthopaedic research, 19(6) (2001) 1027-1034.
[5] R.B.Martin, D.B. Burr, N.A. Sharkey, Skeletal Tissue Mechanics, New York, Springer Verlag, 1998.
[6] J.Y.Rho, L.Kuhn-Spearing, P.Zioupos, Mechanical properties and the hierarchical structure of bone, Medical engineering & physics, 20(2) (1998) 92-102.
[7] W. Voigt, Uber die beziehung zwischen den beiden elasticitats constantan isotroper korper, Annals of Physics, 38 (1889) 185-192.
[8] A. Reuss, Berechnung der fliebgrenze von mischkristallen auf grund der plasticitatsbedingung fur einkristalle, ZAAM, 9 (1929) 49-58.
[9] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechanics and Physics of Solids, 11(2) (1963) 127-140.
[10] K.Piekarski, Analysis of bone as a composite material, International Journal of Engineering Science, 11(6) (1973) 557-565.
[11] I.Jager, P. Fratzl, Mineralized collagen fibrils: A mechanical model with a staggered arrangement of mineral particles, Biophysical Journal, 79(4) (2000) 1737-1746.
[12] H. Gao, B. Ji, I. Jäger, E. Arzt, P. .Fratzl, Materials become insensitive to flaws at nanoscale: lessons from nature, Proceedings of the national Academy of Sciences, 100(10) (2003) 557-600.
[13] S.P. Kotha, N. Guzelsu, The effects of interphase and bonding on the elastic modulus of bone: changes with age-related osteoporosis, Medical Engineering & Physics, 22(8) (2000) 575-585.
[14] T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica, 21(5) (1973) 571-574.
[15] A. G.Reisinger, D. H.Pahr, P.K. Zysset, Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods, Biomechanics and modeling in mechanobiology, 9(5) (2010) 499-510.
[16] C. Hellmich, J.F. Barthélémy, L. Dormieux, Mineral–collagen interactions in elasticity of bone ultrastructure–a continuum micromechanics approach, European Journal of Mechanics-A/Solids, 23(5) (2004) 783-810.
[17] A. Fritsch, C. Hellmich, Universal microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity, Journal of Theoretical Biology, 244(4) (2007) 597-620.
[18] S. Nikolov, D. Raabe, Hierarchical modeling of the elastic properties of bone at submicron scales: the role of extrafibrillar mineralization, Biophysical journal, 94(11) (2008) 4220-4232.
[19] E. Hamed, Y. Lee, I. Jasiuk, Multi-scale modeling of elastic properties of cortical bone, Acta Mechanica, 213(1-2) (2010) 131-154.
[20] B.H. Ji, H.J. Gao, Mechanical properties of nanostructure of biological materials, Journal of the Mechanics and Physics of Solids, 52 (2004) 1963-1990.
[21] T. Siegmund, M.R. Allen, D.B. Burr, Failure of mineralized collagen fibrils: Modeling the role of collagen cross-linking, Journal of Biomechanics, 41(7) (2008) 1427-1435.
[22] J. Ghanbari, R. Naghdabadi, Nonlinear hierarchical multiscale modeling of cortical bone considering its nanoscale microstructure, Journal of Biomechanics, 42(10) (2009) 1560-1565.
[23] F. Yuan, S. Stock, D. Haeffner, J. Almer, D. Dunand, L. Brinson, a new model to simulate the elastic properties of mineralized collagen fibril, Biomechanics and modeling in mechanobiology, 10(2) (2011) 147-160.
[24] J.F. Mammone, S.M. Hudson, Micromechanics of bone strength and fracture, Journal of Biomechanics, 26(4-5) (1993) 439-446.
[25] Q. .Luo, R. .Nakade, X. .Dong, Q. .Rong, X. .Wang, Effect of mineral–collagen interfacial behavior on the microdamage progression in bone using a probabilistic cohesive finite element model, Journal of the mechanical behavior of biomedical materials, 4(7) (2011) 943-952.
[26] E.Hamed, I. Jasiuk, Multiscale damage and strength of lamellar bone modeled by cohesive finite elements, journal of the mechanical behavior of biomedical materials, 28 (2013) 94-110.
[27] A.Vercher-Martínez, E.Giner, C.Arango, F.J. Fuenmayor, Influence of the mineral staggering on the elastic properties of the mineralized collagen fibril in lamellar bone, Journal of the mechanical behavior of biomedical materials, 42 (2015) 243-256.
[28] Y. Wang, A. Ural, Mineralized collagen fibril network spatial arrangement influences cortical bone fracture behaviour, Journal of biomechanics, 66 (2018) 70-77.
[29] L. Lin, J. Samuel, X.Zeng, X. Wang, Contribution of extrafibrillar matrix to the mechanical behavior of bone using a novel cohesive finite element model, Journal of the mechanical behavior of biomedical materials, 65 (2017) 224-235.
[30] P.D. Falco, E. Barbieri, N.Pugno, H.S. Gupta, Staggered fibrils and damageable interfaces lead concurrently and independently to hysteretic energy absorption and inhomogeneous strain fields in cyclically loaded antler bone, ACS Biomaterials Science & Engineering, 3(11) (2017) 2779-2787.
[31] M.Maghsoudi-Ganjeh, L.Lin, X.Wang, X.Zeng, Computational investigation of ultrastructural behavior of bone using a cohesive finite element approach, Biomechanics and modeling in mechanobiology, 18(2) (2019) 463-478.
[32] A. K.Nair, A.Gautieri, S. W.Chang, M.J. Buehler, Molecular mechanics of mineralized collagen fibrils in bone, Nature communications, 4 (2013) 1711-1724.
[33] M.J. Buehler, Atomistic and continuum modeling of mechanical properties of collagen: elasticity, fracture, and self-assembly, Journal of Materials Research, 21(8) (2006) 1947-1961.
[34] M.J.Buehler, Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies, Journal of the mechanical behavior of biomedical materials, 1(1) (2008) 59-67.
[35] M.J.Buehler, Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization, Nanotechnology, 18(29) (2007) 295102.
[36] D.K.Dubey, V.Tomar, Microstructure dependent dynamic fracture analyses of trabecular bone based on nascent bone atomistic simulations, Mechanics Research Communications, 35(1-2) (2008) 24-31.
[37] M. Sadat-Shojai, Calcium Phosphate–Reinforced Polyester Nanocomposites for Bone Regeneration Applications, In Biodegradable Polymeric Nanocomposites (2015) 12-45.
[38] M. Rubin, I. .Jasiuk, J. Taylor, J. Rubin, T. Ganey, R. Apkarian, TEM analysis of the nanostructure of normal and osteoporotic human trabecular bone, Bone, 33(3) (2003) 270-282.
[39] K. Hibbitt, ABAQUS: User's Manual, 2013.
[40] Z.L.Shen, M.R.Dodge, H.Kahn, R.Ballarini, S.J.Eppell, In vitro fracture testing of submicron diameter collagen fibril specimens, Biophysical journal, 99(6) (2010) 1986-1995.
[41] M.Minary-Jolandan, M.F.Yu, Nanomechanical heterogeneity in the gap and overlap regions of type I collagen fibrils with implications for bone heterogeneity, Biomacromolecules, 10(9) (2009) 2565-2570.
[42] L.Yang, K.O.V.d. Werf, C.F.Fitié, M.L.Bennink, P.J.Dijkstra, J.Feijen, Mechanical properties of native and cross-linked type I collagen fibrils, Biophysical journal, 94(6) (2008) 2204-2211.
[43] C.A.Grant, D.J.Brockwell, S.E.Radford, N.H.Thomson, Effects of hydration on the mechanical response of individual collagen fibrils, Applied Physics Letters, 92(23) (2008) 233-902.
[44] M.J.Olszta, X.Cheng, S.S..Jee, R.Kumar, Y.Y.Kim, M.J.Kaufman, E.P.Douglas, L.B.Gower, Bone structure and formation: A new perspective, Materials Science and Engineering: R: Reports, 58(3-5) (2007) 77-116.
[45] M.P.Wenger, L.Bozec, M.A.Horton, P.Mesquida, Mechanical properties of collagen fibrils, Biophysical journal, 93(4) (2007) 1255-1263.
[46] L.Yang, K.O.v.d. Werf, B.F.Koopman, V.Subramaniam, M.L.Bennink, P.J.Dijkstra, J.Feijen, Micromechanical bending of single collagen fibrils using atomic force microscopy, Journal of Biomedical Materials Research Part A, 82(1) (2007) 160-168.
[47] J.A.V.D. Rijt, K.O.V.D. Werf, M.L.Bennink, P.J.Dijkstra, J.Feijen, Micromechanical testing of individual collagen fibrils, Macromolecular bioscience, 6(9) (2006) 697-702.
[48] A.J.Heim, W.G.Matthews, T.J.Koob, Determination of the elastic modulus of native collagen fibrils via radial indentation, Applied physics letters, 89(18) (2006) 181-902.
[49] A.C.Lorenzo, E.R.Caffarena, Elastic properties, Young's modulus determination and structural stability of the tropocollagen molecule: a computational study by steered molecular dynamics, Journal of biomechanics, 38(7) (2005) 1527-1533.
[50] S.J.Eppell, B.N.Smith, H.Kahn, R.Ballarini, Nano measurements with micro-devices: mechanical properties of hydrated collagen fibrils, Journal of the Royal Society Interface, 3(6) (2005) 117-121.
[51] S.Vesentini, C.F.Fitié, F.M.Montevecchi, A.Redaelli, Molecular assessment of the elastic properties of collagen-like homotrimer sequences, Biomechanics and modeling in mechanobiology, 3(4) (2005) 224-234.
[52] N.Sasaki, S.Odajima, Stress-strain curve and Young's modulus of a collagen molecule as determined by the X-ray diffraction technique, Journal of biomechanics, 29(5) (1996) 655-658.
[53] H.Hofmann, T.Voss, K.Kühn, J.Engel, Localization of flexible sites in thread-like molecules from electron micrographs: Comparison of interstitial, basement membrane and intima collagens, Journal of molecular biology, 172(3) (1984) 325-343.
[54] S.Cusack, A.Miller, Determination of the elastic constants of collagen by Brillouin light scattering, Journal of molecular biology, 135(1) (1979) 39-51.
[55] R.Harley, D.James, A.Miller, J.W.White, Phonons and the elastic moduli of collagen and muscle, Nature, 267(5608) (1977) 285-298.
[56] M.Amaral, M.A.Lopes, R.F.Silva, J.D.Santos, Densification route and mechanical properties of Si3N4–bioglass biocomposites, Biomaterials, 23(3) (2002) 857-862.
[57] A.Ravaglioli, A.Krajewski, Bioceramics: Materials· Properties· Applications, Springer Science & Business Media,  (1991).
[58] V.R.Sherman, A.n.W. Yang, M.A. Meyers, The materials science of collagen, journal of mechanical behaviour of biomedical materials,  (2015).
[59] Q. Luo, R. Nakade, X. Dong, Q. Rong, X. Wang, Effect of mineral–collagen interfacial behavior on the microdamage progression in bone using a probabilistic cohesive finite element model, Journal of the mechanical behavior of biomedical materials, 4(7) (2011) 943-952.
[60] F.Hang, A.H. Barber, Nano-mechanical properties of individual mineralized collagen fibrils from bone tissue, Journal of The Royal Society Interface,  (1555) 500-505.
[61] E. Wilson, A. Awonusi, M. Morris, D. Kohn, M. Tecklenburg, L. Beck, Highly ordered interstitial water observed in bone by nuclear magnetic resonance, Journal of bone and mineral research, 20(4) (2005) 625-634.
[62] A.Groetsch, A.Gourrier, J.Schwiedrzik, M.Sztucki, R.J.Beck, J.D.Shephard, J.Michler, P.K.Zysset, U.Wolfram, Compressive behaviour of uniaxially aligned individual mineralised collagen fibres at the micro-and nanoscale, Acta biomaterialia, 89 (2019) 313-329.
[63] W.Wu, J.Owino, A.Al-Ostaz, L.Cai, Applying periodic boundary conditions in finite element analysis, In SIMULIA Community Conference, Providence,  (2014) 707-719.
[64] Y. Liu, R. Ballarini, S.J. Eppell, Tension tests on mammalian collagen fibrils, Interface focus,  (2016).