Abstract
The present paper aims to develop an automatical strategy for generating accurate different-scale microstructures of human tooth enamels (HTEs), and to elaborate a numerical scheme for simulating their elastic responses. At first, the strong governing formulation of these microstructures is briefly constructed, and the relevant weak formulation is then deduced based on the virtual work theorem. Afterwards, a subdividing approach, which cuts the elements intercepted by the interfaces between distinct phases automatically, is established with the aid of the level set method (LSM), and the discrete counterpart of the governing formula is obtained by combining the weak formulation derived and a discretized model. To be noted, two silent merits are found when the elaborated strategy is applied: (1) the continents constituting the microstructures of different scales can be arbitrarily-shaped and conveniently reproduced; (2) the periodic boundary condition commonly employed can be enforced on the external surfaces of representative unit cells (RUCs) with no difficulty. Besides, a boundary value problem (BVP) involving a simplified HTE nanostructure is designed, analytically solved, and hereafter applied to verify the correctness of the proposed strategy. It is observed that both the displacement and stress predictions by the computational approach are in good agreement with the relevant analytical results irrespective of the material combinations applied. Eventually, discussions are made on the influence of material organizations of both the 2D and 3D HTE microstructures at the ultrastructural and repeated rod levels, and some concluding remarks are drawn.
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References
He L H, Swain M V. Understanding the mechanical behaviour of human enamel from its structural and compositional characteristics. J Mech Behav Biomed Mater, 2008, 1: 18–29
Jia Y F, Xuan F Z. Anisotropic wear behavior of human enamel at the rod level in terms of nanoscratching. Wear, 2012, 290–291: 124–132
Bar-On B, Daniel W H. Enamel and dentin as multi-scale bio-composites. J Mech Behav Biomed Mater, 2012, 12: 174–183
Zheng J, Li Y, Shi M Y, et al. Microtribological behaviour of human tooth enamel and artificial hydroxyapatite. Tribol Int, 2013, 63: 177–185
Zhou Z R, Jin Z M. Biotribology: Recent progresses and future perspectives. Biosurface Biotribology, 2015, 1: 3–24
Meyers M A, Chen P Y, Lin A Y M, et al. Biological materials: Structure and mechanical properties. Prog Mater Sci, 2008, 53: 1–206
Ji B, Gao H. Mechanical properties of nanostructure of biological materials. J Mech Phys Solids, 2004, 52: 1963–1990
Nanci A. Ten Cate’s Oral Histology: Development, structure and function (Seventh Edition). Missouri: Mosby, 2007
Bar-On B, Wagner H D. Mechanical model for staggered bio-structure. J Mech Phys Solids, 2011, 59: 1685–1701
E S F, Shi L, Guo Z G, et al. The recent progress of tribological biomaterials. Biosurface Biotribology, 2015, 1: 81–97
Bechtle S, Ang S F, Schneider G A. On the mechanical properties of hierarchically structured biological materials. Biomaterials, 2010, 31: 6378–6385
Gou M, Qu X, Zhu W, et al. Bio-inspired detoxification using 3D-printed hydrogel nanocomposites. Nat Commun, 2014, 5: 3774
Spears I R. A three-dimensional finite element model of prismatic enamel: A re-appraisal of the data on the Young’s modulus of enamel. J Dental Res, 1997, 76: 1690–1697
Ang S F, Bortel E L, Swain M V, et al. Size-dependent elastic/inelastic behavior of enamel over millimeter and nanometer length scales. Biomaterials, 2010, 31: 1955–1963
Xie Z H, Swain M V, Swadener G, et al. Effect of microstructure upon elastic behaviour of human tooth enamel. J Biomech, 2009, 42: 1075–1080
Bar-On B, Daniel W H. Effective moduli of multi-scale composites. Composites Sci Tech, 2012, 72: 566–573
An B, Wang R, Zhang D. Role of crystal arrangement on the mechanical performance of enamel. Acta Biomaterialia, 2012, 8: 3784–3793
Bargmann S, Scheider I, Xiao T, et al. Towards bio-inspired engineering materials: Modeling and simulation of the mechanical behavior of hierarchical bovine dental structure. Comp Mater Sci, 2013, 79: 390–401
Ang S F, Saadatmand M, Swain M V, et al. Comparison of mechanical behaviors of enamel rod and interrod regions in enamel. J Mater Res, 2012, 27: 448–456
Lu C, Nakamura T, Korach C S. Effective property of tooth enamel: Monoclinic behavior. J Biomech, 2012, 45: 1437–1443
Shimizu D, Macho G A, Spears I R. Effect of prism orientation and loading direction on contact stresses in prismatic enamel of primates: Implications for interpreting wear patterns. Am J Phys Anthropol, 2005, 126: 427–434
Sui T, Sandholzer M A, Baimpas N, et al. Hierarchical modelling of elastic behaviour of human enamel based on synchrotron diffraction characterisation. J Struct Biol, 2013, 184: 136–146
Sui T, Lunt A J G, Baimpas N, et al. Hierarchical modelling of in situ elastic deformation of human enamel based on photoelastic and diffraction analysis of stresses and strains. Acta Biomaterialia, 2014, 10: 343–354
Ausiello P, Apicella A, Davidson C L, et al. 3D-finite element analyses of cusp movements in a human upper premolar, restored with adhesive resin-based composites. J Biomech, 2001, 34: 1269–1277
Ausiello P, Apicella A, Davidson C L. Effect of adhesive layer properties on stress distribution in composite restorations—A 3D finite element analysis. Dental Mater, 2002, 18: 295–303
Magne P. Efficient 3D finite element analysis of dental restorative procedures using micro-CT data. Dental Mater, 2007, 23: 539–548
Barani A, Keown A J, Bush M B, et al. Mechanics of longitudinal cracks in tooth enamel. Acta Biomaterialia, 2011, 7: 2285–2292
Wang M, Qu X, Cao M, et al. Biomechanical three-dimensional finite element analysis of prostheses retained with/without zygoma implants in maxillectomy patients. J Biomech, 2013, 46: 1155–1161
Yettram A L, Wright K W J, Pickard H M. Finite element stress analysis of the crowns of normal and restored teeth. J Dental Res, 1976, 55: 1004–1011
Spears I R, van Noort R, Crompton R H, et al. The effects of enamel anisotropy on the distribution of stress in a tooth. J Dental Res, 1993, 72: 1526–1531
He L H, Yin Z H, Jansen van Vuuren L, et al. A natural functionally graded biocomposite coating-Human enamel. Acta Biomaterialia, 2013, 9: 6330–6337
Jeng Y R, Lin T T, Hsu H M, et al. Human enamel rod presents anisotropic nanotribological properties. J Mech Behav Biomed Mater, 2011, 4: 515–522
Sethian J A. A fast marching level set method for monotonically advancing fronts.. Proc Natl Acad Sci USA, 1996, 93: 1591–1595
Wang B R, Liu J T, Gu S T, et al. Numerical evaluation of the effective conductivities of composites with interfacial weak and strong discontinuities. Int J Thermal Sci, 2015, 93: 1–20
Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Meth Eng, 1999, 45: 601–620
Suquet P. Elements of Homogenization Theory for Inelastic Solid Mechanics. Berlin: Spinger-Verlag, 1987. 194–278
Michel J C, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: A computational approach. Comp Methods Appl Mech Eng, 1999, 172: 109–143
Chen P Y, Lin A Y M, Lin Y S, et al. Structure and mechanical properties of selected biological materials. J Mech Behav Biomed Mater, 2008, 1: 208–226
Guan D, Yan G. A triangular prism variable nodes isoparameter element for calculation of stress field in welded structure. Comput Struct Mech Appl, 1994, 11: 107–112
Viswanath B, Raghavan R, Ramamurty U, et al. Mechanical properties and anisotropy in hydroxyapatite single crystals. Scripta Mater, 2007, 57: 361–364
Xie Z, Swain M V, Hoffman M J. Structural integrity of enamel: Experimental and modeling. J Dental Res, 2009, 88: 529–533
Smith B L, Schäffer T E, Viani M, et al. Molecular mechanistic origin of the toughness of natural adhesives, fibres and composites. Nature, 1999, 399: 761–763
Ge J, Cui F Z, Wang X M, et al. Property variations in the prism and the organic sheath within enamel by nanoindentation. Biomaterials, 2005, 26: 3333–3339
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Liu, T., Deng, Q., Yang, D. et al. Automatic modeling together with numerical simulation of the different-scale microstructures of human tooth enamels. Sci. China Technol. Sci. 60, 1381–1399 (2017). https://doi.org/10.1007/s11431-016-9006-4
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DOI: https://doi.org/10.1007/s11431-016-9006-4