Journal of Biomechanics 47 (2014) 1060–1066
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Size dependent elastic modulus and mechanical resilience of dental enamel Simona O'Brien a,b, Jeremy Shaw c, Xiaoli Zhao a, Paul V. Abbott d, Paul Munroe e, Jiang Xu f,g, Daryoush Habibi a, Zonghan Xie a,h,i,n a
School of Engineering, Edith Cowan University, Perth, WA, Australia Perth Institute of Business and Technology, Perth, WA, Australia c Centre for Microscopy, Characterisation and Analysis, The University of Western Australia, Perth, WA, Australia d School of Dentistry, The University of Western Australia, Perth, WA, Australia e School of Materials Science and Engineering, University of New South Wales, Sydney, NSW, Australia f Department of Materials Science and Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of China g School of Mechanical and Electrical Engineering, Wuhan Institute of Technology, Wuhan 430073, People's Republic of China h School of Mechanical Engineering, University of Adelaide, Adelaide, SA, Australia i School of Materials Science and Engineering, Tianjin Polytechnic University, Tianjin 300387, People's Republic of China b
art ic l e i nf o
a b s t r a c t
Article history: Accepted 22 December 2013
Human tooth enamel exhibits a unique microstructure able to sustain repeated mechanical loading during dental function. Although notable advances have been made towards understanding the mechanical characteristics of enamel, challenges remain in the testing and interpretation of its mechanical properties. For example, enamel was often tested under dry conditions, significantly different from its native environment. In addition, constant load, rather than indentation depth, has been used when mapping the mechanical properties of enamel. In this work, tooth specimens are prepared under hydrated conditions and their stiffnesses are measured by depth control across the thickness of enamel. Crystal arrangement is postulated, among other factors, to be responsible for the size dependent indentation modulus of enamel. Supported by a simple structure model, effective crystal orientation angle is calculated and found to facilitate shear sliding in enamel under mechanical contact. In doing so, the stress build-up is eased and structural integrity is maintained. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Tooth enamel Young's modulus Nanoindentation Deformation Finite element analysis
1. Introduction Being the hardest tissue in the human body (Fig. 1(a)), dental enamel is built to last (Lawn et al., 2009). The remarkable resilience of enamel stems from its unique microstructure that comprises highly organised hydroxyapatite (HAP) platelets (Fig. 1b, c); Nanci, 2008; Xie et al., 2009a, 2008). Electron and atomic force microscopy investigations also reveal that a thin protein layer is present between the HAP crystallites in mature enamel (Habelitz et al., 2001; Warshawsky, 1989) and this, though small in quantity, plays an important role in regulating the mechanical properties of enamel to uniquely suit its functions (Gao et al., 2003; Lawn et al., 2010). Young's modulus is commonly used to quantify a material's stiffness. Enamel has a hybrid laminate structure at nanoscale (He and Swain, 2007; Xie et al., 2009b), which means its modulus is determined not only by the
n Corresponding author at: School of Mechanical Engineering, University of Adelaide, SA 5005, Australia. Tel.: þ61 8 8313 3890; fax: þ61 8 8313 4367. E-mail address: [email protected] (Z. Xie).
0021-9290/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.12.030
volume fraction of its constituents but also by the spatial arrangement of the mineral crystals (Xie et al., 2009a). Young's modulus of enamel is routinely measured using the depth-sensing indentation (DSI, also called nanoindentation) method (Oliver and Pharr, 1992, 2004). Values reported in the literature vary by over 100% (Cuy et al., 2002). Such a wide range results from multiple factors, such as variations in location, testing conditions (i.e., dry or wet) or age (Cuy et al., 2002; Habelitz et al., 2001; Lewis and Nyman, 2008; Park et al., 2008; Staines et al., 1981). Moreover, the Young's modulus of enamel was found to decrease with increasing indentation depth (Zhou and Hsiung, 2007). Given the dependence of Young's modulus of enamel on indentation depth, it is important to conduct measurements under depth control when interrogating the stiffness over the thickness of enamel. Unfortunately, most nanoindentation tests have been performed under load control (i.e., at constant load), making it difficult to compare the modulus values obtained from different regions, where indentation depths may differ (He et al., 2006; Xie et al., 2007b; Zhou and Hsiung, 2007). Moreover, although the sample preparation and testing conditions are known to influence the measured mechanical properties of enamel, (Guidoni et al.,
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
Enamel
1061
Dentin
Aqueous putty
1mm
5 μm
Fig. 1. Tooth structure viewed at different length scales. Optical image of (a) tooth macrostructure consisting of enamel and dentin. (b) Schematic illustration of enamel microstructure formed by tightly packed rods (or prisms) that are aligned perpendicular to the enamel–dentin junction. Inside the rods the mineral crystallites are bonded together by proteins (Habelitz et al., 2001; Warshawsky, 1989) (modified from (Nanci, 2008)). (c) Transmission electron micrograph of small enamel section (refer to ‘box’ in (b)) prepared parallel to the c-axis of rods.
2006; Staines et al., 1981) the majority of mechanical tests were carried out under dry condition or on ‘wet’ specimens that had previously been dried during preparation (Cuy et al., 2002; Xu et al., 1998). In this study, we conducted an in-depth investigation of the elastic behaviour of enamel under the influence of mechanical loads. Enamel samples were prepared and tested in a hydrated state. Young's modulus of enamel was determined using a line scan obtained from the occlusal surface to the enamel–dentin junction (EDJ) at various depths. A mechanistic model was used to link crystal orientation angle with the depth dependent elastic behaviour of enamel. Furthermore, the implications of crystal orientation in promoting shear deformation and dissipating indentation energy were examined and clarified. 2. Experimental 2.1. Sample collection and preparation Ethics approval was given by the Ethics Review Committee of Edith Cowan University prior to the collection of teeth. Nine third molars extracted from patients aged 20–30 years were used in this study. The teeth were intact without any decay or other disease and were extracted for clinical reasons and on the treating dentists' advice because of crowding in the jaw and impaction of the teeth preventing them from eruption. Informed consents were obtained from patients involved. Upon extraction, teeth were stored at 4 1C in Hanks' Balanced Salt Solution (HBSS) (Sigma-Aldrich Co., St. Louis, USA) with the addition of 0.02% thymol crystals to prevent demineralisation and inhibit bacterial growth (Habelitz et al., 2002; White et al., 1994). Struers resin and hardener were mixed to form 30 mm diameter by 10 mm high cylindrical blocks inside Struers plastic moulding cups. A 15 mm diameter hole was then drilled through the centreline of the block, which was later filled
from one end with aquatic putty (Selleys Knead-It Aqua, Selleys, Australia). The tooth was inserted root first into the cylinder and pressed into the putty, leaving only the occlusal surface exposed. A precision saw (Isomet 1000, Buehler Ltd., Lake Bluff, IL, USA) was used to section the partially embedded tooth at a distance of 5 mm from its edge in a direction parallel to mesial–distal plane. HBSS was used as a coolant during sectioning. The cut surface was then ground and polished with series of Struers silicon carbide papers up to 4000 grit. The final polishing of the specimens was done on a soft polishing cloth soaked in water (LaboPol 25, Struers, Copenhagen, Denmark). The c-axis of prisms was organised roughly parallel to the polished surface. Once prepared, the specimen was stored in HBSS that contains 0.02% thymol crystals.
2.2. Nanoindentation testing A depth-sensing indentation system (Ultra-Micro Indentation System, UMIS2000, CSIRO, Australia) equipped with a Berkovich indenter was used to measure the Young's modulus of enamel (Oliver and Pharr, 1992, 2004). The indenter tip was calibrated by conducting single-cycling indentation tests on fused silica (a standard material having a known modulus of 72 GPa) at different loads. To conduct measurements in a wet condition, a specially designed stainless steel holder was used, and the specimen was clamped onto the bottom using a slide-screw assembly. A stainless steel plate of 1 mm in thickness was placed on the specimen to distribute the clamping force evenly. The specimen was submerged in HBSS before testing. The polished surface of enamel was indented from the occlusal surface to the enamel–dentin junction (EDJ). Each row was generated along a line parallel to the surface and contains 5 indents with an interval of 50 mm. The spacing between rows is also 50 mm. Load–partial unload tests were run in a closed-loop under load control to determine the elastic modulus of enamel at different depths. A maximum load of 400 mN was applied in eight increments. The loading rate was set to 2.5 mN/s, which represents the static response of the material. Following each increment were 10 decrements, from which the average Young's modulus and standard errors were calculated with the load applied roughly perpendicular to the prism direction (Oliver and Pharr, 1992, 2004) and plotted as a function of indentation depth. Recent nanoindentation studies have shown that prism orientation has little impact on the hardness and modulus of enamel. (Braly et al., 2007;
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Xie et al., 2009a). To ensure that the multi-cycling nanoindentation is a suitable technique for the modulus measurement of materials at different depths, fused silica was tested by following the above procedure. It was found that the measured modulus of fused silica is independent of depth and within 1% of the theoretical value.
simulated for comparison purpose. Then, shear sliding effect was enabled and implemented by defining the material to be elasto-plastic, with the shearing taking place along the vertical direction and the yield strength being that of protein (1.09 GPa) (Xie et al., 2008).
3. Results and discussion
2.3. Microstructure observation Ultra-thin sections through the central head region in a direction parallel to enamel rods were prepared for transmission electron microscopy (TEM). To do this, a dual-electron/focused ion beam (FIB) workstation (Nova Nanolab 200, FEI Company, Hillsboro, OR, USA) was used. The detailed procedure can be found elsewhere (Xie et al., 2007a). Briefly, prior to the milling a 1 μm thick platinum layer was deposited to protect the surface area of interest from ion beam damage. A ‘rough’ sectioning was then performed with a current of 10 nA, in which trenches were created at both sides of the platinum strip to obtain a cross section of 3 μm thickness. A number of ‘fine’ mills were taken at reduced currents (5–1 nA) to thin the section to 1 μm. Final mills were carried out at further reduced currents (300–100 pA) to reduce the thickness down to 100 nm suitable for electron transparency. The TEM specimens were transferred from the FIB sample holder to carbon coated copper TEM grids using a high-precision micromanipulator (Kleindiek Nanotechnik GmbH, Reutlingen, Germany). TEM imaging was performed using a field emission gun-TEM (Philips CM200, Eindhoven, Netherlands).
2.4. Finite element analysis Mineral platelets of enamel start to slide against each other, when the shear stress in the protein matrix exceeds a critical value (He and Swain, 2007). As a result, there were no obvious cracks near the indentation site (Xie et al., 2008). According to this, finite element models were constructed in this work to clarify the effect of the shear deformation of the protein layer on the magnitude and distribution of contact-induced stress in enamel. The simulations were performed using COMSOL software. Two-dimensional axisymmetric models, with the axial coordinate z along the loading direction of the indenter, were constructed. The models consisted of an enamel block measuring 50 50 mm2, loaded by a spherical indenter with a radius of 5 mm. A refined mesh was created around the contact zone within an area of 10 10 mm2 directly underneath the indenter where the stress concentration was expected to rise during indentation. The total number of mesh elements is 17631, which includes the elements within the indenter. Further mesh refinement does not improve the simulation accuracy significantly. The contact between the indenter and the sample was assumed to be frictionless. The boundary conditions are similar to those used in the previous works (Wo et al., 2013; Zhao et al., 2011) and are briefly described as follows: The bottom of the block (z ¼ 50 mm) is fixed in the z direction, while its right edge (r ¼50 mm) is fixed in the x direction. The axisymmetric axis coincides with the left edge of the block (r¼ 0) for obtaining 3D simulation results. The tip of the indenter was positioned at z ¼0 mm (0 displacement) before the simulation started. The indentation process is modelled as a downward displacement of the indenter tip from 0 to 0.4 mm at a step of 0.04 mm. As mentioned above, the main objective for the finite element analysis (FEA) is to clarify the difference between the mechanical response of enamel with and without shear sliding. Hence, the shear sliding effect was treated in a parametric way without introducing structural complexity into the modelling. Specifically the structural mechanics module was applied, using Hook's law as the constitutive relation. As indicated in Fig. 1(c), on a sub-micrometre scale enamel exhibits a ‘bricks-and-mortar’ structure, with the ‘bricks’ being made from mineral crystals which are embedded in a protein matrix. The volume fraction of protein is below 4%, which can be derived from the dimensional parameters listed in Table 1. In the FEA model Young's modulus (129 GPa) and Poisson's ratio (0.25) values were used for the mineral phase (refer to Table 1). First, a perfect elastic response was
We first investigated how Young's modulus of enamel changed across its thickness at various indentation depths. A typical load– partial unload curve obtained from the middle region of enamel is shown in Fig. 2(a), from which Young's modulus of enamel is calculated (Oliver and Pharr, 1992) and found to decrease with increasing depth (Fig. 2(b)). Given the depth-dependent elastic property of enamel, the modulus, measured as a function of indentation depth, should then be applied to simulations, in order to better understand the mechanical design of enamel. Following the same approach, we determined Young's modulus of enamel over its thickness from the occlusal surface to the EDJ and plotted for three indentation depths, namely 0 mm, 0.5 mm and 2 mm (Fig. 2(c)). For each depth, Young's modulus decreased with increasing distance from the occlusal surface, confirming that enamel is indeed a functionally graded material (He and Swain, 2009). The change of the enamel elasticity over its thickness is attributed to the variation in the microstructure and composition from the surface towards the EDJ (Cuy et al., 2002; Low et al., 2008). Moreover, as the indenter tip approaches the enamel specimen, the contact point position, which could be at either mineral platelet or protein, has its strong influence on the load–displacement curve, and thus the value of Young's modulus. Consequently, modulus values obtained for shallower indentation depths, for example at ht ¼0.5 mm, show a more complex undulating pattern than those obtained at a depth of 2 mm, presumably due to the inhomogeneous nature of the enamel structure. A plastic-damage model was developed for enamel and, according to it, the shear deformation of protein (i.e., the extension and break of peptide chains) can cause the reduction in stiffness under indentation loading (An et al., 2012a). Recently, the nonuniform arrangement of the HAP crystallites was also identified to be a key factor responsible for unique mechanical behaviour of enamel (An et al., 2012b). It is, therefore, natural to inquire what roles the crystal arrangement or orientation plays in the depthdependent elastic response of enamel. For this purpose, a mechanistic model was built from the experimental observation that enamel exhibits a staggered hybrid structure (as seen in Fig. 1(c)), in which the mineral platelets can rotate to accommodate the deformation (Zhou and Hsiung, 2007). Notably, such a structural configuration has been successfully adopted by others to understand unique mechanical characteristics of enamel, for instance, rate-dependent deformation (Zhou and Hsiung, 2006) and creep (He and Swain, 2008; Schneider et al., 2008). According to the
Table 1 Structural and mechanical parameters of enamel constituents used in the theoretical analysis and modeling. Parameter
Mineral crystal Symbol
Thickness (nm) Length (nm) Poisson's ratio Young's modulus (GPa) Spacing of mineral crystals (nm) a
hm l νm Em d
Value 50 500 0.25 129 1.5
Protein Reference Kerebel et al. (1979) Driessens and Verbeeck (1990) Liu et al. (2006) Ang et al. (2010)
Symbol
Value
Reference
a
hp
1.5
νp Ep
0.45 2
Liu et al. (2006) Ji and Gao (2004), Spears (1997)
The thickness of protein, hp is calculated from hp/hm ¼ 0.03 which corresponds to a volume fraction of mineral E97% in the enamel.
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
1063
Load, P (mN)
400
200
0 0.0
0.5
1.0 1.5 2.0 Indentation depth, h [μm]
2.5
Young’s modulus, E [GPa]
93
85
77 0.7
1.7 Indentation depth, h [μm]
2.7
Fig. 3. Understanding the depth-dependent Young's modulus of enamel during deformation. (a) Young's modulus of enamel in relation to effective crystal orientation angle. Inset shows the mechanistic model used. (b) Schematic illustrations showing the local responses of mineral crystals to increasing indentation loads (or depths) from P1 to P3 by rotation from θ1 to θ3, resulting in a reduction in Young's modulus of enamel.
120
Young’s modulus, E [GPa]
ht = 0.0 ht = 05 ht = 2.0
where θ is the angle between the c-axis of the mineral platelet and the loading direction (also termed as the crystal orientation angle). The compression modulus, E2, and the shear modulus, G, of the staggered composite are given as (Liu et al., 2006):
90
E2 ¼
60 0.0
0.5
G
1.0
Normalised distance Fig. 2. Young's modulus of dental enamel measured by depth-sensing indentation tests at different depths and locations. (a) Representative indentation load, P, versus displacement (i.e., penetration depth), h, curve produced by a Berkovich indenter in the middle region of the enamel. (b) The average Young's modulus and standard deviations of enamel obtained from the middle region at different penetration depths. (c) The average Young's modulus and standard deviations of enamel corresponding to three indentation depths of 0, 0.5 and 2 μm plotted from the occlusal surface to the EDJ. The normalised distance is calculated as the distance from the surface to the indentation divided by the enamel thickness. The values in the shaded region in (c) are used in plotting (b).
model (inset in Fig. 3), Young's modulus of enamel can be expressed as (Xie et al., 2008) Ee ¼
Ez ¼
ð cos θ sin θÞ 2
2
sin θ 4
2
E2 þ 4
cos θ 2
sin θ 2
1 ð12K p hp ðhm þ hp Þ=αl Gp ð4Gp þ 3K p ÞÞ þ ð1=Ez Þ 2
hp þhm Gp hp
ð2Þ
ð3Þ
where hm and l are the thickness and length of the mineral crystal, respectively, hp and Gp are the thickness and shear modulus of protein layers between mineral platelets, α is the non-uniformed shear strain factor of the composite, Ez the modulus of the composite perpendicular to the c axis of the mineral platelet, and Kp the bulk modulus of protein. Note that Gp, α, Ez and Kp can be derived from (Liu et al., 2006): Ep 2ð1 þ νp Þ
Gp ¼ (
α¼
1þ
4 3
ðhp þ hm Þl hp ð1 þdÞ
ð4Þ 2 )
1d 1 þ dÞ
ð5Þ
ð1Þ
G
1 ðð3hp =ðhp þ hm Þð4Gp þ 3K p ÞÞ þ ð1=Em Þð3K p =ð4Gp þ 3K p ÞÞððhm þ 4νm hm hp þ 2hp 2νm hp Þ=hm ðhp þ hm ÞÞÞ 2
2
2
ð6Þ
Kp ¼
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
Ep 3ð1 2νp Þ
ð7Þ
The structural and mechanical properties of the main constituents of enamel, i.e., the mineral and protein, are given in Table 1. Young's modulus of enamel, measured under different indentation depths within a narrow region halfway between the occlusal surface and EDJ, is shown in Fig. 3. It is worth noting that the effective crystal orientation angle of enamel changes from 451 to 481 when the indentation depth increases from 0.75 mm to 2.42 mm. Also note that the length of the crystallites used in the modelling is 500 nm (Driessens and Verbeeck, 1990). However, there is no consensus on the crystal length in the literature. The possibility that they are considerably longer cannot be ruled out (Nanci, 2008; Waters, 1980). To appreciate the effect that the crystal length may have upon the angle calculation, a wide range of values from 150 nm to 500 mm was modelled. Surprisingly, the results show that the crystal length has little impact on both compression modulus and shear modulus, consequently exerting minimal effect on the angles derived. The size dependent indentation modulus has also been observed for cortical bone. The plastic-damage model was proposed and successfully captured the stiffness loss during unloading (Lucchini et al., 2011; Zhang et al., 2010). However, the damage mechanism remains unclear, which renders the model phenomenological and lacking in a physical interpretation of its parameters. To overcome this problem, a physically sound approach was adopted for simulating the modulus reduction of dental enamel, (An et al., 2012a) based upon the experimental observation (Xie et al., 2008). It suggests that the shear deformation of protein contributes to the degradation of stiffness in the unloading curve. In another study, Zhou and Hsiung postulated that the rotation of mineral crystallites might also be responsible for the change of elastic modulus of enamel under indentation loading (Zhou and Hsiung, 2007). Recently, the crystal arrangement or orientation has been identified, among others, as a critical parameter that regulates the mechanical behaviour of enamel (An et al., 2012b; Xie et al., 2008). From this perspective, the structural basis that underpins the elastic behaviour of enamel was examined in this work; that is, the decrease of the Young's modulus of enamel with increasing indentation depth may also be governed by bending and/or tilting of the mineral crystals. As such, multiple origins of the depth-dependent elastic behaviour were possible. Note that the bending deformation of slender crystals is not explicitly considered during modelling. However, the bending would result in the (partial) tilting of the crystals; further the crystal length was found to have minimal impact on the angles derived. The crystal bending can thus be accommodated by the proposed mechanical model. More interestingly such an adaptive response is able to increase the contact area and, consequently, lower the stress concentration in enamel. For example, with the increase of indentation load from 100 to 250 mN, the contact pressure decreases by about 13%. By doing so, the loadbearing ability of enamel would be enhanced. It is noteworthy that fatigue damage may occur during multi-cycling nanoindentation (Jia and Xuan, 2012), and if so, that would affect the measurement accuracy of the indentation modulus. On the other hand, the relative orientations of the indenter to the enamel rods' direction were found to affect the measured elastic modulus (Cheng et al., 2010). To minimise its impact in the study of depth-dependent modulus in enamel, the multi-cycling indentations were conducted in the present work. This means the depth-dependent elastic property was measured at fixed locations, where the relative orientation of the indenter always remains constant during a multi-cycling test cycle. While the direction of indentation on the enamel may affect the absolute value of modulus, it would have
little impact on its depth-dependent nature and the underlying mechanism discovered here. As mentioned earlier, shear deformation between mineral crystals has been observed in human enamel by using transmission electron microscopy (Xie et al., 2007b). This deformation mode is considered to be essential for the remarkable damage tolerance of enamel (He and Swain, 2007; Xie et al., 2009b). To facilitate such a shear process, the shear stress developed between the mineral crystals under indentation loading should be maximised from the viewpoint of mechanical design. According to the model shown in Fig. 3, the shear stress, τ, can be written as
τ¼
1 s sin 2θ 2
ð8Þ
where s is the normal stress. Therefore, to meet this design requirement, the effective crystal orientation angle would be 451, which is consistent with the modelling result (Fig. 3). Although the shearing is considered to be the predominant deformation mechanism of enamel (He and Swain, 2007; Xie et al., 2009b) a quantitative understanding of its role in mitigating fracture and preventing catastrophic failure remains lacking. To redress this gap, finite elemental analysis (FEA) simulations was performed here to elucidate the benefit of shear deformation for the stress management in enamel. Compared to the ‘purely elastic’ deformation (i.e., not considering the plastic shear process); Fig. 4(a)), a reduction in the maximum shear stress by 9% (i.e., from 5.5 GPa to 5.0 GPa) for an indentation depth of 0.28 mm is observed when the shearing effect is
τ (GPa) 5.50
2.75
0.00
Material volume,(μm3)
1064
s4
s3
s2
s1
Fig. 4. The shear stress distribution generated in enamel by indentation at a depth of 0.28 μm: (a) perfect elastic model, (b) shear sliding enabled. (c) the size of regions under the influence of shear stresses greater than (s1) 3.0 GPa, (s2) 3.5 GPa, (s3) 4.0 GPa and (s4) 4.5 GPa, calculated from the FEA simulations; note that ‘PE’ stands for the perfect elastic model and ‘SS’ for the shear sliding model.
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
taken into account (Fig. 4(b)). The stress distribution pattern is also modified under plastic shear and the volume populated by larger stress is reduced significantly (Fig. 4(c)). For example, the volume influenced by the stress level 43.5 GPa is reduced by more than 70% when the plastic shearing is enabled, and the volume influenced by the stress level 44.5 GPa almost completely diminishes when the shear deformation is considered. Consequently, the decrease of stress level and reduction of material volumes subjected to higher stresses, resulting from the shear sliding between mineral crystals, could further enhance the load-bearing ability of enamel.
4. Conclusion By combining depth-sensing indentation tests and modelling, multiple origins of the depth-dependent indentation stiffness were established in this study with a special focus on the crystal arrangement. The effective crystal orientation angle was calculated and found to facilitate the shear sliding of mineral crystals during indentation loading. Finite element analysis reveals that the shear deformation between crystal platelets serves to (a) lower the stress level and (b) reduce the material volume populated by larger stresses. This leads to a critical appreciation of the stress management ability of enamel during loading. The present work will rekindle interest towards understanding the mechanical responses of mineralised tissues to changing loads. From that, the mechanical design principles of these materials may be better understood.
Conflict of interest statement The authors certify that there is no conflict of interest with any financial organisation regarding the material discussed in the manuscript.
Acknowledgements We would like to thank Professor Mark Hoffman of University of New South Wales for providing constructive comments on the manuscript. The authors also thank Sparkle Dental in Joondalup, Western Australia, for providing fresh human molars. This work was partially supported by an Australian Dental Research Foundation (ADRF) grant (6/2010). The authors acknowledge the facilities, scientific and technical assistance of the Australian Microscopy and Microanalysis Research Facility at the Centre for Microscopy, Characterisation and Analysis, the University of Western Australia, a facility funded by the University as well as State and Commonwealth Governments. S. O'Brien also acknowledges an ECU Postgraduate Scholarship for her Ph.D. study. References An, B.B., Wang, R.R., Zhang, D.S., 2012a. Region-dependent micro damage of enamel under indentation. Acta Mech. Sin. 28, 1651–1658. An, B., Wang, R., Zhang, D., 2012b. Role of crystal arrangement on the mechanical performance of enamel. Acta Biomater. 8, 3784–3793. Ang, S.F., Bortel, E.L., Swain, M.V., Klocke, A., Schneider, G.A., 2010. Size-dependent elastic/inelastic behaviour of enamel over millimeter and nanometer length scales. Biomaterials 31, 1955–1963. Braly, A., Darnell, L., Mann, A., Teaford, M., Weihs, T., 2007. The effect of prism orientation on the indentation testing of human molar enamel. Arch. Oral Biol. 52, 856–860. Cheng, Z.J., Wang, X.M., Ge, J., Yan, J.X., Ji, N., Tian, L.L., Cui, F.Z., 2010. The mechanical anisotropy on a longitudinal section of human enamel studied by nanoindentation. J. Mater. Sci: Mater. Med. 21, 1811–1816. Cuy, J.L., Mann, A.B., Livi, K.J., Teaford, M.F., Weihs, T.P., 2002. Nanoindentation mapping of the mechanical properties of human molar tooth enamel. Arch. Oral Biol. 47, 281–291.
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An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res 7, 1564–1583. Oliver, W.C., Pharr, G.M., 2004. Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J. Mater. Res. 19, 3–20. Park, S., Wang, D.H., Dongsheng, Z., Romberg, E., Arola, D., 2008. Mechanical properties of human enamel as a function of age and location in the tooth. J. Mater. Sci. - Mater. Med 19, 2317–2324. Schneider, G.A., He, L.H., Swain, M.V., 2008. Viscous flow model of creep in enamel. J. Appl. Phys., 103 Spears, I.R., 1997. A three-dimensional finite element model of prismatic enamel: a re-appraisal of the data on the Young's modulus of enamel. J. Dent. Res 76, 1690–1697. Staines, M., Robinson, W.H., Hood, J.A.A., 1981. Spherical indentation of tooth enamel. J. Mater. Sci. 16, 2551–2556. Warshawsky, H., 1989. Organization of crystals in enamel. Anat. Rec. 224, 242–262. Waters, N.E., 1980. Some mechanical and physical properties of teeth. In: Vincent, J. F.V., Currey, J.D. (Eds.), The Mechanical Properties of Biological Materials. Cambridge University Press, Cambridge, UK White, J.M., Goodis, H.E., Marshall, S.J., Marshall, G.W., 1994. Sterilization of teeth by gamma radiation. J. Dent. Res 73, 1560–1567. Wo, P.C., Zhao, X.L., Munroe, P.R., Zhou, Z., Li, K., Habibi, D., Xie, Z., 2013. Extremely hard, damage-tolerant ceramic coatings with functionally graded, periodically varying architecture. Acta Mater. 61, 193–204. Xie, Z.H., Swain, M.V., Swadener, G., Munroe, P., Hoffman, M., 2009a. Effect of microstructure upon elastic behaviour of human tooth enamel. J. Biomech. 42, 1075–1080. Xie, Z., Swain, M.V., Hoffman, M.J., 2009b. Structural integrity of enamel: experimental and modeling. J. Dent. Res. 88, 529–533. Xie, Z.H., Hoffman, M., Munroe, P., Singh, R., Bendavid, A., Martin, P.J., 2007a. Microstructural response of TiN monolithic and multilayer coatings during microscratch testing. J. Mater. Res. 22, 2312–2318. Xie, Z.H., Mahoney, E.K., Kilpatrick, N.M., Swain, M.V., Hoffman, M., 2007b. On the structure–property relationship of sound and hypomineralized enamel. Acta Biomater. 3, 865–872. Xie, Z.H., Swain, M., Munroe, P., Hoffman, M., 2008. On the critical parameters that regulate the deformation behaviour of tooth enamel. Biomaterials 29, 2697–2703. Xu, H.H.K., Smith, D.T., Jahanmir, S., Romberg, E., Kelly, J.R., Thompson, V.P., Rekow, E.D., 1998. Indentation damage and mechanical properties of human enamel and dentin. J. Dent. Res. 77, 472–480.
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Zhou, J., Hsiung, L.L., 2007. Depth-dependent mechanical properties of enamel by nanoindentation. J. Biomed. Mater. Res. Part A 81A, 66–74. Zhou, J.K., Hsiung, L.L., 2006. Biomolecular origin of the rate-dependent deformation of prismatic enamel. Appl. Phys. Lett. 89, 3.
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Size dependent elastic modulus and mechanical resilience of dental enamel Simona O'Brien a,b, Jeremy Shaw c, Xiaoli Zhao a, Paul V. Abbott d, Paul Munroe e, Jiang Xu f,g, Daryoush Habibi a, Zonghan Xie a,h,i,n a
School of Engineering, Edith Cowan University, Perth, WA, Australia Perth Institute of Business and Technology, Perth, WA, Australia c Centre for Microscopy, Characterisation and Analysis, The University of Western Australia, Perth, WA, Australia d School of Dentistry, The University of Western Australia, Perth, WA, Australia e School of Materials Science and Engineering, University of New South Wales, Sydney, NSW, Australia f Department of Materials Science and Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of China g School of Mechanical and Electrical Engineering, Wuhan Institute of Technology, Wuhan 430073, People's Republic of China h School of Mechanical Engineering, University of Adelaide, Adelaide, SA, Australia i School of Materials Science and Engineering, Tianjin Polytechnic University, Tianjin 300387, People's Republic of China b
art ic l e i nf o
a b s t r a c t
Article history: Accepted 22 December 2013
Human tooth enamel exhibits a unique microstructure able to sustain repeated mechanical loading during dental function. Although notable advances have been made towards understanding the mechanical characteristics of enamel, challenges remain in the testing and interpretation of its mechanical properties. For example, enamel was often tested under dry conditions, significantly different from its native environment. In addition, constant load, rather than indentation depth, has been used when mapping the mechanical properties of enamel. In this work, tooth specimens are prepared under hydrated conditions and their stiffnesses are measured by depth control across the thickness of enamel. Crystal arrangement is postulated, among other factors, to be responsible for the size dependent indentation modulus of enamel. Supported by a simple structure model, effective crystal orientation angle is calculated and found to facilitate shear sliding in enamel under mechanical contact. In doing so, the stress build-up is eased and structural integrity is maintained. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Tooth enamel Young's modulus Nanoindentation Deformation Finite element analysis
1. Introduction Being the hardest tissue in the human body (Fig. 1(a)), dental enamel is built to last (Lawn et al., 2009). The remarkable resilience of enamel stems from its unique microstructure that comprises highly organised hydroxyapatite (HAP) platelets (Fig. 1b, c); Nanci, 2008; Xie et al., 2009a, 2008). Electron and atomic force microscopy investigations also reveal that a thin protein layer is present between the HAP crystallites in mature enamel (Habelitz et al., 2001; Warshawsky, 1989) and this, though small in quantity, plays an important role in regulating the mechanical properties of enamel to uniquely suit its functions (Gao et al., 2003; Lawn et al., 2010). Young's modulus is commonly used to quantify a material's stiffness. Enamel has a hybrid laminate structure at nanoscale (He and Swain, 2007; Xie et al., 2009b), which means its modulus is determined not only by the
n Corresponding author at: School of Mechanical Engineering, University of Adelaide, SA 5005, Australia. Tel.: þ61 8 8313 3890; fax: þ61 8 8313 4367. E-mail address: [email protected] (Z. Xie).
0021-9290/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.12.030
volume fraction of its constituents but also by the spatial arrangement of the mineral crystals (Xie et al., 2009a). Young's modulus of enamel is routinely measured using the depth-sensing indentation (DSI, also called nanoindentation) method (Oliver and Pharr, 1992, 2004). Values reported in the literature vary by over 100% (Cuy et al., 2002). Such a wide range results from multiple factors, such as variations in location, testing conditions (i.e., dry or wet) or age (Cuy et al., 2002; Habelitz et al., 2001; Lewis and Nyman, 2008; Park et al., 2008; Staines et al., 1981). Moreover, the Young's modulus of enamel was found to decrease with increasing indentation depth (Zhou and Hsiung, 2007). Given the dependence of Young's modulus of enamel on indentation depth, it is important to conduct measurements under depth control when interrogating the stiffness over the thickness of enamel. Unfortunately, most nanoindentation tests have been performed under load control (i.e., at constant load), making it difficult to compare the modulus values obtained from different regions, where indentation depths may differ (He et al., 2006; Xie et al., 2007b; Zhou and Hsiung, 2007). Moreover, although the sample preparation and testing conditions are known to influence the measured mechanical properties of enamel, (Guidoni et al.,
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
Enamel
1061
Dentin
Aqueous putty
1mm
5 μm
Fig. 1. Tooth structure viewed at different length scales. Optical image of (a) tooth macrostructure consisting of enamel and dentin. (b) Schematic illustration of enamel microstructure formed by tightly packed rods (or prisms) that are aligned perpendicular to the enamel–dentin junction. Inside the rods the mineral crystallites are bonded together by proteins (Habelitz et al., 2001; Warshawsky, 1989) (modified from (Nanci, 2008)). (c) Transmission electron micrograph of small enamel section (refer to ‘box’ in (b)) prepared parallel to the c-axis of rods.
2006; Staines et al., 1981) the majority of mechanical tests were carried out under dry condition or on ‘wet’ specimens that had previously been dried during preparation (Cuy et al., 2002; Xu et al., 1998). In this study, we conducted an in-depth investigation of the elastic behaviour of enamel under the influence of mechanical loads. Enamel samples were prepared and tested in a hydrated state. Young's modulus of enamel was determined using a line scan obtained from the occlusal surface to the enamel–dentin junction (EDJ) at various depths. A mechanistic model was used to link crystal orientation angle with the depth dependent elastic behaviour of enamel. Furthermore, the implications of crystal orientation in promoting shear deformation and dissipating indentation energy were examined and clarified. 2. Experimental 2.1. Sample collection and preparation Ethics approval was given by the Ethics Review Committee of Edith Cowan University prior to the collection of teeth. Nine third molars extracted from patients aged 20–30 years were used in this study. The teeth were intact without any decay or other disease and were extracted for clinical reasons and on the treating dentists' advice because of crowding in the jaw and impaction of the teeth preventing them from eruption. Informed consents were obtained from patients involved. Upon extraction, teeth were stored at 4 1C in Hanks' Balanced Salt Solution (HBSS) (Sigma-Aldrich Co., St. Louis, USA) with the addition of 0.02% thymol crystals to prevent demineralisation and inhibit bacterial growth (Habelitz et al., 2002; White et al., 1994). Struers resin and hardener were mixed to form 30 mm diameter by 10 mm high cylindrical blocks inside Struers plastic moulding cups. A 15 mm diameter hole was then drilled through the centreline of the block, which was later filled
from one end with aquatic putty (Selleys Knead-It Aqua, Selleys, Australia). The tooth was inserted root first into the cylinder and pressed into the putty, leaving only the occlusal surface exposed. A precision saw (Isomet 1000, Buehler Ltd., Lake Bluff, IL, USA) was used to section the partially embedded tooth at a distance of 5 mm from its edge in a direction parallel to mesial–distal plane. HBSS was used as a coolant during sectioning. The cut surface was then ground and polished with series of Struers silicon carbide papers up to 4000 grit. The final polishing of the specimens was done on a soft polishing cloth soaked in water (LaboPol 25, Struers, Copenhagen, Denmark). The c-axis of prisms was organised roughly parallel to the polished surface. Once prepared, the specimen was stored in HBSS that contains 0.02% thymol crystals.
2.2. Nanoindentation testing A depth-sensing indentation system (Ultra-Micro Indentation System, UMIS2000, CSIRO, Australia) equipped with a Berkovich indenter was used to measure the Young's modulus of enamel (Oliver and Pharr, 1992, 2004). The indenter tip was calibrated by conducting single-cycling indentation tests on fused silica (a standard material having a known modulus of 72 GPa) at different loads. To conduct measurements in a wet condition, a specially designed stainless steel holder was used, and the specimen was clamped onto the bottom using a slide-screw assembly. A stainless steel plate of 1 mm in thickness was placed on the specimen to distribute the clamping force evenly. The specimen was submerged in HBSS before testing. The polished surface of enamel was indented from the occlusal surface to the enamel–dentin junction (EDJ). Each row was generated along a line parallel to the surface and contains 5 indents with an interval of 50 mm. The spacing between rows is also 50 mm. Load–partial unload tests were run in a closed-loop under load control to determine the elastic modulus of enamel at different depths. A maximum load of 400 mN was applied in eight increments. The loading rate was set to 2.5 mN/s, which represents the static response of the material. Following each increment were 10 decrements, from which the average Young's modulus and standard errors were calculated with the load applied roughly perpendicular to the prism direction (Oliver and Pharr, 1992, 2004) and plotted as a function of indentation depth. Recent nanoindentation studies have shown that prism orientation has little impact on the hardness and modulus of enamel. (Braly et al., 2007;
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Xie et al., 2009a). To ensure that the multi-cycling nanoindentation is a suitable technique for the modulus measurement of materials at different depths, fused silica was tested by following the above procedure. It was found that the measured modulus of fused silica is independent of depth and within 1% of the theoretical value.
simulated for comparison purpose. Then, shear sliding effect was enabled and implemented by defining the material to be elasto-plastic, with the shearing taking place along the vertical direction and the yield strength being that of protein (1.09 GPa) (Xie et al., 2008).
3. Results and discussion
2.3. Microstructure observation Ultra-thin sections through the central head region in a direction parallel to enamel rods were prepared for transmission electron microscopy (TEM). To do this, a dual-electron/focused ion beam (FIB) workstation (Nova Nanolab 200, FEI Company, Hillsboro, OR, USA) was used. The detailed procedure can be found elsewhere (Xie et al., 2007a). Briefly, prior to the milling a 1 μm thick platinum layer was deposited to protect the surface area of interest from ion beam damage. A ‘rough’ sectioning was then performed with a current of 10 nA, in which trenches were created at both sides of the platinum strip to obtain a cross section of 3 μm thickness. A number of ‘fine’ mills were taken at reduced currents (5–1 nA) to thin the section to 1 μm. Final mills were carried out at further reduced currents (300–100 pA) to reduce the thickness down to 100 nm suitable for electron transparency. The TEM specimens were transferred from the FIB sample holder to carbon coated copper TEM grids using a high-precision micromanipulator (Kleindiek Nanotechnik GmbH, Reutlingen, Germany). TEM imaging was performed using a field emission gun-TEM (Philips CM200, Eindhoven, Netherlands).
2.4. Finite element analysis Mineral platelets of enamel start to slide against each other, when the shear stress in the protein matrix exceeds a critical value (He and Swain, 2007). As a result, there were no obvious cracks near the indentation site (Xie et al., 2008). According to this, finite element models were constructed in this work to clarify the effect of the shear deformation of the protein layer on the magnitude and distribution of contact-induced stress in enamel. The simulations were performed using COMSOL software. Two-dimensional axisymmetric models, with the axial coordinate z along the loading direction of the indenter, were constructed. The models consisted of an enamel block measuring 50 50 mm2, loaded by a spherical indenter with a radius of 5 mm. A refined mesh was created around the contact zone within an area of 10 10 mm2 directly underneath the indenter where the stress concentration was expected to rise during indentation. The total number of mesh elements is 17631, which includes the elements within the indenter. Further mesh refinement does not improve the simulation accuracy significantly. The contact between the indenter and the sample was assumed to be frictionless. The boundary conditions are similar to those used in the previous works (Wo et al., 2013; Zhao et al., 2011) and are briefly described as follows: The bottom of the block (z ¼ 50 mm) is fixed in the z direction, while its right edge (r ¼50 mm) is fixed in the x direction. The axisymmetric axis coincides with the left edge of the block (r¼ 0) for obtaining 3D simulation results. The tip of the indenter was positioned at z ¼0 mm (0 displacement) before the simulation started. The indentation process is modelled as a downward displacement of the indenter tip from 0 to 0.4 mm at a step of 0.04 mm. As mentioned above, the main objective for the finite element analysis (FEA) is to clarify the difference between the mechanical response of enamel with and without shear sliding. Hence, the shear sliding effect was treated in a parametric way without introducing structural complexity into the modelling. Specifically the structural mechanics module was applied, using Hook's law as the constitutive relation. As indicated in Fig. 1(c), on a sub-micrometre scale enamel exhibits a ‘bricks-and-mortar’ structure, with the ‘bricks’ being made from mineral crystals which are embedded in a protein matrix. The volume fraction of protein is below 4%, which can be derived from the dimensional parameters listed in Table 1. In the FEA model Young's modulus (129 GPa) and Poisson's ratio (0.25) values were used for the mineral phase (refer to Table 1). First, a perfect elastic response was
We first investigated how Young's modulus of enamel changed across its thickness at various indentation depths. A typical load– partial unload curve obtained from the middle region of enamel is shown in Fig. 2(a), from which Young's modulus of enamel is calculated (Oliver and Pharr, 1992) and found to decrease with increasing depth (Fig. 2(b)). Given the depth-dependent elastic property of enamel, the modulus, measured as a function of indentation depth, should then be applied to simulations, in order to better understand the mechanical design of enamel. Following the same approach, we determined Young's modulus of enamel over its thickness from the occlusal surface to the EDJ and plotted for three indentation depths, namely 0 mm, 0.5 mm and 2 mm (Fig. 2(c)). For each depth, Young's modulus decreased with increasing distance from the occlusal surface, confirming that enamel is indeed a functionally graded material (He and Swain, 2009). The change of the enamel elasticity over its thickness is attributed to the variation in the microstructure and composition from the surface towards the EDJ (Cuy et al., 2002; Low et al., 2008). Moreover, as the indenter tip approaches the enamel specimen, the contact point position, which could be at either mineral platelet or protein, has its strong influence on the load–displacement curve, and thus the value of Young's modulus. Consequently, modulus values obtained for shallower indentation depths, for example at ht ¼0.5 mm, show a more complex undulating pattern than those obtained at a depth of 2 mm, presumably due to the inhomogeneous nature of the enamel structure. A plastic-damage model was developed for enamel and, according to it, the shear deformation of protein (i.e., the extension and break of peptide chains) can cause the reduction in stiffness under indentation loading (An et al., 2012a). Recently, the nonuniform arrangement of the HAP crystallites was also identified to be a key factor responsible for unique mechanical behaviour of enamel (An et al., 2012b). It is, therefore, natural to inquire what roles the crystal arrangement or orientation plays in the depthdependent elastic response of enamel. For this purpose, a mechanistic model was built from the experimental observation that enamel exhibits a staggered hybrid structure (as seen in Fig. 1(c)), in which the mineral platelets can rotate to accommodate the deformation (Zhou and Hsiung, 2007). Notably, such a structural configuration has been successfully adopted by others to understand unique mechanical characteristics of enamel, for instance, rate-dependent deformation (Zhou and Hsiung, 2006) and creep (He and Swain, 2008; Schneider et al., 2008). According to the
Table 1 Structural and mechanical parameters of enamel constituents used in the theoretical analysis and modeling. Parameter
Mineral crystal Symbol
Thickness (nm) Length (nm) Poisson's ratio Young's modulus (GPa) Spacing of mineral crystals (nm) a
hm l νm Em d
Value 50 500 0.25 129 1.5
Protein Reference Kerebel et al. (1979) Driessens and Verbeeck (1990) Liu et al. (2006) Ang et al. (2010)
Symbol
Value
Reference
a
hp
1.5
νp Ep
0.45 2
Liu et al. (2006) Ji and Gao (2004), Spears (1997)
The thickness of protein, hp is calculated from hp/hm ¼ 0.03 which corresponds to a volume fraction of mineral E97% in the enamel.
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
1063
Load, P (mN)
400
200
0 0.0
0.5
1.0 1.5 2.0 Indentation depth, h [μm]
2.5
Young’s modulus, E [GPa]
93
85
77 0.7
1.7 Indentation depth, h [μm]
2.7
Fig. 3. Understanding the depth-dependent Young's modulus of enamel during deformation. (a) Young's modulus of enamel in relation to effective crystal orientation angle. Inset shows the mechanistic model used. (b) Schematic illustrations showing the local responses of mineral crystals to increasing indentation loads (or depths) from P1 to P3 by rotation from θ1 to θ3, resulting in a reduction in Young's modulus of enamel.
120
Young’s modulus, E [GPa]
ht = 0.0 ht = 05 ht = 2.0
where θ is the angle between the c-axis of the mineral platelet and the loading direction (also termed as the crystal orientation angle). The compression modulus, E2, and the shear modulus, G, of the staggered composite are given as (Liu et al., 2006):
90
E2 ¼
60 0.0
0.5
G
1.0
Normalised distance Fig. 2. Young's modulus of dental enamel measured by depth-sensing indentation tests at different depths and locations. (a) Representative indentation load, P, versus displacement (i.e., penetration depth), h, curve produced by a Berkovich indenter in the middle region of the enamel. (b) The average Young's modulus and standard deviations of enamel obtained from the middle region at different penetration depths. (c) The average Young's modulus and standard deviations of enamel corresponding to three indentation depths of 0, 0.5 and 2 μm plotted from the occlusal surface to the EDJ. The normalised distance is calculated as the distance from the surface to the indentation divided by the enamel thickness. The values in the shaded region in (c) are used in plotting (b).
model (inset in Fig. 3), Young's modulus of enamel can be expressed as (Xie et al., 2008) Ee ¼
Ez ¼
ð cos θ sin θÞ 2
2
sin θ 4
2
E2 þ 4
cos θ 2
sin θ 2
1 ð12K p hp ðhm þ hp Þ=αl Gp ð4Gp þ 3K p ÞÞ þ ð1=Ez Þ 2
hp þhm Gp hp
ð2Þ
ð3Þ
where hm and l are the thickness and length of the mineral crystal, respectively, hp and Gp are the thickness and shear modulus of protein layers between mineral platelets, α is the non-uniformed shear strain factor of the composite, Ez the modulus of the composite perpendicular to the c axis of the mineral platelet, and Kp the bulk modulus of protein. Note that Gp, α, Ez and Kp can be derived from (Liu et al., 2006): Ep 2ð1 þ νp Þ
Gp ¼ (
α¼
1þ
4 3
ðhp þ hm Þl hp ð1 þdÞ
ð4Þ 2 )
1d 1 þ dÞ
ð5Þ
ð1Þ
G
1 ðð3hp =ðhp þ hm Þð4Gp þ 3K p ÞÞ þ ð1=Em Þð3K p =ð4Gp þ 3K p ÞÞððhm þ 4νm hm hp þ 2hp 2νm hp Þ=hm ðhp þ hm ÞÞÞ 2
2
2
ð6Þ
Kp ¼
S. O'Brien et al. / Journal of Biomechanics 47 (2014) 1060–1066
Ep 3ð1 2νp Þ
ð7Þ
The structural and mechanical properties of the main constituents of enamel, i.e., the mineral and protein, are given in Table 1. Young's modulus of enamel, measured under different indentation depths within a narrow region halfway between the occlusal surface and EDJ, is shown in Fig. 3. It is worth noting that the effective crystal orientation angle of enamel changes from 451 to 481 when the indentation depth increases from 0.75 mm to 2.42 mm. Also note that the length of the crystallites used in the modelling is 500 nm (Driessens and Verbeeck, 1990). However, there is no consensus on the crystal length in the literature. The possibility that they are considerably longer cannot be ruled out (Nanci, 2008; Waters, 1980). To appreciate the effect that the crystal length may have upon the angle calculation, a wide range of values from 150 nm to 500 mm was modelled. Surprisingly, the results show that the crystal length has little impact on both compression modulus and shear modulus, consequently exerting minimal effect on the angles derived. The size dependent indentation modulus has also been observed for cortical bone. The plastic-damage model was proposed and successfully captured the stiffness loss during unloading (Lucchini et al., 2011; Zhang et al., 2010). However, the damage mechanism remains unclear, which renders the model phenomenological and lacking in a physical interpretation of its parameters. To overcome this problem, a physically sound approach was adopted for simulating the modulus reduction of dental enamel, (An et al., 2012a) based upon the experimental observation (Xie et al., 2008). It suggests that the shear deformation of protein contributes to the degradation of stiffness in the unloading curve. In another study, Zhou and Hsiung postulated that the rotation of mineral crystallites might also be responsible for the change of elastic modulus of enamel under indentation loading (Zhou and Hsiung, 2007). Recently, the crystal arrangement or orientation has been identified, among others, as a critical parameter that regulates the mechanical behaviour of enamel (An et al., 2012b; Xie et al., 2008). From this perspective, the structural basis that underpins the elastic behaviour of enamel was examined in this work; that is, the decrease of the Young's modulus of enamel with increasing indentation depth may also be governed by bending and/or tilting of the mineral crystals. As such, multiple origins of the depth-dependent elastic behaviour were possible. Note that the bending deformation of slender crystals is not explicitly considered during modelling. However, the bending would result in the (partial) tilting of the crystals; further the crystal length was found to have minimal impact on the angles derived. The crystal bending can thus be accommodated by the proposed mechanical model. More interestingly such an adaptive response is able to increase the contact area and, consequently, lower the stress concentration in enamel. For example, with the increase of indentation load from 100 to 250 mN, the contact pressure decreases by about 13%. By doing so, the loadbearing ability of enamel would be enhanced. It is noteworthy that fatigue damage may occur during multi-cycling nanoindentation (Jia and Xuan, 2012), and if so, that would affect the measurement accuracy of the indentation modulus. On the other hand, the relative orientations of the indenter to the enamel rods' direction were found to affect the measured elastic modulus (Cheng et al., 2010). To minimise its impact in the study of depth-dependent modulus in enamel, the multi-cycling indentations were conducted in the present work. This means the depth-dependent elastic property was measured at fixed locations, where the relative orientation of the indenter always remains constant during a multi-cycling test cycle. While the direction of indentation on the enamel may affect the absolute value of modulus, it would have
little impact on its depth-dependent nature and the underlying mechanism discovered here. As mentioned earlier, shear deformation between mineral crystals has been observed in human enamel by using transmission electron microscopy (Xie et al., 2007b). This deformation mode is considered to be essential for the remarkable damage tolerance of enamel (He and Swain, 2007; Xie et al., 2009b). To facilitate such a shear process, the shear stress developed between the mineral crystals under indentation loading should be maximised from the viewpoint of mechanical design. According to the model shown in Fig. 3, the shear stress, τ, can be written as
τ¼
1 s sin 2θ 2
ð8Þ
where s is the normal stress. Therefore, to meet this design requirement, the effective crystal orientation angle would be 451, which is consistent with the modelling result (Fig. 3). Although the shearing is considered to be the predominant deformation mechanism of enamel (He and Swain, 2007; Xie et al., 2009b) a quantitative understanding of its role in mitigating fracture and preventing catastrophic failure remains lacking. To redress this gap, finite elemental analysis (FEA) simulations was performed here to elucidate the benefit of shear deformation for the stress management in enamel. Compared to the ‘purely elastic’ deformation (i.e., not considering the plastic shear process); Fig. 4(a)), a reduction in the maximum shear stress by 9% (i.e., from 5.5 GPa to 5.0 GPa) for an indentation depth of 0.28 mm is observed when the shearing effect is
τ (GPa) 5.50
2.75
0.00
Material volume,(μm3)
1064
s4
s3
s2
s1
Fig. 4. The shear stress distribution generated in enamel by indentation at a depth of 0.28 μm: (a) perfect elastic model, (b) shear sliding enabled. (c) the size of regions under the influence of shear stresses greater than (s1) 3.0 GPa, (s2) 3.5 GPa, (s3) 4.0 GPa and (s4) 4.5 GPa, calculated from the FEA simulations; note that ‘PE’ stands for the perfect elastic model and ‘SS’ for the shear sliding model.
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taken into account (Fig. 4(b)). The stress distribution pattern is also modified under plastic shear and the volume populated by larger stress is reduced significantly (Fig. 4(c)). For example, the volume influenced by the stress level 43.5 GPa is reduced by more than 70% when the plastic shearing is enabled, and the volume influenced by the stress level 44.5 GPa almost completely diminishes when the shear deformation is considered. Consequently, the decrease of stress level and reduction of material volumes subjected to higher stresses, resulting from the shear sliding between mineral crystals, could further enhance the load-bearing ability of enamel.
4. Conclusion By combining depth-sensing indentation tests and modelling, multiple origins of the depth-dependent indentation stiffness were established in this study with a special focus on the crystal arrangement. The effective crystal orientation angle was calculated and found to facilitate the shear sliding of mineral crystals during indentation loading. Finite element analysis reveals that the shear deformation between crystal platelets serves to (a) lower the stress level and (b) reduce the material volume populated by larger stresses. This leads to a critical appreciation of the stress management ability of enamel during loading. The present work will rekindle interest towards understanding the mechanical responses of mineralised tissues to changing loads. From that, the mechanical design principles of these materials may be better understood.
Conflict of interest statement The authors certify that there is no conflict of interest with any financial organisation regarding the material discussed in the manuscript.
Acknowledgements We would like to thank Professor Mark Hoffman of University of New South Wales for providing constructive comments on the manuscript. The authors also thank Sparkle Dental in Joondalup, Western Australia, for providing fresh human molars. This work was partially supported by an Australian Dental Research Foundation (ADRF) grant (6/2010). The authors acknowledge the facilities, scientific and technical assistance of the Australian Microscopy and Microanalysis Research Facility at the Centre for Microscopy, Characterisation and Analysis, the University of Western Australia, a facility funded by the University as well as State and Commonwealth Governments. S. O'Brien also acknowledges an ECU Postgraduate Scholarship for her Ph.D. study. References An, B.B., Wang, R.R., Zhang, D.S., 2012a. Region-dependent micro damage of enamel under indentation. Acta Mech. Sin. 28, 1651–1658. An, B., Wang, R., Zhang, D., 2012b. Role of crystal arrangement on the mechanical performance of enamel. Acta Biomater. 8, 3784–3793. Ang, S.F., Bortel, E.L., Swain, M.V., Klocke, A., Schneider, G.A., 2010. Size-dependent elastic/inelastic behaviour of enamel over millimeter and nanometer length scales. Biomaterials 31, 1955–1963. Braly, A., Darnell, L., Mann, A., Teaford, M., Weihs, T., 2007. The effect of prism orientation on the indentation testing of human molar enamel. Arch. Oral Biol. 52, 856–860. Cheng, Z.J., Wang, X.M., Ge, J., Yan, J.X., Ji, N., Tian, L.L., Cui, F.Z., 2010. The mechanical anisotropy on a longitudinal section of human enamel studied by nanoindentation. J. Mater. Sci: Mater. Med. 21, 1811–1816. Cuy, J.L., Mann, A.B., Livi, K.J., Teaford, M.F., Weihs, T.P., 2002. Nanoindentation mapping of the mechanical properties of human molar tooth enamel. Arch. Oral Biol. 47, 281–291.
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