Size-dependent elastic/inelastic behavior of enamel over millimeter and nanometer length scales
Introduction
Teeth enamel, irrespective of the host, are materials that exhibit remarkable resilience with their ability to withstand millions of loading events in the oral environment throughout a creatures lifetime. Similar to other biological materials with load-bearing functions such as bone and nacre, characteristics such as hierarchical structure [1], anisotropy [2], [3] and composition [4] are crucial for long-term survival of the materials. A better understanding of enamel's deformation behavior under loading is required for restorative purposes in dentistry.
The composition of human enamel contains ∼90vol% hydroxyapatite (HAP) crystallites, ∼8vol% water and ∼2vol% organic matrix [5]. A schematic of enamel's hierarchical structure is shown in Fig. 1. Microstructurally, it is composed of closely-packed parallel rod-like structures also called prisms that span from the dentino-enamel junction (DEJ) to approximately 6–12 μm below the tooth surface [6]. The cross section of these rods resembles the shape of keyholes with diameters of ∼5 μm. At the nanostructural level, the rods consist of HAP crystallites. Meckel et al. [7] characterized crystallites orientation in the rods by observing ultra-thin sections (500–800 Å) of mature human enamel under electron microscopy. They identified that the crystallites within any single rod are oriented along the rod axis in the core region of a rod, but are gradually tilted away upon moving towards the tail of a rod, to be about 60–70° inclined to the rod axis. The junctions between rods are marked by abrupt changes of crystallite orientations. Although they did not observe evidence of “interprismatic substance” between the rods, this structure prevails in the electron microscopy images from other researchers [8], [9]. The electron microscopy images of cross-sections of human third molar enamel [8] and embryonic bovine enamel [9] show that the crystallites in the interprismatic regions appear to deviate up to 90° from those of the rod cores. As proteins and water accumulate within these less dense packed transition regions they appear as distinct protein-rich structures and are called ‘rod sheaths’ [10]. Adult human enamel crystallites have roughly hexagonal forms and a cross section of ∼30 nm in thickness and 55–90 nm in width [11], [12]. The crystallites are long, some investigators believe that they span over the entire thickness of the enamel layer [13]. The crystallites are believed to be surrounded by 1–2 nm of thin organic layer too [11].
Enamel's hierarchical structure has also made analysis of its mechanical behavior complex. A profound understanding of the structure-behavior relationships necessitates its mechanical characterization at all hierarchical levels. Characterization at the nanoscale helps to probe local origins of macroscale responses. Characterization at all length scales may provide guidance as to how enamel translates the strengths derived from nanostructures and hierarchical structures to macroscale robustness. However, currently research on enamel lacks a comprehensive assessment of the mechanical properties on most important hierarchical length scales which are namely: ‘bulk enamel’ (1–5 mm), ‘multiple-rod’ (∼50 μm), ‘intra-rod’ (∼5 μm) and ‘single-crystallite’ (10–50 nm). Indentation with spherical tipped indenters enables determination of the elastic/inelastic response at these various length scales because of the geometrical self similarity and the ability to select indenters with varying radii.
In this paper, we quantify human and bovine enamel's stress-strain behavior with uni-axial compression at millimeter length scale and spherical indentation with indenter radii of 3 mm, 8.3 μm, 63 nm. The corresponding contact areas are bulk enamel, several rods, multiple HAP crystallites and finally approximately one HAP crystallite. Besides the elastic response we are interested in the limit of elastic deformation which corresponds to an elastic–inelastic transition. The influence of the viscoelastic response on the observed behavior will also be discussed.
Section snippets
Theory: indentation stress-strain relationships
Indentation derived stress-strain curves were first introduced by Tabor [15], and originally developed for metals. The resultant stress-strain responses are different from uni-axial mechanical tests due to the complex 3D- stress–distribution around the indent.
A spherical indenter in contact with a specimen surface is shown in Fig. 2. The indentation contact area, A can be related to the indentation radius, a, or to the indenter radius, R and the contact depth, hc:
The classical
Specimen preparation
For uni-axial compression tests, permanent bovine mandibular incisors were used due to their larger size and amount of enamel compared to human teeth. Bovine enamel also shows very similar microstructure with human enamel in terms of rod size and shape, and also a similar amount of apatite and organic components [18], [19]. Rectangular prisms were first cut from the labial side of bovine incisors, as shown in the upper inset in Fig. 3. With the upper surface bonded by double-sided tape on a
Results
The radii of the indenters were calibrated as 8.3 ± 0.9 μm, 0.86 ± 0.03 μm and 63 ± 11 nm respectively for the 3 indenters. For simplification, these indenters are called R = 8.3 μm indenter, R = 0.86 μm indenter and R = 63 nm indenter throughout this article. For the 8.3 μm indenter, with P = 900 μN, ht = 20 nm, the contact pressure was , much lower than the indentation limit of elastic range of fused silica, observed as ∼9 GPa by using an R = 7 μm spherical indenter [24].
Discussion
To illustrate the reality, the limit of elastic range versus the corresponding contact radius for both enamel and synthetic HAP are plotted in Fig. 8. The limit of elastic range of HAP (∼7 GPa), probed by using R = 10 μm indenter by He and Swain [26] is also included in the figure. With the reported reduced elastic modulus of HAP of 130 GPa, the limit of elastic range of ∼7 GPa and R = 10 μm introduced into Eq. (6), the contact radius is calculated as ∼1270 nm. The limits of elastic range of
Conclusions
A comprehensive assessment of the elastic/inelastic behaviors in enamel for the different lengths scales is presented (Fig. 11). At the smallest length scale, plastic deformation occurs beyond the limit of elastic range, and is unlikely to be caused by proteins alone. At the biggest length scale, micro-crack induced damage ensued after the limit of elastic range was reached. The limit of elastic range of enamel is 0.4–17 GPa from millimeter to nanometer length scales. These length scales
Acknowledgement
The authors wish to express gratitude to German Research Foundation for financial support. Special thanks are extended to Dr. Rami Farah from the University of Otage, Assist. Prof. Stefan Habelitz and Dr. Neda Meshkin from University of California, San Francisco in the early phase of compression sample preparations. We also appreciate Dr. Hans Jelitto's from Hamburg University of Technology for his assistance in the construction of the compression device.
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