Acta Biomaterialia 14 (2015) 146–153

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Mechanics of microwear traces in tooth enamel Oscar Borrero-Lopez a, Antonia Pajares a, Paul J. Constantino b, Brian R. Lawn b,c,⇑ a

Departamento de Ingeniería Mecánica, Energética y de los Materiales, Universidad de Extremadura, 06006 Badajoz, Spain Department of Biology, Saint Michaels College, Colchester, VT 05439, USA c Materials Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA b

a r t i c l e

i n f o

Article history: Received 4 July 2014 Received in revised form 6 November 2014 Accepted 25 November 2014 Available online 4 December 2014 Keywords: Enamel microwear Contact mechanics Microplasticity Microfracture Diet

a b s t r a c t It is hypothesized that microwear traces in natural tooth enamel can be simulated and quantified using microindentation mechanics. Microcontacts associated with particulates in the oral wear medium are modeled as sharp indenters with fixed semi-apical angle. Distinction is made between markings from static contacts (pits) and translational contacts (scratches). Relations for the forces required to produce contacts of given dimensions are derived, with particle angularity and compliance specifically taken into account so as to distinguish between different abrasives in food sources. Images of patterns made on human enamel with sharp indenters in axial and sliding loading are correlated with theoretical predictions. Special attention is given to threshold conditions for transition from a microplasticity to a microcracking mode, corresponding to mild and severe wear domains. It is demonstrated that the typical microwear trace is generated at loads on the order of 1 N – i.e. much less than the forces exerted in normal biting – attesting to the susceptibility of teeth to wear in everyday mastication, especially in diets with sharp, hard and large inclusive intrinsic or extraneous particulates. Published by Elsevier Ltd. on behalf of Acta Materialia Inc.

1. Introduction The wear of teeth by small particulates is of great interest to evolutionary biologists because of its utility as an indicator of diet. This interest is manifest in the extensive field of dental microwear [1–14]. Conventional wisdom has it that hard foods are fractured in compression (normal, axial loading), leaving residual pits on the enamel surface, whereas soft foods are ground down in shear (sliding, translational loading), leaving scratches. Individual microcontact signals typically occur on a width scale of 1–20 lm [5,7]. The ratio of pits to scratches is taken as a measure of diet: some teeth show either scratches or pits, suggesting specialist diets; others show both scratches and pits, suggesting more omnivorous diets. Most advances in interpretation are based on quantifying this ratio, using ever more sophisticated methodologies in imaging and digitizing techniques [14]. Microwear patterns tend to be transient, so that they are indicators of recent food intake rather than a full dietary history. What is missing from dental microwear is a first-principles account of the basic contact micromechanics of pit and scratch formation. This deficiency provides the rationale for the current study, i.e. a strong physically based analytical footing for ⇑ Corresponding author at: Materials Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. Tel.: +1 301 975 5775. E-mail address: [email protected] (B.R. Lawn). http://dx.doi.org/10.1016/j.actbio.2014.11.047 1742-7061/Published by Elsevier Ltd. on behalf of Acta Materialia Inc.

describing individual microwear events in a hitherto empirical field. Wear can be life-limiting in some species of animals, especially grazers and browsers, where the work rate is high and continual. In humans, wear can be sufficiently severe (bruxing) to necessitate treatment by a dentist. A recent paper by Lucas et al. [15] set a precedent for an understanding by examining the nature of individual wear tracks produced by translating individual particles glued to a nanoindenter tip. The principal consideration in that study was the competitive roles of silicate-based phytoliths in vegetable matter [16], quartz dust in the atmosphere [10] and even enamel particles, typically 1–100 lm in diameter. It was suggested that incursive particulates should be at least as hard as enamel and should have a sufficiently high attack angle in order to cause significant tooth wear. However, the Lucas et al. study focused almost exclusively on the ultra-low load (mN) region, whereas the type of material removal process that leads to accelerated tooth wear involves much more deleterious microfracture [17,18]. Such differentiations in removal processes are well understood by tribologists and ceramic machinists as a transition between polishing and abrasion [19], and are a manifestation of an intrinsic ductile–brittle transition with increasing contact load and dimension [20,21]. This transition, along with the roles of particle compliance and sharpness, remains to be quantified in the context of dental microwear. Accordingly, in this paper, microwear in tooth enamel is examined using the well-established methods of indentation mechanics [21–24] as the basis for modeling the action of angulate, compliant

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particles in food diet. First, controlled indentation experiments in axial and translational loading are employed to simulate individual pits and scratches. Distinction is made between polishing and abrasion damage modes by identifying the loading conditions under which the microcontact undergoes a plastic to brittle transition. For modeling, the particles are considered to be ‘‘sharp’’, i.e. with fixed-angle profiles, as opposed to ‘‘blunt’’, with fixed-radius profiles. (Some consideration has been given to blunt contacts in an earlier analysis of the role of food size in tooth fracture [25,26], but sharp contacts are more deleterious and thereby serve as a worst case.) This type of indentation-based analysis has been pre-empted in a preceding model of macrowear, in which tooth occlusal wear rates were determined by integration of individual microwear events over a macroscopic contact area over time [18]. However, that earlier model was generic, without attention to individual contact micromechanics. The roles of particle angularity, compliance, size and number are now elucidated. The prospect of using trace dimensions, specifically track width and depth, to infer corresponding microcontact conditions in naturally occurring microwear tracks is examined. It is demonstrated that the contact loads lie in the region of 1 N, i.e. two to three orders of magnitude below the nominal bite forces for humans and other hominid species. The implications of the analysis concerning dietary characteristics are explored. 2. Morphology of pits and scratches Extracted molars from healthy adults were obtained from local dentists, and were disinfected using an ethanol solution (70%) and stored in distilled water under refrigeration. Those with any surface damage were discarded, and four others were embedded upright in a resin mold, as depicted in Fig. 1. The top surfaces were lightly ground to produce island facets 2–3 mm in diameter on the cusps, and then finished in an automatic polishing machine (Phoenix 4000, Buehler, Lake Bluff, IL) using diamond particle suspensions down to 1 lm. The maximum depth removed below the original occlusal surface was 100 lm, i.e. small compared with the thickness 1.5 mm of the enamel. Preliminary Vickers indentations (HSV-30, Shimadzu, Kyoto, Japan) at a load of 10 N on the polished surfaces yielded hardness values H = 4.0 ± 0.2 GPa (mean and standard deviation, 10 indents). A conical Rockwell diamond indenter with included angle w = 60° and tip radius 25 lm (Nanotest, Micro Materials Ltd.,

Embedded molar

Resin

147

Wrexham, UK) was selected in an attempt to simulate wear events in a particle-rich diet. The indenter had some surface imperfections on a scale of 3 lm in the near-tip region, but, in the context of the irregular geometries of natural particulates, this was considered to add a touch of realism. Tests were run in both normal axial and lateral sliding contact over a load range of 0.1–3 N on the polished surfaces, which were kept moist during the testing. In the axial tests, the loads were ramped up monotonically to maximum load and held for 5 s before unloading. In the sliding tests, the specimens were translated at 2 lm s1, with load applied linearly at 5 mN s1 to its prescribed maximum and then held steady over a further translation distance of 200 lm. The indentation sites were examined and photographed by optical microscopy (Epiphot 300, Nikon, Tokyo, Japan). Some specimens were sputter-coated with a gold layer and then examined by scanning electron microscopy (SEM; QUANTA 3D FEG, FEI Company, Hillsboro, OR) using secondary electrons at low voltage (5 kV). All tests produced residual impressions, indicating that the Rockwell indenter can be considered effectively sharp. Optical microscope images of the indentations in axial contact are shown in Fig. 2 at normal loads of 0.25, 0.5, 1 and 2 N. The impressions have somewhat irregular edges, attributable to the aforementioned imperfections in the shape of the indenter tip, but are otherwise near-circular. Radially extending cracks are clearly evident at the higher loads in Fig. 2, but not at the lower loads, consistent with the existence of a plastic-to-brittle threshold contact dimension [20]. Also faintly visible is some peripheral microcracking at the highest load. Analogous optical images of indentations in translational motion are shown in Fig. 3 at loads of 0.25, 0.5, 1 and 2 N. These images are reminiscent of scratches in a broad range of brittle solids [27]. Once beyond the initial load ramp, the linear traces in Fig. 3 assume a constant steady-state width. Again, outward extending cracks, similar to those observed in Fig. 2, but with a tendency to propagate more in the direction of translation, are evident at the higher loads. Peripheral microcracking is more pronounced than in static loading, coalescing in places into lateral cracks [28]. The incidence of cracking of any type becomes more evident at the higher loads, again indicative of threshold behavior. Note the apparent absence of any cracking within the tracks, consistent with a surface smearing action of the moving contact. Higher magnification SEM images of the contact sites at the upper end of the load range are pictured in Fig. 4. It is apparent that

Occlusal view

Wear facets

Fig. 1. Schematic diagram showing the embedment of a molar tooth into a resin mold. Grinding and polishing of the top surface produces smooth planar test facets for testing.

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Fig. 2. Optical microscope images of static indentations or pits in polished tooth enamel using a Rockwell cone indenter with tip radius 25 lm: loads (a) 0.25 N, (b) 0.5 N, (c) 1 N and (d) 2 N. The same scale is used for all images.

Fig. 3. Optical microscope images of scratches in polished tooth enamel using the same indenter as in Fig. 2, but in sliding mode: loads (a) 0.25 N, (b) 0.5 N, (c) 1 N and (d) 2 N. Note different scales on each image.

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3. Contact micromechanics

(a)

3.1. Microplasticity modes Fig. 5a shows a schematic of a tooth surface contact with a sharp, fixed-profile indenter of included half-angle w, at normal load P and contact half-width a. The contact may be static (axial loading) or sliding (translational loading). A major simplification ensuing from the fixed-profile geometry is that of geometrical similarity [22], in which the contact can be defined by

P=a2 ¼ ap

10 µm (b)

ð1Þ

where the indentation pressure p is independent of load, but dependent on indenter angle w, i.e. p = p(w), with a an indenter shape factor. It is assumed for the moment that a is the same in static and sliding contact. In the case of ideally elastic contacts, a compliant indenter will increase (blunt) the effective half-angle w0 (i.e. w0 > w) according to the Sneddon relation [29]

tan w0 ¼ ð1 þ E=EI Þtan w

ð2Þ

where E and EI are specimen and indenter modulus, respectively. Fig. 5b shows plots of effective half-angle w0 as a function of true half-angle w for selected ratios EI/E, demonstrating how compliant particles blunt the contact. If the contacting particle were to deform plastically, w0 would increase further, so Eq. (2) affords a lower bound to the blunting. The angle dependence of indentation pressure can be approximated by the Johnson equation for ideal elastic/plastic contacts [24], with due allowance for indenter compliance by replacing w with w0 , i.e.

P

(a)

10 µm

EI

Fig. 4. SEM images of portions of tracks from microcontacts with a diamond cone indenter at load P = 2 N in (a) axial loading and (b) translational loading (sliding direction left to right). Note the copious amount of debris from microcracking, with smearing at the sliding interface.

Ψ′

E

h Elastic

a

(b) 90 Effective indenter angle, ψ′

the contacts at these loads have significantly disrupted the enamel microstructure. The static indentation (Fig. 4a) has left a pit within which material has been crushed and partially ejected. Some of the debris is visible on the surface outside the actual contact area. Radial cracks are visible. The sliding indentation (Fig. 4b) shows a smooth, smeared out trace, but with evidence of the same type of crushing and debris at the sides, coalescing into lateral cracks in places. (The surfaces in these images were not subjected to a post-indentation rinse and clean preparation prior to insertion into the SEM, thus preserving the debris field.) It is not difficult to envisage that any such debris itself could contribute significantly to the particulate components responsible for accelerated wear. One advantage of the smearing within the scratches is that it allows for relatively straightforward measurements of the smoothed cross-section profiles. Since a sharper particle will leave a deeper contact, it is apparent that the profile must contain useful information on the effective angle of the indenting particle, even allowing for the shallowing effects of elastic recovery. Accordingly, measurements of the width 2a and residual depth h were made across scratches over the load range 0.25–3 N, using a surface profilometer (Surftest SJ-400, Mitutoyo, Kanagawa, Japan). Such measurements over five scratches yielded h/a = 0.30 ± 0.12, i.e. relatively shallow traces in these tests. In the next section, micromechanical models are developed to predict loads and particle sharpness from contact dimensions.

Plastic

0.01 0.1

60

1

E I /E = 10

30

0

0

30

60

90

True indenter angle, ψ Fig. 5. (a) Idealized geometry of asperity contact. A conical indenter under load P produces a depression of half-width a on the test surface. The non-rigid compliant indenting particle blunts the contact, increasing the true semi-apical angle w to the effective angle w0 . Sliding contact translates the configuration out of the page of diagram. Upon unloading, elastic recovery leaves residual impression of depth h. (b) Plot of w0 vs. w for selected modulus ratios EI/E (Reproduced from Ref. [18].).

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p ¼ ð2Y=3Þ½1 þ lnðE cot w0 =3YÞ

ð3Þ

P ¼ 0:40aHa2 f1 þ ln½0:56ðE=HÞ cot w0 g

ð4Þ

Plots of P vs. a are shown for conical particles (a = p) in Fig. 7a for different indenter angles w and fixed modulus ratio EI/E = 1, and in Fig. 7b for different modulus ratios at mid-range indenter angle 45°. The contact loads generated over a given contact size a diminish systematically at higher w and lower EI. The loads are on the order of 0.1–10 N over the half-width data range a = 1–20 lm. Allusion was made in Section 2 to information about the indenter characteristics contained by the profile of an unloaded impression. The residual impression depth h is somewhat less than that at maximum load, owing to elastic recovery. There are several treatments of elastic recovery beneath sharp indenters, but here resort is made to a semi-empirical relation for the relative residual depth from an earlier study [31]: 1=2

12

8

16

20

16

20

Contact load, P (N)

ψ = 15o

6

30o 45o

4

60o 75o

2 0 10

(b) Indenter angle ψ = 45o 8 EI/E = 10

6

1

4 0.1

2 0 0

12

8

4

Contact half-width, a (µm) Fig. 7. Plots of load P vs. contact half-width a for conical particles (a = p) on tooth enamel from Eq. (4), (a) for different indenter angles w at fixed modulus ratio EI/ E = 1, and (b) for different modulus ratio at mid-range indenter angle 45°.

ð5Þ

where c = 0.91 is a geometrical constant to account for the depression of the indented surface outside the loaded impression. Inserting Poisson’s ratio m = 0.25 and H/E = 4.0 GPa/90 GPa for enamel into Eq. (5) yields the plot of w0 vs. h/a in Fig. 8. The measured value h/a = 0.30 for the scratches in Section 2 corresponds to an effective angle w0 = 69.5°. This is somewhat larger than the nominal angle of 60° for the Rockwell cone indenter, attesting to the blunting effect of the rounded tip. 3.2. Microcracking thresholds

90

60

30

It is well documented in the literature [20], as well as by the observations in Section 2, that in semi-brittle materials there exists a threshold for cracking in contact events. At higher loads, fixed-

0 10-2

10-1

100

101

102

Residual depth/indent half-width, h/a Fig. 8. Effective indenter semi-apical angle w0 vs. relative depth h/a of residual impression.

0.10 Y/E

0.08

p/E

4

(a) Indenter modulus EI/E =1

Effective semi-angle, ψ’

2

h=a ¼ fðc cot w0 Þ  ½2ð1  m2 Þc2 cot w0 H=Eg

0

8

Contact load, P (N)

where Y is the yield stress of the specimen material. To evaluate Y for tooth enamel, it is convenient to resort to data from Vickers indentation hardness tests, in combination with nominal values of enamel material properties [18,30]. Vickers diamond pyramid indenters may be considered to be effectively rigid (EI = 1000 GPa) relative to enamel (E = 90 GPa), so w0  w = 74° in Eq. (2). Fig. 6 plots p/E vs. Y/E from Eq. (3) in this rigid-indenter limit. Identifying the contact pressure p with Vickers hardness H = 4.0 GPa gives H/E = 4.0 GPa/90 GPa = 0.045, corresponding to Y/E = 0.027 on the Johnson curve. This gives Y = 2.4 GPa = 0.60H for tooth enamel. Inserting this value into Eq. (3) yields an expression for applied load P in terms of contact half-width a for any indenting particle of shape parameter a and effective included angle w0

10

Johnson

profile indenters produce cracks extending radially from the plastic impression. In axial loading, these radial cracks have a basic pennylike geometry [32] and satisfy the fracture mechanics relation between load P and radial crack half-width c [21,33]

0.06 H/E (Vickers)

0.04

P=c3=2 ¼ ð1=nÞðH=EÞ1=2 T

0.02

ð6Þ

with T the toughness and an angle-dependent coefficient

0

0

0.025

0.05

0.075

0.10

Y/E Fig. 6. Plot of Johnson Eq. (3) showing contact pressure p vs. yield stress Y, both axes normalized to specimen modulus E. Dashed lines indicate values for Vickers indenter on tooth enamel.

n ¼ n0 ðcot w0 Þ

2=3

ð7Þ

Eq. (6) is the fracture counterpart of Eq. (1) for plastic deformation. To quantify the threshold condition, it is again convenient to use data from Vickers diamond pyramid tests, employing commercial hardness testing machines over load ranges 0.1–10 N (HSV-30,

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c c(P)

102

a a(P)

101

P th

100

P* 10-1 10-3

10-2

10-1

100

102

101

103

Indenter load, P(N) (N) Fig. 9. Vickers contact dimensions a and c as a function of load P for tests on polished tooth enamel. Data are mean values (five indents per point) from experiment, lines from theory. Standard deviations are 5% in a and 10% in c (smaller than data points). Note how the curves cross each other at load P⁄, indicating a brittle–plastic transition. Load Pth indicates threshold below which radial cracks do not initiate.

Shimadzu, Kyoto, Japan) and 50–100 N (MV-1, Matsuzawa, Tokyo, Japan), with a fixed hold time of 5 s at maximum penetration. Accordingly, Vickers indentations were made on polished transverse enamel sections of extracted human molar teeth, taking care to keep the specimens moist at all times [34]. The results are plotted in Fig. 9. The filled symbols are plastic impression size a and radial crack size c for Vickers indentations, measured along indent diagonals, as a function of axial load P (average 5 indents or 10 linear measurements per point). The line through the a(P) data is a plot of Eq. (1), with p = H = 4.0 GPa for enamel and a = 2 for pyramidal indenters. The line through the c(P) data in Fig. 9 is a fit to Eq. (6) with H/E = 4.0 GPa/90 GPa and T = 1.0 MPa m1/2, yielding a coefficient n = 0.020. (Allowing for scatter in the data, this coefficient compares with an earlier value of 0.016 from Vickers tests on a range of ceramic materials [33].) Inserting w0  w = 74° for Vickers indenters into Eq. (7) then yields the angle-independent quantity no = 0.046. The plot in Fig. 9 demonstrates the threshold condition. The a(P) and c(P) curves cross each other at P = P⁄, below which load any radial cracks would be subsumed into the plastic impression. In actuality, the radial cracks disappear at some higher load Pth = kP⁄ (k > 1), because these cracks first have to be initiated [35]. Combining Eqs. (4, 6 and 7) then yields the following threshold load relation, slightly modified from earlier derivations [21] to accommodate indenter shape

Pth ¼ ðk=a3 n40 ÞðT 4 =E2 HÞFðw0 Þ

ð8Þ

with effective angle term

Fðw0 Þ ¼ 15:6ðtan w0 Þ

8=3

3

=f1 þ ln½0:56ðE=HÞcotw0 g

ð9Þ

In Fig. 9, the load Pth = 250 mN, below which radial cracks no longer appear relative to the crossover at P⁄ = 25 mN, defines the quantity k = Pth/P⁄ = 10. It is apparent from inspection of Eq. (9) that the critical load in Eq. (8) is highly sensitive to indenter angle (and compliance), evidenced by a reported reduction in Pth by more than an order of magnitude in going from Vickers to corner-cube (w = 74–54.7°, axis to edge angle) indenters [36]. 4. Discussion An analysis of wear track formation in tooth enamel has been developed, based on long-established micromechanics of

deformation and fracture beneath sharp, fixed-profile indenter tips (Fig. 5a). The assumption of a fixed profile leads to simplifications in modeling, specifically in relation to the relative scales of plasticity and fracture processes in the ensuing wear process. This simplification leads naturally to the concept of a threshold load, below which plasticity modes dominate (mild wear) and above which fracture modes dominate (severe wear). Images of contact sites in axial and translational loading (Figs. 2–4) confirm the essential features of individual microcontact pits and scratches, with plastic impressions and (at higher loads) accompanying cracks and debris. For contact radii in the range a = 1–20 lm, typical of natural microwear signals, corresponding contact loads are in the order of P  0.1–10 N (Fig. 7). A principal outcome of the micromechanics analysis is the explicit inclusion of particle angle and modulus in the contact equations. Recall that the effective indenter angle w0 increases with both increasing particle angle w and decreasing particle modulus EI (Fig. 5b). Of particular concern is how the microcontact force P in Eq. (4) varies with w0 for specified indent half-widths a. This force is plotted in Fig. 10 as the solid curves for fixed a = 1 lm and 10 lm, with a = p for conical indenters. Interestingly, while P is sensitive to the value of a (quadratic dependence in Eq. (4)), it is much less so to w0 (logarithmic dependence in Eq. (4)). Included as the dashed curve in Fig. 10 is the threshold force Pth from Eq. (8), again for a = p. This quantity reflects an especially high sensitivity of the F function in Eq. (9) to w0 . Note that the solid and dashed curves cross each other: the domain Pth < P at low w0 is that of microcrack-dominated severe wear from sharp, hard particles; the domain P < Pth at high w0 is that of plasticitydominated mild-wear from blunt, compliant particles. As an illustrative example, consider a mid-range particle angle w = 45°: the crossovers at w0  43° for a = 1 lm and w0  77° for a = 10 lm in Fig. 10 correspond to minimum particle moduli EI  27 GPa and EI  1 in Eq. (2) needed to exceed the respective microcracking thresholds. In this context, note that silica particles found in natural diets of many mammals (especially herbivores) have an intermediate modulus EI  75 GPa. One factor not included explicitly in the formulation is that of particle size. Yet it is well established in tribology studies that coarser particles can effect the transition from mild (polishing) to severe (abrasion) removal processes in brittle materials. For any given mass of particulates in a wear medium, larger particles will make fewer microcontacts over any given wear facet, meaning that those particles will shoulder higher individual loads. To take the

102

Microcontact force, P (N)

Indentation dimension (µm)

103

a = 10 µm

100 a = 1 µm

10-2

10-4 0

Threshold

30

60

90

Effective indenter angle, Ψ ’ Fig. 10. Plot of microcontact load P as a function of effective indenter half-angle w0 from Eq. (4) for two contact half-widths a (solid curves). Also plotted is threshold load Pth from Eqs. (8) and (9). Calculations for conical indenter (a = p).

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argument further, consider the bounding case of a close-packed array of particles with characteristic dimension l. Then the area occupied by one particle is l2, so the number of particles in unit area is 1/l2. Therefore, the number of microcontacts diminishes rapidly as particle size increases, so again each particle experiences a higher force. For a wear medium with a distribution of particle sizes, e.g. a soft food bolus, the largest will make first contact and thereby assume an inordinate proportion of the net load over a wear facet. Accordingly, diets with coarser incursive grits may be expected to greatly accelerate the enamel abrasion rate by effecting a transition from mild to severe wear. The current analysis pertains to individual microwear events. Integration over all such events within an occlusal facet over time can be used to obtain a macroscopic volume removal rate equation. A model of this type in a previous study [18] provided such an equation, analogous to the well-documented Archard’s law [19], with the efficacy of the removal process quantified by an angledependent wear coefficient K  cot w0 . However, that earlier model circumvented any consideration of the detailed micromechanics of the individual events, other than indicating that K will be much greater for a microcracking than for a microplasticity mode. The current analysis augments that earlier study by outlining the micromechanics of individual events, with specific attention to the loads required to produce pits and scratches of given scale and to the threshold conditions for transition from mild to severe wear. The images in Figs. 2–4 suggest microcracking on the scale of the enamel microstructure, i.e. 5 lm, consistent with the suggestion that the protein sheaths between hydroxyapatite rods can act as favorable weak interface sites for crack paths [37–43]. It is to be acknowledged that the analysis is based on simplifying assumptions that are open to discussion. The first of these relates to the representation of impinging particulates as sharp, fixed-angle indenters. This simplification enables the explicit introduction of indenter angle and modulus via Sneddon’s equation for elastic contacts (Eq. (2)). In reality, microcontacting particles have less well-defined angles, often with rounded tips, and may themselves undergo plastic deformation. However, the fixed-profile geometry with elastic contacts represents a worst case, and thus provides a conservative estimate of the damage process. It is also assumed in Johnson’s Eq. (3) that the indented material is ideally elastic–plastic, i.e. linearly elastic up to a well-defined yield stress Y = 2.4 GPa, whereas tooth enamel is more nonlinear in its initial stress–strain response, with preliminary irreversible deformation reported at stresses as low as 0.33 GPa [40,41]. In actuality, the yield stress Y appears in the present analysis as little more than a fitting parameter, and its physical interpretation is not central to the derivations. Another simplification implicit in the analysis is the focus on a single static or sliding contact event, whereas wear facets are made up of many such interacting events over time. Damage can accumulate from multiple passes, further exacerbating a transition from deformation-controlled to fracture-controlled removal processes [44], so again the present model is conservative in its capacity to account for wear rates. It has further been assumed that the indenter shape factor a in Eq. (1) is the same in static and sliding contact, whereas the contact stresses are likely to redistribute toward the leading edge of a translating particle, thereby enhancing the contact pressure at any given load. Finally, the model ignores the effects of microstructural anisotropy and property gradients within the enamel: nanoindentation tests on cross sections of human molars have shown that the hardness of enamel can diminish by a factor of 2 to 3 with increasing depth below the occlusal surface as the configuration of hydroxyapatite prisms changes [45], while abrasion tests on bovine enamel after erosion in acidic solutions have demonstrated a strong correlation between wear rate and Vickers hardness [46]. However, incorporation of these factors into a more complex model is unlikely to shift

the curves significantly along the (logarithmic) load axis in Fig. 10, so the main conclusions drawn from the analysis remain valid. The chief impact of this study may lie more in biology than materials science, specifically in the insight it can provide into the fundamental mechanisms of microwear [1–6,8–14]. As mentioned in the Introduction, this is a highly active, and sometimes contentious, area of evolutionary science [14,47]. The images in Figs. 2–4 indicate that axial and translational tests with standard indenters can simulate naturally occurring microwear signals from the diets of humans and other mammals [15]. The contact analysis can be used to extract quantitative information on the loads and particle characteristics responsible for such signals. For instance, Eq. (4) can be used to infer loads P  1 N for real-life microcontacts of micrometer-scale half-width. Recall that the tests were made on lightly polished surfaces, less than one-tenth of the enamel thickness, so the estimated load may be slightly higher on pristine occlusal surfaces. However, this value could be lower by a factor of 2 or 3 in severely worn teeth, owing to the diminished hardness in the enamel subsurface [45]. Regardless of such variations, loads of this order are relatively small in comparison with nominal bite forces for humans and other mammals, affirming that some form of enamel wear is not an unexpected occurrence in consumption of foods with a distribution of included sharp particulates. Note again that the effective angle w0 appears as a logarithmic term in Eq. (4), so that estimates of P from this equation are relatively insensitive to true particle angle and modulus. This same insensitivity means that it is difficult to infer anything about the geometry or size of particulates from the widths of the microcontact markings alone. However, depth measurements can provide an estimate of the effective particle angle. Also, as demonstrated in Fig. 10, the appearance of well-developed cracks outside the periphery of a microwear pit or scratch would be clear evidence of contact with sharp, hard particles, although it is also acknowledged that any traces of such cracking could be smeared out on a well-worn surface. Finally, given the appearance of copious enamel debris at the higher load indentations in Fig. 4, it is not difficult to contemplate the extreme wear rates experienced under extreme mastication or bruxing conditions. 5. Conclusions (i) Microwear traces (pits and scratches) from particle contacts in natural tooth enamel can be simulated and quantified by static and sliding microindentation experiments. (ii) Microcontact relations enable forces associated with the production of individual microwear traces to be evaluated. These forces are orders of magnitude lower than those incurred during normal biting activity. (iii) The roles of particle angle and modulus in microwear production are elucidated. (iv) Dietary conditions under which wear undergoes a transition from mild to severe are interpreted in terms of a shift from microplasticity to microfracture damage modes. (v) The impact of the study on evolutionary biology is discussed. Disclosures There are no conflicts of interest. Information of product names and suppliers in this paper is not to imply endorsement by NIST. Acknowledgements We wish to thank David and Oscar Maestre for kindly providing dental samples from their clinic (Maxilodental Maestre, Badajoz, Spain), Maria Carbajo (Facility of Analysis and Characterization of

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Solids and Surfaces, UEx, Badajoz, Spain) for the SEM images in Fig. 4, and Centro Tecnologico Industrial de Extremadura (CETIEX, Badajoz, Spain) for use of their profilometer. Robert Cook (NIST) provided useful comments on the paper. This study was supported in part by the US National Science Foundation (Grant # 1118385) and from NIST funding (administered via Dakota Consulting Inc.). Appendix A. Figures with essential color discrimination Certain figures in this article, particularly Figs. 1–3 and 9, may be difficult to interpret in black and white. The full color images can be found in the on-line version, at http://dx.doi.org/10.1016/ j.actbio.2014.11.047. References [1] Walker A, Hoeck HN, Perez L. Microwear of mammalian teeth as an indicator of diet. Science 1978;201:908–10. [2] Grine FE. Trophic differences between ‘gracile’ and ‘robust’ Australopithecines: a scanning electron microscope analysis of occlusal events. S Afr J Sci 1981;7:203–30. [3] Fortelius M. Ungulate cheek teeth: developmental, functional and evolutionary interrelations. Acta Zool Fennica 1985;180:1–76. [4] Janis CM, Fortelius M. On the means whereby mammals achieve increased functional durability of their dentitions, with special reference to limiting factors. Biol Rev 1988;63:197–230. [5] Teaford MF. A review of dental microwear and diet in modern mammals. Microscopy 1988;2:1149–66. [6] Ungar PS. Dental allometry, morphology, and wear as evidence for diet in fossil primates. Evol Anthrop 1998;6:205–17. [7] Grine FE, Kay RF. Early hominid diets from quantitative image analysis of dental microwear. Nature 1988;333:765–8. [8] Scott RS, Ungar PS, Bergstrom TS, Brown CA, Childs BE, Teaford MF, et al. Dental microwear texture analysis: technical considerations. J Human Evol 2006;51:339–49. [9] Sanson G. The biomechanics of browsing and grazing. Amer J Bot 2006;93:1531–45. [10] Damuth J, Janis CM. On the relationship between hypsodonty and feeding ecology in ungulate mammals, and its utility in palaeoecology. Biol Rev 2011;86:733–58. [11] Teaford MF, Ungar PS. Diet and the evolution of the earliest human ancestors. Proc Natl Acad Sci USA 2000;97:13506–11. [12] Lucas PW. Dental functional morphology: how teeth work. Cambridge: Cambridge University Press; 2004. [13] Ungar PS. Mammal teeth. Baltimore, MD: Johns Hopkins University Press; 2010. [14] Ungar P, Sponheimer M. The diets of early hominins. Science 2011;334:190–3. [15] Lucas PW, Omar R, Al-Fadhalah K, Almusallam AS, Henry AG, Michael S, et al. Mechanisms and causes of wear in tooth enamel: implications for hominin diets. J R Soc Interface 2013;10:20120923. [16] Piperno DR. Phytoliths: a comprehensive guide for archaeologists and paleoecologists. Lanham, MD: AltaMira Press; 2006. [17] Arsecularatne JA, Hoffman M. On the wear mechanisms of human dental enamel. J Mech Behav Biomed Mater 2010;3:347–56. [18] Borrero-Lopez O, Pajares A, Constantino P, Lawn BR. A model for predicting wear rates in tooth enamel. J Mech Behav Biomed Mater 2014;37:226–34.

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