journal of the mechanical behavior of biomedical materials 35 (2014) 85 –92

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Research Paper

Role of multiple cusps in tooth fracture Amir Barania, Mark B. Busha,n, Brian R. Lawna,b a

School of Mechanical and Chemical Engineering, The University of Western Australia, WA 6009, Australia Materials Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg MD 20899, USA

b

art i cle i nfo

ab st rac t

Article history:

The role of multiple cusps in the biomechanics of human molar tooth fracture is analysed.

Received 4 December 2013

A model with four cusps at the bite surface replaces the single dome structure used in previous

Received in revised form

simulations. Extended finite element modelling, with provision to embed longitudinal cracks

18 March 2014

into the enamel walls, enables full analysis of crack propagation from initial extension to final

Accepted 27 March 2014

failure. The cracks propagate longitudinally around the enamel side walls from starter cracks

Available online 3 April 2014

placed either at the top surface (radial cracks) or from the tooth base (margin cracks). A feature

Keywords:

of the crack evolution is its stability, meaning that extension occurs steadily with increasing

Molar teeth

applied force. Predictions from the model are validated by comparison with experimental data

Fracture evolution

from earlier publications, in which crack development was followed in situ during occlusal

Multicusp

loading of extracted human molars. The results show substantial increase in critical forces to

Critical load

produce longitudinal fractures with number of cuspal contacts, indicating a capacity for an individual tooth to spread the load during mastication. It is argued that explicit critical force equations derived in previous studies remain valid, at the least as a means for comparing the capacity for teeth of different dimensions to sustain high bite forces. & 2014 Elsevier Ltd. All rights reserved.

1.

Introduction

The human molar bites with sufficient force to break down and process an enormous variety of foods (Lucas, 2004; Ungar, 2010). However, high bite forces can cause the tooth enamel to fracture by longitudinal cracking along the side walls (Lucas et al., 2008; Lawn and Lee, 2009; Barani et al., 2011; Lee et al., 2011; Lawn et al., 2013). Longitudinal fractures may radiate from the occlusal surface down to the enamel margin (radial R cracks), or conversely from the margin to the occlusal surface (margin M cracks), all the while remaining confined within the enamel coat in the form of ‘channel’ fissures (Chai et al., 2009b; Barani et al., 2011; Keown et al., 2012b). Secondary chipping, and even splitting, can occur n

Corresponding author.

http://dx.doi.org/10.1016/j.jmbbm.2014.03.018 1751-6161/& 2014 Elsevier Ltd. All rights reserved.

(Constantino et al., 2010; Chai et al., 2011). The longitudinal fracture mode is of special interest for its frequency of occurrence and its potential as a precursor to spallation of the enamel from the dentin in occlusal overloads (Popowics et al., 2001; Qasim et al., 2005), i.e. as a ‘failure’ condition. A fundamental understanding of the morphology of longitudinal cracking in multicusp structures, along with the capacity to predict the critical loads as a function of key geometrical parameters, is of particular relevance to dentistry and biology (Lucas et al., 2008). Substantial advances have recently been made in the understanding of longitudinal cracks through experimental observation of fractures in axially loaded epoxy-filled glass-shell models (Qasim et al., 2005, 2006a, 2006b, 2007) and in extracted molar

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journal of the mechanical behavior of biomedical materials 35 (2014) 85 –92

teeth (Lawn et al., 2009; Lee et al., 2009, 2011; Chai et al., 2011; Keown et al., 2012b). Critical load data for longitudinal fractures on cusp-loaded molar teeth show considerable scatter, typically between 500 N and 1000 N, reflecting natural variations in tooth size and shape (Lee et al., 2009; Keown et al., 2012b). Mathematical modelling of the fracture process in shell structures has yielded simple but powerful explicit equations for the prediction of critical fracture loads in terms of key geometrical dimensions, notably tooth radius, tooth height and enamel thickness (Rudas et al., 2005; Chai et al., 2009b; Lawn and Lee, 2009; Barani et al., 2011, 2012b). Predictions from these equations have been used to provide an upper bound to the experimental critical load data on human molars (Lee et al., 2011). A criticism of the structural models used to simulate the cuspal configuration of molar tooth fracture is the simplicity of the geometrical representation, that of a single hemispherical dome on a short cylindrical base. In reality, many teeth have multiple cusps. Human molars in particular have four (sometimes five) cusps, while premolars have two (Kono et al., 2002; Benazzi et al., 2013). A human molar is shown in Fig. 1. Other mammals have different cusp structures, some more and some less convoluted (Ungar, 2010). Previous justification for the use of the simple dome structure is that it accounts for the main features of the experimentally observed crack development, most notably full containment of the crack within the enamel wall (i.e. without penetrating into dentin) and a largely stable extension to full fracture with progressively increasing axial load (Chai et al., 2009b; Barani et al., 2011, 2012b). It also facilitates derivation of the critical load equations with minimal complication, enabling quantitative prediction of bite force (Lee et al., 2011). Nevertheless, the case for a significant role of a multiple cusp geometry remains to be answered. Relevant to the argument is the reported observation on glass shell structures that the critical force for fracture diminishes as the load point is shifted further from the central axis, by a factor of two or more (Qasim et al., 2006b). The presence of multiple cusps also provides the potential for spreading the applied load over

Fig. 1 – Image of human molar tooth, showing cusp geometry at occlusal surface. Courtesy Paul Abbott.

more than a single cusp, and the possibility that an object located between cusps may generate a wedging force driving cracks along the cuspal ‘valleys’ (DeLong and Douglas, 1983). The aim of this paper is to determine the role of multiple cusps in the incidence of longitudinal fracture, and specifically to examine how any differences in crack response affect the previously determined critical load equations. We replace our preceding single-dome model of a human molar with that of a four-cusp enamel cap. In reality, the molar tooth morphology in humans and other primates is more complex (Yamashita, 1998; Kono et al., 2002; Smith et al., 2005; Benazzi et al., 2013; Berthaume et al., 2013; Frunza and Suciu, 2013), but in the interest of computational simplicity we assume the individual cusps to be hemispherical and equispaced. It will be argued that this idealization is not restrictive, since ultimate failure is governed by the stress state over the entire longitudinal crack path and not at local geometrical features. An extended 3D finite element model (XFEM) described in preceding publications, with provision for incremental crack extension embedded into the XFEM code, is used to determine this ultimate failure condition (Barani et al., 2011, 2012b). While traditional finite element analyses have been employed extensively to map out stress distributions in occlusally loaded molar teeth (Yettram et al., 1976; Rubin et al., 1983; Spears et al., 1993; Wakabayashi et al., 2008; Tajima et al., 2009; Anderson et al., 2011; Anderson and Rayfield, 2012; Benazzi et al., 2012, 2014), such analyses lack capacity to predict the critical loads at full failure. Using the XFEM methodology, we examine various loading and crack alignment configurations, and compare and contrast the results with those from previous, single-dome studies.

2.

Methods

The multicusp molar model is shown at left in Fig. 2, along with the former single-dome model at right. In each case the structure consists of an enamel shell of thickness d¼ 1.0 mm, base radius R¼ 5.0 mm and height H¼ 7.5 mm, bonded to a dentin interior. These are nominal values—actual values vary considerably in real molar teeth, by as much as a factor of two (Keown et al., 2012b). Simple scaling relations exist for predicting critical conditions for tooth enamel of different dimensions (Barani et al., 2011), so our choice of values is not limiting. The hemisphere radius in the four-cusp configuration is r¼ 2.5 mm, with smoothed-out valleys between. The structures in Fig. 2 were configured using XFEM implemented in Abaqus (Abaqus 6.9-EF1, Simulia, Providence, RI). Details of the procedure have been described previously (Barani et al., 2011, 2012a, 2012b). A full 3D analysis was necessary to deal with off-axis loading. The structures were mounted on a frictionless flat surface and anchored at the mid-point of the base to resist any lateral forces. A finite element mesh was constructed, with higher density along the prospective crack plane, and refined until the calculations converged. Frictionless contacts were applied using a flat plate loaded normally onto one or more cusps, as shown in Fig. 3. Properties for the enamel (e) and dentin (d) were assumed to be homogeneous and isotropic in each material. Nominal values were taken from previous studies (Barani et al., 2011): Young's modulus and Poisson's ratio, Ee ¼90 GPa, Ed ¼ 18 GPa,

journal of the mechanical behavior of biomedical materials 35 (2014) 85 –92

87

r

3 1 1

3

2

2 H

Y R

d

R

4

1

2

3 1

3

2

X

Fig. 2 – Schematic figure for human molar tooth, showing side and top views of four-cusp model (left) and the singledome model (right). Arrows and filled circles indicate load points. Shaded areas represent enamel thickness (shown in side views only). Dashed line represents crack plane.

νe ¼ 0.22, νd ¼ 0.35; and toughness, Te ¼ 0.7 MPa m1/2. Loading was controlled by displacing the indenter parallel to the axis of the model. When more than one cusp was loaded simultaneously, the indenters were displaced equally. The resulting net bite forces were determined from the normal reaction stresses integrated over the base of the model. Starter cracks were introduced at strategic locations on the cuspal surface or at the margin to imitate R or M cracks. Previous studies have confirmed that the fracture is insensitive to the size of such starter cracks, reflecting the stability in ensuing propagation (Barani et al., 2011). Crack advance around the enamel wall was enabled using a cohesive zone model, with parameters chosen to coincide with the toughness of the enamel (Barani et al., 2011). The crack was allowed to extend within the enamel coat by rupturing mesh elements in a cell-by-cell stepwise fashion, as typified by the mid-load R and M crack section views in Fig. 3 for loading on a single quadrate cusp. Some ancillary XFEM computations were also made to investigate the influence of location of the starter crack on the fracture evolution.

3.

Results

3.1.

Margin cracks

The bulk of the XFEM computations was performed on the multicusp model with a starter crack placed at the tooth base in the symmetry plane of cusp 1 (dashed line, Fig. 2). The flat indenter was applied on either a single cusp (1), two opposite cusps simultaneously (1,2), or all four cusps simultaneously

Fig. 3 – XFEM section views of radial (R) and margin (M) cracks for molar tooth loaded normally on a single cusp (1). View shows mid-load configurations, i.e. at approximately half load required to cause full side-wall fracture. Pixelated crack fronts indicate cell-by-cell rupture process in the XFEM code. (a) Margin crack and (b) Radial crack.

(1,2,3,4). Ensuing margin crack extension up the side wall from the base was measured as the projected vertical coordinate Y (Fig. 3a). Analogous computations were performed on the single dome model, for both off-axis loading at a distance of 3.5 mm from the centre (i.e. closely aligned with the load point in the multicusp model) and axial loading. Generally, the cracks grew slowly at first, then accelerated around the side wall on approaching the load point. The load at which the crack spans the full height of the tooth is designated as the ‘critical’ force for failure. Results of these computations are shown in Fig. 4a, as coordinate Y versus net force P at the tooth base. The following observations may be made: (i) for the multicusp curves, the critical net force increases with number of cusps contacted, although not in a proportional manner; (ii) the critical force for one-cusp loading in the multicusp model is a little higher than for off-axis loading in the dome model; (iii) the critical load for two-cusp loading in the multicusp model is similar to that for axial loading in the dome model. The spread in critical load values shows a strong dependence on the specific contact/load configuration. For comparison, experimental data from previous laboratory tests on extracted molars (Lee et al., 2011; Keown et al., 2012b) are shown in Fig. 4b. These data were produced by

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journal of the mechanical behavior of biomedical materials 35 (2014) 85 –92

3

Crack size,Y (mm)

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C1

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C1234

DA

DO

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5.0

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Crack size,Y (mm)

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900

1200

Bite force,P (kN) Fig. 4 – Margin crack coordinate Y measured from tooth base (cf. Fig. 2a) as a function of net reaction force P over base. (a) XFEM computations for the multicusp model, for loading on a single cusp (C1), two opposite cusps (C12), and all four cusps (C1234); and for the dome model loaded axially (DA) and off-axis (DO). (b) Laboratory data on cusp-loaded extracted molars from (Lee et al., 2011; Keown et al., 2012b).

loading a single prominent cusp of any given tooth with a flat indenter (i.e. analogous to Fig. 3). The various traces shown correspond to individual tests on several teeth spanning a broad range of dimensions, with typical wide scatter. Key points in relation to the XFEM curves are: (i) the experimental data span the computed curves for loading on one and two (opposite) cusps; (ii) the computed curve for off-axis loading of the single-dome model provides a lower bound to the experimental data, while that for axial loading provides an upper bound, as previously reported (Lee et al., 2011). Fig. 5 shows additional XFEM results aimed at determining the sensitivity of the computations to starter crack location. In this particular instance the starter cracks were placed at different locations around the tooth base, still normal to the outer surface but now at prescribed angles relative to the axis through cusp 1. The margin crack trajectories (Fig. 5a) for the different angles all converge on the load point (Qasim et al., 2006b). The corresponding crack-coordinate versus net-force plots (Fig. 5b) demonstrate a marked insensitivity to variation in starter location.

3.2.

Radial cracks

Analogous XFEM computations were performed on the multicusp model with a starter crack placed at the apex of cusp 1

45 o

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60

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0

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o

30 o

600 900 Bite force, P (kN)

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Fig. 5 – (a) Crack trajectories for margin cracks in the multicusp model, for specified angle of starter crack location relative to axis through cusp 1. (b) Corresponding XFEM computations, plotting crack coordinate Y as a function of net base reaction force P for loading.

in the plane containing both the load point and the tooth centre axis. As with the margin cracks, a flat indenter was applied on either a single cusp (1), two opposite cusps simultaneously (1,2), or all four cusps simultaneously (1,2,3,4). In these cases, radial cracks extended down the side wall from the cusp, but also into the saddle at the tooth centre (Fig. 3b). Initially, the cracks extended in both directions simultaneously, but that in the saddle slowed down and arrested as load increased. Radial crack extension down the side wall from the base was measured as the projected vertical coordinate X measured downward from the cusp apex (Fig. 3b). Again, comparative computations were performed on the single dome model, for both off-axis and axial loading. Fig. 6a shows the radial crack development as coordinate X versus net force P. Pertinent results include: (i) the critical net force increases with number of cusps contacted, but not proportionally, as with margin cracks; (ii) the critical force for one-cusp loading in the multicusp model is close to that for off-axis loading in the dome model; (iii) the critical load for two-cusp loading in the multicusp model is similar to that for axial loading in the dome model; (iv) the critical loads for all cases are a little higher for radial cracks than for margin cracks (Fig. 4a), as previously reported (Barani et al., 2011). Sensitivity to the

journal of the mechanical behavior of biomedical materials 35 (2014) 85 –92

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Crack size, X (mm)

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C12

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Crack size, X (mm)

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1800

Fig. 6 – Radial crack coordinate X measured from top of tooth occlusal surface (cf. Fig. 2b) as a function of net reaction force over base. (a) XFEM computations for the multicusp model, for loading on single cusp (C1), two opposite cusps (C12), and all four cusps (C1234); and for the dome model loaded axially (DA) and off-axis (DO). (b) Laboratory data on cusp-loaded extracted molars from (Lee et al., 2011; Keown et al., 2012b).

manner in which the load is spread at the occlusal surface is once more evident. Experimental data from laboratory tests on extracted molars (Lee et al., 2011; Keown et al., 2012b) are shown in Fig. 6b. The same conclusions may be drawn as for margin cracks: (i) the experimental data span the XFEM curves for loading on one and two (opposite) cusps; (ii) the computed curves for off-axis and axial loading of the dome model provide approximate bounds to the experimental data (Lee et al., 2011). Fig. 7 shows radial crack trajectories (Fig. 7a) and crackcoordinate versus net force (Fig. 7b). Starter cracks were placed immediately below cusp 1, but at prescribed angles relative to the cusp axis. The crack trajectories are similar to those for margin cracks in Fig. 5a, albeit for propagation in an opposite direction. Similarly, the crack-coordinate versus net force plots are quite insensitive to the starter orientation.

4.

Discussion

Our study is built on a model of human molars as hemispherical, symmetric, four-fold cuspal structures (Fig. 2). This idealized morphology simplifies the computations, while at the same time enabling us to draw some strong, far-reaching conclusions

45

30 o

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o

o

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600 900 Bite force, P (kN)

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Fig. 7 – (a) Crack trajectories for radial cracks in the multicusp model, for different orientations of starter crack relative to axis through cusp 1. (b) Corresponding XFEM computations, plotting crack coordinate X as a function of net base reaction force P for loading.

concerning the mechanics of low-crowned (‘bunodont’) tooth failure. Many mammals beyond the primates, such as pigs and otters, have a similar, tooth geometry (Ungar, 2010). As indicated earlier, cusp multiplicity in mammals can vary from our simple quadrate representation (Yamashita, 1998; Kono et al., 2002; Smith et al., 2005; Benazzi et al., 2013; Berthaume et al., 2013; Frunza and Suciu, 2013), from five to three in molars down to two in premolars. Individual cusps tend to be unequal in size, irregularly spaced, and elongate with deep valleys, and the tooth base may be ringed with a ridge or ‘cingulum’ (Anderson et al., 2011). The enamel thickness can vary around the tooth walls (Kono, 2004; Smith et al., 2005). Wear can flatten the cusps, with some influence on the propagation of longitudinal cracks depending on the nature of the occlusal contact (Keown et al., 2012a). All these geometrical factors can be handled by suitable elaboration of the XFEM mesh. However, while such factors can be expected to have some effect on local stress fields and hence on the initiation of longitudinal cracks, their influence on subsequent propagation to failure in the far field is likely to be much less pronounced (Whitton et al., 2008; Barani et al., 2011, 2012b). Dealing with more complex cusp geometries may provide useful insight into detailed biological function (Lucas, 2004; Berthaume et al., 2010), but at some expense of generality. The XFEM computations in this study demonstrate the capacity of multicusp geometries in human molar teeth to

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sustain high bite forces. As demonstrated in previous analyses of more simplistic single-dome tooth structures, cracks initiated at the cuspal surface or the margins grow stably with increasing load—fractures may start easily, but are constrained by a damage-tolerant tooth structure (Maas and Dumont, 1999; Lucas et al., 2008; Chai et al., 2009a; Lee et al., 2011). The results in Figs. 4 and 6 show a pronounced increase in net critical force to drive longitudinal radial or margin cracks fully along the enamel side wall as the number of occlusal contacts is increased. However, the scaling is not proportional to the number of loaded cusps, again consistent with the notion that adjacent and remote contacts have a diminishing influence on the fracture net force. These results confirm that multicusp geometries can be effective in spreading bite forces. Mention has been made of the broad scatter between computed crack growth curves in Figs. 4 and 6. It was noted that the computed curves for one- and two-cusp loading provide approximate bounds to the experimental data, whose own scatter is attributable to variations in actual tooth dimensions. This correspondence between calculation and experiment serves to validate the use of previously determined fracture scaling equations for predicting critical bite forces (Barani et al., 2011, 2012b; Lee et al., 2011), at least in relative terms. Comparison of the results for multicusp structures with those for simple single-dome structures provides useful clues as to the important variables responsible for the data shifts shown in Figs. 4 and 6. The curves for one-cusp loading in the multicusp model (C1) are close to those for off-axis loading in the single-dome model (DO), while those for two-cusp loading in the multicusp model (C12) are close to those for axial loading in the dome model (DA). This suggests that location of the load point is the primary factor in the curve shifts for one- and twocusp loading. The effect amounts to a factor of about two, similar to the magnitude observed experimentally in analogous tests in off-axis loading of domed glass shells (Qasim et al., 2006b). Closer inspection of the curves in Figs. 4 and 6 shows slightly higher values of critical load for the one-cusp loaded model relative to the off-axis loaded dome, consistent with an elongation in effective height of the load point shown in Fig. 2 (Barani et al., 2012b). The shift in curves to higher critical loads for four-cusp loading (C1234) relative to the axially-loaded dome (DA) simply reflects a redistribution of occlusal load from the immediate vicinity of the radial or margin cracks (Fig. 3) to the more distant cusps, thereby diminishing the stress intensity on the side wall containing the crack. Another factor in the data shifts shown in Figs. 4 and 6 is the radius of individual cusps. This is evident as apparent differences in the shapes of the crack growth curves relative to those for single domes, especially for radial cracks as shown in Fig. 6a. The results for the multicusp model show a relatively early initial extension, consistent with previous studies demonstrating a scaling relation between applied load and net tooth radius (Barani et al., 2011, 2012b). As the crack extends along the enamel wall, however, the fracture becomes more and more dominated by the overall tooth radius, and thus tends toward the critical load for the singledome configuration. However, these shifts are small compared with those brought about by load offsets, and so may be regarded as of secondary importance to the critical conditions.

The influence of starter crack location or orientation relative to the load axis does not strongly affect the crack trajectories, simply redirecting the cracks along near-linear paths along the enamel wall either to (margin cracks) or from (radial cracks) the load point. The computed crack trajectories shown in Figs. 5 and 7a closely resemble actual trajectories seen in extracted molar teeth in laboratory tests (Lee et al., 2009, 2011; Keown et al., 2012b). Location does not seem to have a strong influence on the critical loads. On the other hand, some ancillary computations for starter cracks located in the valley between two adjacent cusps loaded simultaneously with a flat plate (e.g. cusps 1 and 3 shown in Fig. 2) do show different behaviours. It could have been thought that such valleys would be prime locations for fissures to begin. But these cracks did not extend at all, even at high loads. Examination of the stress field between the cusps revealed a region of strong compression inhibiting crack extension out of the valley. Some further computations with wedge loading between the cusps did eliminate the compression zone, thereby allowing the fracture to proceed down the side walls and even into the dentin. Such loading is more pertinent to the issue of tooth splitting (Chai et al., 2011), a topic which demands sophisticated modifications to the existing XFEM models and which we intend to pursue in a future study. There are other factors pertaining to the susceptibility of teeth to fracture that have not been specifically addressed in the present work, among them the assumption of a homogeneous, isotropic enamel structure. In reality, the mechanical properties of tooth enamel, notably elastic modulus and fracture toughness, experience gradients between the outer and inner surfaces (Cuy et al., 2002; Bajaj and Arola, 2009a). Enamel anisotropy and prism decussation can also influence the fracture behaviour, by directing the cracks in the longitudinal direction as well as enhancing the enamel toughness (Rees and Jacobson, 1995; Bajaj and Arola, 2009b). Incorporation of such factors into the XFEM algorithm has been shown to influence the shape of crack growth curves such as those shown in Figs. 4 and 6b, but not so much the ultimate critical fracture forces (Barani et al., 2012a).

5.

Conclusions

(i) XFEM fracture predictions of longitudinal crack evolution in the enamel of molar teeth with cusps are consistent with experimental data from earlier studies. (ii) Loading on a single cusp in a multicusp molar tooth reduces the critical load for longitudinal fracture relative to axial loading of a single dome model, and thereby diminishes the capacity of the structure to sustain high bite forces. (iii) The diminution in load capacity in single-cusp loading is predominantly attributable to a shift in location of the applied force toward the enamel side wall on which the crack propagates. Details of the local cusp geometry (height and radius) have a secondary effect on the critical load. (iv) Simultaneous loading on two opposite cusps results in critical loads comparable with those for an axially loaded single dome.

journal of the mechanical behavior of biomedical materials 35 (2014) 85 –92

(v) Loading on four cusps substantially increases the critical load to fracture, indicating a capacity for the cusps to spread the load in normal chewing. (vi) The location/orientation of a starting flaw on the enamel margin or on a cusp does not significantly affect the final critical load.

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Acknowledgements This work was funded by the Australian Research Council (DP130101472). We thank Paul Constantino and Paul Abbott for helpful discussions on the role of molar cusps in tooth function, and Paul Abbott for supplying the sample tooth in Fig. 1.

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Role of multiple cusps in tooth fracture.

The role of multiple cusps in the biomechanics of human molar tooth fracture is analysed. A model with four cusps at the bite surface replaces the sin...
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