mechanical dentin review

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DOI: 10.1177/1544111303014001032003 14: 13CROBM

J.H. Kinney, S.J. Marshall and G.W. Marshalle Mechanical Properties of Human Dentin: a Critical Review and Re-evaluation of the Dental Liter

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14(1):13-29 (2003) Crit Rev Oral Biol Med 13

Introduction

Dentin is the most abundant mineralized tissue in thehuman tooth. Therefore, knowledge of its mechanicalproperties is essential for predicting the effects of microstruc-tural alterations due to caries, sclerosis, and aging on toothstrength. In clinical dentistry, knowledge of dentin propertiesis important for understanding the effects of the wide varietyof restorative dental procedures that range from the design ofpreparations to choice of bonding methods.

In spite of this importance, over half a century of researchhas failed to provide consistent values of dentin's mechanicalproperties. The Young's modulus is unknown to within a factorof three; the shear modulus is uncertain by a factor of four; andthe other elastic constants have not been measured. There is asix-fold uncertainty in the ultimate strength, and there have

been few studies of fracture toughness or fatigue.This is a review of the last 50+ years of research into the

mechanical properties of dentin. It is a critical review, in thesense that the prior literature on the mechanical properties ofdentin is often re-examined and re-interpreted within the con-text of dentin microstructure. For example, the widely varyingtensile strength data are re-examined in terms of the Weibulldistribution function; the wide variations in previously report-ed tensile strengths can then be explained by a specimen sizeeffect whose origin lies in the existence of a population of flawsin the dentin.

The bulk of this review focuses on the elastic properties of

dentin. This emphasis is a necessary consequence of both thepaucity of published data on other mechanical properties, andthe fact that an understanding of elastic behavior is essential tothe proper interpretation of physical measurements of failureIt was often necessary to re-evaluate the data on the elasticproperties to reconcile the contradictions in the literature andreach consensus. Because the physical data were frequently notavailable in the more recent publications, indirect methodswere used to test the meaningfulness of the results. Thisinvolved consideration of the entire elastic stiffness matrix tocheck the constraints forced upon all of the elastic constantsfrom the measurement of a single property. In this way, it waspossible to check for self-consistency with all of the elastic con-stants as well as the known microstructure.

After a brief discussion of the composition and microstruc-ture of dentin, the review first considers its elastic and vis-coelastic properties. Then, hardness, strength, fracture tough-ness, and fatigue are discussed in order. This organization loose-ly follows that of the last comprehensive review of dentin prop-erties (Waters, 1980), with the exception that a more thoroughtreatment of fatigue and fracture toughness has been addedThe review concludes by proposing reliable ranges for the mag-nitudes of mechanical properties. It is hoped that these recom-mendations, and the evidence on which they are based, sparkadditional discourse and research on the properties of dentin.

THE MECHANICAL PROPERTIES OF HUMAN DENTIN:

A CRITICAL REVIEW AND RE-EVALUATION

OF THE DENTAL LITERATURE

J.H. Kinney*S.J. MarshallG.W. Marshall

Division of Biomaterials and Bioengineering, Department of Preventive and Restorative Dental Sciences, Mail Stop 0758, University of California, San Francisco, San Francisco, CA 94143-0758*corresponding author, [email protected]

ABSTRACT: The past 50 years of research on the mechanical properties of human dentin are reviewed. Since the body of workin this field is highly inconsistent, it was often necessary to re-analyze prior studies, when possible, and to re-assess them with-in the framework of composite mechanics and dentin structure. A critical re-evaluation of the literature indicates that the mag-nitudes of the elastic constants of dentin must be revised considerably upward. The Young's and shear moduli lie between 20-25GPa and 7-10 GPa, respectively. Viscoelastic behavior (time-dependent stress relaxation) measurably reduces these values atstrain rates of physiological relevance; the reduced modulus (infinite relaxation time) is about 12 GPa. Furthermore, it appearsas if the elastic properties are anisotropic (not the same in all directions); sonic methods detect hexagonal anisotropy, althoughits magnitude appears to be small. Strength data are re-interpreted within the framework of the Weibull distribution function

The large coefficients of variation cited in all strength studies can then be understood in terms of a distribution of flaws withinthe dentin specimens. The apparent size-effect in the tensile and shear strength data has its origins in this flaw distribution, andcan be quantified by the Weibull analysis. Finally, the relatively few fracture mechanics and fatigue studies are discussed. Dentinhas a fatigue limit. For stresses smaller than the normal stresses of mastication, ~ 30 MPa, a flaw-free dentin specimen apparently will not fail. However, a more conservative approach based on fatigue crack growth rates indicates that if there is a pre-exist-ing flaw of sufficient size (~ 0.3-1.0 mm), it can grow to catastrophic proportion with cyclic loading at stresses below 30 MPa.

Key words. Dentin, calcified tissues, mechanical properties, fatigue, fracture toughness.


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Dentin MicrostructureThe microstructure of dentin suggests a hierar-chical approach to the understanding of itsmechanical properties. At the smallest lengthscales are the constituent materials: a carbonatednanocrystalline apatite mineral phase (approxi-mately 50% by volume) and a felt-work of type Icollagen fibrils (see Fig. 1). The collagen fibrilsapproximately 30% by volume, are roughly 50-

100 nm in diameter; they are randomly orientedin a plane perpendicular to the direction of dentinformation (Jones and Boyde, 1984). The mineraloccupies two sites within this collagen scaffoldintrafibrillar (inside the periodically spaced gapzones in the collagen fibril) and extrafibrillar (inthe interstices between the fibrils). The partition-ing between these two sites is uncertain, althoughit is believed that between 70 and 75% of the min-eral may be extrafibrillar (Bonar et al., 1985Pidaparti et al., 1996). The mineral crystallites areneedle-like near the pulp; the shape continuouslyprogresses to plate-like with proximity to theenamel (Kinney et al., 2001b). The crystallite thick-

ness, ~ 5 nm, is invariant with location.At a higher level of organization is an inter-

mediate, or composite, length scale. At thislength scale, dentin can be modeled as a contin-

uous fiber-reinforced composite, with the intertubular dentinforming the matrix and the tubule lumens with their associatedcuffs of peritubular dentin forming the cylindrical fiber rein-forcement (see Fig. 2). The tubules run continuously from thedentin-enamel junction to the pulp in coronal dentin, and fromthe cementum-dentin junction to the pulp canal in the root. Theregular, almost uni-axial, alignment of the tubules has led someto suggest that they play an important function in the orienta-tion dependence of the mechanical properties (Waters, 1980).

At the greatest length scale are the effective, or continuum

properties of dentin. The effective properties of dentin describe theresponse of the tooth to applied loads, and allow for predictions oftooth strength and fracture properties. Young's modulus, tensileand compressive strength, and fracture toughness are examples ofthese effective properties, and reflect the complex interactions ofthe constituent materials and the microstructure. At this largestlength scale, we anticipate that the effective properties willdepend on tubule density, orientation, and the local average den-sity of the mineral phase. Ultimately, measurement of the effectiveproperties, particularly the properties of altered forms of dentin(such as carious or transparent), will benefit from an understand-ing of the material behavior at all of these length scales.

The Elastic Properties of DentinThe elastic properties of dentin are of paramount importance inall discussions of tooth strength. The elastic constants, usuallydefined in terms of a stiffness (Cij) or compliance (Sij) matrixinclude the Young's modulus, shear modulus, and Poisson'sratio. Depending on the symmetry of the microstructure, theelastic constants have different degrees of independence. Forexample, an isotropic structure has only two independent elas-tic constants, while an orthotropic structure has nine indepen-dent elastic constants. Therefore, any study of the elastic prop-erties of dentin must consider its symmetry.

14 Crit Rev Oral Biol Med 14(1):13-29 (2003

Figure 1. (a) SEM image of a fixed, demineralized dentin specimen showing the collagen fib-rils that are randomly oriented in the plane perpendicular to the tubule lumens (after Marshall

et al., 1997). (b)A higher-resolution AFM image of an unfixed specimen obtained in water.The AFM image shows the periodic 67-nm hole and overlap zones characteristic of the TypeI collagen fibrils found in dentin and bone (after Marshall et al., 2001; Habelitz et al., 2002b).

Figure 2. (a) SEM and (b) AFM images of fully mineralized dentinspecimens showing the tubule lumens with surrounding cuffs of peri-tubular dentin. The bar in the SEM photomicrograph is 10 m.Labeling: L = tubule lumen, PT = peritubular dentin, IT = intertubulardentin matrix. SEM image after Kinney et al. (2001a). AFM imageafter Kinney et al. (1993).




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THE YOUNGS MODULUS

The slope of the proportional part of the stress-strain curve pro-vides the Young's modulus, while the yield strength and the ulti-mate strength can be obtained from the nonlinear region of thestress-strain curve. Because uni-axial stress-strain behavior isamong the most straightforward of measurements, it is surprisingthat there is so much uncertainty in the value of Young's modulusfor dentin obtained by this method. This uncertainty in Young'smodulus extends to all measurement techniques, including bend-

ing, indentation, and ultrasound. A graphed representation of themeasurements of Young's modulus with the year in which theywere reported is shown in Fig. 3. The mean and standard devia-tion of these values are 13.2 GPa and 4.0 GPa, respectively.Unfortunately, over the past 50 years there has been an increase inthe dispersion of the reported values; there is no evidence that thistrend is abating. Therefore, it is appropriate to begin discussion ofthe Young's modulus of dentin by considering the constituent andcomposite-level organizational hierarchies in the hope that theymay aid readers in discriminating between valid and invaliddeterminations of the effective modulus.

Constituent materials models

An early attempt to provide reasonable bounds for the contin-uum Young's modulus of dentin was proposed by Katz (1971).Dentin was considered to be a two-phase mixture of apatitemineral and collagen, with the volume fraction and Young'smodulus of the mineral phase given by VA and EA, respective-ly, and the corresponding values for the collagen given by VCand EC. The effective Young's modulus of the mixture (Ee) wasrigorously bounded from above and below by:

Ee < VAEA + VCEC VA(EA - EC) + EC (1)

EA

ECEe >

_____________ (2)VAEC + VCEM

The elastic properties of the constituent materials in theirbulk form have been measured with reasonable accuracy: EA ~110 GPa and EC < 0.001 GPa (Katz and Ukraincik, 1971; Baloochet al., 1998). Substitution of these values into the above equa-tions generates the upper and lower bounds for the Young'smodulus that are shown in Fig. 4. Unfortunately, the large dis-parity between the moduli of the two phases leads to a wideseparation between the upper and lower bounds in the vicinityof the known mineral concentration in dentin (40-50% by vol-ume). The separation between the bounds is narrowed onlyslightly by the application of more restrictive, variational

bounding methods (Hashin, 1983). Clearly, bounds based on theconstituent materials properties are of no use for discriminating

between experimental measurements, all of which lie well with-in the upper and lower bounds predicted in Eqs. 1 and 2.

14(1):13-29 (2003) Crit Rev Oral Biol Med 15

Figure 3. The Young's modulus as reported in the literature over thepast 50 years. The data prior to 1999 appear in tabular form in thereference by Kinney et al. (1999). Data after 1999 are from Huo etal. (2000), Kinney et al. (2001a), Kishen et al. (2000), andPalamara et al. (2000). In the past few years, there has been a four-fold variation in the reported magnitude of the Young's modulus; theuncertainty in its value appears to be expanding with time. Theuncertainty in the magnitude of the Young's modulus is probably notreflective of the actual variations in the mechanical properties ofdentin. Rather, the uncertainty most likely arises from either a vis-coelastic response (stress relaxation) or experimental artifact.

Figure 4. The upper (dashed line) and lower (solid line) theoreticalbounds for the Young's modulus of a composite of hydroxyapatite min-eral and type I collagen fibrils as calculated from Eqs. 1 and 2. Theexperimental data are shown as the solid bar at a volume percent min-eral phase (45%) corresponding to the known composition of dentin. Thegraph emphasizes the difficulties of modeling mineralized tissues withbounds; The large difference in moduli between the two phases separatesthe upper and lower bounds by too great a magnitude to be useful.




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Micromechanics models

Because of the large difference in modulus between the miner-al and organic phases, we do not anticipate that improvementcan be made by further refinements of bounds. Therefore, it isworthwhile considering a micromechanics approach at thecomposite hierarchy of dentin microstructure. At this lengthscale, the dentinal tubules with their surrounding peritubularcuffs are like fibers, providing reinforcement to the intertubu-lar matrix. Assuming that the moduli of the matrix (EM) and

the "fibers" (the lumens plus peritubular cuffs, EF) are known,rough estimates of the bounds can be made by appropriatesubstitutions into Eqs. 1 and 2. However, direct substitution ofEM and EF ignores the possibly complex interactions betweenthe "fibers" and the matrix.

The interactions between fiber and matrix have beenaccounted for in self-consistent generalized micromechanicsapproaches to aligned fiber composites (Christensen, 1990).Analytical solutions have been derived for the case of alignedcylindrical fibers in an isotropic matrix. The fibers introduce atransverse anisotropy and require five equations for the inde-pendent elastic constants to be defined. These equations have

been reformulated for dentin in terms of the Young's modulusof intertubular (Ei) and peritubular (Ept) dentin (Kinney et al.,

1999). The expression for the effective longitudinal (in thedirection of the tubules) Young's modulus, Eel, in terms of thetubule concentration, c, was derived:

AlEel = cEpt(1 - ________ ) + (1 - c)El + (c,vl,vpt) (3)Apt + AlIn the above equation, is a "clamping" factor thataccounts for mismatch in Poisson's ratio between the peri-tubular and intertubular dentins, and Al and Apt are thearea fractions of the intertubular and peritubular dentin,respectively. The area ratio in Eq. 3 is apparently constantwith position in the tooth, and is approximately equal to

0.25 (Pashley, 1989).Only when the ratio of the peritubular and intertubular

dentin moduli is greater than 3 is there a measurable effect ofthe tubules on stiffness for physiologically relevant tubuledensities (5-15%). Thus, a theoretical framework now existsfor estimating the effects of tubule orientation on the elasticproperties: For physiologically relevant tubule densities, theanisotropy caused by the tubules is insignificant (Kinney etal., 1999). This does not mean that dentin is elasticallyisotropic; rather, if dentin proves to be anisotropic, it is

because the intertubular dentin matrix is anisotropic and notbecause of the tubules. Based on observations that the colla-gen fibrils in the intertubular matrix are aligned perpendicu-lar to the tubule axis, we anticipate that dentin might have atransverse isotropic (hexagonal) symmetry (stiffer in theplane of the collagen fibrils).

The micromechanics model offers several advantages overbounding methods. Key is the ability of the model to makeaccurate estimates of the effective elastic constants based on afew simple measurements of the tubule concentration. Themodel has its drawbacks, however. Accurate values of the elas-tic constants of peritubular and intertubular dentin must beavailable, and this requires careful mechanical testing, whichwe will now review in greater detail.

Tensile and compressive measurementsof Young's modulus

There have been numerous experimental attempts to measurethe Young's modulus of dentin. The majority of these measure-ments have been performed in either tension or compression.The primary emphasis of the tensile studies was to establish theultimate tensile strength (UTS) of dentin; the determination ofthe Young's modulus appeared to be secondary to this effort.

The earliest "modern" measurement of Young's modulus in

tension was published in 1962 (Bowen and Rodriguez, 1962). Inthat study, specimens of dentin, nominally 2 mm thick, werecut freehand with a rotary diamond disk under constant waterirrigation. The specimens were narrowed in the middle, with a"radiused" shoulder (fillet) maintained so that stress concentra-tion would be prevented. The knife-edges of an optical straingauge (length, 6 mm) were attached directly to the specimensThe specimens were hydrated at least 1 hr in distilled water

before being tested. The stress-strain curves were linear to fail-ure, and the mean modulus of elasticity was 19.3 GPa (2.8 x 106

psi) with a coefficient of variation of 28%.Five years later, Lehman (1967) measured the tensile prop-

erties of dentin with hollowed, cylindrical specimens from theroot. As in the earlier study by Bowen and Rodriguez, the ten-

sile stress-strain curves were linear to failure. However, theYoung's modulus was 11.0 GPa, almost half that of the earlierstudy, and the coefficient of variation (53%) was almost double

There were three important differences between the twostudies. First, the latter study did not use strain gauges affixedto the specimen, thereby increasing the likelihood of gripeffects like tow-in. Second, the specimens were hollowed alongthe root canal by means of a dental bur, increasing the proba-

bility of undetected flaws in the interior of the specimen. Thirdthe gauge sections were not "radiused", leaving a stress con-centration that increased the chance of failure at grip ends,thereby invalidating the test.

Though it is difficult to reconstruct an experiment decadesafter the fact, we are fortunate that Lehman reported all of hisdata. A close inspection shows that the modulus and ultimatetensile strength were strongly correlated (p < 0.001), with thehigher values of elastic modulus associated with the specimensthat had the highest tensile strength. Furthermore, 75% of thespecimens failed in tension at or below 40 MPa, a low valuesuggesting that flaws introduced during specimen preparationmay have affected the linear stress-strain behavior. When werestrict our focus to those specimens that failed above 40 MPawe obtain a tensile modulus of 16.9 GPa with a coefficient ofvariation of 26%. This is more in line with the earlier work ofBowen and Rodriguez (1962).

Around the same time that the tensile measurements werebeing performed, other researchers were establishing the compressive properties of dentin. The earliest of these measure-ments was reported by Peyton et al. (1952). This study exam-ined the compressive behavior of 1.8 mm in cross-section by4.5-mm-long dentin specimens. Strain gauges were affixed tosteel rods that were then used to apply load to the specimensA Young's modulus of 11.6 GPa was reported.

Concerned that the low value of the Young's modulusobtained from their earlier study might have been caused by acombination of non-parallel alignment of the load platens(tow-in) and possible stress relaxation effects, the same grouprepeated their earlier study with greater attention to experi-

16 Crit Rev Oral Biol Med 14(1):13-29 (2003




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mental variables (Craig and Peyton, 1958). Placing the straingauges directly on the specimen eliminated tow-in; stress relax-ation effects were measured by careful cycling of the compres-sive load well below the proportional limit. These correctionsraised the compressive Young's modulus to 18.5 GPa, in excel-lent agreement with the tensile measurements of the time.

Measurements of the modulus are sensitive to specimenpreparation, experimental design, and stress relaxation; notaccounting for these experimental variables leads to underesti-

mation of the elastic constants. Therefore, the comments ofWaters (1980) in his review article were surprising: "Craig andPeyton (1958) obtained values for the proportional limit andcompressive strength in reasonable agreement with otherworkers, but their value of the modulus is, for some unac-countable reason, considerably higher." This comment reflectsa bias toward favoring a low value for the Young's modulus ofdentina bias perhaps enforced by the large number of lowvalues reported in subsequent years (see Fig. 3).

Testing of dentin in compression and tension was per-formed only infrequently in the years since Craig and Peyton.Two of these studies are particularly noteworthy. In the workreported by Stanford et al. (1960), dentin specimens were testedin compression. Even with corrections for platen deformation,

the compressive modulus was 13.8 GPa, significantly less thanreported in the earlier studies. Though it could be argued thatthe data were not adjusted for the affects of non-parallel align-ment (tow-in), it is unlikely that this could account for all of thediscrepancy. Furthermore, in tensile testing performed by Sanoet al. (1994), similarly low values of Young's modulus werereported (13-15 GPa).

Both of the above studies were performed with great atten-tion to detail, so instrumentation artifacts or flaws in specimenpreparation cannot readily explain the lower values of elasticmodulus. However, one detail worth mentioning is that, in

both studies, the specimens were stored for an undisclosedamount of time in water. In the work reported by Sano et al.(1994), the tensile specimens were stored in 0.9% NaCl water at

4C for about 24 hrs prior to being tested (Pashley, 2001, per-sonal communication). Earlier studies have showed that short-term storage in saline solution degrades bond strengths(Goodis et al., 1993), and that storage in water can reduce bendstrength in bone (Gustafson et al., 1996). Given the small size ofdentin specimens, it is possible that even short-term storage inwater or saline might reduce the elastic modulus by dissolutionof the mineral phase. The effects of water storage will be con-sidered in greater detail later.

In more recent work, Palamara et al. (2000) recorded theresponse of a grid pattern coated on the surfaces of dentinspecimens during compressive loading. The results are of par-ticular interest because, while the Young's moduli measured inorientations both parallel and orthogonal to the tubule axes

(the principal structural directions) were identical, the modulusat 45 off-axis to the tubules was determined to be lower. Theobservation that the off-axis modulus was smaller than thatmeasured in either principal axis requires that the intertubulardentin matrix be anisotropic. Since this is a significant finding,we must consider its veracity in greater detail.

Our analysis begins by a consideration of the simplestdeviation from isotropic symmetry, cubic, which requires threeindependent elastic constants. The reciprocal Young's modulusin a cubic system along the direction of the unit vector li can beexpressed in terms of the compliance matrix Sij (Nye, 1972):

1S11 = S11 - 2(S11 - S12 - __ S44)(l12l22 + l22l32 + l32l12) (4)2

The variation of the Young's modulus with orientation dependsonly on the terms in l, and is zero for the principal orthogonaldirections and a maximum of 1/3 in the directions. For theYoung's modulus to be less in the off-axis directions, (S11-S12-S44/2

must be less than zero. This requires the shear modulus, G, to beless than would be expected in an isotropic system:

EG < _________ (5)

2(1 + v)

With the values provided by Palamara et al. (2000), [E11 =10.4 GPa and E(45) = 7.7 GPa], we can calculate a rigorousupper bound for the shear modulus. For a cubic symmetrymodel, G < 2.9 GPa. This is less than would be required fromisotropic symmetry for v = 0.25 (4.1 GPa).

One can argue that, from an analysis of the dentinmicrostructure, an orthotropic symmetry would be a more rea-sonable alternative to cubic. However, we find that the situation

is only slightly altered from the analysis for cubic symmetry. Fororthotropic symmetry, we begin with the general case of a lam-ina loaded in plane at an arbitrary angle q with respect to a prin-cipal axis, l1, in this case parallel to the tubule axes. The Young'smodulus as a function of q can be expressed as (Jones, 1975)

E1____ = (1 + a - 4b)cos4(q) + (4b - 2a)cos2(q) + a (6)Eq

In Eq. 6, a and b are dimensionless variables given by

E1a = ____E

2

1 E1b = __ (__ - 2v12) (7)4 G12Here, G12 is the shear modulus and v12 is Poisson's ratio. For thedata presented in Palamara et al. (2000), a = 1. The observationthat Eq was less than both E1 and E2 places a rigorous restrictionon the value of the shear modulus:

E1 E1G12

mechanical dentin review

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