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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, kinney3{at}llnl.gov
Abstract Introduction Dentin Microstructure The Elastic Properties of Dentin THE YOUNG’S MODULUS Constituent materials models Micromechanics models Tensile and compressive measurements of Young’s modulus Indentation measurements of Young’s modulus Sonic measurements of the elastic constants Viscoelastic behavior of dentin SUMMARY OF ELASTIC PROPERTIES Hardness of Dentin Ultimate Strength of Dentin Fracture Properties of Dentin Fatigue Properties of Dentin Conclusions REFERENCES
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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-existing 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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In spite of this importance, over half a century of research has failed to provide consistent values of dentin’s mechanical properties. The Young’s modulus is unknown to within a factor of three; the shear modulus is uncertain by a factor of four; and the other elastic constants have not been measured. There is a six-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 the sense that the prior literature on the mechanical properties of dentin is often re-examined and re-interpreted within the context of dentin microstructure. For example, the widely varying tensile strength data are re-examined in terms of the Weibull distribution function; the wide variations in previously reported tensile strengths can then be explained by a specimen size effect whose origin lies in the existence of a population of flaws in the dentin.
The bulk of this review focuses on the elastic properties of dentin. This emphasis is a necessary consequence of both the paucity of published data on other mechanical properties, and the fact that an understanding of elastic behavior is essential to the proper interpretation of physical measurements of failure. It was often necessary to re-evaluate the data on the elastic properties to reconcile the contradictions in the literature and reach consensus. Because the physical data were frequently not available in the more recent publications, indirect methods were used to test the meaningfulness of the results. This involved consideration of the entire elastic stiffness matrix to check the constraints forced upon all of the elastic constants from the measurement of a single property. In this way, it was possible to check for self-consistency with all of the elastic constants as well as the known microstructure.
After a brief discussion of the composition and microstructure of dentin, the review first considers its elastic and viscoelastic properties. Then, hardness, strength, fracture toughness, and fatigue are discussed in order. This organization loosely follows that of the last comprehensive review of dentin properties (Waters, 1980), with the exception that a more thorough treatment of fatigue and fracture toughness has been added. The review concludes by proposing reliable ranges for the magnitudes of mechanical properties. It is hoped that these recommendations, and the evidence on which they are based, spark additional discourse and research on the properties of dentin.
| Dentin Microstructure |
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). The collagen fibrils, approximately 30% by volume, are roughly 50-100 nm in diameter; they are randomly oriented in a plane perpendicular to the direction of dentin formation (Jones and Boyde, 1984). The mineral occupies two sites within this collagen scaffold: intrafibrillar (inside the periodically spaced gap zones in the collagen fibril) and extrafibrillar (in the interstices between the fibrils). The partitioning between these two sites is uncertain, although it is believed that between 70 and 75% of the mineral may be extrafibrillar (Bonar et al., 1985; Pidaparti et al., 1996). The mineral crystallites are needle-like near the pulp; the shape continuously progresses to plate-like with proximity to the enamel (Kinney et al., 2001b). The crystallite thickness, 5 nm, is invariant with location.
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). The tubules run continuously from the dentin-enamel junction to the pulp in coronal dentin, and from the cementum-dentin junction to the pulp canal in the root. The regular, almost uni-axial, alignment of the tubules has led some to suggest that they play an important function in the orientation dependence of the mechanical properties (Waters, 1980).
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| The Elastic Properties of Dentin |
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THE YOUNG’S MODULUS
The slope of the proportional part of the stress-strain curve provides the Young’s modulus, while the yield strength and the ultimate strength can be obtained from the nonlinear region of the stress-strain curve. Because uni-axial stress-strain behavior is among the most straightforward of measurements, it is surprising that there is so much uncertainty in the value of Young’s modulus for dentin obtained by this method. This uncertainty in Young’s modulus extends to all measurement techniques, including bending, indentation, and ultrasound. A graphed representation of the measurements of Young’s modulus with the year in which they were reported is shown in Fig. 3. The mean and standard deviation of these values are 13.2 GPa and 4.0 GPa, respectively. Unfortunately, over the past 50 years there has been an increase in the dispersion of the reported values; there is no evidence that this trend is abating. Therefore, it is appropriate to begin discussion of the Young’s modulus of dentin by considering the constituent and composite-level organizational hierarchies in the hope that they may aid readers in discriminating between valid and invalid determinations of the effective modulus.
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 | (1) |
 | (2) |
The elastic properties of the constituent materials in their bulk form have been measured with reasonable accuracy: EA 110 GPa and EC < 0.001 GPa (Katz and Ukraincik, 1971; Balooch et al., 1998). Substitution of these values into the above equations generates the upper and lower bounds for the Young’s modulus that are shown in Fig. 4. Unfortunately, the large disparity between the moduli of the two phases leads to a wide separation between the upper and lower bounds in the vicinity of the known mineral concentration in dentin (40-50% by volume). The separation between the bounds is narrowed only slightly by the application of more restrictive, variational bounding methods (Hashin, 1983). Clearly, bounds based on the constituent materials properties are of no use for discriminating between experimental measurements, all of which lie well within the upper and lower bounds predicted in Eqs. 1 and 2.
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The interactions between fiber and matrix have been accounted for in self-consistent generalized micromechanics approaches to aligned fiber composites (Christensen, 1990). Analytical solutions have been derived for the case of aligned cylindrical fibers in an isotropic matrix. The fibers introduce a transverse anisotropy and require five equations for the independent elastic constants to be defined. These equations have been reformulated for dentin in terms of the Young’s modulus of intertubular (Ei) and peritubular (Ept) dentin (Kinney et al., 1999). The expression for the effective longitudinal (in the direction of the tubules) Young’s modulus, Eel, in terms of the tubule concentration, c, was derived:
 | (3) |
In the above equation, 
is a "clamping" factor that accounts for mismatch in Poisson’s ratio between the peritubular and intertubular dentins, and Al and Apt are the area fractions of the intertubular and peritubular dentin, respectively. The area ratio in Eq. 3
is apparently constant with 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 of the tubules on stiffness for physiologically relevant tubule densities (5-15%). Thus, a theoretical framework now exists for estimating the effects of tubule orientation on the elastic properties: For physiologically relevant tubule densities, the anisotropy caused by the tubules is insignificant (Kinney et al., 1999). This does not mean that dentin is elastically isotropic; rather, if dentin proves to be anisotropic, it is because the intertubular dentin matrix is anisotropic and not because of the tubules. Based on observations that the collagen fibrils in the intertubular matrix are aligned perpendicular to the tubule axis, we anticipate that dentin might have a transverse isotropic (hexagonal) symmetry (stiffer in the plane of the collagen fibrils).
The micromechanics model offers several advantages over bounding methods. Key is the ability of the model to make accurate estimates of the effective elastic constants based on a few simple measurements of the tubule concentration. The model has its drawbacks, however. Accurate values of the elastic constants of peritubular and intertubular dentin must be available, and this requires careful mechanical testing, which we will now review in greater detail.
Tensile and compressive measurements of Young’s modulus
There have been numerous experimental attempts to measure the Young’s modulus of dentin. The majority of these measurements have been performed in either tension or compression. The primary emphasis of the tensile studies was to establish the ultimate tensile strength (UTS) of dentin; the determination of the 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). In that study, specimens of dentin, nominally 2 mm thick, were cut freehand with a rotary diamond disk under constant water irrigation. The specimens were narrowed in the middle, with a "radiused" shoulder (fillet) maintained so that stress concentration would be prevented. The knife-edges of an optical strain gauge (length, 6 mm) were attached directly to the specimens. The specimens were hydrated at least 1 hr in distilled water before being tested. The stress-strain curves were linear to failure, 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 properties of dentin with hollowed, cylindrical specimens from the root. As in the earlier study by Bowen and Rodriguez, the tensile stress-strain curves were linear to failure. However, the Young’s modulus was 11.0 GPa, almost half that of the earlier study, and the coefficient of variation (53%) was almost double.
There were three important differences between the two studies. First, the latter study did not use strain gauges affixed to the specimen, thereby increasing the likelihood of grip effects like tow-in. Second, the specimens were hollowed along the root canal by means of a dental bur, increasing the probability of undetected flaws in the interior of the specimen. Third, the gauge sections were not "radiused", leaving a stress concentration that increased the chance of failure at grip ends, thereby invalidating the test.
Though it is difficult to reconstruct an experiment decades after the fact, we are fortunate that Lehman reported all of his data. A close inspection shows that the modulus and ultimate tensile strength were strongly correlated (p < 0.001), with the higher values of elastic modulus associated with the specimens that had the highest tensile strength. Furthermore, 75% of the specimens failed in tension at or below 40 MPa, a low value suggesting that flaws introduced during specimen preparation may have affected the linear stress-strain behavior. When we restrict our focus to those specimens that failed above 40 MPa, we obtain a tensile modulus of 16.9 GPa with a coefficient of variation of 26%. This is more in line with the earlier work of Bowen and Rodriguez (1962).
Around the same time that the tensile measurements were being performed, other researchers were establishing the compressive properties of dentin. The earliest of these measurements was reported by Peyton et al. (1952). This study examined the compressive behavior of 1.8 mm in cross-section by 4.5-mm-long dentin specimens. Strain gauges were affixed to steel rods that were then used to apply load to the specimens. A Young’s modulus of 11.6 GPa was reported.
Concerned that the low value of the Young’s modulus obtained from their earlier study might have been caused by a combination of non-parallel alignment of the load platens (tow-in) and possible stress relaxation effects, the same group repeated their earlier study with greater attention to experimental variables (Craig and Peyton, 1958). Placing the strain gauges directly on the specimen eliminated tow-in; stress relaxation effects were measured by careful cycling of the compressive load well below the proportional limit. These corrections raised the compressive Young’s modulus to 18.5 GPa, in excellent agreement with the tensile measurements of the time.
Measurements of the modulus are sensitive to specimen preparation, experimental design, and stress relaxation; not accounting for these experimental variables leads to underestimation of the elastic constants. Therefore, the comments of Waters (1980) in his review article were surprising: "Craig and Peyton (1958) obtained values for the proportional limit and compressive strength in reasonable agreement with other workers, but their value of the modulus is, for some unaccountable reason, considerably higher." This comment reflects a bias toward favoring a low value for the Young’s modulus of dentin—a bias perhaps enforced by the large number of low values reported in subsequent years (see Fig. 3).
Testing of dentin in compression and tension was performed only infrequently in the years since Craig and Peyton. Two of these studies are particularly noteworthy. In the work reported by Stanford et al. (1960), dentin specimens were tested in compression. Even with corrections for platen deformation, the compressive modulus was 13.8 GPa, significantly less than reported in the earlier studies. Though it could be argued that the data were not adjusted for the affects of non-parallel alignment (tow-in), it is unlikely that this could account for all of the discrepancy. Furthermore, in tensile testing performed by Sano et al. (1994), similarly low values of Young’s modulus were reported (13-15 GPa).
Both of the above studies were performed with great attention to detail, so instrumentation artifacts or flaws in specimen preparation cannot readily explain the lower values of elastic modulus. However, one detail worth mentioning is that, in both studies, the specimens were stored for an undisclosed amount of time in water. In the work reported by Sano et al. (1994), the tensile specimens were stored in 0.9% NaCl water at 4°C for about 24 hrs prior to being tested (Pashley, 2001, personal 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 bend strength in bone (Gustafson et al., 1996). Given the small size of dentin specimens, it is possible that even short-term storage in water or saline might reduce the elastic modulus by dissolution of the mineral phase. The effects of water storage will be considered in greater detail later.
In more recent work, Palamara et al. (2000) recorded the response of a grid pattern coated on the surfaces of dentin specimens during compressive loading. The results are of particular interest because, while the Young’s moduli measured in orientations both parallel and orthogonal to the tubule axes (the principal structural directions) were identical, the modulus at 45° off-axis to the tubules was determined to be lower. The observation that the off-axis modulus was smaller than that measured in either principal axis requires that the intertubular dentin 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 simplest deviation from isotropic symmetry, cubic, which requires three independent elastic constants. The reciprocal Young’s modulus in a cubic system along the direction of the unit vector li can be expressed in terms of the compliance matrix Sij (Nye, 1972):
 | (4) |
The variation of the Young’s modulus with orientation depends only on the terms in l, and is zero for the principal orthogonal directions and a maximum of 1/3 in the <111> directions. For the Young’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 be less than would be expected in an isotropic system:
 | (5) |
With the values provided by Palamara et al. (2000), [E11 = 10.4 GPa and E(45°) = 7.7 GPa], we can calculate a rigorous upper bound for the shear modulus. For a cubic symmetry model, G < 2.9 GPa. This is less than would be required from isotropic symmetry for v = 0.25 (4.1 GPa).
One can argue that, from an analysis of the dentin microstructure, an orthotropic symmetry would be a more reasonable alternative to cubic. However, we find that the situation is only slightly altered from the analysis for cubic symmetry. For orthotropic symmetry, we begin with the general case of a lamina loaded in plane at an arbitrary angle q with respect to a principal axis, l1, in this case parallel to the tubule axes. The Young’s modulus as a function of q can be expressed as (Jones, 1975)
 | (6) |
In Eq. 6, a and b are dimensionless variables given by
 | (7) |
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Here, G12 is the shear modulus and v12 is Poisson’s ratio. For the data presented in Palamara et al. (2000), a = 1. The observation that Eq was less than both E1 and E2 places a rigorous restriction on the value of the shear modulus:
 | (8) |
The above restriction on the shear modulus is identical to the one derived for cubic symmetry. With the values reported in Palamara et al. (2000) for the off-axis Young’s modulus, Eq. 6 requires that the magnitude of the shear modulus be less than 2.5 GPa, very similar to the result obtained with a cubic symmetry model. This is an extremely low value for the shear modulus, and is not in agreement with results reported in those studies where the shear modulus has been measured independently. Sound velocity measurements in bovine dentin, to be discussed in a later section, provided a shear modulus of 8.0 GPa (Gilmore et al., 1969), and a torsion pendulum measurement led to a value of 6.1 GPa (Renson and Braden, 1975). [NB: The value that we cite corrects the typographical error in the original paper by Renson and Braden (1975), where the decimal was placed in the wrong location.] These values were all significantly greater than what is required to explain the results of Palamara et al.
Though the largest source of error in measurements with grids lies in the accurate determination of displacement (strain), the absence of any independent measure of displacement (e.g., no strain gauges or cross-head displacements were reported) prevents us from assessing this as a possible source of error. However, the compression tests were conducted between 50 and 150 MPa, which is within the range of the proportional limit in compression (80-140 MPa) that has been established in earlier work (Craig and Peyton, 1958; Stanford et al., 1960). It is therefore possible that Palamara et al. (2000) were operating beyond the yield stress when recording their grid displacements. This factor would explain the lower values of the Young’s modulus, and could easily explain the off-axis anisotropy. Also, considerable relaxation (creep) could have occurred during the time intervals required to obtain the images of the grids. However, in the absence of additional data, the possibility that dentin is elastically anisotropic cannot be ignored. The question of elastic symmetry will be taken up again in a later section.
Indentation measurements of Young’s modulus
Indentation, where a hardened stylus is brought into contact with a surface, has largely been used to measure hardness. The modern theory of indentation (Doerner and Nix, 1986; Oliver and Pharr, 1992), where force and stylus displacement are analyzed to measure Young’s modulus, has been applied to mineralized tissues only in the last decade. The technique measures the indenter load as a function of depth of penetration, from which the contact stiffness, Sc (not to be confused with the compliance matrix), is obtained from the derivative of the unloading curve evaluated at the peak force (Fig. 5). Care must be taken to remove any excess machine compliance, Cm, that is associated with the sample mounting (Kinney et al., 1996):
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 | (9) |
The indentation modulus E*, sometimes referred to as the reduced modulus, is determined from the corrected contact stiffness, Sc, and the contact areas for each indentation, A, that are obtained from a tip shape calibration procedure (Doerner and Nix, 1986):
 | (10) |
From Eqs. 9 and 10, it is possible to obtain the Young’s modulus of the probed specimen, Es:
 | (11) |
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In Eq. 11, vs and vi are the Poisson’s ratios of the specimen and indenter stylus, and Ei is the Young’s modulus of the indenter. The approximation in Eq. 11 is valid because the modulus of the indenter is considerably larger than the modulus of the dentin.
Among the first to apply nanoindentation to the study of dentin and dental materials was van Meerbeek et al. (1993), who measured a Young’s modulus of dentin of 19.3 GPa. This was followed by Kinney et al. (1996), who used nanoindentation to measure the Young’s modulus of intertubular (18-21 GPa) and peritubular (29.8 GPa). Since then, nanoindentation has become a common technique for the determination of local mechanical properties of structural features in biological hard tissues (Rho et al., 1999).
A significant drawback to the widespread use of nanoindentation in mineralized tissues is the inability to indent in water. To overcome this drawback, Balooch et al. (1998) applied the atomic force microscope (AFM) with a specially designed attachment called the Triboscope to perform indentations on fully hydrated specimens. In addition to allowing indentations to be made in water, the device also functioned as an AFM, allowing for precise positioning of the indenter and subsequent imaging of the indentation with microscopic spatial resolution. With this technique, Kinney et al. (1999) measured a pronounced decline in E with submersion in water. These investigators attributed this decrease to a combination of softening of the collagen phase and partial surface demineralization. Without knowing the effects of surface demineralization, they assumed that a plasticizing of the collagen fibrils and other noncollagenous proteins in the tissue caused the majority of the decrease in modulus with time in water.
Although nanoindentation probes only a thin surface layer (< 1 µm), the mechanical properties obtained are assumed to be representative of the bulk material. Chemical changes in the surface layer of mineralized tissues resulting from storage solutions are, thus, important considerations for accurate determination of nanomechanical properties. This aspect of nanoindentation has been recently explored by Habelitz et al. (2002a), who studied changes in nanomechanical properties of dentin and enamel during storage in de-ionized water, calcium-chloride-buffered saline solution, and Hanks’ Balanced Salt Solution (HBSS). The investigators were able to show that storing teeth in de-ionized water or CaCl2 solution resulted in a large decrease in elastic modulus and hardness. After one day of storage, a decrease in the Young’s modulus and hardness of up to 30% was observed. By one week of solution storage, mechanical properties dropped to below half of their initial values. In contrast, storing the specimens in HBSS did not significantly alter the mechanical properties for a time interval of two weeks. The behavior of the Young’s modulus of specimens stored in water and HBSS is reproduced in Fig. 6.
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By eliminating near-surface demineralization through proper specimen storage procedures, one can now obtain consistent values of the Young’s modulus of hydrated peritubular and intertubular dentin. This is an important development, since it will allow for the study of the differences between wet and dry tissues, and provide elastic moduli more representative of in vivo conditions.
The nanoindentation method is limited in that it cannot be used to determine any of the other elastic constants, and it does not provide a direct measure of the continuum Young’s modulus, since micromechanics arguments must be used to combine the separate measures of the inter- and peritubular dentin. Other techniques will now be described that have the potential to determine all of the second-order elastic constants of dentin completely.
Sonic measurements of the elastic constants
Measurement of sound speed is among the most accurate ways of determining the elastic constants of a material. For an isotropic material, the shear (G), bulk (K), and Young’s (E) moduli are related to the longitudinal (Vl) and shear (Vs) wave velocities and the specimen density, 
, by (Love, 1960):
 | (12) |
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Using the technique of ultrasonic interferometry, Gilmore et al. (1969) established the sound speeds in bovine dentin. With these data, and assuming isotropic elasticity, the investigators used Eqs. 12 to derive the second-order elastic constants. To facilitate comparisons with other measurements, we have reproduced the entire range in values of the elastic moduli from this work (Table 1). The moduli, which are significantly higher than many more recent mechanical measurements, are in good agreement with nanoindentation (Habelitz et al., 2002a).
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The main difficulty with this method lies in determining the principal symmetry directions in the tooth from only two orientations. Lees and Rollins assumed hexagonal (transverse isotropic) symmetry based on the microstructure of dentin. However, since the elastic properties were measured in only a single plane, the solution to the stiffness tensor for hexagonal symmetry was underdetermined. Therefore, the investigators assumed that the lower-sound speed values reported earlier by Gilmore et al. were obtained from measurements 45° with respect to the tubule axes. With this assumption, the complete elastic stiffness matrix was estimated (Table 2).
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The Young’s modulus, which can be derived from the stiffness matrix of Lees and Rollins (1972), is graphed in Fig. 7 as a function of tubule orientation. The transverse anisotropy in the critical reflectance data lies in contrast to what has been observed by either nanoindentation or compression testing along orthogonal axes. Also, the magnitude of the Young’s modulus is significantly higher than that measured by Gilmore et al. Finally, the conclusion that the Young’s modulus is higher in the direction of the tubules is at odds with both the micromechanics model, which dictates that the contribution to the elastic stiffness from the peritubular dentin should be negligible, and microstructural observations that the mineralized collagen fibrils lie perpendicular to the tubules.
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Clearly, what is needed is an ultrasonic method suited to small specimens, and which allows for the determination of all of the elastic constants from a single orientation of the specimen and without assumptions as to the symmetry. Indeed, these requirements are met by a technique known as resonant ultrasound spectroscopy.
Resonant ultrasound spectroscopy (RUS) is a method for measuring single-crystal elastic constants with great precision (Migliori et al., 1993). RUS makes use of Hooke’s law and Newton’s second law to predict the resonant modes of vibration of a specimen of known shape (Migliori et al., 1993; Maynard, 1996). The entire stiffness tensor, Cij, can be determined by comparison of the frequency spectrum produced by the resulting eigenvalue problem with the measured resonant frequencies (vibrational eigenmodes) of the specimen. It is important to note a significant difference between RUS and other sonic methods. In RUS, sound speed is not measured. Rather, the resonant frequencies of mechanical vibration are determined. This means that the stiffness tensor can be determined for small specimens from one measurement at a single orientation.
Since the development of RUS, the applications of this technique have been extended to include geological structures and complex particulate and fiber-reinforced composites (Jung et al., 1999). Recently, this technique has been applied to small specimens of human coronal dentin, with interesting results (Kinney et al., 2002). Cubes of dentin, approximately 2 mm on an edge, were mounted on opposing corners between two transducers in the RUS system. The resonant frequencies between 0.5 and 1.4 MHz were measured. The values of the resonant frequencies were calculated from an initial approximation of the stiffness tensor. The experimental and predicted frequency distributions were then compared, and the residuals, F, were calculated. The elastic constants were adjusted from their initial values to minimize these residuals. Isotropic, cubic, and hexagonal material symmetry groups were modeled.
The elastic constants of hydrated and dry dentin, as measured by RUS, are listed in Table 1 for the isotropic, cubic, and hexagonal symmetry models. The best fit for the RUS data was with a hexagonal symmetry model, although deviations from isotropic symmetry were small. The angular deviation of the Young’s modulus with respect to the axis of the dentinal tubules is graphed in Fig. 7. The Young’s modulus was minimum in the direction of the tubules, and increased monotonically to maximum in the plane of the mineralized collagen fibrils. This should be compared with the critical angle results in the same Fig.
The measured anisotropy of the elastic constants of dentin can now be reconciled with its structural anisotropy. As in bone, the symmetry of the elastic constants is determined by the orientation of the collagen fibrils. However, the magnitude of the anisotropy (10%) is not large. This may explain why previous studies have failed to detect anisotropic behavior in either the contact stiffness (Kinney et al., 1999) or microhardness (Wang and Weiner, 1998a), since indentation techniques are known to be less sensitive to anisotropy (Vlassak and Nix, 1994). Furthermore, because the tubule orientation can vary widely across a specimen, a small anisotropy would most likely go undetected in a mechanical test of a larger specimen.
Viscoelastic behavior of dentin
Thus far, we have assumed that dentin is perfectly elastic at small strains, and that the elastic constants do not depend on the strain rate. In other words, we have assumed that small deformations resulting from an imposed stress remain constant with time. In most biological tissues, this is not the case. At a constant stress, these materials continue to deform with time (creep). Therefore, if a constant strain is to be maintained, the applied stress must be continuously reduced (stress relaxation). Materials that exhibit a time-dependent response are called viscoelastic. If the time-dependence of the relaxation does not depend on the magnitude of the applied stress, the material exhibits linear viscoelasticity. On the other hand, if the time response changes with the applied stress, the material exhibits nonlinear viscoelasticity.
In a viscoelastic solid, the stress is out of phase with the strain. The amount of this phase shift is defined by a phase angle, 
, between the applied stress and the resulting strain. A detailed analysis defines the loss tangent of the phase angle in terms of a storage modulus, E', and a loss modulus, E'':
 | (13) |
In a perfectly elastic solid, there would be no energy lost to creep deformation (E'' = 0), and the stress and strain would be in phase ( = 0). With increasing viscoelastic behavior, the phase angle would increase due to an increase in E'' and the corresponding decrease in E'. At extremely high strain rates (characteristic of ultrasound), or when stresses are too low to activate creep (typical of RUS), viscoelastic behavior should be less apparent. Therefore, the observation that the elastic constants determined by both RUS and sound velocity measurements are of similar magnitude, and that they are also higher than those determined by other methods, suggests that viscoelastic effects might contribute to the lower modulus values measured with mechanical testing. Indeed, Craig and Peyton (1958) measured a significant contribution from "stress-relaxation" during their compression loading studies, and nanoindentation measurements display continuous creep during the static load segment prior to unloading (e.g., Kinney et al., 1996; and also Fig. 5 in this review).
The classic analysis of viscoelasticity is based on the Maxwell element, a spring and dashpot arranged in series. The time-dependent relaxation modulus—for example, E(t)—can be written for a single element, i, as:
 | (14) |
where Ei represents the stiffness of the spring element i, and 
i is the relaxation time of the element. A parallel arrangement of a group of Maxwell elements can be treated as a simple sum:
 | (15) |
For an infinite number of Maxwell elements, the relaxation modulus can be written as a function of the continuous relaxation spectrum, H(), and the reduced modulus at infinite relaxation time, Eo (Ferry, 1970):
 | (16) |
Similarly:
 | (17) |
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A few studies have explored stress relaxation in dentin (Korostoff et al., 1975; Tengrove et al., 1995). Still, little is known about its time-dependent behavior. In what appears to have been the most detailed study of viscoelasticity in dentin, Korostoff et al. (1975) measured stress relaxation in specimens prepared from the roots of human incisors, cuspids, and premolars. The cylindrical specimens were machined to dimensions approximately 6.4 mm tall, and had a 3.8-mm outer and a 1.1-mm inner diameter. The specimens were loaded in compression at 37°C to a constant strain, 
c, of 0.6%. The relaxation modulus, Er(t), was determined from the experimentally observed drop in applied load, L(t), and the change in specimen height, 
hc:
 | (18) |
In Eq. 18, ho was the original specimen height, and do and di were the outer and inner specimen diameters, respectively.
Korostoff et al. (1975) observed an exponential decline in the relaxation modulus with time from the shortest time measured (t 30 sec) to a maximum relaxation time (t 104 sec). Beyond 104 sec, no further stress relaxation was detected. They established the fully reduced modulus, Eo, to be 12.0 GPa. The magnitude of the reduced modulus was as much as 40% lower than the instantaneous value at t = 0. This means that the Young’s modulus measured by mechanical testing could range between 12 and 20 GPa, or greater, depending on the strain rate. Thus, it is probable that stress relaxation accounts for much of the discrepancy between mechanical testing and sonic measurements. This possibility can be explored more thoroughly by reconsideration of the AFM data.
It is well-known that the indenter stylus continues to displace inward during the constant load portion of an indentation measurement of dentin. This continued displacement is evidence that creep relaxation is occurring; the standard procedure for analyzing the indenter load/displacement data does not account for this relaxation. Recently, Feng and Ngan (2002) derived a correction formula for creep during indentation measurements. The correction is applied to the contact stiffness, Sm, measured at the beginning of unload:
 | (19) |
In Eq. 19, Sc is the corrected contact stiffness to be used in Eq. 10, and the correction factor is the ratio of the time derivatives of the change in indenter displacement, hh, during the constant hold cycle, and of the load, P, at the beginning of the unloading cycle.
This information has never been reported for indentation measurements on calcified tissues. However, approximate values for the correction factor can be obtained by estimation from the graphed representations of load and displacement with time. A graphed representation of both the stylus displacement and the load as a function of time is shown in Fig. 8. These data were obtained from a typical AFM indentation measurement on wet intertubular dentin (e.g., Kinney et al., 1999). We approximated the creep relaxation rate by taking the slope of the displacement vs. time at the midpoint of the unloading cycle (1.7 nm/sec in this example); dP/dt was approximated at a point midway between the maximum load and the minimum load during the unloading cycle (131 µN/sec in this example). When the correction was applied to this specimen of wet dentin, the calculated Young’s modulus increased from 20.6 GPa to 25.5 GPa, in excellent agreement with the RUS data (23.2-25.0 GPa; Table 1).
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1/t)? This question can be answered only partially at this time.
A complete analysis of the time-dependent equation of state for dentin requires knowledge of the relaxation spectrum, H(), over all time scales. Then, the elastic constants can be formulated by expressions like Eq. 16. In principle, H() can be obtained by analysis of the stress relaxation data. Prior to availability of computers and digitally instrumented load frames, this procedure was complicated; approximations were used to derive H() from E(t). For example, Korostoff et al. (1975) used an approximation method attributed to Alfrey and Doty (1945):
 | (20) |
With this approximation, Korostoff et al. (1975) evaluated the relaxation spectrum for root dentin for relaxation times from 30 sec to 104 sec. The important result was the finding that the relaxation spectrum was constant over the three decades examined. Though the magnitude of H varied among specimens (H = 0.38 MPa; SD = 0.14), it was in all cases independent of the relaxation time. This was an important result; a constant value of H greatly simplifies the evaluation of the integral relationships in Eqs. 16 and 17.
There were admitted shortcomings with the study. Most critical, no information was provided about stress relaxation at time scales shorter than 30 sec. Physiological time scales of greatest relevance lie in the range of 0.1 to 10 sec. There is a serious concern with extrapolating the above results to shorter time scales. In bone, deviation from simple exponential relaxation has been observed; stress relaxation at short time scales has followed a Kohlrausch-Williams-Watts (KWW) functional form (Sasaki et al., 1993). Much more research, particularly at short time scales, is warranted.
No one has measured the frequency dependence of the loss tangent (Eq. 13) for dentin. In the absence of this information, it is instructive to consider bone, which is compositionally similar to dentin. In a series of classic articles, Lakes and Katz (1979) provided the loss tangent in wet cortical bone over eight decades in relaxation time. Of interest was the observation that the loss tangent was a minimum, and also relatively constant, at physiologically relevant frequencies. If similar behavior can be inferred for dentin, then it would be reasonable to treat the dental hard tissues as perfectly elastic in physical models of mechanical deformation at physiologic strain rates, using, of course, the reduced values of the storage and loss moduli. More study is needed.
SUMMARY OF ELASTIC PROPERTIES
The good agreement between indentation and sonic measurements of the Young’s modulus of hydrated dentin allows us to assign its magnitude between 18 and 25 GPa. The bottom of this range (18-20 GPa) is probably more appropriate for strain rates encountered with physiologic loading. This is significantly greater than the previously accepted range (13-16 GPa). However, the larger Young’s modulus is now more consistent with what has been observed in bone, a mineralized tissue of similar composition. Smaller values of the Young’s modulus that are frequently reported can most likely be attributed to a strain-rate-dependent viscoelastic response, non-uniform stress states in small specimens, improper storage conditions, or flaws introduced during specimen preparation.
Sonic methods provide the most direct and precise determination of the elastic constants. They are least affected by stress relaxation. Because of this, we propose that the data from the RUS (Kinney et al., 2002) and sound speed measurements (Gilmore et al., 1969) be used to define the most probable range for all of the elastic constants of dentin in the absence of viscoelastic effects (Table 1). Because RUS is more likely to probe perfectly elastic behavior (the maximum strain in the specimen was 6 x 10-6, and the Mach number was approximately 1.4 x 10-5), and because RUS does not require assumptions as to the material’s symmetry, we believe them to be the most accurate values of the elastic constants yet measured. However, because RUS has been applied only to coronal dentin from a single site, one should refrain from generalizing these results. It is probable that the elastic properties are site-dependent. Much more work remains to be done.
The intrinsic elastic constants provide only part of the story. The viscoelastic behavior of dentin must be explored in greater detail if a constitutive equation of state is to be developed for dentin. In the frequency range of physiological interest (0.1-10 Hz), the phase angle between stress and strain is most likely small, so that dentin can be modeled as a perfectly elastic solid. However, it is likely that the effective elastic moduli are reduced by about 10-20% from the values obtained with sonic measurements. This is consistent with the differences between the RUS measurements and those obtained by indentation and uni-axial testing. However, this observation is still speculative: The mechanisms controlling viscoelastic behavior, and their possible dependence on the stress amplitude, remain unknown in dentin.
As our knowledge of the elastic properties of normal dentin improves, other questions are raised that have significant implications for dentin pathologies. Is there a functional relationship between the mineral density and the elastic moduli? Does age-related transparency alter the elastic properties of root dentin? Does reparative dentin that forms at the dentin/pulp interface have the same elastic properties as the secondary dentin it adjoins? Answers to these questions will be important in the continued development of minimally invasive approaches in restorative dentistry.
| Hardness of Dentin |
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Hardness is defined in units of pressure, or force per unit area of indentation. Unlike Vickers or Brinell methods, which use the contact area of the indenter stylus, the Knoop method uses the projected area, Ap, in the calculation of hardness (KNH):
 | (21) |
where P is the applied load (in kg) and l is the length (in mm) across the long axis of the remnant impression. In SI units, 1 kg/mm2 9.8 MPa.
Many investigators have determined that dentin hardness depends on mineral concentration. Featherstone et al. (1983) developed an analytic expression relating Knoop hardness to the volume percent of mineral (Vm):
 | (22) |
Though the validity of Eq. 22 obviously breaks down for low concentrations of mineral, the expression appears to fit the experimental data over a range of mineral concentrations associated with normal and carious dentin.
In addition to its association with the mineral concentration, hardness has been correlated with location in the tooth. The works of Ogawa (1983) and Wang and Weiner (1998b) show that the mantle dentin immediately subjacent to the enamel (KNH 60 kg/mm2) is softer than the underlying primary dentin (KNH 70 kg/mm2). Also, hardness gradually decreases with proximity to the pulp, falling precipitously in an inner layer of dentin of about 0.5 mm thickness that surrounds the pulp (KNH 30 kg/mm2).
Clearly, Eq. 22 mandates that Vm must be lower near the pulp than in the primary dentin. However, whether this is a result of increased porosity, or whether the intertubular dentin matrix is less mineralized, is an important question. In a careful and methodical study, Pashley and Parham (1985) determined that there was a significant correlation between decreased hardness and increased density of tubule lumens. The authors concluded that the reduced hardness was an end result of the lower mineral concentration brought about by the increased tubule porosity. This conclusion, however, was seriously challenged in a later study by Kinney et al. (1996), who showed that most, if not all, of the decreased hardness near the pulp could be explained by a decrease in the hardness of the intertubular dentin matrix. Thus, it is likely that the intertubular dentin matrix near the pulp is less mineralized.
It is tempting to try to relate dentin hardness to other physical properties like yield stress, tensile strength, or Young’s modulus. In ductile metals, for example, the yield stress and tensile strengths are often observed to scale with hardness. However enticing these scalings, we must remember that they were derived from plasticity theory for materials that display significant yielding. In contrast, mineralized tissues are more brittle, showing little if any yielding prior to failure.
Though we should not anticipate similar relationships with dentin hardness, there is a possibility that its hardness might scale with the Young’s modulus. Eq. 22 describes a relationship between hardness and mineral concentration, and there is strong evidence that the Young’s modulus is also dependent on the amount of mineral. However, because hardness also depends on yielding, microstructural features in the dentin that do not influence the elastic properties might mask any such correlation. In many materials, for example, the grain size controls yield strength by the Hall-Petch relationship. Therefore, the ratio of hardness to modulus need not be constant. Nevertheless, a relationship between hardness and modulus in dentin is worth seeking. Although several papers have reported hardness and modulus values for enamel and dentin, a systematic analysis of their relationship has not been undertaken.
| Ultimate Strength of Dentin |
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Shear strength, measured either by punching or lap-shear, also has been highly variable. Using a shear punch apparatus, Cooper and Smith (1968) obtained values for shear strength that ranged from 64 to 132 MPa, not too dissimilar from later measurements by Roydhouse (1970) (69-147 MPa). More recent measurements by single-plane lap shear produced shear strengths of 36 MPa; this low value may have been due to the specimens having been from dentin closer to the pulp, or to problems with the experimental design, or with bending of the specimen (Gwinnett, 1994).
Our group (Watanabe et al., 1996) demonstrated that some of the disparity in reported values of the shear strength could be attributed to tubule orientation and location within the tooth. The lowest values of lap shear strength were obtained in less mature dentin nearer the pulp, with the tubules oriented parallel to the shear plane (54 MPa). The highest values were obtained in mature cuspal dentin with the shear plane perpendicular to the tubules (92 MPa). Others also have observed a similar orientation dependence with the tensile strength (Lertchirakarn et al., 2001).
The large standard deviations common to all measurements of dentin strength suggest that strength is controlled by the flaw distribution in a specimen. A flaw is similar to the weakest link in a chain: Variations in the flaw size lead to variations in the failure strength. Specimens with large flaws will fail at lower stresses than will specimens containing smaller flaws. For a random distribution of flaws, the tensile strength should obey a Weibull probability distribution function (pdf), where the probability, Pf, that a specimen fails at a stress level 
is given by the expression:
