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Time-dependent properties of human root dentin
dental materials Dental Materials 18 (2002) 486±493
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Time-dependent properties of human root dentin Jeeraphat Jantarat a, Joseph E.A. Palamara b, Cecylia Lindner b, Harold H. Messer b,* a
b
Department of Operative Dentistry, Mahidol University, Bangkok, Thailand School of Dental Science, The University of Melbourne, 711 Elizabeth Street, Melbourne, Victoria 3000, Australia Received 21 September 2000; revised 14 May 2001; accepted 19 June 2001
Abstract Objectives: The aim of this study was to investigate the creep, stress relaxation and strain rate behavior of human root dentin under compressive loading. Methods: Cylindrical root dentin samples of 3.5 mm outer diameter, 1.5 mm internal canal diameter and 6±10 mm long were prepared from freshly extracted teeth. The samples were tested in a closed-loop servohydraulic testing machine at constant load or displacement, and varied strain rate. In vivo strain rates were estimated using strain gauges bonded to human teeth. Results: A family of creep curves, determined at different loads within dentin's elastic region, was found to be consistent with a material having linear viscoelastic behavior. A positive correlation (r 2 0.79, P , 0.001) was found between creep rate and stress. Young's modulus (E) was found to be a function of the strain rate with rates of loading in the range 10±500,000 Ns 21. Loading at constant displacement showed stress to be a decreasing function of time (i.e. stress relaxation). Signi®cance: Dentin showed linear-viscoelastic behavior under various conditions of compressive loading. Time dependent properties of dentin should be taken into account in restorative dentistry. q 2002 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. Keywords: Dentin; Creep; Stress relaxation; Young's modulus; Strain rate; Viscoelasticity; Time-dependent properties
1. Introduction Dentin, as a calci®ed tissue, has been described as a twophase composite of collagen and hydroxyapatite [1,2]. The elastic properties of calci®ed tissues result from the intermixture of organic and mineral constituents [3±5]. Dentin has a lower organic content and slightly higher mineral and water content than bone [5]. Unlike bone, dentin possesses a highly ordered microstructure consisting of dentinal tubules which are ®lled with dentinal ¯uid [1]. Several studies have shown that dentin exhibits time dependent properties [6±9]. Characteristic phenomena of viscoelastic materials include: (1) the strain increases with time if the stress is held constant (creep); (2) if the strain is held constant, the stress decreases with time (relaxation); (3) the stiffness depends on the rate at the load which is applied; (4) hysteresis (a phase lag) occurs if cyclic loading is applied, leading to the dissipation of mechanical energy; (5) acoustic waves experience attenuation; (6) rebound of an object following an impact is less than 100%; and (7) during rolling, frictional resistance occurs [10]. All * Corresponding author. Tel.: 161-3-9341-0414; fax: 161-3-9348-2415. E-mail address: [email protected] (H.H. Messer MDSc, PhD).
materials exhibit a viscoelastic response to some extent. In dentin, the viscoelastic response is potentially clinically signi®cant. The ®rst phenomenon, creep behavior, was demonstrated in dentin in 1958 [6]. Strain increased with time when the load was held for 20 min. After a load within proportional limit was relieved, the major portion of deformation (the Hookian elastic deformation) was recovered immediately. However, the remaining deformation (retarded elastic deformation) was not completely recovered until 50 min later. When loading was above the proportional limit, a similar shaped curve was obtained but a portion of the deformation was not completely recovered (permanent deformation). In the same study, hysteresis was also demonstrated. Other studies have demonstrated stress relaxation behavior [7,8]. Trengrove et al. [9] found that air-dried dentin showed a decreased stress relaxation. Longitudinal sound velocity (LSV) has also been used to demonstrate viscoelastic behavior of dentin [11±13]. The LSV in dentin varies depending on the location. The authors suggested that LSV may be used to characterize physical properties of hard dental tissues. More recently, atomic force microscopy with a nanoindentation technique has also been used to characterize mechanical properties of dentin [14,15].
0109-5641/02/$20.00 + 0.00 q 2002 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. PII: S 0109-564 1(01)00074-4
J. Jantarat et al. / Dental Materials 18 (2002) 486±493
Fig. 1. Orientation and con®guration of the dentin cylinders with respect to the root canal and the long axis of the tooth.
The aim of this study was to characterize the creep and stress relaxation behavior and the strain rate dependence of Young's modulus of human root dentin under compressive loading. The hypotheses tested were that root dentin shows linear viscoelastic behavior under compressive loading and that the value of Young's modulus increases with increasing strain rate.
2. Materials and methods 2.1. Sample preparation Intact non-carious maxillary incisors and canines were obtained from patients undergoing extractions for routine reasons. All teeth were examined under transillumination and magni®cation for possible cracks. Teeth were stored in thymol solution at 48C and were tested within 3 weeks of extraction. Cylindrical annular dentin samples of 3.5 mm outer diameter, 1.5 mm internal canal diameter and 6±10 mm long were prepared using a miniature lathe with water cooling. This specimen shape was based on that used by Duncanson and Korostoff [7] and Korostoff et al. [8], in their stress-relaxation study (Fig. 1). Two horizontal cuts at the cervical and apical third area were made using a slow-speed diamond saw with water coolant system. The root canals were enlarged using endodontic K-®les up to size 40. The specimens were then concentrically mounted in a miniature lathe (Hobby Product Co., Columbus OH). The canals were enlarged with centric
487
Fig. 2. Diagrammatic representation of the testing apparatus. The dentin sample was kept fully hydrated throughout testing.
rotation using Gates Glidden drills #2 and #3 followed by Para-Post drills up to 1.5 mm diameter. After the preparation of the internal diameter, the dentin specimens were mounted on a mandrel and turned in the miniature lathe, to produce cylinders with external diameter of 3.5 mm. Machining was done using a diamond bur with a watercooled high speed air-rotor dental handpiece. Both ends of the dentin sample were carefully polished with a polishing device to make them plano-parallel. The length and diameter of each specimen were measured using a digital caliper. 2.2. Testing system Samples were subjected to compressive loads in a closedloop servohydraulic testing machine (MTS Model 810, MTS Systems, Eden Prairie, MN). The testing machine could be con®gured for load, displacement or strain control. Two platens were constructed from high tensile steel alloy with highly polished surfaces to decrease frictional force. Over the ends of the platens, two nylon rings were ®tted to house the specimen and provide hydration by adding water to the well inside the nylon ring. An extensometer, (MTS model 632.11F-20, MTS Systems, Eden Prairie, MN) with a span of 25 mm was attached to the platens (Fig. 2). All specimens were immersed in water during testing at room temperature. A platen-to-platen test was conducted to ensure that creep due to machine compliance was insigni®cant.
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2.3. Test conditions Dentin samples were subjected to continuous load. Load and extensometer data were recorded simultaneously on a microcomputer using Labview software (National Instruments Corp, Austin, TX). To ensure that testing was conducted within the proportional limit for all samples, the load at the proportional limit was determined from a preliminary study. Nine dentin samples were subjected to a ramped load until fracture occurred, using the servohydraulic testing machine. The mean load at fracture was approximately 1336 N (range: 1160±1685 N) and no sample fractured at less than 1160 N. Mean load at proportional limit was approximately 1060 N (range: 840± 1380 N). It was concluded that loads of up to 700 N could be applied safely, and allow for all testing procedures to occur within the proportional limit. 2.4. Creep Eight dentin samples were subjected to constant loads of 100, 300, 500 and 700 N, with all samples subjected to each load. To ensure the stability of the samples and the servohydraulic testing machine, a pre-load of 10 N was applied. Load was held for 90 min. The extensometer was used to measure change in displacement throughout the loading period and for 60 min after removal of the load (the 10 N `preload' was applied during the ®nal 60 min). The sequence of testing (for each magnitude of load) was randomized and the sample was rested for at least 24 h prior to the next test. A family of creep curves (at 100, 300, 500 and 700 N) was generated for each specimen. Creep rate per hour and creep compliance were calculated. Creep compliance, J(t) is de®ned as the strain e divided by the stress s at a given time [16]. J t
e t so
The specimens were loaded to fracture at the end of the experiment. The following data were calculated; stress at proportional limit, % strain at proportional limit, ultimate compressive strength, % strain at fracture and Young's modulus. The unpaired t-test was used to identify statistically signi®cant differences between these post-test samples and a control group of fresh samples not subjected to repeated testing (nine samples used in the preliminary tests described in the previous section). 2.5. Strain rate dependency of elastic modulus Seven dentin samples were subjected to a load of 700 N with rates of loading at 10, 10 2, 10 3, 10 4, 10 5 and 5 £ 10 5 Ns 21. The highest rate was the upper limit of the MTS machine's capability. The sequence of loading was random, except that 5 £ 10 5 Ns 21 was always tested last to minimize potential damage to the samples. After the test of 5 £ 10 5 Ns 21, the samples were always tested again at 10 Ns 21, to
show that no changes had occurred due to the high strain rates. Young's modulus was computed at all strain rates. The Young's modulus data were plotted against strain rate and the power coef®cient was calculated using curve ®tting to data points. To estimate strain rate during normal function in vivo, a preliminary study was conducted in four volunteers. Strain in dentin was estimated from strain in cervical enamel, since the strain in cervical enamel is very similar to that in adjacent dentin [17,18]. A single-element gauge (CEA-06031MF-120; Micro-Measurements, Raleigh, NC, USA) was attached vertically to the cervical area of one mandibular second premolar of each subject with cyanoacrylate adhesive (M-bond 200; Micro-Measurements, Raleigh, NC, USA). The tooth was etched with 37% phosphoric acid for 30 s prior to bonding. The gauges were then covered with bonding agent to prevent moisture contamination of the gauge. Testing included biting on a bite fork, which simultaneously recorded load, chewing on soft food (paraf®n), hard food (almond nut), and hard candy. Strain was recorded during forceful biting on the buccal cusp in the centric position, and the strain rate estimated from the slope of the (compressive) strain vs. time. The strain rate in adjacent dentin was assumed to the similar to that cervical of enamel. Institutional ethics approval was obtained for this experiment. 2.6. Stress relaxation Seven dentin samples were loaded to 700 N and the displacement was held constant (using displacement control via the extensometer) for 60 min while the load was recorded. Post-load data were recorded for 30 min after the removal of the load (with the 10 N `preload' applied throughout the recovery period). Stress relaxation moduli (stress/strain) and percentage reduction in stress over 60 min were calculated. 3. Results 3.1. Load at fracture Fresh control samples were loaded to fracture, to provide an estimate of the loads that could be used safely during experiments. Experimental dentin samples used in the creep study were also subsequently loaded to fracture. Stress at proportional limit, Young's modulus, % strain at proportional limit, and ultimate compressive strength (MPa) of both fresh samples (non-experimental group) and the experimental group are shown in Table 1. Young's modulus of the control group and post-test group were 13.3 and 12.5 GPa, respectively. Ultimate compressive strengths were 162.9 and 167.7 MPa. There were no statistically signi®cant differences between control and post-experimental groups for the value of Young's modulus and ultimate compressive strength (Student t test). Thus repeated testing
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Table 1 Compressive properties of root dentin samples before and after compressive testing. Standard deviations are given in parenthesis following the mean. No signi®cant differences (P . 0.05) were observed between the two groups Compressive properties
Fresh samples (n 9)
Post-test samples (n 8)
Stress at proportional limit (MPa) % strain at proportional limit Ultimate compressive strength (MPa) Young's modulus (GPa)
129.1 (17.0) 1.06 (0.21) 162.9 (21.5) 13.3 (1.3)
123.24 (12.8) 1.24 (0.17) 167.7 (17.0) 12.5 (1.2)
on the same samples did not appear to result in damage to the dentin specimens. 3.2. Creep A representative family of creep curves for one sample loaded at 100, 300, 500, and 700 N is shown in Fig. 3. The curves show the characteristic pattern for viscoelastic materials of increasing strain over time while the stress is held constant. The extent of creep was found to be proportional to the load. Recovery following removal of the load was biphasic: instantaneous elastic recovery (approximately 90±95%), followed by a gradual return toward baseline. Creep rate per hour was calculated and plotted as % strain/hour vs. stress (Fig. 4(a)). Regression analysis showed a highly signi®cant effect of stress on creep rate (R 2 0.79, P , 0.001). The data from the 100 N group were not included in the regression due to a very small creep rate. The load may be too small for creep to predominate over the `toe effect' which occurs initially when uniformly loading the sample. With higher load, creep rate was found to be higher. Creep compliance (Jt) was also determined (Fig. 4(b)). A non-signi®cant minor slope was found from the regression line between creep compliance per hour and stress, indicating that creep compliance was not affected by stress. Hence dentin was found to be predominantly linearly viscoelastic. 3.3. Strain rate dependency Young's modulus vs. log of strain rate was plotted for
each sample individually (Fig. 5, linear/log plot). Young's modulus increased with faster strain rate, i.e. dentin is stiffer at higher strain rate. All samples showed the same trend, with similar slopes despite moderate variation in modulus value among specimens. The regression coef®cient (R 2) for E vs log10 strain rate ranged from 0.86 to 0.99 for individual samples, and the effect of strain rate on the modulus was highly signi®cant (P , 0.001, one-way ANOVA with repeated measures). Elastic modulus was proportional to strain rate raised to approximately the 0.02 power. Young's modulus values normalized at a strain rate of 0.01 s 21 averaged 14.5 GPa with standard deviation of 1.2 (range 12.1± 16.0) and the power coef®cient averaged 0.017 with standard deviation of 0.001 (range 0.015±0.018). In vivo, the overall mean of strain rate at the cervical region during normal function was 0.004 s 21. The highest strain rate recorded among four subjects was 0.009 s 21 and the lowest was 0.0005 s 21 during biting on of hard food (almond nut) and soft food respectively. 3.4. Stress relaxation The stress value at constant strain was plotted against time (Fig. 6(a), one sample only). The sample was loaded to 700 N and the extensometer was set to control at constant strain. The load began to decrease after the peak load was reached. After 1 h the load had declined by approximately 45 N which is 8% of initial peak load (for this sample). Following removal of the load, a small stress increment typical of viscoelastic recovery was observed [10]. The normalized relaxation modulus Er(t)/Er(t0) plotted against log10t for the same sample is shown in Fig. 6(b). This plot normalized the relaxation modulus based on the initial stress of the sample [7]. The relaxation moduli demonstrated a linear relationship with the logarithm of time during the period of 60 min. Stress relaxation after 60 min averaged 8.0 (4.6 SD) MPa (n 7). The reduction of stress was 9.3% with standard deviation of 4.8% of initial stress. 4. Discussion
Fig. 3. Family of creep curves at different loads (100, 300, 500 and 700 N). The load was held constant for 90 min and strain was recorded for an additional 60 min after the load was removed (except for a 10 N `preload').
Dentin and bone have similar composition and mechanical properties [19]. In contrast to dentin, viscoelastic properties of both cortical bone and trabecular bone have been well characterized [20±28]. Other than demonstrating the
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Fig. 4. (a) Creep rate per hour vs. stress for all samples at all loads. (b) Creep compliance per hour vs. stress.
presence of creep behavior [6] and acoustic attenuation [13], only the stress relaxation behavior of dentin has been examined in detail [7±9]. The samples used in this study were similar to those in previous studies [7,8]. The samples are radially symmetrical. During preparation, the samples were prepared carefully to avoid artifacts due to dehydration, overheating and surface scratches. The dentin samples in the present study were obtained from root dentin, which can provide greater dimensions than coronal dentin. Dentin exhibits a gradation in properties, since the tubule orientation, density and diameter differ according to location [29]. The progressive deposition of peritubular dentin results in a decrease in diameter of the tubules toward the enamel or root surface [30]. A gradient has been shown in hardness and punch
shear strength from the pulpal surface to outer dentin [31]. Newer techniques such as AFM permit the measurement of properties on a much smaller scale [14,15]. A possible contribution of the ¯uid-®lled tubules to time-dependent properties requires the use of larger samples. The result of fracture tests showed values of proportional limit, ultimate compressive strength and Young's modulus (Table 1) are consistent with the results obtained by many authors [32±35]. In contrast, Craig and Peyton [6] and Kinney et al. [36] reported a relatively high value for Young's modulus, in the range of 16.6±18.5 GPa. No effect of repeated experimental loading on elastic properties was found in the present study. Understanding the viscoelastic response of dentin has important implications clinically, such as the response to
J. Jantarat et al. / Dental Materials 18 (2002) 486±493
491
Fig. 5. In¯uence of strain rate on elastic modulus for eight dentin samples, each subjected to six rates of loading.
occlusal load of intact or restored teeth, the effect of aging, and cavity preparation and the design of restorations. The most important phenomena of viscoelastic materials are continued deformation of a material under constant load or creep, and progressive reduction in stress while a material is under constant deformation or relaxation [10]. In the present study, dentin demonstrated both phenomena. Creep is widely recognized as important in engineering design [37]. In clinical function, creep behavior is potentially important during clenching or bruxing, with a reported duration of loading of 8±9 s [38±40] and according to one report up to 5 min [41]. Stress relaxation is signi®cant in clinical situations such as use of a threaded post in the root canal [42], placement of pins during restorations, and polymerization contraction of composite restorations [43±45]. Grimaldi and Hood [46] investigated cuspal de¯ection and recovery following load removal in extracted maxillary premolars. They reported delayed elastic recovery over a 10 min period in teeth with an MOD cavity preparation. Constant load following by removal of the load showed characteristics of viscoelastic materials with creep, elastic recovery, and retarded elastic deformation. Creep compliance (Jt) did not vary with stress (Fig. 4(b)). This result indicates that dentin is linearly viscoelastic, in a similar fashion to bone [47]. Young's modulus of dentin in this study was found to increase with faster strain rate, in a similar relationship to that found for bone. The data in Fig. 5 are highly consistent among samples, with similar slopes of all curves.
Curve ®tting indicated that the modulus is approximately proportional to the strain rate raised to the 0.017 power, which is at the lower end of the range reported for bone (0.018 [48] and 0.06 [20,21]). A low strain rate produces lower strength because the low strain rate allows more time for cumulative damage than a higher strain rate [20]. Part of the variability in reported values for Young's modulus in many studies [36] may be due to the differences in the rate of loading samples during testing. Over the range of strain rates used in this study, the modulus value varied by 2±3 GPa. Strain rates encountered clinically are unknown. In a preliminary study, we have estimated the strain rate in cervical dentin in four subjects, based on strain rates recorded in bucco-cervical enamel [17,18]. The measurements were limited to compressive strains recorded during forceful biting in approximately centric relation. Strain rates in bone have been measured directly, using strain gauges bonded to a human tibial diaphysis [49]. The strain rate during walking was 0.002 s 21 and during running was 0.01 s 21. The preliminary estimates that we recorded (0.0005±0.009 s 21) are lower than the range reported for bone. The wide range of strain rates used in the present study represents those strain rates encountered in daily activities such as chewing (0.004 s 21) as well as strain rates associated with impact trauma in accidents which will be much higher. Stress relaxation is the only area of viscoelastic behavior that has previously been well investigated in dentin, and was not considered in detail in this study. The stress relaxation modulus showed a linear dependence on the logarithm of
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Fig. 6. Stress relaxation curve of root dentin. (a) Stress relaxation curve for one sample: following a ramped load to 700 N in 3.5 s, the strain was held constant for 60 min. The load was then removed (retaining a 10 N load) and strain recorded for an additional 30 min. (b) Decrease in relaxation modulus over 90 min at constant strain, for the same sample.
time similar to that seen in previous studies [7,8]. The reduction of relaxation modulus in this study was 6±17% over 60 min, compared with 10±30% over 6 h reported by Duncanson and Korostoff [7]. The latter study was performed at 378C, compared with the present study at 20±228C. Dentin's modulus of elasticity decreased with increasing temperature [34], therefore it is stiffer at the lower temperature. Hence the relaxation modulus along with other viscoelastic properties of dentin may be also temperature dependent. Acknowledgements This investigation was supported by a grant from the National Health and Medical Council of Australia, Australian
Society of Endodontology, Australian Society of Endodontology (Victorian Branch). Special thanks to Miss Florence Choo, Statistical Consultation Centre, The University of Melbourne, for statistical advice. References [1] Marshall Jr GW. Dentin: microstructure and characterization. Quint Int 1993;24:606±17. [2] Pashley DH. Dynamics of the pulpo-dentin complex. Crit Rev Oral Biol Med 1996;7:104±33. [3] Currey JD. Three analogies to explain the mechanical properties of bone. Biorheol 1964;2:1±10. [4] Katz JL. Hard tissue as a composite material. I. Bounds on the elastic behavior. J Biomech 1971;4:455±73. [5] Waters NE. Some mechanical and physical properties of teeth. Symposia of the Society for Experimental Biology 1980;34:99±135.
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Time-dependent properties of human root dentin Jeeraphat Jantarat a, Joseph E.A. Palamara b, Cecylia Lindner b, Harold H. Messer b,* a
b
Department of Operative Dentistry, Mahidol University, Bangkok, Thailand School of Dental Science, The University of Melbourne, 711 Elizabeth Street, Melbourne, Victoria 3000, Australia Received 21 September 2000; revised 14 May 2001; accepted 19 June 2001
Abstract Objectives: The aim of this study was to investigate the creep, stress relaxation and strain rate behavior of human root dentin under compressive loading. Methods: Cylindrical root dentin samples of 3.5 mm outer diameter, 1.5 mm internal canal diameter and 6±10 mm long were prepared from freshly extracted teeth. The samples were tested in a closed-loop servohydraulic testing machine at constant load or displacement, and varied strain rate. In vivo strain rates were estimated using strain gauges bonded to human teeth. Results: A family of creep curves, determined at different loads within dentin's elastic region, was found to be consistent with a material having linear viscoelastic behavior. A positive correlation (r 2 0.79, P , 0.001) was found between creep rate and stress. Young's modulus (E) was found to be a function of the strain rate with rates of loading in the range 10±500,000 Ns 21. Loading at constant displacement showed stress to be a decreasing function of time (i.e. stress relaxation). Signi®cance: Dentin showed linear-viscoelastic behavior under various conditions of compressive loading. Time dependent properties of dentin should be taken into account in restorative dentistry. q 2002 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. Keywords: Dentin; Creep; Stress relaxation; Young's modulus; Strain rate; Viscoelasticity; Time-dependent properties
1. Introduction Dentin, as a calci®ed tissue, has been described as a twophase composite of collagen and hydroxyapatite [1,2]. The elastic properties of calci®ed tissues result from the intermixture of organic and mineral constituents [3±5]. Dentin has a lower organic content and slightly higher mineral and water content than bone [5]. Unlike bone, dentin possesses a highly ordered microstructure consisting of dentinal tubules which are ®lled with dentinal ¯uid [1]. Several studies have shown that dentin exhibits time dependent properties [6±9]. Characteristic phenomena of viscoelastic materials include: (1) the strain increases with time if the stress is held constant (creep); (2) if the strain is held constant, the stress decreases with time (relaxation); (3) the stiffness depends on the rate at the load which is applied; (4) hysteresis (a phase lag) occurs if cyclic loading is applied, leading to the dissipation of mechanical energy; (5) acoustic waves experience attenuation; (6) rebound of an object following an impact is less than 100%; and (7) during rolling, frictional resistance occurs [10]. All * Corresponding author. Tel.: 161-3-9341-0414; fax: 161-3-9348-2415. E-mail address: [email protected] (H.H. Messer MDSc, PhD).
materials exhibit a viscoelastic response to some extent. In dentin, the viscoelastic response is potentially clinically signi®cant. The ®rst phenomenon, creep behavior, was demonstrated in dentin in 1958 [6]. Strain increased with time when the load was held for 20 min. After a load within proportional limit was relieved, the major portion of deformation (the Hookian elastic deformation) was recovered immediately. However, the remaining deformation (retarded elastic deformation) was not completely recovered until 50 min later. When loading was above the proportional limit, a similar shaped curve was obtained but a portion of the deformation was not completely recovered (permanent deformation). In the same study, hysteresis was also demonstrated. Other studies have demonstrated stress relaxation behavior [7,8]. Trengrove et al. [9] found that air-dried dentin showed a decreased stress relaxation. Longitudinal sound velocity (LSV) has also been used to demonstrate viscoelastic behavior of dentin [11±13]. The LSV in dentin varies depending on the location. The authors suggested that LSV may be used to characterize physical properties of hard dental tissues. More recently, atomic force microscopy with a nanoindentation technique has also been used to characterize mechanical properties of dentin [14,15].
0109-5641/02/$20.00 + 0.00 q 2002 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. PII: S 0109-564 1(01)00074-4
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Fig. 1. Orientation and con®guration of the dentin cylinders with respect to the root canal and the long axis of the tooth.
The aim of this study was to characterize the creep and stress relaxation behavior and the strain rate dependence of Young's modulus of human root dentin under compressive loading. The hypotheses tested were that root dentin shows linear viscoelastic behavior under compressive loading and that the value of Young's modulus increases with increasing strain rate.
2. Materials and methods 2.1. Sample preparation Intact non-carious maxillary incisors and canines were obtained from patients undergoing extractions for routine reasons. All teeth were examined under transillumination and magni®cation for possible cracks. Teeth were stored in thymol solution at 48C and were tested within 3 weeks of extraction. Cylindrical annular dentin samples of 3.5 mm outer diameter, 1.5 mm internal canal diameter and 6±10 mm long were prepared using a miniature lathe with water cooling. This specimen shape was based on that used by Duncanson and Korostoff [7] and Korostoff et al. [8], in their stress-relaxation study (Fig. 1). Two horizontal cuts at the cervical and apical third area were made using a slow-speed diamond saw with water coolant system. The root canals were enlarged using endodontic K-®les up to size 40. The specimens were then concentrically mounted in a miniature lathe (Hobby Product Co., Columbus OH). The canals were enlarged with centric
487
Fig. 2. Diagrammatic representation of the testing apparatus. The dentin sample was kept fully hydrated throughout testing.
rotation using Gates Glidden drills #2 and #3 followed by Para-Post drills up to 1.5 mm diameter. After the preparation of the internal diameter, the dentin specimens were mounted on a mandrel and turned in the miniature lathe, to produce cylinders with external diameter of 3.5 mm. Machining was done using a diamond bur with a watercooled high speed air-rotor dental handpiece. Both ends of the dentin sample were carefully polished with a polishing device to make them plano-parallel. The length and diameter of each specimen were measured using a digital caliper. 2.2. Testing system Samples were subjected to compressive loads in a closedloop servohydraulic testing machine (MTS Model 810, MTS Systems, Eden Prairie, MN). The testing machine could be con®gured for load, displacement or strain control. Two platens were constructed from high tensile steel alloy with highly polished surfaces to decrease frictional force. Over the ends of the platens, two nylon rings were ®tted to house the specimen and provide hydration by adding water to the well inside the nylon ring. An extensometer, (MTS model 632.11F-20, MTS Systems, Eden Prairie, MN) with a span of 25 mm was attached to the platens (Fig. 2). All specimens were immersed in water during testing at room temperature. A platen-to-platen test was conducted to ensure that creep due to machine compliance was insigni®cant.
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2.3. Test conditions Dentin samples were subjected to continuous load. Load and extensometer data were recorded simultaneously on a microcomputer using Labview software (National Instruments Corp, Austin, TX). To ensure that testing was conducted within the proportional limit for all samples, the load at the proportional limit was determined from a preliminary study. Nine dentin samples were subjected to a ramped load until fracture occurred, using the servohydraulic testing machine. The mean load at fracture was approximately 1336 N (range: 1160±1685 N) and no sample fractured at less than 1160 N. Mean load at proportional limit was approximately 1060 N (range: 840± 1380 N). It was concluded that loads of up to 700 N could be applied safely, and allow for all testing procedures to occur within the proportional limit. 2.4. Creep Eight dentin samples were subjected to constant loads of 100, 300, 500 and 700 N, with all samples subjected to each load. To ensure the stability of the samples and the servohydraulic testing machine, a pre-load of 10 N was applied. Load was held for 90 min. The extensometer was used to measure change in displacement throughout the loading period and for 60 min after removal of the load (the 10 N `preload' was applied during the ®nal 60 min). The sequence of testing (for each magnitude of load) was randomized and the sample was rested for at least 24 h prior to the next test. A family of creep curves (at 100, 300, 500 and 700 N) was generated for each specimen. Creep rate per hour and creep compliance were calculated. Creep compliance, J(t) is de®ned as the strain e divided by the stress s at a given time [16]. J t
e t so
The specimens were loaded to fracture at the end of the experiment. The following data were calculated; stress at proportional limit, % strain at proportional limit, ultimate compressive strength, % strain at fracture and Young's modulus. The unpaired t-test was used to identify statistically signi®cant differences between these post-test samples and a control group of fresh samples not subjected to repeated testing (nine samples used in the preliminary tests described in the previous section). 2.5. Strain rate dependency of elastic modulus Seven dentin samples were subjected to a load of 700 N with rates of loading at 10, 10 2, 10 3, 10 4, 10 5 and 5 £ 10 5 Ns 21. The highest rate was the upper limit of the MTS machine's capability. The sequence of loading was random, except that 5 £ 10 5 Ns 21 was always tested last to minimize potential damage to the samples. After the test of 5 £ 10 5 Ns 21, the samples were always tested again at 10 Ns 21, to
show that no changes had occurred due to the high strain rates. Young's modulus was computed at all strain rates. The Young's modulus data were plotted against strain rate and the power coef®cient was calculated using curve ®tting to data points. To estimate strain rate during normal function in vivo, a preliminary study was conducted in four volunteers. Strain in dentin was estimated from strain in cervical enamel, since the strain in cervical enamel is very similar to that in adjacent dentin [17,18]. A single-element gauge (CEA-06031MF-120; Micro-Measurements, Raleigh, NC, USA) was attached vertically to the cervical area of one mandibular second premolar of each subject with cyanoacrylate adhesive (M-bond 200; Micro-Measurements, Raleigh, NC, USA). The tooth was etched with 37% phosphoric acid for 30 s prior to bonding. The gauges were then covered with bonding agent to prevent moisture contamination of the gauge. Testing included biting on a bite fork, which simultaneously recorded load, chewing on soft food (paraf®n), hard food (almond nut), and hard candy. Strain was recorded during forceful biting on the buccal cusp in the centric position, and the strain rate estimated from the slope of the (compressive) strain vs. time. The strain rate in adjacent dentin was assumed to the similar to that cervical of enamel. Institutional ethics approval was obtained for this experiment. 2.6. Stress relaxation Seven dentin samples were loaded to 700 N and the displacement was held constant (using displacement control via the extensometer) for 60 min while the load was recorded. Post-load data were recorded for 30 min after the removal of the load (with the 10 N `preload' applied throughout the recovery period). Stress relaxation moduli (stress/strain) and percentage reduction in stress over 60 min were calculated. 3. Results 3.1. Load at fracture Fresh control samples were loaded to fracture, to provide an estimate of the loads that could be used safely during experiments. Experimental dentin samples used in the creep study were also subsequently loaded to fracture. Stress at proportional limit, Young's modulus, % strain at proportional limit, and ultimate compressive strength (MPa) of both fresh samples (non-experimental group) and the experimental group are shown in Table 1. Young's modulus of the control group and post-test group were 13.3 and 12.5 GPa, respectively. Ultimate compressive strengths were 162.9 and 167.7 MPa. There were no statistically signi®cant differences between control and post-experimental groups for the value of Young's modulus and ultimate compressive strength (Student t test). Thus repeated testing
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Table 1 Compressive properties of root dentin samples before and after compressive testing. Standard deviations are given in parenthesis following the mean. No signi®cant differences (P . 0.05) were observed between the two groups Compressive properties
Fresh samples (n 9)
Post-test samples (n 8)
Stress at proportional limit (MPa) % strain at proportional limit Ultimate compressive strength (MPa) Young's modulus (GPa)
129.1 (17.0) 1.06 (0.21) 162.9 (21.5) 13.3 (1.3)
123.24 (12.8) 1.24 (0.17) 167.7 (17.0) 12.5 (1.2)
on the same samples did not appear to result in damage to the dentin specimens. 3.2. Creep A representative family of creep curves for one sample loaded at 100, 300, 500, and 700 N is shown in Fig. 3. The curves show the characteristic pattern for viscoelastic materials of increasing strain over time while the stress is held constant. The extent of creep was found to be proportional to the load. Recovery following removal of the load was biphasic: instantaneous elastic recovery (approximately 90±95%), followed by a gradual return toward baseline. Creep rate per hour was calculated and plotted as % strain/hour vs. stress (Fig. 4(a)). Regression analysis showed a highly signi®cant effect of stress on creep rate (R 2 0.79, P , 0.001). The data from the 100 N group were not included in the regression due to a very small creep rate. The load may be too small for creep to predominate over the `toe effect' which occurs initially when uniformly loading the sample. With higher load, creep rate was found to be higher. Creep compliance (Jt) was also determined (Fig. 4(b)). A non-signi®cant minor slope was found from the regression line between creep compliance per hour and stress, indicating that creep compliance was not affected by stress. Hence dentin was found to be predominantly linearly viscoelastic. 3.3. Strain rate dependency Young's modulus vs. log of strain rate was plotted for
each sample individually (Fig. 5, linear/log plot). Young's modulus increased with faster strain rate, i.e. dentin is stiffer at higher strain rate. All samples showed the same trend, with similar slopes despite moderate variation in modulus value among specimens. The regression coef®cient (R 2) for E vs log10 strain rate ranged from 0.86 to 0.99 for individual samples, and the effect of strain rate on the modulus was highly signi®cant (P , 0.001, one-way ANOVA with repeated measures). Elastic modulus was proportional to strain rate raised to approximately the 0.02 power. Young's modulus values normalized at a strain rate of 0.01 s 21 averaged 14.5 GPa with standard deviation of 1.2 (range 12.1± 16.0) and the power coef®cient averaged 0.017 with standard deviation of 0.001 (range 0.015±0.018). In vivo, the overall mean of strain rate at the cervical region during normal function was 0.004 s 21. The highest strain rate recorded among four subjects was 0.009 s 21 and the lowest was 0.0005 s 21 during biting on of hard food (almond nut) and soft food respectively. 3.4. Stress relaxation The stress value at constant strain was plotted against time (Fig. 6(a), one sample only). The sample was loaded to 700 N and the extensometer was set to control at constant strain. The load began to decrease after the peak load was reached. After 1 h the load had declined by approximately 45 N which is 8% of initial peak load (for this sample). Following removal of the load, a small stress increment typical of viscoelastic recovery was observed [10]. The normalized relaxation modulus Er(t)/Er(t0) plotted against log10t for the same sample is shown in Fig. 6(b). This plot normalized the relaxation modulus based on the initial stress of the sample [7]. The relaxation moduli demonstrated a linear relationship with the logarithm of time during the period of 60 min. Stress relaxation after 60 min averaged 8.0 (4.6 SD) MPa (n 7). The reduction of stress was 9.3% with standard deviation of 4.8% of initial stress. 4. Discussion
Fig. 3. Family of creep curves at different loads (100, 300, 500 and 700 N). The load was held constant for 90 min and strain was recorded for an additional 60 min after the load was removed (except for a 10 N `preload').
Dentin and bone have similar composition and mechanical properties [19]. In contrast to dentin, viscoelastic properties of both cortical bone and trabecular bone have been well characterized [20±28]. Other than demonstrating the
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Fig. 4. (a) Creep rate per hour vs. stress for all samples at all loads. (b) Creep compliance per hour vs. stress.
presence of creep behavior [6] and acoustic attenuation [13], only the stress relaxation behavior of dentin has been examined in detail [7±9]. The samples used in this study were similar to those in previous studies [7,8]. The samples are radially symmetrical. During preparation, the samples were prepared carefully to avoid artifacts due to dehydration, overheating and surface scratches. The dentin samples in the present study were obtained from root dentin, which can provide greater dimensions than coronal dentin. Dentin exhibits a gradation in properties, since the tubule orientation, density and diameter differ according to location [29]. The progressive deposition of peritubular dentin results in a decrease in diameter of the tubules toward the enamel or root surface [30]. A gradient has been shown in hardness and punch
shear strength from the pulpal surface to outer dentin [31]. Newer techniques such as AFM permit the measurement of properties on a much smaller scale [14,15]. A possible contribution of the ¯uid-®lled tubules to time-dependent properties requires the use of larger samples. The result of fracture tests showed values of proportional limit, ultimate compressive strength and Young's modulus (Table 1) are consistent with the results obtained by many authors [32±35]. In contrast, Craig and Peyton [6] and Kinney et al. [36] reported a relatively high value for Young's modulus, in the range of 16.6±18.5 GPa. No effect of repeated experimental loading on elastic properties was found in the present study. Understanding the viscoelastic response of dentin has important implications clinically, such as the response to
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491
Fig. 5. In¯uence of strain rate on elastic modulus for eight dentin samples, each subjected to six rates of loading.
occlusal load of intact or restored teeth, the effect of aging, and cavity preparation and the design of restorations. The most important phenomena of viscoelastic materials are continued deformation of a material under constant load or creep, and progressive reduction in stress while a material is under constant deformation or relaxation [10]. In the present study, dentin demonstrated both phenomena. Creep is widely recognized as important in engineering design [37]. In clinical function, creep behavior is potentially important during clenching or bruxing, with a reported duration of loading of 8±9 s [38±40] and according to one report up to 5 min [41]. Stress relaxation is signi®cant in clinical situations such as use of a threaded post in the root canal [42], placement of pins during restorations, and polymerization contraction of composite restorations [43±45]. Grimaldi and Hood [46] investigated cuspal de¯ection and recovery following load removal in extracted maxillary premolars. They reported delayed elastic recovery over a 10 min period in teeth with an MOD cavity preparation. Constant load following by removal of the load showed characteristics of viscoelastic materials with creep, elastic recovery, and retarded elastic deformation. Creep compliance (Jt) did not vary with stress (Fig. 4(b)). This result indicates that dentin is linearly viscoelastic, in a similar fashion to bone [47]. Young's modulus of dentin in this study was found to increase with faster strain rate, in a similar relationship to that found for bone. The data in Fig. 5 are highly consistent among samples, with similar slopes of all curves.
Curve ®tting indicated that the modulus is approximately proportional to the strain rate raised to the 0.017 power, which is at the lower end of the range reported for bone (0.018 [48] and 0.06 [20,21]). A low strain rate produces lower strength because the low strain rate allows more time for cumulative damage than a higher strain rate [20]. Part of the variability in reported values for Young's modulus in many studies [36] may be due to the differences in the rate of loading samples during testing. Over the range of strain rates used in this study, the modulus value varied by 2±3 GPa. Strain rates encountered clinically are unknown. In a preliminary study, we have estimated the strain rate in cervical dentin in four subjects, based on strain rates recorded in bucco-cervical enamel [17,18]. The measurements were limited to compressive strains recorded during forceful biting in approximately centric relation. Strain rates in bone have been measured directly, using strain gauges bonded to a human tibial diaphysis [49]. The strain rate during walking was 0.002 s 21 and during running was 0.01 s 21. The preliminary estimates that we recorded (0.0005±0.009 s 21) are lower than the range reported for bone. The wide range of strain rates used in the present study represents those strain rates encountered in daily activities such as chewing (0.004 s 21) as well as strain rates associated with impact trauma in accidents which will be much higher. Stress relaxation is the only area of viscoelastic behavior that has previously been well investigated in dentin, and was not considered in detail in this study. The stress relaxation modulus showed a linear dependence on the logarithm of
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Fig. 6. Stress relaxation curve of root dentin. (a) Stress relaxation curve for one sample: following a ramped load to 700 N in 3.5 s, the strain was held constant for 60 min. The load was then removed (retaining a 10 N load) and strain recorded for an additional 30 min. (b) Decrease in relaxation modulus over 90 min at constant strain, for the same sample.
time similar to that seen in previous studies [7,8]. The reduction of relaxation modulus in this study was 6±17% over 60 min, compared with 10±30% over 6 h reported by Duncanson and Korostoff [7]. The latter study was performed at 378C, compared with the present study at 20±228C. Dentin's modulus of elasticity decreased with increasing temperature [34], therefore it is stiffer at the lower temperature. Hence the relaxation modulus along with other viscoelastic properties of dentin may be also temperature dependent. Acknowledgements This investigation was supported by a grant from the National Health and Medical Council of Australia, Australian
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