Dentin is a viscoelastic tissue that contributes to the load distribution in human teeth and leads to their fracture resistance. Despite previous researches on the time-dependent behavior of dentin, it is not very clear whether the viscoelastic behavior of this tissue is linear or nonlinear, and what viscoelastic constitutive equations mechanically characterize it. Therefore, the aim of this study was to describe the viscoelastic behavior of human dentin and determine the best-fitting viscoelastic model for this tissue.
After preparation of human dentin specimens from 50 subjects, tensile stress relaxation tests were performed at 1%, 3%, 5% and 7% strain amplitudes. We first evaluated the viscoelastic linearity of this tissue and then fitted the experimental data using different constitutive models, namely, 2-, 3- and 4-term Prony series for linear viscoelasticity, Fung’s quasilinear viscoelastic model, and also Schapery and modified superposition models for nonlinear viscoelasticity.
Despite an almost linear trend at small strains up to 5%, the relaxation rate generally depended on strain amplitude, indicating some degree of nonlinearity in dentine viscoelasticity. According to the results of data fitting using different models, the modified superposition formulation could best capture the viscoelastic behavior of human dentin.
In this study, we have quantitatively examined the viscoelastic behavior of human dentin, using a large number of samples. We have obtained the coefficients of various viscoelastic formulations, which can be utilized in subsequent researches on human dentin assuming linear, quasilinear or nonlinear viscoelasticity for this tissue.
As a calcified composite tissue in human tooth, dentin is generally composed of mineral (hydroxyapatite) and organic (collagen) components, in addition to water content. The interactions between the mineral and organic components of dentin contribute to the elastic characteristics of this tissue. The water content of dentin and its connection to other constituents also define the mechanical behavior of dentin tissue, specifically its viscoelasticity. A proper knowledge of the mechanical properties of dentin allows forecasting the impacts of different factors such as loading conditions and aging on this tissue, and also provides a better insight about the influence of different restorative dental treatments [ , ]. The selection of restorative materials is highly influenced by their mechanical properties, since the mismatch between mechanical properties of dentin and restorative materials (either in terms of elastic or viscoelastic behaviors) causes stress concentrations at their interface by loading during physiological function of teeth such as mastication, thermal and occlusal loadings, abrupt impact, or during treatment conditions through polymerization shrinkage loadings [ , ]. In long term, this might lead to fracture due to fatigue loading, crack initiation and propagation in the restorative material, or failure in the interface bond [ , ]. Stress concentration at the dentin-material interface might be either due to elastic parameters or viscoelastic mismatch of time dependent viscous parameters, and even other parameters such as thermal capacity and heat transfer, as well as polymerization of some composite restorative materials during treatment [ , ].
The mechanical characteristics of the dentin tissue have been long the focus of many studies and different properties including shear and elastic moduli, viscoelastic properties, hardness, ultimate strength, fatigue and fracture properties have been examined [ ]. Compared to enamel, as a more brittle and yet harder tissue that mainly plays roles in grinding (crushing) the food and protecting the dentin tissue due to its superior wear resistance, dentin is associated with a greater force resistance and acts to absorb bite forces [ ].
The co-presence of viscous collagenous fibers and hard hydroxyapatite crystals, together with water content, lead to the time-dependent viscoelastic behavior of dentin tissue, which allows load distribution in the tooth and provides this tissue with fracture resistance [ ]. The spatial alterations in dentin microstructure has led some researchers to apply indentation methods to examine the viscoelastic properties of different regions in dentin [ , ]. For instance in a recent study, Montoya et al. have reported that the time-dependent deformation of coronal dentin depends on both composition (the ratio of mineral to organic components) and microstructure, and that the viscous deformation rate is higher in the regions closer to the tooth pulp [ ]. Although AFM based indentation methods provide compressive loading conditions, they are associated with some limitations, since loads are applied on the highly localized surface of the material in micro or nanoscales, which makes them less sensitive to the bulk properties of materials. Features such as the structural fibrous arrangement and internal caries of dentin are highly influential in mechanical characteristics of dentin. To better evaluate, further calculations have been proposed to combine the results obtained from separate measurements performed on the intertubular and peritubular regions of dentin, so as to obtain an overall result for the whole tissue [ ]. In general, compared to test methods in macro level, results of indentation tests on dentin are more sensitive to the existence of caries and extrinsic parameters such as the sample preparation approach and the hydration level of specimens [ ]. The tensile or compressive testing methods in the tissue level are performed on the bulk of the material as a continuum, and variation in the number and size of pores and caries results in different mechanical properties. However, in micro- and nano-indentation methods, due to the small scale of the indented site and application of a very localized force, the measured mechanical properties vary from site to site, depending on the local structure. While micro and nano indentation methods are limited in providing information for the bulk of the material (in tissue level), they provide superb information on the details of fibrous structures and their components.
In pioneer studies, the time-dependent behavior of dentin has been reported in the form of stress relaxation [ ], creep and hysteresis [ ] data. Some of these studies have been conducted on the demineralized dentin that is constituted by collagen, since it is used as the interface for bonding in the restorative dental operations that involve acid etching [ ] and may play a part in energy dissipation during such procedures [ ]. Since viscoelastic behavior is more attributed to the fiber interactions, the effects of viscoelastic behavior is more prominent on demineralized tissue, compared to non-fibrous mineral components.
Duncanson et al. have conducted compressive stress relaxation experiments and reported an almost linear viscoelastic behavior for radicular dentin [ ]. Viscoelastic linearity of mineralized [ , ] and also demineralized [ ] dentin tissues has also been reported in other studies. In contrast, Pashley et al. have observed nonlinear viscoelasticity in compressive creep experiments on demineralized dentin [ ] and Montoya et al. have reported that viscoelastic nonlinearity is higher in the coronal dentin, compared to the radicular dentin tissue, and attributed this difference to the microstructural aspects [ ].
Considering different approaches and models that have been utilized to study the viscoelastic behavior of human dentin, further investigations are still required to better clarify whether the viscoelastic behavior of the whole dentin tissue is linear or nonlinear, and also examine and compare different viscoelastic models to select a reliable model for dentin viscoelasticity. In the current study, we hypothesized nonlinear viscoelasticity for human dentin and investigated the extent of this nonlinearity. We also aimed at discovering to what level of loading, the viscoelastic behavior of dentin may be considered linear. We performed a comprehensive study using stress relaxation tests on 56 samples of human dentin and attempted to determine the constitutive viscoelastic theory that best describes this behavior considering linear, quasilinear and nonlinear assumptions. A precise information on the viscoelastic behavior of dentin can be assistive in the study of stress analysis in human teeth under biological functions such as mastication, as well as under mechanical environment including impact, and application of restorative materials during treatment plans and their long term performance. In this regard, to describe the viscoelastic behavior of dentin, after performing stress relaxation tests on human dentin specimens at various strain levels, we examined the nonlinearity of the viscoelastic behavior of human dentin. The obtained experimental data were used to find the best-fitting model for dentine viscoelasticity, as an appropriate viscoelastic model. The quasilinear viscoelastic model together with the Schapery and also modified superposition models of nonlinear viscoelasticity were employed, in addition to the linear case, to better evaluate dentine viscoelasticity. The well-known quasilinear viscoelastic theory has been introduced in the analysis of viscoelastic behavior of biological tissues to accommodate the well-established nonlinearity in the elasticity of tissues, while assuming a linear behavior for the viscous domain. More complex nonlinear viscoelastic models have been introduced to include nonlinearity in both elastic and viscous domains. These models have been previously used to describe the viscoelastic behavior of some biological tissues. For instance, the quasilinear viscoelastic model has been widely used to describe the viscoelasticity of different tissues such as articular cartilage [ ], ligament [ ], etc. The modified superposition and Schapery models have been both able to appropriately describe the viscoelastic behaviors of tendon and ligament [ ] and also tracheal cartilage, smooth muscle and connective tissue [ ].
Methods and materials
This in vitro experiment used 50 intact human premolars and third molars with equal number for each. Teeth were extracted from healthy young subjects whom the extraction of teeth was in their treatment plan because of space management and malposition, and were referred to the dental clinic of School of Dentistry, Tehran University of Medical Sciences. Informed consent was obtained from the donors according to guidelines of the Medical Ethics Committee. After extraction, teeth were cleaned from blood with saline solution, and soft tissues and debrises were removed using hand scaler and bristle brush. Since dentin has a complex microstructure, location, density and direction of the dentinal tubules and the direction of the collagen fibers can influence the results of mechanical testing [ ]. Therefore, in this study, the protocol for disinfecting and storing the teeth followed the ISO 11405 standard instructions and further by inspecting teeth under ×10 magnification and only specimens with no caries, no fracture and no extensive wear were included in the study. Furthermore, all specimens were obtained from transverse section of occlusal dentin, and prepared samples were assigned to different groups, randomly. Teeth were disinfected by placing them in a 0.5% chloramine T solution for one week followed by storing in distilled water at 4 °C not more than two weeks.
In order to prepare dentin specimens, teeth were embedded in the polyester resin by root to facilitate holding the sample during cutting, then positioned in the cutting machine (Mecatome T 201 A, Presi, France) and transversally sectioned at upper one third of the crown using a diamond disc with running water as the coolant. The first cut aimed to remove the occlusal enamel. After the first section, each tooth was examined under ×10 magnification to ensure removal of all occlusal enamel. Then subsequent cuts were placed to obtain disk specimens from dentin samples with 1.5 mm ± 0.1 mm thickness ( Fig. 1 (A)). All samples were prepared from the same region of teeth for maximum similarity of the tubular micro-structural arrangements.
Subsequently, an I-bean pattern (dumbbell shaped) was drawn on the dentin discs and cut out using a high-speed handpiece with diamond bur and water spray coolant. Due to the size and age of teeth, one specimen per tooth was expected to attain; however, some larger molars yielded two disks. Fig. 1 (B) illustrates the shape and dimensions of specimens. Sample dimensions were in accordance with the order of previous data on tensile tests on human dentin samples [ ]. Each I-beam dimensions were measured three times at upper, mid and lower part of the narrow part of the dumbbell (a total of nine measurements) using digital caliper with the accuracy of 0.01 mm (Mitutoyo C500-196-30, Japan), and the mean of measurements was used for the calculations. The prepared specimens were randomly assigned into test groups. After removal of inspected samples with damage or high variation in dimension, and obtaining two samples from larger third molar teeth, a total of 56 samples were tested.
The intermediate region of the specimen, which would be subjected to tension during the test, had a length to diameter ratio about 2.5, as previously suggested [ ], to allow an almost uniform stress distribution throughout the section under loading.
Tensile stress-relaxation experiments were performed using a Zwick testing machine (Z250, Zwick GmbH & Co. Germany) at 25 °C. Considering the small sizes of dentin specimens, a suitable pair of fixtures was required for appropriate sample holding during the tests. The required fixture was designed in CATIA V5 R ( Fig. 1 (C, D)). After design optimization based on sample dimensions and fixation, steel fixtures were fabricated. The schematics of the fixtures are illustrated in Fig. 1 (C).
At the beginning of each experiment, sample dimensions (length and cross section area) were measured using a digital caliper. Samples were fixed to the grippers with an average gauge length of 4 mm. The gripping parts of the fixture were not polished to make firm gripping without movement of the samples during testing. An initial preload of 0.5 was applied to ensure fixed holding of samples with no gaps or slipping. Samples were observed carefully during testing to avoid slipping which is shown in obtained data with unusual sharp changes. A load cell of 50 N (with a tolerance of 0.01 N) was used and the displacements were measured by the software of the device using the gauge length as the input. The strain was calculated accordingly through the displacement of the grippers and the initial length.
Stress relaxation tests were performed on each dentin sample using strain amplitudes in the order of 1%, 3%, 5% and 7% applied through ramp loading with a 0.015 mm/s strain rate and a relaxation period of 5 min at each strain amplitude, during which the force measured by the load cell was recorded [ ]. Prior to the next relaxation experiment on each specimen, the sample was removed from the fixtures and left to recover for 60 min, more than 10 times longer than the relaxation duration [ ]. As previously suggested, during all stress relaxation tests and subsequent recoveries, dentin specimens were kept hydrated by immersing in the deionized water [ ]. During consecutive tests on each sample, samples were investigated for surface cracks and if any crack had developed during a test, the sample was not subjected to the next test with higher loading. Furthermore, the force data of each test were checked for sudden changes, since rapid crack propagation, either internal or superficial, causes sudden drop in the measured load. Before stress relaxation tests, the tensile tests were performed on a limited number of prepared samples, to measure failure strains, to make sure that the maximum applied strain is not within the scope of fracture. Experimental results (force vs. time data) were used to obtain engineering stress vs. time data via dividing the force data by the cross section area of the sample, considering dentin as a hard tissue undergoing small deformations. At each strain amplitude, the average relaxation data were obtained for the subsequent analysis. Stress relaxation data were presented as mean ± STD.
In linear elastic materials, the constant modulus of elasticity (E) is defined by the slope of the stress-strain (σ-ε) line. In viscoelastic materials, the relationship between stress and strain depends on time, being indicated by a relaxation function. In order to describe the viscoelastic behavior of materials, different models of linear, quasilinear and nonlinear types have been previously proposed. In the present study, these models have been used to select the best-fitting model for viscoelastic characteristics of the studied human dentin samples. In this regard, the average stress relaxation curve at each strain amplitude was fitted to the linear, quasilinear and nonlinear models of viscoelasticity. Fitting was carried out in the Matlab software via the curve fitting toolbox, based on the Levenberg-Marquardt algorithm. The coefficient of determination, R 2 , was used as a measure for the goodness of fitting. Parameter R 2 varies between 0 to 1, such that an R 2 level closer to 1 indicates a better fitting.
Viscoelastic linearity examination
We obtained isochronal stress-strain points at 0.5, 5, 10, 30, 50, 100, 150, 200, 250 and 300 s, to examine the viscoelastic linearity of human dentin. In order to quantitatively check the linearity of dentin viscoelasticity, a power law (σ = At n ) was used [ ] for fitting the experimental relaxation data at the studied strain amplitudes (i.e. 1%, 3%, 5% and 7%). In the obtained log-log plot of the relaxation (stress vs. time) data, the slope of each line equals the rate of relaxation, which is indeed the exponent of the employed power law, i.e. n. We performed statistical analysis to check whether n depends on strain magnitude, implying viscoelastic nonlinearity. We used Q-Q plots and Shapiro-Wilk test to examine the distribution normality of the experimental data sets. Then, analysis of variance (ANOVA) was utilized to compare n levels at the applied strains to examine whether the rate of relaxation depends on the strain level, which is an indication of viscoelastic nonlinearity.
To model linear viscoelasticity, the Boltzmann superposition integral for the stress relaxation was used:
where, E(t), σ(t), ε(t) and τ represent the relaxation function, tensile stress, the applied strain and integration variable, respectively [ ]. Here, we used Prony series for the relaxation function and with the assumption of strain as a step function, the linear viscoelasticity was formulated as the following:
In which τ i represents the relaxation times. The average relaxation data were fitted by the 2-, 3- and 4-term Prony series at each strain level, to find the most appropriate Prony series for dentin tissue, assuming a linear viscoelasticity case (see [ ] for more details).
Since the theory of linear viscoelasticity was not able to adequately capture the nonlinear viscoelastic response of many tissues, Fung proposed the quasilinear viscoelastic (QLV) model, which has been widely applied to describe the time dependent behavior of various tissues. The main advantage and popularity of this model is due to its basic assumption of decoupling the time and dependent components of the relaxation function. In the quasilinear viscoelastic (QLV) model [ ], the relaxation function (E(ε,t)), is expressed based on the product of two functions of time and strain, i.e. E t (t) and G(ε), respectively. Therefore, considering strain as a step function, stress, which depends on both time and strain, can be expressed as the following: