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Modeling of the viscoelastic behavior of dental light-activated resin composites during curing
dental materials Dental Materials 19 (2003) 277±285
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Modeling of the viscoelastic behavior of dental light-activated resin composites during curing Bibi S. Dauvillier*, Maxim P. Aarnts, Albert J. Feilzer Department of Dental Materials Science, Academic Centre of Dentistry Amsterdam (ACTA), Louwesweg 1, NL-1066 EA Amsterdam, The Netherlands Received 16 January 2001; revised 25 August 2001; accepted 4 December 2001
Abstract Objective. Three models consisting of springs and dashpots were investigated to describe the viscoelastic behavior of a commercial lightactivated restorative composite during curing. Methods. Stress±strain data on Z100 were recorded by means of a dynamic test method performed on a universal testing machine. The model was tested by matching its response to experimental data and the material parameters, E (Young's modulus) and h (viscosity), associated with the model were calculated. Results. The universal testing machine generated reliable stress±strain data on the fast curing, light-activated resin composite during curing. The high polymerization rate of Z100 had a negative effect on the viscous ¯ow capability of the material. A predictive model of the viscoelastic behavior of Z100 during curing was carried out, using the Maxwell model for the initial 3 min in the curing process and the Kelvin model for the remainder of the process. Signi®cance. Dental researchers analyzing shrinkage stress problems by mathematical modeling can obtain a good quantitative estimate of the shrinkage stress development of Z100 before the restoration is actually made. q 2003 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. Keywords: Viscoelasticity; Mathematical modeling; Shrinkage stress; Photocuring; Dental composites
1. Introduction Shrinkage stresses generated in dental resin composites during curing are among the major problems in adhesive dentistry, since they interfere with the integrity of the restored tooth. For dental restorative applications, light-activated resin composites have replaced chemically activated resin composites, mainly because of their rapid polymerization reaction and the greater freedom in timing the initiation of polymerization. This `cure on command' allows the dentist to place and contour the restorative material with ease. However, it has been demonstrated that under similar test conditions, light-activated resin composites generate higher polymerization shrinkage stress than the analogous chemically activated composites [1]. This striking difference in shrinkage stress development is caused not only by the difference in shrinkage development, but also by the mechanical behavior of the curing composite, which is by nature viscoelastic [2]. * Corresponding author. Tel.: 131-20-5188235; fax: 131-20-6692726. E-mail address: [email protected] (B.S. Dauvillier).
In recent years, considerable attention has been given to the use of mechanical models to describe the viscoelastic behavior of dental restorative composites during curing [3± 5]. In these studies, several linear viscoelastic models were investigated by applying a modeling procedure to experimental stress±strain data. In addition, the material parameters associated with the model were calculated. The results of these studies contribute to a better understanding of the shrinkage stress problems associated with adhesive composite restoration. Moreover, recent initial modeling studies have shown that both the shrinkage strain, which is associated with the polymerization reaction and is affected by temperature, and the stress ®eld as well, can vary greatly within the restoration preparation [6,7]. Hence, by combining the results from temporal shrinkage strain and mechanical models, the stress ®eld in and around the restoration can be analyzed using ®nite element analysis methods. Up to now, modeling studies have focused on the viscoelastic behavior of chemically activated resin composites. Through introducing adjustments to the specimen mounting device in a specialized universal testing machine, it is possible to monitor the mechanical behavior of light-activated
0109-5641/03/$20.00 + 0.00 q 2003 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. PII: S 0109-564 1(02)00041-6
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Fig. 1. Experimental setup for light-activated resin composites in dynamic test method. An oscillatory axial strain with amplitude 1.00 ^ 0.01 mm and frequency of 1.0 and 0.1 Hz was successively applied to the curing composite around its original height (strain 0%) in experiment (left). In the stress response the sinusoidal stress is superimposed on the shrinkage stress (right).
resin composites during curing [8]. Stress±strain data recorded with a dynamic test method were used to perform a modeling study of the viscoelastic behavior of this type of dental composites. The aim of this study was to identify a mechanical model for the viscoelastic behavior of light-activated resin composites during curing. Experimental stress±strain data were generated with a dynamic test method in which a vertical oscillatory strain was applied to a commercial light-activated resin composite. Three linear viscoelastic models (Kelvin, Maxwell, and Standard Linear Solid) were investigated by means of a validated modeling procedure [5]. On the basis of the modeling and evaluation of results, a suitable model was chosen for light-activated resin composites. 2. Materials and methods 2.1. Dental resin composite The material utilized in this investigation was a commercial light-activated resin composite (Z100 MP A3, LOT: 19981009, 3M Dental Products, St Paul, MN 55144-1000, USA). According to the manufacturer's instructions, the resin composite was light-cured for 40 s (Elipar Highlight, standard mode, ESPE, Seefeld D-82229, Germany). The light intensity at the light exit tip was 600 mW/cm 2 (radiometer, model 100, Demetron Research Corp., Danbury, CT 06810-7377, USA). 2.2. Dynamic test method Stress±strain data on the light-activated resin composite during curing were obtained by means of a dynamic test method performed on a custom-made automated universal testing machine (ACTAIntense, ACTA, Department of Dental Materials Science, Amsterdam NL-1066 EA, The Netherlands). The composite was bounded between a steel disk, which was connected to the crosshead with the load cell (1 kN), and a rectangular glass plate (¯oat glass, Bakker, J & I, Castricum, The Netherlands, 50 £ 50 £ 4 mm 3),
which was connected to the stationary part of the framework (Fig. 1). To ensure the cylindrical shape of the specimen, the composite was inserted into a lightly greased te¯on mold (d 3.1 mm, L 1.6 mm), creating a C-factor for the specimen of 1.0 (d/2L). The steel disk was moved down until it reached the pre-adjusted specimen height. Optimal bonding between the resin composite and the steel disk was achieved by coating the sandblasted surface (Korox w 50 mm, Bego Bremer GoldschlaÈgerei Wilh. Herbst GmbH & Co., 2 min/5 bar pressure) of the disk with silane (Silicoater, Heraeus Kulzer GmbH, Wehrheim D-6393, Germany, 5 min). For the same reason, the glass surface was lightly sandblasted (Korox w 50 mm, Bego Bremer GoldschlaÈgerei Wilh. Herbst GmbH & Co., 10 s/5 bar pressure), primed (RelyX ceramic primer, 3M Dental Products, St Paul, MN 55144-1000, USA), and ®nally coated with a pressurized air-spread adhesive layer (Scotchbond Multipurpose, 3M Dental Products, St Paul, MN 55144-1000, USA). The submicron-thick adhesive layer was lightcured for 40 s (Elipar Highlight, standard mode, ESPE, Seefeld D-82229, Germany). In the dynamic test method, the upper steel disk performed an oscillating vertical sinusoidal deformation on the curing resin composite around its original height (1.6 mm). The test method was programmed to perform two frequencies with amplitude of 1.00 ^ 0.01 mm (0.0625% strain). First, a frequency of 1.0 Hz was applied for 50 s to the fast curing composite, followed by a frequency of 0.1 Hz until termination of the measurement (1 h). The oscillatory deformation was measured with two displacement probes (Solartron LVDT type AX/1/S, Dimed, Antwerpen B-2140, Belgium) at the level of the specimen. The signal of the displacement probes was continuously monitored and fed back into a control loop, thereby ®netuning the movement of the crosshead on the specimen. The light irradiation process was measured with a custom-made light sensor device at the level of the specimen. The distance between the light exit tip was equal to the thickness of the glass plate (Fig. 1), which was 4 mm. During the measurement, the data (time, load, and displacement signal) were
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collected simultaneously by a data acquisition console at a rate of one hundred points (1.0 Hz period) and 18 points (0.1 Hz period) per seconds, respectively. The measurements were started 5 s prior to the light irradiation process and were repeated three times at room temperature (23 ^ 1 8C). One hour after the start of the experiment, the specimen was subjected to tensile loading with a crosshead speed of 60 mm/min until fracture. To verify the polymerization ef®ciency of the resin composite, the cylindrical specimen was fractured and exposed to Astra blue dye test according to the procedure described by De Gee et al. [9]. 2.3. Volumetric shrinkage measurement During the dynamic test measurement, the axial shrinkage strain of the specimen was not measured, because the oscillatory deformation was performed around the original height of the specimen. However, the displacement caused by axial shrinkage must be taken into account when modeling the stress data recorded by means of the dynamic test method. For this reason, volumetric shrinkage measurements (n 3) were performed by means of a mercury dilatometer at 23 ^ 0.1 8C, using the procedure described by De Gee et al. [10]. 2.4. Stress±strain data analysis The data obtained from the dynamic test measurement consisted of an array of load and displacement values for many points in time. The sinusoidal strain (1 sine) and normal stress (s exp) were calculated using the following equations
1 sine
DL L0
1
s exp
F A
2
279
2.5. Mechanical models The mechanical behavior of the resin composite during curing was considered linear viscoelastic [13], because the strain applied to the specimen was within the limit of linear viscosity of polymer-based materials (,0.5%). Linear viscoelasticity can be described using mechanical models consisting of springs, which obey Hooke's law and dashpots, which obey Newton's law of viscosity. The mechanical models were investigated in one dimension only, because the stress±strain data were monitored in one direction. The shrinkage behavior of the composite is considered to be isotropic, as the specimen was made as high as possible without exceeding the depth of cure (2 mm) [14]. The models must be kept simple because by using uni-axial data, only a restricted number of material parameters can be ®tted in a unique way. As a consequence, the validity of the qualitative and quantitative viscoelastic behavior studied is con®ned to the stress and strain range and the strain rate range covered by the experiment. In this study the Maxwell, Kelvin, and Standard Linear Solid model were investigated. The Maxwell model consists of a spring and a dashpot in series. It represents a viscoelastic liquid because the deformation of the dashpot is permanent in the sense that it does not recover upon removal of an external load. The Kelvin model consists of a spring and a dashpot in parallel. As the deformation process is reversible, with a time delay caused by the dashpot, this model represents a viscoelastic solid. Finally, the Standard Linear Solid model consists of a spring and a Maxwell model in parallel. The model is able to exhibit a pure viscoelastic liquid property, as well as a pure viscoelastic solid property. The physical understanding and differential equations governing the behavior of these viscoelastic models are well documented in literature [15,16]. 2.6. Modeling procedure
in which DL is the displacement value measured by the probes (m), L0 the original specimen height (m), A the cross-sectional area of the specimen (m 2), and F is the recorded load response of the specimen (N). In addition to the applied sinusoidal strain, the strain caused by axial shrinkage must be taken into account when modeling the stress data as recorded by means of the test method. The axial shrinkage strain development of the resin composite (C-factor 1.0) in the dynamic test method was calculated by multiplying the mean volumetric shrinkage strain by a factor of 0.45 [11]. The functional expression of the axial shrinkage strain (1 shrinkage) was calculated by means of a cubic spline ®t [12], and the axial shrinkage strain was added to the oscillatory strain (Eq. (1)) for all the points in time of the dynamic test measurement, which resulted in the desired strain (1 tot) for the modeling procedure:
A validated modeling procedure was developed which is capable of evaluating a mechanical model and calculating its material parameters, E (Young's modulus) and h (viscosity), from experimental stress±strain data. The modeling procedure is described in detail by Dauvillier et al. [5], and only a brief description of the method will be given here. The modeling procedure was performed on sinusoidal stress cycles isolated from the stress data recorded by the dynamic test method. The time span (1 or 10 s) of the isolated interval was kept small with respect to the rate of polymerization reaction. As a result, the mechanical behavior of the composite can be assumed to be constant during the isolated interval, which justi®es the use of simple differential equations for the mechanical models. These equations were solved analytically, allowing the stress to be expressed as a function of strain and unknown material parameters, with the known functional form of the strain (Eq. (3))
1tot 1sine 1 1shrinkage
1 t At 1 B sin vt
3
4
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Fig. 2. Stress (F) and strain (A) data of Z100 with C-factor 1.0 collected with the dynamic test method. For clarity, only the initial 75 s of the curing reaction is shown. The total strain on the curing resin composite consists of axial shrinkage (C), derived from volumetric shrinkage measurements, and oscillatory strain (A), applied by the crosshead (note different y scale). For the modeling procedure, the total strain curve was constructed by a linear combination of (A) and (C), resulting in (E). The shrinkage stress (D) and dynamic stress (B) response were isolated from experimental stress data (F) via FFT smoothing and substraction (F±D), respectively. The dotted lines represent the initiation and termination of the irradiation process of the light unit.
in which 1 is the strain, t the time in isolated interval (0±1 or 0±10 s), A the slope of the shrinkage strain (s 21), B the amplitude, and v is the angular frequency (rad s 21) of the oscillatory strain. The shrinkage strain in the isolated time interval was assumed to be linear in time. Except for the Kelvin model, the initial condition (s (0)) for the stress equation of the model was taken from the shrinkage stress data, thereby avoiding the extensive computation involved in evaluating the
initial stress mathematically. By means of the initial parameter values, a least square method was performed at equidistantly spaced k points in time of the isolated interval, to assess how well the model stress (s model) approximates the experimental stress of the interval (s exp):
d
k X i1
s model ti 2 s exp ti 2
5
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Fig. 3. Modeling results for two stress cycles of Z100 during curing at time 4.58 s (A) and time 10.59 s (B) for the Maxwell model (top), Kelvin model (middle) and Standard Linear Solid model (bottom).
The search for the ®nal material parameters values was carried out by minimizing the residual (d ) with an optimization routine based on the Levenberg±Marquardt method [17]. The modeling procedure provides (I) the parameters, (II) the error estimates on the parameters, and (III) the residual (d ), a quantitative measure of the difference between experimental and model stress. The algorithms for the modeling procedure were implemented in MATLAB (version 5.3, The MathWorks, Inc., Natick, USA) on a desktop computer (Windows q 98 platform). 2.7. Evaluation of the viscoelastic model To evaluate how well the model predicts the real shrinkage stress development of Z100, each model was loaded for small intervals in curing time with its mean material parameters, as calculated by analyzing the stress intervals with the modeling procedure, and axial (linear) shrinkage strain. The calculated stress value at the end of each interval was taken as the initial stress condition for the next interval. In addition, an evaluation was performed in which the Maxwell model was used for the initial 3 min in the curing process and the Kelvin model for the remainder of the curing process. The shrinkage stress develop-
ment of Z100 in the experiment was isolated from the recorded stress data (Eq. (2)), using the standard Fast Fourier Transform (FFT) smoothing ®lter in Origin (version 5.0, Microcal Software, Inc., Northampton, MA 01060, USA). The best result was obtained by taking 200 points for smoothing. 3. Results Fig. 2A and F show the applied strain and stress response of Z100 during the initial 75 s of the curing process at room temperature. The positive strain of the sinusoidal cycles represents the crosshead displacement away from the specimen, while the negative strain represents the crosshead displacement towards the specimen. As the oscillatory strain was performed around the original specimen height, the stress response on the oscillatory strain was superimposed on the continuous shrinkage stress development of the specimen. There was no premature debonding from either the glass plate or the steel disk, because subsequent to tensile loading the fracture always occurred in the glass plate, at a stress level 1.5 times the maximum stress (18 MPa) measured in the test method. The mean axial
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Table 1 Material parameters for several cycles during one measurement of Z100 during curing with standard deviation in parenthesis. Material parameters: E(x), Young's modulus; h , viscosity; and d , quantitative measure of the difference between experimental and model stress Kelvin model
Maxwell model
Time (s)
E (GPa)
h (GPa s)
d
E (GPa)
4.6 10.6 20.6 40.6 302 1202 3352
0.24 (0.03) 1.96 (0.02) 3.55 (0.01) 4.70 (0.05) 5.45 (0.01) 6.32 (0.05) 6.57 (0.01)
0.02 (,0.01) 0.07 (,0.01) 0.11 (,0.01) 0.11 (,0.01) 0.75 (,0.01) 0.60 (,0.01) 0.56 (,0.01)
0.746 0.387 0.173 2.87 0.374 0.77 0.496
0.32 (0.02) 1.94 (0.06) 3.50 (0.09) 4.55 (0.09) 5.36 (0.04) 6.02 (0.05) 6.48 (0.04)
Standard Linear Solid model
h (GPa s) 0.57 (,0.01) 29.9 (,0.01) 124 (,0.01) 77.2 (,0.01) 3447 (,0.01) (,0.01) 5807 (,0.01)
shrinkage strain, as derived from the volumetric measurements, and its derivative are shown in Figs. 2C and 4D, respectively. The strain data used for the modeling procedure consisted of oscillatory strain superimposed on the axial shrinkage strain curve (Fig. 2E). The graphic results of the modeling procedure on two stress intervals isolated from one dynamic test experiment are shown in Fig. 3A and B. The continuous black line represents the measured stress, while the dots represents the values computed by the model, using the material parameter values as estimated by the modeling procedure. Table 1 and Fig. 4A±C show the estimated material parameter values for the three models for several stress intervals during one experiment. The evaluation results for the three models, and by using ®rst Maxwell (up to 180 s) and
d
E1 (GPa)
h (GPa s)
E2 (GPa)
d
0.318 3.43 7.37 6.54 5.33 5.27 4.60
0.39 (0.05) 2.28 (0.05) 3.38 (0.44) 4.76 (5.23) 1.59 (1.91) 1.40 (2.20) 1.33 (3.99)
0.10 (,0.01) 0.22 (,0.01) 2.01 (,0.01) 4.92 (,0.01) 7.73 (,0.01) 6.90 (,0.01) 7.51 (,0.01)
, 0.01 (,0.01) 1.85 (0.11) 1.33 (0.59) ,0.01 (5.39) 4.01 (2.15) .91 (2.46) 5.34 (4.34)
0.285 0.163 0.152 1.423 0.312 0.327 0.483
then Kelvin (remainder of cure process), are illustrated on two different time-scales in Fig. 5. Each point on the curves was generated by statistically averaging three individual experiments. 4. Discussion In the choice of a model for describing the viscoelastic behavior of dental resin composites during curing, there is always a trade-off between accuracy and level of mathematics. It will always be possible to develop a perfect model, which gives a perfect prediction of the material behavior by using a complicated mathematical formula, but this is hardly a sensible approach. What is required is
Fig. 4. Parameter values of the Maxwell model (A), Kelvin model (B), and Standard Linear Solid model (C) as a function of curing time of Z100 for one measurement. Error bars indicate the relative standard error in the estimated parameter value. The axial shrinkage strain rate development (D) was derived from volumetric measurements.
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Fig. 5. Axial shrinkage stress development (Ð) measured; (B) Kelvin model; (O) Maxwell model; (V) Standard Linear Solid model; and (X) Maxwell and Kelvin) during the initial 10 min (A) and 1 h (B) curing of Z100. Error bars indicate the relative standard error in the calculated mean (n 3).
a simple model, which will produce an adequate prediction. In this study, the choice of the model and the calculation of its material parameters were determined on the basis of matching the model prediction to experimental data. Therefore, the success of this modeling study depends on the quality of the experimental data. In the course of the research on modeling the viscoelastic behavior of chemically activated resin composites, improvements were made to certain aspects of the dynamic test method which are important in obtaining reliable experimental data [5]. To achieve the same reliability for light-activated resin composites, several modi®cations to the mounting device were necessary. The position of the light source was established by placing it at one end of the cylindrical specimen, thereby simulating a method of curing which is thought to be clinically feasible. The choice of the light source, the duration of light irradiation process, and the distance between the light exit tip to the specimen was found to be adequate for the complete polymerization of Z100, as the Astra blue dye tests on the specimens revealed no visible staining inside the semi-cylindrical surface of the resin composite. Furthermore, the position of the light sensor device was adequate to detect accurately the initiation and duration of the light irradiation process. Finally, the strain applied in the early stage of curing, when the shrinkage rate is high and the mechanical behavior of light-activated resin composites changes rapidly from soft to hard, remain a sinusoid (Fig. 2A). In this way the differential equations of the models could be solved analytically, thereby greatly reducing computational time involved in the modeling procedure. The amplitude of the strain was fractional higher than the preset value (0.0625%), because the feedback system overcompensated the crosshead movement on the specimen as it became stiffer. The higher strain amplitude did not, however, lead to errors in the modeling results, because it was not the preset, but the measured oscillatory sine that was utilized in the modeling procedure. The light irradiation of Z100 was initiated after at least four oscillatory strain cycles has been applied to Z100, because laboratory tests on a control specimen revealed that the experimentally
applied sinusoid, which has a starting point, converges within four cycles to a mathematical sinusoid as described by Eq. (4). An interesting feature of Z100 is that in the initial 2 s of the curing process, the material undergoes 15% of the measured axial shrinkage strain without generating shrinkage stress (Fig. 2C and D). The relationship between shrinkage stress and shrinkage strain displayed by this commercial dental composite contrasts with that of a commercial chemically activated composite where, under closely similar experimental conditions and up to 4 min after the start of polymerization, a large proportion of composite shrinkage (50% or more) failed to contribute to stress development in the material [4]. Although the chemical composition of these commercial resin composites differs, it may be concluded that in general an increase in the rate of polymerisation of resin composites has a negative effect on the viscous ¯ow capability of the material. The viscosity (h ) values of all models, as calculated by analyzing the stress cycles by means of the modeling procedure, were all positive and developed according to the spring-dashpot arrangement in the model with curing time (Table 1, Fig. 4A±C). The Young's modulus (E) values were also positive and increased monotonically with the curing time. The Young's modulus of Z100 after 1 h curing (approximately 6.5 GPa) is not in agreement with the value of 13 GPa provided by the manufacturer. It is known that factors such as the choice of mechanical model, test methods (bending, shear, compression, tension), conditions of the test method (strain rate, curing time, temperature), and light irradiation procedure (light intensity, duration time) have a considerable effect on the ultimate Young's modulus value of viscoelastic resin composites. The Young's modulus of viscoelastic materials generally decreases with a decreasing strain rate [18], as evident in the strain rate transition from 1.0 to 0.1 Hz (Fig. 4A±C). For a valid comparison between Young's modulus values, the variables described earlier must be the same. Any difference between these variables will make comparisons between Young's moduli impossible.
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The plots of two time intervals to which the modeling procedure was applied are shown in Fig. 3. All models failed to predict the experimental stress in the early stage of curing (4.58 s). The lack of modeling capability is not caused by the low signal-to-noise ratio (SNR 2.40) of the experimental stress data. A previous study showed that when the viscoelastic behavior of the resin composite was exactly the same as that of a model, then it was possible to model very precise experimental stress data with practical SNR values as low as 2.20 [5]. All models failed to predict the stress a few seconds after the initiation of polymerization, because the material parameters were kept constant over an interval of time, whereas in reality the material stiffness increases noticeably, as indicated by the asymmetrical sinusoid around the shrinkage stress line (Fig. 3A). Better modeling results should be obtained by isolating smaller time intervals in the stress data, i.e. 0.50 s or less, or by using differential equations for the model in which material parameters vary in time. For further studies, the former is recommended, since solving partial differential equations involves intensive computation work. The Maxwell stress curve at cure time 10.59 s is a poor approximation of the stress response of the dental composite. This is clear not only from the graphical ®t, but also in the d parameter (Table 1)Ðthe quantitative measure of the difference between experimental and model stressÐwhich is the highest value of all models. The phase lag between the model stress curve and the experimental stress curve indicates that the model predicts more viscous ¯ow than the material actually undergoes. As a result, this model predicts for the ®rst 3 min of curing a lower stress development than Z100 generates in the experiment (Fig. 5A). This result is essentially different from chemically activated resin composites, where the Maxwell model predicts the shrinkage stress better and for a longer period in curing time (up to 11 min) [5]. Obviously, high polymerization rates reduce the ability of the composite to ¯ow permanently. This may explain why light-activated resin composites generate higher polymerization shrinkage stresses than analogous chemically activated composites [1], because composite shrinkage is less compensated by permanent viscous ¯ow from the unbonded, outer surface of the material. In the remainder of the curing process, the Maxwell model predicts too much stress relief, which is obviously not the case for Z100 in the experimental situation (Fig. 5B). The better modeling results achieved with the Kelvin and Standard Linear Solid model con®rms that the behavior of light-activated resin composites is more viscoelastic solid-like (reversible ¯ow) than viscoelastic liquid-like (permanent ¯ow). The slightly better approximation by the Standard Linear Solid model, where the development of E2 indicates reversible ¯ow behavior, can be explained by the extra parameter it contains in comparison with the Kelvin model. A closer look at the sinusoid curve at 10.59 s reveals that the Kelvin model responds somewhat more stif¯y in the ®rst section and slightly softer in the
second section of the curve than Z100 in the measurement. As stated for the Maxwell model, this can be explained by the fact that the material parameters were kept constant in time, whereas in reality the material increases in stiffness over time. The small error introduced by assuming constant parameters counterbalanced the intensive computations required by models with timedependent material parameters. Up to 25 s into the light irradiation procedure, the response of the Kelvin model closely resembles the shrinkage stress response of Z100 in the measurement (Fig. 5A). After 25 s, the model generates higher shrinkage stresses, leading ultimately to a shrinkage stress level that is a factor of two higher than measured for Z100 during the experiment (Fig. 5B). There may be two explanations, one or both of which may be responsible for the failure of the Kelvin model in the remainder of the curing process. First, the ¯ow behavior of light-activated resin composite could be non-Newtonian; i.e. the viscosity is not constant, but depends on the strain rate. For this reason, the viscosity values of the Kelvin model may not be valid in this part of the curing process, where the shrinkage rate is close to zero (Fig. 4D), since the values were calculated from experimental stress that depended exclusively on the oscillatory strain (Fig. 2F). Secondly, the failure of the Kelvin model can perhaps be accounted for by the fact that the material undergoes two deformation processes, namely reversible, by viscous ¯ow as predicted by the model, and permanent, by imperfections in the materials, such as voids, crazes, and microcracks [19,20]. These imperfections are usually generated when local stress spots in the material exceed inter-atomic bond strength within the polymer and/or polymer-®ller interface and are likely to be present in light-activated resin composite, where high shrinkage stress developed during curing (Fig. 2D). It is dif®cult to develop a model that is capable of taking into account these two processes, which occur simultaneously but are different in origin. Theoretically, the Standard Linear Solid model is able to describe both the viscoelastic liquid (E2 0) and the viscoelastic solid behavior of resin composites during curing. On a practical level, the evaluation results of the model reveal that the viscoelastic behavior of Z100, as excited by the conditions of the dynamic test method, cannot be adequately predicted by this model. A close look at the error development in the E1 and E2 parameters at curing time 20 s and later con®rms that there is a basic problem in modeling the stress using the Standard Linear Solid model. Fig. 4C and D show that there is a clear relationship between the value of the standard error in E1 and E2 and the shrinkage rate pro®le. As the shrinkage rate declines, and the contribution of shrinkage strain to the applied strain deteriorates as the curing process continues, the stress response becomes more exclusively dynamical, i.e. more dependent on the sinusoidal strain of one frequency alone. In
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this situation the values of the E1 and E2 parameters associated with the Standard Linear Solid model cannot be distinguished from one another, as evidenced by the high error value, because with this type of stress response no more than two independent parameters can be determined. To obtain reliable E1 and E2 values in the remainder of the curing process, the stress to be modeled must be generated by a multi-wave strain, which entails sophisticated dynamic test conditions. This approach requires modi®cations in the application software, which, however, has not been established. 5. Conclusions The automated universal testing machine developed for the dynamic testing of dental restorative material proved capable of generating stress±strain data on light-activated resin composites. The use of a light sensor device made it possible to accurately detect the initiation and duration of the light irradiation process, so that errors in assigning stress±strain data to curing time were reduced to a minimum. An increase in the polymerization rate has a negative effect on the viscous ¯ow capability of dental resin composites. Adequate predictive modeling of the viscoelastic behavior of Z100 can be carried out by using the Maxwell model for the initial 3 min of the curing process and the Kelvin model for the remainder of the curing process. Acknowledgements The authors are grateful to Pierre HuÈbsch for his guidance in mathematical modeling, Prem Pallav for the design and production of the ACTAIntense, Joost Buijs for writing the application software, and Barbara Fasting for her comments on the English text. This investigation was ®nancially supported by the Netherlands Institute for Dental Sciences.
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References [1] Feilzer AJ, De Gee AJ, Davidson CL. Setting stresses in composites for two different curing modes. Dent Mater 1993;9:2±5. [2] Feilzer AJ, De Gee AJ, Davidson CL. Quantitative determination of stress reduction by ¯ow in composite restorations. Dent Mater 1990;6:167±71. [3] HuÈbsch PF. A numerical and analytical investigation into some mechanical aspects of adhesive dentistry. University of Wales, 1995. [4] Dauvillier BS, Feilzer AJ, De Gee AJ, Davidson CL. Visco-elastic parameters of dental restorative materials during setting. J Dent Res 2000;79:818±23. [5] Dauvillier BS, HuÈbsch PF, Aarnts MP, Feilzer AJ. Modeling of viscoelastic behavior of dental chemically activated resin composites during curing. J Biomed Mater Res (Appl Biomater) 2001;58:16±26. [6] Pananakis D, Watts DC. Incorporation of the heating effect of the light source in a non-isothermal model of a visible-light-cured resin composite. J Mater Sci 2000;35:4589±600. [7] Versluis A, Tantbirojn D, Douglas WH. Do dental composites always shrink toward the light? J Dent Res 1998;77:1435±45. [8] Algera TJ, Feilzer AJ, De Gee AJ, Davidson CL. The in¯uence of slow-start polymerization on the shrinkage stress development. J Dent Res 1998;77(685):429. [9] De Gee AJ, ten Harkel-Hagenaar E, Davidson CL. Color dye for identi®cation of incompletely cured composite resins. J Prosthet Dent 1984;52:626±31. [10] De Gee AJ, Davidson CL, Smith A. A modi®ed dilatometer for continuous recording of volumetric polymerization shrinkage of composite restorative materials. J Dent 1981;9:36±42. [11] Feilzer AJ, De Gee AJ, Davidson CL. Increased wall-to-wall curing contraction in thin bonded resin layers. J Dent Res 1989;68:48±50. [12] Schultz MH. Spline analysis. New York: Prentice-Hall, 1973. [13] Ferry JD. Viscoelastic properties of polymers. New York: Wiley, 1970. [14] Rueggeberg FA, Caughman WF, Curtis Jr JW, Davis HC. Factors affecting cure at depths within light-activated resin composites. Am J Dent 1993;6:91±95. [15] Lakes RS. Viscoelastic solids. New York: CRC Press, 1999. [16] Shames IH. Elastic and inelastic stress analysis. New York: PrenticeHall, 1992. [17] Press WH. Numerical recipes: the art of scienti®c computing. New York: Cambridge University Press, 1986. [18] Jacobsen P, Darr A. Static and dynamic moduli of composite restorative materials. J Oral Rehabil 1997;24:265±73. [19] Nielsen LE, Landel RF. Mechanical properties of polymers and composites. 2nd ed. New York: Marcel Dekker, 1994. [20] Wypych G. Handbook of ®llers. Toronto: ChemTec Publishing, 1999.
www.elsevier.com/locate/dental
Modeling of the viscoelastic behavior of dental light-activated resin composites during curing Bibi S. Dauvillier*, Maxim P. Aarnts, Albert J. Feilzer Department of Dental Materials Science, Academic Centre of Dentistry Amsterdam (ACTA), Louwesweg 1, NL-1066 EA Amsterdam, The Netherlands Received 16 January 2001; revised 25 August 2001; accepted 4 December 2001
Abstract Objective. Three models consisting of springs and dashpots were investigated to describe the viscoelastic behavior of a commercial lightactivated restorative composite during curing. Methods. Stress±strain data on Z100 were recorded by means of a dynamic test method performed on a universal testing machine. The model was tested by matching its response to experimental data and the material parameters, E (Young's modulus) and h (viscosity), associated with the model were calculated. Results. The universal testing machine generated reliable stress±strain data on the fast curing, light-activated resin composite during curing. The high polymerization rate of Z100 had a negative effect on the viscous ¯ow capability of the material. A predictive model of the viscoelastic behavior of Z100 during curing was carried out, using the Maxwell model for the initial 3 min in the curing process and the Kelvin model for the remainder of the process. Signi®cance. Dental researchers analyzing shrinkage stress problems by mathematical modeling can obtain a good quantitative estimate of the shrinkage stress development of Z100 before the restoration is actually made. q 2003 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. Keywords: Viscoelasticity; Mathematical modeling; Shrinkage stress; Photocuring; Dental composites
1. Introduction Shrinkage stresses generated in dental resin composites during curing are among the major problems in adhesive dentistry, since they interfere with the integrity of the restored tooth. For dental restorative applications, light-activated resin composites have replaced chemically activated resin composites, mainly because of their rapid polymerization reaction and the greater freedom in timing the initiation of polymerization. This `cure on command' allows the dentist to place and contour the restorative material with ease. However, it has been demonstrated that under similar test conditions, light-activated resin composites generate higher polymerization shrinkage stress than the analogous chemically activated composites [1]. This striking difference in shrinkage stress development is caused not only by the difference in shrinkage development, but also by the mechanical behavior of the curing composite, which is by nature viscoelastic [2]. * Corresponding author. Tel.: 131-20-5188235; fax: 131-20-6692726. E-mail address: [email protected] (B.S. Dauvillier).
In recent years, considerable attention has been given to the use of mechanical models to describe the viscoelastic behavior of dental restorative composites during curing [3± 5]. In these studies, several linear viscoelastic models were investigated by applying a modeling procedure to experimental stress±strain data. In addition, the material parameters associated with the model were calculated. The results of these studies contribute to a better understanding of the shrinkage stress problems associated with adhesive composite restoration. Moreover, recent initial modeling studies have shown that both the shrinkage strain, which is associated with the polymerization reaction and is affected by temperature, and the stress ®eld as well, can vary greatly within the restoration preparation [6,7]. Hence, by combining the results from temporal shrinkage strain and mechanical models, the stress ®eld in and around the restoration can be analyzed using ®nite element analysis methods. Up to now, modeling studies have focused on the viscoelastic behavior of chemically activated resin composites. Through introducing adjustments to the specimen mounting device in a specialized universal testing machine, it is possible to monitor the mechanical behavior of light-activated
0109-5641/03/$20.00 + 0.00 q 2003 Academy of Dental Materials. Published by Elsevier Science Ltd. All rights reserved. PII: S 0109-564 1(02)00041-6
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Fig. 1. Experimental setup for light-activated resin composites in dynamic test method. An oscillatory axial strain with amplitude 1.00 ^ 0.01 mm and frequency of 1.0 and 0.1 Hz was successively applied to the curing composite around its original height (strain 0%) in experiment (left). In the stress response the sinusoidal stress is superimposed on the shrinkage stress (right).
resin composites during curing [8]. Stress±strain data recorded with a dynamic test method were used to perform a modeling study of the viscoelastic behavior of this type of dental composites. The aim of this study was to identify a mechanical model for the viscoelastic behavior of light-activated resin composites during curing. Experimental stress±strain data were generated with a dynamic test method in which a vertical oscillatory strain was applied to a commercial light-activated resin composite. Three linear viscoelastic models (Kelvin, Maxwell, and Standard Linear Solid) were investigated by means of a validated modeling procedure [5]. On the basis of the modeling and evaluation of results, a suitable model was chosen for light-activated resin composites. 2. Materials and methods 2.1. Dental resin composite The material utilized in this investigation was a commercial light-activated resin composite (Z100 MP A3, LOT: 19981009, 3M Dental Products, St Paul, MN 55144-1000, USA). According to the manufacturer's instructions, the resin composite was light-cured for 40 s (Elipar Highlight, standard mode, ESPE, Seefeld D-82229, Germany). The light intensity at the light exit tip was 600 mW/cm 2 (radiometer, model 100, Demetron Research Corp., Danbury, CT 06810-7377, USA). 2.2. Dynamic test method Stress±strain data on the light-activated resin composite during curing were obtained by means of a dynamic test method performed on a custom-made automated universal testing machine (ACTAIntense, ACTA, Department of Dental Materials Science, Amsterdam NL-1066 EA, The Netherlands). The composite was bounded between a steel disk, which was connected to the crosshead with the load cell (1 kN), and a rectangular glass plate (¯oat glass, Bakker, J & I, Castricum, The Netherlands, 50 £ 50 £ 4 mm 3),
which was connected to the stationary part of the framework (Fig. 1). To ensure the cylindrical shape of the specimen, the composite was inserted into a lightly greased te¯on mold (d 3.1 mm, L 1.6 mm), creating a C-factor for the specimen of 1.0 (d/2L). The steel disk was moved down until it reached the pre-adjusted specimen height. Optimal bonding between the resin composite and the steel disk was achieved by coating the sandblasted surface (Korox w 50 mm, Bego Bremer GoldschlaÈgerei Wilh. Herbst GmbH & Co., 2 min/5 bar pressure) of the disk with silane (Silicoater, Heraeus Kulzer GmbH, Wehrheim D-6393, Germany, 5 min). For the same reason, the glass surface was lightly sandblasted (Korox w 50 mm, Bego Bremer GoldschlaÈgerei Wilh. Herbst GmbH & Co., 10 s/5 bar pressure), primed (RelyX ceramic primer, 3M Dental Products, St Paul, MN 55144-1000, USA), and ®nally coated with a pressurized air-spread adhesive layer (Scotchbond Multipurpose, 3M Dental Products, St Paul, MN 55144-1000, USA). The submicron-thick adhesive layer was lightcured for 40 s (Elipar Highlight, standard mode, ESPE, Seefeld D-82229, Germany). In the dynamic test method, the upper steel disk performed an oscillating vertical sinusoidal deformation on the curing resin composite around its original height (1.6 mm). The test method was programmed to perform two frequencies with amplitude of 1.00 ^ 0.01 mm (0.0625% strain). First, a frequency of 1.0 Hz was applied for 50 s to the fast curing composite, followed by a frequency of 0.1 Hz until termination of the measurement (1 h). The oscillatory deformation was measured with two displacement probes (Solartron LVDT type AX/1/S, Dimed, Antwerpen B-2140, Belgium) at the level of the specimen. The signal of the displacement probes was continuously monitored and fed back into a control loop, thereby ®netuning the movement of the crosshead on the specimen. The light irradiation process was measured with a custom-made light sensor device at the level of the specimen. The distance between the light exit tip was equal to the thickness of the glass plate (Fig. 1), which was 4 mm. During the measurement, the data (time, load, and displacement signal) were
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collected simultaneously by a data acquisition console at a rate of one hundred points (1.0 Hz period) and 18 points (0.1 Hz period) per seconds, respectively. The measurements were started 5 s prior to the light irradiation process and were repeated three times at room temperature (23 ^ 1 8C). One hour after the start of the experiment, the specimen was subjected to tensile loading with a crosshead speed of 60 mm/min until fracture. To verify the polymerization ef®ciency of the resin composite, the cylindrical specimen was fractured and exposed to Astra blue dye test according to the procedure described by De Gee et al. [9]. 2.3. Volumetric shrinkage measurement During the dynamic test measurement, the axial shrinkage strain of the specimen was not measured, because the oscillatory deformation was performed around the original height of the specimen. However, the displacement caused by axial shrinkage must be taken into account when modeling the stress data recorded by means of the dynamic test method. For this reason, volumetric shrinkage measurements (n 3) were performed by means of a mercury dilatometer at 23 ^ 0.1 8C, using the procedure described by De Gee et al. [10]. 2.4. Stress±strain data analysis The data obtained from the dynamic test measurement consisted of an array of load and displacement values for many points in time. The sinusoidal strain (1 sine) and normal stress (s exp) were calculated using the following equations
1 sine
DL L0
1
s exp
F A
2
279
2.5. Mechanical models The mechanical behavior of the resin composite during curing was considered linear viscoelastic [13], because the strain applied to the specimen was within the limit of linear viscosity of polymer-based materials (,0.5%). Linear viscoelasticity can be described using mechanical models consisting of springs, which obey Hooke's law and dashpots, which obey Newton's law of viscosity. The mechanical models were investigated in one dimension only, because the stress±strain data were monitored in one direction. The shrinkage behavior of the composite is considered to be isotropic, as the specimen was made as high as possible without exceeding the depth of cure (2 mm) [14]. The models must be kept simple because by using uni-axial data, only a restricted number of material parameters can be ®tted in a unique way. As a consequence, the validity of the qualitative and quantitative viscoelastic behavior studied is con®ned to the stress and strain range and the strain rate range covered by the experiment. In this study the Maxwell, Kelvin, and Standard Linear Solid model were investigated. The Maxwell model consists of a spring and a dashpot in series. It represents a viscoelastic liquid because the deformation of the dashpot is permanent in the sense that it does not recover upon removal of an external load. The Kelvin model consists of a spring and a dashpot in parallel. As the deformation process is reversible, with a time delay caused by the dashpot, this model represents a viscoelastic solid. Finally, the Standard Linear Solid model consists of a spring and a Maxwell model in parallel. The model is able to exhibit a pure viscoelastic liquid property, as well as a pure viscoelastic solid property. The physical understanding and differential equations governing the behavior of these viscoelastic models are well documented in literature [15,16]. 2.6. Modeling procedure
in which DL is the displacement value measured by the probes (m), L0 the original specimen height (m), A the cross-sectional area of the specimen (m 2), and F is the recorded load response of the specimen (N). In addition to the applied sinusoidal strain, the strain caused by axial shrinkage must be taken into account when modeling the stress data as recorded by means of the test method. The axial shrinkage strain development of the resin composite (C-factor 1.0) in the dynamic test method was calculated by multiplying the mean volumetric shrinkage strain by a factor of 0.45 [11]. The functional expression of the axial shrinkage strain (1 shrinkage) was calculated by means of a cubic spline ®t [12], and the axial shrinkage strain was added to the oscillatory strain (Eq. (1)) for all the points in time of the dynamic test measurement, which resulted in the desired strain (1 tot) for the modeling procedure:
A validated modeling procedure was developed which is capable of evaluating a mechanical model and calculating its material parameters, E (Young's modulus) and h (viscosity), from experimental stress±strain data. The modeling procedure is described in detail by Dauvillier et al. [5], and only a brief description of the method will be given here. The modeling procedure was performed on sinusoidal stress cycles isolated from the stress data recorded by the dynamic test method. The time span (1 or 10 s) of the isolated interval was kept small with respect to the rate of polymerization reaction. As a result, the mechanical behavior of the composite can be assumed to be constant during the isolated interval, which justi®es the use of simple differential equations for the mechanical models. These equations were solved analytically, allowing the stress to be expressed as a function of strain and unknown material parameters, with the known functional form of the strain (Eq. (3))
1tot 1sine 1 1shrinkage
1 t At 1 B sin vt
3
4
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Fig. 2. Stress (F) and strain (A) data of Z100 with C-factor 1.0 collected with the dynamic test method. For clarity, only the initial 75 s of the curing reaction is shown. The total strain on the curing resin composite consists of axial shrinkage (C), derived from volumetric shrinkage measurements, and oscillatory strain (A), applied by the crosshead (note different y scale). For the modeling procedure, the total strain curve was constructed by a linear combination of (A) and (C), resulting in (E). The shrinkage stress (D) and dynamic stress (B) response were isolated from experimental stress data (F) via FFT smoothing and substraction (F±D), respectively. The dotted lines represent the initiation and termination of the irradiation process of the light unit.
in which 1 is the strain, t the time in isolated interval (0±1 or 0±10 s), A the slope of the shrinkage strain (s 21), B the amplitude, and v is the angular frequency (rad s 21) of the oscillatory strain. The shrinkage strain in the isolated time interval was assumed to be linear in time. Except for the Kelvin model, the initial condition (s (0)) for the stress equation of the model was taken from the shrinkage stress data, thereby avoiding the extensive computation involved in evaluating the
initial stress mathematically. By means of the initial parameter values, a least square method was performed at equidistantly spaced k points in time of the isolated interval, to assess how well the model stress (s model) approximates the experimental stress of the interval (s exp):
d
k X i1
s model ti 2 s exp ti 2
5
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281
Fig. 3. Modeling results for two stress cycles of Z100 during curing at time 4.58 s (A) and time 10.59 s (B) for the Maxwell model (top), Kelvin model (middle) and Standard Linear Solid model (bottom).
The search for the ®nal material parameters values was carried out by minimizing the residual (d ) with an optimization routine based on the Levenberg±Marquardt method [17]. The modeling procedure provides (I) the parameters, (II) the error estimates on the parameters, and (III) the residual (d ), a quantitative measure of the difference between experimental and model stress. The algorithms for the modeling procedure were implemented in MATLAB (version 5.3, The MathWorks, Inc., Natick, USA) on a desktop computer (Windows q 98 platform). 2.7. Evaluation of the viscoelastic model To evaluate how well the model predicts the real shrinkage stress development of Z100, each model was loaded for small intervals in curing time with its mean material parameters, as calculated by analyzing the stress intervals with the modeling procedure, and axial (linear) shrinkage strain. The calculated stress value at the end of each interval was taken as the initial stress condition for the next interval. In addition, an evaluation was performed in which the Maxwell model was used for the initial 3 min in the curing process and the Kelvin model for the remainder of the curing process. The shrinkage stress develop-
ment of Z100 in the experiment was isolated from the recorded stress data (Eq. (2)), using the standard Fast Fourier Transform (FFT) smoothing ®lter in Origin (version 5.0, Microcal Software, Inc., Northampton, MA 01060, USA). The best result was obtained by taking 200 points for smoothing. 3. Results Fig. 2A and F show the applied strain and stress response of Z100 during the initial 75 s of the curing process at room temperature. The positive strain of the sinusoidal cycles represents the crosshead displacement away from the specimen, while the negative strain represents the crosshead displacement towards the specimen. As the oscillatory strain was performed around the original specimen height, the stress response on the oscillatory strain was superimposed on the continuous shrinkage stress development of the specimen. There was no premature debonding from either the glass plate or the steel disk, because subsequent to tensile loading the fracture always occurred in the glass plate, at a stress level 1.5 times the maximum stress (18 MPa) measured in the test method. The mean axial
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Table 1 Material parameters for several cycles during one measurement of Z100 during curing with standard deviation in parenthesis. Material parameters: E(x), Young's modulus; h , viscosity; and d , quantitative measure of the difference between experimental and model stress Kelvin model
Maxwell model
Time (s)
E (GPa)
h (GPa s)
d
E (GPa)
4.6 10.6 20.6 40.6 302 1202 3352
0.24 (0.03) 1.96 (0.02) 3.55 (0.01) 4.70 (0.05) 5.45 (0.01) 6.32 (0.05) 6.57 (0.01)
0.02 (,0.01) 0.07 (,0.01) 0.11 (,0.01) 0.11 (,0.01) 0.75 (,0.01) 0.60 (,0.01) 0.56 (,0.01)
0.746 0.387 0.173 2.87 0.374 0.77 0.496
0.32 (0.02) 1.94 (0.06) 3.50 (0.09) 4.55 (0.09) 5.36 (0.04) 6.02 (0.05) 6.48 (0.04)
Standard Linear Solid model
h (GPa s) 0.57 (,0.01) 29.9 (,0.01) 124 (,0.01) 77.2 (,0.01) 3447 (,0.01) (,0.01) 5807 (,0.01)
shrinkage strain, as derived from the volumetric measurements, and its derivative are shown in Figs. 2C and 4D, respectively. The strain data used for the modeling procedure consisted of oscillatory strain superimposed on the axial shrinkage strain curve (Fig. 2E). The graphic results of the modeling procedure on two stress intervals isolated from one dynamic test experiment are shown in Fig. 3A and B. The continuous black line represents the measured stress, while the dots represents the values computed by the model, using the material parameter values as estimated by the modeling procedure. Table 1 and Fig. 4A±C show the estimated material parameter values for the three models for several stress intervals during one experiment. The evaluation results for the three models, and by using ®rst Maxwell (up to 180 s) and
d
E1 (GPa)
h (GPa s)
E2 (GPa)
d
0.318 3.43 7.37 6.54 5.33 5.27 4.60
0.39 (0.05) 2.28 (0.05) 3.38 (0.44) 4.76 (5.23) 1.59 (1.91) 1.40 (2.20) 1.33 (3.99)
0.10 (,0.01) 0.22 (,0.01) 2.01 (,0.01) 4.92 (,0.01) 7.73 (,0.01) 6.90 (,0.01) 7.51 (,0.01)
, 0.01 (,0.01) 1.85 (0.11) 1.33 (0.59) ,0.01 (5.39) 4.01 (2.15) .91 (2.46) 5.34 (4.34)
0.285 0.163 0.152 1.423 0.312 0.327 0.483
then Kelvin (remainder of cure process), are illustrated on two different time-scales in Fig. 5. Each point on the curves was generated by statistically averaging three individual experiments. 4. Discussion In the choice of a model for describing the viscoelastic behavior of dental resin composites during curing, there is always a trade-off between accuracy and level of mathematics. It will always be possible to develop a perfect model, which gives a perfect prediction of the material behavior by using a complicated mathematical formula, but this is hardly a sensible approach. What is required is
Fig. 4. Parameter values of the Maxwell model (A), Kelvin model (B), and Standard Linear Solid model (C) as a function of curing time of Z100 for one measurement. Error bars indicate the relative standard error in the estimated parameter value. The axial shrinkage strain rate development (D) was derived from volumetric measurements.
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283
Fig. 5. Axial shrinkage stress development (Ð) measured; (B) Kelvin model; (O) Maxwell model; (V) Standard Linear Solid model; and (X) Maxwell and Kelvin) during the initial 10 min (A) and 1 h (B) curing of Z100. Error bars indicate the relative standard error in the calculated mean (n 3).
a simple model, which will produce an adequate prediction. In this study, the choice of the model and the calculation of its material parameters were determined on the basis of matching the model prediction to experimental data. Therefore, the success of this modeling study depends on the quality of the experimental data. In the course of the research on modeling the viscoelastic behavior of chemically activated resin composites, improvements were made to certain aspects of the dynamic test method which are important in obtaining reliable experimental data [5]. To achieve the same reliability for light-activated resin composites, several modi®cations to the mounting device were necessary. The position of the light source was established by placing it at one end of the cylindrical specimen, thereby simulating a method of curing which is thought to be clinically feasible. The choice of the light source, the duration of light irradiation process, and the distance between the light exit tip to the specimen was found to be adequate for the complete polymerization of Z100, as the Astra blue dye tests on the specimens revealed no visible staining inside the semi-cylindrical surface of the resin composite. Furthermore, the position of the light sensor device was adequate to detect accurately the initiation and duration of the light irradiation process. Finally, the strain applied in the early stage of curing, when the shrinkage rate is high and the mechanical behavior of light-activated resin composites changes rapidly from soft to hard, remain a sinusoid (Fig. 2A). In this way the differential equations of the models could be solved analytically, thereby greatly reducing computational time involved in the modeling procedure. The amplitude of the strain was fractional higher than the preset value (0.0625%), because the feedback system overcompensated the crosshead movement on the specimen as it became stiffer. The higher strain amplitude did not, however, lead to errors in the modeling results, because it was not the preset, but the measured oscillatory sine that was utilized in the modeling procedure. The light irradiation of Z100 was initiated after at least four oscillatory strain cycles has been applied to Z100, because laboratory tests on a control specimen revealed that the experimentally
applied sinusoid, which has a starting point, converges within four cycles to a mathematical sinusoid as described by Eq. (4). An interesting feature of Z100 is that in the initial 2 s of the curing process, the material undergoes 15% of the measured axial shrinkage strain without generating shrinkage stress (Fig. 2C and D). The relationship between shrinkage stress and shrinkage strain displayed by this commercial dental composite contrasts with that of a commercial chemically activated composite where, under closely similar experimental conditions and up to 4 min after the start of polymerization, a large proportion of composite shrinkage (50% or more) failed to contribute to stress development in the material [4]. Although the chemical composition of these commercial resin composites differs, it may be concluded that in general an increase in the rate of polymerisation of resin composites has a negative effect on the viscous ¯ow capability of the material. The viscosity (h ) values of all models, as calculated by analyzing the stress cycles by means of the modeling procedure, were all positive and developed according to the spring-dashpot arrangement in the model with curing time (Table 1, Fig. 4A±C). The Young's modulus (E) values were also positive and increased monotonically with the curing time. The Young's modulus of Z100 after 1 h curing (approximately 6.5 GPa) is not in agreement with the value of 13 GPa provided by the manufacturer. It is known that factors such as the choice of mechanical model, test methods (bending, shear, compression, tension), conditions of the test method (strain rate, curing time, temperature), and light irradiation procedure (light intensity, duration time) have a considerable effect on the ultimate Young's modulus value of viscoelastic resin composites. The Young's modulus of viscoelastic materials generally decreases with a decreasing strain rate [18], as evident in the strain rate transition from 1.0 to 0.1 Hz (Fig. 4A±C). For a valid comparison between Young's modulus values, the variables described earlier must be the same. Any difference between these variables will make comparisons between Young's moduli impossible.
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The plots of two time intervals to which the modeling procedure was applied are shown in Fig. 3. All models failed to predict the experimental stress in the early stage of curing (4.58 s). The lack of modeling capability is not caused by the low signal-to-noise ratio (SNR 2.40) of the experimental stress data. A previous study showed that when the viscoelastic behavior of the resin composite was exactly the same as that of a model, then it was possible to model very precise experimental stress data with practical SNR values as low as 2.20 [5]. All models failed to predict the stress a few seconds after the initiation of polymerization, because the material parameters were kept constant over an interval of time, whereas in reality the material stiffness increases noticeably, as indicated by the asymmetrical sinusoid around the shrinkage stress line (Fig. 3A). Better modeling results should be obtained by isolating smaller time intervals in the stress data, i.e. 0.50 s or less, or by using differential equations for the model in which material parameters vary in time. For further studies, the former is recommended, since solving partial differential equations involves intensive computation work. The Maxwell stress curve at cure time 10.59 s is a poor approximation of the stress response of the dental composite. This is clear not only from the graphical ®t, but also in the d parameter (Table 1)Ðthe quantitative measure of the difference between experimental and model stressÐwhich is the highest value of all models. The phase lag between the model stress curve and the experimental stress curve indicates that the model predicts more viscous ¯ow than the material actually undergoes. As a result, this model predicts for the ®rst 3 min of curing a lower stress development than Z100 generates in the experiment (Fig. 5A). This result is essentially different from chemically activated resin composites, where the Maxwell model predicts the shrinkage stress better and for a longer period in curing time (up to 11 min) [5]. Obviously, high polymerization rates reduce the ability of the composite to ¯ow permanently. This may explain why light-activated resin composites generate higher polymerization shrinkage stresses than analogous chemically activated composites [1], because composite shrinkage is less compensated by permanent viscous ¯ow from the unbonded, outer surface of the material. In the remainder of the curing process, the Maxwell model predicts too much stress relief, which is obviously not the case for Z100 in the experimental situation (Fig. 5B). The better modeling results achieved with the Kelvin and Standard Linear Solid model con®rms that the behavior of light-activated resin composites is more viscoelastic solid-like (reversible ¯ow) than viscoelastic liquid-like (permanent ¯ow). The slightly better approximation by the Standard Linear Solid model, where the development of E2 indicates reversible ¯ow behavior, can be explained by the extra parameter it contains in comparison with the Kelvin model. A closer look at the sinusoid curve at 10.59 s reveals that the Kelvin model responds somewhat more stif¯y in the ®rst section and slightly softer in the
second section of the curve than Z100 in the measurement. As stated for the Maxwell model, this can be explained by the fact that the material parameters were kept constant in time, whereas in reality the material increases in stiffness over time. The small error introduced by assuming constant parameters counterbalanced the intensive computations required by models with timedependent material parameters. Up to 25 s into the light irradiation procedure, the response of the Kelvin model closely resembles the shrinkage stress response of Z100 in the measurement (Fig. 5A). After 25 s, the model generates higher shrinkage stresses, leading ultimately to a shrinkage stress level that is a factor of two higher than measured for Z100 during the experiment (Fig. 5B). There may be two explanations, one or both of which may be responsible for the failure of the Kelvin model in the remainder of the curing process. First, the ¯ow behavior of light-activated resin composite could be non-Newtonian; i.e. the viscosity is not constant, but depends on the strain rate. For this reason, the viscosity values of the Kelvin model may not be valid in this part of the curing process, where the shrinkage rate is close to zero (Fig. 4D), since the values were calculated from experimental stress that depended exclusively on the oscillatory strain (Fig. 2F). Secondly, the failure of the Kelvin model can perhaps be accounted for by the fact that the material undergoes two deformation processes, namely reversible, by viscous ¯ow as predicted by the model, and permanent, by imperfections in the materials, such as voids, crazes, and microcracks [19,20]. These imperfections are usually generated when local stress spots in the material exceed inter-atomic bond strength within the polymer and/or polymer-®ller interface and are likely to be present in light-activated resin composite, where high shrinkage stress developed during curing (Fig. 2D). It is dif®cult to develop a model that is capable of taking into account these two processes, which occur simultaneously but are different in origin. Theoretically, the Standard Linear Solid model is able to describe both the viscoelastic liquid (E2 0) and the viscoelastic solid behavior of resin composites during curing. On a practical level, the evaluation results of the model reveal that the viscoelastic behavior of Z100, as excited by the conditions of the dynamic test method, cannot be adequately predicted by this model. A close look at the error development in the E1 and E2 parameters at curing time 20 s and later con®rms that there is a basic problem in modeling the stress using the Standard Linear Solid model. Fig. 4C and D show that there is a clear relationship between the value of the standard error in E1 and E2 and the shrinkage rate pro®le. As the shrinkage rate declines, and the contribution of shrinkage strain to the applied strain deteriorates as the curing process continues, the stress response becomes more exclusively dynamical, i.e. more dependent on the sinusoidal strain of one frequency alone. In
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this situation the values of the E1 and E2 parameters associated with the Standard Linear Solid model cannot be distinguished from one another, as evidenced by the high error value, because with this type of stress response no more than two independent parameters can be determined. To obtain reliable E1 and E2 values in the remainder of the curing process, the stress to be modeled must be generated by a multi-wave strain, which entails sophisticated dynamic test conditions. This approach requires modi®cations in the application software, which, however, has not been established. 5. Conclusions The automated universal testing machine developed for the dynamic testing of dental restorative material proved capable of generating stress±strain data on light-activated resin composites. The use of a light sensor device made it possible to accurately detect the initiation and duration of the light irradiation process, so that errors in assigning stress±strain data to curing time were reduced to a minimum. An increase in the polymerization rate has a negative effect on the viscous ¯ow capability of dental resin composites. Adequate predictive modeling of the viscoelastic behavior of Z100 can be carried out by using the Maxwell model for the initial 3 min of the curing process and the Kelvin model for the remainder of the curing process. Acknowledgements The authors are grateful to Pierre HuÈbsch for his guidance in mathematical modeling, Prem Pallav for the design and production of the ACTAIntense, Joost Buijs for writing the application software, and Barbara Fasting for her comments on the English text. This investigation was ®nancially supported by the Netherlands Institute for Dental Sciences.
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