Abstract
Objectives
For shrinkage stress simulation in dental restoration, the elastic properties of composite resins should be acquired beforehand. This study proposes a formula to measure the equivalent Young’s modulus of a composite resin through a calculation scheme of the shrinkage stress in dental restoration.
Methods
Two types of composite resins remarkably different in the polymerization shrinkage strain were used for experimental verification: the methacrylate-type (Clearfil AP-X) and the silorane-type (Filtek P90). The linear shrinkage strains of the composite resins were gained through the bonded disk method. A formula to calculate the equivalent Young’s moduli of composite resin was derived on the basis of the restored ring substrate. Equivalent Young’s moduli were measured for the two types of composite resins through the formula. Those values were applied as input to a finite element analysis (FEA) for validation of the calculated shrinkage stress.
Significance
Both of the measured moduli through the formula were appropriate for stress simulation of dental restoration in that the shrinkage stresses calculated by the FEA were in good agreement within 3.5% with the experimental values. The concept of equivalent Young’s modulus so measured could be applied for stress simulation of 2D and 3D dental restoration.
1
Introduction
Composite resins for dental restoration that shrink during a polymerization reaction irradiated by light are apt to suffer considerable tensile stresses in the resin. This generates the risk of interfacial fractures and/or microleakage between the tooth and the resin. Polymerization of the composite resin goes through a pre-gel and a post-gel stages. Shrinkage arising during the pre-gel stage does not generate shrinkage stress since the stress is overcome by the flowability of the resin. During the post-gel stage, the composite resin stiffens as it shrinks, leading to an increase in the shrinkage stress. In this stage, the material properties of the composite resin such as the elastic modulus, viscosity, and volume shrinkage increase nonlinearly with time, and their growth behaviors are not the same during the process. Although the shrinkage stress of the composite resin in dental restoration occurs in accordance with the progress of Young’s modulus and shrinkage strain , an exact calculation of the stress during restoration may also need to account for time-dependent viscoelastic and creep effects accompanying the viscous flow of the composite resin during the curing process.
To date, shrinkage stress distributions in the composite resin for dental restoration have been investigated mostly by using finite element methods with two main areas of focus. The first is the fatigue behavior of a restored tooth under the mastication load and repeated thermal shocks. The second has been on the residual shrinkage stress behavior under early resin shrinkage . In the cases of finite element analysis (FEA), insufficient modeling that deviated from actual dental restoration states has remained giving rise to a considerable number of stress errors.
Several techniques have been used for finite element modeling to ensure similarity to actual experimental conditions. For Young’s modulus as a necessary input to FEA, an arbitrary value or the Young’s modulus data of the finally cured state were assumed. Versluis et al. and Kowalczyk used elastic FE model with a constant Young’s modulus of the cured state and adopted an assumption of effective reduced linear shrinkage for the estimation of the real shrinkage stresses. For applicability of Young‘s modulus to FEA, Li et al. and Barink et al. measured the time-dependent Young’s modulus and time-dependent material properties during chemically activated polymerization with a special test setup in combination with their associated physical simulators . Versluis et al. calculated the Young’s modulus through the relationship between the Knoop hardness and Young’s modulus. Vukicevic et al. measured the Young’s modulus as a function of the resin depth under light curing and applied it differently to FEA according to the resin depth. With two- and three-dimensional (2D, 3D) modeling in the stress analysis, Rodrigues et al. simulated the polymerization shrinkage behavior over light curing by applying the equivalent temperature reduction with a constant thermal expansion coefficient.
This study proposes a formula to measure the equivalent Young’s modulus of a composite resin through a calculation scheme of the shrinkage stress in dental restoration. We used a simple model simulating a composite restoration surrounded by an elastic ring . Methacrylate-type (Clearfil AP-X) and silorane-type (Filtek P90) composite resins, which have remarkably different shrinkage strains, were utilized. The polymerization shrinkage stresses perpendicularly exerted onto the marginal circumference of the composite resins were measured on the basis of the substrate ring and its elastic properties. The linear shrinkage strain was also measured using the bonded disk method with a linear variable differential transformer. The equivalent Young’s modulus of each composite resin was obtained through the formula, and was applied to 2-D and 3-D finite element analyses of the ring substrate model. The calculated and experimentally measured shrinkage stresses were compared to verify the formula.
2
Experiment
2.1
Ring substrate and dental restoration
A ring substrate 6 mm in outer diameter, 4 mm in inner diameter, and 2 mm in height was prepared using polymethyl methacrylate (PMMA), which is often utilized as the main ingredient of a denture. The ring substrate was cleaned with ethyl alcohol. Then, a strain gauge (KFG-1-120-C1-11L1M2R, 1 mm in gauge length, Kyowa, Japan) was attached to the outer side in the circumferential direction of the substrate ring to measure the circumferential strain ε s . The strain gauge was connected to a strain amplifier (DAS-406B, Minebea, Japan).
Adhesive resin was applied to the inner wall of the ring substrate using a micro brush and dried by blowing air for 10 s. With an LED irradiator (Morita Pencure, Japan), light irradiation was conducted for 10 s to partially cure the adhesive. Two types of composite resins, Clearfil AP-X (Kuraray, Japan) and Filtek P90 (3M ESPE, USA), filled up the substrate hole. The LED head was maintained at a distance of 2 mm from the top of the specimen during the irradiation. The light irradiation was conducted at an intensity of 1000 W for 20 s. The light was imposed on the resin part and the inner part of the ring substrate to avoid direct exposure to the strain gauge. Temperature rise at the strain gauge by the exothermic photo polymerization reaction of the resins was measured below 10 °C, which might give negligible error (below +0.08% of normal voltage) for the strain measurement. The mechanical properties of the composite resins are listed in Table 1 .
| Properties | Composite resin | Adhesive layer | Substrate | ||
|---|---|---|---|---|---|
| Clearfil AP-X | Filtek P90 | Clearfil AP-X | Filtek P90 | PMMA | |
| Elastic modulus (GPa) | 16.8 | 14.4 | 4.4 | 2.1 | 3.2 |
| Poisson’s ratio | 0.26 | 0.3 | 0.24 | 0.3 | 0.3 |
| Polymerization shrinkage (vol.%) | −1.9 | −0.88 | – | – | – |
2.2
Measurement of the polymerization shrinkage stress
By using an elastic ring substrate shown in Fig. 1 , the polymerization shrinkage stress set as the contraction stress ( σ cs ) vertically onto the inner surface of the ring can be calculated from the static relationship to the circumferential strain ( ε s ) of the outer surface of the ring as
σ c s = − E s ε s r o 2 − r i 2 2 r i 2
where E s is the Young’s modulus of the ring substrate, and r o and r i are the outer and inner radii of the ring, respectively. The Eq. (1) assumes that the shrinkage process can be approached with elastic axisymmetric relationship derived for internally pressured thick walled cylindrical vessels . This equation was derived under the plane stress that the axial stress is zero. The negative sign represents the circumferential strain acting as compression due to the positive contraction stress σ cs in the shrunk composite resin, which exerted tensile force on the inner surface of the ring. The voltage signal data from the strain gauge was collected through LabVIEW 8.0 at an interval of 0.1 s for 600 s after the light irradiation started.
2.3
Measurement of the polymerization shrinkage strain
The polymerization shrinkage of the resin was also measured using the bonded disk method , which is based on a linear variable differential transformer (LVDT). The composite resin was put on a microscope glass slide (76 × 26 × 1 mm, Marienfeld, Germany). Then, two wire spacers with a diameter of 0.5 mm were placed around the composite resin at a distance of 22.5 mm, which maintained the constant thickness of the resin bond. After that, the composite resin was covered with a flexible thin square cover glass (18 × 18 × 0.14 mm, Marienfeld, Germany). To make a disc-shaped bond, the square cover glass was pressed with the weight of another glass slide. At this time, the C-Factor of the composite resin bond placed between both glasses was set to a range of 5–6 to minimize the shape effect of the resin bond . By placing two edges on the diagonal of the square cover glass on the two wire spacers, the flexibility of the cover glass was expanded to the maximum. The prepared specimen was placed between the LVDT and the light irradiator that were aligned vertically. The axial shrinkage strain ( ε z ) of the resin bond was measured under the same irradiation conditions as the above ring substrate specimen. The radial shrinkage strain was much less than the axial shrinkage strain which was approximate to the volumetric shrinkage as clarified in . The linear shrinkage strain was calculated as one-third of the measured axial shrinkage using the method in .
2
Experiment
2.1
Ring substrate and dental restoration
A ring substrate 6 mm in outer diameter, 4 mm in inner diameter, and 2 mm in height was prepared using polymethyl methacrylate (PMMA), which is often utilized as the main ingredient of a denture. The ring substrate was cleaned with ethyl alcohol. Then, a strain gauge (KFG-1-120-C1-11L1M2R, 1 mm in gauge length, Kyowa, Japan) was attached to the outer side in the circumferential direction of the substrate ring to measure the circumferential strain ε s . The strain gauge was connected to a strain amplifier (DAS-406B, Minebea, Japan).
Adhesive resin was applied to the inner wall of the ring substrate using a micro brush and dried by blowing air for 10 s. With an LED irradiator (Morita Pencure, Japan), light irradiation was conducted for 10 s to partially cure the adhesive. Two types of composite resins, Clearfil AP-X (Kuraray, Japan) and Filtek P90 (3M ESPE, USA), filled up the substrate hole. The LED head was maintained at a distance of 2 mm from the top of the specimen during the irradiation. The light irradiation was conducted at an intensity of 1000 W for 20 s. The light was imposed on the resin part and the inner part of the ring substrate to avoid direct exposure to the strain gauge. Temperature rise at the strain gauge by the exothermic photo polymerization reaction of the resins was measured below 10 °C, which might give negligible error (below +0.08% of normal voltage) for the strain measurement. The mechanical properties of the composite resins are listed in Table 1 .
| Properties | Composite resin | Adhesive layer | Substrate | ||
|---|---|---|---|---|---|
| Clearfil AP-X | Filtek P90 | Clearfil AP-X | Filtek P90 | PMMA | |
| Elastic modulus (GPa) | 16.8 | 14.4 | 4.4 | 2.1 | 3.2 |
| Poisson’s ratio | 0.26 | 0.3 | 0.24 | 0.3 | 0.3 |
| Polymerization shrinkage (vol.%) | −1.9 | −0.88 | – | – | – |
2.2
Measurement of the polymerization shrinkage stress
By using an elastic ring substrate shown in Fig. 1 , the polymerization shrinkage stress set as the contraction stress ( σ cs ) vertically onto the inner surface of the ring can be calculated from the static relationship to the circumferential strain ( ε s ) of the outer surface of the ring as
σ c s = − E s ε s r o 2 − r i 2 2 r i 2
where E s is the Young’s modulus of the ring substrate, and r o and r i are the outer and inner radii of the ring, respectively. The Eq. (1) assumes that the shrinkage process can be approached with elastic axisymmetric relationship derived for internally pressured thick walled cylindrical vessels . This equation was derived under the plane stress that the axial stress is zero. The negative sign represents the circumferential strain acting as compression due to the positive contraction stress σ cs in the shrunk composite resin, which exerted tensile force on the inner surface of the ring. The voltage signal data from the strain gauge was collected through LabVIEW 8.0 at an interval of 0.1 s for 600 s after the light irradiation started.
2.3
Measurement of the polymerization shrinkage strain
The polymerization shrinkage of the resin was also measured using the bonded disk method , which is based on a linear variable differential transformer (LVDT). The composite resin was put on a microscope glass slide (76 × 26 × 1 mm, Marienfeld, Germany). Then, two wire spacers with a diameter of 0.5 mm were placed around the composite resin at a distance of 22.5 mm, which maintained the constant thickness of the resin bond. After that, the composite resin was covered with a flexible thin square cover glass (18 × 18 × 0.14 mm, Marienfeld, Germany). To make a disc-shaped bond, the square cover glass was pressed with the weight of another glass slide. At this time, the C-Factor of the composite resin bond placed between both glasses was set to a range of 5–6 to minimize the shape effect of the resin bond . By placing two edges on the diagonal of the square cover glass on the two wire spacers, the flexibility of the cover glass was expanded to the maximum. The prepared specimen was placed between the LVDT and the light irradiator that were aligned vertically. The axial shrinkage strain ( ε z ) of the resin bond was measured under the same irradiation conditions as the above ring substrate specimen. The radial shrinkage strain was much less than the axial shrinkage strain which was approximate to the volumetric shrinkage as clarified in . The linear shrinkage strain was calculated as one-third of the measured axial shrinkage using the method in .
3
Analysis
3.1
Deduction of the equivalent Young’s modulus during polymerization shrinkage
The polymerization shrinkage stress σ cs acting on the inner surface of the specimen ring can be calculated on the basis of the dental restoration ring model ( Fig. 1 ) as a function of Young’s modulus E c and the linear shrinkage strain ε r of the composite resin. Since the Eq. (1) was derived under plane stress, theoretical σ cs of the composite resin part is also derived under the same stress condition.
Without the outer ring, let the radii of the composite resin part prior to and after the shrinkage be r i and r 1 , respectively. The shrunk radial displacement Δ r i (≤0) arises from the resin shrinkage only. When the outer ring is perfectly bonded with the composite resin part ( Fig. 2 ), the elastic spring-back effect of the ring on the shrunk resin region generates the tensile stress σ cs , and thus the tensile radial displacement of Δ r 1 (≥0) from the shrunk radius r 1 . On account of the geometric compatibility between the outer ring and the composite resin part, σ cs can be represented by the following equations:
σ c s = 1 A ε r − ( 1 − ν c E c ) ( 1 + 1 ε r )
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