Finite element analysis of bonded model Class I ‘restorations’ after shrinkage



The C-Factor has been used widely to rationalize the changes in shrinkage stress occurring at the tooth/resin-composite interfaces. Experimentally, such stresses have been measured in a uniaxial direction between opposed parallel walls. The situation of adjoining cavity walls has been neglected. The aim was to investigate the hypothesis that: within stylized model rectangular cavities of constant volume and wall thickness, the interfacial shrinkage-stress at the adjoining cavity walls increases steadily as the C-Factor increases.


Eight 3D-FEM restored Class I ‘rectangular cavity’ models were created by MSC.PATRAN/MSC . Marc , r2-2005 and subjected to 1% of shrinkage, while maintaining constant both the volume (20 mm 3 ) and the wall thickness (2 mm), but varying the C-Factor (1.9–13.5). An adhesive contact between the composite and the teeth was incorporated. Polymerization shrinkage was simulated by analogy with thermal contraction. Principal stresses and strains were calculated. Peak values of maximum principal ( MP ) and maximum shear ( MS ) stresses from the different walls were displayed graphically as a function of C-Factor. The stress-peak association with C-Factor was evaluated by the Pearson correlation between the stress peak and the C-Factor.


The hypothesis was rejected: there was no clear increase of stress-peaks with C-Factor. The stress-peaks particularly expressed as MP and MS varied only slightly with increasing C-Factor. Lower stress-peaks were present at the pulpal floor in comparison to the stress at the axial walls. In general, MP and MS were similar when the axial wall dimensions were similar. The Pearson coefficient only expressed associations for the maximum principal stress at the ZX wall and the Z axis.


Increase of the C-Factor did not lead to increase of the calculated stress-peaks in model rectangular Class I cavity walls.


The main failures related to teeth restored with resin-composites are microleakage and fracture of the remaining tooth tissue. These failures are the consequences of stresses to which all teeth are subjected. In theory, it is reasonable to think that stresses, of any nature, will always affect the weakest link of the restored tooth complex. If the restoration is well-bonded and the remaining tooth tissue is mechanically resistant, the weakest link is the tooth/composite interface. In this case, there could be higher risks of failure by microleakage and subsequent secondary decay . On the other hand, if the tooth was already decayed, the weakest link will be the dental tissue , with a high risk of dental fracture failure.

The primary origins of stress in a restored tooth usually comes from differences in contraction/expansion between the tooth and the restored material or by occlusal loads . Differences in contraction/expansion may occur during polymerization of the restored material or during heating/cooling (when the coefficient of thermal expansion of the resin-composite is different to that of tooth tissue). Regardless of the origin of stresses, their distribution and consequences will be considerably different, depending on whether the restoration is bonded or not along the entire interface .

Among the different origins of stresses, resin-composite polymerization shrinkage stress has been extensively investigated. Evident and undesirable clinical signs and symptoms such as enamel microcracks adjacent to the restoration, microleakage and post-operative sensitivity have been associated with this shrinkage . However, none of these phenomena is suitable as a measure of polymerization stress. The most common testing assembly for determination of resin-composite shrinkage stress is the so-called “tensilometer” . During this test, the resin-composite is inserted between two flat surfaces of metal, glass or composite and, when polymerization shrinkage takes place, the opposing substrates are pulled toward each other. This causes load cell deformation, which registers the force development during polymerization. An extensometer may be introduced into the system to continuously monitor the change in distance between the walls and trigger a response of the crosshead in the opposite direction. The nominal stress is calculated by dividing the calibrated load cell reading by the specimen cross sectional area. An alternative approach involves the use of cantilever load cells, such as the Bioman device of Watts et al. .

Several studies with tensilometers have shown a positive correlation between shrinkage stress and C-Factor (bonded area/un-bonded area) . Thus, some authors suggested the use of this index to predict the durability of the tooth/restoration interface during or following polymerization. However, the C-Factor is a simple index and the different walls that compose the cavity are not equally represented . Since only the longitudinal axis of measurement is considered, the triaxial stress state is completely ignored . Hence, whereas parallel walls are measured by a tensilometer, stresses can also exist between adjoining walls, like two pages of an open book ( Fig. 1 ). Moreover, extrapolation of the results of tensilometer studies to clinical reality needs to be done carefully because cavity geometries are complex. Stresses generated by a composite adhered within a cavity depend not only on the C-Factor but also on the compliance of the remaining wall structures and also the mass or volume of resin-composite involved. The ease of the wall displacement is analogous to the assembly compliance, which is already recognized as one of the factors affecting the polymerization stress measurements with tensilometers.

Fig. 1
Coordinate system for the axes and the walls of a ‘rectangular’ model cavity.

Cuspal deformations have been used as an indicator of tooth/resin-composite interface shrinkage stress, which can be measured with restored teeth . However, this factor would only be valid for comparisons between cases with exactly the same compliance. Therefore, if a greater deformation was found in a premolar with a Class II cavity when compared to a Class I cavity, it would not necessarily indicate that a higher stress was developed at the tooth/composite interface. It might only be a consequence of greater compliance, caused by the absence of the marginal ridge. Thus, it is possible that a negative correlation between the cuspal deformation and interface stress could occur. But this greater wall movement can cause considerable deformation in certain critical regions. Consequently, high localized stresses may result, increasing the risk of tooth fracture. In this case, the weakest link could be the tooth rather than the interface.

The magnitude of tooth deformation also depends on the mechanical properties of the restorative materials (composite, adhesive), as a high modulus composite causes higher deformation of cavity walls during polymerization than a less rigid one . However, a reduction in composite elastic modulus could lead to low hardness and low wear strength. This could eventually generate poor marginal fit, post-operative sensitivity or dental fracture, compromising the longevity of the restoration. The mode of mechanical loading is also part of the complex relations that exists between these factors. This makes this type of investigation difficult to simulate fully in laboratory studies.

The finite element method (FEM) has been used to evaluate the status of restored tooth . This method permits modeling of the tooth, interface and composite stress and strain under simulated clinical conditions . It is possible to isolate the variables of interest (compliance, C-Factor, etc.) to study their individual or conjoint effects in cases where it is not possible to perform an experiment. Thus, FEM has become an important tool to interpret the failure mechanisms of bonded restorations, and to suggest possible alternative cavity configurations that minimize failures. The aim of this study was to model by FEM the stresses generated by composite shrinkage in different ‘rectangular cavity’ configurations as a function of C-Factor. The hypothesis was that shrinkage stresses at the interface of cavities with adjoining walls ( Figs. 1 and 2 ) increase as a function of C-Factor.

Fig. 2
(A) Class I idealized cavity. Letters indicate dimensions in Table 1 . (B) Simulated 1/4 geometry model.

Material and methods

3D-FEM models of restored Class I idealized rectangular cavities with different C-Factors and constant volume (20 mm 3 ) were created by MSC.PATRAN/MSC . Marc v.r2-2005 (MSC.Software, CA, USA) ( Figs. 2 and 4 and Table 1 ). Using planes of symmetry, quarter models were generated as representative of the whole structure. Some dimensions presented in Table 1 have no evident correlation with normal clinical cases, but they allowed extension of curves, indicating the stress variation over a greater range of C-Factor. The volume was fixed to eliminate this source of variability, since in a low compliance system, the stress does not depend strongly on the C-Factor, but on the volume of the composite . The walls were always 2 mm thick in an attempt to maintain the compliance as constant as possible.

Table 1
C-Factor and dimensions (mm) for the models correspondent to Fig. 2 .
Dimensions (mm) C-Factor
A a ( x ) B a ( z ) C a ( y ) D E F G H I J K L M
5 2 2 3.65 0.8 6 4 9 1 0.45 1.43 3.34 1.22 3.8
2 5 2 6.65 0.8 6 7 6 1 0.45 1.43 3.34 1.22 11.0
3.2 2.5 2.5 4.15 0.8 6.5 4.5 7.2 1 0.45 1.43 3.34 1.22 4.6
2.5 3.2 2.5 4.85 0.8 6.5 5.2 6.5 1 0.45 1.43 3.34 1.22 6.1
10 1 2 2.65 0.8 6 3 14 1 0.45 1.43 3.34 1.22 2.2
10 2 1 3.65 0.8 5 4 14 1 0.45 1.43 3.34 1.22 5.4
4 5 1 3.65 0.8 5 4 8 1 0.45 1.43 3.34 1.22 13.5
4 1 5 2.65 0.8 9 3 8 1 0.45 1.43 3.34 1.22 1.9

a this dimension is associated with the model axis in FE-analysis.

Boundary conditions are presented in Fig. 3 . The mesh of all models consisted of eight-node isoparametric hexahedral elements ( Hex8 ), which were obtained by uniform extrusion of 2D-quadrilateral-four-node elements ( Quad4 ) at the Z axis of the global coordinate, with a 0.15 mm edge length. The nodes of the models were constrained according to corresponding symmetry planes (plane ZY -fixed in X and plane ZX -fixed in Y ). All nodes on the base were constrained in all directions ( X , Y and Z ), in agreement with the literature .

Fig. 3
Boundary conditions of Class I cavity models.

The polymerization shrinkage was simulated using a thermal analogy, in which a coefficient of thermal expansion (3.33 × 10 −3 °C −1 ) was applied to the resin-composite, with a temperature drop of 1 °C, equivalent of 1% volume shrinkage . This makes the composite contract while having no thermal effect on the tooth. The 1% chosen shrinkage is one of the lowest values found in the literature (even without considering the gel shrinkage period), but simulated cavity walls were only 2 mm thick which tends to minimize the stress by facilitating its deformation. An adhesive contact was created for nodes at the tooth-composite interface for all configurations in this study. This condition established perfect bonding between nodes of the two contact bodies (tooth and restoration) at the interface region.

The materials were all considered homogeneous, linear, elastic and isotropic and their properties are presented in Tables 2 and 3 . A high elastic modulus was selected for the composite since this condition would make differences in shrinkage stress more evident. Maximum principal ( MP ) and maximum shear ( MS ) stresses, generated at different walls, as a function of the C-Factor, were calculated. To reduce the effect of artifacts at singular points in the model, such as cavity angles and shape boundaries, the stress level was plotted for the mean value of eleven nodes (representing 1 mm 2 area) subjected to the highest stress instead of one peak value of the maximal principal stress. The same was done for X , Y and Z axes, separately. The scheme presented in Fig. 4 was used to facilitate the visualization of all regions. The cavity walls were denominated accordingly to planes that defined each wall: ZY _wall ( ZY plane), ZX _wall ( ZX plane) and pulpal_wall (or floor; XY plane). Deformations (maximum principal strains) in models with resin-composite and stress peaks (maximum principal stresses) at the enamel region were also evaluated.

Table 2
Elastic properties of the materials.
Material Elastic modulus (GPa) Poisson’s ratios
Enamel a 80.0 0.30
Dentin b 15.0 0.31
Pulp c 0.002 0.45
Resin-composite d 21.0 0.24

a Rees and Jacobsen (1995) .

b Ausiello, Apicella and Davidson (2002) (22).

c Barink et al. (2003) (25).

d Dejak and Mlotkowiski (2008) .

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Nov 28, 2017 | Posted by in Dental Materials | Comments Off on Finite element analysis of bonded model Class I ‘restorations’ after shrinkage
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