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Introduction

Resin composites provide excellent esthetic outcomes and the ability to use minimal intervention approaches to preserve the tooth structure . Resin composites have therefore become commonly used as direct filling materials for both anterior and posterior restorations . However, dental resin composites undergo shrinkage during polymerization. Shrinkage strain is an inherent property of resin composites alongside Poisson’s ratio and Young’s modulus. The magnitude and kinetics of shrinkage strain are influenced by material compositions . Shrinkage is restricted when a resin composite is bonded to the cavity walls. Under such constraints, some or all of the negative shrinkage strain is replaced with positive elastic and creep strains. The stress thus developed may cause interfacial debonding . Although no direct evidence exists to categorically show that shrinkage stress reduces the longevity of composite restorations, numerous in vitro studies have shown that it is still a concern .

To correlate the constraints afforded by different cavity-preparation shapes with shrinkage-stress development, Feilzer et al. introduced the concept of the configuration factor (C-factor), which is defined as the ratio between the free surface area and the bonded surface area of a dental restoration. Their work suggested that the higher the C-factor of a restoration, the higher the shrinkage stress. The instrument they used to measure shrinkage stress was noncompliant, in that the thickness of the specimen is maintained throughout the experiment. Any contraction of the specimen detected by a length gauge, for example, is immediately recovered by a feedback response imposed by the control unit of the instrument .

Such a feedback response is absent in a compliant instrument. Numerous studies have used a cantilever-based instrument (tensometer) to study the influence of the instrument’s compliance and the specimen’s C-factor on the development of polymerization shrinkage stress . The instrument’s compliance can be adjusted by placing the specimen at different positions along the length of the cantilever beam . These later studies showed that the effect of the C-factor actually depends on the compliance of the stress-measuring instrument. Shrinkage stress does increase with the C-factor with noncompliant instruments . However, with compliant instruments, some studies showed that higher C-factors produced lower shrinkage stress , while other studies showed no significant correlation between the two . As a result, the validity of the C-factor as a predictor of shrinkage stress was called into question. This is particularly relevant since, clinically, the remaining tooth housing a dental restoration is not completely rigid; its compliance varies with the cavity geometry and remaining wall thickness.

Based on engineering mechanics, Fok developed a comprehensive model for laboratory shrinkage-stress measurement that took into account the mechanical properties of the resin composite, dimensions of the cylindrical specimen and compliance of the stress-measuring instrument. According to this theory, the shrinkage stress of a composite is proportional to the product of its final shrinkage strain and elastic modulus, modulated by the compliance of the instrument relative to that of the resin composite .

Among the many correct predictions made, the model is able to predict the reduction in shrinkage stress with an increasing C-factor when a compliant stress-measuring instrument is used. However, it is unable to show the opposite for a noncompliant stress-measuring instrument because the model, being only one-dimensional (1-D), ignores the lateral constraints at the bonded surfaces. In a recent study , the above 1-D shrinkage-stress model was extended to 3-D by using 3-D Hooke’s law to include the additional axial strain/stress due to the lateral constraints through the Poisson’s ratio effect. Common materials undergo lateral contraction when stretched and undergo lateral expansion when compressed; Poisson’s ratio is defined as the ratio of the lateral strain to the axial strain. Additional stress will be created in the axial direction when the lateral deformation is constrained. A new dimensionless parameter ( α ) is introduced in the aforementioned theory which defines the relative thickness of the boundary layers that are adjacent to the bonded surfaces and are subjected to multi-axial stresses. The new 3-D model can predict most, if not all, of the shrinkage stress and strain results reported, irrespective of the compliance of the instrument. As an example, the new theory correctly shows that the axial shrinkage strain, ε _st , as measured by the bonded-disk method depends on the specimen’s C-factor ( F _c ), Poisson’s ratio ( ν ) and the relative boundary-layer thickness ( α ) via the following equation:

where ε _{s 0} is the actual linear shrinkage strain of the resin composite.

Central to the theory is the assumption that the boundary-layer thickness is proportional to the specimen’s diameter, which in turn is based on the assumption of geometrical similarity. Provided the specimen is sufficiently long for it to complete its transition from being fully constrained at the boundaries to having no lateral constraints away from the boundaries, its hourglass shape near each of the boundaries should remain the same, irrespective of its size. Using shrinkage-strain data reported in the literature , it was concluded that the effective thickness of the boundary layer was ∼11.5% of the specimen’s diameter . However, the actual thickness of the boundary layer is expected to be larger due to the aforementioned gradual transition.

In this study, experiments were performed to measure the boundary-layer thickness of cylindrical specimens used for shrinkage stress/strain measurement in order to assess the validity of the aforementioned theory . Resin composite specimens with different diameters, thicknesses and, hence, C-factors were used to study their effect on the thickness of the boundary layers. Different resin composites were also used to test the generality of the assumption.

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Methodology