# 1/(2+ R c): A simple predictive formula for laboratory shrinkage-stress measurement

## Objective

This paper presents and verifies a simple predictive formula for laboratory shrinkage-stress measurement in dental composites that can account for the combined effect of material properties, sample geometry and instrument compliance.

## Methods

A mathematical model for laboratory shrinkage-stress measurement that includes the composite’s elastic modulus, shrinkage strain, and their interaction with the sample’s dimensions and the instrument’s compliance has previously been developed. The model contains a dimensionless parameter, R c , which represents the compliance of the instrument relative to that of the cured composite sample. A simplified formula, <SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='1/(2+Rc)’>1/(2+Rc)1/(2+Rc)
1 / ( 2 + R c )
, is proposed here for the normalized shrinkage stress to approximate the original model. The accuracy of the simplified formula is examined by comparing its shrinkage-stress predictions with those given by the exact formula for different cases. These include shrinkage stress measured using instruments with different compliances, samples with different thicknesses and composites with different filler fractions.

## Results

The simplified formula produces shrinkage-stress predictions that are very similar to those given by the full formula. In addition, it correctly predicts the decrease in shrinkage stress with an increasing configuration factor for compliant instruments. It also correctly predicts the value of the so-called flow factor of composites despite the fact that creep is not considered in the model.

## Significance

The new simple formula significantly simplifies the prediction of shrinkage stress for disc specimens used in laboratory experiments without much loss in precision. Its explicit analytical form shows clearly all the important parameters that control the level of shrinkage stress in such measurements. It also helps to resolve much of the confusion caused by the seemingly contradictory results reported in the literature. Further, the formula can be used as a guide for the design of dental composite materials or restorations to minimize their shrinkage stress.

## Introduction

Resin composites are increasingly being used for dental restorations because of their superior tooth-like appearance and improved mechanical properties. However, these polymeric materials shrink during polymerization , which generates stress in a bonded restoration and the surrounding tooth tissues. Debonding caused by the shrinkage stress will subsequently lead to leakage and probably premature failure of the restoration through secondary caries .

Experimentally, shrinkage stress that can be generated by a dental composite is measured by using instruments based on the uniaxial tensile or cantilever bend test . Many experimental studies have been carried out to evaluate factors that may affect the level of shrinkage stress . Examples include those investigating the effects of material properties, such as shrinkage strain and Young’s modulus, on the magnitude of shrinkage stress generated. However, many of the experimental results contradict with each other. For example, Condon and Ferracane showed that a high filler content of the composite led to a high shrinkage stress, but the exact opposite was shown by Goncalves et al. . Geometrically, the ratio between the bonded to the nonbonded area, namely, the configuration or C factor ( F c ), of a restoration has been suggested as a factor that determines the level of shrinkage stress . Specifically, it is claimed that the larger the C factor, the higher the shrinkage stress. Again, other studies have shown the exact opposite .

Most of the confusion mentioned above arises from the lack of understanding in the principles of structural mechanics. According to these principles, mechanical responses such as deformation, stress and failure of a structure depend on three elements: geometrical factors, material properties and loading/boundary conditions. In the case of shrinkage-stress generation, geometrical factors include the size and shape of the specimen or restoration; material properties include Young’s modulus, Poisson’s ratio and shrinkage strain; and loading/boundary conditions are provided by the surrounding constraints in the form of the stress-measuring instrument or remaining tooth tissues. Failure to consider any one of these three elements will lead to an incomplete or even inaccurate understanding of the problem.

It is now recognised that, in addition to the fraction of bonded area, the compliance of the constraint surrounding the resin composite also plays a crucial role in determining the magnitude of the shrinkage stress, and that different instruments or cavity preparations have different compliances. Several experimental studies have investigated the combined effect of the instrument’s compliance, the material’s mechanical properties and the specimen’s dimensions on the maximum shrinkage stress value that can be generated . And to better understand and explain the development of polymerization shrinkage stress, complementary theoretical studies have also been carried out. However, most theoretical treatments of shrinkage-stress development so far provided only numerical solutions for special cases ; the exact relationships between the contributing factors and shrinkage stress were still unclear from these results.

In a recent study , one of the present authors performed a comprehensive theoretical treatment of the laboratory measurement of shrinkage stress using simple disc specimens. By assuming a Maxwell-type viscoelastic model and standard mathematical forms to describe the time-dependency of the material properties, the analytical solution obtained showed clearly the combined effect of specimen geometry, material properties and compliance of the stress-measuring device on the shrinkage stress developed. Central to the solution is the ratio between the compliance of the stress-measuring device and that of the specimen, called the Compliance ratio ( R c ). Shrinkage-stress predictions based on the new solution agreed with many of the experimental results reported, which helped to resolve many of the previous contradictions. The solution given in Ref. was, however, rather complicated mathematically. In this study, we will provide a simplified, approximate form of the solution to ease understanding and numerical calculation. For verification, the simplified solution will be applied to the cases considered previously , as well as new ones involving the effects of the C factor and the supposed flow behaviour of the composite materials .