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, 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.