Abstract
Objectives
The aim of this paper is to develop a comprehensive mathematical model for shrinkage stress development in dental composites that can account for the combined effect of material properties, specimen geometry and external constraints.
Methods
A viscoelastic model that includes the composite’s elastic, creep and shrinkage strains, and their interaction with the sample’s dimensions and the external constraint is developed. The model contains two dimensionless parameters. The first one represents the compliance of the external constraint relative to that of the composite sample, and the second controls the rate of shrinkage stress decay through creep. The resulting differential equation is solved for two special cases: zero compliance and zero creep. Predictions for shrinkage stress measurements are then made using the analytical solutions for instruments with different compliances, samples with different thicknesses and composites with different filler fractions.
Results
The model correctly predicts how shrinkage stress increases with time, its dependence on the interaction between the entire system’s compliance and the material properties, and the effect of the filler fraction on its maximum value. Comparisons with reported shrinkage stress measurements have provided very good agreement between theory and experiments.
Significance
The results provided by the model can help to resolve most, if not all, of the seemingly conflicting experimental observations reported in the literature. They can also provide some useful guidelines for optimizing the mechanical performance of dental composite restorations. The compliance ratio, a new parameter derived from the model, represents a fuller description of the constraints of the system.
1
Introduction
Numerous studies , mostly experimental, have been performed to assess the shrinkage stress generated by different dental composites during curing. The shrinkage stress-measuring instruments reported in these studies use either a uniaxial tensile or cantilever bending configuration. In the former, a feedback control system is sometimes used to maintain the thickness of the composite sample, which is usually in the shape of a disk. Thus, different instruments have different compliances, and many studies have been conducted to understand the interactive effect of the material’s mechanical properties (mainly elastic modulus and shrinkage strain) and the instrument’s compliance on the maximum shrinkage stress value that can be achieved . However, many of the reported results appear to contradict with each other. As a result, debate is still ongoing with regard to the relative importance of these factors to the generation of shrinkage stress. In particular, the presumed benefit of increasing the filler fraction of the composite as an attempt to reduce the shrinkage strain and, hopefully, the shrinkage stress has been called into question, with some experimental results showing the exact opposite .
In contrast to experimental work, there have been few theoretical treatments of the topic of shrinkage stress development in dental composites. Previous attempts only provided numerical solutions and property measurements without fully considering the effect of the compliance of the surrounding constraint on the shrinkage stress. It is the aim of this paper to fill this important gap in our study of this class of restorative materials by providing explicit analytical solutions for the problem using a simple Maxwell viscoelastic model based on engineering mechanics. Such analytical solutions will allow the shrinkage stress kinetics and the effect of the material properties and geometrical constraints to be analyzed explicitly and comprehensively.
2
Methods
Consider the model shown in Fig. 1 , in which the shrinkage of a composite of cross-sectional area A and thickness l is constrained by a spring of stiffness k representing the stress-measuring device (or the surrounding tooth tissues). A one-dimensional viscoelastic model is used to describe the mechanical behavior of the composite material . This is justified because shrinkage stress measurement is most commonly made in a uniaxial (1-D) direction. In classical mechanics, a viscoelastic model consists of two constituent elements: an elastic spring and a viscous dashpot. The two elements can be arranged in series (the Maxwell model) or in parallel (the Kelvin-Voight model). In the former case, the elements have the same stress and their strains combine to give the total strain of the model. In the latter case, the opposite is true. Given that the composite material can undergo permanent deformation, the Maxwell model is more appropriate. However, we need to add a third element to account for polymerization shrinkage. Thus, the net strain ( ɛ ) of the composite at all times consists of the sum of three components: the elastic strain ( ɛ e ), the creep strain ( ɛ c ) and the linear shrinkage strain ( ɛ s ). Incrementally over a short time interval dt ,
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