## Abstract

Zirconia is currently used as a framework material for posterior all-ceramic bridges. While the majority of research efforts have focused on the microstructure and corresponding mechanical properties of this material, clinical fractures appear to be largely associated with the appliance geometry.

## Objective

The objective of this study was to estimate the maximum stress concentration posed by the connector geometry and to provide adjusted estimates of the minimum connector diameter that is required for achieving 20 years of function.

## Methods

A simple quantitative description of the connector geometry in an all-ceramic 4-unit bridge design is used with published stress concentration factor charts to estimate the degree of stress concentration and the maximum stress.

## Results

The magnitude of stress concentration estimated for clinically relevant connector geometries ranges from 2 to 3. Using previously published recommendations for connector designs, adjusted estimates for the minimum connector diameter required to achieve 20 years of clinical function are presented.

## Significance

To prevent clinical fractures the minimum connector diameter in multi-unit bridges designs must account for the loads incurred during function and the extent of stress concentration posed by the connector geometry.

## 1

## Introduction

A recent article describes an investigation aimed at evaluating the fatigue behavior of an yttria-stabilized zirconia (3Y-TZP) and application of the results to develop guidelines for the design of dental bridges that achieve lifetimes exceeding 20 years of clinical function . The authors present an interesting study of the crack propagation behavior of 3Y-TZP achieved under cyclic loading at stress levels that are approximately half those required to initiate fracture under monotonic loading. Experimental results are presented for the fatigue behavior in air and within aqueous conditions. Sub-critical crack growth parameters were determined from fitting power law models to results of the cyclic loading experiments. Then, a Wöhler diagram was constructed using the sub-critical crack growth parameters for the 3Y-TZP, along with results from a determination of the strength under quasi-static loading. The diagram provides a comprehensive description of the number of cycles to failure (i.e. the life) resulting from a particular maximum cyclic stress, and/or the critical cyclic stress for a desired life.

When reviewing the field of dental ceramics it becomes evident that the unique and desirable qualities of zirconia have been recognized for some time . But one outstanding limitation to its widespread application in restorative dentistry is clinical longevity. Indeed, there is now greater emphasis on identifying and adopting clinically relevant *in vitro *test methods that provide more insight on the reliability of dental ceramics in vivo . These comments are relevant to all dental ceramics. One of the many strengths of the presentation by Studart et al. is an application of laboratory results quantifying the mechanical behavior of this ceramic to the clinical setting. First, the authors develop estimates for the maximum tensile stress in dental bridges of 3-, 4- and 5-unit configurations as a function of the connector diameter for a 250 N load on the pontics. The estimated stress distributions are then used with the mechanical properties determined from experiments to estimate the minimum connector diameter ( *d *_{min }) that reduces the probability of failure to <5% over 20 years of function. Of particular note, the estimated *d *_{min }accounts for fatigue induced by chewing and bulk fracture caused by severe clenching or grinding. According to a consideration of both modes of failure, the estimated *d *_{min }ranges from 2.7 mm < for the 3-unit bridge to 4.9 mm < for a 5-unit bridge. This approach has also been used in estimating the *d *_{min }for bridge designs constructed from a Al _{2 }O _{3 }–ZrO _{2 }glass composite and a Li _{2 }O·2SiO _{2 }glass ceramic .

Despite experimental results showing that 3Y-TZP is susceptible to water-assisted cyclic crack growth, the authors provide convincing evidence that it can be used successfully as a framework material for posterior all-ceramic bridges, provided that the connector is appropriately designed. Namely, the study highlights that it is essential to limit the magnitude of maximum stresses that develop under function by using a connector diameter greater than the estimated *d *_{min }. But there is an important aspect of the multiple unit bridge geometry that was overlooked. Studart et al. simplified the calculation of stress by adopting a mechanics of materials approach where the bridge is modeled as a cylindrical beam simply supported between the two abutments. The stress induced by flexure is then estimated using beam theory according to the connector diameter. While this simplified calculation leads to higher stress levels than those estimated from finite element analyses available in the literature , this approach ignores the stress concentration posed by the non-uniform bridge geometry. To obtain truly conservative guidelines for the design of dental bridges, a thorough evaluation of the effect of bridge geometry on stress concentrations is required.

## 2

## Theoretical background

We illustrate the importance of bridge geometry and connector design taking a typical all-ceramic 4-unit bridge as an example ( Fig. 1 (a) ). Features of geometry important to the stress concentration that develops with occlusal loading are indicated in Fig. 1 (b). Occlusal loading of the bridge results in the development of a bending moment ( *M *) acting over the length of the bridge. The simplification adopted by Studart et al. results in a linear stress distribution about the neutral axis, and a maximum tensile stress of *σ *_{nom }at the connector’s surface ( Fig. 1 (c)). However, the abrupt decrease in cross-section area at the connectors causes a stress concentration at the connector root and a maximum stress ( *σ *_{max }) exceeding *σ *_{nom }. There is symmetry in the stress distribution about the neutral axis, but the region of criticality is in tension. By definition, the stress concentration ( *K *_{t }) defines the ratio of maximum stress to either the nominal or net-section stress and *K *_{t }≥ 1. If the stress concentration posed by the connector geometry is small ( *K *_{t }∼ 1), then the estimated maximum stress in the bridge is adequately described by the values presented by Studart et al. . But if the *K *_{t }is appreciable, the maximum stress is substantially larger than *σ *_{nom }and the estimated *d *_{min }required to achieve 20 years of clinical function is insufficient. Of note, several 2D and 3D finite element studies have identified that “high” stress concentrations develop at the connector area . Yet, no study has provided a quantitative description of the magnitude of *K *_{t }in terms of the bridge geometry. Thus, it is essential to look at the issue of stress concentrations in bridge designs a little further.

Stress concentrations that develop at the intersection of the pontics can be described according to some simple geometric features, including the connector diameter ( *d *), the diameter at the pontics ( *D *) and the radius of curvature at their intersection ( *ρ *) as illustrated in Fig. 1 (b). Here, *d *and *ρ *are of primary interest. An examination of manufactured devices provides a representative range of these features and includes 3 mm < *d *< 5 mm and 0.2 mm < *ρ *<0.5 mm; this range is consistent with those used in earlier studies [e.g. ]. Using common stress concentration factor charts for beams subjected to bending, the influence of these geometric features on the *K *_{t }is shown in Fig. 1 (d); the *K *_{t }potentially ranges from approximately 2–3. Thus, using the formulation used by Studart et al. and accounting for the range in *K *_{t }estimated here, the maximum stress on the connector for a 4-unit bridge is determined and presented in Fig. 2 ; the magnitude of maximum stress scales linearly with the *K *_{t }. Note that a maximum stress for *K *_{t }= 1 would be equivalent to that estimated in Fig. 6 of the Studart paper. Also note that this same diagram could be prepared for the 3- and 5-unit configurations as well. Fig. 2 emphasizes the importance of connector geometry on the magnitude of maximum stress. Indeed, fractographic evaluations of all-ceramic bridges have reported that failures typically originate from flaws within the connector region due to the stress concentration . Results from *in vitro *experimental studies of all-ceramic bridges have supported these findings as well .