XFEM results were in good agreement with those of the experimental results.
XFEM enabled us to capture a more accurate load at first sign of a crown’s failure.
Increasing margin thickness will increase fracture resistance of all-ceramic crown.
Decreasing convergence angle is recommended to enhance fracture resistance.
Adding an adhesive layer will enhance fracture resistance according to XFEM results.
This study aimed to individually quantify the effects of various design parameters, including margin thickness, convergence angle of abutment, and bonding conditions on fracture resistance of resin bonded glass dental crown systems (namely, glass simulated crown).
Materials and methods
An in vitro experimental test and an in silico computational eXtended Finite Element Method (XFEM) were adopted to explore crack initiation and propagation in glass simulated crown models with the margin thickness ranging from 0.8 to 1.2 mm, convergence angle from 6° to 12°, and three different bonding conditions, namely non-bonded (NB), partially bonded (PB), fully bonded (FB).
The XFEM modeling results of cracking initiation loads and subsequent growth in the glass simulated crown models were correlated with the experimental results. It was found that the margin thickness has a more significant effect on the fracture resistance than the convergence angle. The adhesively bonded state has the highest fracture resistance among these three different bonding conditions.
Crowns with thicker margins, smaller convergence angle and fully bonded are recommended for increasing fracture resistance of all-ceramic crowns. This numerical modeling study, supported by the experimental tests, provides more thorough mechanical insight into the role of margin design parameters, thereby forming a novel basis for clinical guidance as to preparation of tapered abutments for all-ceramic dental crowns.
All-ceramic dental crowns have become more and more popular attributable to their excellent esthetics, inertness and biocompatibility . However, clinical failure rates of all-ceramic prostheses are still substantially higher than metal–ceramic counterparts particularly for molar teeth . Based on clinical guidelines for preparing all-ceramic crowns, greater attention needs to be paid to geometric features. Margin thickness , degree of convergence angle , cusp angles , configurations of margin , etc. are some typical clinically relevant geometric parameters. Of these, margin design appears to be most important, being of a profound effect on fracture resistance of all-ceramic crowns, which can largely determine the extent of removal of sound tooth tissue during preparation of taper abutment for achieving adequate esthetic outcomes and structural integrity of tooth-crown system. On the other hand, not only does minimal preparation, by preserving as much sound tooth tissue surrounding the cervical area (ferrule effect) as possible, increase long term prognosis of the tooth, but also helps preserve pulp vitality, thereby maintaining health status of the host abutment tooth . The concern remains on how these design parameters affect the structural strength of the crown against mechanical loading. To address this issue, both experimental and numerical studies have been conducted to identify an effective design for all-ceramic crowns, especially for those in the posterior region where the occlusal forces are more demanding .
In the literature, the response of a restored tooth to a single load to failure test is often adopted in the laboratory for its simplicity in evaluating the complex stress state of dental materials loaded onto biological hard tissues . For assessing all-ceramic crowns, typically, an axial load is applied on anatomical crown by controlling some factors of interest to identify how these factors affect the catastrophic fracture loads that the crown can resist . While simple, such a test is considered realistic in order to quantify the effects of various factors on the failure load for different designs and dental materials. However, the test often has several major limitations, such as the scatter of recorded failure loads and the different fracture origins compared with what happens with clinically failed crowns, making the test somewhat less reliable clinical guide for structural design and material selection . The other key limitation is that from such a simple test it is difficult to reveal the fractographic details such as crack initiation and propagation.
To restrict the attention onto the aforementioned critical design parameters, some simplified structural system, e.g. the glass simulated crowns to be used in this study, have proven effective, helping determine the critical conditions for marginal failure of complex ceramic crowns under given loading. There have been a number of studies reported in the literature on use of glass specimens to explore fracture behavior of brittle all-ceramic crowns . It must be pointed out that the fracture behavior of glass and ceramics are slightly different. Glass fracture usually initiates from surface flaws, whereas dental porcelain may start from both surface flaws and internal defects. However, glass specimens have significant benefits for laboratory studies , such as elastic modulus close to that of tooth enamel and dental porcelain, clearer fractographic features, easier visualization of internal cracks and less complexity of fabrication . In the tests, bonding with dental resin has been shown to increase the fracture resistance . Again such tests are limited to the effectiveness of capturing fracture initiation in the specimens .
As one of the most powerful simulation techniques, the finite element method (FEM) has been widely employed to explore detailed stress distributions in all-ceramic crowns, to a certain extent reducing the need for time-consuming and expensive specimen preparation and experimental testing . For example, Qasim et al. conducted a FE study for determining the basic stress states in dome-like polymer-filled thin walled glass structures, and they found a shift in maximum tensile stress from the near-contact apex area to dome edges as a consequence of localized ball contact versus more uniform apex loading with a soft material. As a result, the location of maximum tensile stress changed from the apex contact area to the margins. Ford et al.’s FE study explored the stress fields for assessing contact driven damage in ceramic prostheses; but unfortunately formation of the potential crack was not modeled. Kamposiora et al.’s FE study identified the effects of margin design, cement materials, cement thickness, loading direction and magnitude on stress levels and distributions within luting cement. Nevertheless, there have been limited studies available simulating crack formation in crown-like dental restorations to date .
XFEM has proven to be a promising method to simulate crack initiation and propagation . In comparison with the conventional finite element method (FEM), XFEM offers significant advantages for modeling crack initiation and propagation along an arbitrary, mesh-independent and solution-dependent path, with more flexibility and versatility than other numerical techniques available, such as continuum-to-discrete element approach . Therefore, XFEM, instead of standard FEM, was employed to explore the fracture behavior of glass simulated crown models in this study.
In our previous experimental study on single load to failure , it was found that the margin thickness becomes less influential in the tests for the adhesively bonded glass crown-like specimens. In addition, no clear difference was observed in the failure loads of two different margin thicknesses, i.e. 0.8 or 1.2 mm, with different surface treatments . Simple analytical models were also developed to identify the stress concentration at the crowns’ margin. However, the effects of margin design parameters, especially the supporting roles of the margin and adhesion, were not adequately addressed in the previous analysis. An important hypothesis tested was whether the crown design parameters, specifically the degree of convergence, marginal thickness and bonding conditions, play an equal role in contributing to the fracture strength. This study will provide an in vitro experimental and an in silico numerical study for quantifying the effects of these design parameters on the fracture behavior, thereby establishing a mechanical basis for clinical application.
Materials and methods
Experimental tests on glass simulated dental crowns
The margin thickness and convergence angle of taper abutment were considered as the key factors that determine margin stress concentration and potential crack initiation and extension under the axial load to failure tests. As described previously , 60 glass crown-like specimens were prepared from hollow borosilicate glass tubes (10 mm outer diameter and 2.25 mm wall thickness). The glass tubes were initially cut to the desired length with a precision water-cooled diamond saw (Isomet, Buehler Ltd., Lake Bluff, USA) and the ends were positioned in a special holder for generating a flat surface with a #400 grit diamond wheel (Flexovit Abrasive, Victoria, Australia). The cut edges of the glass tubes were ground and polished in a polishing machine (Beuhler UK Ltd., England) (300 rpm). The taper angles of the tubes were fabricated by a customized milling system which used round-end tapered diamond burs (TR-13, MANI Inc., Japan) mounted in a high-speed hand piece positioned in a small bench lathe. After the milling process, the internal glass surfaces were ultrasonically cleaned and left to dry. The glass tubes (15 for each design) were machined for the two different convergence angles (6° or 12°) and two different marginal thicknesses (0.8 or 1.2 mm).
The resin dies used for supporting the glass specimens were generated using chemically cured high strength modified epoxy (Masterflow ® 622, BASF Construction Chemicals, Australia). One-third of the specimens (20 of them) had a thin layer of Vaseline grease smeared between the glass and resin, representing the situation of no bonding with the underlying support. Another 20 specimens had resin poured directly into the milled glass to represent the situation of partial bonding. The resin dies were burnished using a straight diamond impregnated silicone bur to form a butt joint connection between the glass and resin. The glass–resin complexes were stored dry for 7 days until the maximum strength of the resin was reached before testing at room temperature.
In the bonded group (other 20 specimens), the glass specimens were bonded with the resin die individually using a three-step All Bond 3 adhesive resin (Bisco ® ) via light curing for 40 s. The glass specimens were fitted by hand and excess adhesive resin was removed.
The glass-simulated crown was loaded using a 16 mm diameter steel ball with a 12 mm machined flat in a universal testing machine (Shimadzu, Japan), in which a cross head speed of 0.2 mm/min was used until fracture occurred ( Fig. 1 ). Note that the sample was held from the glass end and loaded from the resin end to maintain the same contact area for loading. While this set-up appears opposite to the clinical situation, the loading condition is actually equivalent due to mechanical equilibrium of action and reaction forces.
The fracture load that induced initial cracking was recorded along with subsequent crack extension using a high-speed video recorder. The post-failure fragments were collected and fractographically analyzed optically (50–100×) and with scanning electron microscope (SEM) (Phillip XL30 CP) to identify the fracture origin and direction of crack propagation .
Finite element models
The finite element (FE) models in this study consisted of glass cylinder supported on a resin core abutment loaded by a flattened hard steel ball. The glass crowns were 10 mm high and had an outer surface radius of 5 mm ( Fig. 2 b ). Two design parameters, namely margin thickness ( t ) and convergence angle ( θ ), are considered here first. The range of design parameters was from 0.8 to 1.2 mm for margin thickness and from 6° to 12° for the convergence angle ( Fig. 2 b), respectively.
The three-dimensional (3D) FE model consisted of three parts for the partially bonded (PB) and non-bonded (NB) groups: namely, glass simulated crown, resin core and steel ball indenter as shown in Fig. 2 b and c. The fully bonded (FB) group, as illustrated in Fig. 2 d, included the four parts: glass crown, resin core, adhesive layer and steel indenter. The adhesive layer thickness was 200 μm.
Three-dimensional tetrahedral elements were employed for the glass crown, adhesive cement and resin core; and rigid tetrahedral elements were used for the ball indenter. The relatively fine mesh was placed in the potential cracking domain of glass, adhesive and resin, whereas a relatively coarse mesh for the rigid ball indenter was used to ensure numerical efficiency and accuracy in calculation . Fig. 2 a displayed the mesh of 3D FE model, the mesh size of each component was 0.3 mm for glass crown, 0.1 mm for adhesive cement, 0.3 mm for resin core, and 0.4 mm for steel indenter, respectively, after a convergence test as reported previously .
Fig. 2 b identifies the loading and boundary conditions. The surface-to-surface contact was applied to simulate the surface interaction in the PB and NB groups. The tie constraint was employed to simulate the surface treatment of the FB group. The contact analysis assured the union and loading transfer between the different components, featuring a proper surface interaction for the PB and NB models (an initial gap of 200 μm was assigned between resin core and glass crown for the NB models, which closed on loading). An adhesive layer that fully bonded between the glass and resin components for FB models was assumed.
All the FE models were kinematically constrained on the bottom surface of the glass crown ( Fig. 2 b) to fully replicate the in vitro experimental conditions. A stainless-steel flat surface indenter with a blunt contact surface area of 78.55 mm 2 was chosen to load the top surface of resin core in an axial direction with a maximum load up to 5000 N ( Fig. 2 b). The contact between the steel ball and the top surface of resin was prescribed with a frictional coefficient 0.3. The XFEM fracture analysis was performed to evaluate the formation of margin cracks.
A Matlab code (Mathworks, Australia), implemented by the authors, was used to create numerical fitting surface for the fracture load responses under the three difference bonding conditions. The response surface function returns the coefficients for a linear polynomial considering margin thickness t and convergence angle θ for a best fit in a least-square sense, for the fracture load at crack initiation ( f ( t , θ )), as formulated in Eq. (1) .
The following assumptions were made in the XFEM model: (1) all the materials were considered homogeneous and isotropic ; (2) small slip was permitted between the resin core and glass crown in the partially bonded or non-bonded conditions; and (3) there were no significant flaws in any component. The material properties are listed in Table 1 .
|Materials||Elastic modulus (GPa)||Poisson’s ratio||Density (kg/m 3 )||Tensile strength (MPa)||Fracture toughness (MPa m ½ )|
|Masterflow ® 622 resin||15||0.35||1690||60||1.94|
|Pyrex ® borosilicate glass||62.72||0.2||2230||35||0.77|
Fundamentals of XFEM
In this study, the extended finite element method (XFEM) was applied for simulating crack initiation and propagation in the glass simulated crown system. The associated damage modeling was achieved through the use of a traction-separation law across the fractured surface . Damage associated with crack initiation is determined by the maximum principal stress criterion as indicated in Eq. (2) ,
σ max 0
represents the maximum allowable principal stress of the materials (glass). σe1
σ 1 e
is the maximum principal stress in element ( e ), f e indicates the stress ratio, determining if cracking would occur in this element. A crack is assumed to initiate when the stress ratio reaches a critical value of one ( f e = 1). Specifically, cracking takes place when the maximum principal stress within an element reaches or exceeds the predefined tensile strength of the glass material (35 MPa).
Subsequently, crack growth in XFEM is based upon strain energy release . The critical strain energy release rate G (J/m 2 ) is a material parameter related to fracture toughness K IC (MPa m ½ ), meaning the energy required to propagate the crack per unit area under mode I fracture. As such, it enables quantification of the fracture load for each model when the crack initiates in the structure. Thus the capacity of fracture resistance can be quantified for the models using different design parameters. More detailed background of the mathematics and mechanics about XFEM fracture analysis can be obtained from the literature .