Numerical Simulation of Crack Formation in All Ceramic Dental Bridge

Key Engineering Materials Vol. 312 (2006) pp. 293-298 online at http://www.scientific.net © 2005 Trans Tech Publications, Switzerland

Numerical Simulation of Crack Formation in All Ceramic Dental Bridge Qing Li1,a, Ionut Ichim2,b, Jeff Loughran1,c, Wei Li3,d, Michael Swain2,e and Jules Kieser2,f 1

School of Engineering, James Cook University, Townsville, QLD 4811, Australia

2

Department of Oral Science, Faculty of Dentistry, University of Otago, New Zealand

3

School of Aerospace and Mechanical Engineering, University of Sydney, NSW 2006, Australia a

[email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

d

Keywords: Dental bridge, Nonlinear finite element method, Discrete element method, Fracture mechanics, Crack growth, Ceramics.

Abstract. Ceramics have rapidly emerged as one of the major dental biomaterials in prosthodontics due to exceptional aesthetics and outstanding biocompatibility. However, a challenging aspect remaining is its higher failure rate due to brittleness, which has to a certain extent prevented the ceramics from fully replacing metals in such major dental restorations as multi-unit bridges. This paper aims at simulating the crack initiation and propagation in dental bridge. Unlike the existing studies with prescriptions of initial cracks, the numerical model presented herein will predict the progressive damage in the bridge structure which precedes crack initiation. This will then be followed by automatic crack insertion and subsequent crack growth within a continuum to discrete framework. It is found that the numerical simulation correlates well to the clinical and laboratory observations. Introduction As a relatively new technique, all-ceramic dental bridges exhibit outstanding aesthetics and excellent biocompatibility. In contrast to metal restoration, ceramics are recognized as superior in translucency and shape resemblances, which are particularly attractive to the patients who have a high esthetic requirement. For this reason, ceramics have emerged as one of the major dental biomaterials in the start-of-the-art prosthonontic clinic. However, a problematic aspect of such ceramic materials is their limited loading capability due to the relatively low fracture toughness [1] and time-dependent strength decrease caused by progressive crack growth [2]. This has become a major barrier limiting the exploitation of ceramic materials to fully replace metals in such major dental restorations as bridges, where tensile stress levels are sizeable. It would be perferred if the crack initiation and growth in the bridge can be predicted and assessed prior to the bridge construction, thus providing criteria for an improved design. From a biomechanical point of view, the all-ceramic bridge exhibits a new stress status and complex damage mode, which drastically challenge the strength theory and design principles established in traditional prosthetic dentistry. In the context of structural analysis, experimental and numerical approaches have been adopted. Kelly et al reported a series of in-vitro and in-vivo test results in the mid 1990’s [3]. They showed that the failures likely occurred in bridge connectors that link the pontic to the abutment, with approximately 70 to 78% originating from the interface between the core and aesthetic ceramics [3]. More recently, Oh et al [4] investigated the effect of bridge design on the fracture resistance, where a range of connector radii is experimentally tested. Substantial effort has also been devoted to the study of fracture mechanism of dental ceramic structures and materials [5]. Rekow et al analysed crack growth and damage accumulation in dental restorations [6]. Baran et al examined the fatigue characteristics of ceramics in various conditions [7]. To monitor the damage process of bridges, Fischer et al adopted a non-destructive means to material testing [8]. Recently Lohbauer et al explored the fatigue life and failure probability of dental ceramics, where fracture mechanics and statistical methods were employed [9]. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 141.212.55.228-05/07/06,16:02:11)

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Fracture of Materials: Moving Forwards

As a powerful numerical tool, the linear finite element (FE) method has been employed to reveal detailed stress distributions in bridges [10-12]. Pospiech’s work [13] clearly demonstrated that the peak tensile stress occurs in the connector region, and showed further evidence for dentists to improve the bridge design. Lang et al [14] adopted a simplified FE model to predict the load to fracture of an all-ceramic bridge and also validated the numerical results via laboratory tests. However, until now there are no published data available in simulating the crack formation in the dental bridges by making use of nonlinear finite/discrete element methods. From a fracture mechanics perspective, there are two important hypotheses that are conventionally adopted to enable various crack-induced numerical simulations; firstly, the prescription of initial cracks and secondly the adoption of linear elastic fracture mechanics theory. This paper will attempt to explore the possibility of not relying on a priori prescription of a preliminary crack. Rather, the numerical modelling will predict the progressive damage which precedes crack initiation. This will then be followed by automatic crack insertion and subsequent crack growth within a continuum-to-discrete framework. The proposed study is expected to model more realistically the crack formation mechanism in an all-ceramic bridge structure. Materials and Methods Fracture Model. The fracture in such brittle materials as ceramics is generally related to anisotropic phenomenon [15]. The formation and growth of micro-cracks within a brittle solid occur in directions that attempt to maximize the subsequent energy release rate and also minimize the strain energy density [16]. For this purpose, the Rankine rotating crack model has been developed for computing the tensile failure in brittle materials [16]. For mode I dominated fracture problems, the initial failure surface for Rankine rotating crack model can be defined by a tensile failure surface as in Fig. 1a). σ2

σ σ1

Initial Yield Surface

ft

Softening associated with micro-fracture σ = E (1 − ω ) ε ED = E (1 − ϖ )

ft E

a) Initial yield surface

Et

ε b) Softening

Fig. 1. Rankine rotating crack model In practice, the damage accumulates in ceramic materials through the growth of voids and microcracks, which over time may lead to fracture. One may argue that the cyclic loading of the bridge during mastication will create condition whereby stress corrosion assisted and extension of such cracks will enable them to grow to a critical size. However, from clinical experience it is known that in vivo failed ceramic crowns exhibit a large amount of micro-crack damage. Following [15,16], a single scalar damage variable ω is assumed to control strain softening due to micro-cracks. The elastic scalar damage for monotonic loading at a non-local level is then given by

σ = (1 − ϖ ) Eε

(1)

where ϖ is the non-local scalar damage, σ, ε the local stress, strain and E the Young’s modulus. Generally, all finite elements in the non-local region are not simultaneously damaged and hence it is not possible to achieve total damage equivalent to ω = 1 . The problem is overcome by writing the non-local scalar damage variable as a simple function of the non-local strain, i.e.

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ϖ=

ϕ (ε ) − 1 ϕ (ε )

0 ≤ϖ ≤1

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(2)

The function ϕ is chosen to ensure complete damage (i.e., ϖ = 1) if

ε≥

ft f + t E Et

(3)

where ft is the tensile strength and Et is the slope of the softening curve. The damage variable, in effect, reduces the apparent stiffness to zero when ω = 1 (Fig. 1b). Using this procedure a continuum with micro-cracks can be treated as an equivalent anisotropic material with degraded properties orthogonal to the crack surface. A rotating crack model is then used to ensure alignment of cracks with the principal tensile stress axes. Upon achievement of complete softening, discrete cracks are inserted into the continuum solids in line with an average failure indicator evaluated at each nodal point. The evolution of cracks in a continuum dictates that topology update procedures be available to ensure an adequate mesh representation at each stage. Finite Element Modeling. As a preliminary study, a two-dimensional plane strain finite element model is created to represent a 3 unit fixed dental bridge as shown in Fig. 2, where a molar and a first premolar are used as the abutments. The detailed geometries of tooth structure including supporting bone, ligament, enamel, dentine and pulp are captured from a three-dimensional CT scan data of a human (cadaver) mandible and the material properties can be refereed to [1,11,12] In this study, two load cases are taken into account: (i) a distributed load applied normal to the center of occlusal surface of the pontic as shown in Fig. 2; and (ii) four distributed loads applied to the occlusal surfaces of both abutments and pontic as shown in Fig. 6, respectively. A far-field clamped kinematic boundary is set around the side and bottom edges of the supporting bone. pi

Fig. 2. finite element model of the three unit bridge The crack formation processes in these two different load cases are respectively investigated herein. The crack presentation is then compared with the existing experimental and clinical studies to validate the numerical model presented. Results

Load Case (i). The first principal stress contours under Load Case (i) are plotted in Figs. 3a) and b), without and with cracks initiated respectively. From Fig. 3a), a higher tensile stress concentration can be clearly observed in the vicinity of the premolar-pontic connector. At a critical stress level, a crack instigates from the boundary in this region as expected (Fig. 3b).

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a) Stress contour (σ11) before the crack initiated

b) Stress contour (σ11) after the crack initiated

Fig. 3. Load Case(i): the first principal stress distribution before (a) and after (b) cracking

crack

a) t = 0.007 (crack initiation)

c) t = 0.0120 (crack growth)

b) t = 0.0085 (crack growth)

d) t = 0.0150 (fracture of bridge)

Fig. 4. Load Case(i): crack formation and fracture of the bridge Fig. 4 depicts the crack initiation and propagation process at several different time steps under Load Case (i). The single distributed load on the pontic occlusal area results in an initial crack from the lower notch in the mesial connector as displayed in Fig. 4a. This evidently reflects a high tensile stress concentration in this region (Fig. 3a). It should be pointed out here that the crack insertion is completely due to the current status of damage and does not rely on any artificial crack prescription. The subsequent computation will iteratively determine the direction and rate of the crack propagation until the bridge fractures (Fig

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4b-d). In addition, the Rankine rotating crack model allows some new cracks to be initiated if the damage status in other parts of the structure reaches the material’s threshold. Thus, cracks form on the occlusal surface as shown in Fig. 4d), where the tensile stress also exceeds the failure strength. It is found that the above modeling result of crack formation is in a good agreement with the experiment by Oh et al [4] (Fig. 5). The fracture paths in both the numerical and experimental analyses indicate a failure band that connects the weakest connector with the occusal loading point.

Fracture path

Fig. 5. experimental observation of fracture path by Oh et al [4] Load Case (ii). In this case the multiple distributed loads pii are applied not only on the pontic surface, but also the occlusal surface of ceramic crowns. As a result, crack initiations become considerably more complex. As shown in Fig. 6a), cracks initiate from two different, yet representative areas in the bridge: (1) the junction between the ceramic crowns and dentine substructure, I1, I 2 and I3; and (2) the boundary notches in the left and right connector, i.e. B1 and B2. It is interesting to note that due to the layered structure of the restored abutment, the cracks initiate near the loading areas from the interface between ceramic and dentine, which experience higher tensile stresses. In fact, the sub-surface cracks beneath ceramic crowns have been observed in many clinical and experimental studies, e.g. [3]. The numerical simulation provides supportive evidence for such the crack initiation and growth. The failure mode under Load Case (ii) differs from that under Load Case (i). The fracture path develops from both the left and right lower connector notches. This reflects a different stress and damage status due to the application of these four distributed loads. It should be noted that although the cracks initiate from the ceramic crown-dentin interface at a very early stage, these original interfacial cracks may not propagate rapidly towards fracture failure in the crowns. Instead, the catastrophic fracture more likely occurs in the gingival parts of the pontic as shown in Fig. 6b). pii

pii I1

B1

B2

I2

a) t = 0.008 (crack initiation)

I3

b) t = 0.0175 (fracture of bridge)

Fig. 6. Load Case (ii): crack formation and fracture of the bridge Discussions

During normal biting functions, oral structures are loaded within their elastic limit, and hence most existing FE studies applied to dental restorations employ a linear solver. Although linear FEA can bring important information about the location and type of potentially dangerous stresses, it cannot realistically follow the cracking process. After cracks initiate, a complex process of crack opening, closing and growth may take place and necessiates a nonlinear FE procedure. As breakage is one of the most typical failure patterns in all-ceramic restorations, use of a nonlinear code able to produce

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crack formation here completely due to the current status of damage, as the one employed in this paper, will be of special significance in improving the design and reducing the clinical failure rate. As a preliminary numerical investigation into the crack formation in the bridge, a representative plain strain model has been adopted in this study, which at first sight may seem unrealistic for the type of geometry and loading of the dental bridges. However, as the first approximation such a simplification has been extensively used in prosthetic dentistry and is hence considered suitable for demonstrating the scope and potential of the Rankine rotating crack model in this paper. The model does produce some meaningful crack analysis results and matches very well the observations of the failure in such bridge structures. With further development it is believed that this approach will enable detailed 3D modeling of the complex loading conditions in multi-unit all ceramic dental bridge in the future. It is also possible to incorporate the influence of stress corrosion and fatigue into the model. Summary This paper employs a nonlinear finite-discrete element modeling method to examine the accumulation of damage in the all-ceramic dental bridge. A Rankine rotating crack model with an automatic mesh topology update procedure is used to evolve damage, which avoids artificial prescription of initial cracks in existing studies. The presentation of crack initiation in the bridge is numerically simulated for two different load cases. The fracture cracks predicted from numerical model show a very good agreement with the existing experimental and clinical observations. The results of this paper and the methodology presented are significant for a better understanding of the fracture strength of all-ceramic bridges and provide important data for the bridge optimization. Acknowledgement The financial support from CDG scheme at James Cook University is grateful. References

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