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Available online at www.sciencedirect.com
Research paper
Modified maximum tangential stress c...
et al. (2014b) examined the fracture toughness of the zirconia/
veneer interface using bi-layered four point bending speci...
2. Analytical method
2.1. Elastic stress field around the interface crack tip
The three point bending specimen tested by Wa...
the mixed mode fracture, and the positive T-stress has
opposite effect, which cannot be considered by the traditional
MTS ...
kinking angles under mixed mode loadings experimentally
obtained by Wang et al. (2014a) were compared with the
predicted v...
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Modified maximum tangential stress criterion for fracture behavior of zirconia veneer interfaces

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The veneering porcelain sintered on zirconia is widely used in dental prostheses, but
repeated mechanical loadings may cause a fracture such as edge chipping or delamination.
In order to predict the crack initiation angle and fracture toughness of zirconia/veneer bilayered
components subjected to mixed mode loadings, the accuracy of a new and
traditional fracture criteria are investigated. A modified maximum tangential stress
criterion considering the effect of T-stress and critical distance theory is introduced, and
compared to three traditional fracture criteria. Comparisons to the recently published
fracture test data show that the traditional fracture criteria are not able to properly predict
the fracture initiation conditions in zirconia/veneer bi-material joints. The modified
maximum tangential stress criterion provides more accurate predictions of the experimental
results than the traditional fracture criteria

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Modified maximum tangential stress criterion for fracture behavior of zirconia veneer interfaces

  1. 1. www.elsevier.com/locate/jmbbm Available online at www.sciencedirect.com Research paper Modified maximum tangential stress criterion for fracture behavior of zirconia/veneer interfaces M.M. Mirsayar, P. Parkn Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USA a r t i c l e i n f o Article history: Received 28 September 2015 Received in revised form 29 November 2015 Accepted 30 November 2015 Available online 23 December 2015 Keywords: Dental prosthesis Zirconia/veneer bi-material joint Interface crack Fracture criteria Modified maximum tangential stress criterion a b s t r a c t The veneering porcelain sintered on zirconia is widely used in dental prostheses, but repeated mechanical loadings may cause a fracture such as edge chipping or delamination. In order to predict the crack initiation angle and fracture toughness of zirconia/veneer bi- layered components subjected to mixed mode loadings, the accuracy of a new and traditional fracture criteria are investigated. A modified maximum tangential stress criterion considering the effect of T-stress and critical distance theory is introduced, and compared to three traditional fracture criteria. Comparisons to the recently published fracture test data show that the traditional fracture criteria are not able to properly predict the fracture initiation conditions in zirconia/veneer bi-material joints. The modified maximum tangential stress criterion provides more accurate predictions of the experi- mental results than the traditional fracture criteria. & 2015 Elsevier Ltd. All rights reserved. 1. Introduction Veneers, made from dental porcelain or composites, are used in dentistry to protect the tooth's surface from damage and to improve the esthetics of a tooth. Since the veneering porce- lain sintered on zirconia has high strength, the zirconia- based bi-layered restorations are widely used in dental prostheses to restore the missing parts of teeth (Mosharraf et al., 2011; Gostemeyer et al., 2010; Dittmer et al., 2009; Guazzato et al., 2004; Fischer et al., 2008; Kim et al., 2011). At the interface of zirconia and veneer, a crack may be created and grow during the service life of the restored tooth, and lead to a fracture such as edge chipping (cohesive failure) or delamination (interfacial failure) (Chai et al., 2014). Recent publications in prosthodontics field showed a vital need of analytical research on fracture mechanics of restored teeth as they undergo a complex combination of mechanical loadings (Chai et al., 2014; Wang et al., 2014a, 2014b; Kosyfaki and Swain, 2014; Kotousov et al., 2011). A literature review reveals that the previous investigations on the zirconia/veneer interface have mostly focused on the improvement of the interfacial bond strength using different surface treatments rather than the analytical modeling and prediction of the interface fracture (Mosharraf et al., 2011; Fischer et al., 2008; Kim et al., 2011). The chipping and delamination at the zirconia/veneer interface is bi-material mixed mode crack problems, and the use of fracture mechanics concepts for the fracture of the dental restorations has increased during the past few years (Kotousov et al., 2011; Gostemeyer et al., 2012). Gostemeyer et al. (2012) and Wang http://dx.doi.org/10.1016/j.jmbbm.2015.11.037 1751-6161/& 2015 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ1 979 847 5690; fax: þ1 979 458 0780. E-mail address: ppark@civil.tamu.edu (P. Park). j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 9 ( 2 0 1 6 ) 2 3 6 – 2 4 0
  2. 2. et al. (2014b) examined the fracture toughness of the zirconia/ veneer interface using bi-layered four point bending speci- mens suggested by Charalambides et al. (1989). Although the fracture test method developed by Charalambides et al. (1989) has widely been used by many researchers, it only covers a narrow range of mixed mode loading conditions at the inter- face crack tip. Wang et al. (2014a) recently conducted a set of experiments on the zirconia/veneer interface using a mod- ified three point bend specimen (Fig. 1). By changing the geometric parameters of this specimen, Wang et al. (2014a) obtained the fracture toughness and crack kinking angles over the wide range of mixed mode loading. Wang et al. (2014a) experimental data showed that the veneer is weaker than the bonded zirconia/veneer interface, which explains the clinical phenomenon that veneer chipping rate is larger than interface delamination rate. In order to estimate the kinking angle of the fracture at the interface crack tip, Wang et al. (2014a) investigated three traditional fracture criteria: maximum tangential stress (MTS) (Yuuki and Xu, 1992), KII¼0 (Cotterel and Rice, 1980), and energy release rate (G) (He and Hutchinson, 1989) criteria. The MTS fracture criterion employed by Wang et al. (2014a), was a simplified version of the original well-known MTS criterion proposed by Erdogan and Sih (1963). This simplified MTS criterion was suggested by Yuuki and Xu (1992) for a mixed mode fracture analysis of interface cracks. The MTS criterion predicts that a crack propagates in the direction of the maximum tangential stress in the vicinity of the crack tip. The application of this criterion was limited to special combination of materials (having a specific bi-material con- stant, ε) because it ignores the role of critical distance in governing stress field equations. In addition, Yuuki and Xu (1992) used only the singular stress field (the terms associated with stress intensity factors) to develop their criterion, and did not consider the effect of non-singular higher order terms. The G criterion, proposed by He and Hutchinson (1989), states that, at a bi-material crack tip, a fracture occurs in the direction where the energy release rate is maximum, and the crack kinking conditions depend on the relative toughness of the materials at the interface. The KII¼0 criterion, proposed by Cotterell and Rice (1980), also assumes that fracture occurs in direction where the mode II stress intensity factor becomes zero. Wang et al. (2014a) predicted the crack kinking angles of the zirconia/veneer interface using the three traditional fracture criteria, but none of the three criteria was capable of successfully predicting the kinking angles with a satisfactory accuracy. Moreover, Wang et al. (2014a) compares the fracture toughness values measured from the various mixed mode loading tests only to the mode I fracture toughness (KIC), while the measured fracture toughness values vary with the mode mixity. Modeling of the bi-material mixed mode fracture is one of the extensively studied topics in the field of fracture mechanics. The recent publications on this topic show that the first non-singular stress term (T-stress) plays a significant role in predicting the kinking angle and the onset of inter- facial crack propagation (Ayatollahi et al., 2010, 2011; Mirsayar et al., 2014; Mirsayar, 2014; Mirsayar and Park, 2015). Since the effect of the T-stress is significant under mixed mode loadings (mode I and II), it is necessary to take into account the non-singular stress term when dealing with the fracture under complex loading conditions such as the dental restorations. Recently, Mirsayar (2014) proposed a modified version of the MTS criterion, called MMTS, to estimate fracture initiation conditions, i.e. the onset of fracture (Mirsayar, 2014) and the crack kinking angle (Mirsayar and Park, 2015), at the bi-material crack tip. The MMTS criterion utilizes the theory of critical distance pre- sented by Taylor (2008), and also takes into account the effect of T-stress in addition to stress intensity factors. Mirsayar (2014) and Mirsayar and Park (2015) showed that the MMTS criterion successfully predicted the experimentally measured fracture initiation conditions of various bi-materials contain- ing cracks with a higher accuracy than Yuuki and Xu’s (1992) simplified MTS criterion. In this study, the MMTS criterion is applied to estimate both the kinking angle and mixed mode fracture toughness of cracks in zirconia/ veneer bi-material joint. The MMTS pre- dictions are compared with the simplified MTS, KII¼0, and G criteria. The estimated fracture conditions are compared to Wang et al. (2014a) experimental data for zirconia/ veneer bi- material specimens. The effect of T-stress on the predictions provided by each fracture criterion is also discussed. Fig. 1 – Configuration of the modified three point bend specimen used by Wang et al. (2014a). The geometry of the specimen is: a¼10, w¼20, l2¼2, l¼20, thickness¼5, s¼0 or 4 (all dimensions in mm). The crack angle, ω, varies to create different mixed mode conditions at the crack tip. j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 9 ( 2 0 1 6 ) 2 3 6 – 2 4 0 237
  3. 3. 2. Analytical method 2.1. Elastic stress field around the interface crack tip The three point bending specimen tested by Wang et al. (2014a) and an expanded view of the interface crack tip are shown in Fig. 1. The linear elastic stress field around the interface crack tip is expressed in terms of a series expansion as given in Eq. (1). In Eq. (1), (m) denotes material number (m¼1 or 2), and (r and θ) are the polar coordinates with the origin at the bi-material crack tip. The parameter R(m) is a material parameter defined for each material, which is a function of elastic properties and bi-material constant, ε (Mirsayar, 2014). The parameter L is a characteristic length (Mirsayar, 2014; Taylor, 2008), and srr (m) , sθθ (m) and τrθ (m) are the radial, tangential, and shear stresses at the material (m), respectively. The parameters frr, fθθ and frθ in Eq. (1) are known functions of ln r=L Â Ã , θ, and ε. The coefficients KI and KII are the stress intensity factors (MPa m0.5 ) associated with mode I (opening) and mode II (sliding), and T is the T-stress. More details about the parameters, given in Eq. (1), can be found in Mirsayar (2014). sðmÞ rr ¼ KIffiffiffiffiffiffi 2πr p fðmÞ rr_1ðln r L Â Ã ; ϵ; θÞ þ KIIffiffiffiffiffiffi 2πr p fðmÞ rr_2ðln r L Â Ã ; ϵ; θÞ þ4 T RðmÞ cos 2 ðθÞ þ H:O:T sðmÞ θθ ¼ KI ffiffiffiffiffiffiffiffi 2πr p fðmÞ θθ_1ðln r L h i ; ϵ; θÞ þ KII ffiffiffiffiffiffiffiffi 2πr p fðmÞ θθ_2ðln r L h i ; ϵ; θÞ þ4 T RðmÞ sin 2 ðθÞ þ H:O:T τðmÞ rθ ¼ KI ffiffiffiffiffiffiffiffi 2πr p fðmÞ rθ_1ðln r L h i ; ϵ; θÞ þ KII ffiffiffiffiffiffiffiffi 2πr p fðmÞ rθ_2ðln r L h i ; ϵ; θÞ À4 T RðmÞ sin ðθÞ cos ðθÞ þ H:O:T ð1Þ 2.2. MMTS criterion According to the MMTS criterion, a crack propagates in the direction where the tangential stress, sðmÞ θθ , reaches its critical value, sðmÞ C , at a critical distance, rðmÞ c , from the crack tip (Mirsayar, 2014). The critical distance, rðmÞ c , defined in Eq. (2) is a material property that is independent from the loading and boundary conditions (Mirsayar, 2014; Taylor, 2008). rðmÞ c ¼ 1 2π KðmÞ IC sðmÞ C !2 ð2Þ where sðmÞ C and KðmÞ IC are the tensile strength and mode I fracture toughness of each material, respectively. The crack kinking angle, θðmÞ 0 , is determined by satisfying the following equations: ∂sðmÞ θθ ∂θ rðmÞ c ;θðmÞ 0 ¼ 0 ∂2 sðmÞ θθ ∂θ2 rðmÞ c ;θðmÞ 0 o0 8 : ð3Þ Replacing the extended form of tangential stress from Eq. (1) into Eq. (3), the crack kinking angle at the bi-material crack tip can be obtained by Eq. (4), respectively (Mirsayar, 2014). While the traditional MTS criterion uses the first two terms of Eq. (1), the MMTS includes the third term to consider the effect of T-stress as shown in Eq. (4). By applying the condition for the crack propagation, sðmÞ θθ ¼ sðmÞ C , with the first three terms of Eq. (1), the onset of the fracture can be predicted as shown in Eq. (5). ∂sðmÞ θθ ∂θ ¼ 0- KIffiffiffiffiffiffiffiffiffiffi 2πrðmÞ c p ∂fðmÞ θθ_1 ∂θ þ KIIffiffiffiffiffiffiffiffiffiffi 2πrðmÞ c p ∂fðmÞ θθ_2 ∂θ þ 4T RðmÞ sin ð2θÞ ¼ 0-θðmÞ 0 ∂2 sðmÞ θθ ∂θ2 rðmÞ c ;θðmÞ 0 o0 8 : ð4Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2πrðmÞ c q sðmÞ c ¼ KIfðmÞ θθ_1 þ KIIfðmÞ θθ_2 ¼ KðmÞ IC À4 T ffiffiffiffiffiffiffiffiffiffiffiffiffi 2πrðmÞ p RðmÞ sin 2 ðθðmÞ 0 Þ ð5Þ 3. Results and discussion The details of the effect of T-stress on the fracture initiation conditions at the interface crack tip can be found in Mirsayar (2014) and Mirsayar and Park (2015). The kinking angles and fracture toughness of the zirconia/ veneer interface measured by Wang et al. (2014a) are compared to the theoretical predictions using the traditional G, KII¼0 and Yuuki and Xu's simplified MTS criteria in Fig. 2a and b. The analytical predictions shown in Fig. 2 do not consider the effect of T- stress. Wang et al. (2014a) pointed out that the traditional fracture criteria without considering T-stress tend to over- estimate the crack kinking angles when compared to the experimentally measured values. This trend can also be observed in Fig. 2a. In addition, Wang et al. (2014a) mentioned that the fracture toughness measured under mixed mode loadings are larger than the mode I fracture toughness (KIC). As shown in Fig. 2a, the experimentally measured fracture toughness are still larger than the predicted values using the traditional MTS criterion. Wang et al. (2014a), also investigated the effect of T-stress on the predictions of kinking angles using G and KII¼0 criteria. However, they did not consider the effect of T- stress on their predictions using the MTS criterion. In fact, it is not possible to bring the T-stress term into the simplified MTS criterion, because of its mathematical limitations by not using the critical distance theory (see Yuuki and Xu (1992) and Mirsayar and Park (2015) for more details). Fig. 3a compares the kinking angles predicted by G, KII¼0, and MMTS criteria considering the effect of T-tress to the experi- mental data. Although considering the T-stress term improves the predictions, the modified G and KII¼0 criteria do not still provide a satisfactory accuracy in predicting the kinking angles. On the other hand, as shown in Fig. 3a, the MMTS criterion estimates the kinking angles with a higher accuracy than other modified fracture criteria. By considering the T-stress term, the MMTS criterion predicts the kinking angles lower than the traditional MTS criterion because of the negative values of the T-stress (see Wang et al. (2014a) for the details of T-stress calculation). The effect of the T-stress sign (positive or negative) on the crack kinking angle is discussed in Mirsayar et al. (2014) and Mirsayar and Park (2015), in detail. According to Mirsayar et al. (2014), the negative T-stress decreases the kinking angles in j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 9 ( 2 0 1 6 ) 2 3 6 – 2 4 0238
  4. 4. the mixed mode fracture, and the positive T-stress has opposite effect, which cannot be considered by the traditional MTS criterion. The comparisons of the predictions by the MTS, MMTS, and test data shown in Fig. 3a demonstrate the effect of the negative T-stress. The experimentally measured fracture toughness are compared with the MTS and MMTS predictions in Fig. 3b. It is obvious that the MMTS criterion successfully predicts the mixed mode fracture toughness with a higher accuracy than the MTS criterion. Based on the MMTS criterion, the negative T-stress has an effect of increasing the mixed mode fracture toughness, and hence, the MTS predictions must be lower than the test data (Mirsayar, 2014). Fig. 3b clearly shows this effect of T-stress on fracture toughness. In Fig. 3b, the mode I fracture toughness of veneer was selected to be 0.92 MPa m0.5 as reported in Wang et al. (2014a), and the critical distance rc¼0.2 mm is selected based on the regular range of the critical distances reported for ceramic materials (Aliha and Ayatollahi, 2012). 4. Conclusion The fracture criteria for predicting the kinking angle and fracture toughness of the zirconia/veneer bi-material cracks were investigated focusing on the role of T-stress. The crack Fig. 2 – Evaluation of the fracture initiation by different fracture criteria without considering the effect of T-stress; (a) the crack kinking angles and (b) the fracture toughness. The experimental data (Wang et al. 2014a) is replotted to show the contribution of each fracture mode, and the analytical predictions are reproduced using the methods suggested by the following papers; MTS criterion (Yuuki and Xu, 1992), G criterion (He and Hutchinson, 1989), and KII ¼0 criterion (Cotterell and Rice, 1980). Fig. 3 – Evaluation of the fracture initiation by different fracture criteria considering the effect of T-stress; (a) the crack kinking angles and (b) the fracture toughness. The analytical predictions are reproduced using the methods suggested by the following papers; MMTS criterion (Mirsayar, 2014; Mirsayar and Park, 2015), MTS criterion (Yuuki and Xu, 1992), G criterionþT (Wang et al., 2014a), and KII ¼0 criterionþT (Wang et al., 2014a). j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 5 9 ( 2 0 1 6 ) 2 3 6 – 2 4 0 239
  5. 5. kinking angles under mixed mode loadings experimentally obtained by Wang et al. (2014a) were compared with the predicted values using the MTS, G, and KII ¼0 criteria with and without considering the T-stress. It turned out that the G and KII¼0 criteria do not properly predict the kinking angles whether those criteria consider T-stress effect or not. The MMTS criterion employing the concept of critical distance and considering the effect of T-stress successfully predicted the crack kinking angles of the zirconia/veneer interface. The mixed mode fracture toughness are predicted by the MTS and MMTS criteria, and compared to Wang et al.'s experimental data. By taking into account the effect of T-stress, the MMTS criterion showed a good agreement with the experimental data. It can be concluded that the MMTS criterion is capable of predicting both the fracture initiation angle and the fracture resistance of zirconia/veneer interface with a higher accuracy than other fracture criteria. While no standard recommendation is currently available in the prediction and measurement of fracture toughness of the bi-material sys- tems, the results of this study will be useful in standardiza- tion of brittle fracture of such layered dental restorations. Acknowledgments The research presented in this paper was supported by Zachry Department of Civil Engineering at Texas AM Uni- versity. Any opinions, findings, conclusions, and recommen- dations expressed in this paper are those of the authors alone and do not necessarily reflect the views of the sponsoring agency. r e f e r e n c e s Aliha, M.R.M., Ayatollahi, M.R., 2012. Analysis of fracture initia- tion angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int. J. 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