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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

http://dx.doi.org/10.1751-6161/& 2015 El

nCorresponding autE-mail address:

Research paper

Modified maximum tangential stress criterionfor 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

1016/j.jmbbm.2015.11.037sevier Ltd. All rights rese

hor. Tel.: þ1 979 847 5690ppark@civil.tamu.edu (P.

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.

rved.

; fax: þ1 979 458 0780.Park).

1. Introduction

Veneers, made from dental porcelain or composites, are usedin dentistry to protect the tooth's surface from damage and toimprove 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 dentalprostheses to restore the missing parts of teeth (Mosharrafet al., 2011; Gostemeyer et al., 2010; Dittmer et al., 2009;Guazzato et al., 2004; Fischer et al., 2008; Kim et al., 2011). Atthe interface of zirconia and veneer, a crack may be createdand grow during the service life of the restored tooth, andlead to a fracture such as edge chipping (cohesive failure) ordelamination (interfacial failure) (Chai et al., 2014). Recentpublications 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

Fig. 1 – Configuration of the modified three point bend specimen used by Wang et al. (2014a). The geometry of the specimenis: a¼10, w¼20, l2¼2, l¼20, thickness¼5, s¼0 or 4 (all dimensions in mm). The crack angle, ω, varies to create different mixedmode 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

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 thefracture test method developed by Charalambides et al. (1989)has widely been used by many researchers, it only covers anarrow range of mixed mode loading conditions at the inter-face crack tip. Wang et al. (2014a) recently conducted a set ofexperiments on the zirconia/veneer interface using a mod-ified three point bend specimen (Fig. 1). By changing thegeometric parameters of this specimen, Wang et al. (2014a)obtained the fracture toughness and crack kinking anglesover the wide range of mixed mode loading. Wang et al.(2014a) experimental data showed that the veneer is weakerthan the bonded zirconia/veneer interface, which explainsthe clinical phenomenon that veneer chipping rate is largerthan interface delamination rate.

In order to estimate the kinking angle of the fracture at theinterface crack tip, Wang et al. (2014a) investigated threetraditional fracture criteria: maximum tangential stress (MTS)(Yuuki and Xu, 1992), KII¼0 (Cotterel and Rice, 1980), andenergy release rate (G) (He and Hutchinson, 1989) criteria. TheMTS fracture criterion employed by Wang et al. (2014a), was asimplified version of the original well-known MTS criterionproposed by Erdogan and Sih (1963). This simplified MTScriterion was suggested by Yuuki and Xu (1992) for a mixedmode fracture analysis of interface cracks. The MTS criterionpredicts that a crack propagates in the direction of themaximum tangential stress in the vicinity of the crack tip.The application of this criterion was limited to specialcombination of materials (having a specific bi-material con-stant, ε) because it ignores the role of critical distance ingoverning stress field equations. In addition, Yuuki and Xu(1992) used only the singular stress field (the terms associatedwith stress intensity factors) to develop their criterion, anddid not consider the effect of non-singular higher orderterms. The G criterion, proposed by He and Hutchinson(1989), states that, at a bi-material crack tip, a fracture occursin the direction where the energy release rate is maximum,and the crack kinking conditions depend on the relativetoughness of the materials at the interface. The KII¼0criterion, proposed by Cotterell and Rice (1980), also assumesthat fracture occurs in direction where the mode II stressintensity 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.

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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 areshown in Fig. 1. The linear elastic stress field around theinterface crack tip is expressed in terms of a series expansionas given in Eq. (1). In Eq. (1), (m) denotes material number(m¼1 or 2), and (r and θ) are the polar coordinates with theorigin at the bi-material crack tip. The parameter R(m) is amaterial parameter defined for each material, which is afunction 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 theradial, tangential, and shear stresses at the material (m),respectively. The parameters frr, fθθ and frθ in Eq. (1) areknown functions of ln r=L

� �, θ, and ε. The coefficients KI and

KII are the stress intensity factors (MPa m0.5) associated withmode I (opening) and mode II (sliding), and T is the T-stress.More details about the parameters, given in Eq. (1), can befound in Mirsayar (2014).

sðmÞrr ¼ KIffiffiffiffiffiffi

2πrp f ðmÞ

rr_1ðln rL

� �; ϵ; θÞ þ KIIffiffiffiffiffiffi

2πrp f ðmÞ

rr_2ðln rL

� �; ϵ; θÞ

þ4T

RðmÞ cos2ðθÞ þH:O:T

sðmÞθθ ¼ KIffiffiffiffiffiffiffiffi

2πrp f ðmÞ

θθ_1ðlnrL

h i; ϵ; θÞ þ KIIffiffiffiffiffiffiffiffi

2πrp f ðmÞ

θθ_2ðlnrL

h i; ϵ; θÞ

þ4T

RðmÞ sin2ðθÞ þH:O:T

τðmÞrθ ¼ KIffiffiffiffiffiffiffiffi

2πrp f ðmÞ

rθ_1ðlnrL

h i; ϵ; θÞ þ KIIffiffiffiffiffiffiffiffi

2πrp f ðmÞ

rθ_2ðlnrL

h i; ϵ; θÞ

�4T

RðmÞ sin ðθÞ cos ðθÞ þH:O:T ð1Þ

2.2. MMTS criterion

According to the MMTS criterion, a crack propagates in thedirection where the tangential stress, sðmÞ

θθ , reaches its criticalvalue, sðmÞ

C , at a critical distance, rðmÞc , from the crack tip

(Mirsayar, 2014). The critical distance, rðmÞc , defined in Eq. (2)

is a material property that is independent from the loadingand boundary conditions (Mirsayar, 2014; Taylor, 2008).

rðmÞc ¼ 1

2πKðmÞIC

sðmÞC

!2

ð2Þ

where sðmÞC and KðmÞ

IC are the tensile strength and mode Ifracture toughness of each material, respectively. The crackkinking angle, θðmÞ

0 , is determined by satisfying the followingequations:

∂sðmÞθθ∂θ

� �rðmÞc ;θðmÞ

0

¼ 0

∂2sðmÞθθ

∂θ2

� �rðmÞc ;θðmÞ

0

o0

8>>>><>>>>:

ð3Þ

Replacing the extended form of tangential stress fromEq. (1) into Eq. (3), the crack kinking angle at the bi-materialcrack 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 considerthe effect of T-stress as shown in Eq. (4). By applying thecondition for the crack propagation, sðmÞ

θθ ¼ sðmÞC , with the first

three terms of Eq. (1), the onset of the fracture can bepredicted as shown in Eq. (5).

∂sðmÞθθ∂θ ¼ 0- KIffiffiffiffiffiffiffiffiffiffi

2πrðmÞc

p ∂f ðmÞθθ_1∂θ þ KIIffiffiffiffiffiffiffiffiffiffi

2πrðmÞc

p ∂f ðmÞθθ_2∂θ þ 4T

RðmÞ sin ð2θÞ ¼ 0-θðmÞ0

∂2sðmÞθθ

∂θ2

� �rðmÞc ;θðmÞ

0

o0

8>>><>>>:

ð4Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi2πrðmÞ

c

qsðmÞc ¼ KIf

ðmÞθθ_1 þ KIIf

ðmÞθθ_2 ¼KðmÞ

IC �4Tffiffiffiffiffiffiffiffiffiffiffiffiffi2πrðmÞ

p

RðmÞ sin 2ðθðmÞ0 Þ ð5Þ

3. Results and discussion

The details of the effect of T-stress on the fracture initiationconditions at the interface crack tip can be found in Mirsayar(2014) and Mirsayar and Park (2015). The kinking angles andfracture toughness of the zirconia/ veneer interface measuredby Wang et al. (2014a) are compared to the theoreticalpredictions using the traditional G, KII¼0 and Yuuki andXu's simplified MTS criteria in Fig. 2a and b. The analyticalpredictions shown in Fig. 2 do not consider the effect of T-stress. Wang et al. (2014a) pointed out that the traditionalfracture criteria without considering T-stress tend to over-estimate the crack kinking angles when compared to theexperimentally measured values. This trend can also beobserved in Fig. 2a. In addition, Wang et al. (2014a) mentionedthat the fracture toughness measured under mixed modeloadings are larger than the mode I fracture toughness (KIC).As shown in Fig. 2a, the experimentally measured fracturetoughness are still larger than the predicted values using thetraditional MTS criterion.

Wang et al. (2014a), also investigated the effect of T-stresson the predictions of kinking angles using G and KII¼0criteria. 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 simplifiedMTS criterion, because of its mathematical limitations by notusing the critical distance theory (see Yuuki and Xu (1992)and Mirsayar and Park (2015) for more details). Fig. 3acompares the kinking angles predicted by G, KII¼0, andMMTS criteria considering the effect of T-tress to the experi-mental data. Although considering the T-stress termimproves the predictions, the modified G and KII¼0 criteriado not still provide a satisfactory accuracy in predicting thekinking angles. On the other hand, as shown in Fig. 3a, theMMTS criterion estimates the kinking angles with a higheraccuracy than other modified fracture criteria. By consideringthe T-stress term, the MMTS criterion predicts the kinkingangles lower than the traditional MTS criterion because of thenegative values of the T-stress (see Wang et al. (2014a) for thedetails of T-stress calculation).

The effect of the T-stress sign (positive or negative) on thecrack kinking angle is discussed in Mirsayar et al. (2014) andMirsayar and Park (2015), in detail. According to Mirsayar et al.(2014), the negative T-stress decreases the kinking angles in

Fig. 2 – Evaluation of the fracture initiation by differentfracture 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 toshow the contribution of each fracture mode, and theanalytical predictions are reproduced using the methodssuggested by the following papers; MTS criterion (Yuuki andXu, 1992), G criterion (He and Hutchinson, 1989), and KII¼0criterion (Cotterell and Rice, 1980).

Fig. 3 – Evaluation of the fracture initiation by differentfracture criteria considering the effect of T-stress; (a) thecrack kinking angles and (b) the fracture toughness. Theanalytical predictions are reproduced using the methodssuggested by the following papers; MMTS criterion(Mirsayar, 2014; Mirsayar and Park, 2015), MTS criterion(Yuuki and Xu, 1992), G criterionþT (Wang et al., 2014a), andKII 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

the mixed mode fracture, and the positive T-stress hasopposite effect, which cannot be considered by the traditionalMTS criterion. The comparisons of the predictions by the MTS,MMTS, and test data shown in Fig. 3a demonstrate the effect ofthe negative T-stress. The experimentally measured fracturetoughness are compared with the MTS and MMTS predictionsin Fig. 3b. It is obvious that the MMTS criterion successfullypredicts the mixed mode fracture toughness with a higheraccuracy than the MTS criterion. Based on the MMTS criterion,the negative T-stress has an effect of increasing the mixedmode fracture toughness, and hence, the MTS predictionsmust be lower than the test data (Mirsayar, 2014). Fig. 3b

clearly shows this effect of T-stress on fracture toughness. InFig. 3b, the mode I fracture toughness of veneer was selectedto be 0.92 MPam0.5 as reported in Wang et al. (2014a), and thecritical distance rc¼0.2 mm is selected based on the regularrange of the critical distances reported for ceramic materials(Aliha and Ayatollahi, 2012).

4. Conclusion

The fracture criteria for predicting the kinking angle andfracture toughness of the zirconia/veneer bi-material crackswere investigated focusing on the role of T-stress. The crack

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kinking angles under mixed mode loadings experimentallyobtained by Wang et al. (2014a) were compared with thepredicted values using the MTS, G, and KII¼0 criteria with andwithout considering the T-stress. It turned out that the G andKII¼0 criteria do not properly predict the kinking angleswhether those criteria consider T-stress effect or not. TheMMTS criterion employing the concept of critical distanceand considering the effect of T-stress successfully predictedthe crack kinking angles of the zirconia/veneer interface. Themixed mode fracture toughness are predicted by the MTS andMMTS criteria, and compared to Wang et al.'s experimentaldata. By taking into account the effect of T-stress, the MMTScriterion showed a good agreement with the experimentaldata. It can be concluded that the MMTS criterion is capableof predicting both the fracture initiation angle and thefracture resistance of zirconia/veneer interface with a higheraccuracy than other fracture criteria. While no standardrecommendation is currently available in the prediction andmeasurement 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 byZachry Department of Civil Engineering at Texas A&M Uni-versity. Any opinions, findings, conclusions, and recommen-dations expressed in this paper are those of the authors aloneand do not necessarily reflect the views of the sponsoringagency.

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