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.
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.
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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
 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
 Ã
; ϵ; θÞ þ KIIffiffiffiffiffiffi
2πr
p fðmÞ
rr_2ðln r
L
 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.
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.
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.
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