journal of the mechanical behavior of biomedical materials 39 (2014) 119–128
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Research Paper
Fracture mechanics analyses of ceramic/veneer interface under mixed-mode loading Gaoqi Wanga, Song Zhanga,n, Cuirong Bianb, Hui Kongb a
Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Ministry of Education), School of Mechanical Engineering, Shandong University, Jinan 250061, PR China b Department of Prosthodontics, Qilu Hospital of Shandong University, Jinan 250012, PR China
art i cle i nfo
ab st rac t
Article history:
Few studies have focused on the interface fracture performance of zirconia/veneer
Received 29 March 2014
bilayered structure, which plays an important role in dental all-ceramic restorations.
Accepted 21 July 2014
The purpose of this study was to evaluate the fracture mechanics performance of zirconia/
Available online 29 July 2014
veneer interface in a wide range of mode-mixities (at phase angles ranging from 01 to 901),
Keywords:
and to examine the effect of mechanical properties of the materials and the interface on
Bi-materials
the fracture initiation and crack path of an interfacial crack. A modified sandwich test
Dental ceramics
configuration with an oblique interfacial crack was proposed and calibrated to choose the
Interface toughness
appropriate geometry dimensions by means of finite element analysis. The specimens
Mixed-mode
with different interface inclination angles were tested to failure under three-point bending
Fracture criterion
configuration. Interface fracture parameters were obtained with finite element analyses. Based on the interfacial fracture mechanics, three fracture criteria for crack kinking were used to predict crack initiation and propagation. In addition, the effects of residual stresses due to coefficient of thermal expansion mismatch between zirconia and veneer on the crack behavior were evaluated. The crack initiation and propagation were well predicted by the three fracture criteria. For specimens at phase angle of 0, the cracks propagated in the interface; whereas for all the other specimens the cracks kinked into the veneer. Compressive residual stresses in the veneer can improve the toughness of the interface structure. The results suggest that, in zirconia/veneer bilayered structure the veneer is weaker than the interface, which can be used to explain the clinical phenomenon that veneer chipping rate is larger than interface delamination rate. Consequently, a veneer material with larger fracture toughness is needed to decrease the failure rate of all-ceramic restorations. And the coefficient of thermal expansion mismatch of the substrates can be larger to produce larger compressive stresses in the veneer. & 2014 Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.: þ86 531 88392746; fax: þ86 531 88392746. E-mail address: [email protected] (S. Zhang).
http://dx.doi.org/10.1016/j.jmbbm.2014.07.019 1751-6161/& 2014 Elsevier Ltd. All rights reserved.
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1.
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
Introduction
Bi-materials have been widely applied in aerospace, automotive, marine, microelectronic structures and other fields. In the prosthodontics field, zirconia-based bilayered restorations have been used extensively because the veneering porcelain sintered on zirconia presents a combination of both high strength and excellent esthetics (Liu et al., 2012). However, the weak interface between zirconia and veneer is one of the important factors leading to chipping and delamination (Roediger et al., 2010; Liu et al., 2012). Therefore, it has great significance to understand the fracture performance of zirconia/veneer interface, which has received little attention. The studies on the zirconia/veneer interface were mainly focused on the improvement of bond strength by means of different surface treatments (Fischer et al., 2008; Mosharraf et al., 2011; Kim et al., 2011). The measurement of bond strength was most prevalently performed with shear or tensile bond tests, which often showed cohesive fracture patterns within the veneer layer (Henriques et al., 2012). Recently, fracture mechanics approaches became more popular in the research of interfacial bonding performance of dental restorations (Soderholm, 2010; Jancar, 2011; Kotousov et al., 2011). A fracture mechanics methods proposed by Charalambides et al. (1989) was used to measure the interface toughness between zirconia and veneer (Göstemeyer et al., 2012; Wang et al., in press). Nevertheless, using this method can only yield a narrow range of mode-mixities (characterized by phase angle), which is defined as the ratio between the shear and opening modes at an interfacial crack tip. The actual occlusal loading scenarios are not just limited to near shear or opening conditions, which means an interfacial crack is always subjected to mixed mode conditions. An interfacial crack subjected to mixed mode loading either kinks into substrates or propagates in the interface. The maximum energy release rate criterion (Gmax criterion) (He et al., 1991; He and Hutchinson, 1988, 1989), the zero KII criterion (KII ¼0 criterion) (He and Hutchinson, 1989), and the maximum tangential stress criterion (MTS criterion) (Yuuki and Xu, 1992) can be used to predict the crack kinking behaviors including selection of crack path, the threshold of crack initiation, and the kinking angle. There are also some criteria for evaluating interface fracture (Charalambides et al., 1992; Hutchinson and Suo, 1992; Thurston and Zehnder, 1995), which present the relation between interface fracture toughness and mode-mixity. In general, the toughness of an interface, which can be characterized by stress intensity factors (SIF) or energy release rate (ERR), increases with the increase of phase angle. Consequently, in order to understand the fracture mechanism of bilayered all-ceramic restorations, it is necessary to investigate the zirconia/veneer interface in a wild range of mode-mixities by an effective method. In the past, various experimental methods and test configurations have been used for determining the mixed-mode fracture performance of bi-material interfaces, such as fourpoint bending specimen (Charalambides et al., 1989), unsymmetric end-notched flexure specimen (Sundararaman and Davidson, 1998), double cantilever beam specimen (Cao and
Evans, 1989), sandwich specimens (Suo and Hutchinson, 1989), Brazil-nut-sandwich specimen (Wang and Suo, 1990), and asymmetric four-point bending specimen (O’dowd et al., 1992a, 1992b). However, most of the mentioned test methods are not appropriate for the current zirconia/veneer combination. Four-point bending specimen and end-notched flexure specimen can only provide limited range of I/II mode-mixities; double cantilever beam specimen and sandwich specimens proposed by Suo and Hutchinson are not appropriate for brittle materials with low fracture toughness; Brazil-nutsandwich specimen and asymmetric four-point bending specimen require complicated preparation process or complicated testing fixtures. The veneering porcelain is sintered on the zirconia under a high temperature to produce bilayered structure, and the veneer material which has relatively small fracture toughness turns to viscous-elastic state under high temperature. Therefore, a simple and valid test configuration that is suitable for the current material combination is needed. In this study, a modified beam test configuration was suggested for investigating the mixed mode fracture performance of the interface. The test configuration was proposed based on a sandwich interfacial fracture specimen (Suo and Hutchinson, 1989) and a SCB specimen (Ayatollahi et al., 2006) that were used for measuring mixed mode fracture toughness of homogeneous materials. The purpose of this study was to evaluate the fracture mechanics performance of zirconia/veneer interface by investigating the propagation of an interfacial crack under a wide range of mode-mixities, and to examine the effect of fracture mechanical properties of the materials and the interface on the crack initiation and crack path. The structure of the paper was as follows. First, the specimen was described, and the relations between specimen geometries and phase angle were calibrated by means of several finite element analyses (FEA) to choose appropriate geometry dimensions. Secondly, the specimens were prepared based on the FEA results, and tested with three point bending method. Failure mode, fracture initiation load of all specimens and crack kinking angle of the kinking specimens were recorded. Finally, the experimental results were compared with theoretical results of the fracture criteria, and the clinical guiding significance of the study was presented.
2.
Material and methods
2.1.
Test specimen and numerical analyses
The oblique crack sandwich specimen can be easily made and tested in conventional testing machines. As shown in Fig. 1a, the specimen consists of two pieces of zirconia bonded with an interlayer of veneer. One interface of the two materials is normal to the horizontal plane, and another interface has an inclination angle ω relative to the horizontal direction. The oblique interface contains an interfacial edge pre-crack of a in length. The specimen has a height of w, a thickness of b, and a length of 2l1þl2. The specimen is loaded by the vertical load P in standard three point bending mode with a span of l. s is the load offset distance from the location right above the open end of the crack. When the parameters
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
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Fig. 1 – Schematic diagram of (a) single edge oblique crack specimen and (b) the interfacial crack configuration (right crack tip). σθ, σr, and τrθ are stress components of a point which has coordinates of (r, θ) in polar coordinates with the origin at the crack tip. Table 1 – Mechanical properties of materials. Materials
Elastic modulus E (GPa)
Poisson ratio ν
Fracture toughness KIC (MPa m1/2)
CTE α ( 10 6/1C)
Lava Zirconia IPS e.max Ceram
210 70
0.31 0.27
5.0–10.0 –
10.0 9.5
All data were provided by manufacturers.
s, ω, a/w, and l/w are changed, different phase angles can be obtained. Therefore, calibrations were conducted with the current material combination to relate the phase angle to the geometries. Fig. 1b gives the schematic diagram of the interfacial crack tip configuration (right crack tip), in which zirconia is on the top and veneer is on the bottom. The stress field at the crack tip depends on the two Dundurs' parameters α, β (Dundurs, 1969), which characterize the elastic mismatch of the two materials. The two Dundurs' parameters are defined as follows: 9 μ ðκ 2 þ 1Þ μ2 ðκ 1 þ 1Þ > > α¼ 1 > μ1 ðκ2 þ 1Þ þ μ2 ðκ 1 þ 1Þ = ; ð1Þ μ ðκ 2 1Þ μ2 ðκ 1 1Þ > > β¼ 1 > ; μ1 ðκ2 þ 1Þ þ μ2 ðκ 1 þ 1Þ where μi ¼Ei/2(1þνi) is the shear modulus, κi ¼3–4νi for the plane strain state. Ei is the elastic modulus, νi is the Poisson's ratio, i¼ 1, 2 represents zirconia and veneer, respectively. ε is the oscillation index that defined as follows: ε¼
1 1 β ln : 2π 1þβ
ð2Þ
For the current material combination (Table 1), the plane strain Dundurs' parameters are calculated to be α¼ 0.509, β¼ 0.170, and the oscillation index ε¼ 0.055. The interface fracture parameters including SIFs, ERR, and phase angle, depend on the geometries of specimen and applied load. The complex SIF is defined as (Rice and Sih, 1965; Rice, 1988) K ¼ K1 þ iK2
ð3Þ
where K1 and K2 represent mode I and mode II interface SIFs, respectively. The phase angle ψ of KLiε is expressed as ψ ¼ arctan
ImðKLiε Þ ReðKLiε Þ
:
ð4Þ
where L is an arbitrarily chosen reference length which has no absolute physical significance. In the present study L was chosen to be 1 mm. The phase angle ψ is a measure of the degree of mixity of shear to opening modes on a crack tip. ψ¼901 or 901 represent a crack having nearly zero opening; whereas ψ¼ 0 means a crack subjected to nearly zero shear. The calibration of specimens was performed by the finite element software ABAQUS. All the numerical models presented in this paper were solved under plane strain conditions. Finite element models with different values of the geometry parameters (as shown in Fig. 1) were analyzed. The mechanical properties of the two materials in the model are given in Table 1. Fig. 2 shows a typical mesh pattern for simulating the specimen. The model was meshed to about 21000 8-node second order elements, and a highly refined mesh was used in the region in the vicinity of the crack tip with 36 wedge shaped 6-node second order singular elements. The load and constraints were identical to the schematic diagram of the specimen as shown in Fig. 1a. The calibrations were conducted by applying load of 100 N in each case. A J-integral-based method (interaction integral method) built into ABAQUS was used to obtain the interface SIFs, ERR, and ψ automatically (Shih and Asaro, 1988; ABAQUS theory manual). First, the specimen was calibrated for s¼0, l2 ¼2 mm, different l/w and a/w. The calibration functions for l/w¼ 0.8, 1.0, and 1.2, and for a/w¼ 0.4, 0.5, and 0.6 are plotted in Fig. 3a and b. The parameters were chosen to be a/w ¼0.5, l/w¼ 1.0, and l¼ 20 mm. The interface inclination angle ω 861, 781, 701, 631, 581, 531, and 491 corresponded to ψ of about 01, 151, 301, 451, 601, 751, and 901, respectively. However, the larger ψ required smaller ω, which needed plenty of veneer material. The loading situation of s ¼4 mm was therefore used to provide data for large ψ. The relation between ω and ψ in this situation is plotted in Fig. 3c. The angle ω was set to be 781,
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journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
Fig. 2 – A typical finite element mesh pattern for a specimen.
Fig. 3 – Calibration functions for specimens. l2 ¼ 2 mm, l1 ¼ l¼ 20 mm, and (a) s ¼0, a/w ¼0.5; (b) s¼ 0, l/w¼ 1; (c) s ¼4 mm, l/w ¼1, and a/w¼0.5. Table 2 – Geometric parameters of the specimens. 731, 701, and 651, corresponding to phase angle of about 451, 601, 751, and 901. Corresponding to each ω, the specimens were divided into 7 groups, which were numbered as A86, A78, and A70 for loading situation of s ¼0, and B78, B73, B70, B65 for loading situation of s ¼4 mm, respectively. All the chosen parameters are summarized in Table 2.
2.2.
Specimen preparation and bending test
A total of 42 specimens (6 in each group) were prepared. Lava zirconia (3M ESPE, Seefeld, Germany) as core and IPS e.max Ceram (Ivoclar Vivadent AG, Schaan, Liechtenstein) as veneer were used, because the material combination is commonly used, and has similar values of coefficient of thermal expansion (CTE) (Table 1). 42 Rectangler pre-sintered zirconia blocks with the
Parameter
Value
a w l1 l2 l s ω
10 mm 20 mm 20 mm 2 mm 20 mm 0 or 4 mm 861(A86), 781(A78), 701(A70) (s ¼0) 781(B78), 731(B73), 701(B70), 651(B65) (s¼ 4 mm) 5 mm
b
same dimensions (20 20 5 mm3) and 42 trapezoidal presintered zirconia blocks with corresponding ω were cut from presintered blocks with a diamond saw. The specimens were
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
sintered and cleaned in a sonic bath filled with ethanol for 5 min and gently air dried. A rectangler block and a trapezoidal block for each bend specimen were aligned by a guide and adjusted to make l2 ¼ 2 mm (Fig. 1). A piece of thin graphite paper with length of 10 mm and thickness of 0.05 mm was pasted on zirconia surface to produce approximate interfacial pre-crack. The pasty veneer were then stacked into the space between the two components of zirconia blocks. The zirconia–veneer interface was bonded at 750 1C for 4 min in vacuum followed by a furnace cool. The glass transition temperature of the veneer is 490 1C, therefore most of the thermal residual stresses were produced mainly due to the CTE mismatch during temperature change from 490 1C to room temperature 25 1C. Three point bending test was conducted on each specimen with a test machine (Instron 8801, Instron Co., Canton, MA), and the load–displacement curve was recorded. Crack kinking angles of the kinking specimens were measured by photography using a digital camera.
2.3.
Calculation of interface fracture parameters
The model described in Section 2.1 was used to obtain ERRs and SIFs by changing the load to corresponding mean fracture load of each group of specimens. T-stress, which is the stress parallel to the interface, is mainly composed of thermal residual stresses for the current specimens. Residual stresses were calculated based on the temperature difference from glass transition temperature of the veneer to room temperature ( 465 1C). Then the T-stress was extracted from stress values of 6 points on the circle with center at the crack tip and radius of 1 mm (Fig. 4).
2.4.
Kinking criteria of interfacial crack
2.4.1.
Gmax criterion
The Gmax criterion proposed by He and Hutchinson (1989) suggests that the kinking occurs in the direction where the ERR is maximum. The criterion establishes the relation between SIFs of the kinked crack and that of interfacial crack. Thereby the ERR of kinked crack at each kinking angle can be obtained from interfacial SIFs. The main calculation process is as follows: The SIFs of the kinked crack KI and KII can be expressed from interfacial SIFs K as (He and Hutchinson, 1989; He et al., 1991) pffiffiffiffi ð5Þ KI þ iKII ¼ cKa0iε þ dKa0 iε þ bσ 0 a0
123
where a0 is the length of kinked crack and chosen to be 1 mm, σ0 is T-stress, parameters c¼c1þic2, d¼ d1þid2, and b¼b1þib2 are dimensionless complex functions of kinking angle θ0 , and Dunder's parameters α and β. He and Hutchinson (1989) calculated values of c, d, b for a limited number of combinations of α and β by solving two integral equations. In the present study, a simply FEA method proposed recently was used to calculate c, d, b of the current material combination as functions of θ0 (Jakobsen et al., 2008; Noijen et al., 2012). The ERR of the interfacial crack can be calculated by K21 þ K22 κ1 þ 1 κ2 þ 1 þ : ð6Þ G0 ¼ μ1 μ2 16cosh2 ðεπ Þ The ERR of the kinked crack is expressed as G¼
K2I þ K2II ; 0 E2
ð7Þ
where for plane strain state, Ei0 ¼Ei/(1 v2i ). Therefore, Gmax, the maximum ERR of the kinked crack, was obtained from KI and KII of interfacial crack by finding out the maximum G value of all possible kinking angles. The interfacial crack initiates to kink other than propagate in the interface, if the following condition is satisfied: G0 ΓðψÞ o ; Γ IC Gmax
ð8Þ
where Γ(ψ) is interface toughness at phase angle ψ. ΓIC is mode I fracture toughness of the veneer. The effect of T-stress on the fracture initiation and kinking behavior was also evaluated by the Gmax criterion. η can be used to characterize the level of T-stress (He et al., 1991) pffiffiffi σ0 8 ffi; ð9Þ η ¼ pffiffiffiffiffiffiffiffiffiffi En G0 where
" # 1 1 1 1 1 ¼ þ : 0 0 2 E1 E2 cosh2 ðπεÞ En
ð10Þ
T-stresses were calculated to be approximately σ0 ¼ 5.3 MPa for each group of specimen, which was equivalent to η ¼ 0.5. Therefore, Gmax criterion was implemented with η ¼ 0.5, 0.5, and 0. η¼ 0.5 was approximately equal to compressive stress of σ0 ¼ 5.3 MPa, while η ¼0.5 was nearly equals to tensile stress of σ0 ¼ 5.3 MPa, and η¼ 0 means no Tstress was determined.
Fig. 4 – Calculation of T-stress, which is represented by σ0.
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2.4.2.
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
KII ¼ 0 criterion
According to KII ¼ 0 criterion, kinking angle is identical to the direction which make mode II SIF of the kinked crack equal to 0. Therefore, kinking angle θ0 can be calculated from Eqs. (5), (6), and (7) with KII ¼ 0. Effect of T-stress was also evaluated by this criterion.
2.5.
σθmax Criterion
Yuuki and Xu (1992) put forward maximum tangential stress (MTS) criterion, which concludes that a crack will initiate in the direction where the stress component σθ at the crack tip is maximum. The tangential stress of a point (r, θ) near an interfacial crack tip (Fig. 1) is expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK21 þ K22 Þ r ð11Þ σ θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bðθ; ε; ψÞ cos ε ln L 2 ð2πrÞcoshðπεÞ B is given in reference (Yuuki and Xu, 1992); r was set to be 1 mm. The kinking angle which gives the maximum σθ can be obtained by ∂σ θ ¼0 ∂θ
ð12Þ
And the fracture criterion was described as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK21 þ K22 Þ Bðθ; ε; ψÞZ KIC : Kθ max ¼ 2 coshðπεÞ
ð13Þ
When the value of Kθmax reaches the fracture toughness of veneer KIC, the fracture will occur.
2.6.
Fracture toughness test of veneer
The kinking criteria make a comparison between the fracture toughness of substrates and that of interface. Therefore, the mode I fracture toughness KIC of the veneer was measured using single edge notch bend (SENB) method. 5 Specimens with 3 mm in thickness, 4 mm in height, and 36 mm in length were sawed notches at the center to a depth about 2 mm in height direction, and then loaded to fracture in three-point bending test. The accurate depth of the notches was determined by optical analysis of the fractured surfaces. The notch root radius ρ was approximately 100 μm, which required larger load to fracture from the notch root than from a sharp crack tip. So the true fracture toughness, KIC was obtained by (Damani et al., 1996) KIC ¼ XKm ;
ð14Þ
where Km is SIF calculated from measured values (ISO 15732, 2003), and the parameter X can be obtained by pffiffiffiffiffiffiffiffiffiffi X ¼ tanhð2Y δa=ρÞ; ð15Þ where Y is a correction factor, δa can be approximated on the basis of defects size. Based on the fractography analysis, many small defects were found. Therefore an edge-crack configuration was assumed, and δa was set to be 20 μm, Y was set to be 1.12.
3.
Results and discussions
3.1.
Failure modes
The fracture mode of the specimens in Group A86 (ψ¼ 0) was interfacial fracture, that the crack propagated in the zirconia/ veneer interface. Fracture loads of the A86 specimens were used to calculate interface toughness at ψ¼0. For other groups, fracture occurred in the form of cracks kinking into the veneer (Table 3).
3.2.
Interface fracture parameters
Fig. 5 shows the typical load–displacement curve of a specimen in Group B65 (ψ¼ 901) with fracture load of 679.2 N. It can be seen that the curve was approximately linear until the specimen catastrophically failed, indicating that no plastic deformation and no stable crack growth were found during loading and the specimens fractured in a brittle manner. The fracture loads obtained from the experiments are shown in Table 3. The interface SIFs and interface ERRs were calculated from the mean fracture loads of corresponding specimens. In aggregate the fracture load was larger for the specimens having larger phase angle. And with the increase of ψ, the value of K1 decreased, whereas K2 increased sharply.
3.3.
Crack initiation and kinking angle
3.3.1.
Crack initiation
The three complex coefficients c, d, b were obtained by the FEA method. Values of the coefficients as functions of kinking angle for the current material combination are shown in Fig. 6. Then the parameters were used in the equations of Gmax criterion. The mean fracture load of the five veneer SENB specimens was 11.2(70.78) N. By using the equations in the international standard (ISO 15732, 2003), Km was calculated to be 1.21 MPa m1/2, therefore, KIC ¼ 0.92 MPa m1/2. Critical ERR of the veneer, ΓIC, was obtained by Eq. (7) with KI ¼KIC and KII ¼0, therefore ΓIC ¼11.2 N/m.
Table 3 – Fracture load and failure mode of specimens, and fracture mechanics parameters including stress intensity factors K1, K2, energy release rate G0, and phase angle ψ of the interfacial cracks. Specimens
ψ
Failure loads (7SD)
Failure modes
K1 (MPa m1/2)
K2 (MPa m1/2)
G0 (N/m)
A86 A78 A70 B78 B73 B70 B65
01 151 301 451 601 751 901
394.2 354.7 420.3 513.3 580.5 648.0 673.2
Interfacial Kinking Kinking Kinking Kinking Kinking Kinking
1.10 0.94 0.87 0.67 0.47 0.26 0.01
0.02 0.23 0.48 0.66 0.81 0.93 0.96
10.23 8.02 8.32 7.55 7.47 7.87 7.87
(755.5) (748.0) (765.0) (791.2) (734.5) (785.4) (771.7)
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
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Fig. 5 – Typical load–displacement curve of a specimen (B65). Fig. 7 – Comparison between G0(ψ)/Gmax and Γ(ψ)/ΓIC (η¼ 0). Interface energy release rates G0 were solved by FEA models using fracture loads; maximum energy release rates of kinked crack Gmax were calculated based on Gmax criteria; interface critical energy release rates Γ(ψ) were obtained using Eq. (16); fracture toughness of the veneer ΓIC was measured using the SENB method.
Fig. 6 – Complex coefficients as functions of kinking angle for the current material combination (α ¼0.509, β¼ 0.170).
It is difficult to obtain interface toughness data for large phase angles by experiment because specimens with large phase angle often show kinking failure (Table 3). Therefore, the interface toughness Γ(ψ) was calculated from a toughness law proposed by Hutchinson and Suo (1992): ΓðψÞ ¼ Γ 1 ð1 þ λ tan 2 ψÞ:
Fig. 8 – Kθmax at different phase angles versus the fracture toughness of the veneer.
ð16Þ
Γ1 is the interface toughness for approximate mode I (ψ¼0). So the value of Γ1 was set to be the mean interface ERR of specimens in Group A86 (ψ¼0, Γ1 ¼ 10.23 N/mm), which showed interface fracture. The Γ(ψ) of the interface at other phase angles were estimated by Eq. (16), with λ ¼0.3. G0(ψ)/Gmax and Γ(ψ)/ΓIC (η¼ 0) as functions of phase angle are shown in Fig. 7. It is obvious that based on Gmax criterion an interfacial crack will kink into the veneer when the phase angle is larger than about 151, while keep advancing in the interface with phase angle smaller than 151. This is consistent with experimental results. Apparently, the crack path selection is governed by the toughness of the interface and that of the veneer which change the toughness ratio of the interface to veneer. It can be seen from Fig. 8 that the kinking occurred when the value of Kθmax reached the fracture toughness of the veneer KIC. This is identical to the σθmax criterion. But the Kθmax values were slightly larger than the KIC, because the fracture initiated harder from the notch introduced by graphite paper than from a sharp crack, which was not taken into consideration. Besides, compressive T-stresses had a blocking effect on the crack kinking. The crack kinked into the veneer for most of the specimens because of the low
fracture toughness of veneer. This is coincident with the clinical phenomenon that veneer chipping rate is larger than interface delamination rate. Consequently, a veneer material with larger fracture toughness is needed to decrease the failure rate of all-ceramic restorations. In summary, it is concluded based on all the criteria that the weaker the toughness of interface and substrates, the easier the crack initiation occurs. Moreover, whether an interfacial crack extends within the interface or advances by kinking into either substrate depends on relative toughness of interface to that of the joining materials. If the substrate's fracture toughness is sufficiently larger than interface toughness, the crack will keep in the interface. However, when the fracture toughness of the substrate is comparable to that of interface, whether or not to kink depends on the mode-mixity. For the current right crack tip configuration with veneer on the bottom (Fig. 1), the crack has the tendency to kink into the veneer at positive phase angles (01–901). The experimental results show that the fracture toughness of veneer is comparable to the interface toughness at small phase angles. Since the interface toughness becomes
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journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
larger with the increase of phase angles, cracks at most of the phase angles tend to kink into the veneer. At negative phase angles ( 901–01), the crack has the tendency to kink into the zirconia. However, the fracture toughness of zirconia is much larger than the interface toughness at most phase angles, so that the crack will keep extending in the interface.
3.3.2.
3.4.
Effect of residual stress
Fig. 11 shows the prediction of crack kinking angle taking into account the effect of T-stress using Gmax criterion and KII ¼ 0 criterion. Compared with previous predictions, tensile T-stress (η¼0.5) increases the kinking angle, whereas compressive Tstress (η¼ 0.5) decreases the kinking angle. The prediction of
Kinking angle
The kinking angles were measured by magnifying the photo by a factor of 5 to obtain the real initial kinking angle (Fig. 9d), because the crack changed direction when propagated away from the interface. Fig. 9a–c shows the typical crack paths of specimens with phase angles of 301, 601, and 901, the kinking angles of which are 46.71, 56.51, and 69.21, respectively. Obviously, kinking angle increases as the phase angle increases. Fig. 10 displays the theoretical curves (η ¼ 0) for kinking angles as functions of ψ using Gmax criterion, KII ¼0 criterion, and MTS criterion. Also shown in the figure is the experimental data obtained in this research. Apparently, the three criteria and experimental results followed the same trend. However, the experimental results were almost smaller than the theoretical ones, which was due to the effect of T-stress.
Fig. 10 – Comparison of experimental kinking angles for zirconia/veneer interface (l¼ 1 mm) with predicted values based on Gmax criterion, KII ¼ 0 criterion, and MTS criterion.
Fig. 9 – Typical propagation path of kinked crack of specimens with (a) ψ ¼301 (A70), (b) ψ ¼601 (B73), and (c) ψ¼ 901 (B65). Kinking angles were measured by magnifying the photo by a factor of 5 (d).
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
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interface fracture mechanics. A test specimen was developed in an oblique crack sandwich configuration and tested under three point bending mode. The following conclusions are drawn:
Fig. 11 – Comparison of experimental kinking angles with predicted values taking into account the effect of T-stress.
(1) Interfacial crack tends to kink into the veneer and propagated away from the interface at most of the phase angles due to the poor fracture toughness of veneer material compared with interface toughness. It partially explains the clinical phenomenon that the rate of minor veneer chipping is larger than the rate of delamination. Moreover, it indicates that the fracture toughness of veneer material should be increased to improve the clinical performance of all-ceramic restorations. (2) Compressive residual stress in the veneer can reduce the probability of fracture and enhance the fracture toughness of the interface structure. Therefore, the CTE mismatch can be slightly larger to produce larger compressive residual stress in the veneer. (3) The fracture initiation and kinking behavior can be well predicted by the three criteria, which can provide guidance for material manufacture and selection.
Acknowledgments
Fig. 12 – Comparison between G0(ψ)/Gmax and Γ(ψ)/ΓIC taking into account the effect of T-stress.
This research is based upon work supported by the Independent Innovation Foundation of Shandong University (Grant no. 2012JC032), and Scientific and Technological Planning Project of Shandong Province (Grant no. 2010GDD20211).
r e f e r e nc e s compressive condition (η¼ 0.5) seems to be coincident with experimental results. Fig. 12 shows the comparison between G0(ψ)/Gmax and Γ(ψ)/ΓIC taking into account the effect of T-stress. Obviously, tensile Tstress (η¼0.5) increases the phase angle range that crack kinking happens, while compressive T-stress (η¼ 0.5) slightly decreases it. Moreover, G0(ψ)/Gmax becomes lower when η¼ 0.5, and becomes larger when η¼ 0.5. Since G0(ψ) are equal in the three situation, Gmax is reduced when η¼ 0.5. Meanwhile, Kθmax is reduced, which means that it needs larger load to make Kθmax reach the fracture toughness of the veneer to induce fracture. Consequently, tensile T-stress tends to destabilize interfacial cracks and flaws, and cause them to kink out of the interface. Whereas, compressive T-stress stabilizes interfacial cracks and flaws, and makes the crack remain in the interface. Clinically, the veneer usually has smaller CTE than zirconia, thereby compressive residual stress is produced in the veneer. The results of the fracture mechanics analyses suggest that, a larger CTE mismatch could increase compressive residual stress in the veneer, which is beneficial to the toughness of the bi-material structure.
4.
Conclusion
The paper presents the results of a combined experimental and theoretical study on fracture mechanics performance of zirconia/ veneer interface, which is used in dental restorations based on
ABAQUS. V6.11 Theory Manual. Ayatollahi, M.R., Aliha, M.R.M., Hassani, M.M., 2006. Mixed mode brittle fracture in PMMA – an experimental study using SCB specimens. Mat. Sci. Eng.: A 417 (1), 348–356. Cao, H.C., Evans, A.G., 1989. An experimental study of the fracture resistance of bimaterial interfaces. Mech. Mater. 7 (4), 295–304. Charalambides, M., Kinloch, A.J., Wang, Y., Williams, J.G., 1992. On the analysis of mixed-mode failure. Int. J. Fract. 54 (3), 269–291. Charalambides, P.G., Lund, J., Evans, A.G., McMeeking, R.M., 1989. A test specimen for determining the fracture resistance of bimaterial interfaces. J. Appl. Mech. 56 (1), 77–82. Damani, R., Gstrein, R., Danzer, R., 1996. Critical notch-root radius effect in SENB-S fracture toughness testing. J. Eur. Ceram. Soc. 16 (7), 695–702. Dundurs, J., 1969. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. J. Appl. Mech. 36 (3), 650–652. Fischer, J., Grohmann, P., Stawarczyk, B., 2008. Effect of zirconia surface treatments on the shear strength of zirconia/ veneering ceramic composites. Dent. Mater. 27 (3), 448–454. Go¨stemeyer, G., Jendras, M., Borchers, L., Bach, F.W., Stiesch, M., Kohorst, P., 2012. Effect of thermal expansion mismatch on the Y-TZP/veneer interfacial adhesion determined by strain energy release rate. J. Prosthodont. Res. 56 (2), 93–101. He, M.Y., Bartlett, A., Evans, A.G., Hutchinson, J.W., 1991. Kinking of a crack out of an interface: role of in-plane stress. J. Am. Ceram. Soc. 74 (4), 767–771.
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He, M.Y., Hutchinson, J.W., 1988. Kinking of a crack out of an interface: tabulated solution coefficients. Harvard University Report MECH-113A. He, M.Y., Hutchinson, J.W., 1989. Kinking of a crack out of an interface. J. Appl. Mech. 56 (2), 270–278. Henriques, B., Gonc¸alves, S., Soares, D., Silva, F.S., 2012. Shear bond strength comparison between conventional porcelain fused to metal and new functionally graded dental restorations after thermal–mechanical cycling. J. Mech. Behav. Biomed. Mater. 13, 194–205. Hutchinson, J.W., Suo, Z., 1992. Mixed mode cracking in layered materials. Adv. Appl. Mech 29, 63–191. ISO, 2003. Fine ceramics (advanced ceramics, advanced technical ceramics) – test method for fracture toughness of monolithic ceramics at room temperature by single edge precracked beam (SEPB) method (ISO 15732: 2003). Jakobsen, J., Andreasen, J.H., Bozhevolnaya, E., 2008. Crack kinking of a delamination at an inclined core junction interface in a sandwich beam. Eng. Fract. Mech. 75 (16), 4759–4773. Jancar, J., 2011. Bond strength of five dental adhesives using a fracture mechanics approach. J. Mech. Behav. Biomed. Mater. 4 (3), 245–254. Kim, H.J, Lim, H.P., Park, Y.J., Vang, M.S., 2011. Effect of zirconia surface treatments on the shear bond strength of veneering ceramic. J. Prosthet. Dent. 105, 315–322. Kotousov, A., Kahler, B., Swain, M., 2011. Analysis of interfacial fracture in dental restorations. Dent. Mater. 27 (11), 1094–1101. Liu, D., Matinlinna, J.P., Pow, E., 2012. Insights into porcelain to zirconia bonding. J. Adhes. Sci. Technol. 26, 1249–1265. Mosharraf, R., Rismanchian, M., Savabi, O., Ashtiani, A.H., 2011. Influence of surface modification techniques on shear bond strength between different zirconia cores and veneering ceramics. J. Adv. Prosthodont. 3, 221–228. Noijen, S.P.M., van der Sluis, O., Timmermans, P.H.M., Zhang, G. Q., 2012. A semi-analytic method for crack kinking analysis at isotropic bi-material interfaces. Eng. Fract. Mech. 83, 8–25.
O’dowd, N.P., Shih, C.F., Stout, M.G., 1992a. Test geometries for measuring interfacial fracture toughness. Int. J. Solids Struct. 29 (5), 571–589. O’dowd, N.P., Stout, M.G., Shih, C.F., 1992b. Fracture toughness of alumina-niobium interfaces: experiments and analyses. Phil. Mag. A. 66 (6), 1037–1064. Rice, J.R., Sih, G.C., 1965. Plane problems of cracks in dissimilar media. J. Appl. Mech. 32 (2), 418–423. Rice, J.R., 1988. Elastic fracture mechanics concepts for interfacial cracks. J. Appl. Mech. 55 (1), 98–103. Roediger, M., Gersdorff, N., Huels, A., Rinke, S., 2010. Prospective evaluation of zirconia posterior fixed partial dentures: fouryear clinical results. Int. J. Prosthodont. 23, 141–148. Shih, C.F., Asaro, R.J., 1988. Elastic-plastic analysis of cracks on bimaterial interfaces: part I-small scale yielding. J. Appl. Mech. 55 (2), 299–316. Soderholm, K.J., 2010. Review of the fracture toughness approach. Dent. Mater. 26 (2), e63–e77. Sundararaman, V., Davidson, B.D., 1998. An unsymmetric endnotched flexure test for interfacial fracture toughness determination. Eng. Fract. Mech. 60 (3), 361–377. Suo, Z., Hutchinson, J.W., 1989. Sandwich test specimens for measuring interface crack toughness. Mater. Sci. Eng.: A 107, 135–143. Thurston, M.E., Zehnder, A.T., 1995. Nickel–alumina interfacial fracture toughness: experiments and analysis of residual stress effects. Int. J. Fract. 76 (3), 221–241. Wang, J.S., Suo, Z., 1990. Experimental determination of interfacial toughness curves using Brazil-nut-sandwiches. Acta Metall. Mater. 38 (7), 1279–1290. Wang, G.Q., Zhang, S., Bian, C.R., Kong, H., 2014. Interface toughness of zirconia/veneer system and effect of liner application. J. Prosthet. Dent. http://dx.doi.org/10.1016/j. prosdent.2013.12.010, in press. Yuuki, R., Xu, J.Q., 1992. Stress based criterion for an interface crack kinking out of the interface in dissimilar materials. Eng. Fract. Mech. 41 (5), 635–644.
Available online at www.sciencedirect.com
www.elsevier.com/locate/jmbbm
Research Paper
Fracture mechanics analyses of ceramic/veneer interface under mixed-mode loading Gaoqi Wanga, Song Zhanga,n, Cuirong Bianb, Hui Kongb a
Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Ministry of Education), School of Mechanical Engineering, Shandong University, Jinan 250061, PR China b Department of Prosthodontics, Qilu Hospital of Shandong University, Jinan 250012, PR China
art i cle i nfo
ab st rac t
Article history:
Few studies have focused on the interface fracture performance of zirconia/veneer
Received 29 March 2014
bilayered structure, which plays an important role in dental all-ceramic restorations.
Accepted 21 July 2014
The purpose of this study was to evaluate the fracture mechanics performance of zirconia/
Available online 29 July 2014
veneer interface in a wide range of mode-mixities (at phase angles ranging from 01 to 901),
Keywords:
and to examine the effect of mechanical properties of the materials and the interface on
Bi-materials
the fracture initiation and crack path of an interfacial crack. A modified sandwich test
Dental ceramics
configuration with an oblique interfacial crack was proposed and calibrated to choose the
Interface toughness
appropriate geometry dimensions by means of finite element analysis. The specimens
Mixed-mode
with different interface inclination angles were tested to failure under three-point bending
Fracture criterion
configuration. Interface fracture parameters were obtained with finite element analyses. Based on the interfacial fracture mechanics, three fracture criteria for crack kinking were used to predict crack initiation and propagation. In addition, the effects of residual stresses due to coefficient of thermal expansion mismatch between zirconia and veneer on the crack behavior were evaluated. The crack initiation and propagation were well predicted by the three fracture criteria. For specimens at phase angle of 0, the cracks propagated in the interface; whereas for all the other specimens the cracks kinked into the veneer. Compressive residual stresses in the veneer can improve the toughness of the interface structure. The results suggest that, in zirconia/veneer bilayered structure the veneer is weaker than the interface, which can be used to explain the clinical phenomenon that veneer chipping rate is larger than interface delamination rate. Consequently, a veneer material with larger fracture toughness is needed to decrease the failure rate of all-ceramic restorations. And the coefficient of thermal expansion mismatch of the substrates can be larger to produce larger compressive stresses in the veneer. & 2014 Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.: þ86 531 88392746; fax: þ86 531 88392746. E-mail address: [email protected] (S. Zhang).
http://dx.doi.org/10.1016/j.jmbbm.2014.07.019 1751-6161/& 2014 Elsevier Ltd. All rights reserved.
120
1.
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
Introduction
Bi-materials have been widely applied in aerospace, automotive, marine, microelectronic structures and other fields. In the prosthodontics field, zirconia-based bilayered restorations have been used extensively because the veneering porcelain sintered on zirconia presents a combination of both high strength and excellent esthetics (Liu et al., 2012). However, the weak interface between zirconia and veneer is one of the important factors leading to chipping and delamination (Roediger et al., 2010; Liu et al., 2012). Therefore, it has great significance to understand the fracture performance of zirconia/veneer interface, which has received little attention. The studies on the zirconia/veneer interface were mainly focused on the improvement of bond strength by means of different surface treatments (Fischer et al., 2008; Mosharraf et al., 2011; Kim et al., 2011). The measurement of bond strength was most prevalently performed with shear or tensile bond tests, which often showed cohesive fracture patterns within the veneer layer (Henriques et al., 2012). Recently, fracture mechanics approaches became more popular in the research of interfacial bonding performance of dental restorations (Soderholm, 2010; Jancar, 2011; Kotousov et al., 2011). A fracture mechanics methods proposed by Charalambides et al. (1989) was used to measure the interface toughness between zirconia and veneer (Göstemeyer et al., 2012; Wang et al., in press). Nevertheless, using this method can only yield a narrow range of mode-mixities (characterized by phase angle), which is defined as the ratio between the shear and opening modes at an interfacial crack tip. The actual occlusal loading scenarios are not just limited to near shear or opening conditions, which means an interfacial crack is always subjected to mixed mode conditions. An interfacial crack subjected to mixed mode loading either kinks into substrates or propagates in the interface. The maximum energy release rate criterion (Gmax criterion) (He et al., 1991; He and Hutchinson, 1988, 1989), the zero KII criterion (KII ¼0 criterion) (He and Hutchinson, 1989), and the maximum tangential stress criterion (MTS criterion) (Yuuki and Xu, 1992) can be used to predict the crack kinking behaviors including selection of crack path, the threshold of crack initiation, and the kinking angle. There are also some criteria for evaluating interface fracture (Charalambides et al., 1992; Hutchinson and Suo, 1992; Thurston and Zehnder, 1995), which present the relation between interface fracture toughness and mode-mixity. In general, the toughness of an interface, which can be characterized by stress intensity factors (SIF) or energy release rate (ERR), increases with the increase of phase angle. Consequently, in order to understand the fracture mechanism of bilayered all-ceramic restorations, it is necessary to investigate the zirconia/veneer interface in a wild range of mode-mixities by an effective method. In the past, various experimental methods and test configurations have been used for determining the mixed-mode fracture performance of bi-material interfaces, such as fourpoint bending specimen (Charalambides et al., 1989), unsymmetric end-notched flexure specimen (Sundararaman and Davidson, 1998), double cantilever beam specimen (Cao and
Evans, 1989), sandwich specimens (Suo and Hutchinson, 1989), Brazil-nut-sandwich specimen (Wang and Suo, 1990), and asymmetric four-point bending specimen (O’dowd et al., 1992a, 1992b). However, most of the mentioned test methods are not appropriate for the current zirconia/veneer combination. Four-point bending specimen and end-notched flexure specimen can only provide limited range of I/II mode-mixities; double cantilever beam specimen and sandwich specimens proposed by Suo and Hutchinson are not appropriate for brittle materials with low fracture toughness; Brazil-nutsandwich specimen and asymmetric four-point bending specimen require complicated preparation process or complicated testing fixtures. The veneering porcelain is sintered on the zirconia under a high temperature to produce bilayered structure, and the veneer material which has relatively small fracture toughness turns to viscous-elastic state under high temperature. Therefore, a simple and valid test configuration that is suitable for the current material combination is needed. In this study, a modified beam test configuration was suggested for investigating the mixed mode fracture performance of the interface. The test configuration was proposed based on a sandwich interfacial fracture specimen (Suo and Hutchinson, 1989) and a SCB specimen (Ayatollahi et al., 2006) that were used for measuring mixed mode fracture toughness of homogeneous materials. The purpose of this study was to evaluate the fracture mechanics performance of zirconia/veneer interface by investigating the propagation of an interfacial crack under a wide range of mode-mixities, and to examine the effect of fracture mechanical properties of the materials and the interface on the crack initiation and crack path. The structure of the paper was as follows. First, the specimen was described, and the relations between specimen geometries and phase angle were calibrated by means of several finite element analyses (FEA) to choose appropriate geometry dimensions. Secondly, the specimens were prepared based on the FEA results, and tested with three point bending method. Failure mode, fracture initiation load of all specimens and crack kinking angle of the kinking specimens were recorded. Finally, the experimental results were compared with theoretical results of the fracture criteria, and the clinical guiding significance of the study was presented.
2.
Material and methods
2.1.
Test specimen and numerical analyses
The oblique crack sandwich specimen can be easily made and tested in conventional testing machines. As shown in Fig. 1a, the specimen consists of two pieces of zirconia bonded with an interlayer of veneer. One interface of the two materials is normal to the horizontal plane, and another interface has an inclination angle ω relative to the horizontal direction. The oblique interface contains an interfacial edge pre-crack of a in length. The specimen has a height of w, a thickness of b, and a length of 2l1þl2. The specimen is loaded by the vertical load P in standard three point bending mode with a span of l. s is the load offset distance from the location right above the open end of the crack. When the parameters
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
121
Fig. 1 – Schematic diagram of (a) single edge oblique crack specimen and (b) the interfacial crack configuration (right crack tip). σθ, σr, and τrθ are stress components of a point which has coordinates of (r, θ) in polar coordinates with the origin at the crack tip. Table 1 – Mechanical properties of materials. Materials
Elastic modulus E (GPa)
Poisson ratio ν
Fracture toughness KIC (MPa m1/2)
CTE α ( 10 6/1C)
Lava Zirconia IPS e.max Ceram
210 70
0.31 0.27
5.0–10.0 –
10.0 9.5
All data were provided by manufacturers.
s, ω, a/w, and l/w are changed, different phase angles can be obtained. Therefore, calibrations were conducted with the current material combination to relate the phase angle to the geometries. Fig. 1b gives the schematic diagram of the interfacial crack tip configuration (right crack tip), in which zirconia is on the top and veneer is on the bottom. The stress field at the crack tip depends on the two Dundurs' parameters α, β (Dundurs, 1969), which characterize the elastic mismatch of the two materials. The two Dundurs' parameters are defined as follows: 9 μ ðκ 2 þ 1Þ μ2 ðκ 1 þ 1Þ > > α¼ 1 > μ1 ðκ2 þ 1Þ þ μ2 ðκ 1 þ 1Þ = ; ð1Þ μ ðκ 2 1Þ μ2 ðκ 1 1Þ > > β¼ 1 > ; μ1 ðκ2 þ 1Þ þ μ2 ðκ 1 þ 1Þ where μi ¼Ei/2(1þνi) is the shear modulus, κi ¼3–4νi for the plane strain state. Ei is the elastic modulus, νi is the Poisson's ratio, i¼ 1, 2 represents zirconia and veneer, respectively. ε is the oscillation index that defined as follows: ε¼
1 1 β ln : 2π 1þβ
ð2Þ
For the current material combination (Table 1), the plane strain Dundurs' parameters are calculated to be α¼ 0.509, β¼ 0.170, and the oscillation index ε¼ 0.055. The interface fracture parameters including SIFs, ERR, and phase angle, depend on the geometries of specimen and applied load. The complex SIF is defined as (Rice and Sih, 1965; Rice, 1988) K ¼ K1 þ iK2
ð3Þ
where K1 and K2 represent mode I and mode II interface SIFs, respectively. The phase angle ψ of KLiε is expressed as ψ ¼ arctan
ImðKLiε Þ ReðKLiε Þ
:
ð4Þ
where L is an arbitrarily chosen reference length which has no absolute physical significance. In the present study L was chosen to be 1 mm. The phase angle ψ is a measure of the degree of mixity of shear to opening modes on a crack tip. ψ¼901 or 901 represent a crack having nearly zero opening; whereas ψ¼ 0 means a crack subjected to nearly zero shear. The calibration of specimens was performed by the finite element software ABAQUS. All the numerical models presented in this paper were solved under plane strain conditions. Finite element models with different values of the geometry parameters (as shown in Fig. 1) were analyzed. The mechanical properties of the two materials in the model are given in Table 1. Fig. 2 shows a typical mesh pattern for simulating the specimen. The model was meshed to about 21000 8-node second order elements, and a highly refined mesh was used in the region in the vicinity of the crack tip with 36 wedge shaped 6-node second order singular elements. The load and constraints were identical to the schematic diagram of the specimen as shown in Fig. 1a. The calibrations were conducted by applying load of 100 N in each case. A J-integral-based method (interaction integral method) built into ABAQUS was used to obtain the interface SIFs, ERR, and ψ automatically (Shih and Asaro, 1988; ABAQUS theory manual). First, the specimen was calibrated for s¼0, l2 ¼2 mm, different l/w and a/w. The calibration functions for l/w¼ 0.8, 1.0, and 1.2, and for a/w¼ 0.4, 0.5, and 0.6 are plotted in Fig. 3a and b. The parameters were chosen to be a/w ¼0.5, l/w¼ 1.0, and l¼ 20 mm. The interface inclination angle ω 861, 781, 701, 631, 581, 531, and 491 corresponded to ψ of about 01, 151, 301, 451, 601, 751, and 901, respectively. However, the larger ψ required smaller ω, which needed plenty of veneer material. The loading situation of s ¼4 mm was therefore used to provide data for large ψ. The relation between ω and ψ in this situation is plotted in Fig. 3c. The angle ω was set to be 781,
122
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
Fig. 2 – A typical finite element mesh pattern for a specimen.
Fig. 3 – Calibration functions for specimens. l2 ¼ 2 mm, l1 ¼ l¼ 20 mm, and (a) s ¼0, a/w ¼0.5; (b) s¼ 0, l/w¼ 1; (c) s ¼4 mm, l/w ¼1, and a/w¼0.5. Table 2 – Geometric parameters of the specimens. 731, 701, and 651, corresponding to phase angle of about 451, 601, 751, and 901. Corresponding to each ω, the specimens were divided into 7 groups, which were numbered as A86, A78, and A70 for loading situation of s ¼0, and B78, B73, B70, B65 for loading situation of s ¼4 mm, respectively. All the chosen parameters are summarized in Table 2.
2.2.
Specimen preparation and bending test
A total of 42 specimens (6 in each group) were prepared. Lava zirconia (3M ESPE, Seefeld, Germany) as core and IPS e.max Ceram (Ivoclar Vivadent AG, Schaan, Liechtenstein) as veneer were used, because the material combination is commonly used, and has similar values of coefficient of thermal expansion (CTE) (Table 1). 42 Rectangler pre-sintered zirconia blocks with the
Parameter
Value
a w l1 l2 l s ω
10 mm 20 mm 20 mm 2 mm 20 mm 0 or 4 mm 861(A86), 781(A78), 701(A70) (s ¼0) 781(B78), 731(B73), 701(B70), 651(B65) (s¼ 4 mm) 5 mm
b
same dimensions (20 20 5 mm3) and 42 trapezoidal presintered zirconia blocks with corresponding ω were cut from presintered blocks with a diamond saw. The specimens were
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
sintered and cleaned in a sonic bath filled with ethanol for 5 min and gently air dried. A rectangler block and a trapezoidal block for each bend specimen were aligned by a guide and adjusted to make l2 ¼ 2 mm (Fig. 1). A piece of thin graphite paper with length of 10 mm and thickness of 0.05 mm was pasted on zirconia surface to produce approximate interfacial pre-crack. The pasty veneer were then stacked into the space between the two components of zirconia blocks. The zirconia–veneer interface was bonded at 750 1C for 4 min in vacuum followed by a furnace cool. The glass transition temperature of the veneer is 490 1C, therefore most of the thermal residual stresses were produced mainly due to the CTE mismatch during temperature change from 490 1C to room temperature 25 1C. Three point bending test was conducted on each specimen with a test machine (Instron 8801, Instron Co., Canton, MA), and the load–displacement curve was recorded. Crack kinking angles of the kinking specimens were measured by photography using a digital camera.
2.3.
Calculation of interface fracture parameters
The model described in Section 2.1 was used to obtain ERRs and SIFs by changing the load to corresponding mean fracture load of each group of specimens. T-stress, which is the stress parallel to the interface, is mainly composed of thermal residual stresses for the current specimens. Residual stresses were calculated based on the temperature difference from glass transition temperature of the veneer to room temperature ( 465 1C). Then the T-stress was extracted from stress values of 6 points on the circle with center at the crack tip and radius of 1 mm (Fig. 4).
2.4.
Kinking criteria of interfacial crack
2.4.1.
Gmax criterion
The Gmax criterion proposed by He and Hutchinson (1989) suggests that the kinking occurs in the direction where the ERR is maximum. The criterion establishes the relation between SIFs of the kinked crack and that of interfacial crack. Thereby the ERR of kinked crack at each kinking angle can be obtained from interfacial SIFs. The main calculation process is as follows: The SIFs of the kinked crack KI and KII can be expressed from interfacial SIFs K as (He and Hutchinson, 1989; He et al., 1991) pffiffiffiffi ð5Þ KI þ iKII ¼ cKa0iε þ dKa0 iε þ bσ 0 a0
123
where a0 is the length of kinked crack and chosen to be 1 mm, σ0 is T-stress, parameters c¼c1þic2, d¼ d1þid2, and b¼b1þib2 are dimensionless complex functions of kinking angle θ0 , and Dunder's parameters α and β. He and Hutchinson (1989) calculated values of c, d, b for a limited number of combinations of α and β by solving two integral equations. In the present study, a simply FEA method proposed recently was used to calculate c, d, b of the current material combination as functions of θ0 (Jakobsen et al., 2008; Noijen et al., 2012). The ERR of the interfacial crack can be calculated by K21 þ K22 κ1 þ 1 κ2 þ 1 þ : ð6Þ G0 ¼ μ1 μ2 16cosh2 ðεπ Þ The ERR of the kinked crack is expressed as G¼
K2I þ K2II ; 0 E2
ð7Þ
where for plane strain state, Ei0 ¼Ei/(1 v2i ). Therefore, Gmax, the maximum ERR of the kinked crack, was obtained from KI and KII of interfacial crack by finding out the maximum G value of all possible kinking angles. The interfacial crack initiates to kink other than propagate in the interface, if the following condition is satisfied: G0 ΓðψÞ o ; Γ IC Gmax
ð8Þ
where Γ(ψ) is interface toughness at phase angle ψ. ΓIC is mode I fracture toughness of the veneer. The effect of T-stress on the fracture initiation and kinking behavior was also evaluated by the Gmax criterion. η can be used to characterize the level of T-stress (He et al., 1991) pffiffiffi σ0 8 ffi; ð9Þ η ¼ pffiffiffiffiffiffiffiffiffiffi En G0 where
" # 1 1 1 1 1 ¼ þ : 0 0 2 E1 E2 cosh2 ðπεÞ En
ð10Þ
T-stresses were calculated to be approximately σ0 ¼ 5.3 MPa for each group of specimen, which was equivalent to η ¼ 0.5. Therefore, Gmax criterion was implemented with η ¼ 0.5, 0.5, and 0. η¼ 0.5 was approximately equal to compressive stress of σ0 ¼ 5.3 MPa, while η ¼0.5 was nearly equals to tensile stress of σ0 ¼ 5.3 MPa, and η¼ 0 means no Tstress was determined.
Fig. 4 – Calculation of T-stress, which is represented by σ0.
124
2.4.2.
journal of the mechanical behavior of biomedical materials 39 (2014) 119 –128
KII ¼ 0 criterion
According to KII ¼ 0 criterion, kinking angle is identical to the direction which make mode II SIF of the kinked crack equal to 0. Therefore, kinking angle θ0 can be calculated from Eqs. (5), (6), and (7) with KII ¼ 0. Effect of T-stress was also evaluated by this criterion.
2.5.
σθmax Criterion
Yuuki and Xu (1992) put forward maximum tangential stress (MTS) criterion, which concludes that a crack will initiate in the direction where the stress component σθ at the crack tip is maximum. The tangential stress of a point (r, θ) near an interfacial crack tip (Fig. 1) is expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK21 þ K22 Þ r ð11Þ σ θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bðθ; ε; ψÞ cos ε ln L 2 ð2πrÞcoshðπεÞ B is given in reference (Yuuki and Xu, 1992); r was set to be 1 mm. The kinking angle which gives the maximum σθ can be obtained by ∂σ θ ¼0 ∂θ
ð12Þ
And the fracture criterion was described as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK21 þ K22 Þ Bðθ; ε; ψÞZ KIC : Kθ max ¼ 2 coshðπεÞ
ð13Þ
When the value of Kθmax reaches the fracture toughness of veneer KIC, the fracture will occur.
2.6.
Fracture toughness test of veneer
The kinking criteria make a comparison between the fracture toughness of substrates and that of interface. Therefore, the mode I fracture toughness KIC of the veneer was measured using single edge notch bend (SENB) method. 5 Specimens with 3 mm in thickness, 4 mm in height, and 36 mm in length were sawed notches at the center to a depth about 2 mm in height direction, and then loaded to fracture in three-point bending test. The accurate depth of the notches was determined by optical analysis of the fractured surfaces. The notch root radius ρ was approximately 100 μm, which required larger load to fracture from the notch root than from a sharp crack tip. So the true fracture toughness, KIC was obtained by (Damani et al., 1996) KIC ¼ XKm ;
ð14Þ
where Km is SIF calculated from measured values (ISO 15732, 2003), and the parameter X can be obtained by pffiffiffiffiffiffiffiffiffiffi X ¼ tanhð2Y δa=ρÞ; ð15Þ where Y is a correction factor, δa can be approximated on the basis of defects size. Based on the fractography analysis, many small defects were found. Therefore an edge-crack configuration was assumed, and δa was set to be 20 μm, Y was set to be 1.12.
3.
Results and discussions
3.1.
Failure modes
The fracture mode of the specimens in Group A86 (ψ¼ 0) was interfacial fracture, that the crack propagated in the zirconia/ veneer interface. Fracture loads of the A86 specimens were used to calculate interface toughness at ψ¼0. For other groups, fracture occurred in the form of cracks kinking into the veneer (Table 3).
3.2.
Interface fracture parameters
Fig. 5 shows the typical load–displacement curve of a specimen in Group B65 (ψ¼ 901) with fracture load of 679.2 N. It can be seen that the curve was approximately linear until the specimen catastrophically failed, indicating that no plastic deformation and no stable crack growth were found during loading and the specimens fractured in a brittle manner. The fracture loads obtained from the experiments are shown in Table 3. The interface SIFs and interface ERRs were calculated from the mean fracture loads of corresponding specimens. In aggregate the fracture load was larger for the specimens having larger phase angle. And with the increase of ψ, the value of K1 decreased, whereas K2 increased sharply.
3.3.
Crack initiation and kinking angle
3.3.1.
Crack initiation
The three complex coefficients c, d, b were obtained by the FEA method. Values of the coefficients as functions of kinking angle for the current material combination are shown in Fig. 6. Then the parameters were used in the equations of Gmax criterion. The mean fracture load of the five veneer SENB specimens was 11.2(70.78) N. By using the equations in the international standard (ISO 15732, 2003), Km was calculated to be 1.21 MPa m1/2, therefore, KIC ¼ 0.92 MPa m1/2. Critical ERR of the veneer, ΓIC, was obtained by Eq. (7) with KI ¼KIC and KII ¼0, therefore ΓIC ¼11.2 N/m.
Table 3 – Fracture load and failure mode of specimens, and fracture mechanics parameters including stress intensity factors K1, K2, energy release rate G0, and phase angle ψ of the interfacial cracks. Specimens
ψ
Failure loads (7SD)
Failure modes
K1 (MPa m1/2)
K2 (MPa m1/2)
G0 (N/m)
A86 A78 A70 B78 B73 B70 B65
01 151 301 451 601 751 901
394.2 354.7 420.3 513.3 580.5 648.0 673.2
Interfacial Kinking Kinking Kinking Kinking Kinking Kinking
1.10 0.94 0.87 0.67 0.47 0.26 0.01
0.02 0.23 0.48 0.66 0.81 0.93 0.96
10.23 8.02 8.32 7.55 7.47 7.87 7.87
(755.5) (748.0) (765.0) (791.2) (734.5) (785.4) (771.7)
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Fig. 5 – Typical load–displacement curve of a specimen (B65). Fig. 7 – Comparison between G0(ψ)/Gmax and Γ(ψ)/ΓIC (η¼ 0). Interface energy release rates G0 were solved by FEA models using fracture loads; maximum energy release rates of kinked crack Gmax were calculated based on Gmax criteria; interface critical energy release rates Γ(ψ) were obtained using Eq. (16); fracture toughness of the veneer ΓIC was measured using the SENB method.
Fig. 6 – Complex coefficients as functions of kinking angle for the current material combination (α ¼0.509, β¼ 0.170).
It is difficult to obtain interface toughness data for large phase angles by experiment because specimens with large phase angle often show kinking failure (Table 3). Therefore, the interface toughness Γ(ψ) was calculated from a toughness law proposed by Hutchinson and Suo (1992): ΓðψÞ ¼ Γ 1 ð1 þ λ tan 2 ψÞ:
Fig. 8 – Kθmax at different phase angles versus the fracture toughness of the veneer.
ð16Þ
Γ1 is the interface toughness for approximate mode I (ψ¼0). So the value of Γ1 was set to be the mean interface ERR of specimens in Group A86 (ψ¼0, Γ1 ¼ 10.23 N/mm), which showed interface fracture. The Γ(ψ) of the interface at other phase angles were estimated by Eq. (16), with λ ¼0.3. G0(ψ)/Gmax and Γ(ψ)/ΓIC (η¼ 0) as functions of phase angle are shown in Fig. 7. It is obvious that based on Gmax criterion an interfacial crack will kink into the veneer when the phase angle is larger than about 151, while keep advancing in the interface with phase angle smaller than 151. This is consistent with experimental results. Apparently, the crack path selection is governed by the toughness of the interface and that of the veneer which change the toughness ratio of the interface to veneer. It can be seen from Fig. 8 that the kinking occurred when the value of Kθmax reached the fracture toughness of the veneer KIC. This is identical to the σθmax criterion. But the Kθmax values were slightly larger than the KIC, because the fracture initiated harder from the notch introduced by graphite paper than from a sharp crack, which was not taken into consideration. Besides, compressive T-stresses had a blocking effect on the crack kinking. The crack kinked into the veneer for most of the specimens because of the low
fracture toughness of veneer. This is coincident with the clinical phenomenon that veneer chipping rate is larger than interface delamination rate. Consequently, a veneer material with larger fracture toughness is needed to decrease the failure rate of all-ceramic restorations. In summary, it is concluded based on all the criteria that the weaker the toughness of interface and substrates, the easier the crack initiation occurs. Moreover, whether an interfacial crack extends within the interface or advances by kinking into either substrate depends on relative toughness of interface to that of the joining materials. If the substrate's fracture toughness is sufficiently larger than interface toughness, the crack will keep in the interface. However, when the fracture toughness of the substrate is comparable to that of interface, whether or not to kink depends on the mode-mixity. For the current right crack tip configuration with veneer on the bottom (Fig. 1), the crack has the tendency to kink into the veneer at positive phase angles (01–901). The experimental results show that the fracture toughness of veneer is comparable to the interface toughness at small phase angles. Since the interface toughness becomes
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larger with the increase of phase angles, cracks at most of the phase angles tend to kink into the veneer. At negative phase angles ( 901–01), the crack has the tendency to kink into the zirconia. However, the fracture toughness of zirconia is much larger than the interface toughness at most phase angles, so that the crack will keep extending in the interface.
3.3.2.
3.4.
Effect of residual stress
Fig. 11 shows the prediction of crack kinking angle taking into account the effect of T-stress using Gmax criterion and KII ¼ 0 criterion. Compared with previous predictions, tensile T-stress (η¼0.5) increases the kinking angle, whereas compressive Tstress (η¼ 0.5) decreases the kinking angle. The prediction of
Kinking angle
The kinking angles were measured by magnifying the photo by a factor of 5 to obtain the real initial kinking angle (Fig. 9d), because the crack changed direction when propagated away from the interface. Fig. 9a–c shows the typical crack paths of specimens with phase angles of 301, 601, and 901, the kinking angles of which are 46.71, 56.51, and 69.21, respectively. Obviously, kinking angle increases as the phase angle increases. Fig. 10 displays the theoretical curves (η ¼ 0) for kinking angles as functions of ψ using Gmax criterion, KII ¼0 criterion, and MTS criterion. Also shown in the figure is the experimental data obtained in this research. Apparently, the three criteria and experimental results followed the same trend. However, the experimental results were almost smaller than the theoretical ones, which was due to the effect of T-stress.
Fig. 10 – Comparison of experimental kinking angles for zirconia/veneer interface (l¼ 1 mm) with predicted values based on Gmax criterion, KII ¼ 0 criterion, and MTS criterion.
Fig. 9 – Typical propagation path of kinked crack of specimens with (a) ψ ¼301 (A70), (b) ψ ¼601 (B73), and (c) ψ¼ 901 (B65). Kinking angles were measured by magnifying the photo by a factor of 5 (d).
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interface fracture mechanics. A test specimen was developed in an oblique crack sandwich configuration and tested under three point bending mode. The following conclusions are drawn:
Fig. 11 – Comparison of experimental kinking angles with predicted values taking into account the effect of T-stress.
(1) Interfacial crack tends to kink into the veneer and propagated away from the interface at most of the phase angles due to the poor fracture toughness of veneer material compared with interface toughness. It partially explains the clinical phenomenon that the rate of minor veneer chipping is larger than the rate of delamination. Moreover, it indicates that the fracture toughness of veneer material should be increased to improve the clinical performance of all-ceramic restorations. (2) Compressive residual stress in the veneer can reduce the probability of fracture and enhance the fracture toughness of the interface structure. Therefore, the CTE mismatch can be slightly larger to produce larger compressive residual stress in the veneer. (3) The fracture initiation and kinking behavior can be well predicted by the three criteria, which can provide guidance for material manufacture and selection.
Acknowledgments
Fig. 12 – Comparison between G0(ψ)/Gmax and Γ(ψ)/ΓIC taking into account the effect of T-stress.
This research is based upon work supported by the Independent Innovation Foundation of Shandong University (Grant no. 2012JC032), and Scientific and Technological Planning Project of Shandong Province (Grant no. 2010GDD20211).
r e f e r e nc e s compressive condition (η¼ 0.5) seems to be coincident with experimental results. Fig. 12 shows the comparison between G0(ψ)/Gmax and Γ(ψ)/ΓIC taking into account the effect of T-stress. Obviously, tensile Tstress (η¼0.5) increases the phase angle range that crack kinking happens, while compressive T-stress (η¼ 0.5) slightly decreases it. Moreover, G0(ψ)/Gmax becomes lower when η¼ 0.5, and becomes larger when η¼ 0.5. Since G0(ψ) are equal in the three situation, Gmax is reduced when η¼ 0.5. Meanwhile, Kθmax is reduced, which means that it needs larger load to make Kθmax reach the fracture toughness of the veneer to induce fracture. Consequently, tensile T-stress tends to destabilize interfacial cracks and flaws, and cause them to kink out of the interface. Whereas, compressive T-stress stabilizes interfacial cracks and flaws, and makes the crack remain in the interface. Clinically, the veneer usually has smaller CTE than zirconia, thereby compressive residual stress is produced in the veneer. The results of the fracture mechanics analyses suggest that, a larger CTE mismatch could increase compressive residual stress in the veneer, which is beneficial to the toughness of the bi-material structure.
4.
Conclusion
The paper presents the results of a combined experimental and theoretical study on fracture mechanics performance of zirconia/ veneer interface, which is used in dental restorations based on
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