Failure Mechanism and Reliability Prediction for Bonded Layered Structure Due to Cracks Initiating at the Interface
Failure mechanism and reliability prediction for bonded layered structure due to cracks initiating at the interface
A Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Yaou Wang, M.S.
*****
The Ohio State University
2009
Dissertation Committee: Approved by Professor Noriko Katsube, Adviser Professor Robert R. Seghi ______Professor Stanislav I. Rokhlin Adviser Professor Mark E. Walter Graduate Program in Mechanical Engineering
ABSTRACT
Bonded layered structures are widely used to meet high performance
requirements that a single layer material cannot satisfy. All layered structures have a
finite service life due to inevitable failures caused by chemical, thermal or mechanical
loadings during operation. The inevitability of failure in a bonded layered structure
demands the prediction of the service life for layered structures, and this requires
understanding of the failure mechanism. The failure of bonded layered structures
often initiates from a crack at surface, a crack at interface, or interfacial delamination.
Failures due to surface fracture or interfacial delamination have been widely studied.
However, brittle fractures originating at the interface have not been extensively
investigated.
In this work, the failures of bonded layered structures due to cracks/flaws
initiating at the interface are investigated. A cracked bilayer domain analysis is first
presented for estimation of the SIF for a 3D half-penny shaped crack originating at a
bonded interface in idealized bilayer geometry subjected to remote constant tensile and
proportional bending loadings. Handbook-type curve-fitted equations are obtained
for the SIF as a function of modulus ratio of bonded dissimilar materials through
extensive finite element parametric studies. The cracked bilayer domain analysis is
-ii- then combined with macro-level stress calculations in a structure without a crack
(uncracked body analysis), and a simplified method is proposed for accurate estimation of the SIF.
The cracked bilayer domain analysis is extended to estimation of the SIF of a half-penny shaped crack normal to the interface in the top layer of a three-layer bonded structure. To obtain a simple estimate of the SIF, the method of reduction of an idealized cracked trilayer domain to that of a corresponding bilayer domain has been introduced based on the notion of an equivalent homogeneous material for the two bottom layers.
Based on the cracked bilayer/trilayer domain analysis, the effect of adhesive layer on the probability of cracks initiation from the interface in bonded layered structures is quantitatively investigated. The notion of a critical stress density function is introduced to account for the bridging mechanism. The failure probability of glass-ceramic disks bonded to simulated dentin subjected to indentation loads is predicted. The theoretical predictions match experimental data suggesting that the bridging mechanism plays an important role for accurate prognostics to occur.
The developed method is useful in predicting brittle failure initiating from interfacial flaws in a layered structure.
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Dedicated to my parents
-iv-
ACKNOWLEDGMENTS
I wish to thank my advisor, Professor Noriko Katsube, for her selfless support,
friendly encouragement, and her patience in correcting my errors.
I appreciate the financial support from the NIHDCR (Grant Number R21
DEO14719-0).
I thank Professor Robert Seghi and Professor Stanislav Rokhlin for stimulating discussions. Thanks to Professor Stanislav Rokhlin for measuring material properties.
Thanks to Professor Robert Seghi for doing failure experiments, and his mastery in experiments is indispensable for this study.
I thank Huseyin Lekesiz for his friendly encouragement and generous support.
I thank Dr. Tim McGreevy, Dr. Weizhou Li and Dr. Huapei Wan in Caterpillar Inc for their thoughtful consideration, generous support and friendly encouragement during my internship.
-v-
VITA
September 3, 1979………………………. Born in Tianjin, China
1998 to 2002…………………………….. B.S., Mechanical Engineering, University of Science and Technology of China
2002 to 2004…………………………….. M.S., Mechanical Engineering, The Ohio State University
2004 to present………………………….. Graduate Research Associate, Mechanical Engineering, The Ohio State University
PUBLICATIONS
Y. Wang, N. Katsube, R.R. Seghi and S.I. Rokhlin, Uncracked body analysis for accurate estimates of mode I stress intensity factor for cracks normal to an interface, Engineering Fracture Mechanics, 2009 (76), P369-385.
Y. Wang, N. Katsube, R.R. Seghi and S.I. Rokhlin, Statistical failure analysis of adhesive resin cement bonded dental ceramics, Engineering Fracture Mechanics, 2007 (74), P1838-1856.
Y. Wang, M. Walter, K. Sabolsky and M.M. Seabaugh, Effects of Powder Sizes and Reduction Parameters on the Strength of Ni-YSZ Anodes, Solid State Ionics, 2006 (177), P 1517-1527.
X. Jing, X. Hu, J. Zhao, Y. Wang and Y. Tian, Application of Synchrotron Computed Microtomography to Study 3D Pore Topology and Density Distribution in Sintered Ceramics, Journal of Materials Science & Engineering, 2003(21), P327-30.
-vi-
FIELDS OF STUDY
Major Field: Mechanical Engineering
Minor Field: Statistics
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TABLE OF CONTENTS
ABSTRACT...... ii ACKNOWLEDGMENTS ...... v VITA...... vi LIST OF TABLES...... xii LIST OF FIGURES ...... xiv
CHAPTER 1 Introduction...... 1 1.1 Research background...... 1 1.2 Objective...... 4 References...... 7
CHAPTER 2 Uncracked body analysis for accurate estimates of mode I stress intensity factor for cracks normal to an interface ...... 9 Abstract...... 9 2.1 Introduction...... 10 2.2 Methodology...... 14 2.2.1 Cracked Domain Analysis ...... 14 2.2.1.1 Cracked domain analysis for 2D interfacial crack...... 15 2.2.1.2 3D interfacial half-penny shaped crack ...... 16 2.2.2 Y factor and equivalent stress...... 17 2.2.2.1 Approximation of stress intensity factor at the crack tip for linear and nonlinear loading stress distribution ...... 18 2.2.2.2 Variation of stress intensity factor along the crack front...... 20 2.3 Results: Configuration correction factor f and scaling factor Y...... 21 2.3.1 Evaluation of FEA computation model ...... 21 2.3.2 Cracked domain analysis for 2D interfacial crack...... 22 2.3.3 Cracked domain analysis for 3D interfacial half-penny shaped crack ...... 24 2.4 Application...... 26 2.4.1 One-layer ball-on-ring biaxial test...... 27 2.4.2 Bilayer indentation test: Perfectly bonded interface...... 30 2.4.3 Bilayer indentation model: Debonded interface with frictionless contact. 32 2.4.4 Bilayer structure subjected to uniform temperature rise: Perfectly bonded interface...... 34
-viii- 2.5 Summary...... 35 References...... 38 Figures...... 40 Tables...... 50
CHAPTER 3 Mode I stress intensity factor estimate for a half-penny-shaped crack in trilayer structure by reduction to crack in bilayer structure...... 53 Abstract...... 53 3.1 Introduction...... 54 3.2 Methodology...... 57 3.2.1 Cracked trilayer domain analysis...... 57 3.2.2 Role of bonding layer thickness...... 59 3.2.3 Replacement of the bonding and bottom layers with a homogeneous effective substrate ...... 60 3.3 FEM results and empirical fitting function for effective layer ...... 61 3.3.1 Numerical determination of effective modulus ...... 61 3.3.2 Curve fitted equation for effective modulus ...... 64 3.3.3 The error of curve-fitted equation for effective modulus ...... 66 3.4 Application...... 66 3.4.1 Approximate method for SIF estimation ...... 67 3.4.2 Comparison with 3D FEM analyses ...... 68 3.4.3 Limitations of the approximate method...... 69 3.5 Summary...... 70 References...... 72 Figures...... 74
CHAPTER 4 Statistical failure analysis of adhesive resin cement bonded dental ceramics ...... 82 Abstract...... 82 4.1 Introduction...... 83 4.2 Experimental methods ...... 87 4.2.1 Biaxial test for determination of ceramic flaw population ...... 87 4.2.2 Material property measurement ...... 89 4.3 Theoretical analysis ...... 91 4.3.1 Critical flaw distribution and local crack stress intensity model ...... 91 4.3.1.1 Critical flaw distribution...... 91 4.3.1.2 Local model for determination of stress intensity factor for system with and without adhesive resin cement ...... 92 4.3.1.3 Determination of parameter k in critical flaw distribution...... 94 4.3.1.4 Determination of parameters m and k in Eq. (4.1) from biaxial indentation experiments...... 95 4.3.2 Effect of resin-cement layer shrinkage on crack bridging...... 97 4.3.3 Prediction of failure probability distributions for trilayer model...... 98 4.4 Results...... 100 4.4.1 Critical stress density distribution function for biaxial tests...... 100 4.4.2 FEA simulation of Residual Curing Stresses...... 101 -ix- 4.4.3 Prediction of failure probability distribution for biaxial tests of ceramic coated with resin cement layer - comparison with experimental data...... 101 4.4.4 Prediction of failure probability distribution for indentation tests on the bonded trilayer onlay model ...... 103 4.4.4.1 Glass-ceramics/resin cement/simulated dentin trilayer model ...... 103 4.4.4.2 Glass/resin cement/simulated dentin trilayer model...... 104 4.5 Discussion...... 105 4.5.1 Determination of parameters m and k...... 105 4.5.2 Curing residual stress versus crack bridging as strengthening mechanisms ...... 106 4.5.3 2D and 3D local crack models...... 107 4.5.4 Discrepancies between the test data and the models ...... 107 4.6 Conclusion ...... 108 References...... 110 Figures...... 112 Tables...... 123
CHAPTER 5 Effective elastic modulus prediction for porous materials based on the identification of microstructures...... 128 Abstract...... 128 5.1 Introduction...... 129 5.2 Microstructure of porous material ...... 132 5.2.1 Microstructure parameters ...... 132 5.2.2 SEM measurement...... 134 5.3 Micromechanics models ...... 135 5.3.1 Mori-Tanaka method ...... 135 5.3.1.1 Individual effects of pores and microcracks on the effective elastic moduli...... 136 5.3.1.2 Combined effect of pores and microcracks on the effective elastic moduli...... 137 5.3.2 Differential scheme method...... 138 5.3.2.1 Individual effects of pores and microcracks on the effective elastic moduli...... 139 5.3.2.2 Overall effect of pores and microcracks on the effective elastic moduli...... 140 5.4 Application...... 143 5.5 Summary...... 145 References...... 147 Figures...... 149 Tables...... 151
APPENDICES
APPENDIX A Details of FEA for effective layer determination...... 153 A.1 Evaluation of FEA computation model ...... 153 A.2 Error related to the determination of effective modulus...... 154 -x-
APPENDIX B Fracture mechanics based failure probability prediction model .... 156 B.1 Statistical theory for failures of brittle materials...... 156 B.2 Calculation of parameters m and k from biaxial experimental data ...... 158
BIBLIOGRAPHY...... 160
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LIST OF TABLES
Table 2.1. Parameters used in the mechanical tests...... 50
Table 2.2. Percent error of the approximate method assessed by 3D FEA for mode I stress intensity factor KIA of an interfacial half-penny shaped crack at the crack tip A in perfectly bonded bilayer indentation model (Fig. 2.7(b))...... 51
Table 2.3. Percent error of the approximate method assessed by 3D FEA for mode I stress intensity factor KIA of an interfacial half-penny shaped crack at the crack tip A in completely debonded bilayer indentation model with frictionless contact between the layers (Fig. 2.7(b))...... 52
Table 4.1. Dimension of samples in the mechanical tests...... 123
Table 4.2 Experimentally measured materials properties...... 124
Table 4.3. Numerically evaluated values of f’ for 2D and 3D perfectly bonded local 2 models (Figs. 4.4b and 4.5b). Recalling that f =1.025× for the 3D local π model (Fig. 4.5a) and f =1.12 for the 2D local model (Fig. 4.4a), note that f ' <1 for both glass/resin cement and glass-ceramic/resin cement systems with f 3D and 2D assumptions. This increases the critical failure stress as shown in Eq. (4.5)...... 125
Table 4.4. Numerical values of parameter k’ of effective critical stress density m m ⎛⎞f ′ distribution Nk()σ Cr′′′= σ Cr calculated from kk' = ⎜⎟ and Table 3 and ⎝⎠f experimentally determined parameters m and k (glass: m=5.21, k=1.75E-12×mm−−25.21 MPa ; glass-ceramics: m=5.92, k=2.25E-14×mm−−25.92 MPa ) ...... 126
-xii- Table 4.5. Average values of residual stress components σ rr and σθθ at the bottom critical surface of glass-ceramics/glass (1mm radius concentric critical area) due to curing process (1% volume shrinkage of resin cement). The results are
calculated by axisymmetric FEA (Biaxial and Trilayer) models. σ rr and σθθ , respectively, represent the normal stresses along the radial and tangential directions...... 127
Table 5.1. Definition of microstructure parameters...... 151
Table 5.2. Microstructure identification results and effective Young’s moduli of AO and CO, DPF wall materials, subjected to tensile loadings. The values in parenthesis represent the standard deviation...... 152
-xiii-
LIST OF FIGURES
Figure 2.1. Schematic representation of proposed simplified method for mode I stress intensity factor calculation for a half-penny shaped crack normal to the interface. Perfect bond or slip boundary conditions are assumed at the interface...... 40
Figure 2.2. 2D and 3D cracked domain analysis to generate the configuration
correction factor f for the mode I stress intensity factor KfI =⋅σ π R...... 41
Figure 2.3. Schematic representation of proposed procedure used to determine an equivalent constant stress for the estimation of mode I stress intensity factor KIA at the crack tip A...... 42
Figure 2.4. Verification of finite element analysis for the 3D interfacial crack model (Fig. 2(d)) utilizing two types of loading. The effect of finite body size on the configuration correction factor f is examined by comparing the results from two models with body size 200×200×200 and 400×400×400, where the crack radius R=1. When the semi-infinite body I is much stiffer than the dissimilar semi-infinite body II (EI/EII=100), solutions of the interfacial crack model (Fig. 2(d)) approach those of the surface crack model (Fig. 2(c))...... 43
Figure 2.5. FEA calculations of f for constant tensile and proportional loadings and Y at the crack tip A of the 2D interfacial crack model as a function of modulus ratio EI/EII. The handbook solutions for special cases (EI/EII is large or 1) and the results by body force method demonstrate excellent match with the FEA data. The Poisson’s ratio is 0.3 (νI=νII=0.3) for all the results...... 44
Figure 2.6. FEA calculations of f for constant tensile and proportional loadings and Y at the crack tip A of the 3D local interfacial half-penny crack model as a function of modulus ratio EI/EII. The Poisson’s ratio is 0.3 (νI=νII=0.3) for all the results except νI=νII=0.1 for ‘Triangle’ and ‘X’...... 45
-xiv- Figure 2.7. (a) A ball-on-ring biaxial flexure test on a ceramic plate with a half-penny shaped crack at the bottom surface. (b) A bilayer structure with a half-penny shaped crack originated at the bottom surface of the top layer subjected to indentation loading, and its finite element model...... 46
Figure 2.8. Comparison of proposed approximate method and 3D FEA for a ball-on-ring biaxial test with two different crack radii...... 47
Figure 2.9. Comparison of the crack opening displacement between 3D FEA bilayer indentation model and proposed approximation method. Crack opening displacement is calculated at different locations along z axis. The top layer (E=60GPa, υ=0.3, thickness h) is not bonded to the substrate (E=200GPa, υ=0.3); the load P (P/Eh2=2/300) is applied through the spherical indenter (radius r=400h) to the top surface. Stress distribution σ obtained from the neighborhood of the imaginary crack in the axi-symmetric model is used as far field in the local model...... 48
Figure 2.10. Percent error in estimating crack tip stress intensity factor KIA of a half-penny shaped crack normal to the interface of a bilayer structure subjected to uniform temperature rise based on the proposed approximate method compared to accurate 3D FEA evaluation...... 49
Figure 3.1. Schematic representation of proposed simplified method for mode I stress intensity factor calculation for a half-penny shaped crack normal to the top-middle interface in a trilayer structure...... 74
Figure 3.2. Effective cracked bilayer domain analysis model versus cracked trilayer domain analysis model. Replace materials II and III with an effective material so that KIA remains the same. Poisson’s ratio is assumed to be 0.3 (νI=νII=νIII=0.3)...... 75
Figure 3.3. Percent error of KIA as a function of dimensionless middle layer thickness (h/R). A trilayer structure is approximated by a bilayer structure. Poisson’s ratio is assumed to be 0.3 (νI=νII=νIII=0.3)...... 76
Figure 3.4. Variation of Eeffective/EII with the change of dimensionless middle layer thickness (h/R). Poisson’s ratio is assumed to be 0.3 (νI=νII=νIII=0.3). As h/R approaches zero, Eeffective significantly deviates from EII. Eeffective was numerically obtained from FEM calculation...... 77
-xv- Figure 3.5. Variation of numerically obtained Eeffective with the change of EI/EII and EII/EIII for the case of (a) h/R=1/5 and (b) 1/10. Linear relationship between log(EIII/Eeffective) and log(EII/EIII) is observed and fitted as shown in solid lines. 78
−α Eeffective ⎛⎞EII Figure 3.6. Distribution of Eeffective. = ⎜⎟, EEII⎝⎠ III ⎡ 2 ⎤ hR⎛⎞EII 1−=−α [1 exp( − 3 )]⎢ 1 + 0.004⎜⎟ log10 ⎥ . Error in KI,A is within 2%. RhE ⎣⎢ ⎝⎠III ⎦⎥ Results are valid for EI/EII , EII/EIII~[0.01,100] and 0.1 Figure 3.7. Application of the proposed method to a 3D trilayer indentation model. The stress intensity factor KIA in (a) can be estimated based on the axisymmetric stress distribution in (c) and the proposed cracked trilayer domain analysis in (b)...... 80 Figure 3.8. Percent effort in KIA estimation for a trilayer indentation model (Fig 3.7a) as a function of EII with various combination of EI and EIII. Accurate KIA is evaluated from complete 3D FEM including a half-penny crack. Approximate KIA is calculated based on the axisymmetric stress calculation and the craked trilayer domain analysis. The parameters are set to be: h/Rcrack=1/10, Rball=5Rcrack, load=400N...... 81 Figure 4.1. Indentation tests on the bonded trilayer onlay model. Average dimension is listed in Table 4.1b...... 112 Figure 4.2. Fractured resin retained ceramic onlay after 3.5 years of service on a maxillary 1st premolar on a patient with a severe bruxing problem. Arrows point to the fractured ceramic surface. The buccal half of the onlay remains bonded to the tooth structure while the lingual portion is missing and the underlying dentin is exposed...... 113 Figure 4.3. Ball-on-ring biaxial experimental tests. Average dimension of samples used for biaxial tests is listed in Table 4.1a...... 114 Figure 4.4. 2D Plane Strain Local Model...... 115 Figure 4.5. 3D Local Model...... 116 -xvi- Figure 4.6. Ball-on-ring biaxial experimental data without resin cement (see Fig. 4.3a) for two materials: glass and glass-ceramics. The straight lines (YX=−5.21 23.81 for glass and YX= 5.92− 32.55 for glass-ceramics), are obtained by fitting the experimental data based on the least-square method. They are used to determine the parameters k and m of the critical stress density m −−25.21 distribution function Nk(σ Cr) = σ Cr . (glass: m=5.21, k=1.75E-12×mm MPa ; glass-ceramic: m=5.92, k=2.25E-14×mm−−25.92 MPa ) ...... 117 m Figure 4.7. Critical stress density distribution N as a function of σ Cr , Nk()σ Cr= σ Cr , at the bottom surface of glass-ceramic disks...... 118 Figure 4.8. Experimental data and predicted failure probability distributions of glass-ceramic disks with and without resin cement layer under biaxial tests (Fig. 4.3). The critical stress density distribution (obtained from Y=5.92X-32.55 in Fig. 4.6) was used to theoretically predict the failure probability of disks with resin cement...... 119 Figure 4.9. Experimental data and predicted failure probability distributions of indentation tests on bonded trilayer onlay model (Fig. 4.1), (glass-ceramics/resin cement/simulated dentin trilayer system)...... 120 Figure 4.10. Experimental data and predicted failure probability distributions of indentation tests on bonded trilayer onlay model (Fig. 4.1), (glass/resin cement/simulated dentin trilayer system)...... 121 Figure 4.11. Deformation caused by shrinkage of the resin cement layer in biaxial tests with resin cement and the trilayer onlay model. The deformations are calculated by AbaqusTM, and 1% volumetric shrinkage of resin cement layer is included. No external force is included...... 122 Figure 5.1. 2D schematics of equating a porous material using homogeneous solid with effective properties (a) through the combination of individual effects of pores (b) and microcracks (c)...... 149 Figure 5.2. An example of microstructure identification on 2D SEM image. The dark parts represent the pores and microcracks, and the white parts are the base materials...... 150 -xvii- CHAPTER 1 Introduction 1.1 Research background Bonded layered structures are widely used in industrial, military and medical fields to meet high performance requirements where a single material cannot satisfy. An anode supported electrolyte laminate composite, for example, is used in the design of solid oxide fuel cells [1-3] to provide the necessary strength to support operational loading. Thermal barrier coatings (TBC) are used to maximize the operational temperature range and prolong the life of substrates in propulsion systems and aerospace applications [4]. Fusion-bond epoxy powder coatings are applied to protect steel pipes from corrosion in pipeline construction [5]. Layered structures are also used in dentistry to protect damaged teeth without loss of aesthetics [6,7]. All layered structures have a finite service life, and a failure under thermal or mechanical loadings usually initiates at surfaces or interfaces due to stress concentrations acting on microdefects that are either preexisting or developed during service. Interface failures of this nature usually include debonding (delamination) of the interface and fracture initiation originating at the interface, which can be an -1- independent event or can follow interface debonding. For example, in solid oxide fuel cell laminates, failure initiating at the anode surface and/or anode-electrolyte interface was found to be due to the residual stresses from a thermal expansion mismatch [1-3]. Similarly, in a thermal barrier coating structure, residual tensions caused interfacial delaminations and/or cracks vertical to the interface [8-9]. In dentisty, failures of dental glass-ceramic restorations initiated from flaws at the interface have been widely observed [10-12] as well. The inevitability of failure demands prediction of the service life of layered structures, and this requires understanding of the failure mechanism of layered structures due to surface or interface fractures. Failures due to surface fracture or interfacial delamination have been widely studied [13-16]. In addition, a large number of solutions for stress intensity factor (SIF) are summarized in handbooks [17,18] for idealized boundary value problems with simple loading conditions. For instance, a half elliptical shaped surface crack in semi-infinite space was studied by Raju and Newman [19] using the finite element method (FEM) and by Murakami et al. [20-22] using the body force method. Hutchinson and Suo [15,16] studied the behavior of two-dimensional mixed mode cracks that either initiate and propagate along the interface or initiate at the free surface of a thin layer in a bi-layer elastic material structure. In addition to the surface fracture and interfacial delamination, the bonded layered structure often fails due to a crack initiating at and approximately perpendicular to the interface. For instance, in dentistry, evaluation of clinically failed glass-ceramic restorations demonstrated that a majority of these fractures (> 90%) are initiated from -2- flaws and stresses originating from the adhesive resin cement interface rather than from the contact surface of the restoration itself [10-12]. The stress intensity factor for a crack initiating at the interface is influenced by the material properties of the layered structure. In this case, the calculation becomes more complicated, and the available solutions are very limited. Noda [23] used the body force method to calculate the SIF for a rectangular shaped crack originating at the interface between two dissimilar semi-infinite materials. Similarly, Lee and Keer [24] analyzed the case of a half-penny-shaped crack with material modulus ratio 1/3. However, no systematic study of a half-penny shaped interfacial crack is available for other modulus ratios. In addition, the existing literature for bonded structures is limited to the case with constant tensile stress, although many solutions with proportional bending stress are available for a surface crack in a homogeneous material. Apart from characterization of the failure mechanism of the layered structure by SIF, failure probability prediction is also important for the survival life prediction and design improvement of layered structures. Statistical prediction of the failure of brittle materials has been investigated in recent decades. Batdorf and Crose [25] proposed a fracture mechanics based statistical model to study the failure probability of brittle materials. Chao and Shetty [26] introduced a critical stress density distribution function to characterize the surface flaw distribution. Wang et al. [27,28] employed the fracture mechanics based failure probability model to predict the failure probability distribution of model dental onlay restorations subjected to indentation load to failure testing. Surface flaw distribution derived from biaxial data was used to predict the -3- theoretical failure probability of the more complex trilayer model. The predicted values for the non-bonded ceramic supported by the simulated dentine substrate fell within the 90% confidence interval of the experimental data. However, the predicted values for the completely bonded case showed a somewhat higher failure probability than the experimentally derived values. It has been suggested that this discrepancy may be attributed to the possible bridging effect by the adhesive resin cement on the interfacial surface defects in the ceramic that was not accounted for in the analytical model [28]. 1.2 Objective In this work, the failure of a bonded tri-layer structure due to a crack initiating at the interface was studied. The effect of the three layers on the stress intensity factor (SIF) of a crack normal to the interface was investigated based on an idealized cracked trilayer domain analysis. Closed-form approximations were obtained for the SIF as a function of modulus ratio of bonded dissimilar materials. The idealized cracked domain analysis was then combined with macro-level stress calculations in a structure without a crack to approximate the SIF of the actual cracked geometry. A simplified method was further proposed where the value of stress at only one location over the position of the crack in an uncracked body analysis is required for evaluating the mode I SIF. The proposed method was based on the simple superposition principle and eliminated the need for curve-fitting the loading conditions. In addition, the solutions for the SIF of the idealized cracked domain analysis were employed to predict the -4- failure probability of a bonded layered structure. First, the mode I SIF is evaluated for the idealized geometry of a crack normal to the perfectly bonded interface between two semi-infinite dissimilar materials; a 2D single edge crack and a 3D half-penny shaped crack are considered. Constant tensile stress and proportional bending loading were considered. For each loading condition, the SIF was explicitly expressed as a curve-fitted function of the modulus ratio of two dissimilar bonded materials through finite element parametric studies. The obtained functions provided closed-form simple solutions for the SIF of a crack normal to the interface for all material combinations. The idealized cracked domain analysis was then combined with macro-level stress calculations in a structure without a crack, i.e., uncracked body analysis, to estimate the SIF of a practical cracked bilayer structure. A simplified method was developed where the value of stress at only one location over the position of the crack in the uncracked body analysis was required for evaluating the SIF. Second, the above work for a crack in a bilayer structure was extended to the case with a trilayer structure, i.e., a top functional material layer, a middle adhesive layer and a bottom substrate layer. In order to address the effect of three layers on the mode I SIF for a crack at the interface, an idealized cracked trilayer domain with a half-penny shaped crack initiating at the top-middle interface was considered. This trilayer structure was replaced by an equivalent bilayer structure equating the SIF for both structures. This lead to an explicit expression for the effective modulus of an equivalent homogeneous material as a function of the moduli of middle and bottom materials and the ratio of the middle layer thickness to the crack size. Combining this -5- result with the work on the cracked bilayer domain analysis, the effect of three layers on the SIF can be expressed in the form of the configuration factor for a crack in the cracked trilayer domain. The mode I SIF for a crack perpendicular to the interface in a bonded trilayer structure can then be evaluated following the similar approximation method for a bilayer structure. Third, based on the cracked bilayer/trilayer domain analysis, the effect of adhesive layer on the probability of cracks initiation from the interface in bonded layered structures was quantitatively examined. The possible crack bridging mechanism, i.e., the constraint effect of adhesive layer on the crack opening displacement, was examined. The method was applied to predict the failure probability of a simplified trilayer dental ‘onlay like’ restoration structure subjected to indentation loadings, and a good match between experimental results and predictions was observed. Forth, we considered the relationship between the effective modulus of porous materials and their microstructures. The available micromechanics analysis solutions were reviewed, and two models (Mori-Tanaka and differential scheme) were used to predict the effective Young’s modulus of diesel particulate filter which is a porous ceramic material. The predictions were compared with experimental results. -6- References [1] Selcuk A., Merere G. and Atkinson A., The influence of electrodes on the strength of planar zirconia solid oxide fuel cells. Journal of Materials Science, 36: 1173-82, 2001. [2] Selcuk A. and Atkinson A., Residual stress and fracture of laminated ceramic membranes. Acta Mater., 47: 867-874, 1999. [3] Wang Y, Walter ME, Sabolsky K and Seabaugh MM. Effects of powder sizes and reduction parameters on the strength of Ni-YSZ anodes. Solid State Ion 2006;177:1517-27. [4] Miller RA. Current status of thermal barrier coatings: An overview. Surf Coat Technol 1987;30:1-11. [5] Kehr JA. Fusion Bonded Epoxy (FBE): A Foundation for Pipeline Corrosion Protection. Houston, Texas: NACE Press; 2003. [6] Mjor IA, Toffenetti OF. Secondary caries: a literature review with case reports. Quintessence Int 2000;31(3):165–79. [7] Wang Y, Katsube N, Seghi RR, Rokhlin SI. Statistical failure analysis of adhesive resin cement bonded dental ceramics. Engng Fracture Mech 2007;74:1838-56. [8] Gruninger M.F., Lawn B.R., Farabaugh E.N. and Wachtman J.J.B., Measurement of residual stresses in coatings on brittle substrates by indentation fracture. J. Am. Ceram. Soc. 70: 344-348, 1987. [9] He M.Y., Evans A.G. and Hutchinson J., Crack deflection at an interface between dissimilar elastic materials: Role of residual stresses. International Journal of Solids and Structures, 31(24): 3443-3455, 1994. [10] Kelly J.R., Campbell S.D. and Bowen H.K., Fracture-surface analysis of dental ceramics. J Prosthet Dent, 62:536-541, 1989. [11] Kelly J.R., Giordano R., Pober R. and Cima M.J.. Fracture surface analysis of dental ceramics: clinically failed restorations. Int J Prosthodont, 3:430-440, 1990. [12] Thompson J.Y., Anusavice K.J., Naman A. and Morris H.F.. Fracture surface characterization of clinically failed all-ceramic crowns. J Dent Res, 73:1824-1832, 1994. [13] Atkinson A, Sun B. Residual stress and thermal cycling of planar solid oxide fuel cells. Mater Sci Technol 2007;23(10):1135-43. [14] Evans A.G. and Hutchinson J., On the mechanics of delamination and spalling in compressed films. International Journal of Solids and Structures. 20: 455-466, 1984. [15] Suo Z. and Hutchinson J., Steady-state cracking in brittle substrates beneath adherent films. International Journal of Solids and Structures, 25(11): 1337-1353, 1989. [16] Suo Z. and Hutchinson J., Interface crack between two elastic layers. International Journal of Fracture, 43: 1-18, 1990. [17] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. New York: ASME Press; 2000. [18] Murakami Y. Stress intensity factors handbook. Amsterdam: Elsevier; 2001;3rd. [19] Raju I.S. and Newman J., Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Engineering Fracture Mechanics, 11: 817-829, 1979. [20] Murakami Y., Analysis of stress intensity factors of modes I, II and III for inclined surface cracks of arbitrary shape. Engineering Fracture Mechanics, 22: 101-114, 1985. [21] Nisitani H. and Murakami Y., Stress intensity factors of an elliptical crack or a -7- semi-elliptical crack subject to tension. International Journal of Fracture, 10: 353-368, 1974. [22] Pommier S., Sakae C. and Murakami Y., An empirical stress intensity factor set of equations for a semi-elliptical crack in a semi-infinite body subjected to a polynomial stress distribution. International Journal of Fatigue, 21: 243-251, 1999. [23] Noda N. Stress intensity formulas for three-dimensional cracks in homogeneous and bonded dissimilar materials. Engng Fracture Mech 2004;71:1-15. [24] Lee JC, Keer LM. Study of a Three-Dimensional Crack Terminating at an Interface. J Appl Mech 1986;53:311-16. [25] Batdorf S.B. and Crose J.G., A statistical Theory for the Brittle Structures Subjected to Nonuniform Polyaxial Stresses. Journal of Applied Mechanics, 41: 459-464, 1974. [26] Chao L.Y. and Shetty D.K., Reliability Analysis of Structural Ceramics Subjected to Biaixal Flexure. Journal of the American Ceramic Society, 74(2): 333-344, 1991. [27] Wang R., Katsube N and Seghi RR, Improved form of a fracture mechanics based failure probability model for brittle materials. Journal of Applied Mechanics, 72(4): 609-612, 2005. [28] Wang R., Katsube N, Seghi RR, Rokhlin SI. Statistical failure analysis of brittle coatings by spherical indentation: theory and experiment. J Mater Sci 2006;41(17):5441-5454. -8- CHAPTER 2 Uncracked body analysis for accurate estimates of mode I stress intensity factor for cracks normal to an interface Abstract A bonded crack model method is presented for estimation of the stress intensity factor (SIF) for a 3D half-penny shaped crack originating at a bonded interface subjected to remote constant tensile and proportional bending loadings. Closed-form approximations are obtained for the SIF as a function of modulus ratio of bonded dissimilar materials. A combination of bonded crack model method and macro-level stress calculations in a structure without a crack (uncracked body analysis) significantly simplifies accurate estimation of SIF. The method was validated using 3D finite element computations. Since the proposed method requires no repetitive stress calculation as crack size changes, it is useful in life predictions. -9- 2.1 Introduction Layered materials are widely used for different products, devices and applications to enhance structural integrity and improve performance. Anode-supported electrolyte laminates, for example, are used in the design of solid oxide fuel cells to provide the necessary strength to support operational loading and residual stresses resulting from manufacturing [1]. Thermal barrier coating layers are employed to maximize the operational temperature range in gas turbines by insulating metallic components from heat [2]. Fusion-bond epoxy powder coatings are applied to protect steel pipes from corrosion in pipeline construction [3]. In dental restorations, layered structures are often used to reinforce damaged teeth without loss of aesthetics [4-5]. All layered structures have a finite service life and the prediction of their useful lifetime is important. A failure under thermal or mechanical loading usually initiates at surfaces or interfaces [6-7] due to stress concentrations acting on microdefects that are either preexisting or developed during service. Interface failures of this nature usually include debonding (delamination) of the interface or fracture initiation originating at the interface that can be an independent event or can follow interface debonding. Failures due to surface fracture or interfacial delamination have been widely studied [8-10]. However, brittle fractures originating at the interface have not been extensively investigated. Examination of brittle failures in a layered structure requires an evaluation of the stress intensity factor (SIF) for a crack originating at the interface. -10- The problem of a crack in a homogeneous body has been widely studied. A large number of solutions for SIF are summarized in handbooks [11-12] for idealized boundary value problems with simple loading conditions. In layered structures, the SIF for a crack initiating at the interface is influenced by the material properties. In this case, the calculation becomes more complicated and the available solutions are very limited. Noda [13] used the body force method to calculate the SIF for a rectangular shaped crack originating at the interface between two dissimilar semi-infinite materials. Similarly, Lee and Keer [14] analyzed the case for a half-penny-shaped crack with material modulus ratio 1/3. However, systematic study of a half-penny shaped interfacial crack is not available for other modulus ratios. In addition, the existing literature for bonded structures is limited to the case with constant tensile stress, while many solutions with proportional bending stress are available for a crack in a homogenous body. The determination of stress intensity factor for a crack by solving boundary value problems of structural components of arbitrary geometry is computationally expensive. In order to avoid modeling cracks, superposition principle can be employed to estimate the stress intensity factors [15]. The predictions of stress from uncracked global structure are used as loading over the crack surface in a local cracked subdomain analysis. The tractions on the intersection surface (between the uncracked global structure and a local cracked subdomain) are applied as the residual forces to the uncracked global structure, and the corresponding stress field for the position of the crack is re-evaluated. This alternating method is repeated until the residual forces become negligible, and the superposition of the obtained local cracked subdomain -11- analysis and the uncracked global structure analysis leads to evaluation of the stress intensity factors of cracks in global structure. This method can be applied to cracks with any size and geometry in a global structure. When the crack size remains much smaller than the size of the overall structure and the existence of the crack does not significantly alter the stress field away from the crack, a crack in an infinite media or a semi-infinite media (a surface crack) can be employed without an alternating procedure. The stress distributions obtained from the uncracked global structure are used as loadings over the crack surface in an infinite media (for an embedded crack) or a semi-infinite media (for a surface crack) and the stress intensity factors can be directly estimated. In many practical cases where this loading condition varies gradually and continuously, solutions for the SIF available in handbooks can be utilized through curve-fitting the loading over the crack face or equivalently the far field stress. In this work, we propose a simplified method where the value of stress at only one location over the position of the crack in an uncracked body analysis is required for evaluating the mode I SIF. The proposed method is based on the simple superposition principle and eliminates the need for curve-fitting the loading conditions. In particular, this method is developed with an emphasis on the bonded structure where systematic solutions for the SIF for a crack developing in a bonded layered geometry are not available in handbooks. The mode I SIF is evaluated for the idealized geometry of a crack normal to the perfectly bonded interface between two semi-infinite dissimilar materials; a 2D single edge crack and a 3D half-penny shaped crack are considered. Constant tensile stress -12- and proportional bending loading are examined in order to elucidate the effect of stress variation along the crack depth on the SIF. For each loading condition, the SIF is explicitly expressed as a curve-fitted function of the modulus ratio of two dissimilar bonded materials. The functions obtained provide closed-form simple solutions for the SIF of a crack normal to the interface for all material combinations. The major steps involved in this method are schematically summarized in Figure 2.1. Given a structure with a small crack (I), the stress distribution in the region of a crack is calculated from an identical structure without a crack (II). The notion of an equivalent constant tensile stress σeq is introduced so that the SIF for the constant stress σeq is identical to that for a nonlinear far filed stress distribution. This stress σeq perpendicular to the crack face is obtained at one location over the crack location in an uncracked body analysis and enables evaluation of the SIF based on the developed closed form function of material modulus ratio of dissimilar materials. By combining the σeq obtained in (II) with the results for the SIF for (III), KI for (I) is estimated. To examine the proposed method, the SIFs for a half-penny-shaped crack located at the bottom interface of the top layer of both a single and bi-layer system subjected to indentation loading are estimated. The estimated SIFs are compared against those determined by accurate 3D FEM calculations including complete crack geometry, and the validity of the proposed method is quantitatively verified through these examples. -13- 2.2 Methodology 2.2.1 Cracked Domain Analysis For simple boundary value problems with idealized crack geometry and loading conditions, the SIF can be calculated as Kf=⋅σ π R, (2.1) where the crack size, the loading stress and the configuration correction factor are respectively denoted by R, σ and f [11]. The configuration correction factor f depends on the crack shape and loading type. It is also referred to as the boundary correction factor [16] or the magnification factor [12]. For example, for a single edge crack subjected to a constant tensile stress σ as shown in Fig. 2.2(a), the value of f is known to be fconstant=1.12 [11-12]. If the crack is subjected to proportional bending stress with the crack tip stress σ, the value of f is fproportional=0.683 [11-12]. There are a number of solutions available for various crack geometries and loading conditions [10-12] including those for a surface half-penny crack. As discussed in section 1, a large number of load bearing structures consist of bilayered materials, and microdefects at interfaces are often the failure initiation sites. However, boundary value problems of a crack originating at the interface between two layered materials with different properties, have not been extensively examined. In this work, the stress intensity factor for a crack originating at an interface between two solids is systematically examined. Finite element analysis (ABAQUS V6.6 [17]) is used to evaluate the configuration correction factor f as a function of the Young’s modulus ratio for a 2D interfacial crack model (Fig. 2.2(b)) and for a 3D interfacial -14- half-penny shaped crack model (Fig. 2.2(d)). Finally simple fitting functions are provided for calculation of the correction factor. 2.2.1.1 Cracked domain analysis for 2D interfacial crack As shown in Fig. 2.2(b), we consider a single edge crack originating at the interface between two perfectly bonded dissimilar semispaces with elastic moduli EI, νI and EII, νII, respectively. The crack length measured from the interface to the crack tip A is R. Constant tensile remote stress σI is applied in the top semispace I and tensile stress σII in the bottom semispace II. The stress σII is chosen so that away from the interface the normal strain parallel to the interface remains the same in both semispaces: σ σ III= . (2.2) EEIII This 2D plane strain interfacial crack model can be used to calculate the SIF of a long rectangular crack originating at the bonded interface subjected to remote normal stress. The effect of Poisson’s ratio on the configuration correction factor f is found [13] to be small, and, in this work, Poisson’s ratio is assumed to be 0.3 for both materials I and II (νI=νII=0.3). The configuration correction factor f then becomes a function of the Young’s modulus ratio EI/EII. The 2D interfacial crack model (Fig. 2.2(b)) subjected to proportional bending stress is also investigated. In this case, linearly-varied tensile stress σI is remotely applied in the top semispace I, while linearly varied compressive stress σII is applied in the bottom semispace II. The slopes of linear tension and compressive stresses are -15- selected so that at the same distance from the interface the equality (2) is satisfied for absolute values of the stresses. Finite element analysis (FEA) with plane strain quadratic elements is performed to evaluate the configuration correction factor f. The body size normalized by the crack depth (R=1) is chosen to be 200 x 200 so that the effect of finite boundary on calculation results can be minimized. The normalized size of an individual finite element is chosen to be smaller than 0.01 around the crack front and the bonding interface so that the stress calculation in the critical area is accurate. A special crack element with one-quarter mid-side node [18, 19] is employed at the crack front. Validation of the FEA models is described in Section 3.1. The J-integral method [20, 21] is used to calculate the stress intensity factor KIA at the tip A. The value of f is then calculated as f AIA= KR/ (σ π ) . (2.3) The Young’s modulus ratio EI/EII is chosen to vary from 1/100 to 100. Based on the FEA results, the configuration correction factor f is fitted as a function of EI/EII for constant tensile and proportional bending loading cases, respectively. 2.2.1.2 3D interfacial half-penny shaped crack The 3D interfacial crack model Fig. 2.2 (d) describes a half-penny shaped crack of radius R originating at the interface between two dissimilar perfectly bonded semispaces (EI, νI and EII, νII). Given νI=νII=0.3, the configuration correction factor f depends on the Young’s modulus ratio EI/EII. As in the 2D interfacial model, two typical loading conditions, i.e. constant tensile and proportional bending stresses are -16- investigated. There is no stress variation in the direction parallel to the crack face (direction y shown in Fig. 2.2 (d) ). Three dimensional FEA with quadratic 20-node cubic elements is performed to evaluate the configuration factor f. As in the 2D model, the body size normalized by the crack radius (R=1) is chosen to be 200 x 200 x 200 and an individual element is chosen to be smaller than 0.01 in critical areas. A special crack element with one-quarter mid-side node [18, 19] is employed at the crack front. The J-integral method [20, 21] is used to calculate the SIF at the tip A, KIA. The value of f is then calculated from Eq. (2.3). Based on the FEA results, the configuration correction factor f is fitted as a function of EI/EII for constant tensile and proportional bending loading cases, respectively. 2.2.2 Y factor and equivalent stress In practice, the location of a possible structure failure is estimated as the location of the maximum stress that can be obtained by solving a corresponding boundary value problem for the identical structure without a flaw or a crack. The actual nonlinear stress field in the neighborhood of a small crack is replaced by an approximate linear stress distribution. This approximate linear stress distribution can then be separated into constant tensile stress and proportional bending loadings using the linear superposition principle. These stress components are used as the far field stress to estimate SIF as in Tada and Paris [11] if the crack size remains significantly smaller than the size of the overall structure, and thus the crack does not affect the global stress distribution. This existing methodology, however, involves a curve-fitting procedure. In this work, we develop an improved estimation method for the SIF for a crack by -17- utilizing the notion of equivalent constant tensile stress and approximately considering stress variation along the crack depth and crack size. 2.2.2.1 Approximation of stress intensity factor at the crack tip for linear and nonlinear loading stress distribution As in Figure 2.3 (a), the linear stress distribution normal to a crack face can be expressed as ⎛⎞zz σ =−⎜⎟1 σσ12 + , (2.4) ⎝⎠R R where σ1 and σ2 respectively represent the normal stress at the interface and at the crack tip of the half-penny shaped crack of radius R; z is measured perpendicular to the bonded interface along the crack depth. This linear stress distribution can be expressed as a combination of a constant and a proportional part σ=σ1 - (σ1-σ2)·z/R. The SIF KIA at the crack tip A is then calculated by adding the SIFs (1) generated from these two stress parts: KfIA=⋅ constantσ112πσσπ Rf − proportional ⋅−( ) R. (2.5) As the result Eq. (2.5) can be rewritten as KfIA=⋅ constantσ eq π R, (2.6) where the equivalent stress σeq is expressed as ⎛⎞ffproportional proportional σ eq =−⎜⎟1 σσ12 + . (2.7) ⎝⎠ffconstant constant The word ‘equivalent stress’ is used because the linear stress distribution along the crack depth is replaced by an equivalent constant tensile stress obtained by equating the SIFs for actual and equivalent stress distributions. -18- By equating σ=σeq from Eq. (2.7) and Eq. (2.4) one can find location z=zeq for the corresponding equivalent constant tensile stress (Fig. 2.3(a)): zeq=Y·R, (2.8) where the parameter Y is used to find the location of the equivalent stress point and is determined as: f Y = proportional , (2.9) fconstant The factor Y is defined as the ratio of fproportional to fconstant and may be considered as a scaling factor. Once the scaling factor Y is known, the SIF KIA can easily be calculated from Eq. (2.6). For the linear stress distribution, the equivalent stress σeq is obtained by directly choosing the stress value at the location zeq=Y·R. In practice, however, the stress distribution may be neither linear along the crack depth direction, nor constant along the width direction. When the crack is much smaller than the structure size (this assumption is fundamental for our model), the distance between points z=R and z=zeq is small, and the nonlinear distribution of the remote stress can be approximated by the linear term (Fig. 2.3(b)). Since the scaling factor Y, Eq(9), is independent of linear stress, the approximate equivalent constant tensile stress, (σeq)App, can be directly obtained from the nonlinear stress distribution curve at the position zeq. Thus the above methodology based on the linear superposition can be approximately applied to the nonlinear stress distribution as illustrated in Fig 2.3(b). The stress intensity factor at the crack tip, KIA, is then approximated by substituting σeq with (σeq)App in Eq. (2.6) as Kf=⋅σ π R (2.10) ()IAApp constant( eq )App -19- In assessing brittle failure initiating from a crack, accurate evaluation of the SIF is critical. Any approximate estimation methodology should require accurate evaluation of far field stress and crack size. In Eqs. (2.8)-(2.10), not only the crack size R but also stress variation along the crack depth are taken into account approximately through the equivalent stress (σeq)App. Since the configuration correction factor f is already available through handbooks [11-12] and the results in section 2.1, this estimation process for SIF requires only computation of the stress field from the boundary value problem in the identical structure without a crack. Therefore, as long as the existence of a crack does not interfere with the far stress field, the proposed method significantly simplifies the SIF estimation for cracks with various sizes. The notion of the equivalent stress similar to Eq. (7) and approximation of nonlinear stress distribution by a corresponding linear stress distribution were discussed by Tada and Paris [11]. However, the use of the equivalent constant stress point introduced in Eq. (8) avoids a curve-fitting procedure and significantly simplifies the SIF estimation procedures. 2.2.2.2 Variation of stress intensity factor along the crack front In section 2.2.1, the formulae are derived for the SIF at the crack tip A (Fig. 2.3). If the configuration correction factor f at angle θ is employed in Eqs. (2.5) and (2.6), the SIF along the crack front can be expressed as a function of θ. Accordingly, the scaling factor Y, determined from Eq. (2.9), becomes a function of angle θ: f ()θ Y()θ = proportional , (2.11) fconstant ()θ and zeq function defined by -20- zeq(θ)=Y(θ)R (2.12) can be determined. The equivalent stress (σeq(θ))App as a function of θ can be determined using zeq(θ) from the stress distribution σxx along the symmetric axis at the crack location. The stress intensity factor along the crack front can then be estimated as: Kf()θ =⋅ ()θσθ () π R . (2.13) ()IApp constant( eq )App This approximation method is developed based solely on 1) the linear superposition and 2) the approximation that the deviation of the actual stress distribution from an idealized linear stress distribution remains relatively small in the region of a small crack. Therefore, the proposed method can be extended to a crack with shape other than a half-penny such as an elliptical crack. The SIF as a function of the crack front can be obtained as long as the configuration correction factors fconstant and fproportional are available. 2.3 Results: Configuration correction factor f and scaling factor Y 2.3.1 Evaluation of FEA computation model The effect of finite body size on the configuration correction factor f is examined by comparing the results from two models with normalized body size 200 x 200 x 200 (with R=1) and 400 x 400 x 400. In Fig. 2.4, f as a function of θ for EI/EII=100 is plotted by the dashed line for 200 x 200 x 200 body and by circles for 400 x 400 x 400 -21- body, respectively. These two results coincide with each other very well for both constant tensile and proportional bending loadings. Thus the selected body size is sufficient for the cracks considered and it does not affect the f factor calculation. Since the mesh sizes were chosen independently for both cases, this validates the independence of mesh size. The above was also verified for other two representative modulus ratios of EI/EII=1 and EI/EII=1/100. When the semi-infinite body I with the crack is much stiffer than the bottom semi-infinite body II (i.e. EI>>EII), the constraint of the bottom medium on the crack opening displacement diminishes. With reduction of the ratio EII / EI the configuration correction factor f for the interfacial crack model (Fig. 2.2(b) and (d)) asymptotically approaches that for the surface crack model (Fig. 2.2(c)). For this case, as shown in Fig. 2.4, the finite element results for the interfacial crack model (dashed line and circles) are almost identical to that for the surface crack model (solid lines) for both constant tensile and proportional bending loadings. 2.3.2 Cracked domain analysis for 2D interfacial crack Following the same procedure as in section 3.1, the dependence on both finite element mesh and finite body size are eliminated in computations of the configuration correction factor f for the 2D local interfacial crack model (Fig 2.2(b)). The FEA calculations of f for constant tensile and proportional bending loadings, respectively, are plotted as functions of the modulus ratio EI/EII by circles in Fig. 2.5(a). The data are then curve fitted as a function of EI/EII for constant tensile and proportional bending loading cases, respectively. Different forms of functions (such as simple polynomial, power, and Weibull equation) are used to curve fit the obtained -22- FEA results. By trial and error, the form of Weibull equation with some modifications is shown to be the most suitable function to reflect the relationship between factor f and the ratio EI/EII, and it is employed for both cases as follows. 6.7 ⎛⎞⎧ ⎡⎤lnEE /+ 9.5 ⎫ ⎜⎟⎪ ()III ⎪ fconstant =−⋅−1.12 1 0.45 exp ⎨ ⎢⎥⎬ (2.14) ⎜⎟12.1 ⎝⎠⎩⎭⎪ ⎣⎦⎪ for constant tensile loading and 5.3 ⎛⎞⎧ ⎡⎤lnEE /+ 7.5 ⎫ ⎜⎟⎪ ()III ⎪ f proportional =−⋅−0.683 1 0.283 exp ⎨ ⎢⎥⎬ (2.15) ⎜⎟9.6 ⎝⎠⎩⎭⎪ ⎣⎦⎪ for proportional bending loading. The curve fitting error is smaller than 1%. Based on Eq. (2.9), the scaling factor Y can also be expressed as a function of modulus ratio EI/EII: ⎧ 5.3 ⎫ ⎪ ⎡⎤ln()EEIII /+ 7.5 ⎪ 1−⋅− 0.283 exp ⎨ ⎢⎥⎬ ⎪ ⎣⎦9.6 ⎪ Y =×0.61 ⎩⎭. (2.16) ⎧ 6.7 ⎫ ⎪ ⎡⎤ln()EEIII /+ 9.5 ⎪ 1−⋅ 0.45 exp ⎨ −⎢⎥⎬ 12.1 ⎩⎭⎪ ⎣⎦⎪ Eq. (2.16) is shown by the solid line in Fig. 2.5(b). When EI/EII is large, the factor f approaches that for the 2D local surface crack model (a single edge crack model). In this case, it is well known [11-12] that fconstant=1.12 and fproportional=0.683; from Eq. (2.9), the scaling factor is calculated to be Y=0.61. When the two semi-infinite bodies I and II are identical, i.e. EI=EII, the 2D local interfacial model is equivalent to a 2D crack embedded in a homogeneous infinite body with fconstant=0.707 and fproportional=0.53 [11-12]; the scaling factor is calculated to be Y=0.75. These values are shown by squares in Fig. 2.5(a,b), demonstrating excellent match with our FEA data (open circles). -23- Based on the body force method [22, 23], Noda [13] calculated the configuration correction factor f under constant tensile loading, fconstant, for some values of modulus ratio EI/EII. These data are plotted by triangles in Fig. 2.5(a). Again, there is good agreement between the FEA data (open circles) and the semi-analytical results (triangles) [13]. 2.3.3 Cracked domain analysis for 3D interfacial half-penny shaped crack The finite element results for the configuration correction factor fA at tip A of a half-penny crack versus modulus ratio EI/EII are plotted by circles in Fig. 2.6(a) where the solid lines show fitting curves from modified Weibull functions as in 2D case: 12 ⎛⎞⎧ ⎡⎤lnEE /+ 20 ⎫ ⎜⎟⎪ ()III ⎪ f A, constant =−⋅−0.65 1 0.21 exp ⎨ ⎢⎥⎬ (2.17) ⎜⎟22 ⎝⎠⎩⎭⎪ ⎣⎦⎪ for constant tensile loading and 12 ⎛⎞⎧ ⎡⎤lnEE /+ 20 ⎫ ⎜⎟⎪ ()III ⎪ f A, proportional =−⋅−0.465 1 0.10 exp ⎨ ⎢⎥⎬ (2.18) ⎜⎟22 ⎝⎠⎩⎭⎪ ⎣⎦⎪ for proportional bending loading. The curve fitting error is less than 1%. Based on Eq. (2.9), the scaling factor Y is again expressed as a function of modulus ratio EI/EII as ⎧ 12 ⎫ ⎪ ⎡⎤ln()EEIII /+ 20 ⎪ 1−⋅ 0.1 exp ⎨ −⎢⎥⎬ ⎪ ⎣⎦22 ⎪ Y =×0.715 ⎩⎭. (2.19) ⎧ 12 ⎫ ⎪ ⎡ln()EEIII /+ 20⎤ ⎪ 1−⋅ 0.21 exp ⎨ −⎢ ⎥ ⎬ 22 ⎪⎩⎭⎣ ⎦ ⎪ Eq. (2.19) is shown by the solid line in Fig. 2.6(b). -24- As for the 2D case, when EI/EII is large, the factor f approaches that for the 3D local surface crack model (a half-penny-shaped surface crack model). The results for this case (Fig. 2.2(c)) are well known fA,constant=0.65 and fA,proportional=0.46 [9-10], and the scaling factor from the data can be calculated as Y=0.71. These values are shown by squares in Fig. 2.6(a,b) and are in good agreement with the FEA results. For an interfacial crack model with EI/EII=1/3 and νI=νII=0.1, Lee and Keer [14] obtained by the body force method [22, 23] that fA,constant=0.533 for the same crack model. This point is shown by a triangle in Fig. 2.6(a). Since our FEA computations are performed for the νI=νII=0.3, the FEA data are also obtained for the case EI/EII=1/3 and νI=νII=0.1. This result is plotted by ‘X’ in Fig. 2.6(a). The triangle point coincides well with the ‘X’ point, and both points (νI=νII=0.1) are in good agreement with the circle point for νI=νII=0.3. This not only verifies our FEA calculation against other methods but also confirms that the factor f is not sensitive to Poisson’s ratio as pointed out by Noda [13]. Noda [13] generated the shape-independent SIF calculation formula * K I = f σ π area with 20% accuracy based on the calculation of rectangular shaped interfacial cracks under uniform tensile loading. In order to compare our obtained results against their approximate formula, Noda’s equation [13] is converted to the corresponding equation for a half-penny shaped crack as in K I = fσ πR , utilizing the relationship of area=πR2/2. His results for 3D model subjected to uniform tensile loading are shown in a dashed line in Figure 2.6(a). A good agreement between two results are observed for the region ln(E1/E11) less than 1. -25- For the region beyond ln(E1/E11) greater than 1, the difference between our FEA results and Noda’s approximate results become significant. Scaling factor Y as a function of modulus ratio EI/EII for 3D cracked domain analysis is shown in solid curve in Figure 2.6(b). For comparison purpose, the result of 2D local model is also plotted in dotted line. While the trend is somewhat similar, there exists significant difference in factor Y between 2D and 3D. By comparing Figure 2.5(a) and Figure 2.6(a), factors f for 2D for both uniform tensile loading and proportional loading are larger than those for 3D. This can be attributed to the fact that a 2D crack is a through crack instead of a half-penny crack in 3D. 2.4 Application In this section, we will demonstrate the validity of the proposed method for the estimation of the stress intensity factor of a crack (summarized in section 2.2) using the biaxial test model as shown in Fig. 2.7 (a) and the bi-layer indentation test model as shown in Fig. 2.7 (b). The bi-layer system in Fig. 2.7 (b) will also be used to examine the SIF of a crack when it is subjected to uniform temperature increase. For each example, simulations of the SIF are performed using the proposed simplified method and the results compared to the complete 3D finite element method. The commercially available software package ABAQUS V6.6 [17] is used for FEA simulations. By comparing the results of these two methods, the validity of the proposed method is quantitatively examined. -26- 2.4.1 One-layer ball-on-ring biaxial test In a ball-on-ring biaxial test, a ceramic plate is supported by a rigid ring as shown in Fig. 2.7 (a); an indentation load is applied to its top surface through a rigid spherical indenter. A half-penny shaped crack is assumed to be located at the center of the bottom surface of the ceramic disc. The contact surface between the indenter and the ceramic and that between the ceramic and the support ring are both assumed to be free from friction. The material properties and the dimensions are summarized in Table 2.1a. In order to examine the effect of crack size on the validity of the proposed method, we choose normalized crack radii R/h to be 0.1 and 0.015 and an normalized indentation load of P/Eh2 = 2/300 where h is the thickness of the plate. In both cases, the crack size remains significantly smaller than the thickness of the ceramic disc. For the biaxial test model, a complete 3D finite element simulation is employed to accurately calculate the mode I SIF of the half-penny shaped crack. Only one-quarter of the loading structure is modeled due to the symmetry. Quadratic 20-node cubic elements are used to model the ceramic disc. The size of individual finite elements around the crack front and in the neighborhood of contact area is carefully chosen so that the error of the numerical calculation remains within 1%. A special crack element with one-quarter mid-side node [18, 19] is employed at the crack front, and the J-integral method [20, 21] is used to calculate the SIF. The proposed method outlined in section 2.2 is employed to approximate the SIF of the half-penny crack in the biaxial test model. First, the biaxial test structure without a crack is simulated by a 2D axisymmetric finite element model. A far field normal stress distribution in the region of the crack location is obtained from the radial -27- normal stress component along the axis of symmetry at the bottom of the disc. Second, for the 3D local surface half-penny crack model, the scaling factor Y is found as described in section 3.3 (it is found to be Y=0.71). Based on this, an approximate equivalent stress (σeq)App is chosen at the location Zeq=0.71·R. And third, the approximate SIF is calculated based on Eq. (2.10). In Fig. 2.8, the results for the SIF obtained by the proposed method (points: open circles and asterisks) are compared to that obtained by the 3D finite element simulations (solid lines). The open circles represent the local model results using a constant Y value which is independent of angle θ along the crack tip. For the case of a small crack of radius (R/h = 0.015), the results of the 3D FEA (solid line) are in good agreement with the proposed approximate results using a constant Y at all angles (open circles). However, for the case of a large crack (R/h = 0.1), the results of the 3D FEA are in good agreement with the proposed approximate results at small angles θ (θ=0o at the crack tip A) but gradually diverged as the angle θ increased. The deviation of the proposed approximate method with constant Y (Y=0.71) from the 3D FEA for the larger crack occurs since Y is actually a function of the angle θ. As the angle θ increases from zero to ninety degrees, the value of fconstant increases but that of fproportional decreases [11-12]. This causes a monotonic decrease of factor Y and the location zeq. In this biaxial test example, the decrease of the location zeq leads to the increase of the equivalent stress and consequently the increase of the stress intensity factor as θ increases. For a small crack this increase is negligible; however, for a crack with R/h = 0.1, the increase of the equivalent stress is larger and the approximate result deviates from the 3D FEM results. As was described in section 2.2.2 the θ -28- dependence of the stress intensity factor can be taken into account. The equivalent stress, (σeq(θ))App, is determined at the location zeq(θ) based on Eq. (2.11) for the case with a large crack with radius R/h = 0.1. Using the handbook [11-12] values of fconstant and fproportional as a function of θ, new estimates of the stress intensity factor along the crack front are then made from Eq. (2.13); the results are represented in Fig. 2.8 as asterisks. This new estimate is in good agreement with the results derived from the 3D FEA calculations for all angles. The proposed approximate method outlined in sections 2.2.1 and 2.2.2 for determination of the SIF along the crack front of a half-penny shaped surface crack is thus validated by comparing the results with the 3D FEM results. When the stress variation along the crack front remains small, as in the case of a small crack with radius R/h = 0.015 in this example, the equivalent stress along the entire crack front can be approximated by that at the crack tip. However, when the stress varies significantly along the crack front, as in the case of a large crack with radius R/h = 0.1, the equivalent stress as a function of angle needs to be employed as discussed in Section 2.2.2. We have found that as long as the existence of a crack does not significantly change the stress distribution in the region away from the crack (as in the case of crack depth below 10% of thickness), the proposed approximate method can be utilized in estimating the stress intensity factor regardless of the crack size. In assessing brittle failure, one is mostly interested in the SIF of a relatively small crack since failure is imminent for larger cracks. For this reason, the proposed method is very effective in evaluating the stress intensity factor for various crack sizes since it only requires the knowledge of the stress distribution without a crack. -29- Further validation will be needed to extend the method outlined in section 2.2.2 to a crack with shape other than a half-penny such as a half-elliptical crack. However, since the method outlined in section 2.2.2 does not involve any assumption related to crack shape, it is anticipated that the stress intensity factor along the crack front may be approximated by the method as long as the configuration correlation factors fconstant and fproportional along the crack front are known. 2.4.2 Bilayer indentation test: Perfectly bonded interface In a bilayer indentation test, a plate is perfectly bonded to a dissimilar solid substrate as shown in figure 2.7 (b), and indentation load P is applied to its top surface through a rigid spherical indenter. If the indenter is blunt and the contact area large enough, a half-penny shaped crack originates from the center of the bonded interface (bottom surface of the top layer) below the indentation load. No contact friction is assumed between the indenter and the top layer. The material properties and the dimensions are summarized in Table 2.1b. The results are illustrated with examples of normalized crack radii R/h = 0.1, 0.015 where h is the thickness of the top layer. A 3D finite element simulation is again employed to calculate the mode I SIF of the half-penny-shaped crack, as shown in Fig. 2.7 (b), following the same procedure described in section 4.1. A 2D axisymmetric finite element model was used to compute the radial normal stress distribution in the structure without a crack. For a 3D local interfacial half-penny crack, the configuration correction factor f and the scaling factor Y are given as a function of the modulus ratio EI/EII in section 3.3. An approximate equivalent stress (σeq)App is chosen at the location Zeq based on Eq. (2.8), and the approximate SIF is calculated based on Eq. (2.10). -30- The results for the SIF at the crack tip obtained by the proposed method are then compared with the 3D finite element simulations. The relative error (in percent) is calculated by ()KK− () %error =×IA Approximate IA3 D FEA 100 . (2.20) ()KIA3 D FEA The error remains smaller than 1% for all combinations of Young’s modulus, indenter size, loading and crack size as summarized in Table 2.2. For some parameters a negative equivalent stress of crack closure is predicted; these cases are indicated by ‘NA’. This example demonstrates that the proposed method can be successfully utilized in bi-layered structures, composed of materials with various combinations of Young’s modulus, to estimate the SIF for a 3D interfacial half-penny crack model. As in the case of a surface crack in section 4.1, as long as the existence of a crack does not significantly affect the far field stress distribution, the proposed method can successfully be utilized. The proposed method can also be used to approximate the SIF of a crack originating at an interface in a multilayered system involving an intermediate adhesive layer. Our preliminary investigations indicate that this method can also be applied to indentation tests of a tri-layer system (top/adhesive/bottom layers) with a crack initiating at the interface between the top and adhesive layers. For a crack size as large as the thickness of the adhesive layer, it is found that the interfacial crack model with dissimilar materials for all layers can be effectively employed. This will be reported in future publications. -31- 2.4.3 Bilayer indentation model: Debonded interface with frictionless contact A crack originating at the interface often results from imperfect bonding between the two layers. In order to assess the applicability of the proposed method, we consider an extreme case of complete debonding between the top and bottom layers in a bilayer indentation test model as described in section 4.2 (Fig. 2.7 (b)). If the layers are not bonded, it is reasonable to assume frictionless contact between top and bottom layers (slip boundary conditions at the interface) i.e. continuity of normal components of displacement and stress and vanishing of shear stress. The frictionless contact between two layers is incorporated into a 3D finite element model (Fig 2.7 (b)) with a crack radius R/h = 0.1 (10% of the top layer thickness) and material properties and dimensions summarized in Table 2.1b. The stress distribution in the structure without a crack is calculated using a 2D axisymmetric finite element model also with frictionless contact between the two layers. As a cracked domain analysis model, we select a 3D surface crack model Fig 2.2(c) with a traction-free bottom surface. While for frictionless contact the normal stress is obviously not zero at the bottom surface of the top layer (Fig 2.7(b)) we assume that the constraint by the bottom layer on the crack opening displacement is small compared to that for the perfectly bonded case (Fig. 2.2(d)), and thus it is reasonable to evaluate the applicability of the 3D surface crack model Fig 2.2(c). (It is essential that, as is stated above, in the cracked domain analysis the far field distribution of stress normal to the crack surface is equal to the actual stress in the structure without a crack.) In the local model the scaling factor Y=0.71 obtained in -32- Section 3.3 is used to determine an approximate equivalent stress (σeq)App and the SIF KIA by Eq. (10). The results for the SIF at the crack tip obtained using the cracked domain analysis were compared with those obtained by the 3D finite element simulations. The relative error estimated by Eq. (20) for the SIF is shown in Table 2.3. For a crack with normalized radius R/h = 0.1 it is equal to or less than 2% for all investigated combinations of Young’s modulus, indenter size and loads. The error is smaller for the compliant bottom layers, which is expected since the local model has a traction free bottom surface. For the perfectly bonded interface case described in Section 4.2, the perfectly bonded interfacial conditions are accurately modeled through the 3D interfacial model (Figure 2.2(d)). Comparing the debonded case to the perfectly bonded case, it is surprising that a simple model with a traction-free condition (Figure 2.2(c )) can be successfully used to accurately calculate KI for the debonded interface with a frictionless contact without modeling exact interfacial conditions. To further our discussion of the applicability of this local surface crack model, the crack opening displacement as a function of distance from the interface (z mm) obtained by the 3D FEA bilayer indentation model (solid line) and local model (dashed line) are compared in Figure 2.9. Crack opening displacement is calculated at different locations along the z axis for the bottom layer. For this particular plot, the top layer (E=60 GPa, υ=0.3) is not bonded to the substrate (E=200GPa, υ=0.3). This layered configuration results in the largest error for the stress intensity factor (2%) as shown in Table 2.3. The error between the actual and the surface crack opening displacement in the approximate method is below 5% near the crack tip (z= 0.8-1.0) and increases to 25% at the crack -33- mouth (on the interface). The error of crack opening displacement is attributed to the fact that the normal stress exerted by the substrate to the top layer at the interfacial surface is neglected in our approximate method through traction free assumption. The normal stress at the interface tends to constraint crack opening displacement, and actual crack opening displacement based on the 3D analysis is smaller than those based on our approximate method. Since the normal stress is exerted at the interfacial surface, its effect on the crack reaches the maximum at the interface and decreases at the crack tip away from the interfacial surface. It can be seen that even for the stiffest substrate the error in the opening crack displacement is small near the crack tip leading to a very good approximation by a surface crack model of the stress intensity factor. Since growth of microdefects at the interface between two layers is often accompanied by local debonding, extension of the proposed method to the debonding case is particularly useful. 2.4.4 Bilayer structure subjected to uniform temperature rise: Perfectly bonded interface The SIF of a half-penny shaped crack normal to the interface of a bilayer structure subjected to uniform temperature rise is estimated based on the proposed approximate method. The finite element mesh for a bilayer structure shown in Figure 2.7(b) is used for this purpose. The top layer is perfectly bonded to the bottom layer, and the dimensions are summarized as shown in Table 2.1c. The top layer is chosen to be ceramics with coefficient of thermal expansion (CTE) as 1×10-5/˚C, and the bottom layer is chosen to be with CTE as 1.3×10-5/˚C. These values are chosen based on practical examples related to thermal barrier coating [24-29] and parametric studies are -34- conducted by varying Young’s modulus of top layer and substrate. The temperature is assumed to rise from 20˚C to 1020˚C to simulate the working environment of thermal barrier coating. The stress state at 20˚C is assumed to be free. The results are illustrated with examples of normalized crack radii R/h, 0.1 and 0.015. Combining 2D axisymmetric model (without a crack) and 3D cracked domain analysis, the SIFs at the crack tip are calculated, and the obtained results are compared against those based on accurate 3D calculations following the same procedure outlined in section 4.2. The percent error in estimating the SIF is plotted as a function of modulus ratio. As shown in Figure 2.10, the error is less than 1% for all the cases considered, and the validity of our proposed approximate method is demonstrated. 2.5 Summary Multilayer materials have been widely used, and the failure prediction for such structures is important. This often requires the calculation of SIF for a crack originating from the interfaces between layers. The solutions for these configurations, however, are not widely available. In this work, SIFs as a function of modulus ratio of dissimilar materials are produced for a 3D half-penny shaped and a 2D single-edged crack originating at the bonded interface. The solutions are obtained for both constant tensile and proportional bending loadings and are expressed in terms of handbook-type curve-fitted functions. Since typical flaws have shapes similar to a half-penny or single edge crack, the functions obtained have significant practical implications. -35- In addition, an effective method for the estimation of SIF for a crack originating at a surface or an interface is proposed. This method requires only stress calculation for an identical structure without a crack. The stress variation along the crack depth and the crack size are approximated by introducing the notion of σeq within a local model. The proposed methods lead to improved estimations of stress intensity factor. The combination of macro-level stress calculation without a crack and detailed micro-level information through local models significantly simplify SIF estimations. The evaluation of the SIF with various crack sizes is often required in predicting structure failure particularly in the case of fatigue life. Since the proposed method only requires stress calculation without a crack, no repetitive stress calculations are needed as crack size changes. The method is validated by comparison with 3D finite element computations for a half-penny shaped crack in biaxial tests and in perfectly bonded bi-layer indentation tests. For a crack smaller than 1/10 of the top layer thickness, the method is found to be accurate within 1% error. Further application of the proposed method to layered structures that are completely debonded at the interface is similarly validated using the bi-layer indentation tests. In this case the two nonbonded layers are assumed to have frictionless contact between them. It is found that a simple 3D local surface crack model can be successfully applied to this case for the calculations of the SIF despite the fact that the normal stress is not zero at the interface for this case. As the crack size increases, its existence may alter the far field stress in the structure and the proposed method may no longer be applicable. In many practical -36- applications, however, most of the fatigue life is spent on the growth of relatively small flaws and the proposed method can be utilized. -37- References [1] Wang Y, Walter ME, Sabolsky K, Seabaugh MM. Effects of powder sizes and reduction parameters on the strength of Ni-YSZ anodes. Solid State Ion 2006;177:1517-27. [2] Miller RA. Current status of thermal barrier coatings: An overview. Surf Coat Technol 1987;30:1-11. [3] Kehr JA. Fusion Bonded Epoxy (FBE): A Foundation for Pipeline Corrosion Protection. Houston, Texas: NACE Press; 2003. [4] Mjor IA, Toffenetti OF. Secondary caries: a literature review with case reports. Quintessence Int 2000;31(3):165–79. [5] Wang Y, Katsube N, Seghi RR, Rokhlin SI. Statistical failure analysis of adhesive resin cement bonded dental ceramics. Engng Fracture Mech 2007;74:1838-56. [6] Wang R., Katsube N, Seghi RR, Rokhlin SI. Statistical failure analysis of brittle coatings by spherical indentation: theory and experiment. J. Mater. Sci. 2006;41(17):5441-5454. [7] Wang R., Katsube N, Seghi RR, Rokhlin SI. Failure probability of borosilicate glass under Herz indentation load. J. Mater. Sci. 2003;38:1589-1596. [8] Atkinson A, Sun B. Residual stress and thermal cycling of planar solid oxide fuel cells. Mater Sci Technol 2007;23(10):1135-43. [9] Evans AG, Hutchinson JW. On the mechanics of delamination and spalling in compressed films. Int J Solids Struct 1984;20:455-66. [10] Suo Z, Hutchinson JW. Steady-state cracking in brittle substrates beneath adherent films. Int J Solids Struct 1989;25(11):1337-53. [11] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. New York: ASME Press; 2000. [12] Murakami Y. Stress intensity factors handbook. Amsterdam: Elsevier; 2001;3rd. [13] Noda N. Stress intensity formulas for three-dimensional cracks in homogeneous and bonded dissimilar materials. Engng Fracture Mech 2004;71:1-15. [14] Lee JC, Keer LM. Study of a Three-Dimensional Crack Terminating at an Interface. J Appl Mech 1986;53:311-16. [15] Han ZD, Atluri SN. SGBEM (for cracked local subdomain) – FEM (for uncracked global structure) alternating method for analyzing 3D surface cracks and their fatigue-growth. Computer Modeling in Engng & Sciences 2002;3 (6):699-716. [16] Newman JC, Raju IS. An empirical stress-intensity factor equation for the surface crack. Engng Fracture Mech 1981;15:185-192. [17] Abaqus v6.6. ABAQUS, Inc 2006. [18] Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. Int J Numer Meth Engng 1975;9:495-507. [19] Barsoum RS. On the use of isoparametric finite elements in linear fracture mechanics. Int J Numer Meth Engng 1976;10:25-37. [20] deLorenzi HG. On the energy release rate and the J-integral for 3-D crack configuration. Int J Fracture 1982;19:183-93. [21] deLorenzi HG. Energy release rate calculations by the finite element method. Engng Fracture Mech 1985;21:129-43. [22] Nisitani H. The two-dimensional stress problem solved using an electric digital computer. Bull Japan Soc Mech Engrs 1968;11(43):14-23. [23] Nisitani H, Murakami Y. Stress intensity factors of an elliptical crack or a semi-elliptical crack subject to tension. Int J Fract 1974;10(3):353-368. -38- [24] Ray AK, Goswami B, Singh MP, Das DK, Roy N, Dash B, Ravi KB, Ray AK, Das G, Karuna Purnapu Rupa P, Parida N, Das A, Swaminathan J and Dwarakadasa E. Characterization of bond coat in a thermal barrier coated superalloy used in combustor liners of aero engines. Mater Characterization 2006;57(3):199-209. [25] Matejicek J, Sampath S, Brand P and Prask H. Quenching, thermal and residual stress in plasma sprayed deposits: NiCrAlY and YSZ coatings. Acta Materialia 1999;47(2):607-617. [26] Ray A, Roy N and Godiwalla KM. Crack propagation studies and bond coat properties in thermal barrier coatings under bending. Bull Mater Sci 2001;24(2):203-209. [27] Atkinson A and Sun B. Residual stress and thermal cycling of planar solid oxide fuel cells. Mater Sci Technol 2007;23(10):1135-1143. [28] Singh RN. High-Temperature Seals for Solid Oxide Fuel Cells (SOFC). J Mater Engng Performance 2006;15:422-426. [29] Weil KS, Hardy JS and Koeppel BJ. New Sealing Concept for Planar Solid Oxide Fuel Cells. J Mater Engng Performance 2006;15:427-432. -39- Figures How to estimate KI for a 2D or 3D structure with a crack? Structure with a crack: I Uncracked body analysis: computation of stress distribution based on identical structure without a crack II z σ Determination of eq z equivalent constant θ tensile stress σeq (normal y to the crack face) in the R region of crack location equating KI for structure y I and correspondingx III geometry III estimation of KI for structure I Figure 2.1. Schematic representation of proposed simplified method for mode I stress intensity factor calculation for a half-penny shaped crack normal to the interface. Perfect bond or slip boundary conditions are assumed at the interface. -40- z σ σI E,ν EI, νI A A R R x perfectly EII, νII bonded σII σ σ εε=⇒=I II IIIfar away EEI II (a) 2D surface crack model – (b) 2D interfacial crack model – single edge crack with length R in a single edge crack with length R originating at semi-infinite body (E, ν). the interface between two perfectly bonded semi-infinite bodies (EI, νI and EII, νII). z Crack Tip A σ σ θ εε=⇒=I II z IIIfar away R EEI II y σI z σ Half-penny Crack EI, νI σII E, ν EII, νII y x Perfectly bonded y x (d) 3D interfacial crack model – (c) 3D surface crack model – a half penny crack with crack radius R a half penny surface crack with crack originating at the interface between two radius R in a semi-infinite space. perfectly bonded semi-infinite bodies (EI, νI f = f ()θ and EII, νII). The crack in body I is perpendicular to the bonded interface. f = fEE(,θ III / ) Figure 2.2. 2D and 3D cracked domain analysis to generate the configuration correction factor f for the mode I stress intensity factor KfI =⋅σ π R. -41- Crack Tip A z z z R R θ zeq R y An example of a half-penny 0 σ2 σeq σ1 σx0x σeq σxx shaped crack = z z R R KIA=⋅fR constan tσ eq π _ 0 σ1 σxx 0 σ1- σ2 σxx (a). If the far field stress distribution in the structure without a crack is linear along crack depth in the structure with virtual crack, it is replaced by an equivalent constant tensile stress σeq such that the stress intensity factor KIA at the crack tip A remains the same. The equivalent tensile stress is located at zeq, which is defined by the scaling factor Y as zeq= Y·R. z z actual stress z Crack Tip A distribution R R θ zeq=Y·R R y σxx σxx (σeq)App (σeq)App An example of a half-penny shaped crack KfIA≈⋅ constan t()σ eq App π R (b) An actual nonlinear far field stress distribution in the structure without a crack can be approximately replaced by an equivalent constant tensile stress (σeq)App at coordinate z=zeq along the crack depth. Figure 2.3. Schematic representation of proposed procedure used to determine an equivalent constant stress for the estimation of mode I stress intensity factor KIA at the crack tip A. -42- Figure 2.4. Verification of finite element analysis for the 3D interfacial crack model (Fig. 2(d)) utilizing two types of loading. The effect of finite body size on the configuration correction factor f is examined by comparing the results from two models with body size 200×200×200 and 400×400×400, where the crack radius R=1. When the semi-infinite body I is much stiffer than the dissimilar semi-infinite body II (EI/EII=100), solutions of the interfacial crack model (Fig. 2(d)) approach those of the surface crack model (Fig. 2(c)). -43- (a). Configuration correction factor f versus modulus ratio EI/EII. Solid lines are fitted curves. (b). Scaling factor Y as a function of modulus ratio EI/EII. Figure 2.5. FEA calculations of f for constant tensile and proportional loadings and Y at the crack tip A of the 2D interfacial crack model as a function of modulus ratio EI/EII. The handbook solutions for special cases (EI/EII is large or 1) and the results by body force method demonstrate excellent match with the FEA data. The Poisson’s ratio is 0.3 (νI=νII=0.3) for all the results. -44- (a). Configuration correction factor f versus modulus ratio EI/EII for 3D cracked domain analysis. FEA results for factor f for 3D cracked domain subjected to uniform (constant) tensile loading and those with proportional loading are respectively indicated by circles, and the corresponding fitted curves are shown in solid lines. The handbook solutions (shown in squares) for special cases (EI/EII is large) and the results by body force method (shown in triangle) demonstrate excellent match with the FEA data (shown in X). (b). Scaling factor Y as a function of modulus ratio EI/EII for 3D cracked domain analysis. Figure 2.6. FEA calculations of f for constant tensile and proportional loadings and Y at the crack tip A of the 3D local interfacial half-penny crack model as a function of modulus ratio EI/EII. The Poisson’s ratio is 0.3 (νI=νII=0.3) for all the results except νI=νII=0.1 for ‘Triangle’ and ‘X’. -45- z Crack tip A Load P θ R r Half-penny crack Ball Load P Ceramic disc Ball (a) (b) Figure 2.7. (a) A ball-on-ring biaxial flexure test on a ceramic plate with a half-penny shaped crack at the bottom surface. (b) A bilayer structure with a half-penny shaped crack originated at the bottom surface of the top layer subjected to indentation loading, and its finite element model. -46- Figure 2.8. Comparison of proposed approximate method and 3D FEA for a ball-on-ring biaxial test with two different crack radii. -47- Figure 2.9. Comparison of the crack opening displacement between 3D FEA bilayer indentation model and proposed approximation method. Crack opening displacement is calculated at different locations along z axis. The top layer (E=60GPa, υ=0.3, thickness h) is not bonded to the substrate (E=200GPa, υ=0.3); the load P (P/Eh2=2/300) is applied through the spherical indenter (radius r=400h) to the top surface. Stress distribution σ obtained from the neighborhood of the imaginary crack in the axi-symmetric model is used as far field in the local model. -48- Crack top tip A top z σ =(σ ) h I eq App approximate I σII II y bottom x bottom ×100% uniform uniform temperature temperature increase increase 3D FEA ) IA Rcrack/ h =0.015 3D FEA - (K ) IA (K roximate Rcrack/ h =0.1 pp a ) IA (K Figure 2.10. Percent error in estimating crack tip stress intensity factor KIA of a half-penny shaped crack normal to the interface of a bilayer structure subjected to uniform temperature rise based on the proposed approximate method compared to accurate 3D FEA evaluation. -49- Tables Table 2.1. Parameters used in the mechanical tests. a. Parameters for a ball-on-ring biaxial test (Fig. 2.7(a)). Ball Support ring Ceramic layer Young's modulus (GPa) rigid rigid 60 Poisson’s ratio rigid rigid 0.3 Radius/h(1) 5 3 4 (1): h is the thickness of the ceramic layer. b. Parameters for a bilayer indentation test (Fig. 2.7(b)). Ball Top layer Bottom layer Young's modulus (GPa) rigid 60 or 200 20 or 60 or 200 Poisson’s ratio rigid 0.3 0.3 Radius/h(2) 5 or 400 4 4 Thickness/h(2) NA 1 3 (2): h is the thickness of top layer. c. Parameters for a bilayer structure subjected to uniform temperature increase (Fig. 2.10). Top layer Bottom layer Young's modulus (GPa) 60, 100, 150 or 200 210 Poisson’s ratio 0.3 0.3 Radius/h(3) 4 4 Thickness/h(3) 1 3 (3): h is the thickness of top layer. -50- Table 2.2. Percent error of the approximate method assessed by 3D FEA for mode I stress intensity factor KIA of an interfacial half-penny shaped crack at the crack tip A in perfectly bonded bilayer indentation model (Fig. 2.7(b)). (a) small indenter % error in KIA (1) (rball/h =5) (1) (1) Rcrack/h =0.1 Rcrack/h =0.015 2 2 2 2 Etop Ebottom P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) (GPa) (GPa) =2/300 =5/300 =2/300 =5/300 20 <1 <1 <1 <1 60 <1 <1 <1 <1 60 200 NA(2) NA(2) NA(2) NA(2) 0(3) <1 <1 2 2 2 2 P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) =1/500 =1/200 =1/500 =1/200 200 20 <1 1 <1 <1 60 <1 <1 <1 <1 200 <1 <1 <1 <1 (b) large indenter % error in KIA (1) (rball/h =400) (1) (1) Rcrack/h =0.1 Rcrack/h =0.015 2 2 2 2 Etop Ebottom P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) (GPa) (GPa) =1/300 =2/300 =1/300 =2/300 20 <1 <1 <1 <1 60 60 NA(2) NA(2) NA(2) NA(2) 200 NA(2) NA(2) NA(2) NA(2) 2 2 2 2 P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) =1/1000 =1/500 =1/1000 =1/500 200 20 <1 <1 <1 <1 60 <1 <1 <1 <1 200 <1 NA(2) <1 NA(2) Note: (1) h is the thickness of top layer. (2) ‘NA’ indicates negative value of the equivalent stress. (3) Ball-on-ring biaxial test (Fig. 2.7(a)). (4) The error of FEA calculations is smaller than 1%. -51- Table 2.3. Percent error of the approximate method assessed by 3D FEA for mode I stress intensity factor KIA of an interfacial half-penny shaped crack at the crack tip A in completely debonded bilayer indentation model with frictionless contact between the layers (Fig. 2.7(b)). % error in KIA (1) Rcrack/h =0.1 small indenter large indenter (1) (1) (rball/h =5) (rball/h =400) 2 2 2 2 Etop Ebottom P/(Etop⋅h )= P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) (GPa) (GPa) 2/300 =5/300 =1/300 =2/300 20 <1 <1 <1 <1 60 60 <1 <1 1 1 200 2 2 2 2 2 2 2 2 P/(Etop⋅h )= P/(Etop⋅h ) P/(Etop⋅h ) P/(Etop⋅h ) 1/500 =1/200 =1/1000 =1/500 200 20 <1 <1 <1 <1 60 <1 <1 <1 <1 200 <1 <1 <1 1 Note: (1) h is the thickness of top layer. (2) The error of FEA calculations is smaller than 1%. -52- CHAPTER 3 Mode I stress intensity factor estimate for a half-penny-shaped crack in trilayer structure by reduction to crack in bilayer structure Abstract The stress intensity factor (SIF) of a half-penny shaped crack normal to the interface in the top layer of a three-layer bonded structure is obtained by the finite element method in a wide range of parameters. To obtain a simple estimate of the SIF, the method of reduction of an idealized cracked trilayer domain to that of a corresponding bilayer domain has been introduced based on the notion of an equivalent homogeneous material for the two bottom layers. The results obtained are utilized in estimating the SIF of a small crack at the interface in a trilayer structure subjected to an -53- indentation load based on the stress calculations in a corresponding uncracked structure. The method may be useful in predicting brittle failure initiating from interfacial flaws in a layered structure. 3.1 Introduction Layered structures are widely used in industrial and medical fields [1, 2] to enhance structural integrity and improve performance. For example, thermal barrier coatings are used in gas turbines to increase the operational temperature [1]. Adhesive-retained ceramic restorations are employed to restore the mechanical performance and esthetics of remaining teeth [2]. Layered structures, however, are prone to failures initiating at the interface due to stress concentration acting on microdefects that are either pre-existing or developed during service [3, 4]. Interface failures of this nature usually include debonding (delamination) of the interface and fracture initiation originating at the interface which can be an independent event or can follow interface debonding. Failures in a layered structure due to interfacial delamination have been widely studied [5-7]. Brittle fractures originating at the interface in a layered structure, however, have not been extensively investigated despite the fact that such failures have been observed [8-12]. In predicting such failures, evaluation of the stress intensity factor (SIF) for a crack perpendicular to the interface is important. Some SIF solutions for a crack at the bonded interface between two semi-infinite dissimilar materials subjected to simple remote loading conditions are -54- readily available [13-17]. Noda [15] calculated the SIF for a rectangular shaped crack perpendicular to the interface subjected to a constant tensile loading. Similarly, Wang et al. (Chapter 2 [17]) analyzed the SIF of a half-penny-shaped interfacial crack subjected to constant tensile and proportional bending loadings. These results are limited to bilayer structures. A layered structure, however, often consists of three layers [1-3]: a top functional material layer, a middle adhesive/bonding layer and a substrate/base material layer at the bottom. In addition, the thickness of middle layer is usually much smaller than that of the top and bottom layers. The evaluation of the SIF for a crack in a trilayer structure requires the consideration of the material properties of all three layers and the crack size relative to the middle layer thickness. In general, the evaluation of the SIF for a crack in structural components with arbitrary geometry is computationally challenging. An iterative method can be employed to overcome this difficulty [18]. In this method the stress distribution in the uncracked global structure is used as loading over the crack surface in a local subdomain analysis and the traction at the intersection between the global structure and the subdomain are applied in the global structure as the residual forces. The corresponding stress in the crack position is re-evaluated and this method is applied iteratively until the residual forces become negligible. The superposition of uncracked body analysis and subdomain analysis leads to evaluation of the SIF of a crack of any size in a global structure with arbitrary geometry. For cracks which remain much smaller than the size of the overall structure, however, this iterative method may not be necessary because the effect of cracks on the stress distribution away from the crack can be neglected. For this case, the stress -55- distribution based on the uncracked body analysis can be used as loading over the crack surface in an infinite medium (for an embedded crack), and the SIF solutions available in handbooks [13, 14] can be utilized based on the simple superposition principle. In estimating the SIF of a crack perpendicular to the interface for bonded global structures, some SIF solutions available in the existing literature [15-17] for bilayer structures can be readily utilized in combination with uncracked body analysis. Currently, however, there is no systematic analysis of the three-layer effect on the SIF of a crack perpendicular to the interface despite the fact that most bonded structures are in fact trilayers including relatively thin adhesive/bonding layers. In this work, the effect of three layers (the material properties of all three layers and the crack size relative to the middle layer thickness) on the stress intensity factor of a half-penny shaped crack perpendicular to the interface Fig. 3.1(i) is examined. Briefly our methodology is as follows. Given a trilayer structure with a small crack shown in Fig. 3.1(i), the stress distribution in the region of a crack is calculated from an identical structure without a crack as shown in Fig. 3.1 (ii) and viewed as an approximate far field stress applied to a small crack in Fig. 3.1(i) (note that this approximation is valid only when the crack size remains much smaller than the size of the overall structure and the existence of the crack does not significantly alter the stress field away from the crack). In order to address the effect of three layers on the mode I SIF for a crack at the interface, an idealized cracked trilayer domain shown schematically in Fig. 3.1(iii) is considered. This trilayer structure is replaced by an equivalent bilayer structure in Fig. 3.1(iv) equating the SIF for both structures. This leads to an explicit expression for the effective modulus of an equivalent bottom layer -56- in the bilayer structure (Fig. 3.1(iv)). The effective modulus is obtained as a function of the moduli of the middle and bottom materials and the ratio of the middle layer thickness h to the crack size R. Combining this reduction to the two layer structure with our prior method (Chapter 2 [17]) for SIF estimation in such a structure, the effect of three layers on the SIF is obtained. In order to examine the validity of the proposed approximation method, the SIFs for a half-penny shaped crack located at the bottom interface of the top layer of a trilayer cylindrical system subjected to an indentation loading are estimated. The estimated SIFs are compared against those determined by 3D FEA calculations including complete crack geometry, and the validity of the proposed method is verified quantitatively through this comparison. 3.2 Methodology 3.2.1 Cracked trilayer domain analysis We consider an idealized cracked trilayer domain as shown in Fig 3.2(a). The top material I and the bottom material III are semi-infinite, and both are perfectly bonded to the middle layer material II. Young’s modulus and Poisson’s ratio for materials I, II and III respectively are EI, νI, EII, νII and EIII, νIII, and h is thickness of the middle layer. A half-penny shaped crack with radius R is located at the bottom of the top layer and is normal to the interface. A tensile stress field normal to the crack face is applied to the cracked trilayer domain. Two types of stress distributions (constant and proportional) are considered. -57- First, uniformly distributed constant tensile stresses σI, σII, σIII are respectively applied to the top, middle and bottom layers I, II, III. The values of these stresses are chosen so that away from the interface the normal strain parallel to the interface remains the same in all three layers: σ σσ IIIIII==. (3.1) EEIIIIII E The second type is proportional bending stress distribution. In this case, linearly varied tensile stress σI is applied to the top material I, while linearly varied compressive stresses σII and σIII are respectively applied to the middle and bottom materials II and III. The applied stress is zero at the interface between materials I and II. The slopes of the linearly varied tensile and compressive stresses are selected so that Eq. (3.1) is satisfied for absolute values of the stresses (σI, σII, σIII) at the same distance from the interface of materials I and II. The mode I stress intensity factor (SIF) at the tip A of the half-penny-shaped crack in the trilayer domain (Fig. 3.2) can be expressed as KfIA,, trilayer=⋅ A trilayer σ π R, (3.2) where R, σ and fA,trilayer represent the radius of the crack, the loading stress and the configuration correction factor [13] respectively. The configuration factor is also referred to as the magnification factor [14] or boundary correction factor [19]. The loading stress σ is the same as the constant tensile stress σI in the case of constant tensile stress distribution. In the case of proportional bending stress distribution, the loading stress σ is the value of the tensile stress σI at the location z=R (crack tip). -58- The configuration correction factor fA,trilayer depends on the middle layer thickness normalized by the crack radius h/R; Poisson’s ratios of materials I, II and III, νI, νII, νIII; and the Young’s modulus ratios: EI/EII and EII/EIII. The effect of Poisson’s ratio on the configuration correction factor has been found [15] to be small, and in this work Poisson’s ratio is assumed to be 0.3 for all three materials I, II and III (νI=νII=νIII=0.3). The configuration correction factor fA,trilayer thus depends on h/R, EI/EII and EII/EIII: f A,, trilayer= fhREEEE A trilayer(/ , I / II , II / III ). (3.3) 3.2.2 Role of bonding layer thickness When the middle layer material II has the same material properties as the bottom material III (EII=EIII), the trilayer structure reduces to a bilayer structure and the configuration correction factor f becomes a function of the ratio EI/EII only. For such a bilayer structure the SIF of the cracked bilayer domain has been calculated in a wide range of material parameters, and the curve-fitted equation of factor fA,bilayer(EI/EII) has been obtained (Chapter 2 [17]). Similarly, when the thickness of the middle (bonding) layer h in the trilayer structure is much larger than the crack radius R the bottom layer III is far from the crack and its effect on the SIF is insignificant. In this case (h>>R), the SIF of the crack in the trilayer structure can again be replaced by the cracked bilayer domain analysis with materials I and II. However, when the thickness of the middle layer II is smaller than the crack radius, the effect of both the bonding and the bottom layers on the SIF needs to be considered. This is demonstrated numerically in Fig. 3.3 by comparing the SIF for bilayer and trilayer systems for the case of EI/EII=10 and EII/EIII =1/10. Percent error -59- of the SIF, KIA, is obtained as a function of h/R when a trilayer structure is approximated by a bilayer structure with materials I and II. For h/R larger than 0.5, the error is within 1%. When h/R is smaller than 0.5, the error is larger than 1% and increases significantly as h/R approaches zero. As discussed in the Introduction, evaluation of the SIF of a crack larger than the thickness of the bonding layer is often required to assess the structural integrity of bonded materials. For this purpose, the evaluation of the configuration factor fA,trilayer for the cracked trilayer domain becomes desirable. Given the value of h/R, EI/EII and EII/EIII, the value of the factor fA,trilayer can be calculated numerically, and the relationship between fA,trilayer and the three parameters can be tabulated. However, for practical applications it is convenient to have simple approximate explicit equations for SIF, and this is achieved by replacing of the bonding and bottom layers by a single effective layer. 3.2.3 Replacement of the bonding and bottom layers with a homogeneous effective substrate The concept of the effective substrate explored in this work is illustrated in Fig. 3.2. The effective material is defined as a homogeneous semi-infinite material to replace materials II and III in the cracked trilayer domain in such a way that the SIF at a crack tip A remains the same for both structures (Fig. 3.2). This reduces the cracked trilayer domain analysis to the effective cracked bilayer domain analysis with material I and the effective material, leading to fhREEEEfEEA,, trilayer(/ , I / II , II / III )≈ A bilayer ( I / effective ). (3.4) -60- The configuration correction factor for the effective cracked bilayer domain fA,bilayer (Fig 3.2(b)) is readily obtained by replacing EII with Eeffective in Eqs. (2.17) and (2.18) in Chapter 2. We search for Eeffective in Eq (4) as a function of h/R, EI/EII and EII/EIII: EEhREEEEeffective= effective (/ , I / II , II / III ). (3.5) 3.3 FEM results and empirical fitting function for effective layer In this section, the procedure to obtain Eeffective for given values of h/R, EI/EII and EII/EIII is summarized, and a possible form of curve-fitting of Eq. (3.5) is discussed. The physical basis for such a replacement is that in the cracked trilayer domain problem, the constraint effect of the middle and bottom layers on the crack opening displacement must remain the same as that of the two layer structure with the effective material and thus the interfacial tractions and displacements are nearly the same for both structures. 3.3.1 Numerical determination of effective modulus Given h/R, EI/EII and EII/EIII, finite element analysis (FEA) with quadratic 20-node cubic elements is performed to evaluate KIA,trilayer for the cracked trilayer domain. The commercially-available software package ABAQUS v6.6 [20] is used for this purpose. The body size normalized by the crack radius (R=1) is chosen to be 200×200×200, and the individual finite element is chosen to be smaller than 0.01 in critical areas (See Appendix A.1 for further details of FEA calculations). Special -61- crack elements with one-quarter mid-side node [21, 22] are employed at the crack front. The J-integral method [23, 24] is used to calculate the SIF at the tip A in the cracked trilayer domain. Based on the SIF (KIA,trilayer) obtained, the value of the corresponding configuration correction factor fA,trilayer , is determined from Eq. (3.2). Next, using Eq. (4) and equating fA,trilayer to the corresponding fA,bilayer (EI/Eeffective) for the bilayer system, the value of the corresponding Eeffective is determined by use of Eqs. (2.17) and (2.18) from Chapter 2. In the above procedure, EI/EII and EII/EIII are chosen to vary from 1/100 to 100 so that most practical material variation can be evaluated. The ratio h/R is chosen to vary from 1/10 to 10 in order to avoid numerical difficulties associated with extensive meshing in the FEA. Fig. 3.4 shows Eeffective/EII versus h/R for the two cases: (1) EI/EII=10 and EII/EIII=1/10; (2) EI/EII=1/10 and EII/EIII=10. The middle layer is chosen to be softer than both the top and bottom layers in the first case while it is chosen to be stiffer in the second case. In both cases as the middle layer thickness increases Eeffective approaches EII (Eeffective/EII → 1 as h/R → 10). As h/R approaches zero Eeffective tends to approach EIII. For the first case Eeffective/EII increases with decrease of h/R since EIII is larger than EII in this case. In the second case, since EIII is smaller than EII, the value of Eeffective/EII is smaller than one and decreases with decrease of h/R. For all other combinations of EI/EII and EII/EIII, it is also observed that Eeffective approaches EII for large h/R and tends to approach EIII for small h/R. Since Eeffective for h/R larger than one is nearly equal to EII for all combinations of the material considered, we will discuss below in detail the cases of h/R less than -62- one (h/R=1/5 and h/R=1/10). In Fig. 3.5 log10 (EIII/Eeffective) is plotted versus log10 (EII/EIII) for both constant and proportional loadings in the range of 1/10 (h/R=1/5 in (a) and h/R=1/10 in (b) ). Further details related to the range of parameter 1/10 The same trend of decrease in the deviation from linearity for thicker middle layers is similarly observed for all other values of h/R (1/10 Analysis of a scatter of the FEA data from the trend in Fig 3.5 shows a higher scatter in log10 (EIII/Eeffective) versus EI/EII ratio for parameter EII/EIII>10 with the larger scatter for h/R=1/10 in Fig. 3.5(b). For reasons described below this scatter is attributed to numerical error (maximum 1% error in FEA calculations of KIA). In FEA calculations, for a given value of h/R, the same mesh is used for all the combinations of EI/EII and EII/EIII. Given specific values of h/R, EI and EII, the constraint effect of bottom layer III on the crack opening displacement (COD) decreases with the decrease of EIII. Thus the COD for small values of EIII is larger than that for large values of EIII. This leads to high displacement gradients in the neighborhood of a crack and larger numerical errors for large values of EII/EIII. Also, the displacement gradients in the neighborhood of a crack are larger for the case h/R=1/10 than those for h/R=1/5 thus leading to larger numerical errors at h/R=1/10 (this is because the smallest element size and thus the distance between nodal points are similar for both cases, due to the limitation of the total number of elements). -63- 3.3.2 Curve fitted equation for effective modulus From the dependences of Eeffective versus EII/EIII shown in Fig 3.5, one can see that the values obtained are nearly independent of EI/EII and thus can be fitted by a single master curve with h/R and EII/EIII as parameters. Also this dependence on log10 (EIII/Eeffective) of log10 (EII/EIII) is nearly linear. Here one should note that in indentation tests [25-27] the replacement of a layered substrate by an effective medium has also been employed. The objective of these studies was to measure top layer properties by the indentation method. Gao et al. [25] have found theoretically the expression for Eeffective under the assumption that the surface load and displacement relationship remains the same when the film-substrate layers are replaced by an effective homogeneous semi-space. Kim et al. [27] obtained a simple curve-fitted equation for Eeffective to replace adhesive/substrate layers by a single equivalent layer employing an empirical equation for Eeffective related to indentation tests [26]. Their approach is justified since the interfacial surface traction and displacement relationship at the coating/adhesive interface should not be altered due to the use of the effective layer. Although applications and definitions of Eeffective in their work are different from ours some similarities of the approaches exist. In order to account for the small deviation from the linearity in Fig. 3.5, the third degree polynomial in the form, 23 EEEIII II⎛⎞⎛⎞ II E II log10=+ab log 10⎜⎟⎜⎟ log 10 + c log 10 , (3.6) EEEEeffective III⎝⎠⎝⎠ III III -64- is employed to fit the numerical results in Fig 3.5 for different h/R. Using the least square curve-fitting parameters a, b and c are found to be h a =−[1 − exp( − 3 )] , (3.7) R b = 0 , (3.8) and hR c =−[1 − exp( − 3 )] ⋅ 0.004 . (3.9) R h Eqs. (3.6)-(3.9) lead to the following equation for Eeffective −α Eeffective ⎛⎞E = ⎜⎟II , (3.10) EEII⎝⎠ III where ⎡ 2 ⎤ hR⎛⎞EII α =−1 [1 − exp( − 3 )]⎢ 1 + 0.004⎜⎟ log10 ⎥ . (3.11) RhE ⎣⎢ ⎝⎠III ⎦⎥ Eq. (3.10) with h/R=1/5 and 1/10 are plotted as solid lines in Figs. 3.5(a) and (b), respectively. Fig. 3.6 shows a 3D surface calculated using Eqs. (3.10), (3.11) for Eeffective/EII versus EII/EIII and h/R for the range of parameters 1/100 When h/R is in the neighborhood of one, the value of Eeffective/EII remains almost one for all the values of EII/EIII. This confirms the prior results shown in Fig. 3.4 where for large values of h/R the effect of EIII on Eeffective is negligible. Knowing EI and Eeffective one can evaluate the configuration correction factor for both constant and proportional loadings based on Eqs. (2.17) (2.18) (see Chapter 2). It is important to note that the combination of Eq. (3.4) with Eqs. (2.17) (2.18) (3.10) -65- (3.11) explicitly determines the configuration correction factor for a trilayer system, fA,trilayer, as a function of h/R, EI/EII and EII/EIII (Eq. (3.3)). The SIF at the crack tip A, KIA, in the cracked trilayer domain can then be estimated by Eq. (3.2). 3.3.3 The error of curve-fitted equation for effective modulus The error of the proposed method for the estimation of KIA for the trilayer structure is found to be less than 2% for the parameter range of 1/10 1/100 2 R ⎛⎞EII If the term 0.004⎜⎟ log10 is ignored in Eq. (3.11), Eq. (3.10) for Eeffective hE⎝⎠III will reduce to a form similar employed by Kim et al. [27]. The maximum error of KIA, however, will increase from 2% to 3% due to the simplification of the equation. 3.4 Application In this section, the results obtained in Section 3 are employed for the SIF estimation of a crack in a 3D trilayer indentation model shown in Fig. 3.7(a). Top layer I and bottom layer III are both perfectly bonded to middle layer II, where a -66- half-penny shaped crack of radius R is located at the bottom of top layer I and normal to the interface between the top and middle layers. An indentation load is applied to the top surface through a spherical indenter. The SIF at the crack tip A, KIA, is estimated based on the method schematically shown in Fig. 3.1, and the results obtained are verified against complete 3D FEA results. 3.4.1 Approximate method for SIF estimation The outline of the proposed estimation method is schematically shown in Figs. 3.7(a), (b) and (c). The cracked trilayer domain analysis shown in Fig. 3.7(b) is employed as the idealized local cracked geometry for the structure shown in Fig. 3.7(a). Given the values of EII/EIII and h/R, the values of Eeffective are determined based on Eqs. (3.10), (3.11). Combining this with Eq. (3.4), the values of fA,constant and fA,proportional for the cracked trilayer domain analysis are respectively estimated from Eqs. (2.17) and (2.18) in the Chapter 2. As shown in Fig. 3.7(c), the trilayer indentation test without a crack is simulated by a 2D axisymmetric finite element model (ABAQUS v6.6 [20]). The radial normal stress distribution along the axis of symmetry at the bottom of top layer I can be viewed as a far field normal stress distribution applied to top layer I of the cracked trilayer domain analysis. Using a linear superposition principle, as has been shown in Chapter 2 [17], the obtained far field stress distribution can be replaced by an approximate equivalent constant tensile stress, (σeq)App, such that the SIF (KIA) remains the same. This equivalent constant stress (σeq)App is shown to be located at the position Zeq=Y·R (Fig. 3.7c) where the factor Y is given by Y=fA,proportional/fA,constant (Chapter 2 [17]). -67- The approximate equivalent stress (σeq)App is directly chosen at the location Zeq on the axis of symmetry using the stress calculated by the axisymmetric FEA for a structure without the crack. The obtained (σeq)App is applied as the far field normal tensile stress σI in the top layer I of the 3D cracked trilayer domain, Fig. 3.7(b). This leads to the estimation of approximate SIF, KIA, as follows, KfIA,,tan App≈⋅ A cons t()σ eq App π R. (3.12) 3.4.2 Comparison with 3D FEM analyses In order to examine the validity of the proposed method, the values of the estimated KIA,App are compared against those of KIA,3 D FEA at the crack tip of a half-penny shaped crack inside the cracked trilayer structure for an indentation test (Fig. 3.7a). A complete 3D finite element simulation with a crack is employed to calculate the mode I SIF of the crack tip A, KIA,3 D FEA . The commercially-available software package ABAQUS v6.6 [20] is used for this purpose. Only one-quarter of the loading structure is modeled due to the symmetry. Quadratic 20-node cubic elements are used to mesh the trilayer model. The size of individual finite elements around the crack front is carefully chosen so that the error of the numerical calculation remains within 1%. A special crack element with one-quarter mid-side node [21, 22] is employed at the crack front, and the J-integral method [23, 24] is used to calculate the SIF, KIA,3 D FEA . The ranges of Young’s moduli of the top, middle and bottom material are chosen to be EI=60/200GPa, EII=20~210GPa and EIII=20/60/200GPa, respectively, to -68- represent properties of materials often used in the industrial field [2, 8-12, 28-31]. Poisson’s ratios of 0.3 are chosen for all materials. The ratio of middle layer thickness over the crack radius is set as 1/10, which represents the smallest normalized thickness for the cracked trilayer domain analysis summarized in Section 3. Given the crack radius R, a thickness of 10R is chosen for the top layer. The spherical indenter radius is chosen to be 50R, and an indentation load of P=400N is applied to the top surface. To compare the approximate and the FEM results, the percentage error of KIA,App, [(KIA,App- KIA,3 D FEA )/ KIA,3 D FEA ]×100%, is plotted as a function of EII for different values of EI and EIII as shown in Fig. 3.8. The maximum error is found to be within 2.5%. 3.4.3 Limitations of the approximate method If the crack size normalized by the thickness of top layer I is larger than 1/10, the existence of a crack may significantly influence the stress distribution in the region away from the crack in top layer I. In this case, the stress distribution obtained from the axisymmetric FEA model without a crack cannot be used to approximate the far field stress. The proposed method, therefore, can be applied to the cases where cracks are small relative to the overall structure. In assessing brittle failure, however, one is mostly interested in the SIF of relatively small cracks since failure is imminent for larger cracks. When applicable, the proposed method is very effective in evaluating the SIF of a crack inside a trilayer structure since it only requires knowledge of the stress distribution without a crack. -69- 3.5 Summary In many layered structures, a relatively thin bonding layer exists between two dissimilar materials and the structure may be viewed as a trilayer. Brittle fracture of the layered structure often initiates from an interfacial flaw and evaluation of the SIF of a flaw or crack perpendicular to the interface is critical in assessing the structural integrity. In this work, the effect of middle and substrate layers on the SIF of a crack in the top layer is examined based on an idealized cracked trilayer domain analysis. Equating the SIF in the cracked trilayer domain analysis with those in the cracked bilayer domain analysis (Chapter 2 [17]), the middle and substrate layers are replaced by an effective homogeneous layer. The corresponding effective modulus is expressed as a curve-fitted function of moduli of middle and substrate layers and the middle layer thickness normalized by the crack radius. This leads to determination of the configuration correction factor where the effect of three layers on the SIF is considered. The configuration correction factor obtained for a crack in the idealized trilayer system can now be utilized in estimating the SIF of a crack in an actual trilayer structure. In demonstrating its practical use, the SIF is estimated for a half-penny shaped crack perpendicular to the interface between the top and the middle layer in a trilayer structure subjected to an indentation load. The stress distribution based on an identical indentation boundary value problem without a crack is combined with the idealized cracked trilayer domain analysis. This combination of macro-level stress calculation and detailed micro-level information related to trilayer structure -70- significantly simplifies the SIF estimation since the method requires only stress calculation without modeling a crack. The results obtained are verified against SIF calculations based on the complete 3D FEA model including crack geometry, and the overall error is found to be less than 3%. The proposed method is applicable to the cases where the crack radius R is smaller than ten times the middle layer thickness (R<10h) and the size of the overall structure. As the crack size increases, the existence of a crack may significantly alter the stress field away from the crack, and the proposed method may no longer be applicable. However, evaluation of the SIF with various sizes is often required in fatigue life prediction and most of the fatigue life is spent on the growth of relatively small flaws. For this reason, the proposed method can be useful in fatigue life predictions for layered structures. -71- References [1] Miller RA. Current status of thermal barrier coatings: An overview. Surf Coat Technol 1987;30:1-11. [2] Mjor IA, Toffenetti OF. Secondary caries: a literature review with case reports. Quintessence Int 2000;31(3):165–79. [3] Wang Y, Katsube N, Seghi RR, Rokhlin SI. Statistical failure analysis of adhesive resin cement bonded dental ceramics. Engng Fracture Mech 2007;74:1838-56. [4] Wang R., Katsube N, Seghi RR, Rokhlin SI. Statistical failure analysis of brittle coatings by spherical indentation: theory and experiment. J Mater Sci 2006;41(17):5441-5454. [5] Evans AG, Hutchinson JW. On the mechanics of delamination and spalling in compressed films. Int J Solids Struct 1984;20:455-66. [6] Suo Z, Hutchinson JW. Steady-state cracking in brittle substrates beneath adherent films. Int J Solids Struct 1989;25(11):1337-53. [7] Suo Z, Hutchinson JW. Interface crack between two elastic layers. Int J Fracture 1990;43:1-18. [8] Ray A, Roy N and Godiwalla KM. Crack propagation studies and bond coat properties in thermal barrier coatings under bending. Bull Mater Sci 2001;24(2):203-209. [9] Chen WR, Wu X, Marple BR and Patnaik PC. The growth and influence of thermally grown oxide in a thermal barrier coating. Surface Coatings Technology 2006;201:1074-1079. [10] Anusavice KJ, Hojjatie B. Tensile stress in glass-ceramic crowns: Effect of flaws and cement voids. Int J Prosthodont. 1992;5:351-358. [11] Malament KA, Socransky SS. Survival of Dicor glass-ceramic dental restorations over 14 years: Part I. Survival of Dicor complete coverage restorations and effect of internal surface acid etching, tooth position, gender, and age. J Prosthet Dent 1999;81(1):23-32. [12] Pagniano RP, Seghi RR, Rosenstiel SF, Wang R, Katsube N. The effect of a layer of resin luting agent on the biaxial flexure strength of two all-ceramic systems. J Prosthet Dent 2005;93(5):459-466. [13] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. New York: ASME Press; 2000. [14] Murakami Y. Stress intensity factors handbook. Amsterdam: Elsevier; 2001;3rd. [15] Noda N. Stress intensity formulas for three-dimensional cracks in homogeneous and bonded dissimilar materials. Engng Fracture Mech 2004;71:1-15. [16] Lee JC, Keer LM. Study of a Three-Dimensional Crack Terminating at an Interface. J Appl Mech 1986;53:311-16. [17] Wang Y, Katsube N, Seghi RR, Rokhlin SI. Uncracked body analysis for accurate estimates of mode I stress intensity factor for cracks normal to an interface. Engng Fracture Mech 2009 (76):369-385. [18] Han ZD, Atluri SN. SGBEM (for cracked local subdomain) – FEM (for uncracked global structure) alternating method for analyzing 3D surface cracks and their fatigue-growth. Computer Modeling in Engng & Sciences 2002;3 (6):699-716. [19] Newman JC, Raju IS. An empirical stress-intensity factor equation for the surface crack. Engng Fracture Mech 1981;15:185-192. [20] Abaqus v6.6. ABAQUS, Inc 2006. [21] Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. Int J Numer Meth Engng 1975;9:495-507. -72- [22] Barsoum RS. On the use of isoparametric finite elements in linear fracture mechanics. Int J Numer Meth Engng 1976;10:25-37. [23] deLorenzi HG. On the energy release rate and the J-integral for 3-D crack configuration. Int J Fracture 1982;19:183-93. [24] deLorenzi HG. Energy release rate calculations by the finite element method. Engng Fracture Mech 1985;21:129-43. [25] Gao YF, Xu HT, Oliver WC and Pharr GM. Effective elastic modulus of film-on-substrate systems under normal and tangential contact. J Mech Phys Solids 2008;56:402-416. [26] Hu XZ and Lawn BR. A simple indentation stress-strain relation for contacts with spheres on bilayer structures. Thin Solid Films 1998;322:225-232. [27] Kim JH, Miranda P, Kim DK and Lawn BR. Effect of an adhesive interlayer on the fracture of a brittle coating on a supporting substrate. J Mater Res 2003;18(1):222-227. [28] Matejicek J, Sampath S, Brand P and Prask H. Quenching, thermal and residual stress in plasma sprayed deposits: NiCrAlY and YSZ coatings. Acta Materialia 1999;47(2):607-617. [29] Ray AK, Goswami B, Singh MP, Das DK, Roy N, Dash B, Ravi KB, Ray AK, Das G, Karuna Purnapu Rupa P, Parida N, Das A, Swaminathan J and Dwarakadasa E. Characterization of bond coat in a thermal barrier coated superalloy used in combustor liners of aero engines. Mater Characterization 2006;57(3):199-209. [30] Singh RN. High-Temperature Seals for Solid Oxide Fuel Cells (SOFC). J Mater Engng Performance 2006;15:422-426. [31] Weil KS, Hardy JS and Koeppel BJ. New Sealing Concept for Planar Solid Oxide Fuel Cells. J Mater Engng Performance 2006;15:427-432. -73- Figures How to estimate KI for a crack in a trilayer structure? A trilayer structure with a Trilayer structure: I crack normal to I-II interface: h II z III θ z i R y σeq Uncracked body analysis based on an I h identical structure without a crack: II computation of stress distribution ii x III y iii Determination of equivalent z σeq constant tensile stress σeq Replace a trilayer structure (normal to the crack face) in with a bilayer structure, the region of crack location, I equating KI for structure iii equating KI for structures i and structure iv and iv Effective y x iv estimation of KI for structure i Figure 3.1. Schematic representation of proposed simplified method for mode I stress intensity factor calculation for a half-penny shaped crack normal to the top-middle interface in a trilayer structure. -74- σI/EI=σII/EII=σIII/EIII z z σI/EI=σeff/Eeff by by εI=εII=εIII | far away εI=εeff | far away σI I h σI I II x y x Effective σII III y z Crack σeff σ tip A III θ R KI [ f (EI/ Eeffective), σI, R] KI [ f (h/R, EI/EII, EIII/EII), σI, R] y (b) Effective cracked bilayer domain (a) Cracked trilayer domain Equate KIA in (a) and (b) Effective layer, Eeffective Figure 3.2. Effective cracked bilayer domain analysis model versus cracked trilayer domain analysis model. Replace materials II and III with an effective material so that KIA remains the same. Poisson’s ratio is assumed to be 0.3 (νI=νII=νIII=0.3). -75- z Crack tip A z θ σ R I z r σI I σII h II I σII σIII y VS x III II y trilayer x bilayer Figure 3.3. Percent error of KIA as a function of dimensionless middle layer thickness (h/R). A trilayer structure is approximated by a bilayer structure. Poisson’s ratio is assumed to be 0.3 (νI=νII=νIII=0.3). -76- Figure 3.4. Variation of Eeffective/EII with the change of dimensionless middle layer thickness (h/R). Poisson’s ratio is assumed to be 0.3 (νI=νII=νIII=0.3). As h/R approaches zero, Eeffective significantly deviates from EII. Eeffective was numerically obtained from FEM calculation. -77- (a). when h/R=1/5. (b). when h/R=1/10. Figure 3.5. Variation of numerically obtained Eeffective with the change of EI/EII and EII/EIII for the case of (a) h/R=1/5 and (b) 1/10. Linear relationship between log(EIII/Eeffective) and log(EII/EIII) is observed and fitted as shown in solid lines. -78- E −α effective ⎛⎞EII Figure 3.6. Distribution of Eeffective. = ⎜⎟, EEII⎝⎠ III ⎡⎤2 hR⎛⎞EII 1−=−α [1 exp( − 3 )]⎢⎥ 1 + 0.004⎜⎟ log10 . Error in KI,A is within 2%. RhE ⎣⎦⎢⎥⎝⎠III Results are valid for EI/EII , EII/EIII~[0.01,100] and 0.1 -79- KIA ? z Load σI z z Crack σI tip A θ R I y σeffective I I h σII h II II σ effective y III y x x III III determination of Eeffective calculation of fA,constant & fA,proportional (a) 3D trilayer indentation (b) Idealized cracked geometry – determination of fA,constant & model with a crack of radius fA,proportional for 3D cracked trilayer domain analysis based on effective R, A is the crack tip. bilayer domain analysis stress distribution from top layer I. approximate equivalent Load Influenced by both layer II and III constant stress (σ ) z z eq App R R zeq=Y·R σxx σxx (σeq)App (σeq)App (c) Axisymmetric model without a crack to calculate stress distribution in top layer I. KfIA≈⋅ A,tan cons t()σ eq App π R Figure 3.7. Application of the proposed method to a 3D trilayer indentation model. The stress intensity factor KIA in (a) can be estimated based on the axisymmetric stress distribution in (c) and the proposed cracked trilayer domain analysis in (b). -80- Figure 3.8. Percent effort in KIA estimation for a trilayer indentation model (Fig 3.7a) as a function of EII with various combination of EI and EIII. Accurate KIA is evaluated from complete 3D FEM including a half-penny crack. Approximate KIA is calculated based on the axisymmetric stress calculation and the craked trilayer domain analysis. The parameters are set to be: h/Rcrack=1/10, Rball=5Rcrack, load=400N. -81- CHAPTER 4 Statistical failure analysis of adhesive resin cement bonded dental ceramics Abstract The goal of this work is to quantitatively examine the effect of adhesive resin cement on the probability of crack initiation from the internal surface of ceramic dental restorations. The possible crack bridging mechanism and residual stress effect of the resin cement on the ceramic surface are examined. Based on the fracture-mechanics-based failure probability model, we predict the failure probability of glass-ceramic disks bonded to simulated dentin subjected to indentation loads. The theoretical predictions match experimental data suggesting that both resin bridging and shrinkage plays an important role and need to be considered for accurate prognostics to occur. -82- 4.1 Introduction The structure of a natural tooth consists of a hard, rigid, translucent ceramic-like outer enamel shell composed primarily of densely packed calcium phosphate crystals which is intimately attached to a softer, more flexible, polymer/crystalline composite structure known as dentin. In their natural unrestored state, teeth are surprisingly durable structures able to withstand repeated loads during mastication without catastrophic failure. Exactly how the enamel is attached to the underlying dentin, and how this intimate attachment to the dentin substrate influences the response of the enamel to external stimuli is still not completely understood. During service, teeth are subjected to any number of external stimuli that can contribute to their partial or complete degradation. Mechanical, chemical and bacterial stimuli can contribute to the degradation process and lead to some need for repair or replacement of the damaged tissues. Once the defective tooth structure has been removed and the area restored, the tooth can return to normal function. Numerous materials and techniques are currently used to restore the missing tooth structure. Each presents its own unique set of benefits and risks, however, each can provide the patient with adequate function over some useful lifespan. It has been estimated that over 60% of all dental treatment performed involves the re-restoration of teeth [1]. It is clear that all restorations have a finite service life. The inevitability of failure for all restorations demands that we study the process to -83- better understand the principal failure mechanisms with the hope that this knowledge can aide to improve the design, performance and even predictability of restorations. An ideal restorative technique would minimize any unnecessary removal of healthy tissue, replace the missing structure with a material that performs like the tooth structure it replaces, and protects the remaining tooth structure from further degradation or fracture. Over the past decade, new restorative materials and techniques have been introduced that generally fulfill these specifications and have challenged the clinical performance of the more traditional restorative paradigms. Adhesive resin retained ceramic restorations are among the most biomemetic restorative techniques that have emerged from these recent developments and warrant careful evaluation and examination. Glasses and glass-ceramics can be made to be optically compatible with natural enamel and dentin and are among the most stable, biologically inert materials used in restorative dentistry. In their most common dental application form, they are used as a veneering material sintered directly to a metal substructure. In this form, fractures are minimized by the strengthening effect of the underlying metal support [2], but the optical benefits of the ceramic are often compromised by its presence. The unwanted optical effects of the underlying metal must be hidden with a layer of opaque porcelain that can render the restoration “lifeless” due to its lack of translucency. The routine clinical application of substantially glassy ceramic materials has been hindered by their high fracture rates and poor clinical longevity when used in conjunction with conventional acid based cementation techniques [3,4]. However, there is significant laboratory and clinical evidence to suggest that the application of -84- adhesive resin cement can act to reinforce ceramic restorations and improve clinical longevity [3-7]. The clinical performance of resin bonded ceramic inlays, onlays, crown and veneers have been well documented [8-15] and in some cases have been shown to compete favorably with cast metal restorations [16]. Additionally, the adhesive nature of some restorations have been shown to reinforce the remaining tooth structure by increasing stiffness and fracture strength [17-19]. There is evidence to suggest that the adhesive resin cements may contribute to the clinical longevity of ceramic restorations by decreasing the probability of crack initiation from the internal surface [3,4,15,20-22]. A similar strengthening effect has been observed in a model trilayer “onlay-like” system (Fig 4.1) [23]. The fracture mechanics based failure probability model [24] was employed to predict the failure probability distribution of model onlay restorations subjected to indentation load to failure testing. Surface flaw distribution derived from biaxial data was used to predict the theoretical failure probability of the more complex trilayer model. The predicted values for the non-bonded ceramic supported by the simulated dentine substrate fell within the 90% confidence interval of the experimental data. However, the predicted values for the completely bonded case showed a somewhat higher failure probability than the experimentally derived values. It has been suggested that this discrepancy may be attributed to the possible bridging effect by the adhesive resin cement on the interfacial surface defects in the ceramic that was not accounted for in the analytical model [23]. The most common failure mode reported for ceramic restorations of all types is bulk fracture in the ceramic material (Figure 4.2). It has been demonstrated by -85- evaluation of clinically failed glass-ceramic restorations that a majority of these fractures (> 90%) are initiated from flaws and stresses originating from the adhesive resin cement interface rather than from the contact surface of the restoration itself [25-27]. This suggests that for this restoration type, Hertzian damage [28] is of less concern. The goal of this work is to quantitatively investigate the effect of adhesive resin cement on the probability of crack initiation from the internal cemented surface of ceramic dental restorations. We hypothesize that the resin cement on the ceramic surface influences failure probability of adhesive ceramic dental restorations through crack bridging mechanism and residual stress due to its curing process. A simplified trilayer onlay model such as dental ceramic disk bonded to simulated dentin (Figure 4.1) is used to represent the more complex clinical situation as shown in Figure 4.2. Distribution of the surface flaws is assumed to be in the form of m Nk(σ Cr) = σ Cr (4.1) where σ Cr , N, k and m respectively represents the critical stress, critical stress distribution function, and the scale and shape parameters k and m [29]. The k and m parameters are obtained by curve-fitting the experimental biaxial flexural test data on ceramic disks or plates. Given the same stress level, the crack opening displacement is restrained by the adhesive resin cement layer and the resulting reduction of the stress intensity factor is calculated by the simplified 2D and 3D local models. This bridging mechanism is incorporated into the critical stress distribution function. The curing process is simulated through linearly elastic FEA models with constant 1% volume shrinkage [30]. Based on the experimentally determined parameters and numerical -86- simulations, failure probability distributions for biaxial tests with adhesive resin cement coated ceramic disks and indentation tests on simple trilayer models are theoretically predicted. 4.2 Experimental methods 4.2.1 Biaxial test for determination of ceramic flaw population Biaxial data reported previously [6,23] using borosilicate glass and a new set of data using a dental glass-ceramic were utilized to verify the validity of our analytical model. Briefly, glass disks of about 1 mm thickness were formed by sectioning a 5/8 inch diameter borosilicate glass rod (item # 8849K62 McMaster Carr Supplies, Chicago, IL) with a slow speed diamond wheel saw and water coolant. Similarly, leucite-reinforced glass-ceramic (ProCAD, Ivoclar, Amherst, NY) plates were fabricated by sectioning the preformed 12.5 mm X 14.5 mm rectangular CAD (Computer Aided Design) blocks into 1.6 mm thick sections. All surfaces of the disks were hand ground on a flat glass plate with 600 grit SiC powder slurrie, so as to assure a consistent surface finish. One surface of each disk was etched with 5% hydrofluoric acid-etching gel (IPS Ceramic Etching Gel, Ivoclar USA, Amherst NY) for 60 seconds, rinsed with water and air dried with an air/water syringe. The dental ceramic disks were arbitrarily separated into two sub groups: one was tested as is while the other group was treated with a silane coupling agent (Silane Primer; SDS Kerr, Orange, CA) and coated with an approximately 100 µm resin cement film [23]. -87- Failure behavior of the treated disks was characterized by the ball-on-ring biaxial flexure test. A schematic of the test apparatus is shown in Fig. 4.3. The specimens were centered on a 12.5mm diameter support ring with the treated surface placed face down. A vertical load was applied to the center of the disks with a spherical indenter load at a crosshead speed of 0.5 mm/min. Detailed dimensions are summarized in Table 4.1a. There was no observed indentation of the ring into the surface of the cement during the test. To study the fracture behavior of ceramic restorations bonded to a compliant substructure such as dentin, a simplified trilayer onlay model was used. As shown in Fig. 4.1, the dental glass-ceramic plates and glass disks were bonded onto the top of dentin-like substrate using standard dental bonding techniques and resin cement (Nexus 2, Kerr Dental, Orange CA). The dentin-like substrate consisted of a glass fiber reinforced polymer (Garolite G10, McMaster Carr Supplies, Chicago IL) that has an elastic modulus similar to dentin. The bonded surfaces were treated in a manner similar to those used to cement dental ceramic restoration to dentin. The ceramic surface was etched and silanated as described previously. The dentin-like substrate was etched with 35% phosphoric acid for 30 sec to clean the surface, rinsed with water and lightly air dried prior to a 15 second application of the adhesive (Optibond Solo Plus, Kerr, Orange, CA). The adhesive was then air thinned to drive off the volitiles and light cured for 20 sec prior to cement application and light curing (Nexus 2 dual cure cement, Kerr Dental, Orange CA). The adhesive was applied to assure good wetting between fiber glass/epoxy substrate and the resin cement. -88- The completed trilayer specimens were loaded from the top surface with a spherical WC indenter using a universal testing machine. Detailed dimension is summarized in Table 4.1b. In order to protect the ceramic surfaces from contact damage and increase the probability of interface initiated cracks, the indenter was covered with a thin adhesive polyethylene tape. Loading and observation of the crack initiation at the interface were achieved using two slightly different methods. For the transparent glass specimens, crack initiation could be viewed directly from the side of the specimen through a 10 x binocular microscope with transillumination of the specimen during continual loading. For the translucent dental ceramic materials, this could not be achieved. Therefore, the specimens were step loaded at increasing 50 N increments and observations were periodically made from the top surface using the 13.5 x binocular dental loupes and transillumination to detect cracks. This was considered to be analogous to a clinical exam. The breaking loads for the cracks initiating at the bottom surfaces were recorded. 4.2.2 Material property measurement The elastic moduli of some of the composite materials were obtained by λ + 2μ μ ultrasonic measurements of longitudinal ν = and shear ν = wave l ρ t ρ velocities, where ρ is the density, λ is the Lame’s parameter and μ is the shear modulus. 22 μ(3λμ+ 2 ) ⎛⎞⎛⎞ννll The Young’s modulus E = and Poisson ratio υ = ⎜⎟⎜⎟22−−11 λμ+ ⎝⎠⎝⎠2ννtt were calculated from measured longitudinal and shear wave velocities. The density -89- was determined by Archimedes’s method. The ultrasonic velocity measurements were performed at 10MHz by the pulse-echo method using both immersion and contact techniques for longitudinal wave velocity and by the contact method for shear wave velocity. A Panametrics 5073 PR pulser/receiver and a Hewlett Packard 54504-A 400 MHz digital oscilloscope were used for time delay measurements by the signal overlapping technique. The sample thickness was measured by a micrometer. The precision of the measurement is limited by the flatness and parallelism of the sample surfaces and by the couplant effect for the shear wave velocity measurement. We estimate at least three correct digits in the determination of the Young’s and shear moduli on our samples. The results are summarized in Table 4.2. Quantitative assessment of the surface roughness was accomplished using a WYCO optical profilometer (Veeco NT3300, Veeco Metrology, Tucson, AZ). All scans were performed utilizing a 10X objective under vertical scanning interferometry (VSI) corrected for curvature and tilt. Ten scans in pre-determined areas were taken from select prepared glass and glass-ceramic surfaces to provide measurements of arithmetic mean roughness (Ra = 0.49 ± 0.09 μm ), and root-mean square roughness (Rq = 0.64 ± 0.10 μm ). -90- 4.3 Theoretical analysis 4.3.1 Critical flaw distribution and local crack stress intensity model 4.3.1.1 Critical flaw distribution In both the biaxial test (Fig. 4.3) and the trilayer indentation tests (Fig. 4.1), the fracture initiates from a flaw located on the bottom surface of the top layer in contact with the indenter. We define as the critical stress σCr the given stress state σ at which a crack with critical length aCr begins to propagate. The surface flaws from which fracture initiates are the result of sample preparation and surface treatment prior to bonding and can be related to surface roughness. In analyzing the stress distribution at surface flaws it is reasonable to use the following assumptions: (1) the critical flaw distribution is isotropic; (2) each critical flaw is considered as an equivalent small surface crack; (3) critical flaws are sufficiently far apart (the spacing between critical flaws is much larger than the depth) and thus the interaction between them is neglected. The hand ground 600 grit SiC slurry produces an average surface Ra of about 0.5 μm and the roughness is found to be similar in two directions. Assumption (1), therefore, can be justified. Since stress state depends on radius we subdivide the top layer on regions where the stress assumed to be approximately constant and therefore the corresponding critical flaw size is different for different regions. If there exists a region containing a flaw which is equal to the critical flaw size, the failure starts from that region. Therefore, assumption (3) that the critical flaws are far apart is reasonable. A distribution N(σCr) gives the number of critical flaws per unit area for which the critical stress varies between zero and σCr, and can be interpreted as the crack density -91- function. As in Chao and Shetty [29] and Wang et al [23,24], the critical stress density distribution N is assumed to be in the form of Equation (4.1). 4.3.1.2 Local model for determination of stress intensity factor for system with and without adhesive resin cement Since the critical flaws are non-interacting, one may obtain critical stress levels in distribution (4.1) by an analysis of stress intensity factors of isolated cracks. To compute crack stress intensity factors at a given level of indentation load in the models of Fig. 4.1 and Fig. 4.3, one needs to perform 3D finite element analysis of these models which is computationally expensive. For this reason we have employed the method of local crack analysis by calculating the stress distribution in the models (Figs. 4.1 and 4.3) without a crack and taking the stress distribution obtained in the vicinity of an isolated crack to determine the stress intensity factor. This approach was shown to be sufficiently accurate by comparison with parametric study performed by 3D finite element analysis, which will be described elsewhere. In general, the variation of normal stress along the crack depth is found to be negligible if the crack size is small. Therefore, only tension loading is used in local models. Deviation from this assumption will be reported in detail elsewhere. In order to quantitatively address the possible bridging effect of the adhesive resin cement, we introduce 2D and 3D local crack models with and without resin cement as shown in Figs. 4.4 and 4.5. It is assumed that only mode I failure occurs. The system without resin cement is considered first. In the 2D plane-strain local model without resin cement, a single edge crack inside a semi-infinite body is -92- introduced (Fig. 4.4a). The body is under a constant tensile stress σ. Similarly, in the 3D local model without resin cement, a half-penny crack inside a semi-infinite body is subjected to a constant tensile stress σ normal to the crack face as shown in Fig. 4.5a. For both local models without resin cement, the crack intensity factor KI at the crack tip (point A) can be represented as: KfI =×(σ π a) , (4.2) where a represents the radius of a half-penny crack in the 3D model, and the crack length in the 2D model. It is well known that the value of the parameter f is 1.025×2/π for the 3D model and 1.12 for the 2D model [31]. When a thin layer of adhesive resin cement is bonded to the ceramic surface, the crack opening displacement of a surface flaw is constrained by the resin. Therefore, given the same constant tensile stress σ normal to the crack surface, the stress intensity factor will be reduced. This effect of resin cement on the stress intensity factor in our local model leads to modification of factor f in Eq. (4.2). In order to simulate this effect, a perfectly bonded semi-infinite body representing the resin cement is included in the 2D and 3D local models as shown in Figs. 4.4b and 4.5b. The far tensile stress σ g in the resin cement region is chosen to satisfy σ /EE= σ g / g so that the far interfacial displacement field is compatible. For these local models the values of factor f for both 2D and 3D models are not readily available. Hence, we employ the FEA model to calculate the value of f. The results are summarized in Table 4.3 where f and f’ represent the value without and with resin cement respectively. -93- 4.3.1.3 Determination of parameter k in critical flaw distribution The critical stress σ Cr in distribution (4.1) for a given flaw size, a, is determined from Eq. (4.2): KIC=× f(σπ Cr a) without adhesive , (4.3) where KIC is the critical stress intensity factor (fracture toughness) which is a material property. Therefore, after substitution of Eq. (4.3) into Eq. (4.1) it describes the flaw size distribution for the surface without resin cement. With use of the local model, the corresponding equation for the crack bridged by resin cement is: ′′ KIC=× f(σπ Cr a) with adhesive , (4.4) where fracture toughness KIC and crack size a remain unchanged. Eqs. (4.3) and (4.4) lead to the relation: f ′ σ = σ ′ , (4.5) Crf Cr where f’/f is less than 1 as shown in Table 4.3. Thus σCr is less than σ’Cr,, which means that the resin cement layer constrains the crack opening and high level of critical stress is required for the sample failure. Inserting Eq. (4.5) into Eq. (4.1), we obtain the effective critical stress density distribution N for the surface with resin cement: m Nk(σ Cr′ ) = ′′σ Cr , (4.6) where m ⎛⎞f ′ kk' = ⎜⎟ . (4.7) ⎝⎠f -94- In the analysis below, Eq. (4.1) is used to predict the failure probability for the surface without resin cement, while Eq. (4.6) is used for the surface with resin cement. 4.3.1.4 Determination of parameters m and k in Eq. (4.1) from biaxial indentation experiments In order to determine experimentally the parameters m and k in Eq. (4.1), we employ the fracture-mechanics-based statistical method developed by Wang et al. [24]. First, experimental data points of cumulative failure probability distribution are obtained, and second they are curve-fitted with the theoretical failure distribution equation with appropriate value of m and k. In extracting physically meaningful data from actual experimental data, we take into account of failure load variation due to thickness variation. For this purpose, we introduce the effective fracture initiation load Pi of specimen i given by 2 o ⎛⎞t PPii= ⎜⎟, (4.8) ⎝⎠ti o where ti is the thickness of specimen i , t is the average thickness, and Pi is the experimental obtained fracture initiation load of specimen i . Eq. (4.8) is based on the assumption of linear elasticity where the maximum stress σ Max is proportional o 2 o to Ptii/π [32]. By using Pi instead of Pi , we eliminate the thickness variation of individual specimens from the raw experimental data. In this work, the effective failure load Pi is sorted by order of magnitude and the cumulative failure probability Pif ()of crack initiation at the i-th fracture load Pi is assumed to be -95- Pif ()= i /(1+ n ) (4.9) where n is the number of specimens. The parameters m and k are determined by curve-fitting the data from the biaxial flexure tests by the following equations [6,23,24]: ln⎡⎤− ln 1−=PmPB ln + (4.10) ⎣⎦( f ) () and ⎡ 2 ⎤ R mkI B = ln ⎢ D ⎥ , (4.11) ⎢ 2 m ⎥ ⎣ ()πt ⎦ where P and Pf represent the effective load and cumulative failure probability respectively and R is the radius of the support ring. The definition of I D and derivation of Eq. (4.11) are summarized in the Appendix B. As shown in the Appendix B, ID is independent of the indentation load for the biaxial test, since within the limits of linear elasticity the stress distribution is proportional to the indentation load (Eq. (B.6)). The I D value for the biaxial tests on ceramic plates or disks without resin cement is determined for a stress distribution induced by a unit indentation load obtained using a linear-elastic axisymmetric FEA model. In the computations, quadratic axisymmetric stress elements are used (100 elements along the radius direction and 20 elements along the thickness of the glass-ceramics); the spherical indenter and supporting ring are assumed to be rigid, and the contact surfaces are assumed to be friction free. The specimen dimension and relevant material properties used in the computation are summarized in Tables 4.1a and 4.2, respectively. -96- 4.3.2 Effect of resin-cement layer shrinkage on crack bridging The curing process of bonding resin is a complicated polymerization process with rate and temperature dependent deformation and material properties [33-35] and volumetric shrinkage. The strengthening effect of the resin layer on ceramics, resulting in crack bridging, may be increased due to shrinkage of the resin layer during curing on the stiff substrate. The shrinkage leads to residual tensile stresses in resin cement and compressive stresses in the substrate, which results in additional strengthening. Since the residual stress is influenced by the rheological properties and curing kinetics of the resin cement [36], it is difficult to simulate accurately. The curing of dental composites is usually accompanied by a volumetric shrinkage in the range from 1.5 to 5% [33]. Sakaguchi et al. [30] developed a strain gage method to measure the shrinkage and isolated the net post-gel shrinkage, which was within the range from 0.66% to 0.87% for commonly used dental composites. To examine the residual stress distribution due to resin shrinkage, axisymmetric FEA computations are performed for the biaxial and the trilayer models without mechanical load. Since the volumetric shrinkage measured by Sakaguchi et al. [30] is performed in the post-gel phase, the elastic modulus of resin cement with a reasonable approximation may be assumed to be constant. The volume fraction of the filler for the resin cement used in this work is much less than that of the composites used by Sakaguchi et al. [30]. Because of this, the volumetric shrinkage rate is expected to be larger than those measured by Sakaguchi et al. [30]. We employ a 1% volumetric shrinkage as an approximation for the resin cement layer. To simulate such a level of shrinkage a thermal expansion coefficient of 0.0033 o C −1 and decrease of 1 o C in -97- temperature are assigned for the resin cement in the FEA simulation so that 1% volumetric shrinkage will be produced. There are two competing factors governing the volume shrinkage. One is the increase of residual stresses due to higher volume shrinkage. In general, a cement material is less filled, so it flows easily during the cementation process thus can exhibit 2-3 times as great a shrinkage as a restorative composite. Therefore, instead of only 1% of the cement volume shrinkage, 2~3% is possible. The competing factor is the residual stresses reduction caused by the viscoelastic behavior of the cement. This effect is changing during the curing process since the material viscosity/stiffness is a function of curing time (the cement shrinkage in the beginning of the curing process does not result in the residual stresses). These two competing factors should be theoretically examined in order to make an accurate residual stress analysis. The current study, based on the linearly elastic material assumption, is limited in this regard and provides only an approximate analysis. 4.3.3 Prediction of failure probability distributions for trilayer model The failure probability distribution for biaxial tests of ceramic disk and plate samples coated with a thin resin cement layer (Fig. 4.3b) is predicted theoretically using biaxial test data on uncoated samples (Fig. 4.3a). For analyses, the fracture mechanics based failure probability model [24] (see Eq. (B.4) in Appendix B) is used. In order to examine the effects of crack bridging by the resin layer on the failure probability, four different cases were investigated. First, the resin layer effect was determined with direct use of the experimentally determined parameters m and k. -98- Curing residual stress was not included and only the strengthening effect of the resin cement layer is accounted for but without its effect on reduction of the crack stress intensity factor. The second simulation includes the curing residual stress. Third, to account for the modification of the crack stress intensity factor in the 3D local model, the value of parameter k was replaced by k’ (Eq. (4.7)) to include the resin layer bridging in addition to curing residual stress. Fourth, the parameter k was replaced by k’ which was modified by a 2D local model and curing residual stress was included. To evaluate the stress fields in biaxial tests of ceramic specimens coated with resin cement, an axisymmetric linearly elastic finite element (ABAQUS Version 6.5) analysis was employed. Quadratic axisymmetric stress elements (100 elements along the radius direction and 20 elements along the thickness of the glass-ceramics, 6 elements along the thickness of the resin cement) were used. The spherical indenter and supporting ring were assumed to be rigid and the contact interface between indenter and sample was assumed to be friction free. Tied constraint is employed in the interface between the glass-ceramic and resin cement layers. The dimension and material properties are summarized in Table 4.1a and Table 4.2 respectively. The same method is employed to predict the failure probability distribution for the indentation tests on the trilayer model (ceramic/glass bonded to simulated dentin, Fig. 4.1). In the axisymmetric finite element model to simulate the indentation tests on the bonded trilayer systems, 40 and 80 elements along the radial direction, respectively, are used for glass-ceramics and for glass. Along the thickness direction, 20 and 6 elements are respectively used in the top/substrate layers and the resin cement -99- layer for both glass-ceramics and glass. The dimension and material properties are summarized in Table 4.1b and Table 4.2 respectively. 4.4 Results 4.4.1 Critical stress density distribution function for biaxial tests Experimentally obtained failure distributions of ball-on-ring biaxial tests without resin cement are shown in Fig. 4.6 for both glass and glass-ceramics. The experimental data are least-square-fitted and the fitting results are shown in Figure 4.6. The parameters m and k of the critical stress density distribution function N, obtained based on the method of section 3.1.4, are shown in Figure 4.6. The parameter k’ of the effective critical stress density distribution evaluated from Eq. (4.7) and Table 4.3 is summarized in Table 4.4. The resulting critical stress density distribution functions are plotted in Fig. 4.7. The dotted line represents the critical stress density distribution curve-fitted from the biaxial test without a resin cement layer. The dashed line and solid line, respectively, represent the effective critical stress density distribution modified by the 3D and 2D local models. The differences, among these three critical stress density distribution functions, are attributed to the differences in the parameter k’ as summarized in Table 4.4. For all of these curves, the critical stress density distribution N rapidly increases with the critical stress, indicating that the results are dominated by small flaws. Since large critical stresses correspond to small cracks, the experimentally determined curve shows that the number of small cracks per unit area is much larger than that of large -100- cracks. Modified distributions (dashed and solid curves) are lower than the experimentally determined distribution (dotted curve); high critical stress is required for failure initiated from the same flaw size. As shown in Fig. 4.7, the resin cement layer provides stronger bridging effect in the 2D case than in the 3D case. 4.4.2 FEA simulation of Residual Curing Stresses As discussed in Section 3.3, the residual stress distribution at the bottom surface of the ceramic layer due to curing of the resin cement coating has been simulated for both the biaxial test model (Fig. 4.3b) and the trilayer onlay indentation model (Fig. 4.1). The normal residual stress components σ rr and σθθ are found to be nearly constant for most of the region. The average values of the residual stress components for 1mm radius concentric area (the critical central area at the bottom of the ceramic layer below the indenter) are summarized in Table 4.5. The calculated level of residual compressive stress is very small compared to the 122 MPa Weibull characteristic fracture stress for glass-ceramics [6]. 4.4.3 Prediction of failure probability distribution for biaxial tests of ceramic coated with resin cement layer - comparison with experimental data Experimentally obtained failure probability distribution of glass-ceramic disks subjected to biaxial tests are plotted in Fig. 4.8. Stars and circles represent biaxial test -101- data with and without resin cement, respectively. The biaxial test data of the glass-ceramic disks with and without the resin cement coating were processed as discussed above. The 95% confidence interval [37] of the experimental data with resin cement is also shown as thin solid lines. The fitted curve for tests of uncoated samples is also shown in Fig. 4.8. Four distinct theoretical predictions of failure probability distribution for the biaxial test with resin cement are plotted by dotted, dot-dashed, dashed and solid lines, respectively. Dot-dashed and dotted lines represent the theoretical failure probability distribution prediction based on the stress calculation from the axisymmetric FEA model with and without curing residual stresses respectively. For these two theoretical predictions, the curve-fitted parameters m and k are used. Solid and dashed lines represent the theoretical failure probability prediction with values of the parameter k’ modified by the 3D and 2D local models respectively. Inclusion of the residual stress due to the curing process significantly shifts the theoretical prediction towards higher loads. Inclusion of both 3D and 2D bridging effects also shifts the failure probability prediction towards higher loads. The dot-dashed curve (the prediction based on the stress reduction and the curing residual stress) falls within the 95% confidence interval for the biaxial test data, and the dashed curve (the prediction based on the stress reduction, the residual stress and the 3D local model) is mostly within this interval. However, the solid curve (the prediction based on stress reduction, residual stress and the 2D local model) and especially the dotted line (the prediction based on the stress reduction by the resin cement layer) mostly fall outside the 95% confidence interval. -102- 4.4.4 Prediction of failure probability distribution for indentation tests on the bonded trilayer onlay model 4.4.4.1 Glass-ceramics/resin cement/simulated dentin trilayer model The experimentally obtained failure probability distribution for the indentation tests on the trilayer model (glass-ceramic/resin cement/dentin) is shown by triangle points in Fig. 4.9. The 95% confidence intervals [37] are represented by thin solid lines. As in the previous theoretical failure probability prediction of the biaxial test for ceramic coated by a resin cement layer, four distinct theoretical predictions of the failure probability distribution for the indentation tests on the three layer model are included in Fig. 4.9. The predictions are performed using the stress calculation for the indentation tests of this system using the axisymmetric FEA model. Dot-dashed and dotted lines are obtained with and without accounting for residual stresses respectively. For these two simulations, the experimentally determined values of parameters m and k are used. Inclusion of the residual stress in the theoretical prediction shifts the failure probability distribution towards higher loads. However, the shift is almost negligible compared to the significant shift seen in the biaxial tests. The prediction based on the stress reduction by the resin cement layer (dotted line) and the prediction based on the stress reduction and the residual stress (dot-dashed line) are inside the 95% confidence interval for the lower fracture initiation load region. -103- Solid and dashed lines represent theoretical predictions with parameter k’ modified by the 2D and 3D local models respectively, thus accounting for change of crack stress intensity factors. Inclusion of both the 3D and 2D bridging effects significantly shifts the failure probability towards higher loads. When compared to the results in biaxial tests, this shift did not help to improve failure probability predictions. The predicted cumulative failure probability curve based on the stress reduction, the residual stress and the 3D local model (dashed line) falls within the 95 % confidence interval of the experimental curve for the lower fracture initiation load region. The predicted curve based on the stress reduction, the residual stress and the 2D local model (solid line) falls within the 95% confidence interval only for a small region of lower loads. 4.4.4.2 Glass/resin cement/simulated dentin trilayer model The glass/resin cement/simulated dentin system experimental data and the corresponding theoretical predictions are shown in Fig. 4.10. Following the same procedure as above computer simulations of the indentation tests of the system have been performed. The prediction based on the stress reduction (dotted line) and the prediction based on the stress reduction and the residual stress (dot-dashed line) fall mostly outside of the 95% confidence interval of the experimental data. The predicted probability curve based on the stress reduction, the residual stress and the 3D local model (dashed line) mostly falls within the 95 % confidence interval. The prediction based on the stress reduction, the residual stress and the 2D local model (solid line) -104- falls within or very close to the 95% confidence interval for the lower load region. The analysis corresponding to the dotted line (far left; prediction based on the stress reduction) which is outside the 95% confidence interval, was also previously performed by Wang et al. [23]. 4.5 Discussion 4.5.1 Determination of parameters m and k For the glass/resin cement/simulated dentin system, the failure probability distribution has already been examined by Wang et al. [23]. In this work, we consider the bridging effect and the residual shrinkage stress of the resin cement layer to improve the predictive capability of the analysis as shown Fig. 4.10. Parameters m and k for the critical stress density distribution shown in Figure 4.6 are different than those by Wang et al. [23] although the same experimental data were used for the analysis. The difference is partly attributed to the modification of the experimental fracture initiation load by considering the variation of thickness of each specimen based on Eq. (4.8). In addition, the stress distribution is more accurately calculated by the FEA model, rather than using the approximate analytical equations employed by Wang et al. [23]. Therefore, in this work, the same experimental data was used to determine more accurately the model parameters. -105- 4.5.2 Curing residual stress versus crack bridging as strengthening mechanisms In this work, we quantitatively examined the strengthening mechanism attributed to the influence of the resin cement layer by both crack bridging and curing residual stress. In the biaxial tests (Fig. 4.8), both the bridging effect and the residual stress appear to be important factors. For the trilayer ‘simulated onlay’ model, however, curing residual stress appears to play an insignificant role as shown in Figs. 4.9 and 4.10. This difference can be attributed to the difference in the calculated residual stress level as shown in Table 4.5. In the trilayer model, the average residual stress at the bottom glass-ceramics surface is 1.9 MPa, which is much smaller than the 12.4 MPa for the biaxial model. In order to consider this difference, the deformation due to resin cement shrinkage for the biaxial test and the trilayer onlay models is shown in Fig. 4.11. With enlargement of displacement by a factor of 20, a bending deformation can be observed in the biaxial test specimen, while there is no such deformation in the ceramic bonded to a thick substrate. The substrate in the trilayer model restricts the contraction of the resin cement, while in the biaxial test model there is no similar constraint on the ceramic. As a result, larger compressive stresses are produced due to the bending deformation at the bottom surface of the ceramics in the biaxial test model. -106- 4.5.3 2D and 3D local crack models The surface flaws vary in their shapes and sizes, and they are different from specimen to specimen. In order to simulate the surface flaws, we developed 2D and 3D local crack models assuming the crack shape as shown in Figs. 4.4 and 4.5. If the surface flaw is much longer than its depth, it can be locally approximated by a 2D surface or interface crack. If the flaw length and depth are comparable, it can be approximated by a 3D local crack model. The shape of an actual flaw can be between these two limiting cases described approximately by our 2D and 3D local crack models. Our results show that combination of 2D and 3D local models results in reasonable agreement with experimental data. 4.5.4 Discrepancies between the test data and the models Despite generally good agreement between the test data and the model predictions as shown in Figures 4.8-4.10, there are some discrepancies. For both trilayer onlay systems as shown in Figures 4.9 and 4.10, the slopes for experimental data with 95% confidence interval curves are steeper than those for theoretical predictions while in biaxial tests, the difference in the slope between experimental data curves and theoretical predictions are small. In addition, in the glass ceramics/resin cement/simulated dentin system, better agreement between experimental data and theoretical predictions are obtained for the lower load portion of the curve, while in glass/resin cement/ simulated dentin system, better agreement are obtained for the higher load range portion of the curve. Possible causes of these discrepancies may be -107- attributed to simplified model assumptions such as uniform 1% volume shrinkage without modeling dynamic curing process and idealized surface flaw geometries. In addition to these simplified assumptions, some physical phenomena which are not included in our current modeling effort may play an important role. For example, during the curing process, the resin cement may partially fill up the surface flaws in the direction of their depth and enhance the bridging effect. In addition to purely mechanical factors, differences in the nature of chemical bonding at the interface may limit predictive capabilities of the current model. 4.6 Conclusion In this work, the influence of the resin cement layer on the strengthening mechanism and fracture behavior of dental ceramics has been quantitatively examined by employing the fracture mechanics based failure probability model proposed by Wang et al. [24]. To account for the bridging effect of the resin cement layer we have introduced the notion of a critical stress density distribution function based on 2D and 3D local crack models. The residual stress induced in the system due to curing of the resin cement layer is simulated by FEA assuming 1% volumetric shrinkage. Good agreement has been obtained between the prediction of the failure probability distribution and experimental data for both the biaxial tests with a resin cement layer and the indentation tests on the bonded trilayer onlay model. The model developed has not only improved the predictive capability but also provided physical insight as to the relative importance of different strengthening -108- mechanisms. In the biaxial test, both the curing residual stress and the crack bridging effects play important roles in strengthening mechanisms. However, in the bonded trilayer onlay model, the bridging effect plays a crucial role while the contribution of the curing residual stress is negligible. This difference in the strengthening mechanisms is attributed to the constraint from the bottom substrate which results in relatively small magnitudes of curing residual stress in the top onlay layer. The strengthening mechanism for the bonded trilayer onlay model has important implications in the clinical application: the strengthening mechanism for the complex clinical restoration structures may be attributed to the bridging effect. Although our analysis is limited to simplified structure, it is expected that the same mechanism exists for more complicated clinical dental restoration structures. Therefore, in order to improve clinical performance, it is more important to increase the resin cement modulus and assure good bonding. -109- References [1] Mjor IA, Toffenetti OF. Secondary caries: A literature review with case reports. 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Fracture-surface analysis of dental ceramics. J Prosthet Dent. 1989;62:536-541. [26] Kelly JR, Giordano R, Pober R, Cima MJ. Fracture surface analysis of dental ceramics: clinically failed restorations. Int J Prosthodont. 1990;3(430-440). [27] Thompson JY, Anusavice KJ, Naman A, Morris HF. Fracture surface characterization of clinically failed all-ceramic crowns. J Dent Res. 1994;73:1824-1832. [28] Wang R, Katsube N, Seghi RR, Rokhlin SI. Failure probability of borosilicate glass under Hertz indentation load. J Mater Sci. 2003;38(8):1589-1596 [29] Chao LY, Shetty DK. Reliability Analysis of Structural Ceramics Subjected to Biaxial Flexure. J Am Ceram Soc. 1991;74(2):333-344. [30] Sakaguchi RL, Versluis A, Douglas WH. Analysis of strain gage method for measurement of post-gel shrinkage in resin composites. Dent Mater. 1997;13:233-239. [31] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. New York: ASME Press, 2000. [32] Shetty DK, Rosenfield AR, McGuire P, Bansal GK, Duckworth WH. Biaxial Flexure Tests for Ceramics. Am Ceram Soc Bull. 1980;59(12):1193-1197. [33] Ferracane JL. Developing a more complete understanding of stresses produced in dental composites during polymerization. Dental Materials. 2005;21(Dent Mater):36-42. [34] Ruddell DE, Thompson JY, Stoner BR. Mechanical properties of a dental ceramic coated by RF magnetron sputtering. J Biomed Materi Res. 2000;51:316-320. [35] Sakasuchi RL, Wiltbank BD, Murchison CF. Cure induced stresses and damage in particulate reinforced polymer matrix composites: a review of the scientific literature. Dent Mater. 2005;21:43-46. [36] Sham M-L, Kim J-K. Curing Behavior and Residual Stresses in Polymeric Used for Encapsulanting Electronic Packages. J Appl Polym Sci. 2005;96:175-182. [37] Dodson B. Weibull Analysis. Milwaukee, Wisconsin: ASQC Quality Press, 1994. [38] Batdorf SB, Crose JG. A Statistical Theory for the Brittle Structures Subjected to Nonuniform Polyaxial Stresses. J Appl Mech. 1974;41(1974):459-464. [39] Standard Test Method for Monotonic Equibiaxial Flexural Strength of Advanced Ceramics at Ambient Temperature. Annual Book of ASTM Standards. 2006;15.01:760-770. [40] Holand W, Beall G. Glass-ceramic technology. Westervill, Ohio: The American Ceramic Society, 2002. -111- Figures load spherical rigid ball with radius R top layer (glass-ceramics/glass) htop hadhesive hsubstrate adhesive resin cement Substrate (simulated dentin) diameter D Figure 4.1. Indentation tests on the bonded trilayer onlay model. Average dimension is listed in Table 4.1b. -112- Figure 4.2. Fractured resin retained ceramic onlay after 3.5 years of service on a maxillary 1st premolar on a patient with a severe bruxing problem. Arrows point to the fractured ceramic surface. The buccal half of the onlay remains bonded to the tooth structure while the lingual portion is missing and the underlying dentin is exposed. -113- load spherical rigid ball load spherical rigid ball with radius R with radius R glass-ceramics/ glass-ceramics/ glass glass h h hadhesive diameter D diameter D adhesive resin cement (a) without adhesive resin cement (b) with adhesive resin cement Figure 4.3. Ball-on-ring biaxial experimental tests. Average dimension of samples used for biaxial tests is listed in Table 4.1a. -114- σ σ σg perfect a a A A bond ε = ε g σ σ ceramics/ ceramics/ resin cement = g glass glass (Eg, υg ) EEg (E, υ) (E, υ) (a) single edge crack with (b) single edge crack with length a in a length a in a semi-infinite semi-infinite body (E, ν) reinforced by a body (E, ν) dissimilar semi-infinite body (Eg, νg) Figure 4.4. 2D Plane Strain Local Model. -115- σ σ glass-ceramics/ σg glass (E, υ) perfect A bond A a a glass-ceramics/ ε = ε g glass σ σ (E, υ) = g resin cement E Eg (Eg, υg ) σ σg σ (b) half penny crack initiated perpendicular to the interface (a) half penny surface crack in a in a semi-space with Young’s modulus E and Poisson’s ratio semi-space; a is crack radius; E is ν. The second semi-space with Young’s modulus Eg and Young’s modulus and ν is Poisson’s Poisson’s ratio υg is bridging the crack. a is crack radius. Figure 4.5. 3D Local Model. -116- Figure 4.6. Ball-on-ring biaxial experimental data without resin cement (see Fig. 4.3a) for two materials: glass and glass-ceramics. The straight lines (YX= 5.21− 23.81 for glass and YX=−5.92 32.55 for glass-ceramics), are obtained by fitting the experimental data based on the least-square method. They are used to determine the m parameters k and m of the critical stress density distribution function Nk()σ Cr= σ Cr . (glass: m=5.21, k=1.75E-12×mm−−25.21 MPa ; glass-ceramic: m=5.92, k=2.25E-14×mm−−25.92 MPa ) -117- m Figure 4.7. Critical stress density distribution N as a function of σ Cr , Nk()σ Cr= σ Cr , at the bottom surface of glass-ceramic disks. -118- Figure 4.8. Experimental data and predicted failure probability distributions of glass-ceramic disks with and without resin cement layer under biaxial tests (Fig. 4.3). The critical stress density distribution (obtained from Y=5.92X-32.55 in Fig. 4.6) was used to theoretically predict the failure probability of disks with resin cement. -119- Figure 4.9. Experimental data and predicted failure probability distributions of indentation tests on bonded trilayer onlay model (Fig. 4.1), (glass-ceramics/resin cement/simulated dentin trilayer system). -120- Figure 4.10. Experimental data and predicted failure probability distributions of indentation tests on bonded trilayer onlay model (Fig. 4.1), (glass/resin cement/simulated dentin trilayer system). -121- The deformation was enlarged by 20 times for both models symmetric axis glass-ceramic adhesive resin cement simulated dentin substrate (a) biaxial tests with adhesive (Fig 3b) (b) trilayer onlay model (Fig 1) Figure 4.11. Deformation caused by shrinkage of the resin cement layer in biaxial tests with resin cement and the trilayer onlay model. The deformations are calculated by AbaqusTM, and 1% volumetric shrinkage of resin cement layer is included. No external force is included. -122- Tables Table 4.1. Dimension of samples in the mechanical tests. a. Average dimension of laboratory samples used for biaxial tests (Fig. 4.3). sample material glass-ceramics glass diameter: D (mm) 14.8(1) 15 thickness of disk: h (mm) 1.6 1.1 thickness of resin cement layer: hadhesive (mm) 0.1 NA radius of spherical indentation ball: R (mm) 12.5 5 diameter of support ring (mm) 12.5 12.5 (1) : The diameter is the equivalent diameter calculated according to ASTM standard C1499-05 [39]. b. Average dimension of laboratory trilayer onlay models used for indentation tests (Fig. 4.1). top layer material glass-ceramics glass substrate layer material simulated dentin simulated dentin diameter: D (mm) 12.5 13 thickness of top disk: htop (mm) 1.6 1.1 thickness of resin cement layer: hadhesive (mm) 0.1 0.1 thickness of substrate: hsubstrate (mm) 5 4 radius of spherical indentation ball: R (mm) 6.25 20 -123- Table 4.2 Experimentally measured materials properties. Young's modulus E (GPa) Poisson's ratio υ glass 62.5 0.19 glass-ceramics(1) 62 0.25 adhesive resin cement 10.21 0.33 simulated dentin 12.6 0.35 (1) : material property obtained from reference [6,40]. -124- Table 4.3. Numerically evaluated values of f’ for 2D and 3D perfectly bonded local 2 models (Figs. 4.4b and 4.5b). Recalling that f =1.025× for the 3D local model π f ' (Fig. 4.5a) and f =1.12 for the 2D local model (Fig. 4.4a), note that <1 for both f glass/resin cement and glass-ceramic/resin cement systems with 3D and 2D assumptions. This increases the critical failure stress as shown in Eq. (4.5). f' for glass-ceramics/resin cement f' for glass/resin cement system system 3D local model 0.971×2/π 0.978×2/π 2D local model 0.951 0.958 -125- Table 4.4. Numerical values of parameter k’ of effective critical stress density m m ⎛⎞f ′ distribution Nk()σ Cr′′′= σ Cr calculated from kk' = ⎜⎟ and Table 3 and ⎝⎠f experimentally determined parameters m and k (glass: m=5.21, k=1.75E-12×mm−−25.21 MPa ; glass-ceramics: m=5.92, k=2.25E-14×mm−−25.92 MPa ) local models employed for materials k' modification 3D 1.32E-12×mm−−25.21 MPa glass 2D 7.46E-13×mm−−25.21 MPa 3D 1.70E-14×mm−−25.92 MPa glass-ceramics 2D 0.89E-14×mm−−25.92 MPa -126- Table 4.5. Average values of residual stress components σ rr and σθθ at the bottom critical surface of glass-ceramics/glass (1mm radius concentric critical area) due to curing process (1% volume shrinkage of resin cement). The results are calculated by axisymmetric FEA (Biaxial and Trilayer) models. σ rr and σθθ , respectively, represent the normal stresses along the radial and tangential directions. materials and structure σ rr (MPa) σθθ (MPa) glass-ceramics/resin cement -12.4 -12.4 glass-ceramics/resin cement/simulated -1.9 -1.9 dentin glass/resin cement/simulated dentin -2.5 -2.5 -127- CHAPTER 5 Effective elastic modulus prediction for porous materials based on the identification of microstructures Abstract Porous materials are widely used in medical and industrial fields, and they are often treated as homogeneous materials with effective moduli when analyzing the structural integrity. Knowledge of the relationship between the effective modulus and the microstructure of a porous material is very important in material design. A porous material often includes microcracks in addition to pores. In this work, the combined effect of oblate spheroid shaped pores and penny shaped microcracks on the effective elastic modulus is investigated for porous ceramics widely used as filters in diesel engines. The micromechanics analysis methods, Mori-Tanaka method and differential-scheme method, are applied to predict the effective Young’s modulus. The predictions are compared with experimental data. -128- 5.1 Introduction Porous materials are widely used in medical and industrial fields. For example, porous materials are recognized as the ideal human bone substitutes due to the good biocompatibility and osteoconductivity [1]. In diesel engines, porous ceramics are used as filters to remove the particulate matter and soot from engine exhaust gas so as to satisfy diesel emission standards [3]. In addition, the natural human tissues, such as bones and dentins, have porous structure as well. A porous structure may fail due to a stress concentration caused by environmental thermal, mechanical or chemical stimuli in daily use. In order to analyze its structural integrity, the material properties of a porous material are required. However, it is computationally difficult and often not feasible to model the details of porous structures directly. Instead, the porous material is often considered as an equivalent homogenous material with effective properties which has the same response to thermal and mechanical loadings. Experimental methods can be used to measure the effective modulus of a specific porous material sample. However, the measurement of effective modulus for some porous materials can not be easily performed. In addition, knowledge of the relationship between the effective modulus and the porous structure can help improve the design or reduce the weight while maintaining the necessary -129- strength of porous materials. For these reasons, an analytical model to predict the effective modulus based on the porous structure is desirable. Porous structures often include microcracks in addition to pores. The effect of pores and that of microcracks on the effective modulus have been previously analyzed. The non-interaction approximation and self-consistent scheme methods were initially developed [4,5]. Those methods have too large an error in the case of materials with large porosity, and hence more advanced methods have been sought. Currently available advanced methods mainly include the differential scheme [6,7], the Mori-Tanaka effective field method [8-10], the Hashin-Shtrikman variation principle [11], the R & A composite sphere method [12], and the generalized self-consistent method [13]. For the Hashin-Shtrikman variation principle, R & A composite sphere and generalized self-consistent methods, the available results are limited to cases where the shape of the pore is spherical. In practice, however, the pores often have shapes close to ellipsoids. For the differential scheme and Mori-Tanaka effective field methods, results are available for ellipsoid shaped pores [14]. In addition, the differential scheme and Mori-Tanaka effective field methods are also used to model the effects of microcracks on the effective modulus of a cracked body [4,10,15]. Those results, however, are limited to the individual effects of pores and microcracks on the effective moduli of porous materials. Kachanov et al. [4] and Sevostianov et al. [16] investigated the combined effect of pores and microcracks on the effective Young’s modulus of porous materials, and their solutions are based on the Mori-Tanaka method. -130- In this work, the combined effect of pores and microcracks on the effective elastic modulus is investigated for a ceramic wall material from a diesel particular filter (DPF) used in diesel engines. In order to characterize the 3D microstructure, the 2D transverse image measurement method used by Sevostianov et al. [16] is employed. The micromechanics analysis methods, Mori-Tanaka method and differential-scheme method, are applied to predict the effective Young’s modulus. For Mori-Tananka method, the solutions obtained by Sevostianov et al. [16] is used; while for differential-scheme method, the results on the individual effects of pores and microcracks [14,15] are combined together as shown in Fig. 5.1. Both methods are used to predict the Young’s moduli of two types of DPF materials with the same component but different microstructures. Based on the experimental results, it is determined that the predictions overestimate the effective Young’s modulus of porous materials. Possible reasons for this mismatch may be attributed to (1) the idealized pore and crack shapes, and (2) the connected pores and cracks due to high porosity. -131- 5.2 Microstructure of porous material 5.2.1 Microstructure parameters In order to quantitatively characterize the porous structure, some simplifications are needed. Sevostianov et al. [16] simplified the microstructure by assuming pores to be oblate spheroid in shape and microcracks to be penny-shaped planar cracks, and used three parameters (porosity p, aspect ratio γ and crack density ρ) to describe the simplified microstructures. In this work, the same simplified method is employed, and the definitions of the three parameters are shown in Table 5.1. The porosity p is defined as the ratio of the volume of both pores and microcracks to that of the overall material (solid+pores+microcracks). The aspect ratio γ is used to describe the oblate spheroid shaped pore and defined as the ratio of the semiminor axis to the semimajor axis. The microcracks are described by the crack density ρ, which is defined as the sum of the cube of crack length normalized by the total volume of the overall material. To identify these three microstructure parameters of a porous material, knowledge of the 3D internal structure is required. However, it is very difficult to measure the 3D internal structure directly. An alternative method is to obtain the information by measuring 2D cross section images [17-19]. In 2D cross section images, the porosity p2D is defined as the area ratio of pores to the whole cross section. The aspect ratio γ2D is defined as the ratio of the semiminor axis to the semimajor axis -132- of an ellipse and the crack density ρ2D is defined as the sum of the square of crack length normalized by the total cross sectional area. The relationship between the parameters of a 3D solid and those of the 2D cross sections has been investigated previously [17-19]. When a large number of arbitrarily chosen cross sections are measured, the average value of the parameters identified from the 2D cross sections are found [17-19] to be related to the parameters of the 3D solid by ⎧ pp2D =1 ⎪ 2D ⎨γ γπ= 4/ . (5.1) ⎪ 2D ⎩ ρρ =1 The 2D definitions of the microstructure parameters (p2D, γ2D and ρ2D) are summarized and compared with those in 3D (p, γ and ρ), as shown in Table 5.1. In a porous material, pores often have different shapes and their aspect ratios, γ’s, are different. The average aspect ratio is generally used to characterize shapes of 2D 2D pores. Let γ i and Ai respectively represent the aspect ratio and the area of i-th ellipse measured in the 2D image as follows 2D Aabiii= π 2D , (5.2) γ iii= ab where ai and bi respectively represent the length of the semiminor axis and the semimajor axis of the i-th ellipse. Then the average aspect ratio for the 3D porous material, γ , can be calculated [14, 17, 18] as 22D D 4 Aiiγ γ = ∑i , (5.3) π A2D ∑i i where 4/π is the conversion factor between the 2D and 3D aspect ratios, as shown in -133- Table 5.1. 5.2.2 SEM measurement In this work, 2D SEM backscatter images are used to measure the microstructure of porous materials. The material sample is arbitrarily cut and polished. SEM backscatter images are then taken from more than 10 different locations of each sample. The commercial software Clemex Visual PE is used to automatically identify the pores and microcracks, and to fit the pores to the closest approximation of the ellipses. One example of the SEM image is shown in Fig. 5.2. The porosity p, average aspect ratio γ and crack density ρ, are first measured on each image, and then the average value is calculated. It is important to note that when the aspect ratio and crack density are measured, the microcracks need to be distinguished from pores. Kachanov and Sevostianov [20] have demonstrated that when the aspect ratio of oblate pores is smaller than 0.1-0.15 the crack density rather than the porosity becomes the characterizing parameter. In this work, the value of 0.15 is used as the threshold to distinguish pores and cracks. That is, if the aspect ratio of a void is smaller than 0.15, it is treated as a cracks; otherwise, it is treated as a pore. -134- 5.3 Micromechanics models In this work, the micromechanics based Mori-Tanaka and differential scheme methods are respectively employed to model the relationship between the effective elastic modulus and the microstructure (pores and microcracks) of porous materials. Both pores and microcracks are assumed to be randomly distributed. Therefore, the equivalent homogeneous solid representing the porous material is isotropic with Young’s modulus E and shear modulus G. In addition, the cracks are assumed to be open under tensile loads and closed under compressive loads. The effect of closed cracks on the modulus is assumed to be negligible, and only tensile loads are considered in this work. 5.3.1 Mori-Tanaka method The Mori-Tanaka method was proposed by Mori and Tanaka [8] with the basic idea to study the average internal image stress in the matrix of a material containing inclusions (such as pores and microcracks) with transformation strain. This method is appropriate when the mutual positions of pores and microcracks are random [4]. In this section, the work based by Benveniste [10], Kachanov et al. [4], Prokopiev and Sevostianov [14] are first summarized for the individual effects of pores and -135- microcracks on the effective elastic moduli. Then a closed form equation proposed by Kachanov et al. [4] and Sevostianov et al. [16] is presented for the combined effect. 5.3.1.1 Individual effects of pores and microcracks on the effective elastic moduli. The influence of the pores on the elastic modulus of a porous material has been widely investigated by the Mori-Tanaka method. By assuming the pores to be oblate spheroid in shape, Prokopiev and Sevostianov [14] obtained the analytical expressions for the effective Young’s modulus E and shear modulus G as follow: −1 ⎧ Ep⎡ ⎤ ⎪ =+⎢1(,)αγν0 ⎥ ⎪ Ep0 ⎣ 1− ⎦ , (5.4) ⎨ −1 ⎪ Gp⎡ ⎤ ⎪ =+⎢12(,)βγν0 ⎥ ⎩Gp0 ⎣ 1− ⎦ where ⎧ (1+++++ν )(28hh 14 24 hhh 12 8 ) αγν(, )= 01 2 3 56 ⎪ 0 30 ⎨ , (5.5) 2114hhhhh+−++ 8 2 ⎪ βγν(, )= 12356 ⎩⎪ 0 30 ()κ ff− 1 and h = 01 , h = , 1 ⎡⎤2 2 21−− (2κ )f −f 2(4κκ−−−− 1)⎣⎦ 2(f01ff ) (4 κ 1) 0 []01 −−+(2κ ff 2 f ) 1 hh== 00 1 , h = , 34 ⎡⎤2 5 f + 4 f 4(4κκ−−−− 1)⎣⎦ 2(f01ff ) (4 κ 1) 0 01 4162κ −−kf + f − 2 f γ 2 (1− g ) h = 001 , f = , 6 ⎡⎤2 0 2(γ 2 − 1) 4(4κκ−−−− 1)⎣⎦ 2(f01ff ) (4 κ 1) 0 -136- κγ 2 1 1−γ 2 fg=+−⎡⎤(2γ 2 1) 3 , g = arctan , κ =1/(2− 2ν ). 1 22⎣⎦ 0 4(γ − 1) γγ1− 2 γ The Young’s modulus, shear modulus and Poisson’s ratio of the base material are respectively represented by E0, G0 and ν 0 . The porosity is denoted by p, and γ is the average aspect ratio for the 3D oblate spheroids. Benveniste [10] and Kachanov et al. [4] determined the effect of microcracks on the elastic modulus based on the Mori-Tanaka method as follows −1 ⎧ E ⎧ 8(1−−νν2 )(10 3 ) ⎫ ⎪ =+⎨1 ρ 00⎬ ⎪ E00⎩⎭45(1−ν / 2) ⎨ −1 , (5.6) ⎧ ⎫ ⎪ G ⎪ ⎡32(1−−νν00 )(5 )⎤⎪ ⎪ =+⎨1 ρ ⎢ ⎥⎬ G 45(2−ν ) ⎩ 00⎪⎩⎭⎣ ⎦⎪ where the parameter ρ is the crack density. 5.3.1.2 Combined effect of pores and microcracks on the effective elastic moduli. Kachanov et al. [4] investigated the effects of pores and microcracks on the effective elastic modulus of porous materials, and obtained an equation for the case of spheroid shaped pores and penny shaped microcracks via Mori-Tanaka method. Following a procedure similar to that used by Kachanov et al. [4], Sevostianov et al. [16] obtained an equation for the combined effect of oblate spheroid shaped pores and penny shaped cracks on the effective modulus -137- −1 ⎧ 2 E ⎧⎫8(1−−νν00 )(10 3 ) ⎪ =+⋅⎨⎬1(,)p αγν0 + ρ ⎪ E00⎩⎭45(1−ν / 2) ⎨ −1 . (5.7) ⎧⎫ ⎪ G ⎪⎪⎡32(1−−νν00 )(5 )⎤ ⎪ =+⋅⎨⎬12(,)p βγν0 + ρ⎢ ⎥ G 45(2−ν ) ⎩ 00⎩⎭⎪⎪⎣ ⎦ In the equation of Sevostianov et al. [16], the interaction between pores and microcracks are neglected. Under the framework of Mori-Tanaka method [4,16], the average stress field is averaged over the solid phase and is related to the macroscopic stress by a factor (1-p)-1. To include the interaction among pores and microcracks under the framework of Mori-Tanaka method [4,16], the parameters p and ρ in Eq. (5.7) need to be multiplied by the factor (1-p)-1, and the resultant equation is −1 ⎧ 2 Ep⎧⎫ρ 8(1−−νν00 )(10 3 ) ⎪ =+⎨⎬1(,)αγν0 + ⎪ Ep00⎩⎭1145(1/2)−−− pν ⎨ −1 . (5.8) ⎧⎫ ⎪ Gp⎪⎪ρ ⎡32(1−−νν00 )(5 )⎤ ⎪ =+⎨⎬12(,)βγν0 + ⎢ ⎥ Gp1145(2)−−− pν ⎩ 00⎩⎭⎪⎪⎣ ⎦ 5.3.2 Differential scheme method Differential scheme method is to study the effect of an infinitesimal concentration of inclusions (pores or microcracks) on the variation of the effective modulus [7]. This method often leads to a procedure of solving differential equations. In this section, the prior work by Hashin [15], Prokopiev and Sevostianov [14] is first summarized for the individual effects of pores and microcracks on the effective elastic -138- moduli. Then a closed form equation for the combined effect is obtained by the combination of the prior solutions of Hashin [15] and Prokopiev and Sevostianov [14]. 5.3.2.1 Individual effects of pores and microcracks on the effective elastic moduli. For the individual effect of the pores on the elastic modulus, Prokopiev and Sevostianov [14] assumed the pores to be oblate spheroid in shape and generated an analytical equation. Because an accurate solution to the analysis is computationally difficult, Prokopiev and Sevostianov [14] further simplified the form of their results for practical applications. Their simplified model is employed in this work as ⎧ EE≅−(1 p ) αγν(,0 ) 0 , (5.9) ⎨ 2(,)βγν0 ⎩GG≅−0 (1 p ) where the functions α and β are defined by Eq. (5.5). The error of the model due to simplification is within 1% when the aspect ratio of the ellipsoid is larger than 0.1. Hashin [15] determined the effect of microcracks on the elastic modulus as follows. Let ν 0 represent the Poisson’s ratio of base material. For a material with crack density ρ , the effective Poisson’s ratio ν is obtained by solving the equation 515145153ν −ν +−νν ρ =+ln0 ln + ln + ln . (5.10) 8ν 64 1−ννν00 128 1+− 128 3 0 Then the effect Young’s modulus E and shear modulus G are calculated by -139- ⎧ 10/9 1/9 E ⎛⎞ν ⎛⎞3−ν 0 ⎪ = ⎜⎟⎜⎟ ⎪ E νν⎝⎠3− ⎨ 00⎝⎠ . (5.11) ⎪ GE1+ν 0 ⎪ = ⎩ GE001+ν 5.3.2.2 Overall effect of pores and microcracks on the effective elastic moduli. Based on knowledge of the individual effects of pores and microcracks, their mixed effect including their interactions can now be modeled. When combining the effects of pores and microcracks, two issues need to be considered in regard to the differential scheme method. First, the differential scheme model will generate a path dependent result [22] when the effects of two factors are considered. For instance, the results for the following two paths will be different. Path 1 is to first consider the effect of pores and then add the effect of microcracks, and path 2 is to first consider the effect of microcracks and then add the effect of pores. The difference in results is not physically acceptable, and is an inherent flaw in the differential scheme model [22]. In this work, in order to evaluate the error due to this inherent flaw, the above two different paths are taken into consideration and their results are compared. The other issue we need to consider is the relative crack density when consider the pores and microcracks separately. For path 1, when considering the individual effect of microcracks, the relative crack density is not the same as the overall crack -140- density ρ . This is because that the microcracks exist only in the solid phase which occupies (1-p) of the overall volume. For this case, the relative crack density is (ρ ⋅−=−VpVp )/(1)/(1)ρ . However, when considering the individual effect of pores, the relative porosity is still p since the volume of microcrack is neglected. For the same reason, the relative crack density in path 2 is ρ /(1− p ) for the individual effect of microcracks as well. With these two considerations addressed, we now can model the overall effect of pores and microcracks on the elastic modulus. For path 1, we first consider only the effect of pores, which leads to a transition material of effective moduli (Et, Gt) based on Eq. (5.9): ⎧ EE≅−(1 p )αγν(,0 ) t 0 . (5.12) ⎨ 2(,)β γν0 ⎩GGt ≅−0 (1 p ) The Poisson’s ratio of the transition material can be obtained by the relationship Et ν t = −1. (5.13) 2Gt Then the additional effect of microcracks is applied to the transition material (Et, Gt, ν t ) via Eqs. (5.10) and (5.11) to generate the overall effective Young’s modulus E, shear modulus G and Poisson’s ratio ν : -141- ⎧ ρ 515145153ν −ννν+− =+lnt ln + ln + ln ⎪1−−+−p 8ν 64 1ννν 128 1 128 3 ⎪ tt t ⎪ 10/9 1/9 ⎪ E ⎛⎞ν ⎛⎞3−ν t ⎨ = ⎜⎟ ⎜⎟ . (5.14) ⎪ Ett⎝⎠νν⎝⎠3− ⎪ GE1+ν ⎪ = t GE1+ν ⎩⎪ tt For path 2, we first consider only the effect of microcracks, which leads to a transition material of effective moduli (Et, Gt, ν t ) based on Eqs. (5.10) and (5.11), ⎧ ρ 515455ν 113−ννν+− =+ln0 lntt + ln + ln t ⎪1−−+−p 8ν 64 1ννν 128 1 128 3 ⎪ t 00 0 10/9 1/9 ⎪ E ⎛⎞⎛νν3− ⎞ ⎨ tt= ⎜⎟⎜0 ⎟ . (5.15) ⎪ E00⎝⎠⎝νν3− t ⎠ ⎪GE1+ν ⎪ tt= 0 GE1+ν ⎩⎪ 00t Then the additional effect of the pores is applied to the transition material (Et, Gt, ν t ) via Eq. (5.9) to generate the overall effective moduli (E, G), ⎧ EE≅−(1 p )αγν(,t ) t . (5.16) ⎨ 2(,)β γνt ⎩GG≅−t (1 p ) Since the simplified form of Prokopiev and Sevostianov [14] is used, our results (for path 1 and path 2) will have a maximum error within 1% as long as the aspect ratio of the oblate spheroid is larger than 0.1. -142- 5.4 Application The models summarized in Section 3 were applied to predict the Young’s modulus of a diesel particulate filter (DPF) wall material, which was a porous ceramics with porosity of around 50%. Two types of materials from the same manufacturer, Corning, denoted by sample AC and sample CO, were used. These two samples were the same in components and porosity, but their microstructures and elastic moduli were different. The Young’s modulus of sample AC was much smaller than that of sample CO. The Young’s moduli of the base materials (p=0) of both samples were not available, but we can assume they were the same according to the manufacturer’s report. In the analysis, E0 was used to represent the Young’s modulus of the base materials for both samples. Their microstructures were first characterized through 2D SEM cross section measurement; the results (porosity p, aspect ratio γ and crack density ρ) are summarized in Table 5.2. The porosities of both samples were identical and their aspect ratios were very close to each other. However, the crack density of sample AC was much higher than that of sample CO. Based on the microstructure parameters, the Young’s modulus was predicted by the Mori-Tanaka and differential-scheme methods, respectively. The predicted Young’s modulus E normalized by E0 is shown in Table 5.2. The predictions for path 1 and path 2 based on differential-scheme method were almost identical. Therefore, -143- the path dependent effect can be neglected for this case. The reported experimental measurements of Young’s moduli (unit: GPa) are also summarized in Table 5.2, so as to compare them against the predicted results. According to the micromechanics model, in either the Mori-Tanaka or the differential-scheme method, the Young’s modulus will decrease as the crack density increases for a specific value of porosity and aspect ratio. As shown in Table 5.2, the predicted Young’s modulus of sample AC was much smaller than that of sample CO, and this agrees with the experimental data. The materials AC and CO are composed mainly of O (wt50%), Mg (wt8%), Al (wt16%) and Si (wt23%) based on reported experimental results. Ceramics of similar composition, such as glass, cordierite (2MgO-2Al2O3-5SiO2) and glass-ceramics, often have Young’s modulus in a range of 60-150GPa [23-25] when the porosity is close to zero. In order to compare the prediction with the reported experimental results, the value of E0 is chosen to be 60GPa and then the experimental results of Young’s modulus is normalized by E0 as shown in Table 5.2. The prediction of E/E0 is found to be much larger than the experimental results. The overestimation of the effective Young’s modulus is caused by the dilute assumption of both the differential scheme and Mori-Tanaka methods, that is pores and microcracks are assumed to be separate and far away from each other. This assumption is valid when the density of pores and microcracks is small. In previous study of Sevostianov [14] on the effective Young’s modulus of porous material, the -144- possibility to replace irregular shaped porous space by isolated spheroidal pores is examined and shown to be reasonable when the porosity is small. When measuring the microstructure parameters (p,γ and ρ) in this work, the connected pores and microcracks are simply separated and the influence of the connections on the effective modulus is neglected as well. For the samples AC and CO, however, the porosity is large and connected pores and microcracks are commonly observed in the cross section image as shown in Fig. 2. As a result, the effective Young’s modulus, E/E0, are overestimated. In the future, we may treat the connected pores and microcracks as one pore or microcrack with an equivalent shape. 5.5 Summary Porous materials are widely observed or applied in medical and industrial fields. When assessing the integrity of a structure containing a porous material, the porous material is often treated as a homogeneous material with the effective properties. Knowledge of the relationship between the effective modulus and the microstructure of the porous material is very important since it can be used to predict the effective properties and/or improve the design of materials. In this work, the porous materials are assumed to include oblate spheroid shaped pores and penny shaped microcracks, and the microstructures was characterized by the -145- three parameters of porosity p, aspect ratio γ and crack density ρ. The micromechanics method, the differential scheme and Mori-Tanaka methods, were used to study the relationship between effective elastic modulus and the microstructure of porous materials. The methods were employed to predict the Young’s moduli of two diesel particulate filter wall materials, which were ceramic materials with large porosity. Based on the experimental results, it is determined that the predictions overestimate the effective Young’s modulus of porous materials. 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Phys. Solids, 1990;38(3):379-404. [14] Prokopiev, O. and Sevostianov, I. On the possibility of approximation of irregular porous microstructure by isolated spheroidal pores. International Journal of Fracture 2006;139:129-136. [15] Hashin, Z. The differential scheme and its application to cracked materials, J. Mech. Phys. Solids, 1988;36(6):719-734. [16] Sevostianov, I., Gorbatikh, L. and Kachanov, M. Recovery of information on the microstructure of porous/microcracked materials from the effective elastic/conductive properties. International Journal of Materials Science and Engineering A, 2001;318:1-14. [17] Laraia, V.J., Rus, J.L., and Heuer, A.H. Microstructural shape factors: relation of -147- random planar sections to three-dimensional microstructures, J. Amer. Ceram. Soc., 1995;78:1532-1536. [18] Kibbel, B., Heuer, A. Anisometric shape factors for microstructures. Journal of American Ceramic Society, 1989;72:517-19. [19] Sevostianov, I., Agnihotri, G. and Garay, J.F. On connections between 3-D microstructures and their 2-D images. International Journal of Fracture 2004;126:65-72. [20] Kachanov, M. and Sevostianov, I. On quantitative characterization of microstructures and effective properties. International journal of solids and structures 2005;42:309-336. [21] Sevostianov, I., Gorbatikh, L. and Kachanov, M. Recovery of information on the microstructure of porous/microcracked materials from the effective elastic/conductive properties. International Journal of Materials Science and Engineering A, 2001;318:1-14. [22] Nemat-Nasser, S. and Hori, M. Micromechanics: overall properties of heterogeneous materials. Published by North-Holland, c1999, 2nd ed. P.361-367, P373. [23] Lee, W.E. and Rainforth, W.M. Ceramic Microstructures: Property Control by Processing, Chapman & Hall, London, 1994. [24] Shao, H. Liang, K., Zhou, F., Wang, G. and Hu, A. Microstructure and mechanical properties of MgO-Al2O3-SiO2-TiO2 glass-ceramics. Materials Research Bulletin 2005;40:499-506. [25] Dimitrijevic, M.; Posarac, M.; Majstorovic, J.; Volkov-Husovic, T. and Matovic, B. Behavior of silicon carbide/cordierite composite material after cyclic thermal shock. Ceramics International 2009;35(3):1077-81. -148- Figures Porous material Equivalent homogeneous solid (E0 represents modulus of base material) (E is the effective modulus) Analytical Model E0 E (a) Equate a porous material by a homogeneous solid. Homogeneous solid (b) The effect of pores on the effective moduli. Homogeneous solid (c) The effect of microcracks on the effective moduli. Figure 5.1. 2D schematics of equating a porous material using homogeneous solid with effective properties (a) through the combination of individual effects of pores (b) and microcracks (c). -149- lj approximated ρ=(Σ l 2) / A by an ellipse 2D j j total cross sectional area 2ai γ i = ai/bi lj is the length of crack j 2bi Ai = πaibi Figure 5.2. An example of microstructure identification on 2D SEM image. The dark parts represent the pores and microcracks, and the white parts are the base materials. -150- Tables Table 5.1. Definition of microstructure parameters. 3D structure 2D image Ratio (3D/2D) ()VVpore+ crack ( AApore+ crack ) ∑ ∑ Porosity p ~1 ∑()VVpore++ crack V solid ∑()AApore++ crack A solid (V : volume) ( A : area) a/b a/b a a Aspect ratio γ b b ~ 4/π 3 2 ∑lcrack ∑lcrack Crack density VV++ V AA++ A ~1 ρ ∑()pore crack solid ∑()pore crack solid (l :length ; V : volume) (l :length ; A : area) -151- Table 5.2. Microstructure identification results and effective Young’s moduli of AO and CO, DPF wall materials, subjected to tensile loadings. The values in parenthesis represent the standard deviation. SEM image identified Experiments Predicted E/E0 Correspond E/E 0 DS (1,2 to p γ ρ E (GPa) (E =60GPa, MT 0 same) assumed) 0.015 5.3 0.301 0.208 CO 53% 0.89 0.089 ±0.007 ±0.22 (0.299-0.304) (0.202-0.213) 0.09 3.5 0.28 0.16 AC 53% 0.87 0.058 ±0.036 ±0.15 (0.27-0.29) (0.14-0.18) -152- APPENDIX A Details of FEA for effective layer determination The FEA determination of the effective layer parameters were obtained by equating SIFs for trilayer and bilayer cracked domains as illustrated in Fig 3.2 in Chapter 3. A.1 Evaluation of FEA computation model For FEA model validation, the dependence on both finite element mesh and finite body size on the SIF were analyzed. It was established that, in the range of parameters 1/10 200×200×200 with R=1 and the mesh size less than 0.01 (R=1) in critical regions limit all the numerical errors to within 1%. The FEA computation model is further examined by considering two extreme cases. By specifying EII=EIII in the trilayer model all the results for the cracked bilayer domain analysis (see Fig. 2.2(d) in Chapter 2) were recovered. By simulating the cases with extremely soft middle and bottom layers (EI/EII and EI/EIII -153- larger than 50), the SIF of a surface crack in a semi-infinite homogeneous material [52,85] (Fig. 2.2(c) in Chapter 2) was recovered. A.2 Error related to the determination of effective modulus Since the effective material is used to replace middle layer II and bottom material III, the value of Eeffective should be between EII and EIII. However, when EII/EIII<1/10 and h/R>1, the numerically calculated Eeffective is found to be slightly smaller than EII in some cases. This underestimation of the Eeffective is attributed to the numerical error (maximum 1%) in the FEA analysis of the cracked trilayer domain and the error (maximum 2%) in the curve-fitted function of the factor fA,bilayer. In order to avoid these errors, the values of Eeffective are chosen to be those of EII for these cases. In the analysis of a cracked bilayer domain subjected to constant tensile loading (Fig. 2.2(d) in Chapter 2), as EI/EII becomes extremely large, the value of fA,constant approaches a well known limiting value of fA,constant(∞)=0.65. For ln(EI/EII) larger than 3.9, the percent error of fA,constant relative to fA,constant(∞) is comparable to the maximum error of the FEA calculations (1%). In this case, for a specific value of fA,trilayer, a wide range of Eeffective values can be utilized as possible solutions to Eq. (3.4). Since unique determination of Eeffective becomes difficult in this extreme case -154- it was excluded from the curve-fitting procedure and only the data in the range of 1/10 The accuracy of the obtained curve-fitted function in estimating the SIF was verified in the range of 1/100 -155- APPENDIX B Fracture mechanics based failure probability prediction model B.1 Statistical theory for failures of brittle materials In order to analyze strength variation of brittle materials, Batdorf et al. [6] introduced the notion of solid angle Ω containing the normals to all directions for which the normal stress is larger than the critical stress σ Cr . The solid angle Ω varies from zero to 2π for surface cracks based on the definition. For Ω PfCr= Ω(,σ σπ )/2 , (B.1) where σ is the applied stress. For Ω /2π = 1, propagation of a crack is independent of its orientation and it is solely determined by the size of the crack [91]. Combining the two cases (Ω (σ ,σπCr )< 2 and Ω (σ ,σπCr )= 2 ), Wang et al. [91] derived the general expression of failure probability for surface cracks as follows. -156- III PPPfff=−11()() − 1 − σ Max I ⎧⎫Cr dN ()σ Cr PddAfCrCr=−1exp⎨⎬Min ln1⎡⎤ −Ω()σ ,σπ /2 σ , Ω< 2 π, (B.2) ∫∫A σ ⎣⎦ Cr ⎩⎭dσ Cr M ⎧⎫σCr dN ()σ PddAII =−1exp −Cr σπ , Ω= 2 fCr⎨⎬∫∫A 0 ⎩⎭dσ Cr M ax Min where σ Cr and σ Cr , respectively, represent the values of the maximum and minimum critical stress and N (σ Cr ) represents the critical stress density distribution function. The critical stress density distribution function can be assumed to have the form proposed by Chao et al. [11]. m Nk(σσCr) = Cr (B.3) Substituting Eq. (B.3) into Eq. (B.2), the failure probability for surface cracks can be expressed as R σσ ⎧⎫⎡Ω⎤21mm−−11⎛⎞ Pmkrdddrf=−1exp⎨⎬ − 2πσσ CrCr − ln1 − σσ CrCr , (B.4) ∫∫00⎢⎥ ∫σ ⎜⎟ ⎩⎭⎣⎦2 ⎝⎠2π where R is the radius of the effective area (support ring radius for ball-on-ring test, and sample radius for indentation test) and −1 ⎡2σ Cr −−σσ12⎤ Ω=2cos ⎢ ⎥ . (B.5) ⎣ σσ12− ⎦ -157- B.2 Calculation of parameters m and k from biaxial experimental data The statistical parameters m and k for the critical stress density distribution m function Nk()σ Cr= σ Cr (Eq.(B.3)) can be calculated by curve-fitting the biaxial experimental data. In biaxial flexural tests, the stress distribution at the bottom of the disk is proportional to Pt/π 2 within the linearly elastic theory, where P and t, respectively, represent an applied load and thickness. If we introduce a non-dimensional factor I D as σ 2 ⎡ 2 m−1 1 ⎛⎞r ⎢ Pt/π ⎛⎞⎛⎞σσ Id= 2π Cr Cr D ∫∫⎜⎟⎢ ⎜⎟⎜⎟22 rR/0= ⎝⎠RPtPtσ ⎝⎠⎝⎠//ππ ⎢ Cr =0 ⎣ Pt/π 2 , (B.6) σ1 2 m−1 ⎤ Pt/π ⎛⎞Ω ⎛⎞⎛⎞σσ⎥ ⎛⎞r −−ln 1 Crdd Cr ∫ ⎜⎟⎜⎟⎜⎟22⎥ ⎜⎟ σ σ ⎝⎠2/ππ⎝⎠⎝⎠Pt Pt / π ⎝⎠ R Cr = 2 ⎥ Pt//ππ22 Pt ⎦ and B defined by 2 ⎡ R mkI D ⎤ B = ln ⎢ 2 m ⎥ , (B.7) ⎣ ()πt ⎦ then ID and B are independent of the load P for biaxial test. They can be calculated if the dimensions of the samples are known. Correspondingly, the failure probability in Eq. (B.4) can be expressed as PeP=−1exp −B m , (B.8) f ( ()) or -158- ln⎡⎤− ln 1−=PmPB ln +. (B.9) ⎣⎦( f ) () If we plot the biaxial experimental data using ln⎡−− ln 1 P ⎤ and ln P , m ⎣ ( f )⎦ () and B can be determined by curve-fitting the data point to a straight line for Eq. (B.9), and k can be calculated by Eq. (B.7). -159- BIBLIOGRAPHY [1] Anusavice KJ, Hojjatie B. Tensile stress in glass-ceramic crowns: Effect of flaws and cement voids. Int J Prosthodont. 1992;5:351-358. 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