Measurement of J-Integral Values of Dental Ceramics by Digital Image Correlations

Home , Fracture mechanics, Stress intensity factor

MEASUREMENT OF J-INTEGRAL VALUES OF DENTAL CERAMICS BY

DIGITAL IMAGE CORRELATION

YANXIA JIANG

Submitted in partial fulfillment of the requirements

for the degree of Master of Science

Thesis Adviser: Dr. Ozan Akkus

Department of Mechanical Engineering

CASE WESTERN RESERVE UNIVERSITY

May, 2016

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

Yanxia Jiang

candidate for the Master of Science degree*.

(signed) Dr. Ozan Akkus

(chair of the committee)

Dr. Anna Akkus

Dr. Bo Li

(date) 1/8/2016

*We also certify that written approval has been obtained for any proprietary material contained therein.

Table of contents

List of Tables...... I

List of Figures ...... II

Acknowledgements ...... IV

Abstract ...... V

Chapter 1 Introduction ...... 1

1.1 Fracture Mechanics ...... 1

1.1.1 Griffith energy release rate ...... 1

1.1.2 Stress intensity factor theory ...... 3

1.1.3 CTOD theory ...... 5

1.1.4 J-integral ...... 7

1.2 Dental crowns ...... 11

1.2.1 A review of dental ceramics ...... 13

1.3 Digital Image Correlation (DIC) ...... 16

1.3.1 Basic principle ...... 16

1.3.2 DIC for fracture analysis ...... 17

2.1 Objectives ...... 20

2.2 Materials and Methods ...... 20

2.2.1 Materials ...... 20

2.2.2 Sample preparation ...... 21

2.2.3 Strain-stress relations ...... 22 2.2.4 Three-point bending test and DIC set up ...... 23

2.2.5 J-integral calculation (formula, integral path, path selection) ...... 24

2.2.6 Conventional calculation of KIC ...... 25

2.2.7 Finite element model ...... 25

2.2.8 Statistical analyses ...... 26

Chapter 3 Results ...... 28

3.1 Strain-stress relations ...... 28

3.2 FEM ...... 28

3.3 Integral path size ...... 29

3.4 Integral path location ...... 30

3.5 Critical stress intensity factor ...... 32

Chapter 4 Discussion and Conclusion ...... 33

Appendix A ...... 35

Appendix B ...... 39

References ...... 84

List of Tables

Table 1: Ceramics materials ...... 43

Table 2: Composite of Vita Mark II ...... 44

Table 3: Composite of IPS E.MAX ...... 45

Table 4: Material properties of IPS E.MAX blue, E.MAX, VM II and MZ 100 .. 46

Table 5: Average strain component in the integral path located area ...... 47

Table 6: J-integral of full integral path and partial integral path ...... 48

Table 7: KIC , 퐾퐽 and Kreported for MZ 100, E-MAX, VM II and E-MAX blue . 49

I List of Figures

Figure 1: The infinite plate geometry in Inglis approach...... 50

Figure 2: Angular coordinate...... 51

Figure 3: Effective crack length of Irwin model ...... 52

Figure 4: Effective crack length of Dugdale model ...... 53

Figure 5: A notch lays parallel to x axis, Γ is an arbitrary curve surrounding the

crack tip, and n is the normal direction of Γ ...... 54

Figure 6: HRR field and K field ...... 55

Figure 7: Geometry for J-integral path-independency...... 56

Figure 8: A closed curve used to prove the path-independency of the J-integral. 57

Figure 9: Specimen geometries used in fracture mechanics tests...... 58

Figure 10: Crown replacement procedure...... 59

Figure 11: Schematic diagram depicting various fracture modes in crown...... 60

Figure 12: The CEREC system...... 61

Figure 13: Micro structure of three different types of ceramics ...... 62

Figure 14: Schematic diagram of DIC matching algorism...... 63

Figure 15: Sample preparation process...... 64

Figure 16: Speckle pattern...... 65

Figure 17: Experimental set up...... 66

Figure 18: Light position with respect to sample...... 67

Figure 19: Rectangular integral path used to calculate J-integral...... 68

Figure 20: Sample dimensions...... 69

Figure 21: Boundary conditions in FEM...... 70

Figure 22: Strain-stress relations of MZ 100, E-MAX blue, and VM II ...... 71

Figure 23: Load-displacement curve of MZ 100, E-MAX blue, VM II and E-MAX

notched beam sample under a three-point bending test...... 72

Figure 24: Comparison of FEM and DIC strain component a) εxx, b) εyy, and c) εxy.

...... 73

Figure 25: Points position used for L2 norm...... 74

Figure 26: εxx mapping of MZ 100, E-MAX blue, VM II and E-MAX before

fracture ...... 75

Figure 27: εyy mapping of MZ 100, E-MAX blue, VM II and E-MAX before

fracture ...... 76

Figure 28: 휀푥푦 mapping of MZ 100, E-MAX blue, VM II and E-MAX before

fracture ...... 77

Figure 29: Integral paths for a different size...... 78

Figure 30: J-integral for different integral path size...... 79

Figure 31: KJ calculated from J-integral for different integral path size...... 80

Figure 32: Path integral with the same size at different locations...... 81

Figure 33: J-integral of MZ 100, E-MAX blue, VM II and E-MAX from different

integral path location...... 82

Figure 34: KJ of MZ 100, E-MAX blue, VM II and E-MAX from different integral

path location...... 83

III

Acknowledgements

I would like to thank my family for unconditional support both financially and spiritually throughout my degree.

I would like to express my deepest gratitude to my supervisor Dr. Ozan Akkus for insightful comments, guidance, encouragement and engagement though the research and the learning process of this master thesis. When my steps faltered, he is always there to help. I am also thankful to Dr. Anna Akkus for her valuable knowledge and experience that enlighten me throughout the project. My sincere thanks also go to the rest of my thesis committee, Dr. Bo Li, for his support and critical improvement suggestions.

I am grateful to Dr. Roperto and Dr. Porto for provision of expertise, technical support and sample preparation that greatly assisted the research, although they may not agree with all of the interpretations/conclusions of this paper.

Last but not the least, I want to thank all the labmates and friends for training me using the facilities and offering help whenever I have difficulties. I am grateful to have the chance to work with them.

Measurement of J-integral Values of Dental Ceramics by Digital Image Correlations

Abstract

YANXIA JIANG

J-integral can be used to characterize the crack initiation and propagation process and

requires a smaller sample dimensions compare to stress intensity factor (SIF) based test. Yet no studies have investigated the J-integral of dental restoration materials. The

aim of this study is to measure J-integral value for dental restoration materials with

Digital Image Correlation (DIC) method. Three materials (IPS E-MAX, Vita Mark II and Paradigm MZ100) are used to prepare beam samples for three-point bending test.

And a finite element model of VM II was built to validate the strain accuracy of DIC.

The critical SIF are then calculated from J-integral (KJ) and compared with the ones calculated from geometry dimensions and load at fracture (KIC). Generally 퐾IC has a

higher value than KJ. Yet there is a good consistency between KJ and KIC, thus proving DIC is an effective way to obtain J-integral for dental restoration materials.

Chapter 1 Introduction

1.1 Fracture Mechanics

1.1.1 Griffith energy release rate

Modern fracture mechanics developed in various directions ever since 1920 when

Griffith (Griffith 1921) first developed his theory. Based on Inglis’ (Inglis 1997) linear solutions of infinite plate around an ellipse hole in 1913, he proposed an energy-based failure criterion. He made the assumption that the strain is elastic and validated his theory on glass, which is a brittle material and satisfied this assumption.

According to the theorem of minimum energy, the system tends to be more stable when it has a smaller potential energy, and will reach an equilibrium state when the potential energy is minimum. Therefore, the potential energy of the system will continuously decrease during the crack propagation, while the surface energy will increase due to formation of the new surfaces during crack propagation. When the energy rate of forming a new surface equals the strain energy release rate, the system is accepted to reach the equilibrium state. When the strain energy release rate is higher than the energy rate of forming a new surface, the crack grows; when strain energy release rate is smaller than the energy rate of forming a new surface, the crack is stable.

Surface energy is defined as follows:

Us =4aB (1-1)

B is the thickness of the plate, while γ is surface energy per unit area. Note that crack

1 generates 2 new surfaces, the bottom and top faces of crack edges, when the crack propagating. Following Inglis’s study, Griffith converged the semi-minor axis of the ellipse to zero. The strain energy released of the plate under stress σ because of the spread of the crack is 1 U  a22 B (1-2) 1 E'  E plain stress  Where E'   E plain strain 1 2

E is Young’s modulus, υ is Poison’s ratio. And for a linear material, the original strain energy of the plate under stress σ is

1  2V UV (1-3) 0 22E'

Thus the strain energy with a crack length a is

 2V 1 U U  U    a22 B (1-4) 012EE''

When the system is under equilibrium state, dU dU  s (1-5) da da Then it can be shown that

2E' 2 E' GE' a  ,  c (1-6) c  2 f aa

Gc  2 (1-7) where 푎푐 is the critical crack size under stress σ, and 휎푓 is the stress at failure, 퐺푐 is the critical energy release rate.

Griffith’s theory applies to elastic materials. In 1948, Irwin (Irwin 1948) and Orowax 2

(Orowax 1949) modified Griffith’s work independently. So that under the condition that the plastic deformation zone in the front of the crack tip is smaller than the critical size, it also can be applied to materials that deform plastically. The modified Griffith’s critical energy release rate is

Gcp2 (1-8)

훾푝 is plastic work per unit area. And the critical crack size and stress becomes

E' (2 ) E（' 2 + ) a  p ,   p (1-9) c  2 f a

1.1.2 Stress intensity factor theory

Irwin (Irwin 1997) came to a definition of stress intensity factor based on Westergaard’s

Airy stress function (Westergaard 1939), which is

=ReZ y Im Z (1-10)

Z is an analytic function with respect to z, where z=x+iy. And Z̅ and Z̿ are the first and second integrals of Z. 푍′ is the derivative of Z. Namely,

dZ Z Z Z '  , Z  , Z  (1-11) dz dz dz

Then we can determine the stress in the x and y axis and the shear stress from the Airy stress function:

2   ReZ  y Im Z ' (1-12) x y2

2   ReZ  y Im Z ' (1-13) y x2

2     yZRe ' (1-14) xy xy

For an infinite plate has a crack of 2a in the x axis, under the tension σ, then the function

 Z(z)= (1-15) a 1 ( )2 z can be used to solve the problem. Irwin introduced a variable

z a rei (1-16) where (r, θ) are the polar coordinates with the origin located at the crack tip as shown in figure 2. Substituting (1-16) into (1-15), and considering that 푎2 ≫ 푎푟 ≫ 푟2, all ar and 푟2 terms can be neglected, we can find that (1-15) becomes

 a i Z() z  e 2 (1-17) 2r

And according to Euler identity, eix= cos x +i sin x, (1-17) can be transformed to

a  Z( z ) (cos i sin ) (1-18) 2r 2 2

Then according to (1-12) to (1-14), we can get the normal and shear stresses

 a  3   cos (1 sin sin ) (1-19) x 2r 2 2 2

 a  3   cos (1 sin sin ) (1-20) y 2r 2 2 2

 a  3    cos sin cos (1-21) xy 2r 2 2 2

And for the displacement:

 ar    3   u 1  2  1 cos  cos (1-22) x         2E  2  2   2 

 ar    3   u 1  2  1 sin  sin (1-23) y         2E  2  2   2  4

Where

34  plane strain     3 (1-24) plane stress  1

Irwin noticed that there is a same term σ√πa in those stress expressions above, and defined stress intensity factor as

Ka (1-25)

And when the stress intensity factor is bigger than the critical value KIC, the fracture initiates.

Relation between K and G

From (1-6) we know that,

2 a G  (1-26) E'

Accordingly it can be shown that

K 2 G  (1-27) E'

1.1.3 CTOD theory

The concept of crack tip opening displacement (CTOD) was first proposed by Wells in

1963 (Wells 1963). CTOD theory extended the LEFM to elastic-plastic fracture mechanics. CTOD is mathematically defined as

(r)uy (r,    )  u y (r,     )  2 u y (r) (1-28)

Irwin model:

Irwin (Irwin and Wit 1983) proposed that the plastic zone is needed to be take into account, and an effective crack length is proposed.

2 1 K a a  r  a  I (1-29) eff p  2ys

Substituting (1-23) and (1-29) into (1-28) and defining 훿푡 = 훿(푥 = 푎),

2 4 KI t  plane stress (1-30) E ys

2 2 1 4(1 ) KI t   plane strain (1-31) 3 E y

Dugdale model

Dugdale’s strip yield model suggests that the yielding will occur on a narrow alignment with the crack (Dugdale 1960). And the terms in the stress expression which will lead to singularity should vanish. The effective crack length of this model is:

2  K aa I (1-32) eff  8  ys

And the corresponding CTOD is:

8 a    ys Insec (1-33) t  E 2 ys

When the applied stress σ is much less than the yield stress σys, (1-33) is reduced to

2 KI t  (1-34) E ys

1.1.4 J-integral

In 1968, (Rice 1968) proposed a path-independent integral, which became an important fracture parameter. It is defined as:

u J () Wdy  T  ds (1-35)  x where Γ is an arbitrary curve surrounding the crack tip, starting from one edge of the crack and ending at the other edge, as shown in figure 4. W is the strain energy density,

ij Wd  (1-36)  ij ij 0

T⃑⃑ is the traction vector, and Ti=σijnj, u⃑ is the displacement vector, ds is the small increasement along Γ. This energy perspective way to describe the crack propagation can be applied not only to elastic process but also can be applied to elastic-plastic process.

J dominance

Hutchinson (Hutchinson 1968), and Rice and Rosengren (Rice and Rosengren 1968), independently worked on the crack tip stress field in 1968, now known as the HRR field.

The domain that is valid for the HRR field is J dominance. The valid domain for K solution, (1-12) to (1-14), is K domain. The K solution has a linear assumption that it only be satisfied when away from the plastic zone. Irwin found that the radius of the plastic zone is

2 1 K r  (1-37) p  2ys

Consequently for a valid K based test, we require that a≫rp. ASTM standard E 399

(E399-09 2009) requires

2 K a,b,B 2.5 IC (1-38)   ys

The HRR field is smaller than the K domain field, therefore the sample size requirement is smaller than K based test. For J based test, the HRR solution is valid for the region away from crack tip greatly than the crack tip opening displacement. Calculating

J-integral for the Dugdale model as discussed in CTOD section, Rice and Rosengren obtained the relation of δ and J as follows:

Jm ys (1-39)

Where m is a constant factor between 1 to 2. For plane stress, m=1.

ASTM requires

J a, b , B  25 IC (1-40)  ys

Proof of Path-independency of J-integral

To prove the path independency, an arbitrary closed curve Γ*, and an area A* is the area enclosed by the closed curve is defined as shown in figure 7. From Green’s theory,

(1-35) becomes

W u J() i dxdy (1-41)  ij * x  x  x

Let’s see the first term. Considering (1-36), and assuming small strain, i.e.

1 ε = (u +u ), and neglecting the body force, i.e. σ =0 yields: ij 2 i,j j.i ij,j

WWij ij     ij x ij  x  x 1 u u  i j ij   2 x  xji  x  x  (1-42)

 ui   ij  xxj 

 ui   ij xxj 

When we substitute (1-42) into (1-41):

u  Wdy T  ds  0 (1-43) * x for any closed curve. Considering a closed curve as shown in Figure 8:

   12    (1-44)

JJJJJ     0 (1-45) 12

Because the paths on crack edge, Γ+ and Γ- , there is no traction and dy is zero, therefore J +and J will vanish. Thus (1-45) becomes Γ Γ-

JJ (1-46) 12

And like the SIF based criterion, when J>JIC, a fracture begins.

Methods of measuring the J-integral

As shown in Figure 9, commonly used fracture test specimen types are compact tension

(CT) specimen, single edge notched (SEN) bending specimen and middle-cracked tension (MT) specimen. Generally, crack width ratio should be 0.45< a⁄W <0.55 for

KIC fracture test and 0.45< a⁄W < 0.70 for J and δ based fracture test. Width to depth ratio W⁄B is 2. Alternative ratio may be from 2 to 4 for CT specimen and 1 to 4 for

SEN specimen. And span to width ratio S⁄W is 4. (E1820-01 2001)

In 1973, Rice, Paris et al. (Rice, Paris et al. 1973) showed an easier way to determine

J-integral with a SEN specimen, in the following form:

1 P  1  P J  dp or Jd   (1-47) Ba0  Ba0  where P is the load, and ∆ is the load line displacement. And J-integral found to be the work done to the cracked body, which can farther simplify (1-52) to 2A J  (1-48) Bb where A is the area under the load-displacement curve.

Relationship of J with G and K

Generally when evaluating J-integral, we separate it into elastic and plastic part.

JJJel pl (1-49)

For the elastic part, Jel=G, and from (1-27) we can know that

K 2 J  (1-50) el E'

Jpl can be obtained from the load-displacement curve.

 A J  pl pl (1-51) pl Bb

Where ηpl is a geometry parameter, which is a function of a⁄w and will not change from material to material. Apl is the area under the load-displacement curve. B is the thickness of the sample. For the CT and SEB samples, studies have been done to determine ηpl factor by fitting an approximate function. CT sample’s plastic parameter

ηpl is:

b  2 0.522 (1-52) pl W And for the SEB sample:

 2aW / 0.282  23  pl   a a   a  (1-53) 0.32 12  49.5   99.8   aW /  0.282  WWW   

Numerical approaches also prove to be satisfactory. The finite element analysis (FEA) software in the market, such as ABAQUS (Hu, Liu et al. 2011) and ANSYS, can be used to calculate J-integral with build in CONTOUR INTEGRAL command (Brocks,

Scheider et al. 2001). The results show a path-independent property of J-integral.

1.2 Dental crowns

Caries and fractured teeth are the top reasons for people to have tooth restorations (Lin,

Chang et al. 2010). After examining the tooth root and surrounding bone, a dentist will take an impression of the tooth that is going to receive the dental crown. And the impression of the opposite tooth also will be made to ensure the bite functions properly.

Then the impression will be sent to a dental lab for crown fabrication. This usually will take 2 to 3 weeks. During this time, the dentist will give the patient a temporary crown.

Relatively low fracture strength and fracture toughness, and microcracks introduced by manufacturing process that make fracture of the restoration is one of the main reasons for failure (Chen, Hickel et al. 1999; Sun, Zhou et al. 2014). The real failure mode is complex, it is a mix mode of tension, compression and shear. High loads can cause fracture to initiate from the contact surface. This is the most common fracture mode

11 observed in vitro. Many articles have shown that clinically failed teeth are more likely to have a radical crack that initiate from interface of between crown and cementation and cervical margin (Øilo, Kvam et al. 2013). Alumina and zirconia-based ceramics are stronger than glass-based ceramics, but they are more opaque than glass-based ceramic.

To satisfy the esthetic requirements, an alumina and a zirconia crown is usually bonded with a glass-based ceramic veneer. The mismatch of the thermal expansion coefficient of the veneer and the core, and the weak link of the veneer that lead cohesive failure and chip off the veneer to the main failure reasons for alumina and zirconia-based crown

(Zhang, Sailer et al. 2013; Preis, Dowerk et al. 2014; Sonza, Della Bona et al. 2014).

Recently, the CAD/CAM technology has become popular. With this system, the patient can get their crown in a single appointment. The CAD/CAM system embodies two units:

1.) data acquisition unit with a scanner that can generate 3D model of the crown within seconds. 2.) Manufacturing unit, where two burrs are employed to mill the block into the desired crown. The CEREC (computer-assisted CERamic REConstruction) system

(Sirona Dental Systems, Bensheim, Germany) was the first CAD/CAM system available for dental office (see figure 12). After being developed in 1980, the third generation CEREC III was introduced in 2000. The newest version is able to fabricate various types of dental restorations such as veneer, inlay, onlay, partial crown and full crown. And occlusion adjustment was introduced by the new software in 2005 (Madani

2010). Now more and more open systems are available. In the open system, people gain a greater flexibility to choose each component with a variety of functions. Other scanner options are E4D (D4D Technologies LLC, Richardson, TX), iTero (Cadent), LAVA 12

COS (3M ESPE, St Paul, MN). Unlike the CEREC system, the iTero system does not need any powder. And LAVA COS only requires a lighter amount of powder to locate reference point. This CAD/CAM method also has some limitations. One is that this subtraction method will remove the part from block according to the design. Thus when comparing the final crown and the original block, we can see that there is a lot of material being wasted. And the milling process may generate flaws on the surface which will lead to failure of the crown. (van Noort 2012; Zandparsa 2014)

1.2.1 A review of dental ceramics

The demands for esthetics as well as recent development of CAD/CAM technology drive the development of dental ceramic. There are three major categories of ceramics that are used in the practice, where the classification is derived from crystalline phase and fabrication techniques of the material: glass ceramic, glass ceramics with fillers and glass free ceramics. Glass ceramics include feldspathic ceramics, glass ceramics with fillers includes leucite-reinforced ceramics, lithium-disilicate ceramic and glass-infiltrated ceramics. Glass free ceramics includes alumina ceramics and zirconia ceramics. The trend of making new ceramics is combining different techniques that can achieve proper strength and toughness, esthetic requirements and long-tern in mouth performance. (Conrad, Seong et al. 2007; Kelly and Benetti 2011; Sadaqah 2014;

Zandparsa 2014)

The glass ceramic has a glass matrix. Feldspathic ceramics mainly contains silica

13 dioxide and aluminum oxide. The glass dominant glass ceramics can mimic the human enamel and dentin better (Zandparsa 2014). Vita Mark II (Vita Zahnfabrik, Bad

Sackingen, Germany) is a widely used feldspathic ceramic, introduced in 1991 for

CEREC I. It is a monochromatic block. For esthetic purposes, a multicolored Vita

TriLuxe and Vita Esthetic Line (Vita Zahnfabrik, Bad Sackingen, Germany) were introduced (Fasbinder 2010). Adding appropriate filler to the glass matrix can increase the mechanical properties and control the appearance features such as transparency. The glass matrix is infiltrated by micron size crystals of leucite and lithium disilicate, creating a highly filled glass matrix. As shown in table 1, lithium-disilicate fillered ceramics, such as IPS E.MAX and IPS Empress 2 (Ivoclar-Vivadent, Schaan,

Liechtenstein), have a higher flexural toughness of 2.0-2.5 MPa√m (Ivoclar Vivadent

2009) and 2.5 MPa√m than feldspathic ceramic. A higher crystallinity can improve the mechanical properties and thermal expansion. For instance IPS ProCAD and IPS

Empress are the examples of leucite fillered ceramics. They slightly improved the mechanical property (Bindl, Lüthy et al. 2006; Ritzberger, Apel et al. 2010) and have a higher expansion/contraction coefficients when compare to feldspathic ceramics (Kelly and Benetti 2011). But a higher crystallinity also will lead to a higher opacity, which is not desirable for esthetic purposes.

Polycrystalline ceramic or glass free ceramic has no glass matrix and all the atoms are packed into regular crystalline arrays. This structure is more difficult to crack

(Zandparsa 2014). As such they are generally tougher and stronger than glass ceramic.

Lava (3M ESPE Dental Products, St. Paul, MN) and DC-Zirkon (DCS Dental AG,

Allschwil, Switzerland) are the examples of zircronia based ceramic, the flexural toughness are 10 MPa√m (3M-ESPE 2002) and 7.4 MPa√m (Guazzato, Albakry et al.

2004) respectively, which is much higher than other types of ceramic (All-Ceramic).

Vita In-Ceram alumina has a toughness of 3.55 MPa√m and Vita In-Ceram Spinell has a toughness of 2.5 MPa√m (Vita Zahnfabrik 2009), are the examples of alumina based ceramic.

A ceramic crown is designed to last for at least 10 years (Reiss and Walther 2000). A lot of studies have been done to investigate the factors that will contribute to longevity, investigating the structure of the restorations, such as convergence angle (Corazza,

Feitosa et al. 2013) of tooth preparation, cusp coverage (Krifka, Stangl et al. 2009), wall thickness (Shibata, Gondo et al. 2014) of an inlay, and the cementing process.

According to the fracture load or microleakage level, we can have an idea which structure is stronger or more resilient to thermal and mechanical fatigue. Moreover,

SEM fractography of the failed samples allows understanding fracture mechanism and fracture modes. Wear effects are characterized by surface gloss and toughness. These parameters can be obtained with a glossmeter and inductive surface profilometer. The vertical loss of the enamel by physical wear is about 20-38 μm per year. An ideal ceramic tooth should have similar wear level with enamel. If the wear resistance is smaller than enamel, then the tooth restoration may wear out quickly. If the wear resistance is greater than enamel, then the ceramic tooth may damage the opposite tooth

(Mörmann, Stawarczyk et al. 2013). Margin fit can be evaluated by the gap between ceramic tooth and the prepared tooth along the circumferential margin (Yeo, Yang et al.

2003).

1.3 Digital Image Correlation (DIC)

1.3.1 Basic principle

DIC is a non-contact method to measure strain and displacement. Basically, DIC is tracking the same subset or pixels in the reference picture and matching them with the picture after loading. Therefore the specimen must have a random feature to make sure that the system can keep tracking these particular subsets and distinguish one subset from another. In practice, people are likely to use spray to generate speckle on the specimen surface. If the area of interest is very small, blowing toner directly onto the specimen has also proved to be effective. However, it may generate clusters. Another option is airbrush. Although the accuracy of the DIC analysis can be improved a little by keeping the system (i.e. cameras, cables and light) from moving or even slight vibration and good lighting conditions, the speckle pattern and quality is the most critical factor.

The speckle pattern can influence the result greatly. In a good speckle pattern: (1) the size of each dot should be uniform. (2)The speckle dots must have a sharp edge. (3)

There is a good contrast between the background and the speckles. (4)The speckle pattern has a 50% coverage.

A classic matching algorithm is as follows:

n 2 Cxyuv(,,,) ((, Ixiyi   )(  Ixuiyvj*2   ,   )) (1-54) n ij,  2

C is the correlation function. The smaller the value of C, the better the similarity. x, y is the coordinate of the pixel. u, v is the displacement. n is the subset size. I is the grey level at particular pixel before deformation, 퐼∗ represents that after deformation. DIC will consider the displacement candidate who has the minimal correlation value to be the result. Thus an overly fine speckle pattern will lead to a mismatching, which will cause an error to displacement and strain calculation.

1.3.2 DIC for fracture analysis

Vanlanduit, Vanherzeele et al. (Niendorf, Dadda et al. 2009) used DIC to visualize the crack. Even though the DIC is designed for analyzing a continuous area, according to

(Shih and Sung 2013), they successfully observed the crack initiation and strain concentration at the crack tip of the concrete beam with von Mises strain field. This can not only help in analyzing the crack behavior, but also can give an early warning before the crack is visible to the naked human eye.

Another topic is using full field displacement to determinate the SIF. One of the difficulties is to locate the crack tip position. There are some ways to achieve it.

Vanlanduit, Vanherzeele et al. (Vanlanduit, Vanherzeele et al. 2009) used the fact that the displacement discontinuity in the crack zone to find the crack length and crack tip.

Helm (Helm 2008) used two cutoff values to detect multiple cracks on a concrete specimen. The cutoff factors are calculated from average, standard and maximum deviation of correlation function. Fagerholt, Dørum et al. (Fagerholt, Dørum et al. 2010) define a weight factor (between 0 and 1) according to the grey scale. Where there is a crack, the corresponding pixel’s weight factor is 0. And the pixel will be excluded from

DIC analysis. Sometimes, we even can determine the crack tip using visual observation

(Shah and Kishen 2011). As we talked in section 1.1.2, (1-22) and (1-23) (Malakooti and Sodano 2014) give the relationship between SIF and displacement. Accordingly once we obtain the displacement filed from DIC, we can determine the SIF inversely

(Mogadpalli and Parameswaran 2008). Accordingly, we also can get energy release rate

G from (1-27).

Another application is inverse analysis. Inverse problem is when we know the fracture behavior and we want to know the constitutive parameters. Traditional methods have difficulties measuring the non-uniform strain field and full strain displacement. DIC provides a promising way to study cohesive zone, which is an ideal model to describe the mode I and mixed-mode fracture. Shen and Paulino (Shen and Paulino 2011) import the full displacement field to FEA model for an inverse study of the cohesive zone. By conducting least square optimization, they were able to identify the cohesive property.

DIC also proved to be effective when studying the relation between the crack initiation and microstructure (Niendorf, Dadda et al. 2009; Haldar, Gheewala et al. 2011). The

18 crack initiation is likely to occur where there is a high strain. Then those suspected zones will be further studied using microscopy, atomic force microscopy and electron backscatter diffraction. To validate the result from the DIC, some attempts proved to be effective. For example, Fagerholt, Dørum et al. (Fagerholt, Dørum et al. 2010) compared the DIC result with the numerical calculated result (i.e. LS-DYNA) under modes I and II. Even though the replicated test showed a marked difference, the result from both methods showed an agreement.

For the study of determination of SIF, Dehnavi, Khaleghian et al. (Dehnavi, Khaleghian et al. 2014) used the photoelasticity method to validate the DIC result; there also is an agreement between the two methods. Another commonly used method is strain gage

(Chehab, Seo et al. 2007)

Chapter 2 Measurement of J-integral by DIC

2.1 Objectives

1. Calculate J-integral of SEN dental ceramic samples under a three point bending

test.

2. Report and compare 퐽퐼퐶 of different dental ceramics.

3. Investigate the integral path selection.

2.2 Materials and Methods

2.2.1 Materials

2.2.1.1 Vita Mark II

Vita Mark II (VM II) is a feldspathic ceramic. The major composites are silica dioxide

(56-64%), and aluminum oxide (20-23%). Table 2 lists the main chemical composite. It has been used as a CAD/CAM ceramics for more than 20 years. The survival rate of the

VM II molar crown milled by CEREC III was more than 90% up to 12 years. VM II has a large variety of color choices that can satisfy different clinical needs. It can be fabricated to veneer, inlay, onlay, partial crown and crown. Young’s modulus is 63.0 ±

0.5 GPa (Vita Zahnfabrik 2012), which is comparable to human enamel. Toughness and flexural strength are lower compared to glass ceramic with filler and glass free ceramic.

2.2.1.2 E-Max

IPS E-MAX (E-MAX) is a lithium-disilicate ceramic, which was introduced in 2005. 20

As an improved material compared to IPS Empress, with higher mechanical properties and high translucency (Conrad, Seong et al. 2007). E-MAX comes in a partially crystallized state, to ensure an easier and faster machining process. In this state, the IPS

E-MAX block consist about 40% 퐿푖2푆푖푂3, colored in blue. After it gets sintered, it reaches to the final state. At this time it consists of about 70% 퐿푖2푆푖2푂5, and the flexural strength increases about 3 times, and toughness increases about 2 times. The color also changes to more human enamel like color.

2.2.1.3 Composite resin

3M™ Paradigm™ MZ100 (MZ 100) has been available on the market since 1991. MZ

100 consists of 85 % by weight ultrafine (0.6 micron) zirconia-silica, reinforced in a highly crosslinked polymeric matrix of bisGMA (Bisphenol A diglycidyl ether dimethacrylate) and TEGDMA (3M Dental Products 2014), Young’s modulus is 15 to

20 GPa (Madani 2010), which is much lower than ceramics. Toughness is 1.35 MPa√m, and flexural strength is 140-160 MPa (3M Dental Products 2014), which is comparable to VM II. MZ100 can be fabricated to veneer, inlay, onlay, partial crown and crown.

2.2.2 Sample preparation

The ceramic block was cut into 3 mm×3 mm×14 mm beams (notch samples) and

2.5 mm×3 mm×14 mm beams (un-notched samples) with a low-speed saw (Buehler,

Evanston, IL). And then the beams were polished with 240, 600 and 800 grid paper

(Buehler, Evanston, IL) successively. Then a notch was added with low-speed saw

21 again for notch groups. The thickness of the saw was 0.2 mm. The notch length was 1.5 mm. For the scale of the specimen and strain expectation of this study, airbrush was chosen to make speckle on the specimen surface. The ink used for generating speckle should be matte, because a glare can seriously affect the results, especially for strain calculations. After adding a notch, a white base ink (High flow acrylics titanium white,

#8549-6, Golden Artist Colors, Inc., New Berlin, NY) was sprayed. After the white ink dries fully, then black ink (High flow acrylics carbon black, #8524-6, Golden Artist

Colors, Inc., New Berlin, NY) was used to add speckles with an air brush (HARDER &

STEENBECK GMBH & CO. KG, Norderstedt, Germany) at a distance of about 20 cm away from specimen. Five samples of each material were prepared for notch samples to measure J-integral, and three un-notched samples of each materials were prepared for flexural bending test to check the materials’ elasticity property.

2.2.3 Strain-stress relations

This test was conducted to estimate the young’s modulus of E-MAX blue and check the elastic properties of the materials. Un-notched beam samples (2.5×3×14 mm) of MZ

100, VM II, E-MAX blue and E-MAX are prepared for 3 point bending. From the load-displacement curve we can get strain-stress curve by 3PS σ = 0 (1-55) f 2WB2 6DW εf = 2 (1-56) S0

Where σf (MPa) and εf are the stress and strain in outer fibers at midpoint, P (N) is load, S0 (mm) is the span, W (mm) is the height, B (mm) is the depth and D (mm) is

22 maximum deflection of the center of the beam..

2.2.4 Three-point bending test and DIC set up

The principle of choosing a camera is that one should be able to make the target or the area of interest fill up the screen as much as possible. Other factors, such as resolutions and the configuration (the angle between their optic axis) of the cameras and the DIC algorithm (Orteu, Cutard et al. 2007) also will affect the accuracy of the results. Good lighting can increase the contrast of the background and the speckle pattern, which will help decrease the noise. While if the lighting condition is too bright, it will also generates problem. Vic-Snap 8 (Correlated Solutions, Inc., Columbia, SC) is used in this study to record pairs of images. When a particular region is too bright for DIC analysis, the screen will show red warning for those places. A good lighting condition should be as bright as possible, but no red parts occur for both the images of sample and calibration grid. The accuracy of the displacement and strain depends on many factors as we talked above, many literatures have explored this problem. For Vic-3D

(Correlated Solutions, Inc., Columbia, SC) DIC software, the accuracy of displacement is up to 0.01 pixel, and 200 μ strain for strain (Lichtenberger and Schreier 2005). And comparable accuracy reported for other DIC systems (Becker, Splitthof et al. 2006;

Orteu, Cutard et al. 2007; Shen and Paulino 2011; Becker, Mostafavi et al. 2012).

A mechanical test machine (Testresources Shakopee, MN) was used for the 3 point bending test. Under the load control mode, the head went down at a force rate of 5N/s.

The span was set to 12 mm. Pictures are taken by GX3300 cameras about every 50 ms

23 until fracture. The resolution of the camera is 3296 x 2472. Vic-3D 7 (Correlated

Solutions, Inc., Columbia, SC) was employed for DIC analysis. The last picture was used to calculateJIC.

2.2.5 J-integral calculation

(1-35) was used to calculate J-integral. For an elastic material, σ=Dε, where

 1- 0 E  D  1- 0 (1-57) (1 2 )(1 )  1-2 00 2 for plane strain problem. E is Young’s modulus, ν is Poisson’s ratio. And (1-36) equals

1 to σ ε . And we know that T =σ n . Substituting above equations into (1-35), we can 2 ij ij i ij j get that

1 u J()  dy   n  i ds (1-58)  ij ij ij i  2 x

For simplicity, a rectangular integral path was chosen (Figure 17). Then we can easily get the integral path normal of each segment. The J-integral along this integral path can be obtained from the sum of J-integral on each segment, as

JJJJJJ     (1-59)  1  2  3  4  5

Poisson’s ratio was taken to be 0.25 (Yilmaz, Aydin et al. 2007), the standard value for conventional ceramics (Kang, Chang et al. 2013), 0.35 for MZ 100 composite resin

(Chung, Yap et al. 2004). Elastic modulus used for VM II is 63 GPa, 95 GPa for

E-MAX and 15 GPa for MZ 100. As far as the author knows, there is no information of elastic modulus of E-MAX blue. From the strain-stress test conducted in this study, we

24 know that the elastic modulus of E-MAX blue is comparable to VM II. Therefore in this study, we assume an elastic modulus of 60 GPa for E-MAX blue (from section 3.1).

Isolated points are chosen along the integral rectangular about every 20 μm. Strain εxx,

du dv ε , ε and displacement gradient and of each point are exported. A Matlab yy xy dx dx

(R2013a version, MathWorks, Inc, Novi, MI) code was used to calculate the J-integral.

2.2.6 Conventional calculation of 퐊퐈퐂

As shown in Figure 20, S0 (m) is the span of the sample, a (m) is the notch length, W

(m) is the height, B (m) is the depth of the sample. Then KIC can be calculated from

(Della Bona, Corazza et al. 2014):

1 6  PS10 3/aW2 Kg max 0  (1-60) IC 33 BW 22  2 1 a / W  where Pmax (N) is the load at fracture. And

a gg  W (1-61) 2 1.99a / W  1  a / W 2.15  3.93 a / W  2.7 a / W    1 2(aW / )

2.2.7 Finite element model

A 2D plane strain finite element model (FEM) was built using ABAQUS (Dassault

Systems, Velizy-Villacoublay, France) to validate the accuracy of strain measurement in

DIC and to find the effects of the missing area at the edge to J-integral. The geometry

25 of the specimens was the same with the actual specimen used in this study as shown in

Figure 15. Structured mesh was used, and the average mesh size is about 0.05 mm. For a static problem, the displacement in x-axis was set to 0 at the support points, as shown in

Figure 21. And a displacement boundary condition was applied on a length of 0.3 mm on the center of the beam, where was the contact position of mechanical loading pin and the beam specimen. The value of the displacement condition was according to the DIC displacement results. The displacement boundary condition values were 0.003 mm,

0.0019 mm, 0.0025 mm and 0.0038 mm respectively for MZ 100, E-MAX blue, VM

II and E-MAX. Young’s modulus and Poisson’s ratio came from Table 4.

To numerically describe the difference between FEM and DIC results, a L2 norm was calculated, which is defined as follows:

2 FEM-  DIC =  FEM -  DIC dv (1-62) L2  

Where Ω is the area where the difference of FEM and DIC strain results compared. v is the small area in the Ω.

2.2.8 Statistical analyses

A Kruskal-Wallis statistical analysis was used in this study for the influence of integral path size and locations. An α of 0.05 was considered statistical significant (R2013a version, MathWorks, Inc, Novi, MI). If the statistical results shown a significant difference among different integral path size or location, then a Mann-Whitney statistical analysis would be performed to compare each size or location with the one

26 has the highest average value. An α of 0.05 was considered to be statistically significant.

Chapter 3 Results

3.1 Strain-stress relations

The strain-stress curve from load-displacement curve revealed MZ 100, E-MAX blue and VM II behave as with elastic. One example of each material’s strain-stress relation is shown in figure 22. Because the inappropriate experiment set up, the data of E-MAX un-notched group is not included. From Kok, Kleverlaan et al work, we know that

E-MAX also behaves as with elastic (Kok, Kleverlaan et al. 2015). We can see from

Figure 22 that E-MAX blue has a similar Young’s modulus as VM II. And MZ 100 is smaller than these two materials.

For a notched beam sample during a 3-point bending test, the load-displacement curve also showed linear relation. Figure 23 showed an example of load-displacement curves of notched samples. Consequently, for all the groups we can use (1-50) to calculate K from J-integral.

3.2 FEM

The FEM computed strain fields are compared with experimentally obtained strain field to verify the accuracy of the developed method. A detailed comparison is shown in

Figure 23. From the color mapping, we can see a good strain distribution pattern agreement between DIC and FE results. Because of the DIC algorithm, we could not measure the strain field from notch edge. The missing area has a width of half subset length, which is about 0.2 mm in this study. Accordingly the highest strain of DIC is 28 smaller than FEM one. But generally, DIC results are higher than FEM ones.

A L2 norm was calculated for each strain component (εxx, εyy and εxy) on the area where the integral path located (Figure 25). Each small subset was about 0.4 mm ×

FEM DIC −6 2 FEM DIC -6 2 0.24 mm. ‖εxx -εxx ‖ =1.09× 10 푚푚 , ‖εyy -εyy ‖ =5.92×10 mm , and L2 L2

FEM DIC -6 2 ‖εxy -εxy ‖ =4.41×10 mm . Table 5 also lists the average strain component in this L2 area for FEM and DIC.

J-integral of full integral path and partial integral path are calculated (See Figure 21).

Detailed results are shown in the Table 6. The effects of the missing area is up to

16.4%.

3.3 Integral path size

Figure 26-28 show the strain component of each dental restoration materials before fracture. In order to investigate the path size effects, we have chosen 8 rectangular paths.

Figure 29 shows an example. The path which is closest to the notch is the first path, and the outmost one is the 8th one.

Figure 30 shows the mean and standard deviation of the JIC results for MZ 100,

-4 E-MAX blue, VM II and E-MAX. JIC of MZ 100 ranges from 1.27±0.33 10 MPa∙m

-4 at integral path 2 to 0.79±0.06 10 MPa∙m at path 8. JIC of E-MAX blue ranges from

-4 -4 0.21±0.05 10 MPa∙m at path 1 to 0.02±0.01 10 MPa∙m. JIC of VM II ranges from

-4 -4 0.22±0.08 10 MPa∙m at path 2 to 0.10±0.05 10 MPa∙m. JIC of E-MAX ranges from

0.72±0.31 10-4 MPa∙m at path 3 to 0.48±0.23 10−4 MPa∙m. The ANOVA revealed there is no statistical significant influence of integral path size for MZ 100 (p=0.09),

E-MAX (p=0.83) and VM II (p=0.27) group. It has a statistical significant influence of integral path size for E-MAX blue group (p=0.0001). The Mann-Whitney test of comparing the JIC value of each integral path of E-MAX blue group with the one on integral path 1 showed that the JIC value on integral path 6, 7, and 8 have a significant difference with the one on path 1. And the JIC value on integral path 2, 3, 4, and 5 have no significant difference with the one on path 1.

KJ of each path are then calculated, as shown in figure 31. KJ of MZ 100 ranges from

1.46±0.19 MPa√m at integral path 2 to 1.13±0.06 MPa√m at path 8. KJ of E-MAX blue ranges from 1.14±0.13 MPa√m at integral path 1 to 0.31±0.12 MPa√m at path 8.

KJ of VM II ranges from 1.14±0.25 MPa√m at integral path 3 to 0.79±0.18 MPa√m at path 8. KJ of E-MAX ranges from 2.82±0.36 MPa√m at integral path 2 to 2.12±0.59

MPa√m at path 8.

3.4 Integral path location

For path location effect, 4 different locations were chosen as shown in figure 32. From the lowest rectangular path represent 1 to upper most 4.

Figure 33 showed the shows the mean and standard deviation of the JIC results for MZ

-4 100, E-MAX blue, VM II and E-MAX. The JIC of MZ 100 are 1.05±0.24 10 MPa∙m,

0.88 ±0.20 10-4 MPa∙m, 0.74 ±0.17 10-4 MPa∙m and 0.62 ±0.11 10-4 MPa∙m

−4 respectively from path 1 to path 4. The JIC of E-MAX blue are 0.12±0.04 10 MPa∙m,

0.11 ±0.03 10-4 MPa∙m, 0.11 ±0.03 10-4 MPa∙m and 0.09 ±0.03 10-4 MPa∙m

-4 respectively from path 1 to path 4. The JIC of VM II are 0.17±0.08 10 MPa∙m,

0.14 ±0.06 10-4 MPa∙m, 0.13 ±0.05 10-4 MPa∙m and 0.12 ±0.04 10-4 MPa∙m

-4 respectively from path 1 to path 4. The JIC of E-MAX are 0.72±0.31 10 MPa∙m,

0.62 ±0.26 10-4 MPa∙m, 0.51 ±0.20 10-4 MPa∙m and 0.40 ±0.14 10-4 MPa∙m respectively from path 1 to path 4. The mean JIC values of these four groups have the same decreasing trend from path 1 to path 4. The statistical study showed that there is no statistical significance among each integral path location for each material.

Figure 34 shows 퐾퐽 of MZ 100, E-MAX blue, VM II and E-MAX from different integral path location. KJ of MZ 100 ranges from 1.28±0.14 MPa√m, 1.22±0.14

MPa√m, 1.12±0.13 MPa√m and 1.03±0.09 MPa√m respectively from path 1 to path 4.

KJ of E-MAX blue ranges from 0.85±0.14 MPa√m, 0.85±0.12 MPa√m, 0.81±0.12

MPa√m and 0.75±0.11 MPa√m respectively from path 1 to path 4. KJ of VM II ranges from 1.03±0.25 MPa√m, 0.94±0.21 MPa√m, 0.93±0.12 MPa√m and 0.88±0.15

MPa√m respectively from path 1 to path 4. KJ of E-MAX ranges from 2.62±0.61

MPa√m, 2.45±0.56 MPa√m, 2.22±0.48 MPa√m and 1.97±0.39 MPa√m respectively from path 1 to path 4.

3.5 Critical stress intensity factor

The results of KIC calculated from geometry and fracture load values are listed in Table

7. KJ calculated from J-integral based on (1-50) used the result from path 1 for the path location study. KIC is generally bigger than KJ . Kreported are from Table 4. The methods used in those studies to calculate SIF is similar to this study. All of them used a

SEN bending sample.

Chapter 4 Discussion and Conclusion

When the integral path gets bigger, and away from the notch, the mean J-integral value tends to decrease for all the materials. When the integral path gets close enough to the notch, the mean J-integral value starts to hold. For E-MAX, the mean JIC value gradually increases from path 1 to path 3, and then starts to decrease from path 4 to path

8. The statistical study showed that only E-MAX blue group is sensitive to the integral path size selection. When the integral size is bigger than integral path size path 5, the JIC values start to have a significant difference with the one from integral path 1. One possible reason may be the different microstructure of the materials showed different sensitivity to the integral path size. Therefore, when a J-integral value is measured for a dental ceramic or composite resin, it is more practical to place the integral path that is smaller than integral path 5 (1.27 mm×0.70 mm), which the length and width is about

23.3% and 42.3% of the sample width respectively Because there is the possibility that a certain dental restoration material is sensitive to the path size when the integral path is bigger than path 5. And MZ 100 has the biggest mean JIC value, following by E-MAX, and then E-max blue and VM II have a comparable mean JIC value when we compare the JIC of different materials on the same path size. However, when we look at the mean KJ value, E-MAX has the highest value, and then MZ 100, E-MAX blue and VM

II have a mean KJ about half of E-MAX.

From the integral path location test, as there is no statistical significant difference JIC among different path locations, JIC is not sensitive to the path locations. 33

When we compare the critical SIF from different resources we can see that values from each resource is comparable, yet the difference between KJ and KIC may come from the Young’s modulus and Poisson’s ratio. As we used the reposted value of these parameters, it may have a gap between the real values of the samples that we use in this study. Also, because of the algorithm of the DIC method, it cannot plot the results on the edge. Therefore, the integral path is not start from the crack edge. If the speckle pattern is finer, we then can chose a smaller subset, and we can get closer to the crack edge. For the difference between KIC and Kreported, it may because different studies used different formulas to calculate KIC. After all, J-integral and SIF is depend on experiment set up and load conditions, thus a different specimen configuration and loading method will give a different value.

Even though the SIF from all methods are comparable, the results from J-integral are more reliable when compare to the geometrically based calculated KIC values. Since

J-integral based test also works for elastic-plastic material, it can be applied to a wider range of materials. For dental ceramic and composite resin under a three-point bending test, J-integral of E-MAX blue is sensitive to integral path size, while other groups are not. For the future research, it would be very interesting to measure J-integral of the materials under their functional geometry.

Appendix A

A.1 Matlab code

%J-integral calculation for dental restoration materials three-point bending test and K_IC from J-integral % MZ 100

E=15000; %MPa, Young’s modulus of MZ 100 r=0.35; % Poisson’s ratio D=E./((1-2.*r).*(1+r)); l=247; % # of points in total l1=23; % # of points in Γ 1 l2=86; % # of points in Γ 2 l3=158; % # of points in Γ 3 l4=221; % # of points in Γ 4 l5=246; % # of points in Γ 5

% load data data=xlsread('composite sample 1.xlsx'); coordinate=xlsread('composite sample 1 _coordinate.xlsx');

The data explored from Vic-3D has all the strain and displacement gradient together. The following code is for assigning each component.

%strain & displacement gradient for i=1:l x(i)=coordinate(3,1+3.*(i-1)); % x coordinate of each point y(i)=coordinate(3,2+3.*(i-1)); % y coordinate of each point exx(:,i)=(data(:,2+5.*(i-1))); % strain 휀푥푥 of each point eyy(:,i)=(data(:,3+5.*(i-1))); % strain 휀푦푦 of each point exy(:,i)=(data(:,4+5.*(i-1))); % strain 휀푥푦 of each point 푑푢 du_dx(:,i)=data(:,5.*i); % displacement gradient 푑푥 % of each point 푑푣 dv_dx(:,i)=data(:,6+5.*(i-1)); % displacement gradient 푑푥 % of each point end

Then we can calculate stress from strain as σ = 퐃ε

% stress sxx=D.*((1-r).*exx+r.*eyy); syy=D.*(r.*exx+(1-r).*eyy); sxy=D.*(((1-2.*r)./2).*exy);

%Γ 1 for m=1:l1

x_bar(m)=10.^(-3).*(x(m+1)-x(m); % Δx = x(m + 1) − x(m) y_bar(m)=10.^(-3).*(y(m+1)-y(m)); % Δy = y(m + 1) − y(m) end

exx_1=exx(:,2:l1+1); % strain 휀푥푥 of each point on Γ 1 eyy_1=eyy(:,2:l1+1); % strain 휀푦푦 of each point on Γ 1 exy_1=exy(:,2:l1+1); % strain 휀푥푦 of each point on Γ 1 sxx_1=sxx(:,2:l1+1); % stress 휎푥푥 of each point on Γ 1 syy_1=syy(:,2:l1+1); % stress 휎푦푦 of each point on Γ 1 sxy_1=sxy(:,2:l1+1); % stress 휎푥푦 of each point on Γ 1 du_dx_1=du_dx(:,2:l1+1); % displacement gradient 푑푢 % of each point on Γ 1 푑푥 dv_dx_1=dv_dx(:,2:l1+1); % displacement gradient 푑푣 % of each point on Γ 1 푑푥

% strain energy density for Γ 1 W1=(1/2).*(sxx_1.*exx_1+syy_1.*eyy_1+2.*sxy_1.*exy_1);

%Γ 2 for m=l1+1:l2

x_bar2(m-l1)=10.^(-3).*(x(m+1)-x(m)); y_bar2(m-l1)=10.^(-3).*(y(m+1)-y(m)); end exx_2=exx(:,l1+2:l2+1); eyy_2=eyy(:,l1+2:l2+1); exy_2=exy(:,l1+2:l2+1); sxx_2=sxx(:,l1+2:l2+1); syy_2=syy(:,l1+2:l2+1); sxy_2=sxy(:,l1+2:l2+1); du_dx_2=du_dx(:,l1+2:l2+1);

dv_dx_2=dv_dx(:,l1+2:l2+1);

W2=(1/2).*(sxx_2.*exx_2+syy_2.*eyy_2+2.*sxy_2.*exy_2);

%Γ 3 for m=l2+1:l3

x_bar3(m-l2)=10.^(-3).*(x(m+1)-x(m)); y_bar3(m-l2)=10.^(-3).*(y(m+1)-y(m));

end exx_3=exx(:,l2+2:l3+1); eyy_3=eyy(:,l2+2:l3+1); exy_3=exy(:,l2+2:l3+1); sxx_3=sxx(:,l2+2:l3+1); syy_3=syy(:,l2+2:l3+1); sxy_3=sxy(:,l2+2:l3+1); du_dx_3=du_dx(:,l2+2:l3+1); dv_dx_3=dv_dx(:,l2+2:l3+1);

W3=(1/2).*(sxx_3.*exx_3+syy_3.*eyy_3+2.*sxy_3.*exy_3);

%Γ 4 for m=l3+1:l4

x_bar4(m-l3)=10.^(-3).*(x(m+1)-x(m)); y_bar4(m-l3)=10.^(-3).*(y(m+1)-y(m)); end exx_4=exx(:,l3+2:l4+1); eyy_4=eyy(:,l3+2:l4+1); exy_4=exy(:,l3+2:l4+1); sxx_4=sxx(:,l3+2:l4+1); syy_4=syy(:,l3+2:l4+1); sxy_4=sxy(:,l3+2:l4+1); du_dx_4=du_dx(:,l3+2:l4+1); dv_dx_4=dv_dx(:,l3+2:l4+1); W4=(1/2).*(sxx_4.*exx_4+syy_4.*eyy_4+2.*sxy_4.*exy_4);

%Γ 5

37 for m=l4+1:l5

x_bar5(m-l4)=10.^(-3).*(x(m+1)-x(m)); y_bar5(m-l4)=10.^(-3).*(y(m+1)-y(m)); end exx_5=exx(:,l4+2:l5+1); eyy_5=eyy(:,l4+2:l5+1); exy_5=exy(:,l4+2:l5+1); sxx_5=sxx(:,l4+2:l5+1); syy_5=syy(:,l4+2:l5+1); sxy_5=sxy(:,l4+2:l5+1); du_dx_5=du_dx(:,l4+2:l5+1); dv_dx_5=dv_dx(:,l4+2:l5+1); W5=(1/2).*(sxx_5.*exx_5+syy_5.*eyy_5+2.*sxy_5.*exy_5);

%J_Γ i J1=(W1+sxx_1.*du_dx_1+sxy_1.*dv_dx_1)*y_bar'; J2=(W2*y_bar2(1,:)')+(sxy_2.*du_dx_2+syy_2.*dv_dx_2)*x_bar2(1,:)'; J3=(W3-sxx_3.*du_dx_3-sxy_3.*dv_dx_3)*y_bar3'; J4=(W4*y_bar4(1,:)')-(sxy_4.*du_dx_4+syy_4.*dv_dx_4)*x_bar4(1,:)'; J5=(W5+sxx_5.*du_dx_5+sxy_5.*dv_dx_5)*y_bar5';

J=J1+J2+J3+J4+J5

K=sqrt((J.*E)./(1-r.^2))

Appendix B

B.1 Average and standard deviation value of J-integral of MZ 100 of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 1.26 1.27 1.25 1.18 1.05 0.94 0.83 0.79 (10-4MPa∙m) SD 0.31 0.33 0.35 0.29 0.24 0.19 0.15 0.06 (10-4MPa∙m)

B.2 Average and standard deviation value of J-integral of E-MAX blue of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 0.21 0.20 0.18 0.15 0.14 0.09 0.05 0.02 (10-4MPa∙m) SD 0.05 0.04 0.04 0.04 0.04 0.03 0.03 0.01 (10-4MPa∙m)

B.3 Average and standard deviation value of J-integral of VM II of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 0.19 0.22 0.20 0.19 0.16 0.13 0.11 0.10 (10-4MPa∙m) SD 0.07 0.08 0.09 0.09 0.08 0.07 0.06 0.05 (10-4MPa∙m)

B.4 Average and standard deviation value of J-integral of E-MAX of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 0.59 0.68 0.72 0.70 0.66 0.62 0.57 0.48 (10-4MPa∙m) SD 0.28 0.30 0.31 0.28 0.26 0.24 0.23 0.23 (10-4MPa∙m)

B.5 Average and standard deviation value of KJ of MZ 100 of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 1.45 1.46 1.46 1.41 1.33 1.26 1.18 1.13 (MPa∙√푚) SD 0.17 0.19 0.19 0.18 0.16 0.13 0.11 0.06 (MPa∙√푚)

B.6 Average and standard deviation value of KJ of E-MAX blue of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 1.14 1.14 1.08 0.99 0.93 0.76 0.53 0.32 (MPa∙√푚) SD 0.13 0.11 0.11 0.11 0.14 0.14 0.14 0.12 (MPa∙√푚)

B.7 Average and standard deviation value of 퐾퐽 of VM II of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 1.11 1.14 1.13 1.09 1.01 0.91 0.81 0.79 (MPa∙√푚) SD 0.23 0.25 0.23 0.27 0.27 0.26 0.24 0.18 (MPa∙√푚)

B.8 Average and standard deviation value of 퐾퐽 of E-MAX of different integral path sizes. Path # 1 2 3 4 5 6 7 8 Average 2.37 2.54 2.63 2.60 2.54 2.45 2.35 2.12 (MPa∙√푚) SD 0.39 0.36 0.63 0.57 0.54 0.52 0.53 0.59 (MPa∙√푚)

B.9 Average and standard deviation value of J-integral of MZ 100 of different integral path locations. Path # 1 2 3 4 Average 1.05 0.88 0.74 0.62 (10-4MPa∙m) SD 0.24 0.20 0.17 0.11 (10-4MPa∙m)

B.10 Average and standard deviation value of J-integral of E-MAX blue of different integral path locations. Path # 1 2 3 4 Average 0.12 0.11 0.11 0.09 (10-4MPa∙m) SD 0.04 0.03 0.03 0.03 (10-4MPa∙m)

B.11 Average and standard deviation value of J-integral of VM II of different integral path locations. Path # 1 2 3 4 Average 0.17 0.14 0.13 0.12 (10-4MPa∙m) SD 0.08 0.06 0.05 0.04 (10-4MPa∙m)

B.12 Average and standard deviation value of J-integral of E-MAX of different integral path locations. Path # 1 2 3 4 Average 0.72 0.62 0.51 0.40 (10-4MPa∙m) SD 0.31 0.26 0.20 0.14 (10-4MPa∙m)

B.13 Average and standard deviation value of 퐾퐽 of MZ 100 of different integral path locations. Path # 1 2 3 4 Average 1.05 0.88 0.74 0.62 (MPa∙√푚) SD 0.24 0.20 0.17 0.11 (MPa∙√푚)

B.14 Average and standard deviation value of 퐾퐽 of E-MAX blue of different integral path locations. Path # 1 2 3 4 Average 0.85 0.85 0.81 0.75 (MPa∙√푚) SD 0.14 0.12 0.12 0.11 (MPa∙√푚)

B.15 Average and standard deviation value of 퐾퐽 of VM II of different integral path locations. Path # 1 2 3 4 Average 1.03 0.94 0.93 0.88 (MPa∙√푚) SD 0.25 0.21 0.17 0.15 (MPa∙√푚)

B.16 Average and standard deviation value of 퐾퐽 of E-MAX of different integral path locations. Path # 1 2 3 4 Average 2.62 2.45 2.22 1.97 (MPa∙√푚) SD 0.61 0.56 0.48 0.39 (MPa∙√푚)

Table 1: Ceramics materials

SIF Clinical Ceramic type Brand name E (GPa) (MPa√m) application*

IPS Empress 62(1) 1.3(1) AC Glass leucite IPS ProCAD 70(2) 1.3(2) -- ceramics Lithium- IPS E-MAX 95±5(3) 2.0-2.5(3) V,I,O,P,C with fillers disilicate IPS Empress 2 96(4) 2.9(10)

Vita Mark II 63.0 ± 0.5(5) 0.92-1.7 (2)(5) V,I,O,P,C

Glass Vita TriLuxe 45±0.5(6) -- V,P,C feldspathic ceramic Vita Esthetic -- 1.2(7) V,C Line

Vita In-Ceram 270(4) 3.55(8) V,C,B alumina Alumina Glass free Vita In-ceram -- 2.5(7) AC ceramics Spinell

Lava 210(9) 10(9) I,O,B,C Zircronia DC-Zirkon 220(10) 7.4(10) --

(1)(Ritzberger, Apel et al. 2010), (2)(Bindl, Lüthy et al. 2006), (3)(Ivoclar Vivadent 2009), (4) (Goetzen, Natt et al. 2003), (5)(Vita Zahnfabrik 2003), (6) (Vita Zahnfabrik 2012), (7)(Zahnfabrik 2000), (8)(Vita Zahnfabrik 2009), (9)(3M-ESPE 2002), (10)(Guazzato, Albakry et al. 2004)

Table 2 Composite of Vita Mark II (Vita Zahnfabrik 2003)

Composite Percentage

푺풊퐎ퟐ 56-64%

푨풍ퟐ푶ퟑ 20-23%

푲ퟐ푶 6-8%

푵풂ퟐ푶 6-9%

퐂퐚푶 0.3-0.6%

푻풊푶ퟐ 0-0.1%

Note: Some low concentration composite i.e. required for coloring is not included.

Table 3: Composite of IPS E.MAX (Ivoclar Vivadent 2009)

Composite Percentage

푺풊퐎ퟐ 57-80%

푳풊ퟐ푶 11-19%

푲ퟐ푶 0-13%

푷ퟐ푶ퟓ 0-11%

풁풓푶ퟐ 0-8%

풁풏푶 0-0.1%

Note: Some low concentration composite i.e. required for coloring is not included.

Table 4: Material properties of IPS E.MAX blue, E.MAX, VM II and MZ 100

Young’s Vickers Toughness Flexural modulus Hardness (MPa√m) Strength (MPa) (GPa) MPa

E-MAX blue -- 0.9-1.25(1) 130 ± 30(1) 5400±100(1)

E-MAX 95± 5(1) 2-2.5(1) 360±60(1) 5800±100(1)

VM II 63.0 ± 0.5(2) 0.92-1.7(2)(3) 154 ± 15(2) 6200(2)

MZ 100 15-20(4) 1.35(5) 140-160(5) --

(1)(Ivoclar Vivadent 2009), (2)(Vita Zahnfabrik 2003), (3)(Bindl, Lüthy et al. 2006), (4)(Madani 2010), (5)(3M Dental Products 2014)

Table 5: Average strain component in the integral path located area

εxx FEM εxx DIC Diff. Diff. (%)

MZ 100 -3.98E-5 -6.36e-6 -3.34E-5 84

E-MAX blue -6.09E-6 5.80E-6 -1.19E-5 195

VM II -2.45E-6 -1.27e-5 1.03E-5 -418

E-MAX -3.18E-5 3.07E-6 -3.49E-5 110

εyy FEM εyy DIC Diff. Diff. (%)

MZ 100 1.02E-4 1.34E-4 -3.20E-5 31

E-MAX blue 1.26E-4 7.11E-5 5.49E-5 43

VM II 1.02E-4 5.05E-5 5.15E-5 50

E-MAX 1.06E-4 1.80E-5 8.80E-5 83

εxy FEM εxy DIC Diff. Diff. (%)

MZ 100 5.46E-6 -9.29E-6 1.48E-05 270

E-MAX blue -2.44E-5 5.06E-7 -2.49E-05 102

VM II 2.15E-5 3.45E-5 -1.30E-05 60

E-MAX 9.72E-6 -3.31E-5 4.28E-05 441

Table 6: J-integral of full integral path and partial integral path

Full integral Partial integral MPa∙m Diff. Diff. (%) path path

MZ 100 1.24E-6 1.17E-6 7.00E-8 5.6

E-MAX blue 2.73E-6 2.55E-6 1.80E-7 6.6

VM II 4.54E-6 4.43E-6 1.10E-7 2.4

E-MAX 5.30E-6 4.43E-6 8.7E-7 16.4

Table 7: KIC , KJ and Kreported for MZ 100, E-MAX, VM II and E-MAX blue

MPa√푚 Vita Mark II E-Max MZ 100 E-Max blue

퐾퐽 1.13 2.82 1.46 1.14

퐾푔푒표푚 1.20 3.84 1.85 1.88

퐾푟푒푝표푟푡푒푑 0.92-1.70 2.00-2.50 1.35 0.90-1.25

Figure 1: The infinite plate geometry in Inglis’ approach. A plate has an elliptical crack with the length of 2a in the middle. Traction of T is applied in tension. (Griffith 1921)

Figure 2: Angular coordinate. The origin is located at the crack tip

Figure 3: Effective crack length of Irwin model

Figure 4: Effective crack length of Dugdale model

Figure 5: A notch lays parallel to x axis, Γ is an arbitrary curve surrounding the crack tip, and n is the normal direction of Γ

Figure 6: HRR field and K field

Figure 7: Geometry for J-integral path-independency.

Figure 8: A closed curve used to prove the path-independency of the J-integral. Γ is counter-clock 1

wise, Γ is clock wise, Γ and Γ are on the crack edges. 2 + -

Figure 9: Specimen geometries used in fracture mechanics tests. The left one is a compact tension (CT) specimen, the middle one is a single edge-notched bend (SEN) specimen, and the right one is a middle-cracked tension (MT) specimen.(Zhu and Joyce 2012)

Figure 10: Crown replacement procedure. The nature damaged crown will be removed and shaped as shown in the middle section for accommodation of the ceramic crown. (Tourmedical n.d.)

Figure 11: Schematic diagram depicting various fracture modes in crown. O: occlusal contact crack; R: radical crack; C: chipping.

Figure 12: The CEREC system. From left to right is tooth geometry acquisition system, monitor and design unit, milling unit and ceramic block respectively. (Vita Zahnfabrik 2003; Jurim Dental Group 2014; Central Valley Dentistry n.d.; Wateree Family Dentistry n.d.)

Figure 13: Micro structure of three different types of ceramics (Kelly 2008)

Figure 14: Schematic diagram of DIC matching algorism. The red dash line shows one of the subsets used as a unit to match the reference image and deformed image

Figure 15: Sample preparation process.

Figure 16: Speckle pattern.

Figure 17: Experimental set up

Figure 18: Light position with respect to sample.

Figure 19: Rectangular integral path used to calculate J-integral.

Figure 20: Sample dimensions.

Figure 21: Boundary conditions in FEM. The geometry is the same with the specimen used in three point bending test. Displacement in x-axis is set to be 0 for the support points. And a displacement boundary condition is added on the center, as the red arrows shown. The solid blue line represent partial integral path, and full integral path include the dash line and solid line.

MZ 100 E-MAX blue VM II

160

120

40 Flexural Flexural (MPa) Stress

0 0.00 0.01 0.02 0.03 0.04 Strain

Figure 22: Strain-stress relation of MZ 100, E-MAX blue, and VM II

MZ 100 E-MAX blue VM II E-MAX

Load(N) 20

0 0.00 0.02 0.04 0.06 0.08 0.10 Displacement (mm)

Figure 23: Load-displacement curve of MZ 100, E-MAX blue, VM II and E-MAX notched beam sample under a three-point bending test.

Figure 24: Comparison of FEM and DIC strain component a) εxx, b) εyy, and c) εxy.

Figure 25: Points position used for L2 norm.

Figure 26: 휀xx mapping of MZ 100, E-MAX blue, VM II and E-MAX before fracture

Figure 27: εyy mapping of MZ 100, E-MAX blue, VM II and E-MAX before fracture

Figure 28: εxy mapping of MZ 100, E-MAX blue, VM II and E-MAX before fracture

Figure 29: Integral paths for a different size.

MZ 100 E-MAX blue VM II E-MAX 1.8 1.6

1.4

)

m 1.2

∙ a

1.0

MP 4 4

- 0.8

(10

J 0.6 0.4 0.2 0.0 1 2 3 4 5 6 7 8

Figure 30: J-integral for different integral path size. (See Appendix B)

MZ 100 E-MAX blue VM II E-MAX 3.5 3.0

2.5

) m

√ 2.0

a (MP

1.5

J K 1.0 0.5 0.0 1 2 3 4 5 6 7 8

Figure 31: KJ calculated from J-integral for different integral path size. (See Appendix B)

Figure 32: Path integral with the same size at different locations. (See Appendix B)

MZ 100 E-MAX blue VM II E-MAX 1.4

1.2

1.0

)

m ∙

a 0.8

4 -

0.6

(10

IC J 0.4

0.2

0.0 1 2 3 4

Figure 33: J-integral of MZ 100, E-MAX blue, VM II and E-MAX from different integral path location. (See Appendix B)

MZ 100 E-MAX blue VM II E-MAX 3.5

3.0

2.5

)

m √

a 2.0

(MP

K 1.5

1.0

0.5

0.0 1 2 3 4

Figure 34: KJ of MZ 100, E-MAX blue, VM II and E-MAX from different integral path location. (See Appendix B)

References

3M-ESPE (2002). "Lava all-ceramic system technical product profile."

3M Dental Products (2014). "3M Paradigm™ MZ100 Block Technical Product Profile."

All-Ceramic, M. E. L. T. "System, Technical Product Profile, Dental Products, 3M Center, St." Paul, MN, USA.

Becker, T., M. Mostafavi, et al. (2012). "An approach to calculate the J‐integral by digital image correlation displacement field measurement." Fatigue & Fracture of Engineering Materials & Structures 35(10): 971-984.

Becker, T., K. Splitthof, et al. (2006). Error estimations of 3D digital image correlation measurements. Speckle06: Speckles, From Grains to Flowers, International Society for Optics and Photonics.

Bindl, A., H. Lüthy, et al. (2006). "Strength and fracture pattern of monolithic CAD/CAM-generated posterior crowns." Dental Materials 22(1): 29-36.

Brocks, W., I. Scheider, et al. (2001). "Numerical aspects of the path-dependence of the j-integral in incremental plasticity." GKSS Forschungszentrum, Geesthacht, Germany, Technical Report No. GKSS/WMS/01/08.

Central Valley Dentistry (n.d.). "CEREC Crowns."

Chehab, G. R., Y. Seo, et al. (2007). "Viscoelastoplastic damage characterization of asphalt–aggregate mixtures using digital image correlation." International Journal of Geomechanics.

Chen, H. Y., R. Hickel, et al. (1999). "Effects of surface finish and fatigue testing on the fracture strength of CAD-CAM and pressed-ceramic crowns." The Journal of prosthetic dentistry 82(4): 468-475.

Chung, S. M., A. U. J. Yap, et al. (2004). "Measurement of Poisson's ratio of dental composite restorative materials." Biomaterials 25(13): 2455-2460.

Conrad, H. J., W.-J. Seong, et al. (2007). "Current ceramic materials and systems with clinical recommendations: a systematic review." The Journal of prosthetic dentistry 98(5): 389-404.

Corazza, P. H., S. A. Feitosa, et al. (2013). "Influence of convergence angle of tooth preparation on the fracture resistance of Y-TZP-based all-ceramic restorations." Dental Materials 29(3): 339-347.

Dehnavi, M. Y., S. Khaleghian, et al. (2014). "Utilizing digital image correlation to determine stress intensity factors." Polymer Testing 37: 28-35.

Della Bona, A., P. H. Corazza, et al. (2014). "Characterization of a polymer-infiltrated ceramic-network material." Dental Materials 30(5): 564-569.

Dugdale, D. (1960). "Yielding of steel sheets containing slits." Journal of the Mechanics and Physics of Solids 8(2): 100-104.

E399-09, A. (2009). "Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness K Ic of Metallic Materials." ASTM International.

E1820-01, A. (2001). "Standard Test Method for Measurement of Fracture Toughness." ASTM International.

Fagerholt, E., C. Dørum, et al. (2010). "Experimental and numerical investigation of fracture in a cast aluminium alloy." International Journal of Solids and Structures 47(24): 3352-3365.

Fasbinder, D. J. (2010). "Materials for chairside CAD/CAM restorations." Compend Contin Educ Dent 31(9): 702-704.

Goetzen, N., G. Natt, et al. (2003). Numerical stress and reliability analysis of all-ceramic fixed-partial dentures. Proceedings of the 2003 Summer Bioengineering Conference, ASME, Key Biscayne.

Griffith, A. A. (1921). "The phenomena of rupture and flow in solids." Philosophical transactions of the royal society of london. Series A, containing papers of a mathematical or physical character: 163-198.

Guazzato, M., M. Albakry, et al. (2004). "Strength, fracture toughness and microstructure of a selection of all-ceramic materials. Part II. Zirconia-based dental ceramics." Dental Materials 20(5): 449-456.

Haldar, S., N. Gheewala, et al. (2011). "Multi-scale mechanical characterization of palmetto wood using digital image correlation to develop a template for biologically-inspired polymer composites." Experimental mechanics 51(4): 575-589.

Helm, J. D. (2008). "Digital image correlation for specimens with multiple growing cracks." Experimental mechanics 48(6): 753-762.

Hu, J., F. Liu, et al. (2011). "Failure analysis based on J-integral values: A case study of hydrogen blistering defect." Engineering Failure Analysis 18(3): 924-932.

Hutchinson, J. (1968). "Singular behaviour at the end of a tensile crack in a hardening material." Journal of the Mechanics and Physics of Solids 16(1): 13-31.

Inglis, C. E. (1997). "Stresses in a plate due to the presence of cracks and sharp corners." SPIE

MILESTONE SERIES MS 137: 3-17.

Irwin, G. and D. Wit (1983). "A summary of fracture mechanics concepts." Journal of Testing and Evaluation 11: 56-65.

Irwin, G. R. (1948). "fracture dynamics." fracturing of metals: 147-166.

Irwin, G. R. (1997). "Analysis of stresses and strains near the end of a crack traversing a plate." SPIE MILESTONE SERIES MS 137: 167-170.

Ivoclar Vivadent, A. (2009). Technical notes: IPS e. max CAD, instructions for use, Lichtenstein.

Jurim Dental Group (2014). "Single Visit Dentistry Service with CEREC Omnicam."

Kang, S.-H., J. Chang, et al. (2013). "Flexural strength and microstructure of two lithium disilicate glass ceramics for CAD/CAM restoration in the dental clinic." Restorative dentistry & endodontics 38(3): 134-140.

Kelly, J. and P. Benetti (2011). "Ceramic materials in dentistry: historical evolution and current practice." Australian dental journal 56(s1): 84-96.

Kelly, J. R. (2008). "Dental Ceramics." The Journal of the American Dental Association 139: S4-S7.

Kok, P., C. J. Kleverlaan, et al. (2015). "Mechanical performance of implant-supported posterior crowns." The Journal of prosthetic dentistry.

Krifka, S., M. Stangl, et al. (2009). "Influence of different cusp coverage methods for the extension of ceramic inlays on marginal integrity and enamel crack formation in vitro." Clinical oral investigations 13(3): 333-341.

Lichtenberger, R. and H. Schreier (2005). "Contactless and full field 3D-deformation measurement for impact and crash test." Article, Limess Messtechnik und Software GmbH.

Lin, C. L., Y. H. Chang, et al. (2010). "Finite element and Weibull analyses to estimate failure risks in the ceramic endocrown and classical crown for endodontically treated maxillary premolar." European journal of oral sciences 118(1): 87-93.

Mörmann, W. H., B. Stawarczyk, et al. (2013). "Wear characteristics of current aesthetic dental restorative CAD/CAM materials: two-body wear, gloss retention, roughness and Martens hardness." journal of the mechanical behavior of biomedical materials 20: 113-125.

Madani, D. A. (2010). Fracture Resistance of Non-metallic Molar Crowns Manufactured with CEREC 3D, University of Toronto.

Malakooti, M. H. and H. A. Sodano (2014). "Toughening response of a crack-tip surrounded by a local elastic gradient." Smart Materials and Structures 23(3): 035009.

Mogadpalli, G. P. and V. Parameswaran (2008). "Determination of stress intensity factor for cracks in orthotropic composite materials using digital image correlation." Strain 44(6): 446-452.

Niendorf, T., J. Dadda, et al. (2009). "Monitoring the fatigue-induced damage evolution in ultrafine-grained interstitial-free steel utilizing digital image correlation." Materials Science and Engineering: A 517(1): 225-234.

Ø ilo, M., K. Kvam, et al. (2013). "Clinically relevant fracture testing of all-ceramic crowns." Dental Materials 29(8): 815-823.

OROWAX, E. (1949). "Fracture and strength of solids." Rep. Prog. Phys 12: 185.

Orteu, J. J., T. Cutard, et al. (2007). "Application of stereovision to the mechanical characterisation of ceramic refractories reinforced with metallic fibres." Strain 43(2): 96-108.

Preis, V., T. Dowerk, et al. (2014). "Influence of cusp inclination and curvature on the in vitro failure and fracture resistance of veneered zirconia crowns." Clinical oral investigations 18(3): 891-900.

Reiss, B. and W. Walther (2000). "Clinical long-term results and 10-year Kaplan-Meier analysis of Cerec restorations." International Journal of Computerized Dentistry 3(1): 9-23.

Rice, J., P. C. Paris, et al. (1973). "Some further results of J-integral analysis and estimates." ASTM special technical publication(536): 231-245.

Rice, J. and G. Rosengren (1968). "Plane strain deformation near a crack tip in a power-law hardening material." Journal of the Mechanics and Physics of Solids 16(1): 1-12.

Rice, J. R. (1968). "A path independent integral and the approximate analysis of strain concentration by notches and cracks." Journal of applied mechanics 35(2): 379-386.

Ritzberger, C., E. Apel, et al. (2010). "Properties and clinical application of three types of dental glass-ceramics and ceramics for CAD-CAM technologies." Materials 3(6): 3700-3713.

Sadaqah, N. R. (2014). "Ceramic Laminate Veneers: Materials Advances and Selection." Open Journal of Stomatology 2014.

Shah, S. and J. C. Kishen (2011). "Fracture properties of concrete–concrete interfaces using digital image correlation." Experimental mechanics 51(3): 303-313.

Shen, B. and G. Paulino (2011). "Direct extraction of cohesive fracture properties from digital image correlation: a hybrid inverse technique." Experimental mechanics 51(2): 143-163.

Shen, B. and G. H. Paulino (2011). "Identification of cohesive zone model and elastic parameters of fiber-reinforced cementitious composites using digital image correlation and a hybrid inverse technique." Cement and Concrete Composites 33(5): 572-585.

Shibata, S., R. Gondo, et al. (2014). "Influence of surrounding wall thickness on the fatigue resistance of molars restored with ceramic inlay." Brazilian oral research 28(1): 01-08.

Sonza, Q. N., Á. Della Bona, et al. (2014). "Effect of the infrastructure material on the failure behavior of prosthetic crowns." Dental Materials 30(5): 578-585.

Sun, T., S. Zhou, et al. (2014). "Load-bearing capacity and the recommended thickness of dental monolithic zirconia single crowns." journal of the mechanical behavior of biomedical materials 35: 93-101.

Tourmedical (n.d.). "Zirconium Crowns." van Noort, R. (2012). "The future of dental devices is digital." Dental Materials 28(1): 3-12.

Vanlanduit, S., J. Vanherzeele, et al. (2009). "A digital image correlation method for fatigue test experiments." Optics and Lasers in Engineering 47(3): 371-378.

Vita Zahnfabrik (2003). "Vitablocs Mark II. Materials Science and Clinical Stuties." es. Brea, California: Vident.

Vita Zahnfabrik (2009). "VITA In-Ceram for CEREC/ CEREC inLab Directions for use Fabrication of the substructure."

Vita Zahnfabrik (2012). "VITABLOCS working instructions."

Wateree Family Dentistry (n.d.). "Cerec."

Wells, A. (1963). " Application of fracture mechanics at and beyond general yielding." British Welding Research Association Report: 563-570.

Westergaard, H. (1939). "Bearing Pressures and Cra; c];< si." Journal of applied mechanics.

Yeo, I.-S., J.-H. Yang, et al. (2003). "In vitro marginal fit of three all-ceramic crown systems." The Journal of prosthetic dentistry 90(5): 459-464.

Yilmaz, H., C. Aydin, et al. (2007). "Flexural strength and fracture toughness of dental core ceramics." The Journal of prosthetic dentistry 98(2): 120-128.

Zahnfabrik, V. (2000). "VITA CEREC Products Edition 11/00."

Zandparsa, R. (2014). "Digital imaging and fabrication." Dental Clinics of North America 58(1): 135-158.

Zhang, Y., I. Sailer, et al. (2013). "Fatigue of dental ceramics." Journal of dentistry 41(12): 1135-1147.

Zhu, X.-K. and J. A. Joyce (2012). "Review of fracture toughness (G, K, J, CTOD, CTOA) testing and standardizationXian-Kui Zhu."