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Quantitative Fracture Analysis of Etched Soda-Lime Silica Glass: Evaluation of the Blunt Crack Hypothesis


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QUANTITATIVE FRACTURE ANALYSIS OF ETCHED SODA LIME SILICA GLASS: EVALUATION OF THE BLUNT CRACK HYPOTHESIS By SALVATORE ALEXANDER RUGGERO A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL F ULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003

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Copyright 2003 by Salvatore Alexander Ruggero

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Dedicated to: My parents and my family

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iv ACKNOWLEDGEMENTS There are many people that I would like to thank for their help and support during the research and writing of my masters thesis. First of all, I would like to thank my advisor, Dr. John J. Mecholsky, for all of h is help and guidance with this project. He always steered me in the right direction, no matter how confusing my data appeared. I would also like to thank the other members of my committee, Dr. Darryl Butt and Dr. E. Dow Whitney, for their time and critiq ue of my thesis. In addition, I would like to thank Mr. Ben Lee of the dental biomaterials lab for his help with the mechanical testing for this project. I would also like to acknowledge my fellow lab mates in Rhines 227 for putting up with me for these p ast couple of years. On a more personal note, I would not be at this juncture without my parents. I could not ask for two better parents, as they have been a constant source of love and motivation. I thank my sisters, Adrianne and Lisa, for putting up w ith my antics and ballyhoo. I also thank my girlfriend, Melissa, for supporting me and enduring my incessant typing as of late to produce this document in a timely manner. Lastly, I would like to thank all my friends and family who have made this possible

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v TABLE OF CONTENTS page ACKNOWLEDGEMENTS........iv LIST OF TABLES.....vii LIST OF FIGURES......viii ABSTRACT.....x CHAPTERS 1 PUPROSE... 1.1 Research Rationale......1 1.2 Research Objectives....1 1.2.1 Specific Aim 1.......2 1.2.2 Specific Aim 2. .. 2 BACKGROUND. 2.1 Acid Etching of Glass. 2.1.1 Crack Tip Blunting....3 2.1.2 Principles of Etching......4 2.2 Fracture Mechanics Approaches to Failure........5 2.2.1 Inglis Analysis...5 2.2.2 Griffith Energy Balance Approach....7 2.2.3 Linear Elastic Fracture Mechanics Stress Analysis... 2.3 Fracture Toughness Measurements .......12 2.3.1 Fracture Surface Analysis.....12 2.3.2 Strength Indentation Technique...13 3 MATERIALS AND METHODS.15 3.1 Specimen Properties and Preparation... 3.1.1 Specimen Properties.....15 3.1.2 Microindentation...... 3.1.3 Annealing. 3.1.4 Etching.

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vi 3.2 Mechanical Property Determination. 3.2.1 Flexure Strength... 3.2.2 Fracture Toughness..23 3.3 Characterization of Fracture Surfaces...25 3.3.1 Light Microscopy.....25 3.3.2 Sca nning Electron Microscopy.... 4 RESULTS AND DISCUSSION.27 4.1 Mechanical Properties....27 4.1.1 Flexure Test Strength Results..27 4.1.2 Fracture Toughness. .....30 4.2 Fracture Mechanics Approaches Applied to Blunt crack.......33 4.2.1 Inglis Solution..33 4.2.2 Stress and Energy Criteria....35 4.3 Characterization of Fracture Surfaces. ...38 4.3.1 Fractographic Analysis.....38 4.3.2 Optical Microscopy......43 4.3.3 Scanning Electron Microscopy....45 5 CONCLUSIONS.52 APPENDIX TA BULATED DATA LIST OF REFERENCES BIOGRAPHICAL SKETCH..62

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vii LIST OF TABLES Table page 3.1 Physical and mechanical properties of Corning glass slides. 3.2 Preliminary 4 point flexure testing results to determine effect of annealing on strength..19 3.3 Preliminary 4 point flexure testing results to determine effect of acid etching on strength ..20 4.1 Comparison of fracture toughness data..32 4.2 Strength, fracture toughness, and crack size of unetched and etched samples A 1 4 point flexure results for unetched samples.54 A 2 4 point flexur e results for etched samples.....55 A 3 Fractography measurements and fracture toughness of unetched samples...56 A 4 Fractography measurements and fracture toughness of etched samples...57 A 5 Mirror size and mirror constant values f or unetched samples...58 A 6 Mirror size and mirror constant values for etched samples...59

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viii LIST OF FIGURES Figure page 2.1 Schematic of conditions for Inglis solution: Elliptical hole in a semi infinite plate... 2.2 Schematic of Inglis approach as ellipse becomes more slender and approximates a sharp crack.....6 2.3 Location of local stresses near a crack tip in cylindrical coordinates 2.4 Three loading modes..10 2.5 Schematic of a fracture surface of a material that failed in a brittle manner.12 2.6 Schematic of 4 point flexure test with side containing indentation as the tensile surface 3.1 Optical images of unetched and etched Vickers indentations... 3.2 Schematic of 4 point flexure apparatus used to do mechanical testing.....22 4.1 Flexure test strength results for 30 unetched and 30 etched samp les ordered from low to high..29 4.2 Fracture toughness of unetched samples using fractography method... 4.3 Effective fracture toughness of etched samples using fractography method. 4.4 Agreement between fractography method and strength indentation method for etched samples....32 4.5 Estimation of crack tip radius of etched samples using Inglis solution.. 4.6 Y factor projections for etched s amples versus strength....36 4.7 Y factor projection plotted versus estimated crack tip radii from Inglis

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ix 4.8 Blunting energy of etched samples....38 4.9 Fracture surfaces of unetched and etched samples showing stable crack growth... 4.10 Mirror to flaw size ratio for unetched and etched samples 4.11 Schematic of fracture surface features... 4.12 Fracture surfaces of unetched and etched samples showing failures at different stress levels....46 4.13 SEM images of etched radial surface cracks. 4.14 SEM images of unetched radial surface cracks.48 4.15 SEM fracture surface of etched sample at different magnifications..49 4.16 SEM fracture surface of etched sample at different magnifications..50 4.17 Comparison of crack tip radius calculated from Inglis and measured from fracture surface..51

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x Abstract of Thesis Presented to the Graduate School Of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science QUANTITATIVE FRACTURE ANALYSIS OF ETCHED SODA LIME SILICA GLASS: EVALUATION OF THE BLUNT CRACK HYPOTHESIS By Salvatore Alexander Ruggero May, 2003 Chairman: John J. Mecholsky, Jr. Major Department: Materials Science and Engineering The purpose of this study was to analyze the stre ngthening that occurs upon acid etching of soda lime silica glass. In particular, the hypothesis of crack tip blunting was analyzed and related to basic fracture mechanics equations. Based on flexure testing and subsequent fractography measurements, i t was determined that there was strengthening of the acid etched glass samples without a significant decrease in flaw size. Measurements of the mirror size confirmed an equal mirror constant for all samples, and a linear decrease in mirror to flaw size ra tio with an increase in strength for the non uniformly strengthened etched samples. In addition, the Inglis equation was used to estimate the crack tip radii of both the unetched and etched samples, and the blunting hypothesis was explained in terms of th e critical stress intensity factor as well as the Griffith fracture energy. Direct measurement of the radius of

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xi curvature of the critical indentation flaws was performed where possible, and estimates of the crack tip radius due to blunting at the surface were obtained.

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1 CHAPTER 1 PURPOSE 1.1 Research Rationale The strengthening of glass due to acid etching has been a topic of research in the ceramics community for nearly half a century, yet many questions s till remain unanswered concerning the mechanism of this strengthening. A spirited debate occurred in the literature during the 1970s and early 1980s concerning the possibility of blunting of the crack tip due to acid etching of glass, however the topic has been mainly absent from the literature for the last 15 years with no definitive answers to the fundamental topics at hand. The main obstacle when conducting this research had been the difficulty in finding definitive visual evidence of the blunting. This task is certainly more difficult than it seems, as evidenced by failure to do so in past research, as well as in the present study. Fractographic analysis offers an alternative method to study the differences between unetched and etched samples. The features on the fracture surfaces can provide unique information to determine if the crack tip blunting hypothesis is a reasonable assumption. 1.2 Research Objectives The main objective of this study was to confirm that acid etching caused a strength increase in indented soda lime silica glass flexure specimens, and to determine if the mechanism causing the strength increase was blunting of the crack tip. A secondary

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2 objective was to determine ways of accounting for the blunting mechanism in terms of fracture mechanics equations describing brittle fracture. 1.2.1 Specific Aim 1 Specific aim 1 was to characterize the fracture surfaces of both the unetched and etched samples to determine if evidence exists for the crack tip blunting mechanism. Crack size and mirror size measurements were made on the fracture surfaces. Also, measurements of the blunt crack tip radius were made on surface indentation cracks and critical cracks where possible. 1.2.2 Specific Aim 2 Specific aim 2 was to use measurements of fr acture surface features along with fracture mechanics equations to propose different ways of accounting for blunted cracks in brittle fracture. The different techniques used were the Inglis equation, the critical stress intensity parameter, and Griffiths energy equation.

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3 CHAPTER 2 BACKGROUND 2.1 Acid Etching of Glass 2.1.1 Crack Tip Blunting During the 1960s many researchers became interested in examining the strengthening due to acid etching of glass. Early work by Mould 1 and Proctor 2 first presented the idea of crack tip blunting as a possible explanation for the observed strengthening of soda lime glass due to etching. Pavelchek et al. performed etching experiments in hydrofluoric acid and water on abraded rods of soda lime silica glass. 3 4 They also came to the conclusion that some sort of crack tip blunting was occurring, and they made use of the Inglis analysis 5 to approximate the crack tip radius for both the sharp and blunted cracks; the radius was estimated to be about 2 nm for the unetched samples. Ito and Tomozawa performed more mechanical testing on abraded soda lime silica glass rods using water and Si(OH) 4 as etchants. 6 They proposed a dissolution and precipitation mechanism for the blunting of the crack tip and estimated the radius of curvature of the blunted cracks at about 3 nm using Inglis as well. In addition, Mecholsky et al. did further work on abraded soda lime silica glass rods. 7 Instead of only looking at strength, this research utilized crack si ze measurements and fracture energy estimates to explain the difference between the unetched and etched samples. The idea that the strengthening could be attributed to the blunted flaw requiring an initial energy to produce a sharp crack was

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4 proposed in this paper. Also, the notion that there might be evidence of blunting on the fracture surface was suggested. In opposition to the numerous advocates of the idea of crack tip blunting mentioned above, there were and are some drawbacks to the theory. Fir st of all, this idea seemingly contradicts the concepts of linear elastic fracture mechanics for sharp crack tips. Secondly, it is very difficult to directly observe a crack tip (rounded or not), so most of the evidence presented for crack tip blunting is indirect evidence. Thirdly, there are other phenomena that could explain the strength increase of etching depending on experimental conditions, such as residual stress release and slow crack growth. Lawn et al. 8 9 performed indentation studies on soda l ime silica glass, and came to the conclusion that the indentation induced cracks extend during etching, and that this extension relaxes residual stresses associated with the indentation. No matter which explanation is perceived as true, there still appear s to be no definitive answer to this question. 2.1.2 Principles of Etching The strength of materials that behave in a brittle manner is controlled by the flaws that are present in the material. These flaws can either be inherent in the microstructure of the material, or can be introduced during processing, machining, or through a number of other actions. The main point is that the strength of a material is inversely proportional to the square root of the size of the crack that initiates failure. In t he case of ceramic materials such as glass which are inherently brittle, decreasing the size and number of flaws present in a given amount of material becomes important. Acid etching of glass has traditionally been used to decrease the size of surface fla ws in glass materials, thus creating materials that fail at a higher stress. The acid reacts with the SiO 2

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5 in the glass and dissolves material from the surface. 10 This idea is especially important in industrial and commercial applications where a part mu st meet minimum stress requirements. 2.2 Fracture Mechanics Approaches to Failure 2.2.1 Inglis Analysis The paper published in 1913 by C.E. Inglis 5 entitled Stresses in a Plate Due to the Presence of Cracks and Sharp Corners laid the foundation for the theories of fracture mechanics for materials that fail in a brittle manner. The significance of the Inglis Analysis was that it provided an exact solution for the stress around an arbitrarily shaped elliptical flaw in a plate of material. Inglis stat ed his intentions well in his paper: The methods of investigation employed for this problem are mathematical rather than experimental. The main work of the paper lies in the determination of the stresses around a hole in a plate, the hole being elliptic i n form. The results are exact, and are consequently applicable to the extreme limits of form which an ellipse can assume. If the axes of the ellipse are equal, a circular hole can be obtained; by making one axis very small the stresses due to the existen ce of a fine straight crack can be investigated. (p.219) The elliptical flaw mentioned above has a major axis of 2a, a minor axis of 2b, and a radius of curvature of r. The flaw is contained in a large plate with an applied tensile stress, s ( Figure 2.1) Figure 2.1 Schematic of conditions for Inglis solution: Elliptical hole in a semi infinite plate

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6 From these starting conditions Inglis solved for the stress intensity at point A, and obtained the following general result: + = b a A 2 1 s s (2.1) He then assumed that if the ends of the flaw appeared to be elliptical, it was reasonable to approximate the shape of the cavity with that of an ellipse, and therefore the substitution b=(a r) 1/2 could be made. Making this substitution into equation 2.1 yields: + = r s s a A 2 1 (2.2) The general equation (2.2) can be simplified as the ellipse becomes more slender and starts to approximate a sharp crack (Fig 2.2). In this ca se 2b and r approach 0 as the ellipse becomes more slender and starts to resemble a sharp crack. Figure 2.2 Schematic of Inglis approach as ellipse becomes more slender and approximates a sharp crack

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7 First equation 2.2 can be written in the following f orm to show the simplification: + = r s s a A 2 1 2 (2.3) Since (a/ r) is much larger than the term is negligible and can be dropped out of the equation. The equation then simplifies to the following for a slender ellipse: = r s s a A 2 (2.4) The Inglis solution can be used to approximate the radius of curvature of a crack in a brittle material, and has been done in prior studies with varying success. 3 7 The idea is that in the absence of plasticity, the minimum radius ( r ) at the crack tip may be close to atomic equilibrium spacing, so the stress at point A can be approximated with the theoretical strength of the material. Using the theoretical strength of the material as s A the radius of a crack tip can be estimated because the crack size, a, and the global failure stress, s can be measured. 2.2.2 Griffith Energy Balance Approach A.A Griffith, a British scientist prominent in the early 1920s, performed groundbreaking work in the area of crack propagation and fra cture of brittle materials. His paper entitled The Phenomena of Rupture and Flow in Solids published in 1921 formulated a way of describing brittle fracture using strain energy concepts. 11 These ideas were predicated on the observations Griffith made reg arding the actual strength of materials compared to the theoretical strength. The prevailing idea at this time in regards to rupture of elastic materials was that an elastic solid ruptured when a maximum tensile

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8 stress was reached. A scratch in the mater ial created a stress concentration that amplified the stress according to the shape of the scratch, but the maximum stress was independent of the size of the scratch. Griffith disagreed with the afore mentioned idea, and proposed a new idea in his paper: If, as it is usually supposed, the materials concerned as substantially isotropic, there is but one hypothesis which is capable of reconciling all these apparently contradictory results. The theoretical deduction that rupture of an isotropic elastic materi al always occurs at a certain maximum tension is doubtless correct; but in ordinary tensile and other tests designed to secure uniform stress, the stress is actually far from uniform so that the average stress at rupture is much below the true strength of the material. The general conclusion may be drawn that the weakness of isotropic solids, as ordinarily met with, is due to the presence of discontinuities, or flaws, as they may be more correctly called, whose ruling dimensions are large compared with mole cular distances. The effective strength of technical materials might be decreased 10 or 20 times at least if these flaws could be eliminated. (p.180) Griffith performed experiments with elastic materials that showed that the resistance to rupture of a ma terial was in fact related to the size of the flaws present in the material. He did these experiments by rupturing glass test tubes with internal pressure. Griffith derived the net change in the potential energy of a plate from introduction of a crack as well as the surface energy of the crack. He then defined the condition for critical crack growth, and solved for the fracture stress. The results of this derivation are shown below: a E 2 s p g = s (2.5) Orowan 12 extended Griffith's theory to include the plastic surface energy term for the fracture of ductile materials:

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9 2 / 1 P s a ( E 2 p g + g = s (2.6) 2.2.3 Linear Elastic Fracture Mechanics Stress Analysis In addition to the energy approach to fracture discussed earlier, there was co nsiderable work done on deriving closed form expressions for the stress field around a crack tip. In particular, Westergaard, 13 Sneddon, 14 Williams, 15 and Irwin 16 were among the first people to publish such solutions. It is generally regarded that Irwins paper entitled Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate was the basis for modern fracture mechanics theory. In this paper Irwin showed that the local stresses near the crack tip are of the general form as follows: 16 .... ) ( f + = q s ij ij r K (2.7) In the above equation r and q are cylindrical coordinates of a point with respect to the crack tip (see Fig 2.3), and K is the stress intensity factor. He further showed that the strain energy app roach is equivalent to the stress intensity approach and that crack propagation occurs when a critical stress intensity K C is achieved Linear elastic fracture mechanics (LEFM) is based on the application of the theory of elasticity to bodies containing cracks or defects. The assumptions used in elasticity are also inherent in the theory of LEFM: small displacements and general linearity between stresses and strains. The general form of the LEFM equations is given in Eq. 2.8. As seen, a singularity exist s such that as r, the distance from the crack tip, tends toward zero, the stresses go to infinity. Since materials plastically deform as the yield stress is exceeded, a plastic zone will form near the crack tip. The basis of LEFM

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10 remains valid, though, if this region of plasticity remains small in relation to the overall dimensions of the crack and cracked body. Figure 2.3 Location of local stresses near a crack tip in cylindrical coordinates There are generally three modes of loading, which involve different crack surface displacements (Figure 2.4). The three modes are: Mode 1: opening or tensile mode (the crack faces are pulled apart) Mode 2: sliding or in plane shear (the crack surfaces slide over each other) Mode 3: tearing or anti plane shear (the crack surfaces move parallel to the leading edge of the crack and relative to each other) The following discussion deals with Mode 1 since t his is the predominant loading mode in most engineering applications. Similar treatments can be performed for Modes 2 and 3. Figure 2.4 Three loadi ng modes

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11 The stress intensity factor, K, which was introduced in Eq. 1, defines the magnitude of the local stresses around the crack tip. These local stresses can be related to the global stress and the geometry of the sample. Many closed form solutions have been derived for K concerning simple geometric situations, and more complex situations have been determined experimentally and numerically. 17 18 In general K depends on loading, crack size, crack shape, and geometric boundaries, with a general form given by: = w a f a K p s (2.8) Equation 2.8 contains the following terms that are defined as follows: s is the remote stress applied to component (not to be confused with the local stresses, s ij in Equation 7), a is crack length, and f ( a/w) is a correction factor that depends on specimen and crack geometry. Stress intensity factors for a single loading mode can be added algebraically. Consequently, stress intensity factors for complex loading conditions of the same mode can be determine d from the superposition of simpler results, such as those readily obtainable from handbooks. Of particular relevance to this study is the closed form solution for a semi elliptical surface crack. 19 2 / 1 2 1 c K a IC s f p = (2.9) Equation 2.9 contains the term f which is an elliptical integral of the second kind and goes from 1 for a slit crack to 1.57 for a semi circular crack. Mecholsky et al. showed that a semi elliptical crack could be approximated as an equivalent sem i circular crack for glass fractures. 20 When inserting the equivalent semi circular crack dimension, c=(ab) 1/2 and the value of f =1.57, the equation becomes:

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12 c Y K IC s = (2.10) In equation 2.10, Y becomes 1.24 for a surface crac k in a material with no residual stress. 2.3 Fracture Toughness Measurements 2.3.1 Fracture Surface Analysis Fracture surface analysis is an important tool that allows measurements to be taken at the fracture surface that can be related to the overal l fracture behavior of the material. As a crack initiates and propagates through a material, characteristic markings are left on the fracture surface that are related to the stress and energy of fracture. A schematic of a fracture surface is shown in figu re 2.5. Figure 2.5 Schematic of a fracture surface of a material that failed in a brittle manner The features of the fracture surface are as follows: The fracture surface origin is where the fracture originated. The a and 2b values on the diagram repr esent the depth and width of the initial and critical flaw. The mirror, mist, and hackle regions of the fracture surface represent the crack propagating at first in a comparatively smooth plane, and

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13 eventually gaining enough energy to go in a different di rection and branch. The boundaries labeled on the diagram as r 1 r 2 and r CB represent the mirror mist, mist hackle, and crack branching boundaries. These distances have been shown to be related to the global failure stress of the material with the follow ing empirical relationship 21 where M j is the mirror constant and K Bj is the stress intensity factor associated with the mirror/mist boundary: j B j j f K M Y r = = 2 / 1 s (2.11) The utility of fracture surface measurements is that the fracture toughn ess can be determined for a material from mechanical testing and subsequent crack size measurements. Mechanical testing such as flexure or tensile testing provides the stress at which the sample failed, the crack size is measured with a microscope, and th e fracture toughness can be solved for in an equation like 2.10 depending of the geometry and load application. 2.3.2 Strength Indentation Technique The strength indentation technique is another method of estimating the fracture toughness of a material. This method is based on Vickers micro indentation and the subsequent stress field that surrounds the indentation. Using fracture mechanics analysis for cracks in tensile loading, a strength formula can be derived to estimate K C The advantage of this m ethod is that it requires no fracture surface measurements. Instead, the information necessary is the failure stress of the sample, the elastic modulus and hardness, and the indentation load. The strength indentation equation developed by Lawn et al. 22 i s shown below, and has a calibration constant ( h v R ) included that was determined experimentally for a range of ceramic materials.

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14 ( ) 4 / 3 3 / 1 8 / 1 P H E K R v c s h = (2.12) The common method of performing this analysis is by indenting a s ample at a given load, and then fracturing the sample with either 3 or 4 point bending. A common experimental setup is shown below in Figure 2.6. The sample fails from a crack introduced by the indent, the failure stress is measured during the mechanical test, and the fracture toughness of the material can be estimated using equation 2.12. This value obtained with equation 2.12 can then be compared to the value obtained using equation 2.10 with a Y factor of 1.65 to account for the residual stress create d by the indentation. Figure 2.6 Schematic of 4 point flexure test with side containing indentation as the tensile surface

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15 CHAPTER 3 MATERIALS AND METHODS 3.1 Specimen Properties and Preparation 3.1.1 Specimen Properties The test sa mples eventually chosen for this study were Corning #0215 soda lime glass microscope slides with dimensions of 75mm 25mm 1 mm. The properties of the Corning glass slides are shown below (as received from Corning). Table 3.1 Physical and mechanical pr operties of Corning glass slides Common Names Standards Soda Lime Glass Glass commonly referred to as Flint Glass Type II Soda Lime Glass per A.S.T.M E 438 federal spec DD G 541b Composition ( percent app rox) Properties SiO 2 ...73% Coeff. Of Exp.89 x 10 7 cm/cm/C Na 2 O..14% Strain Point C CaO..% Anneal point..545 C MgO.% Soften Point....724 C Al 2 O 3 % Density....4 g/cm 3 Y oungs Mod.70 GPa Refract. Index.515 @ Sodium D Line Preliminary testing was originally started with Fisher microscope slides, until it was discovered that Fisher purchases their slides from other manufacturers. They receive the pre made slides and package them, so there could be variability in the composition of the slides. Corning was contacted, and it was determined that they produced all of their slides with soda lime glass of the same composition

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16 Glass slides were chosen for this research because they were of suitable composition and size for 4 point flexure testing, and they were readily available with no machining or cutting necess ary. The alternative of cutting and polishing samples would have been time consuming and could have introduced defects into the samples, which could have competed with the introduced indentation crack as the fracture origin. 3.1.2 Microindentation Mic roindentation can be used to introduce controlled cracks in glass in order to perform mechanical testing. 8,9 The appropriate indentation loads were determined through preliminary experiments. This indentation crack had to be large enough so that during th e 4 point flexure testing the sample would fracture from the introduced crack; if the introduced crack was too large, then the resulting crack could interact with the bottom surface of the sample, producing a flattened crack as opposed to the desired semi elliptical crack geometry. If this was the case, a correction factor would have to be introduced into the fracture mechanics equations, thus complicating the analysis. A total of 8 samples were indented at various loads using a Buhler Micromet 3 microhar dness indenter with a Vickers diamond tip. Two samples were indented each at loads of 100g, 500g, 1kg, and 2kg. The indentations were first examined with an optical microscope to evaluate the indent, and then the samples were fractured in 4 point bending using an Instron testing machine. The testing conditions used were a loading rate of 0.5mm/min, a lower span length of 20mm, and an upper span length 6.3mm. The fracture surfaces were then examined with the optical microscope to determine whether the sa mple failed from the crack introduced by the indent, and whether the crack had a fully rounded or flattened shape. From examining the indents, it was determined that the

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17 2 kg indent produced significant lateral cracking which meant that the load was too l arge for the specimen. Examining the fracture surfaces of the 2 specimens with 1 kg indent loads, it appeared that the crack was semi elliptical and was not interacting with the bottom surface of the specimen. Unfortunately, after indenting and fracturing 60 samples with the 1 kg indent load, it was determined that there was in fact an interaction with the bottom surface of the samples. This was confirmed by looking at the a/b ratio for these cracks. The a/b ratio came out to be 0.7 for the samples with the 1 kg indent load, as opposed to the normal value of 0.8. Another set of 60 samples then had to be prepared using the 500g indent load. The samples indented with the 500g load had cracks that were large enough to initiate failure, but were not so larg e as to have interaction with the bottom surface of the sample. Figure 3.1 shows optical images of the 500g Vickers indentation of both an unetched and an etched sample. Sharp radial cracks are apparent in the unetched sample, while it appears that the a cid has blunted the radial surface cracks in the etched sample. a) Unetched b) Etched Figure 3.1 Optical images of unetched and etched Vickers indentations 50 m m m 50 m m m

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18 The procedure for indentation was as follows. The sample was removed f rom the packaging, and using a t square the middle of the sample was marked with a fine sharpie marker. The sample was then placed in the microhardness indenter with the sharpie mark facing down, the surface was determined with the 20x magnification to the point where the mark was on the opposite side, and the 500g load was released with a dwell time of 20 seconds. The sample was then carefully placed in a ziploc bag and numbered. Each sample was indented in this manner to await further preparation and fl exure testing. 3.1.3 Annealing Annealing is the process of removing residual stresses in glass by re heating to a suitable temperature followed by controlled cooling. 10 The purpose of annealing the samples after they were indented was to remove the lo cal residual stress around the crack created by the indent. Preliminary experiments were performed to determine which (if any) annealing schedule would be suitable. The preliminary experiments were performed on samples that had been indented with a 1 kg l oad. Using the annealing point of the glass provided by Corning ( C), the following annealing schedule was set up: 25 C 390 C at 3 C per minute 390 C 495 C at 3 C per minute 495 C 545 C at 1 C per minute Hold at 545 C for 15 minutes 545 C 25 C at 1 C per minute A Box furnace was used with the samples placed on ceramic boats crack side up. 2 samples were annealed from 25 C following the above schedule, 2 samples were annealed starting from an elevated temperature of 390 C, and 2 samples were no t annealed to see if the residual stress would resolve itself. The 6 samples were then

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19 fractured with the Instron in 4 point bending with the same test conditions as stated before. The samples were analyzed with the optical microscope to determine crack sizes and corresponding fracture toughnesses. From examining the strength data in table 2 it appeared that there was no difference in the strength between the room temperature anneal and the elevated temperature anneal. The annealed samples did show high er strength than the unannealed samples, indicating that there was some residual compressive stress left in the unannealed samples. From this data it was decided that all samples would be annealed from room temperature following the indent. Table 3.2 Pre liminary 4 point flexure testing results to determine effect of annealing on strength Sample Annealing Condition Max Load (N) Max Stress (MPa) 1 Room Temp Anneal 71.9 56.7 2 Room Temp Anneal 67.9 53.5 3 Elevated Temp Anneal 68.6 54.0 4 Elevated Temp An neal 67.4 53.1 5 No Anneal 56.6 44.6 6 No Anneal 53.7 42.3 Unfortunately, no preliminary annealing experiments were done with etching included, so the adverse effect that annealing had on etching of the crack tip was not anticipated until after the fir st batch of 60 samples had been fractured. No strength increase was observed for the annealed etched samples, so it was determined that annealing somehow inhibited the acid from penetrating to the crack tip. Therefore, in addition to changing the indent l oad from 1 kg to 500 grams, the second batch of samples did not include annealing in the preparation process.

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20 3.1.4 Etching Acid etching of glass surfaces has traditionally been used as a surface treatment to remove or reduce flaws inherent in the glass. The strengthening effect of acid etching has mainly been attributed to the decrease in flaw size, but in fact there may be other processes at work. The acid etching of the indented glass samples in this study was designed to show that the increas e in fracture strength and fracture toughness of the samples was due to rounding of the crack tip, with the ultimate goal of developing a quantitative relationship between crack tip blunting and fracture toughness. The etching procedure in this study enta iled submerging the sample in dilute hydrofluoric acid. Preliminary experiments were performed to determine which concentration and submergence time was optimal. 6 samples were indented with a 1 kg load, etched at various concentrations and times, and fr actured in 4 point bending. The results are shown below in Table 3. Table 3.3 Preliminary 4 point flexure testing results to determine effect of acid etching on strength Sample Etching Procedure Max Load (N) Max Stress (MPa) 1 5% HF for 1 minute 55.9 44. 0 2 5% HF for 15 minutes 80.5 63.4 3 2.5% HF for 1 minute 55.0 43.4 4 2.5% HF for 15 minutes 114 90.0 5 1% HF for 1 minute No test No test 6 1% HF for 15 minutes 62.6 49.3 The results from this preliminary experiment showed that a 1 minute etch time was not enough to promote crack tip blunting. The 15 minute etch time did produce the desired strength increase, and the increase was most dramatic in the 2.5% HF solution. From this data it was decided that an etch solution of 2.5% HF with an etch time o f 15 minutes would sufficient to show the effect of etching.

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21 The etching protocol was a fairly simple procedure. Once the etching conditions were set at 2.5% HF for 15 minutes, a large batch of 2.5 % HF was mixed using 50% HF solution and distilled wate r. All of the samples were subsequently etched from this one batch of dilute HF. Standard polystyrene Petri dishes were used to house the etching procedure. The Petri dish was filled with the HF, the sample was dropped in with the indented surface facin g upwards, and the lid was placed on the Petri dish for the 15 minute duration of the etch. Once the 15 minutes was up, the sample was transferred to a Petri dish with distilled water and allowed to soak for 30 seconds, and then transferred to a similar P etri dish and allowed to soak for an additional 30 seconds. The sample was then removed from its bath and then rinsed thoroughly with distilled water from a squirt bottle over the sink. In this way all the acid was removed and the samples were able to be handled. This process was repeated for all etched samples. The HF acid solution was changed after 5 samples had passed through, and the resulting strength data showed no correlation between strength and the number of times the acid had been used. At thi s point the samples were now indented, annealed and etched and were ready for 4 point bend testing. 3.2 Mechanical Property Determination 3.2.1 Flexure Strength The flexure strength for all samples was performed using an Instron model 5500R with a 5 kN load cell. The testing configuration for all samples was a 4 point bend fixture with a 20 mm outer span and an inner span of 6.3 mm. The loading rate for all samples tested was 0.5mm/min. A schematic of the 4 point bend arrangement is shown in figur e 3.2 below.

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22 Figure 3.2 Schematic of 4 point flexure apparatus used to do mechanical testing The fracture stress of the samples was calculated using beam theory and the familiar equation for the 4 point flexure testing with th e loading span having one third the length of the support span: 2 bh PL f = s (3.1) The samples were prepared for testing by first locating the indent (either by eye of with the optical microscope). The loc ation of the indent was marked on the edge of the sample with a sharpie, and two lines were drawn across the width of the sample, each located 10mm from the indent. This allowed the sample to be placed easily in the 4 point bend fixture with the indentati on facing down, and centered within the fixture. Using this

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23 technique all the samples (with a few exceptions) failed from the introduced crack from the indent. For the samples that were etched, and therefore had a higher fracture stress, scotch tape was placed on the compressive side of each sample. This preservation technique kept the samples intact (for the most part). 3.2.2 Fracture Toughness The fracture toughness of the samples was evaluated using two techniques, the fractography method and the s trength indentation method. For the fractographic technique, the samples were analyzed using the light microscope with the filar eyepiece, and the critical crack size was measured for each sample. In addition, the mirror mist boundary size was measured. These values for the critical crack sizes were then used to calculate the fracture toughness of the samples with: c Y K f C s = (3.2) In the above equation, c is an equivalent semicircular crack of the semi elliptical crack with a minor axis radius of a and major axis diameter of 2b, where c=(ab) 1/2 s is the stress of the sample at failure, and Y is the geometric factor that depends on crack geometry and residual stress. For annealed samples the geometric factor used was 1.24, while fo r the non annealed samples that had local residual stress due to the indentation process, the geometric factor used was 1.65. All of these terms were tabulated in a spreadsheet for each set of samples, and in this way the average fracture toughness values were calculated (where applicable). In addition to the critical crack sizes, the mirror sizes were also measured and provided useful information. From the mirror size and the failure strength of the sample the mirror constant was calculated using the fo llowing equation: 21

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24 j B j j f K M Y r Y = = 2 / 1 s (3.3) In addition to the mirror constant, the mirror to flaw size ratio was calculated for all samples. These two important pieces of data obtained from the mirror size were tabulated in a spreadsheet f or both the unetched and etched samples. A second technique for calculating the fracture toughness of the samples was the strength indentation technique. Since the samples were indented before fracture with a specified load, it was very easy to apply the strength indentation technique to determine the fracture toughness, and check to see if it was in agreement with the fractography method This technique uses no measurement of crack features, but instead uses a radial indentation flaw as the origin of frac ture for a specimen in bending. The necessary strength formulas can then be derived to calculate K c based on the analysis of the indentation flaw in flexural failure: 22 ( ) 4 / 3 3 / 1 8 / 1 P H E K R v c s h = (3.4) The only terms needed in the above equa tion were E and H, since R v h is a calibration constant. The elastic modulus used was the standard modulus for glass of 72 GPa, the hardness was taken from Vickers microhardness measurements taken from the glass slides, and was approxim ately 5GPa. The other terms in the equation were known, and they were all tabulated into the same spreadsheet so a comparison of the fractography vs. strength indentation fracture toughness values could be made.

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25 3.3 Characterization of Fracture S urfaces 3.3.1 Light Microscopy The fracture surface analysis was done with an Olympus BHMJ stereomicroscope using an Olympus TGH light source. The microscope included a 10x eyepiece and 5x, 10x, 20x, and 50x objective lenses. A calibrated filar eyepie ce was used to take all measurements from the fracture surfaces of the samples. In addition, an Olympus digital camera was used in conjunction with the microscope to enable digital capture of the images. Each fractured sample was mounted in putty so the fracture origin was facing upwards, and was examined with the microscope. Each sample was first analyzed under low magnification (50x) to find the fracture origin, and then the magnification was increased (200x) to focus on the origin and make the measur ement with the filar eyepiece. The mirror mist boundary measurements were taken in a similar fashion with the filar eyepiece, except the magnification used was usually 100x. The samples that broke at a relatively low stress had a large mirror region, so it was necessary to move the field of view of the microscope while taking the measurement to obtain an accurate value. In order to do this, a feature had to be picked on the sample to serve as a reference point for measurement. In this manner, the size o f the fracture origin and the size of the mirror mist boundary were determined. 3.3.2 Scanning Electron Microscopy Additional analysis of the fracture surfaces as well as analysis of the blunted indentation surface cracks was performed with a JEOL 6400 SEM. The samples were prepared for the SEM by cleaning with ethanol, mounting on aluminum stubs with putty,

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26 covering the putty with carbon paint, and sputter coating with Gold Palladium. Pictures were taken at various magnifications and angles of tilt t o get an idea of how the fractures occurred at all perspectives.

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27 CHAPTER 4 RESULTS AND DISCUSSION In the following sections, the majority of the data is presented in graphical format. First the strength an d fracture toughness of the unetched and etched samples is presented. Next, the application of fracture mechanics to analyze the cracks at the origin of failure of the etched glass is discussed. Finally, an analysis of the fracture surfaces along with ob servations of the crack tip radii at the surface and on the fracture surface (where possible) is presented. The data for the individual specimens can be found in the appendix. 4.1 Mechanical Properties 4.1.1 Flexure Test Strength Results The streng th data for the unetched and etched samples is summarized in figure 4.1. Based on the 30 unetched samples, the average stress at failure was 52 3 MPa. The etched samples had a much higher failure stress on average, and also showed a great deal of variab ility. The average stress observed based on the 30 etched samples was 190 80 MPa. These strength values clearly indicate that the etched samples in general had a greater fracture stress than the unetched samples. It is also important to analyze the var iability of strength in the etched samples. Some etched samples failed at a stress very near to the baseline unetched samples (about 52 MPa), while some etched samples failed at over 300 MPa. In assessing these strength results it is clear that the HF ac id etching

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28 acted as a strengthening agent in the glass samples. Three alternative ideas to account for strengthening of etched glass should be mentioned. The first idea is that the etching process dissolves away material at the surface, thus making the cracks smaller and increasing the fracture strength. This idea will be addressed later when fractography is used to measure the size of the cracks. The second idea is that the material could have a local tensile residual stress located at the indentation This tensile stress would then be relieved due to etching and the fracture strength would increase accordingly. This idea could possibly explain a strength increase due to etching, but not the magnitude of strengthening observed in this study. Previou s studies have shown that the local residual tensile stress of an indentation has the effect of lowering the strength of the order of 10 20 MPa, and can be relieved with annealing. 8,9 Preliminary annealing studies presented in this study agree with previo us studies. The relief of local residual stress might contribute to a small portion of the strengthening, but it is unlikely that it accounts for the magnitude and variability of strengthening observed here. Consider that relief of residual stress can acco unt for at most a strengthening of 20 MPa, whereas increases of strength of over 200 MPa were observed here. The third idea and the primary purpose of this research is to determine if the strengthening of glass due to acid etching is caused by rounding of the crack tip. This hypothesis can explain the large variability in the strength data of the etched samples. Although the indentation load was the same for each specimen, there was variability due to local surface variability, contact variability, an d a number of other factors inherent in the indentation process. Therefore, in this sense, each indentation is unique. Since each sample had a unique indentation flaw introduced into it, it is reasonable to assume that

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29 the etching process did not round th e crack tips uniformly. Thus, the strength increase due to etching should not be expected to be the same. The hypothesis is that samples that fractured at a lower stress had less rounding of the crack tip, while the samples that fractured at a higher st ress had more crack tip rounding. To support this hypothesis, the apparent fracture toughness, the apparent blunting energy, and the rounding of the radial surface cracks were examined. 0 50 100 150 200 250 300 350 400 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Sample Stress (MPa) Unetched Etched Figure 4.1 Flexure test strength results for 30 unetched and 30 et ched samples ordered from low to high

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30 4.1.2 Fracture Toughness The fracture toughness of all samples was determined using a quantitative fractographic method (FM) and the strength indentation method (SIM). Figures 4.2 and 4.3 summarize the fracture toughn ess of the etched and unetched samples respectively using the fractography method. The average FM fracture toughness of the unetched samples was 0.75 .05 MPa m 1/2 while the effective FM fracture toughness of the etched samples ranged from 0.8 to 3 MPa m 1 /2 Note that this is called the effective fracture toughness because it is expected that the structure of the glass is not changing, but rather the crack is being affected. Hence, the true toughness of the glass is not expected to change. The average FM fracture toughness value for the unetched samples of 0.75 Mpa m 1/2 agrees very well with the fracture toughness of soda lime silica glass reported in the literature. 23 24 Figure 4.2 shows the fracture toughness is independent of the fracture strengt h of the samples. This is the expected result because the fracture toughness of an isotropic brittle material should be constant. On the other hand, figure 4.3 shows that the effective fracture toughness of the etched samples depends strongly on failure stress. This graph suggests that the crack size has not changed, while the strength of the samples increased. The strength indentation fracture toughness values agreed well with the fractography fracture toughness values for both etched and unetched sampl es. The average strength indentation fracture toughness for the unetched samples was 0.75 .06 Mpa m 1/2 while the effective fracture toughness of the etched samples ranged from 0.9 to 3 Mpa m 1/2 Table 4.1 summarizes the fracture toughness data from both techniques:

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31 Figure 4.2 Fracture toughness of unetched samples using fractography method Figure 4.3 Effective fracture toughness of etched samples using fractography method 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 100 200 300 400 Stress (MPa) K eff (MPa *m 1/2 ) 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 40 45 50 55 60 65 Stress (MPa) K c (MPa *m 1/2 )

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32 Table 4.1 Comparison of Fracture Toughness Data Fractography Strength Indentation Unetched K C = 0.75 .05 MPa*m 1/2 K C = 0.75 .06 MPa*m 1/2 Etched K eff : 0.8 3 MPa* m 1/2 K eff : 0.9 3 MPa* m 1/2 Figure 4.4 shows the agreement of the two methods for calculating fracture toughness of the etched samples. The line drawn shows a 1:1 correlation, so the two methods are in good agreement. Figure 4.4 Agreement between fractography method and strength indentation method fracture toughness for etched samples The fracture toughness of the etched samples is referred to as K e ff as opposed to K C since the formulas derived for K C were based on the stress distribution around a sharp crack 13 16 Since a blunted crack is the proposed idea for the strength increase, it would be incorrect to call the resulting fracture toughness K C It is not clear how the blunted crack should be handled in terms of fracture mechanics equations. It is clear that the etched samples still fail in a brittle manner from observing the fracture surface features

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33 and looking at the load displacement curves, so presumably the crack starts out blunted, and then becomes sharp before failure. There are a number of possible ways to account for crack tip blunting in the fracture mechanics equations. The first way, as already described, is to use the fracture me chanics equations as they are, and call the resulting fracture toughness the effective fracture toughness. Alternatively, the Y factor which accounts for crack geometry and loading application could be adjusted to account for crack tip blunting. Finally, the blunted crack can be accounted for in the energy balance approach to fracture. 4.2 Fracture Mechanics Approaches Applied to Blunt Crack 4.2.1 Inglis Solution One way to estimate the radius of curvature of the blunted cracks is to use the Inglis so lution. Solving the Inglis equation for the radius of curvature gives the following expression: 2 2 4 Th f a s s r = (4.1) The radius of curvature of both the unetched and etched samples was estimated in this way where Th is taken to be the theoretical stress of the material, and f is the failure stress of the sample. Using the Inglis equation the radius of curvature of the unetched samples was calculated to be 3.2 .3 nm, while the radius of curvature of the cracks of the etched samples ra nged from 3.4 nm to 60 nm. Figure 4.5 shows how the failure stress of the samples increases with increasing radius of curvature. This relationship followed a linear trend and makes sense in terms of the proposed blunting mechanism, since equation 4.1 pred icts that the failure stress of the sample is proportional to the square root of the radius of curvature. The estimated radius of curvature calculated for the etched

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34 samples was greater than that for the unetched samples, which is what was expected in ter ms of the blunting mechanism. When considering the magnitudes of the crack tip radii calculated from the Inglis equation, the estimates seem fairly reasonable. The unetched samples contained cracks that were presumably atomically sharp, and performing a rough estimate that each atom and interatomic bond is approximately 0.3 nm, the 3.2 nm crack tip radius calculated for the unetched samples corresponds to about 5 atoms at the crack tip. This estimate is similar to the 2 nm crack tip radius calculated b y Pavelchek and Doremus. 3 However, they did not calculate a large increase in the radius of curvature of their etched samples, and attributed this to the etchants not being able to penetrate to the crack tip. This is not the case in the present study, as the estimated radii of curvature for the etched samples became quite large compared to the unetched samples. We must keep in mind that the Inglis equation was developed for an elliptical hole in an infinite plate of material, so this calculation can only provide a rough estimate for the crack tip curvature in this situation. Since the equivalent semicircular crack size was used when calculating the Inglis estimates, it might be appropriate to introduce a correction factor to account for this difference. This correction factor would appear as (a) 1/2 =( f e / f s )(c) 1/2 where f e and f s are elliptical integrals of the second kind for the elliptical crack and circular crack, respectively. Applying this correction factor to the unetched samples that had an average a/b ratio of 0.6 0.1 (which is a measure of the ellipticity of the crack), the resulting ( f e / f s ) ratio would be 1.39/1.57, or approximately 0.9. Applying this correction factor to the unetched samples, we see that the crack tip radius is reduced by the factor of 0.9, resulting in a crack tip radiu s of 2.9 .3 nm. The radius of curvature for the etched

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35 samples would change in the same manner according to the correction factor associated with the etched flaws. This correction factor makes sense in terms of making the radius of curvature of the unet ched flaw smaller, and thus more reasonable in terms of atomic dimensions. Actual measurements of the sharp and blunted crack tips are needed to compare to these approximations. This topic will be addressed later. Figure 4.5 Estimation of crack tip r adius of etched samples using Inglis solution 4.2.2 Stress and Energy Criteria Fracture mechanics equations for materials that fail in a brittle manner should be able to account for the crack tip blunting theory if the blunted crack sharpens before frac ture. Based on this premise, there are several ways to factor crack tip blunting into fracture mechanics equations. The first method is to incorporate the degree of blunting into the Y factor of the fracture mechanics stress equation K C =Y(c) 1/2 Since the critical 0 50 100 150 200 250 300 350 400 0 2 4 6 8 10 (Crack tip radius) 1/2 nm 1/2 Strength (MPa) 4 16 36 64 100 nm

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36 stress intensity factor should be a material property, we can set K C = 0.75 MPa*m 1/2 and calculate the appropriate Y factor for the etched samples to satisfy the equation. Figure 4.6 shows graphically how the calculated Y fac tor varies for the etched samples. The etched samples that failed at lower stresses and had less crack tip blunting had Y factors close to the Y factor of 1.65 used for the unetched samples. The Y factors then dropped off linearly with 1/ s f as predicted by the fracture mechanics equation. The goal of this approach would be to incorporate the initial crack tip radius into the formulation of the Y factor. TheY factor is plotted versus 1/( r ) 1/2 in Figure 4.7, and again follows a linear trend as predicted by the Inglis equation. Figure 4.6 Y factor projections for etched samples versus strength 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 0.000 0.005 0.010 0.015 0.020 1/ s s f (1/MPa) Y-factor projection

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37 Figure 4.7 Y factor projection plotted versus estimated crack tip radii from Inglis Another way to account for crack tip blunting using fracture mechanics equa tions is to the extra energy necessary to sharpen a blunted crack. For the purposes of this discussion, we will assume plane stress conditions: f C E K g 2 = (4.2) Where E is the elastic modulus, and g f is the fracture energy. Equatio n 4.2 was then used to estimate the fracture surface energy of the material by using K C =0.75 and E=70 GPa; the resulting fracture surface energy of the glass was estimated to be 4 N/m. Since the material failed in a brittle manner, it is assumed that in t he etched samples the blunted crack was sharpened before failure. This sharpening of the blunted crack can be thought of as an added energy barrier to failure, and could be described in a modified version of equation 6 which is similar to Orowans equatio n 12 for failure of ductile materials: 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 0 0.1 0.2 0.3 0.4 0.5 0.6 1/( r r ) 1/2 (1/nm 1/2 ) Y-Factor Projection

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38 ( ) B f eff E 2 K g + g = (4.3) The B term in equation 4.3 can be thought of as the energy necessary to sharpen the blunted crack before failure. Samples with more highly blunted cracks will require more ene rgy to sharpen the crack and will therefore have a higher effective toughness. Figure 4.8 shows graphically the so called blunting energy plotted against effective fracture toughness for the etched samples. The effective fracture toughness increased linea rly with ( g B ) 1/2 as predicted by equation 4.3. The samples that failed at higher stresses had higher effective fracture toughness, and therefore required more energy to failure in the form of sharpening the blunted crack tip. Figure 4.8 Blunting en ergy of etched samples 4.3 Characterization of Fracture Surfaces 4.3.1 Fractographic Analysis Upon examination of the fracture surfaces it eventually became clear that there was a degree of stable crack growth during 4 point flexure testing. This stable crac k growth was observed in the unetched samples as well as the etched samples that failed at 0 1 2 3 4 5 6 7 8 9 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 K eff (MPa m 1/2 ) (g (g B B ) ) 1/2 1/2 (N/ (N/ m ) ) 1/2 1/2

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39 lower stresses, and presumably whose crack tips stayed relatively sharp. Figure 4.9 shows optical images of fracture surfaces of both unetched and etched samples. T he evidence for stable crack growth can just be made out between the arrows in the unetched sample. The etched samples that failed at higher stresses did not exhibit noticeable stable crack growth, possibly because residual stress was relieved, and/or the crack tip blunting mechanism inhibited any stable crack growth. Using the optical microscope to assess the fracture surface, the initial flaws were easy to find, but the critical flaws due to stable crack growth were more difficult to ascertain. a) Unetc hed, s f = 50 MPa b) Etched, s f = 145 Mpa Figure 4.9 Fracture surfaces of unetched and etched samples showing stable crack growth Crack size was an important factor in the fractography measurements. All of the semi elliptical flaws were converted to an equivalent semi circular crack size by the relation c=(ab) 1/2 so we can compare equivalent semi circular crack sizes for the unetched and etched samples. 20 The unetched samples had an average crack size of 74 6 m, while the average crack size for t he etched samples was 48 5 m. This difference in critical crack size can mainly be attributed to two factors. First, there was more stable crack growth in the unetched samples, which made the critical crack size

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40 larger. For the most part the etched s amples were able to resist stable crack growth which could be due to the blunting mechanism. Secondly, the etching process might have dissolved away material at the surface, thus making the surface flaws smaller to start with. This process would manifest itself by creating smaller critical cracks in both width and depth. Table 4.2 provides crack size along with strength and fracture toughness for both unetched and etched samples. The table provides average values for the unetched samples, and values for selected etched samples to show the range of the data. Table 4.2 Strength, fracture toughness, and crack size of unetched and etched samples Crack size (m) Strength (MPa) Fracture toughness (MPa*m 1/2 ) a/b ratio Unetched 74 6 52 3 0.75 .05 0.6 0.1 Etched 56 66 .82 0.4 Etched 47 150 1.69 0.8 Etched 45 232 2.55 0.9 In addition to crack size, the a/b ratio is a useful measure that provides information about the shape of the crack. A normal a/b ratio for semi elliptical cracks is 0.8. 19 Tabl e 4.2 shows a/b values for both unetched and etched samples, and stable crack growth is a possible explanation why an average a/b ratio of 0.6 was obtained for the unetched samples, as opposed to the normal 0.8. Less stable crack growth was observed in th e etched samples, which is why the average a/b ratio for the etched samples was 0.7. This means that the stable crack growth occurred mainly in one direction, creating elongated critical flaws. Now that the strength and fracture toughness results have b een presented, we can assess the different possibilities for the observed strength increase. There are several possibilities for the observed strength increase. If the flaw size decreases, then the

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41 strength increases. If tensile residual stresses are reli eved, there would be a strength increase. If the crack tip becomes less sharp, then the strength would increase. We see in Table 4.2 that the crack sizes of the etched specimens were slightly smaller than those of the unetched specimens. To assess the s trengthening that should occur due to the decrease in critical flaw size of the etched samples, the familiar fracture mechanics equation can be used: c Y K c f = s (4.4) Assuming a sharp crack with c=48m, K c =.75 MPa*m 1/2 and Y=1.65, the strength should only increase to about 66 MPa, as compared to 52 MPa for the unetched samples, which is a strength increase of about 14 MPa. This is obviously much smaller than the strengthening observed for most of the etched samples. Even for the smallest observed crack size of 39 m m, the strength increase would be about 20 MPa, which is clearly not enough to account for the observed strength increase with etching. Relief of residual stress is another possibility for the strength increase. Howeve r, in examining previous studies and preliminary experiments from this study, the strength increase due to relief of residual stress is approximately 15 20 MPa. Again, this value is much less than the strength increase observed in most of the etched sampl es. In fact, if the decrease in crack size and relief of residual stress are combined to provide a net strength increase of about 40 MPa, they still do not come close to accounting for the observed results. This leads to the conclusion that the majority of the strength increase is most likely due to blunting of the crack tip.

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42 In addition to measuring the crack size, the mirror size was measured for each sample. This distance is represented in Figure 4.11 as r 1 (r i where i=1), and has been shown to be r elated to the global stress at failure through the following equation: 21 j B j j f K M Y r Y = = 2 / 1 s (4.5) This equation implies that the mirror region ends and the mist region begins once a certain stress intensity (K Bj ) is reached. Thus the mirror c onstant, M 1 should be constant for a given material just like K C The only difference is that the mirror constant is associated with the mirror/mist transition and is not in the vicinity of the crack. Because M 1 is not in the vicinity of the crack then it would be reasonable to assume that the unetched and etched samples should have similar mirror constants. When looking at the mirror sizes of the samples it was determined that the unetched samples had a mirror constant of 1.9 0.1 MPa*m 1/2 and the etc hed samples had a mirror constant of 1.7 0.2 MPa*m 1/2 This value for the mirror constant was consistent with findings in previous studies. 25,26 Unbalanced ANOVA was performed to see if there was a statistically significant difference between the mirro r constants of the unetched and etched samples. The test showed that there was no statistical difference between the mirror constants of the unetched and etched samples. The next important relationship to consider is the mirror to flaw size ratio (r 1 /c) for the unetched and etched samples. From looking at the fracture surfaces, it was clear that the mirror to flaw size ratio was decreasing for the etched samples. This decrease in mirror to flaw size ratio also appeared to occur with samples that failed at higher stresses. This pattern can be seen graphically in figure 4.10. The unetched samples had a mirror to flaw size ratio of 14 1 which was similar to the value obtained in a previous study 25

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43 while the etched samples ranged from a mirror to flaw s ize ratio of about 10 to 1.8. As the graph indicates, the mirror size was measured for only 11 of the relatively lower strength samples out of the 30 etched samples. This is because for the higher strength samples the mirror was too small to be measured w ith the optical microscope, or the fracture surface was lost due to shattering of the high energy fracture. As noted before, the crack size did not change significantly among the etched samples. Thus, the change in the mirror to flaw size ratio is due t o the decrease in mirror size corresponding to the increase in strength. The extrapolation of the line in figure 4.7 should be extended to an r/c of 1, because obviously the mirror size cannot get smaller than the crack. What happens in reality is that s omewhere near the r/c ratio of 1, the original crack no longer becomes the dominant crack and the fracture originates at another location. The mirror to flaw size ratio for the crack at this new location should then return to the normal value of about 10 15. 4.3.2 Optical Microscopy Originally the atomic force microscope was designated as the instrument best suited to measure the crack tip radius of the samples, but after many failed attempts it was abandoned partly due to my inexperience on the instrumen t and partly due to the difficulty of the task. First of all it was difficult to obtain a decent image of the sample, and once a reliable image was produced it was difficult to discern where on the fracture surface the tip of the AFM was scanning. In add ition, the nature of the task was unclear at best. Since the facture surface was examined, the flaw had already propagated through

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44 Figure 4.10 Mirror to flaw size ratio for unetched and etched samples Figure 4.11 Schematic of fracture surfa ce features 0 20 40 60 80 100 120 140 160 1 4 7 10 13 16 19 r m /c Stress (MPa) etched unetched

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45 the material, so the best we could do would be to see if there was a difference between the fracture surfaces of the unetched and etched samples on the atomic level that could be attributed to crack tip blunting. Since there is no way to obse rve the critical crack in a material before failure, the other techniques we had available to obtain indirect visual evidence of crack tip blunting were optical microscopy and SEM. Light microscopy was used to measure the crack size and mirror size of th e unetched and etched samples. Figure 4.12 shows optical images of 3 fracture surfaces; one unetched sample and two etched samples that failed at varying stress levels. The unetched sample failed at a stress of 50 MPa, and is characterized by a large mir ror region. The etched samples clearly show a decrease in the fracture mirror size, and hence a decrease in mirror to flaw size ratio with increasing failure stress. The mist and hackle region can be seen in the first etched sample, while the final etched sample exhibited such a high stress and high energy failure that the origin was not found. 4.3.3 Scanning Electron Microscopy Scanning electron microscopy was also used to look for differences between the unetched and etched fracture surfaces, and was a ble to do so at higher magnifications. It was also used to examine unetched and etched radial cracks associated with the indentations. This was done after the samples were fractured by placing the sample halves flat on the aluminum stubs and locating wha t remained of the indentations. The radial cracks emanating from the indentations were located for both the unetched and etched samples. Figure 4.13 shows radial surface cracks from an etched sample, and figure 4.14 shows an unetched sample. From these p ictures the etched radial cracks were estimated to have a crack tip radius of approximately 4000 nm, while the unetched radial

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46 cracks were estimated to have a crack tip radius of about 15 nm. These values are very different than the values obtained using the Inglis solution, as shown in Figure 4.17. The etched crack tip curvature for the critical flaw with Inglis was about 30 60 nm, while the etched surface radial cracks were measured to be about 4000 nm. This discrepancy makes sense since you would expe ct the flaws at the surface to be rounded a great deal more than the critical flaw. The unetched crack tip radius using Inglis was about 3 nm compared with the SEM estimate of 15 nm. This discrepancy is related more to the limitations in the resolution of the SEM. Figures 4.15 and 4.16 show two etched fracture surfaces. The mirror/mist boundary can clearly be seen in both fracture surfaces. a) Unetched, s f = 50 MPa a) Etched, s f = 145 Mpa a) Etched, s f = 290 Mpa Figure 4.12 F racture surfaces of unetched and etched samples showing failures at different stress levels

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47 Figure 4.13 SEM images of etched surface radial cracks

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48 Figure 4.14 SEM images of unetched radial surface crack

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49 Figure 4.15 SEM fracture surface of etched sample at different magnifications

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50 Figure 4.16 Fracture surface of etched sample at different magnifications

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51 Figure 4.17 Comparison of crack tip radius calculated from Inglis and measured from fracture surface 0 50 100 150 200 250 300 350 400 0 2 4 6 8 10 (Crack tip radius) 1/2 nm 1/2 Strength (MPa) 4 16 36 64 100 nm Inglis Measured

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52 CHAPTER 5 CONCLUSIONS The objectives of this study were to 1) char acterize the fracture surfaces of both the unetched and etched samples to determine if evidence for crack tip blunting exists, and 2) use measurements of fracture surface features along with fracture mechanics equations to propose different ways of account ing for blunted cracks in brittle fracture. Based on the results of this study, the following conclusions can be made. Relief of tensile residual stress, and decrease in flaw size due to etching only accounts for a small portion of the observed strengthe ning of soda lime silica glass. The majority of the strengthening is thought to be due to rounding of the crack tip. The crack tip rounding hypothesis can explain the variability in the strength data because each unique critical indentation crack was not etched the same amount, so each crack was rounded a different amount. Direct measurements of the radius of curvature of the critical indentation crack after fracture were obtained where possible, and the radius of curvature of both unetched and etched radial surface cracks was estimated using the SEM. The fracture mechanics approaches to describe the blunted crack appear to be realistic. The Inglis solution provided an estimate of the radius of curvature of both unetched and etched cracks that appea red reasonable in terms of atomic dimensions. The crack tip radii estimated with Inglis along with the calculated Y factors provides a starting point to account for crack tip blunting in the stress intensity factor. In terms of

PAGE 64

53 energy, the Griffith equati on can be modified to include an extra energy term for the blunted crack to become sharp before fracture. A direct correlation was found between the mirror to flaw size ratio and the strength of the etched samples. Thus, it is possible to relate the mir ror to flaw size ratio to the crack tip curvature of the blunted crack.

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54 APPENDIX CALCULATED DATA Table A 1 4 point flexure results for unetched samples Sample Max Load (N) Max Stress (Mpa) 65 65 52 66 67 54 67 63 50 68 63 51 69 61 49 70 63 51 71 64 51 72 63 50 73 66 53 74 65 52 75 62 50 76 63 50 77 64 51 78 65 52 79 67 54 80 69 55 81 69 55 82 73 58 83 67 54 84 65 52 85 67 53 86 64 51 87 74 60 88 65 52 89 71 57 90 63 51 91 67 54 92 69 55 93 62 50 94 63 51 Mean 66 52 St. Dev. 3 3 Min 61 49 Max 74 60

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55 Table A 2 4 point flexure results for etched samples Sample Max Load (N) Max Stress (Mpa) 95 231 185 96 89 71 97 82 66 98 140 112 99 83 67 100 246 196 101 290 232 102 366 293 103 144 115 104 188 150 105 253 203 106 203 162 107 182 145 108 473 378 109 175 140 110 330 264 111 360 288 112 330 264 113 255 204 114 247 197 115 336 269 116 98 78 117 210 168 118 209 167 119 234 187 120 356 285 121 154 123 122 3 45 276 123 381 304 124 136 109 Ave 237 190 St. Dev. 100 81 Min 82 66 Max 473 378

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56 Table A 3 Fractography measurements and fracture toughness of unetched samples Fractography Strength Stress Kc Indentation Kc a/b Sample a(um ) 2b(um) c(um) Mpa (MPa*m.5) (MPa*m.5) Ratio 65 52 197 71 52 0.73 0.75 0.52 66 56 193 74 54 0.76 0.76 0.59 67 61 240 86 50 0.76 0.73 0.51 68 56 212 77 51 0.73 0.73 0.53 69 66 235 88 49 0.75 0.71 0.56 70 56 216 78 51 0.74 0.73 0.52 71 61 221 82 51 0.76 0.73 0.55 72 61 226 83 50 0.76 0.73 0.54 73 52 202 72 53 0.74 0.76 0.51 74 66 188 79 52 0.76 0.74 0.7 75 61 235 85 50 0.75 0.72 0.52 76 56 197 75 50 0.72 0.73 0.57 77 61 212 80 51 0.76 0.74 0.58 78 61 202 79 52 0.76 0.74 0.6 79 52 183 69 54 0. 74 0.77 0.56 80 47 179 65 55 0.73 0.78 0.53 81 52 183 69 55 0.76 0.78 0.56 82 47 155 60 58 0.75 0.82 0.61 83 56 183 72 54 0.75 0.76 0.62 84 47 212 71 52 0.72 0.74 0.44 85 52 188 70 53 0.73 0.76 0.55 86 56 207 76 51 0.74 0.74 0.55 87 47 188 66 60 0. 8 0.83 0.5 88 56 197 75 52 0.74 0.74 0.57 89 56 150 65 57 0.76 0.8 0.75 90 56 197 75 51 0.72 0.73 0.57 91 52 188 70 54 0.74 0.76 0.55 92 61 165 71 55 0.76 0.78 0.74 93 56 202 75 50 0.71 0.72 0.56 94 56 212 77 51 0.74 0.73 0.53 Mean 56 199 74 52 0. 75 0.75 0.6 St. Dev. 5 21 6 3 0.05 0.06 0.1

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57 Table A 4 Fractography measurements and fracture toughness of etched samples. Blank samples indicate crack size could not be measured Strength Stress Fractography Indentation a/b Sampl e a(um) 2b(um) c(um) MPa Kc (MPa*m.5) Kc (MPa*m.5) Ratio 95 38 103 44 185 2.02 1.93 0.73 96 38 160 55 71 0.87 0.94 0.47 97 38 169 56 66 0.82 0.89 0.44 98 42 103 47 112 1.27 1.33 0.82 99 33 174 53 67 0.8 0.9 0.38 100 38 99 43 196 2.13 2.02 0.76 101 42 94 45 232 2.55 2.29 0.9 102 293 2.73 103 38 99 43 115 1.25 1.35 0.76 104 42 103 47 150 1.69 1.66 0.82 105 33 94 39 203 2.1 2.07 0.7 106 42 99 46 162 1.81 1.75 0.86 107 42 103 47 145 1.64 1.61 0.82 108 378 3.31 109 42 108 48 140 1.59 1. 57 0.78 110 264 2.53 111 288 2.7 112 264 2.53 113 204 2.08 114 197 2.03 115 269 2.56 116 38 146 52 78 0.93 1.01 0.52 117 168 1.8 118 42 103 47 167 1.88 1.79 0.82 119 42 103 47 187 2.11 1.95 0.82 120 38 99 43 28 5 3.08 2.67 0.76 121 47 165 62 123 1.6 1.42 0.57 122 276 2.61 123 304 2.81 124 42 103 47 109 1.23 1.3 0.82 Mean 40 117 48 190 0.7 St. Dev. 4 28 5 80 0.2

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58 Table A 5 Mirror size and mirror constant values for unetched samples Sample r m c r m /c Mirror Constant mm mm (MPa m 1/2 ) 65 1410 99 14 2.0 66 1222 96 13 1.9 67 1692 120 14 2.1 68 1222 106 12 1.8 69 1504 118 13 1.9 70 1410 108 13 1.9 71 1504 110 14 2.0 72 1410 113 13 1.9 73 1222 101 12 1.9 74 1504 94 16 2.0 7 5 1410 118 12 1.9 76 1410 99 14 1.9 77 1222 106 12 1.8 78 1504 101 15 2.0 79 1504 92 16 2.1 80 1410 89 16 2.1 81 1222 92 13 1.9 82 1128 78 15 2.0 83 1316 92 14 1.9 84 1410 106 13 1.9 85 1410 94 15 2.0 86 1316 103 13 1.9 87 1175 94 13 2.0 88 14 10 99 14 1.9 89 1128 75 15 1.9 90 1410 99 14 1.9 91 1316 94 14 1.9 92 1316 82 16 2.0 93 1363 101 13 1.8 94 1363 106 13 1.9 Ave 1361 14 1.9 St. Dev 129 1 0.1

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59 Table A 6 Mirror size and mirror constant values for etched samples Sample r m c r m /c Mirror Constant (MPa m 1/2 ) 65 66 705 80 9 1.9 67 846 85 10 1.9 68 212 52 4 1.6 69 752 87 9 1.8 70 71 72 73 141 49 3 1.4 74 94 52 2 1.5 75 76 77 118 52 2 1.6 78 79 16 5 54 3 1.8 80 81 82 83 84 85 86 423 73 6 1.6 87 88 89 90 91 282 82 3 2.1 92 93 94 282 52 5 1.8 Ave 1.7 St. Dev 0.2

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60 RE FERENCES 1. R.E. Mould, Strength of and Static Fatigue of Abraded Glass Under Controlled Ambient Conditions: III, J. Am. Ceram. Soc., 43 [3] 160 167 (1960) 2. B.A. Proctor, The Effects of Hydrofluoric Acid Etching on the Strength of Glasses, Phys. Chem. Gl asses, 3 [1] 7 27 (1962) 3. E.K. Pavelchek and R.H. Doremus, Fracture Strength of Soda-Lime Glass After Etching, J. Mater. Sci., 9 1803 1808 (1974) 4. E.K. Pavelchek and R.H. Doremus, Griffith Fracture Equation -An Experimental Test, Journal of Applied Phy sics, 46 [9] 4096 4097 (1975) 5. C.E. Inglis, Stresses in a Plate Due to the Presence of Cracks and Sharp Corners, Trans. Inst. Naval Arch., 55 219 -240 (1913) 6. S. Ito and M. Tomozawa, Crack Blunting of High Silica Glass, J. Am. Ceram. Soc., 65 [8] 368 371 (1982) 7. J.J. Mecholsky, S.W. Frieman, R.W. Rice, Effect of Surface Finish on the Strength and Fracture of Glass, Proc. XI International Congress on Glass, 489 497 (1977) 8. B.R. Lawn, K. Jakus, A.C. Gonzalez, Sharp vs Blunt Crack Hypothesis in the Streng th of Glass: A Critical Study Using Indentation Flaws, J. Am. Ceram. Soc., 68 [1] 160 167 (1985) 9. B.R. Lawn and T.P. Dabbs, Acid Enhanced Crack Initiation in Glass, Comm. of the American Ceramic Society, 65 [8] C37C38 (1982) 10. J. Mencik, Glass Science and Tech nology 12: Strength and Fracture of Glass and Ceramics Elsevier Publishing, New York (1992) 11. A.A. Griffith, The Phenomena of Rupture and Flow in Solids, Philosophical Trans. of Royal Soc. of London, A221 163 198 (1921) 12. E. Orwan, Energy Criteria of Fra cture, Welding Journal Resesarch Supplement, 34 [3] 147 160 (1955)

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61 13. H.M. Westergaard, Bearing Pressures and Cracks, Journal of Applied Mechanics, 6 49 53 (1939) 14. I.N. Sneddon, The Distribution of Stresses in the Neighborhood of a Crack in an Elastic Solid, Proc. of the Royal Society A, 187 229 260 (1946) 15. M.L. Williams, On the Stress Distribution at the Base of a Stationary Crack, Journal of Applied Mechanics, 24 109 114 (1957) 16. G.R. Irwin, Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate, Journal of Applied Mechanics, 24 361 364 (1957) 17. P.N. Randall, Plane Strain Fracture Toughness Testing of High Strength Metallic Materials, in ASTM STP #410, 89 129 (1967) 18. J.C. Newman and I.S. Raju, Analysis of Surface Cracks in Finite Plates Under Tension or Bending Loads, NASA Technical Paper 1578, 1979 19. J.J. Mecholsky, Quantitative Fracture Surface Analysis of Glass Materials, In Experimental Techniques of Glass Science Edited by C.J. Simmons and O.H. El Bayoumi, Ame rican Ceramic Society, Westerville, OH. 483 517 (1993) 20. J.J. Mecholsky, S.W. Freiman, R.W.Rice, Effect of Grinding on Flaw Geometry and Fracture of Glass, J. Am. Ceram. Soc., 60 [3 4] 114 117 (1977) 21. H.P. Kirchner and J.C. Conway, Criteria For Crack Branching in Cylindrical Rods:II, Flexural, J. Am. Ceram. Soc., 70 [6] 419 425 (1987) 22. B.R. Lawn, P. Chantikul, G.R. Anstis, D.B. Marshall, A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness: II, Strength Method, J. Am. Ceram. Soc., 64 [9] 539 544 (1982) 23. J.J. Mecholsky, Fracture Surface Analysis of Glass Surfaces, In Strength of Inorganic Glasses, Edited by C.R. Kurkjan, Plenum Publishing, 569 590 (1985) 24. T. Rouxel and J.C. San gleboeuf, The Brittle to Ductile Transition in a Soda Lime Silica Glass, Journal of Non Crystalline Solids, 271 224 235 (2000) 25 B. Marshall, B.R. Lawn, J.J. Mecholsk y, Effect of Residual Contact Stress on Mirror/Flaw Size Relations, J. Am. Ceram. Soc. Disc. And Notes, 63 [5 6] 358 360 (1980) 26. J.J. Mecholsky and J.C. Conway, Use of Crack Branching Data for Measuring Near Surface Residual Stresses in Tempered Glass, J. Am. Ceram. Soc., 72 [9] 1584 1587 (1989)

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62 BIOGRAPHICAL SKETCH Salvatore Ruggero was born February 18 th 1978, in Staten Island, New York. He moved to Ormond Beach, Florida, with his parents when he was 12 year s old, and graduated from Flagler Palm Coast High School in 1996. He enrolled in the honors program at the University of Florida and spent two years living in Weaver Hall and experiencing the intellectual and social culture of Gainesville. After a brief hiatus in Milan, Italy, for an exchange program in international business during the fall of 1998, Salvatore returned to Gainesville still confused about the course of his educational path. Nevertheless, he began a course of study in materials science and engineering, and graduated with a BS in May 2001. After a summer spent backpacking through Europe, Salvatore began his graduate education at the University of Florida in August 2001, and eventually joined Dr. John J. Mecholskys research group. His rese arch interests include fracture of materials that behave in a brittle manner, and mechanical properties of biomaterials. In his spare time, Salvatore enjoys playing sports such as tennis, basketball, and racquetball, and also enjoys reading the occasional novel.


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Title: Quantitative Fracture Analysis of Etched Soda-Lime Silica Glass: Evaluation of the Blunt Crack Hypothesis
Physical Description: Mixed Material
Copyright Date: 2008

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QUANTITATIVE FRACTURE ANALYSIS OF ETCHED SODA-LIME SILICA
GLASS: EVALUATION OF THE BLUNT CRACK HYPOTHESIS












By

SALVATORE ALEXANDER RUGGERO


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2003
































Copyright 2003

by

Salvatore Alexander Ruggero

































Dedicated to:

My parents and my family















ACKNOWLEDGEMENTS

There are many people that I would like to thank for their help and support during

the research and writing of my master's thesis. First of all, I would like to thank my

advisor, Dr. John J. Mecholsky, for all of his help and guidance with this project. He

always steered me in the right direction, no matter how confusing my data appeared. I

would also like to thank the other members of my committee, Dr. Darryl Butt and Dr. E.

Dow Whitney, for their time and critique of my thesis. In addition, I would like to thank

Mr. Ben Lee of the dental biomaterials lab for his help with the mechanical testing for

this project. I would also like to acknowledge my fellow lab mates in Rhines 227 for

putting up with me for these past couple of years.

On a more personal note, I would not be at this juncture without my parents. I

could not ask for two better parents, as they have been a constant source of love and

motivation. I thank my sisters, Adrianne and Lisa, for putting up with my antics and

ballyhoo. I also thank my girlfriend, Melissa, for supporting me and enduring my

incessant typing as of late to produce this document in a timely manner. Lastly, I would

like to thank all my friends and family who have made this possible.















TABLE OF CONTENTS
page


A CKN OW LED GEM EN TS ....................................................................... iv

LIST OF TABLES.................. ................... .............. ...................... vii

L IST O F F IG U R E S ................................. ................................... ......viii

ABSTRACT .......... ...... ........................... ................ ............x

CHAPTERS

1 PUPROSE...................................... .................. .............. 1

1.1 R research R ationale................................... ........... ......... ......... 1
1.2 Research Objectives..................................................... ........ 1
1.2.1 Specific Aim 1........... ............................................ 2
1.2.2 Specific Aim 2 ............ .... ................................... 2

2 BACKGROUND ........... ................... ...................... ........ 3

2.1 A cid Etching of G lass................................ .................. ......... 3
2.1.1 Crack Tip Blunting............. .......................... ............ 3
2.1.2 Principles of Etching .............. ...................... ......................4
2.2 Fracture Mechanics Approaches to Failure............................................5
2.2.1 Inglis A nalysis................. ............. .. .. ........ .... ......... 5
2.2.2 Griffith Energy Balance Approach............... ...... .............
2.2.3 Linear Elastic Fracture Mechanics Stress Analysis.......................
2.3 Fracture Toughness Measurements............... ......... ...............12
2.3.1 Fracture Surface A nalysis................................. .. .......... ........ 12
2.3.2 Strength Indentation Technique........... ............... ................ 13

3 MATERIALS AND METHODS.........................................................15

3.1 Specimen Properties and Preparation.......................... ...... ..............15
3.1.1 Specimen Properties ............ ........... ...... ..........15
3.1.2 M icroindentation.............................................. .......... 16
3.1.3 Annealing........... ............................................... 18
3.1.4 Etching................... ............. ........................ ....... 20



V









3.2 M echanical Property Determination................................... .............21
3.2.1 Flexure Strength................................. ....... ..... ............ 21
3.2.2 Fracture Toughness................ .......................... ...............23
3.3 Characterization of Fracture Surfaces......... ......................... ...........25
3.3.1 Light M icroscopy................... ........................... ..............25
3.3.2 Scanning Electron M icroscopy............ .............. ...............25

4 RESULTS AND DISCUSSION .. ............ ............ ... .........................27

4.1 M mechanical Properties ...................... ........................ ......... 27
4.1.1 Flexure Test Strength Results............ ............ .............. 27
4.1.2 Fracture Toughness .................................. ...................30
4.2 Fracture Mechanics Approaches Applied to Blunt crack..........................33
4.2.1 Inglis Solution.............. ............... ...................... ........ 33
4.2.2 Stress and Energy Criteria................................... ...............35
4.3 Characterization of Fracture Surfaces ............ .......................... 38
4.3.1 Fractographic Analysis .............. ........................................38
4.3.2 Optical M icroscopy ........... ............................ ..................43
4.3.3 Scanning Electron Microscopy.........................................45

5 CONCLUSIONS ............ ......................................... 52

APPENDIX TABULATED DATA................... ................... 54

LIST OF REFERENCES............... ................................. ............60

BIOGRAPHICAL SKETCH ........... ........................ ............62

























vi















LIST OF TABLES


Table page


3.1 Physical and mechanical properties of Corning glass slides............ ............ 15

3.2 Preliminary 4-point flexure testing results to determine effect of
annealing on strength.................. .................... ............. ...... ...... 19

3.3 Preliminary 4-point flexure testing results to determine effect of
acid etching on strength ................ ......................................20

4.1 Comparison of fracture toughness data....................... ....... ...........32

4.2 Strength, fracture toughness, and crack size of unetched
and etched samples ............. .. ......................................40

A-1 4-point flexure results for unetched samples ........................................54

A-2 4-point flexure results for etched samples........................ ................55

A-3 Fractography measurements and fracture toughness of unetched samples.........56

A-4 Fractography measurements and fracture toughness of etched samples............57

A-5 Mirror size and mirror constant values for unetched samples........... .............58

A-6 Mirror size and mirror constant values for etched samples........................59















LIST OF FIGURES


Figure page


2.1 Schematic of conditions for Inglis solution: Elliptical hole
in a semi-infinite plate....................................... ............. ............ 5

2.2 Schematic of Inglis approach as ellipse becomes more slender
and approximates a sharp crack ................ ................. .. ..............6

2.3 Location of local stresses near a crack tip in cylindrical coordinates ............10

2.4 Three loading m odes................. ............................................ 10

2.5 Schematic of a fracture surface of a material that failed in a brittle manner.......12

2.6 Schematic of 4-point flexure test with side containing indentation
as the tensile surface............................................................... 14

3.1 Optical images of unetched and etched Vickers indentations........... ............ 17

3.2 Schematic of 4-point flexure apparatus used to do mechanical testing ...........22

4.1 Flexure test strength results for 30 unetched and 30 etched samples
ordered from low to high.................................................................29

4.2 Fracture toughness of unetched samples using fractography method .............31

4.3 Effective fracture toughness of etched samples using
fractography method............................... ...........................31

4.4 Agreement between fractography method and strength indentation
method for etched samples ...........................................................32

4.5 Estimation of crack tip radius of etched samples using Inglis solution ...........35

4.6 Y-factor projections for etched samples versus strength............... ............36

4.7 Y-factor projection plotted versus estimated crack tip radii from Inglis............37









4.8 Blunting energy of etched samples ............ ........ .. ................38

4.9 Fracture surfaces of unetched and etched samples showing
stable crack growth......................................................... 39

4.10 Mirror to flaw size ratio for unetched and etched samples..........................44

4.11 Schematic of fracture surface features........... .............. ..............44

4.12 Fracture surfaces of unetched and etched samples showing failures
at different stress levels............. ............ ............. ............ 46

4.13 SEM images of etched radial surface cracks........... .... ..........47

4.14 SEM images of unetched radial surface cracks.....................................................48

4.15 SEM fracture surface of etched sample at different magnifications...............49

4.16 SEM fracture surface of etched sample at different magnifications ...............50

4.17 Comparison of crack tip radius calculated from Inglis and
measured from fracture surface ........... ...... ................... ................51















Abstract of Thesis Presented to the Graduate School
Of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

QUANTITATIVE FRACTURE ANALYSIS OF ETCHED SODA-LIME SILICA
GLASS: EVALUATION OF THE BLUNT CRACK HYPOTHESIS


By

Salvatore Alexander Ruggero

May, 2003


Chairman: John J. Mecholsky, Jr.
Major Department: Materials Science and Engineering

The purpose of this study was to analyze the strengthening that occurs upon acid

etching of soda-lime silica glass. In particular, the hypothesis of crack tip blunting was

analyzed and related to basic fracture mechanics equations.

Based on flexure testing and subsequent fractography measurements, it

was determined that there was strengthening of the acid etched glass samples without a

significant decrease in flaw size. Measurements of the mirror size confirmed an equal

mirror constant for all samples, and a linear decrease in mirror to flaw size ratio with an

increase in strength for the non-uniformly strengthened etched samples. In addition, the

Inglis equation was used to estimate the crack tip radii of both the unetched and etched

samples, and the blunting hypothesis was explained in terms of the critical stress intensity

factor as well as the Griffith fracture energy. Direct measurement of the radius of









curvature of the critical indentation flaws was performed where possible, and estimates of

the crack tip radius due to blunting at the surface were obtained.















CHAPTER 1
PURPOSE

1.1 Research Rationale

The strengthening of glass due to acid etching has been a topic of research in the

ceramics community for nearly half a century, yet many questions still remain

unanswered concerning the mechanism of this strengthening. A spirited debate occurred

in the literature during the 1970's and early 1980's concerning the possibility of blunting

of the crack tip due to acid etching of glass, however the topic has been mainly absent

from the literature for the last 15 years with no definitive answers to the fundamental

topics at hand. The main obstacle when conducting this research had been the difficulty

in finding definitive visual evidence of the blunting. This task is certainly more difficult

than it seems, as evidenced by failure to do so in past research, as well as in the present

study. Fractographic analysis offers an alternative method to study the differences

between unetched and etched samples. The features on the fracture surfaces can provide

unique information to determine if the crack tip blunting hypothesis is a reasonable

assumption.

1.2 Research Objectives

The main objective of this study was to confirm that acid etching caused a

strength increase in indented soda-lime silica glass flexure specimens, and to determine if

the mechanism causing the strength increase was blunting of the crack tip. A secondary






2


objective was to determine ways of accounting for the blunting mechanism in terms of

fracture mechanics equations describing brittle fracture.

1.2.1 Specific Aim 1

Specific aim 1 was to characterize the fracture surfaces of both the unetched and

etched samples to determine if evidence exists for the crack tip blunting mechanism.

Crack size and mirror size measurements were made on the fracture surfaces. Also,

measurements of the blunt crack tip radius were made on surface indentation cracks and

critical cracks where possible.

1.2.2 Specific Aim 2

Specific aim 2 was to use measurements of fracture surface features along with

fracture mechanics equations to propose different ways of accounting for blunted cracks

in brittle fracture. The different techniques used were the Inglis equation, the critical

stress intensity parameter, and Griffith's energy equation.














CHAPTER 2
BACKGROUND

2.1 Acid Etching of Glass

2.1.1 Crack Tip Blunting

During the 1960's many researchers became interested in examining the strengthening

due to acid etching of glass. Early work by Mould1 and Proctor2 first presented the idea of

crack tip blunting as a possible explanation for the observed strengthening of soda-lime

glass due to etching. Pavelchek et al. performed etching experiments in hydrofluoric acid

and water on abraded rods of soda-lime silica glass.3-4 They also came to the conclusion

that some sort of crack tip blunting was occurring, and they made use of the Inglis

analysis5 to approximate the crack tip radius for both the sharp and blunted cracks; the

radius was estimated to be about 2 nm for the unetched samples. Ito and Tomozawa

performed more mechanical testing on abraded soda-lime silica glass rods using water

and Si(OH)4 as etchants.6 They proposed a dissolution and precipitation mechanism for

the blunting of the crack tip, and estimated the radius of curvature of the blunted cracks at

about 3 nm using Inglis as well. In addition, Mecholsky et al. did further work on

abraded soda-lime silica glass rods.7 Instead of only looking at strength, this research

utilized crack size measurements and fracture energy estimates to explain the difference

between the unetched and etched samples. The idea that the strengthening could be

attributed to the blunted flaw requiring an initial energy to produce a sharp crack was









proposed in this paper. Also, the notion that there might be evidence of blunting on the

fracture surface was suggested.

In opposition to the numerous advocates of the idea of crack tip blunting

mentioned above, there were and are some drawbacks to the theory. First of all, this idea

seemingly contradicts the concepts of linear elastic fracture mechanics for sharp crack

tips. Secondly, it is very difficult to directly observe a crack tip (rounded or not), so most

of the evidence presented for crack tip blunting is indirect evidence. Thirdly, there are

other phenomena that could explain the strength increase of etching depending on

experimental conditions, such as residual stress release and slow crack growth. Lawn et

al.8-9 performed indentation studies on soda-lime silica glass, and came to the conclusion

that the indentation induced cracks extend during etching, and that this extension relaxes

residual stresses associated with the indentation. No matter which explanation is

perceived as true, there still appears to be no definitive answer to this question.

2.1.2 Principles of Etching

The strength of materials that behave in a brittle manner is controlled by the flaws

that are present in the material. These flaws can either be inherent in the microstructure

of the material, or can be introduced during processing, machining, or through a number

of other actions. The main point is that the strength of a material is inversely

proportional to the square root of the size of the crack that initiates failure. In the case of

ceramic materials such as glass which are inherently brittle, decreasing the size and

number of flaws present in a given amount of material becomes important. Acid etching

of glass has traditionally been used to decrease the size of surface flaws in glass

materials, thus creating materials that fail at a higher stress. The acid reacts with the SiO2









in the glass and dissolves material from the surface.10 This idea is especially important in

industrial and commercial applications where a part must meet minimum stress

requirements.

2.2 Fracture Mechanics Approaches to Failure

2.2.1 Inglis Analysis

The paper published in 1913 by C.E. Inglis5 entitled "Stresses in a Plate Due to

the Presence of Cracks and Sharp Corners" laid the foundation for the theories of fracture

mechanics for materials that fail in a brittle manner. The significance of the Inglis

Analysis was that it provided an exact solution for the stress around an arbitrarily shaped

elliptical flaw in a plate of material. Inglis stated his intentions well in his paper:

The methods of investigation employed for this problem are mathematical rather
than experimental. The main work of the paper lies in the determination of the
stresses around a hole in a plate, the hole being elliptic in form. The results are
exact, and are consequently applicable to the extreme limits of form which an
ellipse can assume. If the axes of the ellipse are equal, a circular hole can be
obtained; by making one axis very small the stresses due to the existence of a fine
straight crack can be investigated. (p.219)

The elliptical flaw mentioned above has a major axis of 2a, a minor axis of 2b,

and a radius of curvature of p. The flaw is contained in a large plate with an applied

tensile stress, o (Figure 2.1).






1 i





Figure 2.1 Schematic of conditions for Inglis solution: Elliptical hole in a semi-
infinite plate









From these starting conditions Inglis solved for the stress intensity at point A, and

obtained the following general result:


(2.1)


He then assumed that if the ends of the flaw appeared to be elliptical, it was reasonable to

approximate the shape of the cavity with that of an ellipse, and therefore the substitution

b=(ap)1/2 could be made. Making this substitution into equation 2.1 yields:


Q 1+ 2


(2.2)


The general equation (2.2) can be simplified as the ellipse becomes more slender and

starts to approximate a sharp crack (Fig 2.2). In this case 2b and p approach 0 as the

ellipse becomes more slender and starts to resemble a sharp crack.


I I


Figure 2.2 Schematic of Inglis approach as ellipse becomes more slender and
approximates a sharp crack


(7A 2a\
b )










First equation 2.2 can be written in the following form to show the simplification:


2A = 2 + (2.3)


Since (a/p) is much larger than 1/2, the /2 term is negligible and can be dropped out of the

equation. The equation then simplifies to the following for a slender ellipse:



2A =2 (2.4)


The Inglis solution can be used to approximate the radius of curvature of a crack

in a brittle material, and has been done in prior studies with varying success.3-7 The idea

is that in the absence of plasticity, the minimum radius (p) at the crack tip may be close

to atomic equilibrium spacing, so the stress at point A can be approximated with the

theoretical strength of the material. Using the theoretical strength of the material as cA,

the radius of a crack tip can be estimated because the crack size, a, and the global failure

stress,a, can be measured.


2.2.2 Griffith Energy Balance Approach

A.A Griffith, a British scientist prominent in the early 1920's, performed

groundbreaking work in the area of crack propagation and fracture of brittle materials.

His paper entitled "The Phenomena of Rupture and Flow in Solids" published in 1921

formulated a way of describing brittle fracture using strain energy concepts.11 These ideas

were predicated on the observations Griffith made regarding the actual strength of

materials compared to the theoretical strength. The prevailing idea at this time in regards

to rupture of elastic materials was that an elastic solid ruptured when a maximum tensile









stress was reached. A scratch in the material created a stress concentration that amplified

the stress according to the shape of the scratch, but the maximum stress was independent

of the size of the scratch. Griffith disagreed with the afore mentioned idea, and proposed

a new idea in his paper:

If, as it is usually supposed, the materials concerned as substantially isotropic,
there is but one hypothesis which is capable of reconciling all these apparently
contradictory results. The theoretical deduction-that rupture of an isotropic elastic
material always occurs at a certain maximum tension-is doubtless correct; but in
ordinary tensile and other tests designed to secure uniform stress, the stress is
actually far from uniform so that the average stress at rupture is much below the
true strength of the material. The general conclusion may be drawn that the
weakness of isotropic solids, as ordinarily met with, is due to the presence of
discontinuities, or flaws, as they may be more correctly called, whose ruling
dimensions are large compared with molecular distances. The effective strength
of technical materials might be decreased 10 or 20 times at least if these flaws
could be eliminated. (p. 180)

Griffith performed experiments with elastic materials that showed that the

resistance to rupture of a material was in fact related to the size of the flaws present in the

material. He did these experiments by rupturing glass test tubes with internal pressure.

Griffith derived the net change in the potential energy of a plate from introduction of a

crack as well as the surface energy of the crack. He then defined the condition for critical

crack growth, and solved for the fracture stress. The results of this derivation are shown

below:


Yo= E' (2.5)
Sa a

Orowan12 extended Griffith's theory to include the plastic surface energy term for the


fracture of ductile materials:










S 2E (Ys yp (2.6)
(2.6)
7ra

2.2.3 Linear Elastic Fracture Mechanics Stress Analysis

In addition to the energy approach to fracture discussed earlier, there was

considerable work done on deriving closed form expressions for the stress field around a

crack tip. In particular, Westergaard,13 Sneddon,14 Williams,15 and Irwin16 were among

the first people to publish such solutions. It is generally regarded that Irwin's paper

entitled "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate"

was the basis for modern fracture mechanics theory. In this paper Irwin showed that the

local stresses near the crack tip are of the general form as follows:16


0 = K f () +.... (2.7)


In the above equation r and 0 are cylindrical coordinates of a point with respect to the

crack tip (see Fig 2.3), and K is the stress intensity factor. He further showed that the

strain energy approach is equivalent to the stress intensity approach and that crack

propagation occurs when a critical stress intensity, Kc, is achieved.

Linear elastic fracture mechanics (LEFM) is based on the application of the

theory of elasticity to bodies containing cracks or defects. The assumptions used in

elasticity are also inherent in the theory of LEFM: small displacements and general

linearity between stresses and strains. The general form of the LEFM equations is given

in Eq. 2.8. As seen, a singularity exists such that as r, the distance from the crack tip,

tends toward zero, the stresses go to infinity. Since materials plastically deform as the

yield stress is exceeded, a plastic zone will form near the crack tip. The basis of LEFM









remains valid, though, if this region of plasticity remains small in relation to the overall

dimensions of the crack and cracked body.


Figure 2.3 Location of local stresses near a crack tip in cylindrical coordinates

There are generally three modes of loading, which involve different crack surface

displacements (Figure 2.4). The three modes are:

Mode 1: opening or tensile mode (the crack faces are pulled apart)
Mode 2: sliding or in-plane shear (the crack surfaces slide over each other)
Mode 3: tearing or anti-plane shear (the crack surfaces move parallel to the
leading edge of the crack and relative to each other)
The following discussion deals with Mode 1 since this is the predominant loading mode

in most engineering applications. Similar treatments can be performed for Modes 2 and 3.


Figure 2.4 Three loading modes









The stress intensity factor, K, which was introduced in Eq. 1, defines the

magnitude of the local stresses around the crack tip. These local stresses can be related to

the global stress and the geometry of the sample. Many closed form solutions have been

derived for K concerning simple geometric situations, and more complex situations have

been determined experimentally and numerically.17-18 In general K depends on loading,

crack size, crack shape, and geometric boundaries, with a general form given by:


K =-0 f (2.8)


Equation 2.8 contains the following terms that are defined as follows: o is the remote

stress applied to component (not to be confused with the local stresses, yij, in Equation

7), a is crack length, and f (a/w) is a correction factor that depends on specimen and crack

geometry. Stress intensity factors for a single loading mode can be added algebraically.

Consequently, stress intensity factors for complex loading conditions of the same mode

can be determined from the superposition of simpler results, such as those readily

obtainable from handbooks. Of particular relevance to this study is the closed form

solution for a semi-elliptical surface crack.19


K, C1/2 (2.9)


Equation 2.9 contains the term 4 which is an elliptical integral of the second kind and

goes from 1 for a slit crack to 1.57 for a semi-circular crack. Mecholsky et al. showed

that a semi-elliptical crack could be approximated as an equivalent semi-circular crack

for glass fractures.20 When inserting the equivalent semi-circular crack dimension,

c=(ab)1/2, and the value of 4=1.57, the equation becomes:









KIc = Ya-c (2.10)
In equation 2.10, Y becomes 1.24 for a surface crack in a material with no residual stress.


2.3 Fracture Toughness Measurements

2.3.1 Fracture Surface Analysis

Fracture surface analysis is an important tool that allows measurements to be

taken at the fracture surface that can be related to the overall fracture behavior of the

material. As a crack initiates and propagates through a material, characteristic markings

are left on the fracture surface that are related to the stress and energy of fracture. A

schematic of a fracture surface is shown in figure 2.5.


'Hackle
IMist




.Mirror
Origin








r *-' 1


Figure 2.5 Schematic of a fracture surface of a material that failed in a brittle
manner

The features of the fracture surface are as follows: The fracture surface origin is where

the fracture originated. The a and 2b values on the diagram represent the depth and width

of the initial and critical flaw. The mirror, mist, and hackle regions of the fracture

surface represent the crack propagating at first in a comparatively smooth plane, and









eventually gaining enough energy to go in a different direction and branch. The

boundaries labeled on the diagram as rl, r2, and rCB represent the mirror-mist, mist-

hackle, and crack branching boundaries. These distances have been shown to be related

to the global failure stress of the material with the following empirical relationship21,

where Mj is the mirror constant and KBj is the stress intensity factor associated with the

mirror/mist boundary:

f rj 2 = YM = KBj (2.11)

The utility of fracture surface measurements is that the fracture toughness can be

determined for a material from mechanical testing and subsequent crack size

measurements. Mechanical testing such as flexure or tensile testing provides the stress at

which the sample failed, the crack size is measured with a microscope, and the fracture

toughness can be solved for in an equation like 2.10 depending of the geometry and load

application.

2.3.2 Strength Indentation Technique

The strength indentation technique is another method of estimating the fracture

toughness of a material. This method is based on Vickers micro indentation and the

subsequent stress field that surrounds the indentation. Using fracture mechanics analysis

for cracks in tensile loading, a strength formula can be derived to estimate Kc. The

advantage of this method is that it requires no fracture surface measurements. Instead,

the information necessary is the failure stress of the sample, the elastic modulus and

hardness, and the indentation load. The strength indentation equation developed by Lawn

et al.22 is shown below, and has a calibration constant (rvR ) included that was determined

experimentally for a range of ceramic materials.









K, =f V (a /3)34 (2.12)

The common method of performing this analysis is by indenting a sample at a

given load, and then fracturing the sample with either 3 or 4 point bending. A common

experimental setup is shown below in Figure 2.6. The sample fails from a crack

introduced by the indent, the failure stress is measured during the mechanical test, and the

fracture toughness of the material can be estimated using equation 2.12. This value

obtained with equation 2.12 can then be compared to the value obtained using equation

2.10 with a Y-factor of 1.65 to account for the residual stress created by the indentation.


P/2 P/2
L/3











P/2 P/2

Figure 2.6 Schematic of 4-point flexure test with side containing indentation as
the tensile surface
















CHAPTER 3
MATERIALS AND METHODS

3.1 Specimen Properties and Preparation

3.1.1 Specimen Properties

The test samples eventually chosen for this study were Coming #0215 soda-lime

glass microscope slides with dimensions of 75mm x 25mm x 1 mm.* The properties of

the Coming glass slides are shown below (as received from Corning).

Table 3.1 Physical and mechanical properties of Coming glass slides

Common Names Standards
Soda Lime Glass- Glass commonly referred to as
Flint Glass Type II Soda Lime Glass per
A.S.T.M E-438 federal spec
DD-G-541b


Composition (percentapprox) Properties
SiO ......... ...... ..........73% Coeff Of Exp..........89 x 10-7 cm/cm/C
Na20...... ...... .... .. .... 14% Strain Point........... 511 C
CaO ......... ... .............7% Anneal point...... ....545 C
M gO........... .............. 4% Soften Point...... .......724 C
A1203.................. ......... 2% Density ..................2.4 g/cm3
Youngs Mod..........70 GPa
Refract. Index......... 1.515 @ Sodium D Line


*Preliminary testing was originally started with Fisher microscope slides, until it was
discovered that Fisher purchases their slides from other manufacturers. They receive the
pre-made slides and package them, so there could be variability in the composition of the
slides. Coming was contacted, and it was determined that they produced all of their
slides with soda-lime glass of the same composition









Glass slides were chosen for this research because they were of suitable

composition and size for 4-point flexure testing, and they were readily available with no

machining or cutting necessary. The alternative of cutting and polishing samples would

have been time consuming and could have introduced defects into the samples, which

could have competed with the introduced indentation crack as the fracture origin.


3.1.2 Microindentation

Microindentation can be used to introduce controlled cracks in glass in order to

perform mechanical testing.8'9 The appropriate indentation loads were determined

through preliminary experiments. This indentation crack had to be large enough so that

during the 4-point flexure testing the sample would fracture from the introduced crack; if

the introduced crack was too large, then the resulting crack could interact with the bottom

surface of the sample, producing a flattened crack as opposed to the desired semi-

elliptical crack geometry. If this was the case, a correction factor would have to be

introduced into the fracture mechanics equations, thus complicating the analysis.

A total of 8 samples were indented at various loads using a Buhler Micromet 3

microhardness indenter with a Vickers diamond tip. Two samples were indented each at

loads of 100g, 500g, 1kg, and 2kg. The indentations were first examined with an optical

microscope to evaluate the indent, and then the samples were fractured in 4-point bending

using an Instron testing machine. The testing conditions used were a loading rate of

0.5mm/min, a lower span length of 20mm, and an upper span length 6.3mm. The

fracture surfaces were then examined with the optical microscope to determine whether

the sample failed from the crack introduced by the indent, and whether the crack had a

fully rounded or flattened shape. From examining the indents, it was determined that the









2 kg indent produced significant lateral cracking which meant that the load was too large

for the specimen. Examining the fracture surfaces of the 2 specimens with 1 kg indent

loads, it appeared that the crack was semi elliptical and was not interacting with the

bottom surface of the specimen. Unfortunately, after indenting and fracturing 60 samples

with the 1 kg indent load, it was determined that there was in fact an interaction with the

bottom surface of the samples. This was confirmed by looking at the a/b ratio for these

cracks. The a/b ratio came out to be 0.7 for the samples with the 1 kg indent load, as

opposed to the normal value of 0.8. Another set of 60 samples then had to be prepared

using the 500g indent load. The samples indented with the 500g load had cracks that were

large enough to initiate failure, but were not so large as to have interaction with the

bottom surface of the sample. Figure 3.1 shows optical images of the 500g Vickers

indentation of both an unetched and an etched sample. Sharp radial cracks are apparent

in the unetched sample, while it appears that the acid has blunted the radial surface cracks

in the etched sample.


50ILM

a) Unetched b) Etched

Figure 3.1 Optical images of unetched and etched Vickers indentations











The procedure for indentation was as follows. The sample was removed from the

packaging, and using a t-square the middle of the sample was marked with a fine sharpie

marker. The sample was then placed in the microhardness indenter with the sharpie mark

facing down, the surface was determined with the 20x magnification to the point where

the mark was on the opposite side, and the 500g load was released with a dwell time of

20 seconds. The sample was then carefully placed in a ziploc bag and numbered. Each

sample was indented in this manner to await further preparation and flexure testing.

3.1.3 Annealing

Annealing is the process of removing residual stresses in glass by re-heating to a

suitable temperature followed by controlled cooling.10 The purpose of annealing the

samples after they were indented was to remove the local residual stress around the crack

created by the indent. Preliminary experiments were performed to determine which (if

any) annealing schedule would be suitable. The preliminary experiments were performed

on samples that had been indented with a 1 kg load. Using the annealing point of the

glass provided by Corning (C), the following annealing schedule was set up:


250C 3900C at 30C per minute
3900C 4950C at 3C per minute
4950C 5450C at 1C per minute
Hold at 5450C for 15 minutes
5450C 250C at 1C per minute

A Box furnace was used with the samples placed on ceramic boats crack side up. 2

samples were annealed from 250C following the above schedule, 2 samples were

annealed starting from an elevated temperature of 3900C, and 2 samples were not

annealed to see if the residual stress would resolve itself. The 6 samples were then









fractured with the Instron in 4-point bending with the same test conditions as stated

before. The samples were analyzed with the optical microscope to determine crack sizes

and corresponding fracture toughnesses. From examining the strength data in table 2 it

appeared that there was no difference in the strength between the room temperature

anneal and the elevated temperature anneal. The annealed samples did show higher

strength than the unannealed samples, indicating that there was some residual

compressive stress left in the unannealed samples. From this data it was decided that all

samples would be annealed from room temperature following the indent.

Table 3.2 Preliminary 4-point flexure testing results to determine effect of annealing on
strength
Max Load Max Stress
Sample Annealing Condition (N) (MPa)
1 Room Temp Anneal 71.9 56.7
2 Room Temp Anneal 67.9 53.5
3 Elevated Temp Anneal 68.6 54.0
4 Elevated Temp Anneal 67.4 53.1
5 No Anneal 56.6 44.6
6 No Anneal 53.7 42.3

Unfortunately, no preliminary annealing experiments were done with etching

included, so the adverse effect that annealing had on etching of the crack tip was not

anticipated until after the first batch of 60 samples had been fractured. No strength

increase was observed for the annealed etched samples, so it was determined that

annealing somehow inhibited the acid from penetrating to the crack tip. Therefore, in

addition to changing the indent load from 1 kg to 500 grams, the second batch of samples

did not include annealing in the preparation process.









3.1.4 Etching

Acid etching of glass surfaces has traditionally been used as a surface treatment to

remove or reduce flaws inherent in the glass. The strengthening effect of acid etching has

mainly been attributed to the decrease in flaw size, but in fact there may be other

processes at work. The acid etching of the indented glass samples in this study was

designed to show that the increase in fracture strength and fracture toughness of the

samples was due to rounding of the crack tip, with the ultimate goal of developing a

quantitative relationship between crack tip blunting and fracture toughness.

The etching procedure in this study entailed submerging the sample in dilute

hydrofluoric acid. Preliminary experiments were performed to determine which

concentration and submergence time was optimal. 6 samples were indented with a 1 kg

load, etched at various concentrations and times, and fractured in 4-point bending. The

results are shown below in Table 3.

Table 3.3 Preliminary 4-point flexure testing results to determine effect of acid etching on
strength
Max Load Max Stress
Sample Etching Procedure (N) (MPa)
1 5% HF for 1 minute 55.9 44.0
2 5% HF for 15 minutes 80.5 63.4
3 2.5% HF for 1 minute 55.0 43.4
4 2.5% HF for 15 minutes 114 90.0
5 1% HF for 1 minute No test No test
6 1% HF for 15 minutes 62.6 49.3

The results from this preliminary experiment showed that a 1 minute etch time was not

enough to promote crack tip blunting. The 15 minute etch time did produce the desired

strength increase, and the increase was most dramatic in the 2.5% HF solution. From this

data it was decided that an etch solution of 2.5% HF with an etch time of 15 minutes

would sufficient to show the effect of etching.









The etching protocol was a fairly simple procedure. Once the etching conditions

were set at 2.5% HF for 15 minutes, a large batch of 2.5 % HF was mixed using 50% HF

solution and distilled water. All of the samples were subsequently etched from this one

batch of dilute HF. Standard polystyrene Petri dishes were used to house the etching

procedure. The Petri dish was filled with the HF, the sample was dropped in with the

indented surface facing upwards, and the lid was placed on the Petri dish for the 15

minute duration of the etch. Once the 15 minutes was up, the sample was transferred to a

Petri dish with distilled water and allowed to soak for 30 seconds, and then transferred to

a similar Petri dish and allowed to soak for an additional 30 seconds. The sample was

then removed from its bath and then rinsed thoroughly with distilled water from a squirt

bottle over the sink. In this way all the acid was removed and the samples were able to

be handled. This process was repeated for all etched samples. The HF acid solution was

changed after 5 samples had passed through, and the resulting strength data showed no

correlation between strength and the number of times the acid had been used. At this

point the samples were now indented, annealed, and etched and were ready for 4-point

bend testing.

3.2 Mechanical Property Determination

3.2.1 Flexure Strength

The flexure strength for all samples was performed using an Instron model 5500R

with a 5 kN load cell. The testing configuration for all samples was a 4-point bend

fixture with a 20 mm outer span and an inner span of 6.3 mm. The loading rate for all

samples tested was 0.5mm/min. A schematic of the 4-point bend arrangement is shown in

figure 3.2 below.









P/2 P/2

4 I JI

















P12 P/2

Figure 3.2 Schematic of 4-point flexure apparatus used to do mechanical testing

The fracture stress of the samples was calculated using beam theory and the

familiar equation for the 4-point flexure testing with the loading span having one-third

the length of the support span:

PL
PL (3.1)
bh2

The samples were prepared for testing by first locating the indent (either by eye of with

the optical microscope). The location of the indent was marked on the edge of the

sample with a sharpie, and two lines were drawn across the width of the sample, each

located 10mm from the indent. This allowed the sample to be placed easily in the 4-point

bend fixture with the indentation facing down, and centered within the fixture. Using this









technique all the samples (with a few exceptions) failed from the introduced crack from

the indent. For the samples that were etched, and therefore had a higher fracture stress,

scotch tape was placed on the compressive side of each sample. This preservation

technique kept the samples intact (for the most part).

3.2.2 Fracture Toughness

The fracture toughness of the samples was evaluated using two techniques, the

fractography method and the strength indentation method. For the fractographic

technique, the samples were analyzed using the light microscope with the filar eyepiece,

and the critical crack size was measured for each sample. In addition, the mirror-mist

boundary size was measured. These values for the critical crack sizes were then used to

calculate the fracture toughness of the samples with:

Kc =Yao (3.2)

In the above equation, c is an equivalent semicircular crack of the semi elliptical crack

with a minor axis radius of a and major axis diameter of 2b, where c=(ab)1/2. C is the

stress of the sample at failure, and Y is the geometric factor that depends on crack

geometry and residual stress. For annealed samples the geometric factor used was 1.24,

while for the non-annealed samples that had local residual stress due to the indentation

process, the geometric factor used was 1.65. All of these terms were tabulated in a

spreadsheet for each set of samples, and in this way the average fracture toughness values

were calculated (where applicable). In addition to the critical crack sizes, the mirror sizes

were also measured and provided useful information. From the mirror size and the

failure strength of the sample the mirror constant was calculated using the following

equation:21










Yoafr(, = YM, = K (3.3)

In addition to the mirror constant, the mirror to flaw size ratio was calculated for all

samples. These two important pieces of data obtained from the mirror size were

tabulated in a spreadsheet for both the unetched and etched samples.

A second technique for calculating the fracture toughness of the samples was the

strength indentation technique. Since the samples were indented before fracture with a

specified load, it was very easy to apply the strength indentation technique to determine

the fracture toughness, and check to see if it was in agreement with the fractography

method This technique uses no measurement of crack features, but instead uses a radial

indentation flaw as the origin of fracture for a specimen in bending. The necessary

strength formulas can then be derived to calculate Kc based on the analysis of the

indentation flaw in flexural failure22


Kc, ((P1/3)3/4 (3.4)


The only terms needed in the above equation were E and H, since rf is a calibration

constant. The elastic modulus used was the standard modulus for glass of 72 GPa, the

hardness was taken from Vickers microhardness measurements taken from the glass

slides, and was approximately 5GPa. The other terms in the equation were known, and

they were all tabulated into the same spreadsheet so a comparison of the fractography vs.

strength indentation fracture toughness values could be made.









3.3 Characterization of Fracture Surfaces

3.3.1 Light Microscopy

The fracture surface analysis was done with an Olympus BHMJ stereomicroscope

using an Olympus TGH light source. The microscope included a 10x eyepiece and 5x,

10x, 20x, and 50x objective lenses. A calibrated filar eyepiece was used to take all

measurements from the fracture surfaces of the samples. In addition, an Olympus digital

camera was used in conjunction with the microscope to enable digital capture of the

images.

Each fractured sample was mounted in putty so the fracture origin was facing

upwards, and was examined with the microscope. Each sample was first analyzed under

low magnification (50x) to find the fracture origin, and then the magnification was

increased (200x) to focus on the origin and make the measurement with the filar

eyepiece. The mirror-mist boundary measurements were taken in a similar fashion with

the filar eyepiece, except the magnification used was usually 100x. The samples that

broke at a relatively low stress had a large mirror region, so it was necessary to move the

field of view of the microscope while taking the measurement to obtain an accurate

value. In order to do this, a feature had to be picked on the sample to serve as a reference

point for measurement. In this manner, the size of the fracture origin and the size of the

mirror- mist boundary were determined.

3.3.2 Scanning Electron Microscopy

Additional analysis of the fracture surfaces as well as analysis of the blunted

indentation surface cracks was performed with a JEOL 6400 SEM. The samples were

prepared for the SEM by cleaning with ethanol, mounting on aluminum stubs with putty,






26


covering the putty with carbon paint, and sputter coating with Gold-Palladium. Pictures

were taken at various magnifications and angles of tilt to get an idea of how the fractures

occurred at all perspectives.















CHAPTER 4
RESULTS AND DISCUSSION

In the following sections, the majority of the data is presented in graphical format.

First the strength and fracture toughness of the unetched and etched samples is presented.

Next, the application of fracture mechanics to analyze the cracks at the origin of failure of

the etched glass is discussed. Finally, an analysis of the fracture surfaces along with

observations of the crack tip radii at the surface and on the fracture surface (where

possible) is presented. The data for the individual specimens can be found in the

appendix.

4.1 Mechanical Properties

4.1.1 Flexure Test Strength Results

The strength data for the unetched and etched samples is summarized in figure

4.1. Based on the 30 unetched samples, the average stress at failure was 52 3 MPa. The

etched samples had a much higher failure stress on average, and also showed a great deal

of variability. The average stress observed based on the 30 etched samples was 190 + 80

MPa.

These strength values clearly indicate that the etched samples in general had a

greater fracture stress than the unetched samples. It is also important to analyze the

variability of strength in the etched samples. Some etched samples failed at a stress very

near to the baseline unetched samples (about 52 MPa), while some etched samples failed

at over 300 MPa. In assessing these strength results it is clear that the HF acid etching









acted as a strengthening agent in the glass samples. Three alternative ideas to account for

strengthening of etched glass should be mentioned. The first idea is that the etching

process dissolves away material at the surface, thus making the cracks smaller and

increasing the fracture strength. This idea will be addressed later when fractography is

used to measure the size of the cracks. The second idea is that the material could have a

local tensile residual stress located at the indentation. This tensile stress would then be

relieved due to etching and the fracture strength would increase accordingly. This idea

could possibly explain a strength increase due to etching, but not the magnitude of

strengthening observed in this study. Previous studies have shown that the local residual

tensile stress of an indentation has the effect of lowering the strength of the order of 10-

20 MPa, and can be relieved with annealing.8'9 Preliminary annealing studies presented

in this study agree with previous studies. The relief of local residual stress might

contribute to a small portion of the strengthening, but it is unlikely that it accounts for the

magnitude and variability of strengthening observed here. Consider that relief of residual

stress can account for at most a strengthening of 20 MPa, whereas increases of strength of

over 200 MPa were observed here.

The third idea and the primary purpose of this research is to determine if the

strengthening of glass due to acid etching is caused by rounding of the crack tip. This

hypothesis can explain the large variability in the strength data of the etched samples.

Although the indentation load was the same for each specimen, there was variability due

to local surface variability, contact variability, and a number of other factors inherent in

the indentation process. Therefore, in this sense, each indentation is unique. Since each

sample had a unique indentation flaw introduced into it, it is reasonable to assume that








the etching process did not round the crack tips uniformly. Thus, the strength increase

due to etching should not be expected to be the same.

The hypothesis is that samples that fractured at a lower stress had less rounding of

the crack tip, while the samples that fractured at a higher stress had more crack tip

rounding. To support this hypothesis, the apparent fracture toughness, the apparent

blunting energy, and the rounding of the radial surface cracks were examined.


400

350

300

250

200

150

100

50

0


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Sample


Figure 4.1 Flexure test strength results for 30 unetched and 30 etched samples ordered
from low to high


- Unetched
* Etched
.*.









4.1.2 Fracture Toughness

The fracture toughness of all samples was determined using a quantitative

fractographic method (FM) and the strength indentation method (SIM). Figures 4.2 and

4.3 summarize the fracture toughness of the etched and unetched samples respectively

using the fractography method. The average FM fracture toughness of the unetched

samples was 0.75.05 MPa m1/2, while the effective FM fracture toughness of the etched

samples ranged from 0.8 to 3 MPa m1/2. Note that this is called the effective fracture

toughness because it is expected that the structure of the glass is not changing, but rather

the crack is being affected. Hence, the "true" toughness of the glass is not expected to

change. The average FM fracture toughness value for the unetched samples of 0.75

Mpa m1/2 agrees very well with the fracture toughness of soda-lime silica glass reported

in the literature.23-24 Figure 4.2 shows the fracture toughness is independent of the

fracture strength of the samples. This is the expected result because the fracture

toughness of an isotropic brittle material should be constant. On the other hand, figure

4.3 shows that the effective fracture toughness of the etched samples depends strongly on

failure stress. This graph suggests that the crack size has not changed, while the strength

of the samples increased.

The strength indentation fracture toughness values agreed well with the

fractography fracture toughness values for both etched and unetched samples. The

average strength indentation fracture toughness for the unetched samples was 0.75.06

Mpa m1/2, while the effective fracture toughness of the etched samples ranged from 0.9 to

3 Mpa m1/2. Table 4.1 summarizes the fracture toughness data from bothtechniques:












0.85

0.80

0.75

0.70

0.65

0.60

0.55

0.50


Stress (MPa)


Figure 4.2 Fracture toughness of unetched samples using fractography method


Stress (MPa)


Figure 4.3 Effective fracture toughness of etched samples using fractography method


34


"I-


I 1

4I




IJII











Table 4.1 Comparison of Fracture Toughness Data
Fractography Strength Indentation
Unetched Kc = 0.75 .05 MPa*m2 Kc = 0.75 .06 MPa*m/2
Etched K e : 0.8 3 MPa* m1 Keff: 0.9 3 MPa* m

Figure 4.4 shows the agreement of the two methods for calculating fracture toughness of

the etched samples. The line drawn shows a 1:1 correlation, so the two methods are in

good agreement.


3.00 -

C 2.50

C 2.00 -

12.00- --
: 1.oo --- 4 ^ -- -------


0.50 -

0.50 1.00 1.50 2.00 2.50 3 1:11: 3.50
Fractography Kff


Figure 4.4 Agreement between fractography method and strength indentation method
fracture toughness for etched samples

The fracture toughness of the etched samples is referred to as Keff as opposed to

Kc since the formulas derived for Kc were based on the stress distribution around a sharp

crack13'16. Since a blunted crack is the proposed idea for the strength increase, it would

be incorrect to call the resulting fracture toughness Kc. It is not clear how the blunted

crack should be handled in terms of fracture mechanics equations. It is clear that the

etched samples still fail in a brittle manner from observing the fracture surface features









and looking at the load-displacement curves, so presumably the crack starts out blunted,

and then becomes sharp before failure.

There are a number of possible ways to account for crack tip blunting in the

fracture mechanics equations. The first way, as already described, is to use the fracture

mechanics equations as they are, and call the resulting fracture toughness the effective

fracture toughness. Alternatively, the Y-factor which accounts for crack geometry and

loading application could be adjusted to account for crack tip blunting. Finally, the

blunted crack can be accounted for in the energy balance approach to fracture.

4.2 Fracture Mechanics Approaches Applied to Blunt Crack

4.2.1 Inglis Solution

One way to estimate the radius of curvature of the blunted cracks is to use the

Inglis solution. Solving the Inglis equation for the radius of curvature gives the following

expression:

2
P = 2 (4.1)


The radius of curvature of both the unetched and etched samples was estimated in this

way where 6Th is taken to be the theoretical stress of the material, and 6f is the failure

stress of the sample. Using the Inglis equation the radius of curvature of the unetched

samples was calculated to be 3.2 + .3 nm, while the radius of curvature of the cracks of

the etched samples ranged from 3.4 nm to 60 nm. Figure 4.5 shows how the failure stress

of the samples increases with increasing radius of curvature. This relationship followed a

linear trend and makes sense in terms of the proposed blunting mechanism, since

equation 4.1 predicts that the failure stress of the sample is proportional to the square root

of the radius of curvature. The estimated radius of curvature calculated for the etched









samples was greater than that for the unetched samples, which is what was expected in

terms of the blunting mechanism.

When considering the magnitudes of the crack tip radii calculated from the Inglis

equation, the estimates seem fairly reasonable. The unetched samples contained cracks

that were presumably atomically sharp, and performing a rough estimate that each atom

and interatomic bond is approximately 0.3 nm, the 3.2 nm crack tip radius calculated for

the unetched samples corresponds to about 5 atoms at the crack tip. This estimate is

similar to the 2 nm crack tip radius calculated by Pavelchek and Doremus.3 However,

they did not calculate a large increase in the radius of curvature of their etched samples,

and attributed this to the etchants not being able to penetrate to the crack tip. This is not

the case in the present study, as the estimated radii of curvature for the etched samples

became quite large compared to the unetched samples. We must keep in mind that the

Inglis equation was developed for an elliptical hole in an infinite plate of material, so this

calculation can only provide a rough estimate for the crack tip curvature in this situation.

Since the equivalent semicircular crack size was used when calculating the Inglis

estimates, it might be appropriate to introduce a correction factor to account for this

difference. This correction factor would appear as (a)1/2=(_ e/s)(C)1/2, where 4e and 4s are

elliptical integrals of the second kind for the elliptical crack and circular crack,

respectively. Applying this correction factor to the unetched samples that had an average

a/b ratio of 0.6 + 0.1 (which is a measure of the ellipticity of the crack), the resulting

(4,e/s) ratio would be 1.39/1.57, or approximately 0.9. Applying this correction factor to

the unetched samples, we see that the crack tip radius is reduced by the factor of 0.9,

resulting in a crack tip radius of 2.9 .3 nm. The radius of curvature for the etched









samples would change in the same manner according to the correction factor associated

with the etched flaws. This correction factor makes sense in terms of making the radius

of curvature of the unetched flaw smaller, and thus more reasonable in terms of atomic

dimensions. Actual measurements of the sharp and blunted crack tips are needed to

compare to these approximations. This topic will be addressed later.


4 16 36 64 100 nm
4 16 36 64 100 nm


'qW

(LL






w615
0100





50

0


0 2 4 6 8 10

(Crack tip radius)l rml

Figure 4.5 Estimation of crack tip radius of etched samples using Inglis solution

4.2.2 Stress and Energy Criteria

Fracture mechanics equations for materials that fail in a brittle manner should be

able to account for the crack tip blunting theory if the blunted crack sharpens before

fracture. Based on this premise, there are several ways to factor crack tip blunting into

fracture mechanics equations. The first method is to incorporate the degree of blunting

into the Y factor of the fracture mechanics stress equation Kc=Y6(c)1/2. Since the critical


Oeolooor









stress intensity factor should be a material property, we can set Kc = 0.75 MPa*m1/2 and

calculate the appropriate Y-factor for the etched samples to satisfy the equation. Figure

4.6 shows graphically how the calculated Y-factor varies for the etched samples. The

etched samples that failed at lower stresses and had less crack tip blunting had Y-factors

close to the Y-factor of 1.65 used for the unetched samples. The Y-factors then dropped

off linearly with 1/of, as predicted by the fracture mechanics equation. The goal of this

approach would be to incorporate the initial crack tip radius into the formulation of the

Y-factor. TheY-factor is plotted versus 1/(p)1/2 in Figure 4.7, and again follows a linear

trend as predicted by the Inglis equation.



1.80
1.60-
c 1.40
o
S1.20
*A 1.00-

o
0.80 0
0.60
0.40
0.20
0.00 I
0.000 0.005 0.010 0.015 0.020

1/q (1/MPa)


Figure 4.6 Y-factor projections for etched samples versus strength












1.80
1.60
1.40
S1.20
1.00
0.80
S0.60
0.40
0.20
0.00 .
0 0.1 0.2 0.3 0.4 0.5 0.6

1/(p)12 (/nm12)

Figure 4.7 Y-factor projection plotted versus estimated crack tip radii from Inglis

Another way to account for crack tip blunting using fracture mechanics equations

is to the extra energy necessary to sharpen a blunted crack. For the purposes of this

discussion, we will assume plane stress conditions:

Kc = Ey (4.2)

Where E is the elastic modulus, and yf is the fracture energy. Equation 4.2 was then used

to estimate the fracture surface energy of the material by using Kc=0.75 and E=70 GPa;

the resulting fracture surface energy of the glass was estimated to be 4 N/m. Since the

material failed in a brittle manner, it is assumed that in the etched samples the blunted

crack was sharpened before failure. This sharpening of the blunted crack can be thought

of as an added energy barrier to failure, and could be described in a modified version of

equation 6 which is similar to Orowan's equation2 for failure of ductile materials:










Keff= V2E(f + YB) (4.3)

The dB term in equation 4.3 can be thought of as the energy necessary to sharpen the

blunted crack before failure. Samples with more highly blunted cracks will require more

energy to sharpen the crack and will therefore have a higher effective toughness. Figure

4.8 shows graphically the so-called blunting energy plotted against effective fracture

toughness for the etched samples. The effective fracture toughness increased linearly with

(yB)1/2, as predicted by equation 4.3. The samples that failed at higher stresses had higher

effective fracture toughness, and therefore required more energy to failure in the form of

sharpening the blunted crack tip.


9
8
7
%-,6

4









Figure 4.8 Blunting energy of etched samples

4.3 Characterization of Fracture Surfaces

4.3.1 Fractographic Analysis

Upon examination of the fracture surfaces it eventually became clear that there

was a degree of stable crack growth during 4-point flexure testing. This stable crack
growth was observed in the unetched samples as well as the etched samples that failed at
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50
Keff(MPa in12)

Figure 4.8 Blunting energy of etched samples

4.3 Characterization of Fracture Surfaces

4.3.1 Fractographic Analysis

Upon examination of the fracture surfaces it eventually became clear that there

was a degree of stable crack growth during 4-point flexure testing. This stable crack

growth was observed in the unetched samples as well as the etched samples that failed at









lower stresses, and presumably whose crack tips stayed relatively sharp. Figure 4.9

shows optical images of fracture surfaces of both unetched and etched samples. The

evidence for stable crack growth can just be made out between the arrows in the unetched

sample. The etched samples that failed at higher stresses did not exhibit noticeable stable

crack growth, possibly because residual stress was relieved, and/or the crack tip blunting

mechanism inhibited any stable crack growth. Using the optical microscope to assess the

fracture surface, the initial flaws were easy to find, but the critical flaws due to stable

crack growth were more difficult to ascertain.
















a) Unetched, of = 50 MPa b) Etched, of = 145 Mpa

Figure 4.9 Fracture surfaces of unetched and etched samples showing stable crack growth

Crack size was an important factor in the fractography measurements. All of the

semi-elliptical flaws were converted to an equivalent semi-circular crack size by the

relation c=(ab)1/2, so we can compare equivalent semi-circular crack sizes for the

unetched and etched samples.20 The unetched samples had an average crack size of 74 +

6 rm, while the average crack size for the etched samples was 48 5 im. This

difference in critical crack size can mainly be attributed to two factors. First, there was

more stable crack growth in the unetched samples, which made the critical crack size









larger. For the most part the etched samples were able to resist stable crack growth

which could be due to the blunting mechanism. Secondly, the etching process might

have dissolved away material at the surface, thus making the surface flaws smaller to

start with. This process would manifest itself by creating smaller critical cracks in both

width and depth. Table 4.2 provides crack size along with strength and fracture

toughness for both unetched and etched samples. The table provides average values for

the unetched samples, and values for selected etched samples to show the range of the

data.

Table 4.2 Strength, fracture toughness, and crack size of unetched and etched samples
Crack size Strength Fracture toughness a/b ratio
(Lm_) (MPa) (MPam1"2)
Unetched 74 6 52 3 0.75 .05 0.6 + 0.1
Etched 56 66 .82 0.4
Etched 47 150 1.69 0.8
Etched 45 232 2.55 0.9

In addition to crack size, the a/b ratio is a useful measure that provides

information about the shape of the crack. A normal a/b ratio for semi-elliptical cracks is

0.8.19 Table 4.2 shows a/b values for both unetched and etched samples, and stable crack

growth is a possible explanation why an average a/b ratio of 0.6 was obtained for the

unetched samples, as opposed to the normal 0.8. Less stable crack growth was observed

in the etched samples, which is why the average a/b ratio for the etched samples was 0.7.

This means that the stable crack growth occurred mainly in one direction, creating

elongated critical flaws.

Now that the strength and fracture toughness results have been presented, we can

assess the different possibilities for the observed strength increase. There are several

possibilities for the observed strength increase. If the flaw size decreases, then the









strength increases. If tensile residual stresses are relieved, there would be a strength

increase. If the crack tip becomes less sharp, then the strength would increase. We see in

Table 4.2 that the crack sizes of the etched specimens were slightly smaller than those of

the unetched specimens. To assess the strengthening that should occur due to the

decrease in critical flaw size of the etched samples, the familiar fracture mechanics

equation can be used:



Sf = K (4.4)


Assuming a sharp crack with c=48im, K,=.75 MPa*m'/2, and Y=1.65, the strength

should only increase to about 66 MPa, as compared to 52 MPa for the unetched samples,

which is a strength increase of about 14 MPa. This is obviously much smaller than the

strengthening observed for most of the etched samples. Even for the smallest observed

crack size of 39 |tm, the strength increase would be about 20 MPa, which is clearly not

enough to account for the observed strength increase with etching. Relief of residual

stress is another possibility for the strength increase. However, in examining previous

studies and preliminary experiments from this study, the strength increase due to relief of

residual stress is approximately 15-20 MPa. Again, this value is much less than the

strength increase observed in most of the etched samples. In fact, if the decrease in crack

size and relief of residual stress are combined to provide a net strength increase of about

40 MPa, they still do not come close to accounting for the observed results. This leads to

the conclusion that the majority of the strength increase is most likely due to blunting of

the crack tip.









In addition to measuring the crack size, the mirror size was measured for each

sample. This distance is represented in Figure 4.11 as ri (ri where i=l), and has been

shown to be related to the global stress at failure through the following equation:21

Yor712 = YMJ = KBJ (4.5)

This equation implies that the mirror region ends and the mist region begins once a

certain stress intensity (KBj) is reached. Thus the mirror constant, Mi, should be constant

for a given material just like Kc. The only difference is that the mirror constant is

associated with the mirror/mist transition and is not in the vicinity of the crack. Because

M1 is not in the vicinity of the crack then it would be reasonable to assume that the

unetched and etched samples should have similar mirror constants. When looking at the

mirror sizes of the samples it was determined that the unetched samples had a mirror

constant of 1.9 0.1 MPa*ml/2, and the etched samples had a mirror constant of 1.7 0.2

MPa*ml/2. This value for the mirror constant was consistent with findings in previous

studies.25'26 Unbalanced ANOVA was performed to see if there was a statistically

significant difference between the mirror constants of the unetched and etched samples.

The test showed that there was no statistical difference between the mirror constants of

the unetched and etched samples.

The next important relationship to consider is the mirror to flaw size ratio (ri/c)

for the unetched and etched samples. From looking at the fracture surfaces, it was clear

that the mirror to flaw size ratio was decreasing for the etched samples. This decrease in

mirror to flaw size ratio also appeared to occur with samples that failed at higher stresses.

This pattern can be seen graphically in figure 4.10. The unetched samples had a mirror to

flaw size ratio of 14 1 which was similar to the value obtained in a previous study25,









while the etched samples ranged from a mirror to flaw size ratio of about 10 to 1.8. As

the graph indicates, the mirror size was measured for only 11 of the relatively lower

strength samples out of the 30 etched samples. This is because for the higher strength

samples the mirror was too small to be measured with the optical microscope, or the

fracture surface was lost due to shattering of the high energy fracture. As noted before,

the crack size did not change significantly among the etched samples. Thus, the change

in the mirror to flaw size ratio is due to the decrease in mirror size corresponding to the

increase in strength. The extrapolation of the line in figure 4.7 should be extended to an

r/c of 1, because obviously the mirror size cannot get smaller than the crack. What

happens in reality is that somewhere near the r/c ratio of 1, the original crack no longer

becomes the dominant crack and the fracture originates at another location. The mirror to

flaw size ratio for the crack at this new location should then return to the normal value of

about 10-15.

4.3.2 Optical Microscopy

Originally the atomic force microscope was designated as the instrument best

suited to measure the crack tip radius of the samples, but after many failed attempts it

was abandoned partly due to my inexperience on the instrument and partly due to the

difficulty of the task. First of all it was difficult to obtain a decent image of the sample,

and once a reliable image was produced it was difficult to discern where on the fracture

surface the tip of the AFM was scanning. In addition, the nature of the task was unclear

at best. Since the facture surface was examined, the flaw had already propagated through











160
140 -
120 F etched

a100

80"; 8 unetched
60
40
20
0
1 4 7 10 13 16 19

rrrr r c c

Figure 4.10 Mirror to flaw size ratio for unetched and etched samples


Hackle
SII Mist


Mirror


Origin


r2b rch
Figure 4.11 Schematic of fracture surface features
Figure 4.11 Schematic of fracture surface features









the material, so the best we could do would be to see if there was a difference between

the fracture surfaces of the unetched and etched samples on the atomic level that could be

attributed to crack tip blunting. Since there is no way to observe the critical crack in a

material before failure, the other techniques we had available to obtain indirect visual

evidence of crack tip blunting were optical microscopy and SEM.

Light microscopy was used to measure the crack size and mirror size of the

unetched and etched samples. Figure 4.12 shows optical images of 3 fracture surfaces;

one unetched sample and two etched samples that failed at varying stress levels. The

unetched sample failed at a stress of 50 MPa, and is characterized by a large mirror

region. The etched samples clearly show a decrease in the fracture mirror size, and hence

a decrease in mirror to flaw size ratio with increasing failure stress. The mist and hackle

region can be seen in the first etched sample, while the final etched sample exhibited such

a high stress and high energy failure that the origin was not found.

4.3.3 Scanning Electron Microscopy

Scanning electron microscopy was also used to look for differences between the

unetched and etched fracture surfaces, and was able to do so at higher magnifications. It

was also used to examine unetched and etched radial cracks associated with the

indentations. This was done after the samples were fractured by placing the sample

halves flat on the aluminum stubs and locating what remained of the indentations. The

radial cracks emanating from the indentations were located for both the unetched and

etched samples. Figure 4.13 shows radial surface cracks from an etched sample, and

figure 4.14 shows an unetched sample. From these pictures the etched radial cracks were

estimated to have a crack tip radius of approximately 4000 nm, while the unetched radial









cracks were estimated to have a crack tip radius of about 15 nm. These values are very

different than the values obtained using the Inglis solution, as shown in Figure 4.17. The

etched crack tip curvature for the critical flaw with Inglis was about 30-60 nm, while the

etched surface radial cracks were measured to be about 4000 nm. This discrepancy

makes sense since you would expect the flaws at the surface to be rounded a great deal

more than the critical flaw. The unetched crack tip radius using Inglis was about 3 nm

compared with the SEM estimate of 15 nm. This discrepancy is related more to the

limitations in the resolution ofthe SEM. Figures 4.15 and 4.16 show two etched fracture

surfaces. The mirror/mist boundary can clearly be seen in both fracture surfaces.


a) unetcnea, of = 3u iv


a) Etched, of = 145 Mpa a) Etched, of = 290 Mpa

Figure 4.12 Fracture surfaces of unetched and etched samples showing failures at
different stress levels
























































Figure 4.13 SEM images of etched surface radial cracks



























































Figure 4.14 SEM images of unetched radial surface crack


























































Figure 4.15 SEM fracture surface of etched sample at different magnifications
























































Figure 4.16 Fracture surface of etched sample at different magnifications













4 16 36 64 100


400

350

300

250

200

150

100

50

0


Inglis







01 -A-easured



0 2 4 6 8

(Crack tip radius)112 nm112


Figure 4.17 Comparison of crack tip radius calculated from Inglis and measured from
fracture surface


nm















CHAPTER 5
CONCLUSIONS

The objectives of this study were to 1) characterize the fracture surfaces of both

the unetched and etched samples to determine if evidence for crack tip blunting exists,

and 2) use measurements of fracture surface features along with fracture mechanics

equations to propose different ways of accounting for blunted cracks in brittle fracture.

Based on the results of this study, the following conclusions can be made.

Relief of tensile residual stress, and decrease in flaw size due to etching only

accounts for a small portion of the observed strengthening of soda-lime silica glass. The

majority of the strengthening is thought to be due to rounding of the crack tip. The crack

tip rounding hypothesis can explain the variability in the strength data because each

unique critical indentation crack was not etched the same amount, so each crack was

rounded a different amount.

Direct measurements of the radius of curvature of the critical indentation crack

after fracture were obtained where possible, and the radius of curvature of both unetched

and etched radial surface cracks was estimated using the SEM.

The fracture mechanics approaches to describe the blunted crack appear to be

realistic. The Inglis solution provided an estimate of the radius of curvature of both

unetched and etched cracks that appeared reasonable in terms of atomic dimensions. The

crack tip radii estimated with Inglis along with the calculated Y-factors provides a

starting point to account for crack tip blunting in the stress intensity factor. In terms of









energy, the Griffith equation can be modified to include an extra energy term for the

blunted crack to become sharp before fracture.

A direct correlation was found between the mirror to flaw size ratio and the

strength of the etched samples. Thus, it is possible to relate the mirror to flaw size ratio

to the crack tip curvature of the blunted crack.















APPENDIX
CALCULATED DATA

Table A-1 4-point flexure results for unetched samples


Sample Max Load (N) Max Stress
(Mpa)


Mean 66 52
St. Dev. 3 3
Min 61 49
Max 74 60









Table A-2 4-point flexure results for etched samples


Sample Max Load (N) Max Stress
(Mpa)


231
89
82
140
83
246
290
366
144
188
253
203
182
473
175
330
360
330
255
247
336
98
210
209
234
356
154
345
381
136


185
71
66
112
67
196
232
293
115
150
203
162
145
378
140
264
288
264
204
197
269
78
168
167
187
285
123
276
304
109


Ave 237 190
St. Dev. 100 81
Min 82 66
Max 473 378










Table A-3 Fractography measurements and fracture toughness of unetched samples

Fractography Strength
Stress Kc Indentation a/b
Kc
Sample a(um) 2b(um) c(um) Mpa (MPa*m.5) (MPa*m.5) Ratio


197
193
240
212
235
216
221
226
202
188
235
197
212
202
183
179
183
155
183
212
188
207
188
197
150
197
188
165
202
212


0.73
0.76
0.76
0.73
0.75
0.74
0.76
0.76
0.74
0.76
0.75
0.72
0.76
0.76
0.74
0.73
0.76
0.75
0.75
0.72
0.73
0.74
0.8
0.74
0.76
0.72
0.74
0.76
0.71
0.74


0.75
0.76
0.73
0.73
0.71
0.73
0.73
0.73
0.76
0.74
0.72
0.73
0.74
0.74
0.77
0.78
0.78
0.82
0.76
0.74
0.76
0.74
0.83
0.74
0.8
0.73
0.76
0.78
0.72
0.73


0.52
0.59
0.51
0.53
0.56
0.52
0.55
0.54
0.51
0.7
0.52
0.57
0.58
0.6
0.56
0.53
0.56
0.61
0.62
0.44
0.55
0.55
0.5
0.57
0.75
0.57
0.55
0.74
0.56
0.53


Mean 56 199 74 52 0.75 0.75 0.6
St. Dev. 5 21 6 3 0.05 0.06 0.1










Table A-4 Fractography measurements and fracture toughness of etched samples. Blank
samples indicate crack size could not be measured

Strength
Stress Fractography Indentation a/b
Sample a(um) 2b(um) c(um) MPa Kc (MPa*m.5) Kc Ratio
(MPa*m.5)


185
71
66
112
67
196
232
293
115
150
203
162
145
378
140
264
288
264
204
197
269
78
168
167
187
285
123
276
304
109


2.02
0.87
0.82
1.27
0.8
2.13
2.55

1.25
1.69
2.1
1.81
1.64

1.59







0.93

1.88
2.11
3.08
1.6


1.23


1.93
0.94
0.89
1.33
0.9
2.02
2.29
2.73
1.35
1.66
2.07
1.75
1.61
3.31
1.57
2.53
2.7
2.53
2.08
2.03
2.56
1.01
1.8
1.79
1.95
2.67
1.42
2.61
2.81
1.3


0.73
0.47
0.44
0.82
0.38
0.76
0.9

0.76
0.82
0.7
0.86
0.82

0.78







0.52

0.82
0.82
0.76
0.57


0.82


Mean 40 117 48 190 0.7
St. Dev. 4 28 5 80 0.2










Table A-5 Mirror size and mirror constant values for unetched samples


Mirror
Sample rm c rm/c Constant
mm mm (MPa m1/2)
65 1410 99 14 2.0
66 1222 96 13 1.9
67 1692 120 14 2.1
68 1222 106 12 1.8
69 1504 118 13 1.9
70 1410 108 13 1.9
71 1504 110 14 2.0
72 1410 113 13 1.9
73 1222 101 12 1.9
74 1504 94 16 2.0
75 1410 118 12 1.9
76 1410 99 14 1.9
77 1222 106 12 1.8
78 1504 101 15 2.0
79 1504 92 16 2.1
80 1410 89 16 2.1
81 1222 92 13 1.9
82 1128 78 15 2.0
83 1316 92 14 1.9
84 1410 106 13 1.9
85 1410 94 15 2.0
86 1316 103 13 1.9
87 1175 94 13 2.0
88 1410 99 14 1.9
89 1128 75 15 1.9
90 1410 99 14 1.9
91 1316 94 14 1.9
92 1316 82 16 2.0
93 1363 101 13 1.8
94 1363 106 13 1.9
Ave 1361 14 1.9
St. Dev 129 1 0.1










Table A-6 Mirror size and mirror constant values for etched samples

Mirror
Sample rm c rm/c Constant
(MPa m/2)
65
66 705 80 9 1.9
67 846 85 10 1.9
68 212 52 4 1.6
69 752 87 9 1.8
70
71
72
73 141 49 3 1.4
74 94 52 2 1.5
75
76
77 118 52 2 1.6
78
79 165 54 3 1.8
80
81
82
83
84
85
86 423 73 6 1.6
87
88
89
90
91 282 82 3 2.1
92
93
94 282 52 5 1.8
Ave 1.7
St. Dev 0.2















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BIOGRAPHICAL SKETCH


Salvatore Ruggero was born February 18th, 1978, in Staten Island, New York. He

moved to Ormond Beach, Florida, with his parents when he was 12 years old, and

graduated from Flagler Palm Coast High School in 1996. He enrolled in the honors

program at the University of Florida and spent two years living in Weaver Hall and

experiencing the intellectual and social culture of Gainesville. After a brief hiatus in

Milan, Italy, for an exchange program in international business during the fall of 1998,

Salvatore returned to Gainesville still confused about the course of his educational path.

Nevertheless, he began a course of study in materials science and engineering, and

graduated with a BS in May 2001. After a summer spent backpacking through Europe,

Salvatore began his graduate education at the University of Florida in August 2001, and

eventually joined Dr. John J. Mecholsky's research group. His research interests include

fracture of materials that behave in a brittle manner, and mechanical properties of

biomaterials. In his spare time, Salvatore enjoys playing sports such as tennis, basketball,

and racquetball, and also enjoys reading the occasional novel.