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Elastic-Plastic Fracture Mechanics of Compact Bone

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Elastic-Plastic Fracture Mechanics of Compact Bone
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YAN, JIAHAU ( Author, Primary )
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2008

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Bones ( jstor )
Cattle ( jstor )
Femoral fractures ( jstor )
Femur ( jstor )
Fracture mechanics ( jstor )
Fracture strength ( jstor )
J integral ( jstor )
Manatees ( jstor )
Ribs ( jstor )
Specimens ( jstor )

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University of Florida
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University of Florida
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Copyright Jiahau Yan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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8/31/2006
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495636956 ( OCLC )

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ELASTIC-PLASTIC FRACTURE MECHANICS OF COMPACT BONE By JIAHAU YAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Jiahau Yan

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To my parents and Paoyun

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ACKNOWLEDGMENTS I would like to take this opportunity to express my great appreciation to all the people who made this dissertation possible. I would like to thank my advisor (Dr. John J. Mecholsky, Jr.) for his kindly guidance and for all of the inspiring conversations. It has been a wonderful journey for me to be a part of his research group. I would like to thank my supervisory committee members (Dr. Kenneth J. Anusavice, Dr. Bhavani V. Sankar, Dr. Laurie B. Gower, and Dr. Wolfgang Sigmund) for their helpful suggestions and instructions. I worked with Kari Clifton researching bone materials for more than 4 years. I have learned a lot from her about manatees and bone biology. I also benefited greatly from learning her attitude toward research. I would also like to thank the people in the University of Florida (UF) Dental Biomaterials labs (especially Mr. R. Ben Lee) for their constant generous help; the people in Major Analytical Instrumentation Center (MAIC) (especially Mr. Wayne Acree) for their instructions and help in instruments; the people in the UF Meat Lab (especially Tommy Estevez) for their kindness in providing bovine bone tissues for my study; Mr. Amit Daga for his help with TGA/DTA analysis; and Ms. Maria Palazuelos for her instructions in using the software to measure porosity. I also received help from many other people throughout my learning career at UF. I am grateful to them for all the knowledge, help, and advice they provided. Without them, this dissertation could not have been written. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT.........................................................................................................................x CHAPTER 1 INTRODUCTION........................................................................................................1 2 LITERATURE REVIEW.............................................................................................6 2.1 Bone Tissue Structure and Classification..............................................................6 2.1.1 Compact Bone and Cancellous Bone.........................................................6 2.1.2 Lamellar Bone and Woven Bone...............................................................6 2.1.3 Primary Bone and Secondary Bone............................................................8 2.2 Compositions of Bone and Their Roles in Bone.................................................10 2.2.1 Bone Mineral............................................................................................10 2.2.2 Organics in Bone......................................................................................11 2.2.3 Water........................................................................................................12 2.3 Fracture Mechanics..............................................................................................13 2.3.1 Griffith Strength Relation.........................................................................14 2.3.2 Critical Energy Release Rate and Critical Stress Intensity Factor...........14 2.3.3 Elastic-Plastic Fracture Mechanics and J Integral....................................16 2.4 Fracture Mechanics of Compact Bone................................................................19 2.5 Effect of Temperature on the Mechanical Behavior of Compact Bone..............23 3 SAMPLE PREPARATION AND EXPERIMENTAL METHODS..........................25 3.1 Compact Bone from Bovine Femur and Manatee Rib........................................25 3.2 Test the Effect of Temperature on the Fracture Toughness of Compact Bone...27 3.2.1 Single-Edge V-Notched Beam Method....................................................27 3.2.2 Notching Procedure and Determination of Fracture Toughness..............29 3.2.3 Temperature Control and Testing Apparatus...........................................30 3.2.4 Post Test Measurement and Analysis.......................................................32 3.3 Measuring the J Integral......................................................................................33 v

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3.4 Fracture Toughness of Water-free and Organics-free Compact Bone................37 3.5 Experimental Design and Statistical Analysis.....................................................39 4 RESULTS...................................................................................................................41 4.1 Effect of Temperature on Fracture Toughness of Compact Bone.......................41 4.1.1 Results of Pilot Studies Using PMMA.....................................................41 4.1.2 Results of Bovine Femur and Manatee Rib..............................................46 4.2 Measuring the J Integral of Compact Bone.........................................................53 4.3 Fracture Toughness of Water-free and Organics-free Compact Bone................60 5 Discussion...................................................................................................................65 5.1 Effect of Temperature on Fracture Toughness of Compact Bone.......................65 5.2 Measuring the J Integral of Compact Bone.........................................................74 5.3 Effect of Water and Organics on Fracture Toughness of Bone...........................83 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK............................88 6.1 Conclusions..........................................................................................................88 6.2 Suggestions for Future Work...............................................................................90 LIST OF REFERENCES...................................................................................................92 BIOGRAPHICAL SKETCH...........................................................................................100 vi

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LIST OF TABLES Table page 1-1. Mechanical properties of cortical bone and selected materials....................................3 2-1. Selected studies measuring fracture toughness of compact bone...............................21 3-1. Information of the six groups in the J integral measurement.....................................36 4-1. Fracture toughness of PMMA using the SEVNB method..........................................42 4-2. Fracture toughness of bovine femur and manatee rib at different temperatures........47 4-3. Porosity, density, and composition of bovine femur and manatee rib.......................51 4-4. Fracture toughness, elastic moduli and the J integral of the specimens.....................53 4-5. Fracture toughness of water-free bovine femur and manatee rib...............................60 4-6. Fracture toughness of organics-free bovine femur and manatee rib..........................61 5-1. Results of 5 previous studies on the fracture toughness of bovine femur..................71 5-2. Fracture toughness of bovine femur and manatee rib at 23 and 37C........................75 vii

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LIST OF FIGURES Figure page 1-1. An SEM image showing an osteon in a bovine femur.................................................2 2-1. Compact bone and cancellous bone..............................................................................7 2-2. Plexiform bone and an osteon with cement line...........................................................9 2-3. Collagen fibrils and pores in bones............................................................................12 2-4. Three common load-extension curves of materials under tension or flexure.............16 2-5. Two-dimensional plate with an introduced crack.......................................................17 3-1. Single-edge V-notch in a bovine bone sample...........................................................28 3-2. SEVNB specimens from bovine femur and manatee rib............................................30 3-3. Instron testing machine and 4-point flexure test........................................................31 3-4. Deep-notched specimen in flexure.............................................................................35 3-5. Specimen color change after different heating temperatures.....................................38 4-1. Results of the pilot studies using the SEVNB method on PMMA.............................43 4-2. Two SEM images of the V-notch in PMMA..............................................................45 4-3. Effect of temperature on fracture toughness of bovine femur and manatee rib.........48 4-4. SEM images of the fracture surfaces of bovine and manatee specimens...................49 4-5. Porosity measurements using Image Pro software.....................................................50 4-6. The TGA and DTA profiles of bovine femur and manatee rib..................................52 4-7. Average Jel, Jpl, and Jtotal values of the six studied groups.........................................56 4-8. Comparison of J integrals for 4 materials...................................................................57 4-9. Typical load-extension curves of the deep-notched flexure specimens.....................58 viii

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4-10. SEM images of the fracture surfaces of bovine and manatee specimen..................59 4-11. Fracture toughness of bovine femur and manatee rib at three states........................62 4-12. Load-extension curves of 3 bovine SEVN specimens..............................................63 5-1. Load-extension curves of 3 bovine specimens tested at different temperature..........68 5-2. Load-extension curves of 3 manatee specimens tested at different temperature.......70 5-3. Two bovine specimens with different amounts of slow, stable crack growth............79 5-4. Angle view of the fracture surfaces of 3 bovine specimens.......................................81 5-5. Fracture surfaces of two bovine specimens after different heating temperatures......85 ix

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ELASTIC-PLASTIC FRACTURE MECHANICS OF COMPACT BONE By Jiahau Yan August 2005 Chair: John J. Mecholsky, Jr. Major Department: Materials Science and Engineering Bone is a composite composed mainly of organics, minerals and water. Most studies on the fracture toughness of bone have been conducted at room temperature. Considering that the body temperature of animals is higher than room temperature, and that bone has a high volumetric percentage of organics (generally, 35-50%), the effect of temperature on fracture toughness of bone should be studied. Single-edged V-shaped notched (SEVN) specimens were prepared to measure the fracture toughness of bovine femur and manatee rib in water at 0, 10, 23, 37 and 50C. The fracture toughness of bovine femur and manatee rib were found to decrease from 7.0 to 4.3 MPam1/2 and from 5.5 to 4.1 MPam1/2, respectively, over a temperature range of 50C. The decreases were attributed to inability of the organics to sustain greater stresses at higher temperatures. We studied the effects of water and organics on fracture toughness of bone using water-free and organics-free SEVN specimens at 23C. Water-free and organics-free specimens were obtained by placing fresh bone specimen in a furnace at different temperatures. Water and organics significantly affected the fracture toughness of bone. x

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Fracture toughness of the water-free specimens was 44.7% (bovine femur) and 32.4% (manatee rib) less than that of fresh-bone specimens. Fracture toughness of the organics-free specimens was 92.7% (bovine femur) and 91.5 % (manatee rib) less than that of fresh bone specimens. Linear Elastic Fracture Mechanics (LEFM) is widely used to study bone. However, bone often has small to moderate scale yielding during testing. We used J integral, an elastic-plastic fracture-mechanics parameter, to study the fracture process of bone. The J integral of bovine femur increased from 6.3 KJ/mm2 at 23C to 6.7 KJ/mm2 at 37C. Although the fracture toughness of bovine bone decreases as the temperature increases, the J integral results show a contrary trend. The energy spent in advancing the crack beyond the linear-elastic deformation was much greater than the energy spent in linear-elastic deformation. This could be because bone has at least four toughening mechanisms and a high volumetric percentage of organics (approximately 42% for bovine femur). The J integral is shown to better describe the fracture process of bovine femur and manatee rib. xi

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CHAPTER 1 INTRODUCTION Bone is a composite, composed mainly of collagen matrix, apatite minerals, and water (Martin et al., 1998). The hierarchical assembly of bone’s organic and inorganic constituents makes bone very efficient in terms of stopping crack propagation. Many studies have described the microstructure of bone (Vincent, 1990; Rho et al., 1998; Weiner and Wagner, 1998). This hierarchical assembly scales from nanoscale (apatite crystals and tropocollagen), to microscale (collagen fibril and fibers), to macroscale (osteons and lamellae) (Weiner and Wagner, 1998). Microscopically, collagen fibers in bone comprise microfibrils that have nanosize bone crystals in between the tropocollagen network. These fibers are usually aligned along the major axis of bone. Macroscopically, cortical bone shows anisotropy with osteons aligned longitudinally for long bones. Figure 1-1 shows a scanning electron microscope (SEM) image of an osteon with laminae around the Haversian canal in a bovine femur. The structure of bone is mainly responsible for bone’s toughness to fracture from mechanical loads. Several toughening mechanisms, such as microcracking (Burstein et al., 1975; Zioupos et al., 1995), crack deflection (Yeni and Norman, 2000), and fiber bridging (Nalla et al., 2003; Yan et al., 2005a) have been observed in the fracture of compact bone. The hierarchical structure of bone spans many scales of length, which makes these toughening mechanisms work at many levels, and leads to a profound effect. Because of its hierarchical structure and toughening mechanisms, bone has a much greater fracture toughness despite its relatively 1

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2 Figure 1-1. An SEM image showing an osteon in a bovine femur. Layered structure (called lamellae) can be seen around the Haversian canal. low elastic modulus (usually between 15-30 GPa in longitudinal direction), than one of its main constituents, hydroxyapatite (90-110 GPa for synthetic hydroxyapatite). Table 1-1 shows typical elastic moduli, tensile strength, elongation at fracture, and fracture toughness of 6 common materials. For material scientists, natural materials (such as bone) offer clues to optimizing the properties and design of composites. Fracture mechanics (analyzing the structural behavior of an object in terms of applied stress, crack size and component geometry) has been applied extensively to studies of failure for many materials, and to the design of products (Hertzberg, 1996; Saxena, 1998). Linear-elastic fracture mechanics (LEFM), which is based on the assumption of no (or limited) plasticity in the material during the fracture process, is

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3 Table 1-1. Mechanical properties of cortical bone and selected materials*. Elastic Modulus (GPa) Tensile Strength (MPa) Elongation at fracture (%) Fracture Toughness (MPam1/2) Cortical Bone (longitudinal) 15-30 70-150 0-8 1.6-11.2 Alumina (Al2O3) 393 275-550 ~ 0 2.7-4.2 Soda-lime Glass 69 69 ~ 0 0.75 Synthetic Hydroxyapatite 90-110 60-80 ~ 0 0.9-1.2 316 Stainless Steel 210 600 20-55 45-70 Polymethyl Methacrylate (PMMA) 2.2-3.2 48-72 2-5.5 0.7-1.6 * Ratner and Hoffman, 1996; Callister, 1997; Ruys et al., 1995; Wang and Puram, 2004; Gross and Bhadang, 2004 widely used to quantify the properties of different bones (Melvin and Evans, 1973; Robertson et al., 1978; Behiri and Bonfield, 1984 & 1989; Norman et al., 1995; Wang and Agawal, 1996; Currey et al., 1996; Yeni et al., 1997 & 1998; Zioupos and Currey, 1998; Feng et al., 2000; Lucksanasombool et al., 2001; Wang et al., 2002; Nalla et al., 2003; Yan et al., 2005b). Fracture toughness (KC), which measures a material’s resistance to brittle fracture when a sharp crack is present, may be a better index of the mechanical performance of bone than strength or elongation at fracture (Bonfield, 1987; Lucksanasombool et al., 2001). Scientists have investigated different effects on the fracture toughness of bone: crack velocity and bone density (Behiri and Bonfield, 1984), crack orientation (Behiri and Bonfield, 1989; Feng et al., 2000), age (Zioupos and Currey, 1998; Wang et al., 2002), and mineral density (Currey et al., 1996). We first addressed the effect of temperature on fracture toughness of bone. Previous fracture toughness studies on bone were mostly carried out at room temperature (usually 20-23C). However, body temperatures of mammals are usually much higher

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4 than room temperature (36.5C for manatees, 37C for humans and 38.5-39C for cows). Bone generally contains 55 to 65 vol. % of water and organics (Martin et al., 1998; Boskey, 2001). The effect of temperature has been shown to have a significant effect on the mechanical behavior of polymers (Rosen, 1993). In this study, we tested specimens from manatee rib and bovine femur at 0, 10, 23, 37, and 50C to examine the hypothesis that temperature has a significant effect on the fracture toughness of bone. Next we applied elastic-plastic fracture mechanics (EPFM) to bone. Almost all previous studies on fracture toughness of bone (including the first part of this dissertation) used LEFM to study bone fracture for two main reasons. Bone has a high weight percentage of mineral content (generally, 50-70 wt.% (Boskey, 2001)). Bone mostly fails at an elongation between 0 and 8% (Ratner and Hoffman, 1996); for most cases less than 3% (Currey, 1970). Elongation at fracture in this range is usually considered brittle or quasi-brittle fracture. However, in some cases, bone exhibits substantial plastic deformation before it fails. Our previous 3-point flexure tests on manatee ribs show that extension in the plastic deformation region (beyond the yield stress) sometimes equals or exceeds extension in a linear-elastic region. In EPFM, the J integral is a parameter that can be used to quantify the energy consumed in plastic deformation of a material before fracture. In this dissertation, the J integral was used to study the fracture energy consumed in plastic deformation and compared with the energy consumed in the linear-elastic part of bovine femur under quasi-static loading. Finally we compared the fracture toughness of bovine femur and manatee rib in three different states: wet (submerged in water), water-free and organics-free. Our goal here was to quantify the effects of water and organics on the fracture toughness of bone

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5 and to test the hypothesis that fracture toughness is not greater for water-free and organics-free bovine bone than for water-free and organics-free manatee bone.

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CHAPTER 2 LITERATURE REVIEW 2.1 Bone Tissue Structure and Classification 2.1.1 Compact Bone and Cancellous Bone Bone (a general term that describes most of the skeletons of most vertebrates) is by no means the same in different animals, in different stages of an animal’s life, nor in different parts of an animal. Macroscopically (depending on the porosity of the structure), mammalian bones are of two main types: compact bone and cancellous bone. Compact bone (also called cortical bone) is dense with porosity generally under 10%. It can be found on the outer wall of all bones and is largely responsible for the supportive and protective function of the skeleton. Cancellous bone (also called trabecular or spongy bone), which usually can be found on the inner part of bones and at the ends of long bones, is highly porous with porosity normally between 75% and 95% (Martin et al., 1998). Figure 2-1 shows examples of compact bone and cancellous bone. 2.1.2 Lamellar Bone and Woven Bone On a finer scale (sub-millimeters), bone can be further classified into different types. Lamellar and woven bones are different in their rate of formation and their structural organization (Currey, 2002). Lamellar bone is laid down slower than woven bone (less than 1 m per day compared to more than 4 m per day for woven bone (Boyde, 1980)) and has a highly organized structure described as plywood-like (Giraud-Guille, 1988). Lamellar bone is the primary building blocks of most compact bone that 6

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7 Figure 2-1. Compact bone and cancellous bone. A) Fracture surface of a manatee rib showing compact bone with very few pores. B) Typical cancellous bone structure.

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8 can be seen in osteons. Woven bone is less organized than lamellar bone (in which collagen fibers and apatite crystals are laid down somewhat randomly). Woven bone is often found in a fetus and in the areas around bone fracture healing sites. It is a provisional material that is eventually resorbed and replaced by lamellar bone (Jee, 2001). 2.1.3 Primary Bone and Secondary Bone Compact bone can be further classified as primary or secondary according to the times these bones are formed. As indicated by the name, primary bone is laid down as newly formed bone either from an ossification center, or on an existing bone surface. As a result, primary lamellar bone often shows a concentric pattern around the marrow cavity. Primary bone in some animals (such as humans and other primates) is predominantly circumferential lamellae with a few scattered primary osteons. For many fast-growing animals (such as cows, manatees, and elephants) plexiform bone is the predominant phase when the animals are young. Plexiform bone is characterized by its brick wall appearance (due to woven bone being laid down first, then lamellar bone filling the gaps). Figure 2-2A shows plexiform bone from the femur of a 2-year-old cow. Secondary bone may replace primary bone on its surface or by forming new Haversian systems in it (Enlow, 1969). These new Haversian systems are also called secondary osteons (since they have concentric lamellae around the Haversian canals, as do primary osteons). However, one major difference between primary osteons and secondary osteons is that a boundary (known as the cement line) exists between the secondary osteons and the bone matrix (Martin and Burr, 1989). This collagen-deficient boundary is a product of bone remodeling. Figure 2-2B shows an osteon with its circular cement line. In adult humans and in cows, secondary osteons replace primary bone and

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9 Figure 2-2. Plexiform bone and an osteon with cement line. A) Plexiform bone from the femur of a 2-year-old cow is shown. The parallel channels were where blood vessels ran through. B) An osteon with its circular-like cement line (indicated by arrows).

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10 becomes the predominant phase in long bones. Our study used bovine femurs and manatee ribs from sub-adult and young adult animals for compact bone specimens. 2.2 Compositions of Bone and Their Roles in Bone Generally, the composition of compact bone in human and bovine long bones (by weight percentage) is 60-70% apatite minerals, 18-25% organics, and 8-15% water (Eastoe and Eastoe, 1954; Martin et al., 1998). The volumetric percentage (using the densities, 3.2 g/cm3 for minerals, 1.1 g/cm3 for organics, and 1.0 g/cm3 for water, suggested by Currey (1990)), bone is about 37-45% apatite minerals, 33-40% organics, and 15-23% water. Bone is basically a tri-phase composite, with each phase making a unique contribution to the mechanical properties of bone. 2.2.1 Bone Mineral Bone mineral is generally described as hydroxyapatite, chemical formula, (Ca10(PO4)6(OH)2). However, many cations (such as Mg2+, Sr2+, and Na+) and anions (such as CO32-, Fand H2PO4-) substitute into hydroxyapatite and make it non-stoichiometric (Driessens and Verbeeck, 1990). Because of these impurities, some other calcium phosphate phases, such as carbonate apatite, have also been suggested as the apatite minerals in bone (Posner and Beebe, 1975; Lowenstam and Weiner, 1989). Apatite crystals in human cortical bone are from several nano-meters to about 50 nm (Ziv and Weiner, 1994; Rho et al., 1998). The plate-like crystals of apatite fill the gaps between tropocollagen molecules, for which the c axes of the crystals are roughly parallel to the long axes of the collagen molecules (Kuhn-Spearing et al., 1996; Rho et al., 1998). Apatite minerals serve as fillers in an organic template, giving rigidity and stiffness to bone. It is difficult to quantify the mechanical properties of natural apatite minerals since

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11 they are on the nanometer length scale. Some mechanical properties of synthetic hydroxyapatite are shown in Table 1-1. 2.2.2 Organics in Bone Organics are the second major phase in bone. In terms of volumetric percentage, the organics in bone closely resemble apatite minerals. The organics in bone are mostly type I collagen (~ 90% of all organics (Glimcher, 1976)). Noncollagenous proteins, proteoglycans and phospholipids make up the remaining 10% of organics. Tropocollagens (triple-helix-structure molecules made of three polypeptide chains) line up periodically with gaps about 40 nm wide (Rho et al., 1998). These gaps are filled with apatite minerals to form the most fundamental building blocks in bone. These blocks then build up to form microfibrils. A bunch of microfibrils then aggregates to form fibrils that range from 0.1-3 m (Currey, 2002). Figure 2-3A shows several collagen fibrils on the canal surface of bovine femur. These collagen fibrils can then build up to form woven bone or lamellar bone. Mechanically, the organic matrix is one key to the toughness of bone (Zioupos et al., 1999). Stringent sterilization techniques using radiation can be employed to decrease the infection risk of implanted bone minerals. However, this sterilization has a detrimental effect on the collagen component of bone and adversely affects bone mechanical properties (Hamer et al., 1996). Structurally, organics keep the integrity of bone since they are much more pliable than the apatite minerals and can hold the inorganics together at greater strain without failing. Organics are also responsible for the viscoelastic and creep behaviors of bone (Yamashita et al., 2001; Bowman et al., 1999).

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12 Figure 2-3. Collagen fibrils and pores in bones. A) Collagen fibrils with diameters 1-3 m can be seen on the canal surface of bovine femur (some of them indicated by arrows). B) Many canals and lacunae can be seen throughout compact bone. These are the sites where free water exists. 2.2.3 Water Water is the third major constituent of bone. It exists in three states: (1) free water in blood that runs through canals, vascular vessels, and lacunae (where bone cells reside); (2) bonded to collagen; and (3) in the hydration shells of apatite minerals. Figure 2.3B shows a canal in a bovine femur with several lacunae close to it. Though compact bone in cows appears dense and highly compact macroscopically, it does have many canals for blood to run through providing nutrition to bone cells and providing many lacunae for

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13 bone cells to reside in. Robinson and Elliott (1957) analyzed the volumetric composition of whole bone in dogs and found that 60% of water was bonded to collagen. The hydration of collagen plays an important role in the mechanical properties of bone. Bella et al. (1995) used a collagen peptide model to show that absorbed-water molecules form a highly ordered network. This network gives extra stability to the triple helix of tropocollagen by forming additional water-mediated hydrogen bonds among all remaining backbone peptide groups. Without water, these additional hydrogen bonds would not exist, because of spatial restraints (Beck and Brodsky, 1998). 2.3 Fracture Mechanics When will a material break apart? Is it when the material suffers a stress higher than its strength? Or, is it when the applied strain is larger than the material’s strain limit? For years, scientists have found that testing materials in a standard tensile, compressive or flexural test allows them to consistently estimate the fracture stress (strength) of many materials. Thus, many people believe that strength is a material property, and that stress level is the key to failure of a material. However, certain materials (such as glass and many ceramics) tend to have much greater data scatter in strength when tested. Moreover, many large ships in the early 1900s and during World War II failed at a stress level much lower than they were designed to withstand (Saxena, 1998). All of these factors led people to question the validity of the universally assumed relationship between fracture stress and failure. The breakthrough came when Griffith (1920) applied an energy-balance concept of thermodynamics to the fracture of glass. His work and the work of Irwin led to the development of fracture mechanics (Hertzberg, 1996). Fracture mechanics is now widely used to analyze and quantify the relationship of stress level, existing cracks, and crack propagation mechanisms in materials.

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14 2.3.1 Griffith Strength Relation Griffith found that the critical stress, c, causing the crack to propagate is aEc2 (2.1) where E is the elastic modulus and is the fracture surface energy. Equation 2.1 is the Griffith strength relation. One breakthrough implication of this relation is that c (instead of being constant) is inversely proportional to the square root of a, where a is the crack size. Griffith tested Equation 2.1 using glass specimens with different crack lengths introduced by a glass cutter. After fracturing the specimens, he found a constant relation between c and a1/2. Griffith also used the theoretical cohesive strength between atoms of brittle elastic solids to determine the strength of a flawless material. He used 0.5 nm as the tip radius of the bond and estimated that this strength is on the order of 0.1E. For glass, this strength would correspond to 7000 MPa! However, the strengths of the as-received glass specimens he tested fell well short of this level (with the highest being 170 MPa). Griffith concluded that brittle solids must contain many submicroscopic flaws. 2.3.2 Critical Energy Release Rate and Critical Stress Intensity Factor Equation 2.1 was based on the assumption of brittle fracture without plastic deformation. Orowan (1950) recognized that, for metals and polymers, the fracture energy consumed in plastic deformation is much greater than the surface energy. Orowan suggested that, at any instance before reaching the critical stress, Equation 2.1 can be modified as shown in Equation 2.2, aEP)(2 (2.2)

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15 where P is plastic deformation energy. In the 1950s, Irwin chose to incorporate the energy sources into a single term, G, as shown in Equation 2.3, )(2PG (2.3) G, the elastic energy release rate, measures the change of elastic energy per change of crack length. It reaches a critical value, Gc, known as critical energy release rate, when fracture occurs. Gc is a material property and can be evaluated both analytically or experimentally using standard test specimens. Another widely used parameter in fracture mechanics is the critical stress intensity factor, Kc, which is also called fracture toughness. There are three basic modes of crack propagation. Mode I is called opening mode (or tensile mode), mode II is called sliding mode (or in-plane shearing mode) and mode III is called tearing mode (or out-of-plane shearing mode). Mode I fracture is encountered most encountered and will be discussed mostly in this dissertation. The stress intensity factor represents the amplitude of the crack tip stress singularity and is related to the applied stress, the crack length and crack geometry. It can be expressed in a general form, as shown in Equation 2.4, aWafK)/( (2.4) where f(a/W) is geometric factor and is a function of crack length, a, and object width, W. Different materials may sustain different critical stresses, c, and a critical value of K for each of them exists. This critical stress intensity factor is also called fracture toughness and can be expressed as shown in Equation 2.5, aWafKcc)/( . (2.5)

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16 Kc and Gc are the two most used parameters that quantify how tough a material is before fracture. In fact, they are related through (Irwin, 1957) Equation 2.6, E KGcc2 (2.6) where E’ = E for plane stress and E’ = E/(1-2) for plane strain. A variety of testing configurations and standards has been developed to determine Kc and Gc for all kinds of materials (Anderson, 1995; ASTM E399; ASTM C1421; ASTM E1820). 2.3.3 Elastic-Plastic Fracture Mechanics and J Integral One of the most important assumptions of the analysis in the previous section is that the material has no, or small-scale, yielding before fracture. The analysis based on this assumption is called linear-elastic fracture mechanics (LEFM). However, if the material has certain amount of plastic deformation and the stress distribution ahead of crack tip can no longer be accurately described by the stress intensity factor analysis (K field), elastic-plastic fracture mechanics (EPFM) should be employed. Figure 2-4 shows three commonly observed load-extension curves. For materials fracture in a brittle manner, their load-extension curves may be similar to the ones in Figures 2-4A and 2-4B. Figure 2-4. Three common load-extension curves of materials under tension or flexure. A) Brittle fracture with no yielding. B) Brittle fracture with small-scale yielding. C) Moderate-scale yielding before fracture. While LEFM is applicable to cases A and B, EPFM is more appropriate to case C.

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17 For materials having moderate yieldings before fracture, their load-extension curves may be similar to the one in Figure 2-4C. While LEFM is applicable to cases A and B, EPFM is more appropriate for case C. Two of the most important parameters in EPFM are J integral and crack-tip-opening displacement (CTOD). The J integral, derived by J. R. Rice (1968), is a path-independent integral that characterizes the fracture energy in both linear-elastic and nonlinear-elastic materials. Rice showed that for a stressed object with a notch (Figure 2.5), a line integral along an arbitrary circular path , which he defined as the J integral, can be derived by Equation 2.7, )(dsxuTWdyJ (2.7) where W is strain energy density, T is the traction vector defined according to the outward normal along path , u is the displacement vector, and ds is an element of arc length along path . In the same paper, Rice also showed some unique properties of the J integral. First, for any closed path, the J integral is 0. Second, the J integral around the Figure 2-5. Two-dimensional plate with an introduced crack. The J integral, a line integral along the circular path, , is found to be constant as long as the path covers the crack tip.

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18 crack notch is path independent. The significance of this property is that we can choose an easier path, as long as the path covers the crack tip, to evaluate the J integral. Third, for objects with small-scale yielding under mode I crack propagation, the J integral is found to be E KJI2 (2.8) where E’ is defined as in Equation 2.6. Comparing Equation 2.8 with Equation 2.6, one can find that Jc is equal to Gc if the material behaves linear-elastically before fracture. Thus, the J integral can also be interpreted as the energy release rate for nonlinear elastic materials. The CTOD concept is based on the assumption that, for ductile material behavior, the fracture process is no longer controlled by stress intensity, but by plastic deformation in front of the crack tip. A measure of this is the widening at the crack tip, called the crack-tip-opening displacement (Grellmann, 2001). CTOD, t, can be directly measured, or, calculate from the J integral. For materials that behave linear-elastically, a simple relationship between the J integral and CTOD can be derived using the Dugdale model (Dugdale, 1960) tJ (2.9) where is the applied stress. While LEFM and EPFM are two of the most powerful analytical tools in fracture mechanics, there are others dealing with different types of fracture problems. Among them, time-dependent fracture mechanics deals with fatigue and creep in materials and dynamic fracture mechanics focuses on rapid crack propagation and crack branching. One should always keep in mind that although most materials have characteristic

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19 behavior, i.e., at room temperature metals are usually ductile, and ceramics are mostly brittle, due to the nature of their atomic bonding as well as the microstructure, almost all materials may have brittle fracture as well as large-scale yielding for they are subjected to a wide range of stressing rates or temperatures. For example, while LEFM may be used to determine fracture toughness of PMMA accurately at room temperature, one may want to choose EPFM to study fracture of PMMA at a high temperature since fracture strain of PMMA increases from 0.06 at 20C to 0.24 at 50C (Carswell and Nason, 1944). 2.4 Fracture Mechanics of Compact Bone Before 1970, most researchers reported the mechanical properties of bone using elastic modulus or strength (Currey, 1970). This may not be surprising since the inception of fracture mechanics was around the early 1950s. However, since bone contains many microchannels and sometimes microcracks due to repeated loading in daily life (Burr and Stafford, 1990), the strength of bone specimens may be affected by these structural irregularities that can be considered mechanical flaws. To better understand the mechanical behavior of bone, it is important to understand the fracture mechanisms and different effects on the fracture toughness of bone. Also, it is important to find out a parameter that better describes the toughness of bone. For scientists working on building a better prosthetic bone implant, one of the key issues is having a mechanically potent material. However, to match the superb properties of bone, simple modulus and strength tests of the implants may not be enough since they don’t tell you much about the actual fracture process of the materials. To my best knowledge, the first published paper using fracture mechanics concepts to study the fracture toughness, Kc, of compact bone is by Melvin and Evans (1973). They used single-edged notched (SEN) specimens to estimate the fracture toughness of

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20 bovine femur. They found for transverse fracture, in which the crack propagation is perpendicular to the long axis of bone, the average Kc ranged from 5.58-7.69 MPam1/2. For longitudinal fracture, in which the crack propagation is parallel to the long axis of bone, the average Kc ranged from 3.21-5.05 MPam1/2. They also found that Kc for wet specimens was 60% greater than that for the dry specimens. Since then, different techniques have been applied to measure the fracture toughness of different mammalian bones. Table 2-1 lists the testing methods, bone source, fracture direction relative to the long axis of bone, and Kc values of selected studies. Some of the notable testing methods include single-edge notched beam (Melvin and Evans, 1973; Zioupos and Currey, 1998; Lucksanasombool et al., 2001; Wang et al., 2002), compact tension (Wright and Hayes, 1977; Bonfield et al. 1978; Behiri and Bonfield, 1984; Norman et al., 1992; Vashishth et al., 1997; Feng et al., 2000), compact sandwich (Wang et al., 1998), chevron-notched short rod (De Santis et al., 2000), fatigue-precracked beam (Nalla et al., 2003) and chevron-notched beam (Yan et al., 2005a). Some of the popular bone sources include bovine femur and tibia, and human femur because they are relatively large compared to other sites of bone sources in animals. The Kc values shown in Table 2-1 vary between 1.6 and 12.6 MPam1/2. Though the scatter in data seem large, it may not be surprising considering the difference of bone sources, crack propagation direction and the natural variability (age, sex, etc.) in animals. Other possible reasons causing the scattering could be testing methods, storage media (Lucksanasombool et al., 2001) and testing states. Nevertheless, at least two points can be made about the estimated Kc values in these studies. First, Kc values of compact bone vary mostly between 3 and 6 MPam1/2. Second, Kc of transverse fracture is about 50-100% greater than that of longitudinal fracture.

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21 Table 2-1. Selected studies measuring fracture toughness of compact bone. Investigators Testing Method Bone Source Fracture Direction Fracture Toughness (MPam1/2) Melvin and Evans (1973) Single-edge notched beam Bovine femur Longitudinal Transverse 3.210.43 5.580.52 Behiri and Bonfield (1984) Compact tension Bovine tibia Longitudinal 2.8-6.3 Norman et al. (1992) Compact tension Bovine tibia Transverse 4.93-12.64 Vashishth et al. (1997) Compact tension Human tibia Longitudinal 1.6-2.5 Wang et al. (1998) Compact sandwich Baboon femur Longitudinal 2.250.18 Zioupos and Currey (1998) Single-edge notched beam Human femur Longitudinal 6.41 Wang et al. (2002) Single-edge notched beam Human femur Transverse 5.09.98 De Santis et al. (2000) Chevron-notched short rod Bovine femur Longitudinal 4.80.5 Nalla et al. (2003) Fatigue precracked flexure Human humerus Longitudinal Transverse 3.530.13 5.330.41 Yan et al. (2005a) Chevron-notched beam Manatee rib Bovine femur Transverse Transverse 4.50.5 5.80.5 1Average value at the age of 35 Many effects on the fracture toughness (Kc) of bone have been studied. Besides the above mentioned paper by Melvin and Evans and the effect of crack propagation direction, other notable results are described below. Wright and Hayes (1977) studied the effect of apparent density and found a five percent increase in bone density (from 1.92 to 2.02 g/cm3) that resulted in a 30 percent increase in the fracture toughness. Bonfield et al. (1978) studied the effect of crack velocity and found the faster the crack propagated (from 2.1 to 27-5 m/sec) the greater the facture toughness (from 2.4 to 5.2 MPam1/2). Yeni et al. (1997) studied the influence of bone morphology and found the critical strain

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22 energy release rate (Gc) of human femur decreased as porosity of the bone increased. They also found GIc of human femur increased with osteon density, defined as number of osteons per mm2, increased. Zioupos and Currey (1998) studied the effect of age and found the fracture toughness (Kc) of human femur decreased from 6.4 MPam1/2 (at age 35 years) to 4.9 MPam1/2 (at age 92 years). Recently, several researchers have reported rising R-curve (crack extension resistance) behavior in cortical bone (Vashishth et al., 1997; Malik et al., 2003; Vashishth, 2004; Nalla et al., 2004). R-curve behavior characterizes the resistance to fracture of materials during incremental stable, slow crack extension and results from growth of the process zone as the crack extends from a sharp notch [ASTM E561]. For materials with rising R-curve behavior, the greater stability results in additional crack extension and this may lead to greater measured fracture toughness values than the initiation fracture toughness values (Salem et al., 1992; Zehnder et al., 1997). Vashishth et al. (1997) used compact tension specimens to measure R-curves of young bovine and adult human tibiae along longitudinal fracture. Both materials showed that R-curve behavior of bovine bones had greater overall fracture toughness values and greater growth toughness, defined as K/mm, as the crack grew. They demonstrated that the toughening mechanism was microcracking. Malik et al. (2003) also used compact tension specimens to estimate the R-curve behavior of equine cortical bone along transverse fracture. They found that specimens from lateral cortex and dorsal cortex both exhibited rising R-curve behavior with the fracture mechanisms acting in these specimens to be regionally dependent. They pointed out besides microcracking; mechanisms like osteon pullout, fiber bridging and crack deflection may also have effects on the increment of

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23 fracture toughness. Nalla et al. (2004) evaluated the effect of aging on the fracture toughness of human humerus using R-curve behavior. They found both the initiation toughness and the growth toughness decreased as the age of the donors increased (34-99 years). They showed the growth toughness in bone was principally because of uncracked ligament functioned as crack bridging. The R-curve behavior in bone may be one of the reasons for the large variation in the reported fracture toughness values since different techniques could employ different initial crack lengths and different stressing rates could affect the amount of stable crack growth. 2.5 Effect of Temperature on the Mechanical Behavior of Compact Bone As pointed out in chapter one, to accurately estimate the mechanical properties of bone, the testing should be carried out at a temperature that matches the animal’s body temperature. To date, most of the fracture toughness studies on bone were carried out at room temperature with no published study on the relationship between temperature and bone fracture toughness. Several studies on the relationship between temperature and other mechanical properties of bone have been reported. Bonfield and Li (1968) measured the deformation characteristics of bovine femur in the temperature range from -58 to 90C. They found at temperatures below 25C, elastic, anelastic (the time-dependent recoverable part of the nonlinear deformation) and plastic contributions to the strain could be distinguished. At temperatures higher than 50C, the deformation was a mixture of nonequilibrium recovery and an irreversible change in the structure of bone. They also found that the elastic modulus increased as the temperature decreased. Carter and Hayes (1976) studied the effect of temperature on the fatigue life of adult bovine bone. They tested the specimens in the temperature range from 21 to 45C and found the fatigue life decreased as the temperature increased (by approximately a factor of three

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24 over the temperature range examined). They also found that the primary compact bone (which is mostly plexiform bone in cows) has a longer fatigue life than secondary Haversian bone. The fatigue life was also shown to be positively correlated to bone density. Rimnac et al. (1993) studied the effect of temperature on the creep of bovine femur. They tested the specimens at temperatures 25, 37, and 43C and found a positive, significant correlation between mean steady-state creep rate and temperature. They hypothesized that the reason the bone specimens failed faster at higher temperatures is that the damage mechanisms were associated with dislocations in the hydroxyapatite mineral lattice structure.

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CHAPTER 3 SAMPLE PREPARATION AND EXPERIMENTAL METHODS 3.1 Compact Bone from Bovine Femur and Manatee Rib Bone specimens were cut from bovine femurs and manatee ribs. Fresh bovine femur bone tissues were obtained from three young cows (approximately 24 months old) slaughtered at the University of Florida Meat Processing Center. Manatee rib bone tissues from an adult manatee carcass were obtained from the Florida Fish and Wildlife Conservation Commission's (FWC) Marine Mammal Pathobiology Laboratory (MMPL). Collection and use of manatee bone tissues for research were conducted under U.S. Fish and Wildlife Service permit #MA-773494-7, issued to the Florida FWC. All bone tissues and specimens were kept in a freezer or stored in physiological buffered saline solution under 4C at all times until tested. Bovine femur is one of the most studied compact bones. It is popular for bone mechanical property studies since the availability is relatively great compared to other types of bones in cows and most common animals. This is important since most fracture mechanics tests have certain requirements for specimen dimensions. Larger sizes also mean more specimens can be cut from one femur and it helps relieve the problem of natural variability between animals. Technically, larger bones make the handling of sample preparation and the measurement of dimensions easier. Bovine femur compact bone is mostly plexiform bone in young cows (Martin and Burr, 1989). Gradually, secondary osteonal bone replaces plexiform bone as cows grow older. Robertson et al. (1978) showed for young adult cows (18 to 24 months) that plexiform bone is 25

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26 predominantly in lateral and medial sections, while osteonal bone is mostly in anterior and posterior sections through mid-diaphysis of the femur. In adult manatees, ribs are relatively large compared to other animals’. Unlike human ribs, which consist of highly vascular cancellous tissue and are enclosed in a thin layer of compact bone (Gray, 1973), manatee ribs are mostly primary plexiform bone with fewer osteons in adults. They are almost all compact bone (Figure 2-1A) and this makes the sample preparation easier than using other animal’s bone. Manatee rib also possesses unusual characteristics compared to those of other marine mammals. Compared to terrestrial mammals, the general trend in marine mammals has been a reduction of bone mass and density (de Buffrenil et al., 1990). In contrast, the manatee skeleton exhibits pachyostosis, characterized by thickening of bone tissue, replacement of cancellous bone with compact bone, and the absence of a free medullary cavity (Fawcett, 1942; de Buffrenil et al., 1990; Domning and de Buffrenil, 1991). Specimens from manatee ribs and bovine femurs were then cut into the dimensions according to standards (exact dimensions of the specimens are described in the testing methods below). During cutting and polishing, a constant spray of water was supplied to keep the bone from heating and to keep it wet. Pilot studies using Polymethyl methacrylate (PMMA) and poly-bisphenol-A carbonate (PBC) were also conducted to test the applicability of the testing methods to bone since they were primarily developed for ceramics and metals. The data from PMMA and PBC were also used to compare with the data from bone. PMMA is relatively brittle with its elongation at fracture similar to that of compact bone at room temperature (Table 1-1). PBC is a much tougher material with its elongation at fracture greater than 100% (Callister, 1997). One reason for

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27 choosing these two materials is that the same notching method used on bone can be applied to them as well. 3.2 Test the Effect of Temperature on the Fracture Toughness of Compact Bone 3.2.1 Single-Edge V-Notched Beam Method To date, there is no standard on estimating the fracture toughness of bone. Many of the previous fracture toughness studies of compact bone employed techniques that have been successfully applied to metals or ceramics. In this study, the technique used to estimate the fracture toughness of bone was the single-edge V-notched beam (SEVNB) method (Kbler, 2002). This method is essentially a derivative of the single-edge notched-beam (SENB) method. The major difference between SENB and SEVNB is the way the final sharp notch is induced. For metals, ASTM E399 (ASTM International, 2001a) suggests using a fatigue pre-cracking procedure to induce sharp cracks from the initial notches in SENB specimens. One problem for bone in using this procedure is that the crack will tend to turn towards, and propagate, along the longitudinal direction of the bone. For transverse fracture tests, this is a serious problem since the crack propagates perpendicular to the direction in which the loading is directed. In the SEVNB method, the final sharp crack is induced by a razor blade and has a ‘V’ shape. Figure 3-1 shows a V-notch in a bovine bone sample. The radius of the crack tip is close to 1 m. One may argue that, in fracture mechanics tests, the radius of the crack tips should be close to the atomic scale. However, Wang (2000) used different radii of crack tip in glass and alumina and found the measured fracture toughness values of the specimens were high when the radii were large. The values decreased as the radii decreased and reached what others reported when the radius was smaller than 10 m. The SEVNB method is now under development as an ISO standard for the determination of fracture toughness for

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28 Figure 3-1. Single-edge V-notch in a bovine bone sample. A) A U-shape notch was first induced using a diamond wheel and then a V-notch was then extended from the U-notch using a razor blade. B) The radius of the crack tip is approximately 1 m.

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29 fine ceramics (ISO/WD 23146). One advantage of using the SEVNB method is that for low elastic modulus materials (such as bone and polymers), a razor blade can effectively cut through the material and make the notching process efficient. Also, it is easy to control the notch depth using a razor blade, if the elastic modulus of the specimen is low. 3.2.2 Notching Procedure and Determination of Fracture Toughness For the transverse fracture test, rectangular bars with dimensions 3 x 4 x 45 mm (thickness x height x length) were cut from the mid-sections of bovine femur and manatee rib. Figure 3-2 shows the location from which the specimens were cut. On each specimen, a U-shape notch was first introduced using a diamond wheel (the upper portion of the notch in Figure 3-1A). The depth of the notch was 0.8 mm and the width was ~ 0.3 mm. A V-notch was then extended from the U-notch using a razor blade. The depth of the V-notch was 0.2 mm with the radius of the notch tip close to 1 m (Figure 3-1B). The total initial crack length (ao) was 1 mm, which corresponds to an ao/W ratio (W is specimen height) of 0.25. Kbler (2002) suggested using an initial notch length 0.8-1.2 mm, which corresponds to an ao/W ratio of 0.2-0.3. For a single-edge V-notch beam fractured in four-point flexure, the fracture toughness can be estimated using (Kbler, 2002): *)1(235.1YWaWaWSSWBPKooioc (3.1) where P is fracture load, B is specimen thickness, So is the outer span of the four-point flexure, Si is the inner span, and ao is the initial crack length. Y* is the stress intensity factor coefficient given by ASTM C1421 (ASTM International, 2002). If we let = ao/W, then. 22)1)(1()35.168.049.3(326.19887.1*Y

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30 Figure 3-2. SEVNB specimens from bovine femur and manatee rib. A) A diagram shows the location of the transverse fracture specimens. B) Dimensions of the specimen. The position of the single-edge V-notch is also shown (solid lines in the middle of the specimen). 3.2.3 Temperature Control and Testing Apparatus In this study, all specimens were fractured inside a water tank. Figure 3-3 shows the universal testing machine (Instron Model 1125, Instron Corporation, Canton, MA), temperature controller, and water tanks used in this study. A close-up view of a specimen inside the testing tank is also shown. The outer span (So) was 40 mm and the inner span

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31 Figure 3-3. Instron testing machine and 4-point flexure test. A) The Instron testing machine and the device for controlling testing temperature. Specimens were placed in the testing water tank. B) A close-up view of a 4-point-flexure-test specimen inside the testing tank.

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32 (Si) was 13.3 mm. The traveling speed of the load cell was 1 mm/min, which corresponds to a strain rate of approximately 0.01/min. This quasi-static loading rate is commonly used in fracture mechanics tests. The specimens were tested at five different temperatures, 0, 10, 23, 37 and 50C. A temperature controller and a circulation system were used to maintain the water at testing temperatures during the test. A digital thermometer was used to check the temperature of the testing water tank frequently to make sure the temperature was within 1C of the testing temperature. Ice was put into the tank, or, the heater of the temperature controller was turned on if the temperature was outside the range. Six specimens were tested at each temperature and a total of 30 specimens were fractured for both bovine femur and manatee rib. 3.2.4 Post Test Measurement and Analysis After the specimens were fractured, an optical microscope was used to measure the actual initial crack length, ao, of each specimen. Fracture load (P), specimen dimensions, span sizes and ao of each specimen were then put into Equation 3.1 to determine the fracture toughness. Two or three out of the six specimens at each temperature were selected for taking SEM images. The SEM images were used to study bone microstructure and determine the porosity of the specimens using Image Pro software (Media Cybernetics, Silver Spring, MD) (Yan et al., 2005a). For each specimen, only the fracture surface but not the notched surface was used to determine porosity. Across the fracture surface, there are many small holes (canaliculi or lacunae) that were occupied by bone cells or fluids. These holes were not counted since there are usually hundreds of them per mm2 in compact bone. To count each one of them would require a substantial amount of work. Besides, they represented only a very low percentage of bone porosity (2.3% of the total in human long bones (Evans, 1973)).

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33 Additional samples from both the same bovine femur and manatee rib were used to determine their apparent densities and the weight percentage of the water, organics and minerals. Apparent densities were determined using the mass of the samples divided by the volume of the samples. Three samples from each animal were used. These samples were then placed in the furnace at 110C for 2 h. The decrease in specimen mass was attributed to the loss of water. The samples were then placed in the furnace at 600C for 24 h. The decrease in specimen mass was attributed to the loss of organics. The rest of the bone mass was identified as the minerals. This procedure is similar to the approach that Boskey (2001) outlined in the gravimetric analysis of bone. Thermogravimetric analysis (TGA) and differential thermal analysis (DTA) (Netzsch STA 449C, Netzsch Instruments, Burlington, MA) of bovine femur and manatee rib were also carried out to identify different weight loss stages in the materials as the temperature increased. The analyses were carried out from 23C (room temperature) to 700C at a heating rate 2 C/min. It was believed the organics in bone were mostly burned when reaching 700C. 3.3 Measuring the J Integral The J integral measurement of bone followed the testing procedure suggested in ASTM E1820-99a (ASTM International, 2001b). Rice (1968) showed that the J integral around a notch tip in an object can be defined as )(dsxuTWdyJ (2.7) Rice et al. (1973) further derived the J integral for a rectangular beam (Figure 3-4A) with a deep edge notch subjected to pure bending: voPdvBbJ02 (3.2)

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34 where B is the thickness of the beam, bo is the length of the uncracked ligament (i.e., bo = Wao, where W is the specimen height and ao is the crack length ), P is load and v is the load-line displacement. To obtain an accurate estimation of the J integral from Equation 3.2 for three-point flexure specimens, ao/W should be 0.5 (Saxena, 1998). For materials fracture in a brittle manner, the load-extension curves resemble the one in Figures 2-4A. As stated in section 2.3.3, the J integral for these materials is E KGJccc2 . (3.3) For materials with moderate yielding before fracture (such as the one in Figure 2-4C), the J integral can be broken down into two parts: J integral of elastic (Jel), and plastic (Jpl), deformation. The integral on the right-hand side of Equation 3.2 is the total area under the load-extension curve. While Jel can be estimated using Equation 3.3, Jpl can be determined using Equation 3.4, ploplABbJ2 (3.4) where Apl is the area of the plastic deformation part. Two different estimations of Apl can be made (Figures 3-4B & 3-4C). However, since the load used to calculate Kc in Equation 3.3 is the maximum load, ASTM E1820 suggests using the configuration shown in 3-4(c) to determine Apl. The total J integral of materials with moderate yielding can be estimated using Equation 3.5, plocpleltotalABbEKJJJ22 (3.5) In this study, rectangular bars with dimensions 4 x 4 x 45 mm (thickness x height x length) or 4 x 4 x 25 mm for the three-point flexure tests, were cut from the mid-sections

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35 Figure 3-4. Deep-notched specimen in flexure. A) A rectangular specimen subjected to a 3-point flexure. B) and C) show two different ways of estimating Apl. ASTM E1820 suggests using the one in C since the load used to calculate the fracture toughness of the specimens the maximum load. of bovine femur and manatee rib (Figure 3-2). ASTM E1820 suggests using a support span, S, equal to four times the width, W, for metals. However, greater S/W ratios, 5 and 10, were chosen because of the composite nature of bone (Zweben, 1985). Besides transverse fracture, longitudinal fracture specimens were also prepared. However, due to the curvature of the bone shaft, only an S/W ratio of 5 could be cut for these specimens. Six groups of specimens were prepared to compare the different effects. Table 3-1 lists the bone type, specimen dimensions, S/W ratio, fracture direction, testing temperature and number of specimens of the six groups. The initial notch of each specimen was prepared

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36 Table 3-1. Information of the six groups in the J integral measurement. The bone type, specimen dimensions, S/W ratio, fracture direction, testing temperature and number of specimens of the six groups are listed in the table. Bone Type Dimensions [mm] (B x W x L) S/W ratio Fracture Direction Temperature Number of Specimens Bovine Femur 4 x 4 x 45 10 Transverse 37C 10 Bovine Femur 4 x 4 x 45 10 Transverse 23C 6 Bovine Femur 4 x 4 x 25 5 Transverse 37C 8 Bovine Femur 4 x 4 x 25 5 Longitudinal 37C 10 Manatee Rib 4 x 4 x 45 10 Transverse 37C 6 Manatee Rib 4 x 4 x 45 10 Transverse 23C 6 using the same procedure stated in section 3.2, except for the depth of the U-shape notch, which was 1.8 mm. The total depth of the single-edge V-notch was 2 mm with ao/W equal to 0.5, which is a requirement for Equation 3.2 to be valid. After the V-notch was cut, side grooves were also induced on both sides of the specimen as suggested in ASTM E1820 to ensure that the crack propagate straight across the specimen. Each groove was made of a 0.4 mm U-notch and a 0.1 mm V-notch. The total depth of the two side grooves was 1 mm, which corresponded to 0.25B. Because of the side grooves, the thickness of the specimen, B, used in Equation 3.5 was changed to BN, the thickness of the uncracked ligament. Besides the J integral, the fracture toughness values of the specimens were also calculated using the ASTM E1820 Equation, )()(2/32/1fWBBPSKNc (3.6) where = ao /W and 2/122/1)1()21(2)]7.293.315.2()1(99.1[3)(f . Elastic moduli of the specimens were determined using the slope of the stress-strain curves of

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37 unnotched specimens. All testing was conducted in water using the device shown in Figure 3-3 with a loading rate 1.0 mm/min. After the specimens were fractured, an optical microscope was used to measure ao and BN of each specimen. The testing load-extension curves were exported. Apl of each specimen was determined by the summation of the step sizes of displacement (v) multiplied by the magnitude of the load (Pv) of the load-displacement curve; Ael was then subtracted. Ael was estimated using the fracture load timed its corresponding displacement and then divided by two (Figure 3-4C). Fracture load (P), support span (S), specimen dimensions and ao of each specimen were then put into Equation 3.6 to determine its fracture toughness. Kc, E’, BN, bo, and Apl were then applied in Equation 3.5 to determine the J integral of each specimen. At least half of the specimens from each group were selected to take SEM images. The SEM images were used to study bone microstructure and determine the porosity of the specimens. 3.4 Fracture Toughness of Water-free and Organics-free Compact Bone SEVN specimens of transverse fracture bovine femur and manatee rib (such as the ones shown in section 3.2) were also prepared to estimate the fracture toughness of water-free and organics-free bone. Two methods were used to remove the water in bone. First, specimens were placed in a conventional furnace at 110C for 2 h. It has been shown (Boskey, 2001) that at 110C water can be effectively removed from bone. However, at this temperature, the structure of the collagen in bone may have been altered. Therefore, a second method, in which the specimens were placed in a vacuum oven (Lindberg/Blue M Vacuum Ovens VO914SA, Lindberg/Blue M, Asheville, NC) at 60C for 24 h, was used. Because of the high vacuum, free water in the bone was expected to gradually evaporate from bone and the collagen was expected to be less altered since the

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38 temperature was much lower. To remove the organics in bone, specimens were placed in a furnace at relatively high temperatures. Inevitably, the water was also removed. Three different temperatures were used, 500C for 24 h, 600C for 24 h and 800C for 3 h. Though it was shown at 600C that organics can be mostly removed from bone, a pilot study showed (Figure 3-5) that some organic residue might be still in bone. Therefore, a higher temperature (800C) was used. At 800C, it was found the specimens shrunk greatly and bent when they were placed in a furnace for over 4 h. After heating at 800C for 3 h, the samples turned white and the organics in bone were believed to be burned Figure 3-5. Specimen color change after different heating temperatures. A) At 400C for 24 h. B) At 500C for 24 h. C) At 600C for 24 h. D) At 800C for 3 h. At 600C, the specimen turned light gray with little amount of organics residue left.

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39 away. For the five groups (two for water-free and three for organics-free) tested in this study, six specimens were used in each group and a total of 30 specimens were prepared from both bovine femur and manatee rib. The weight of each specimen was measured before and after heating. All specimens were fractured using a four-point flexure test at room temperature (23C) in air under a loading rate 1 mm/min. After the specimens fractured, fracture toughness values were determined using Equation 3.1. 3.5 Experimental Design and Statistical Analysis To test the effect of temperature on the fracture toughness of compact bone, six specimens were used in each temperature. A total of 30 bovine specimens were prepared from two bovine femurs. To avoid natural variability from femur to femur and site to site, the six specimens in each temperature were chosen randomly. All 30 manatee specimens were prepared from the same manatee. In the J integral study, the minimum number of specimens in each group (Table 3-1) was chosen as six. The bovine specimens were prepared from two femurs. In some groups, the femurs were large enough that an additional two to four specimens were prepared. A total of 34 bovine specimens were tested. All 12 manatee specimens were prepared from the same manatee. To measure the fracture toughness of water-free and organics-free bone, six specimens were used in each process. A total of 30 bovine specimens were prepared from two bovine femurs. To avoid natural variability from femur to femur and site to site, the six specimens in each process were chosen randomly. All 30 manatee specimens were prepared from the same manatee.

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40 The results of each study were compared using a one-way ANOVA with = 0.05 (95% confidence level) to examine whether the mean values of each group are significantly different from one another.

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CHAPTER 4 RESULTS The results of the effect of temperature on the fracture toughness of bovine femur and manatee rib first are presented in section 4-1. In order to establish testing conditions, pilot studies were performed. The results of the pilot studies showed the SEVNB method gave consistent measurements on the fracture toughness of PMMA. In section 4-2, the results of the J integral measurements are presented. Other measurements, such as elastic modulus and fracture toughness, are also presented. The results of measuring fracture toughness of water-free and organics-free bone specimens are presented in section 4-3. 4.1 Effect of Temperature on Fracture Toughness of Compact Bone 4.1.1 Results of Pilot Studies Using PMMA Pilot studies using a SEVNB method were carried out to determine four effects on the fracture toughness of PMMA: (1) were temperature (10, 23, 37 & 50C), (2) crack length-to-height (ao/W) ratio (0.05, 0.25, 0.375 & 0.5), (3) loading rate (0.1, 1.0 & 10.0 mm/min), and (4) radius of notch tip (approximately 1, 80 & 160 m). A total of 14 groups were studied with six specimens per group. Table 4-1 shows the average fracture toughness values and the standard deviations of all groups. The results from these studies were used to determine which testing parameters to use for the compact bone study and to compare the results with the results for bone specimens. Figure 4-1A shows a column graph of the average fracture toughness values of PMMA at 4 tested temperatures. During the testing, a digital thermometer was used to 41

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42 Table 4-1. Fracture toughness of PMMA using the SEVNB method. Standard deviations of each group are also listed. The data in bold indicate the same group and was the group having the same testing parameters as for compact bone specimens. Temperature: (tested in water; ao/W=0.25; 1.0 mm/min; razor tip) 10C 23C 37C 50C Actual temperature 10.5C 23.3C 37.1C 50.5C Kc [MPam1/2] 1.620.09 1.470.06 1.390.05 1.350.07 ao/W Ratio: (tested in air; @ 23C; 1.0 mm/min; razor tip) 0.05 0.25 0.375 0.5 Actual ao/W 0.06 0.24 0.36 0.49 Kc [MPam1/2] 1.19.06 1.32.05 1.380.04 1.400.11 Loading Rate: (tested in air; ao/W=0.25; @ 23C; razor tip) Rate (mm/min) 0.1 1.0 10.0 Kc [MPam1/2] 1.36.05 1.32.05 1.43.07 Radius of Notch Tip & Blade Used: (tested in air; ao/W=0.25; 1.0 mm/min; @ 23C) 305 m wheel + razor blade 152 m wheel 305 m wheel Measured radius of notch tip razor (~1 m) ~80 m ~160 m Kc [MPam1/2] 1.32.05 1.680.09 1.760.10 monitor the temperature of the testing tank to make sure the temperature was within 1 C of the target temperature. Actual temperatures during the test are listed in Table 4-1. All the specimens were fractured in the water tank (Figure 3-3) under a loading rate 1.0 mm/min. The initial crack length was 1.0 mm with ao/W = 0.25, which is in the middle of the crack length range suggested by Kbler (2002). Average fracture toughness values of PMMA decreased from 1.62 MPam1/2 at 10C, to 1.47 MPam1/2 at 23C, to 1.39 MPam1/2 at 37C, and to 1.35 MPam1/2 at 50C. Using a one-way ANOVA with = 0.05, the differences between mean Kc values of all groups are significantly different from one another except for groups 37 and 50C. Figure 4-1B shows a column graph of the average fracture toughness values of PMMA at 4 different ao/W ratios. In the group with ao/W = 0.05 (ao = 0.2 mm and W =

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43 Figure 4-1. Results of the pilot studies using the SEVNB method on PMMA. A) Effect of temperature, B) effect of initial crack length, ao, C) effect of loading rate, and D) effect of notch tip radius. Testing parameters of these studies are listed in Table 4-1. 4 mm), only a razor blade was used. In the other 3 groups with ao/W = 0.25, 0.375, and 0.5, a U-notch with depths 0.8, 1.3, and 1.8 mm, respectively, was cut first followed by a 0.2 mm V-notch to produce crack lengths 1.0, 1.5, and 2.0 mm. All the specimens were

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44 tested in air at 23C under a loading rate 1.0 mm/min. Average fracture toughness values of PMMA increased from 1.19 MPam1/2 at ao/W = 0.05, to 1.32 MPam1/2 at ao/W = 0.25, to 1.38 MPam1/2 at ao/W = 0.375, and to 1.40 MPam1/2 at ao/W = 0.5. Using a one-way ANOVA with = 0.05, Kc value of the group with ao/W = 0.05 is significantly less than that of the other 3 groups. Kc values of the groups with ao/W = 0.25, 0.375, and 0.5 are not significantly different from one another. Figure 4-1C shows a column graph of the average fracture toughness values of PMMA under 3 different loading rates, i.e., 0.1, 1.0, and 10.0 mm/min. All specimens were prepared with ao/W = 0.25 and tested in air at 23C. Average fracture toughness values of PMMA were 1.36 MPam1/2 at 0.1 mm/min, 1.32 MPam1/2 at 1.0 mm/min, and 1.43 MPam1/2 at 10.0 mm/min. Using a one-way ANOVA with = 0.05, Kc value of the group tested at 0.1 mm/min is not significantly different from the other two groups, however, Kc value of the group tested at 1.0 mm/min is significantly less than that of the group tested at 10.0 mm/min. Figure 4-1D shows a column graph of the average fracture toughness values of PMMA with 3 different notch tip radii. The first group had sharp V-notches extended from the U-notches created by a razor blade (Figure 4-2A). The radii of the tips were approximately 1 m (Figures 4-2B). The other two groups only had U-notches made by two diamond wheels with different thickness. The nominal thicknesses of the wheels are 152 m (0.006”) and 305 m (0.012”). Actual tip radii of the specimens in the two groups, estimated by SEM images, were approximately 80 m and 160 m, respectively. All the specimens were tested in air at 23C under a loading rate 1.0 mm/min. Average fracture toughness values of PMMA increased from 1.32 MPam1/2 for the group having

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45 Figure 4-2. Two SEM images of the V-notch in PMMA. A) A sharp V-notch extended from a wheel-cut U-notch in a PMMA specimen. B) A close up view of the V-notch. The radius of the tip is approximately 1 m. V-notches, to 1.68 MPam1/2 of the group using the 152 m wheel, to 1.76 MPam1/2 for the group using the 305 m wheel. Using a one-way ANOVA with = 0.05, the mean Kc value of the group with V-notches is significantly lower than that of the other two groups. There is no significant difference in Kc values between the 2 groups with U-notches.

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46 In these pilot tests, all PMMA specimens fractured in a brittle manner similar to the ones in Figure 2-4A & B. Callister (1997) listed Kc values of PMMA between 0.7-1.6 MPam1/2 (Table 1-1). Choi and Salem (1993) used a SENB method and found Kc of PMMA to be 1.10.04 MPam1/2. The results of these pilot tests were mostly close to these reported data. The testing parameters suggested by Kbler (2002), ao/W = 0.2-0.3 with a razor-cut V-notch, were found to give consistent results on the fracture toughness of PMMA. For the effect of loading rate, there was no significant difference between rates 0.1 and 1.0 mm/min. Rate 1.0 mm/min was used in all the later tests on bone specimens. For the effect of temperature, the trend of Kc values of PMMA was used to compare with those of bovine femur and manatee rib. 4.1.2 Results of Bovine Femur and Manatee Rib As shown in Figure 3-1, bone specimens with sharp crack tips could also be prepared using a razor blade. All the bone specimens were fractured in water under a loading rate 1.0 mm/min. Fracture toughness of bovine femur and manatee rib were estimated at five temperatures, i.e., 0, 10, 23, 37 and 50C, with six specimens at each temperature. The average crack length (ao) of the 30 bovine specimens was 0.94 mm with a standard deviation 0.06 mm. The average crack length of the 30 manatee specimens was 0.96 mm with a standard deviation 0.07 mm. Table 4-2 shows the average fracture toughness values and the standard deviations of the 10 groups. Each of the10 groups is designated by a capital letter, as shown in Table 4-2. Figure 4-3 shows the column graphs of the results. Average fracture toughness of bovine bone decreased from 7.0 MPam1/2 at 0C, to 4.3 MPam1/2 at 50C. Overall, the fracture toughness of bovine bone decreased 0.54 MPam1/2 for every 10C from 0 to 50C. Using a one-way ANOVA with = 0.05,

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47 Table 4-2. Fracture toughness of bovine femur and manatee rib at different temperatures. Six specimens were tested in each group. Standard deviations are also listed [unit: MPam1/2]. 0C 10C 23C 37C 50C Bovine Femur 7.0.7 (A) 6.2.5 (B) 5.5.6 (C) 4.8.4 (D) 4.3.5 (E) Manatee Rib 5.5.2 (F) 5.2.4 (G) 4.7.3 (H) 4.5.3 (I) 4.1.1 (J) all groups are significantly different from each other except for the neighboring groups (e.g., group C is statistically different from A and E, but not B and D). Average fracture toughness of manatee rib decreased from 5.5 MPam1/2 at 0C, to 4.1 MPam1/2 at 50C. Overall, fracture toughness of manatee bone decreased 0.28 MPam1/2 for every 10C from 0 to 50C. Using a one-way ANOVA with = 0.05, all groups are significantly different from each other except for groups F & G, and H & I. Fracture toughness values of bovine femur are greater than that of manatee rib at all five temperatures. However, the difference between the two animals decreased from 1.5 MPam1/2 (27%) at 0C to 0.2 MPam1/2 (5%) at 50C. Figure 4-4 shows the SEM images of two bovine specimens and two manatee specimens. For bovine specimens, fracture surfaces show a predominantly plexiform bone structure (Figure 4-4A), or a predominantly secondary osteonal bone structure (Figure 4-4B), or a combination of the two. Of the 13 bovine specimens used to take the SEM images, 6 of them showed a predominantly plexiform bone structure, 3 of them showed a predominantly secondary osteonal bone structure, and 4 of them showed a combination of two. Most manatee specimens have a similar microstructure. Of the 12 manatee specimens used to take SEM images, 8 of them showed rough fracture surfaces

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48 Figure 4-3. Effect of temperature on fracture toughness of bovine femur and manatee rib. As the temperature increased, both bone specimens showed a decreasing trend in fracture toughness. with many osteons (Figure 4-4C). The other 4 specimens have relatively smooth fracture surfaces (Figure 4-4D). Porosity measurements were carried out on the 13 bovine and 12 manatee specimens used to take SEM images. Average porosity of the 13 bovine specimens was 4.1% with a standard deviation 1.1%. For the 6 specimens showing a predominantly plexiform bone, average porosity was 3.4% with a standard deviation 0.5%. For the 3 specimens showing a predominantly secondary osteonal bone, average porosity was 5.5% with a standard deviation 1.2%. For the 4 specimens showing a combination of the two, average porosity was 3.9% with a standard deviation 1.0%. Average porosity of the 12 manatee specimens was 7.4% with a standard deviation 1.8%. Figure 4.5 shows two

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49 Figure 4-4. Four SEM images of the fracture surfaces of bovine femur SEVN specimens (A and B) and manatee rib SEVN specimens (C & D). The fracture surface in 4-4A has a predominantly plexiform bone structure and the one in 4-4B has a predominantly secondary osteonal bone structure.

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50 Figure 4-5. Porosity measurements using Image Pro software. A) A bovine specimen showing a combination of plexiform and secondary osteonal bones. The measured porosity was 3.8%. B) A manatee specimen with several large vascular channels (which are more commonly seen in manatee ribs). The measured porosity was 7.7%

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51 Table 4-3. Porosity, density, and composition of bovine femur and manatee rib. The standard deviations of all measurements are also listed. Porosity (%) Apparent Density (g/cm3) Water (%) Organics (%) Minerals (%) Bovine Femur 4.11.1 2.050.02 11.40.2 24.80.4 63.80.3 Manatee Rib 7.41.8 1.980.03 12.50.5 25.10.3 62.40.6 examples of the porosity measurements. Average porosities of the bovine and manatee specimens are listed in Table 4-3. Three additional samples from the same bovine femur and manatee rib were used to determine their apparent densities and the weight percentage of water, organics, and minerals in bone. Table 4-3 shows the results of the measurements. The apparent density of the bovine femur was measured to be 2.05.02 g/cm3. The apparent density of the manatee rib was measured to be 1.98.03 g/cm3. The weight percentage of water, organics, and minerals in bovine femur were 11.4%, 24.8%, and 63.8%, respectively. The weight percentage of water, organics, and minerals in manatee rib were 12.5%, 25.1%, and 62.4%, respectively. Figure 4-6 shows the TGA and DTA curves of a bovine femur sample (a) and a manatee rib sample (b). The analyses were carried out from 23C (room temperature) to 700C at a heating rate 2 C/min. Two materials show very similar results. TGA curves (curve 1 in the figure) show that two major weight loss periods occurred around 40-120C and 300-450C. DTA curves (curve 2 in the figure) show that an exothermic phase transition occurred around 100C, which indicates the evaporation of water. As the temperature increased, an endothermic phase transition occurred between 270C and 460C, which indicates the burnout of the organics.

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52 Figure 4-6. The TGA and DTA profiles of bovine femur A) and manatee rib B) samples. TGA curves (curve 1 in the figure) show that two major weight loss periods occurred around 40-120C and 300-450C. DTA curves (curve 2 in the figure) show an exothermic phase transition around 100C and an endothermic phase transition between 270 and 460C.

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53 4.2 Measuring the J Integral of Compact Bone Thirty-four bovine specimens and 12 manatee specimens were fractured to measure the J integral of compact bone. All specimens were fractured in water under a loading rate of 1.0 mm/min. Average crack length of the 34 bovine specimens was 2.03.08 mm with ao/W = 0.51. Average depth of the two side grooves in bovine specimens was 1.06.10 mm. Average crack length of the 12 manatee specimens was 2.02.09 mm with ao/W = 0.51. Average depth of the two side grooves in manatee specimens was 1.09.09 mm. Table 4-4 lists the fracture toughness values (calculated using Equation 3.6), elastic moduli, the J integral of elastic (Jel), and plastic (Jpl) deformations, and the total J integral of the six studied groups. Average fracture toughness of the 10 transverse fracture bovine specimens with S/W=10 at 37C is 5.1.5 MPam1/2. Average fracture toughness of the 6 transverse fracture bovine specimens with S/W=10 at 23C is 5.4.5 MPam1/2. Table 4-4. Fracture toughness, elastic moduli and the J integral of the specimens. Standard deviations of all measurements are also listed (except for the elastic moduli). [Units: Kc– MPam1/2; E–GPa; Jel, Jpl, and Jtotal–KJ/m2] Bovine Femur (S/W=10; Transverse; 37C) [n=10] Bovine Femur (S/W=10; Transverse; 23C) [n=6] Bovine Femur (S/W=5; Transverse; 37C) [n=8] Bovine Femur (S/W=5; Longitudinal; 37 C) [n=10] Manatee rib (S/W=10; Transverse; 37C) [n=6] Manatee rib (S/W=10; Transverse; 23C) [n=6] Kc 5.10.5 5.40.5 4.90.7 2.60.3 4.10.2 4.40.3 E 17.5 18.4 17.5 12.6 15.5 16.2 Jel 1.40.3 1.50.3 1.30.3 0.50.1 1.00.1 1.10.1 Jpl 5.30.8 4.80.6 6.20.6 2.00.3 5.10.5 5.90.6 Jtotal 6.70.8 6.30.5 7.50.7 2.50.3 6.10.5 7.00.6

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54 Average fracture toughness of the 8 transverse fracture bovine specimens with S/W=5 at 37C is 4.9.7 MPam1/2. Average fracture toughness of the 10 longitudinal fracture bovine specimens with S/W=5 at 37C is 2.6.3 MPam1/2. Average fracture toughness of the 6 transverse fracture manatee specimens with S/W=10 at 37C is 4.1.2 MPam1/2. Average fracture toughness of the 6 transverse fractured manatee specimens with S/W=10 at 23C is 4.4.4 MPam1/2. Elastic moduli of each group were determined using 2 additional unnotched flexure specimens. For bovine femur, the mean elastic moduli of the transverse fracture specimens at 37 and 23C are 17.5 GPa and 18.4 GPa, respectively. The mean elastic modulus of the longitudinal fractured bovine specimen is 12.6 GPa. For manatee rib, the mean elastic moduli of the transverse fracture specimens at 37 and 23C are 15.5 GPa and 16.2 GPa, respectively. Jel was estimated using Equation 3.3. To determine whether the specimens were fractured under a plane stress or plane strain condition and therefore E’, experimentally, the following equation is used (Brown and Strawley, 1966), 2)(5.2ysIcKB (4.1) where B is the thickness of the specimen, ys is the yield strength of the material. If the thickness of the specimen is greater than the threshold thickness, i.e., at which B = 2.5 (KIc/ys)2, it is considered to be in a plane strain state and E’=E/(1-2). Otherwise the specimen is considered to be in a plane stress state and E’=E. In this study, the average yield strengths (from the stress-strain curves of the 2 unnotched flexure specimens in each group) of the bovine specimens were 161 MPa at 37C and 175 MPa at 23C. The average yield strengths of the manatee specimens were 144 MPa at 37C and 156 MPa at

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55 23C. Using the fracture toughness values shown in Table 4-4, the threshold thicknesses of the 4 groups mentioned above were 2.51 mm (bovine specimens at 37C), 2.38 mm (bovine specimens at 23C), 2.03 mm (manatee specimens at 37C), and 1.99 mm (manatee specimens at 23C), respectively. The thicknesses of all specimens were 4 mm and therefore, according to Equation 4.1, they were fractured under a plane strain condition with E’=E/(1-2). Poisson’s ratio of compact bone was reported between 0.2 and 0.4 (Black and Hastings, 1998; Yan, 2002). An average value 0.3 was used in all the calculations in estimating E’. E’ and the fracture toughness values of each specimen were then put into Equation 3.3 to determine the Jel of the specimens. To estimate the Jpl of the fractured specimens, BN (the thickness of the uncracked ligament), bo (the length of the uncracked ligament), and Apl (the area of the plastic deformation part) of each specimen were inserted into Equation 3.4. The total J integral (Jtotal) of each specimen was then estimated using Equation 3.5. The average Jel, Jpl, and Jtotal values and their corresponding standard deviations of the six studied groups are listed in Table 4-4. Figure 4-7 shows a column graph of all the average J integral values of the 6 groups. For transverse fracture bovine specimens at 37C with S/W=10, the average Jel, Jpl, and Jtotal values were 1.4 KJ/m2, 5.3 KJ/m2, and 6.7 KJ/m2, respectively. For transverse fracture bovine specimens at 23C with S/W=10, the average Jel, Jpl, and Jtotal values were 1.5 KJ/m2, 4.8 KJ/m2, and 6.3 KJ/m2, respectively. For transverse fracture bovine specimens at 37C with S/W=5, the average Jel, Jpl, and Jtotal values were 1.3 KJ/m2, 6.2 KJ/m2, and 7.5 KJ/m2, respectively. For longitudinal fracture bovine specimens at 37C with S/W=5, the average Jel, Jpl, and Jtotal values were 0.5 KJ/m2, 2.0 KJ/m2, and 2.5 KJ/m2, respectively. For transverse fracture manatee specimens at 37C with S/W=10,

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56 Figure 4-7. Average Jel, Jpl, and Jtotal values of the six studied groups. The data are listed in Table 4-4. the average Jel, Jpl, and Jtotal values were 1.0 KJ/m2, 5.1 KJ/m2, and 6.1 KJ/m2, respectively. For transverse fracture manatee specimens at 23C with S/W=10, the average Jel, Jpl, and Jtotal values were 1.1 KJ/m2, 5.9 KJ/m2, and 7.0 KJ/m2, respectively. The J integral of PMMA and PBC specimens were also measured using the same method as for the bone specimens. These specimens were fractured at 37C in water under a loading rate of 1.0 mm/min. Figure 4-8 shows the average J integral values of PMMA, and PBC specimens along with the results of the transverse fracture and longitudinal fractured bovine specimens. The average Jel, Jpl, and Jtotal values of the PMMA specimens were 0.8 KJ/m2, 0.1 KJ/m2, and 0.9 KJ/m2, respectively. The average Jel, Jpl, and Jtotal values of the PCB specimens were 3.3 KJ/m2, 5.1 KJ/m2, and 8.4 KJ/m2, respectively.

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57 Figure 4-8. Comparison of J integrals for 4 materials (PMMA, PCB, transverse fractured bovine specimens and longitudinal fractured bovine specimens). The numbers inside the square parentheses are the numbers of specimens tested. The average values of all groups are shown in the table don upper right corner. Figure 4-9 shows the typical load-extension curves of the deep notched flexure specimens of 4 different materials at 37C. PMMA (curve I) deformed mostly elastically with little plastic deformation. PCB (curve II) had substantial plastic deformation before catastrophic failure. For transverse fracture, the bovine specimen (curve III) also had substantial plastic deformation before unstable fracture. For longitudinal fracture, the bovine specimen (curve IV) had substantial slow, stable crack growth before the specimen fractured. Although Figure 4-9 provides an illustration of the behavior of these deep notched flexure specimens, it does not serve the purpose of showing the general behavior of the four materials. One should be careful in using this graph for any

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58 Figure 4-9. Typical load-extension curves of the deep-notched flexure specimens of 4 different materials, i.e., PMMA, PCB, transverse fracture bovine femur, and longitudinal fracture bovine femur, at 37C under a loading rate 1.0 mm/min. quantitative comparison of the J integral between the four materials since their elastic moduli vary (e.g., 13-18 GPa for bovine bone and about 3 GPa for PMMA and PCB) Figure 4-10 shows the SEM images of the fracture surfaces of three bovine and one manatee specimens. Figures 4-10A & 4-10B are bovine specimens with predominantly plexiform bone. The plexiform bone specimens mostly had a catastrophic failure after the crack extended 5-25% of the uncracked ligament with their load-extension curves such as curve II in Figure 4-9. Figure 4-10C is a bovine specimen that has predominantly secondary osteonal bone and was fractured after a substantial slow, stable crack growth (similar to curve III in Figure 4-9). Figure 4-10D is a manatee specimen showing many

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59 Figure 4-10. SEM images of the fracture surfaces of 3 bovine specimens, A-C, and 1 manatee specimen, D, used in the J integral measurements. Specimens A and B have predominantly plexiform bone. Specimen C has predominantly secondary osteons. Specimen D shows many large vascular channels with diameters from 100-300 m. large vascular channels with diameters between 100 and 300 m. Most manatee specimens had load-extension curves similar to curve III in Figure 4-9. The porosity of 12 bovine and eight manatee specimens used to estimate their J integral were also measured. Average porosity of the bovine specimens was 3.5.0%. Of the 12 bovine specimens, seven of them are predominantly plexiform bone, two of them are

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60 predominantly secondary osteonal bone, and three of them have a combination of the two types. For the seven specimens showing a predominantly plexiform bone, the mean porosity was 3.1.3%. The porosity of the two specimens showing a predominant secondary osteonal bone are 5.3 % and 4.6%. For the three specimens showing a combination of the two types, the mean porosity was 3.6.4%. The mean porosity of the eight manatee specimens was 6.5.2%. 4.3 Fracture Toughness of Water-free and Organics-free Compact Bone Water-free SEVNB specimens were obtained using two processes: (1) heated at 110C for 2 h, and (2) placed in a 60C vacuum oven for 24 h. Table 4-5 shows the weight loss of the specimens after the two processes and the average fracture toughness values of the two materials. For process (1), bovine specimens lost an average of 10.9% of weight and had an average fracture toughness 3.7.2 MPam1/2. Manatee specimens lost an average of 12.3% of weight and had an average fracture toughness 3.5.2 MPam1/2. For process (2), bovine specimens lost an average of 10.4% of weight and had an average fracture toughness 3.9.3 MPam1/2. Manatee specimens lost an average of 11.5% of weight and had an average fracture toughness 3.6.2 MPam1/2. Table 4-5. Fracture toughness of water-free bovine femur and manatee rib. Two different processes, i.e., (1) heated at 110C for 2 h, and (2) placed in a 60C vacuum oven for 24 h were used to remove the water in bone. (1) 110C for 2 h (2) 60C vacuum oven for 24 h Weight loss (%) 10.9.3 10.4.2 Bovine Femur Kc [MPam1/2] 3.7.2 3.9.3 Weight loss (%) 12.3.4 11.5.4 Manatee Rib Kc [MPam1/2] 3.5.2 3.6.2

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61 Using a one-way ANOVA with = 0.05, weight losses of the bovine and manatee specimens heated at 110C furnace are significantly greater than those of specimens heated in the 60C vacuum oven. Kc values of the bovine and manatee specimens are not significantly different from each other after both water-removal processes. The specimens were found to shrink about 2% linearly after the two processes with no statistical difference (one-way ANOVA, =0.05). All the specimen dimensions and the crack lengths were corrected for the 2% difference before they were put into fracture toughness calculations (e.g., the crack length was 2% shorter than initially cut). Organics-free SEVNB specimens were obtained using three processes: (1) heating at 500C for 24 h, (2) heating at 600C for 24 h, and (3) heating at 800C for 3 h. Table 4-6 shows the weight loss of the specimens after the three processes and the average fracture toughness values of the two materials. For process (1), bovine specimens lost an average of 35.1% of weight and had an average fracture toughness 0.38.03 MPam1/2. Manatee specimens lost an average of 36.5% of weight and had an average fracture toughness 0.41.04 MPam1/2. For process (2), bovine specimens lost an average of 36.4% of weight and had an average fracture toughness 0.32.03 MPam1/2. Table 4-6. Fracture toughness of organics-free bovine femur and manatee rib. Three different processes, i.e., (1) heated at 500C for 24 h, (2) heated at 600C for 24 h, and (3) heated at 800C for 3 h, were used to remove the organics in bone. (1) 500C for 24 h (2) 600C for 24 h (3) 800C for 3 h Weight loss (%) 35.1.6 36.4.5 37.5.2 Bovine Femur Kc [MPam1/2] 0.38.03 0.32.03 0.44.05 Weight loss (%) 36.5.8 37.5.4 38.4.3 Manatee Rib Kc [MPam1/2] 0.41.04 0.36.03 0.42.03

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62 Manatee specimens lost an average of 37.5% of weight and had an average fracture toughness 0.36.03 MPam1/2. For process (3), bovine specimens lost an average of 37.5% of weight and had an average fracture toughness 0.44.05 MPam1/2. Manatee specimens lost an average of 38.4% of weight and had an average fracture toughness 0.42.03 MPam1/2. Using a one-way ANOVA with = 0.05, weight losses of the bovine specimens after the 3 processes are significantly different from one another. Weight loss of the manatee specimens heated at 800C is significantly greater than those heated at 500 and 600C. Weight losses of the manatee specimens heated at 500 and 600C are not significantly different from each other. The differences in mean fracture toughness of the bovine and manatee specimens heated at 600C are significantly lower than those heated at 500C and 800C. It was found the bovine and manatee specimens shrunk linearly about 4% after process (1), 5% after process (2), and 12% after process Figure 4-11. Fracture toughness of bovine femur and manatee rib at three states (fresh bone in water, water-free bone, and organics-free bone). These specimens were fractured at 23C.

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63 (3). The shrinkage of the specimens heated at 500 and 600C are not significantly different from each other. The shrinkage of the specimens heated at 800C is significantly greater than the specimens heated at 500 and 600C. All the specimen dimensions and the crack lengths were corrected for the corresponding shrinkage before they were put into fracture toughness calculations. Figure 4-11 shows the comparisons between the fracture toughness values of bovine femur and manatee rib at three states, i.e., fresh bone in water, water-free bone, and organics-free bone. For bovine femur, fracture toughness decreased from 5.5 MPam1/2 for fresh bone in 23C water, to 3.7 MPam1/2 for specimens heated at 110C for 2 h, and to 0.32 MPam1/2 for specimens heated at 600C for 24 h. For manatee rib, fracture toughness decreased from 4.7 MPam1/2 for fresh bone in 23C water, to Figure 4-12. Load-extension curves of 3 bovine SEVN specimens. Curve I is a fresh bone specimen tested in water. Curve II is a water-free specimen (heated at 110C for 2 h). Curve III is an organics-free specimen (heated at 600C for 24 h). The 3 specimens were fractured at 23C.

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64 3.5 MPam1/2 for specimens heated at 110C for 2 h, and to 0.36 MPam1/2 for specimens heated at 600C for 24 h. Figure 4-12 shows the load-extension curves of 3 bovine SEVN specimens at different states. Curve I is a fresh bone specimen tested in water at 23C. Curve II is a specimen heated at 110C for 2 h. Curve III is a specimen heated at 600C for 24 h. All water-free and organics-free specimens fractured in a manner as the curves II & III in Figure 4-12 with virtually no plastic deformation.

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CHAPTER 5 DISCUSSION The effect of temperature on fracture toughness of bovine femur and manatee rib are first discussed in section 5-1. The trends of the results are compared with the trend of PMMA and one previous study on polypropylene. The differences between the fracture toughness of bovine femur and manatee are also discussed. In section 5-2, the results of the J integral measurements of bovine femur and manatee rib are discussed. The amounts of energy consumed in plastic deformation (Jpl) and elastic deformation (Jel) are compared. The reasons that compact bone has a greater amount of Jpl than Jel are discussed. In section 5-3, the effect of water and organics on fracture toughness of bovine femur and manatee are discussed. The results of the water-free and organics-free bone specimens are compared with the fracture toughness of fresh bone specimens at 23C. 5.1 Effect of Temperature on Fracture Toughness of Compact Bone Fracture toughness of PMMA was found to decrease as the temperature increased. Average fracture toughness values of PMMA decreased from 1.62.09 MPam1/2 at 10C, to 1.35.07 MPam1/2 at 50C. Overall, fracture toughness of PMMA decreased 0.07 MPam1/2 (approximately 4.2%) for every 10C from 10 to 50C. Grellmann (2001) used compact tension specimens to study the effect of temperature on the fracture toughness of polypropylene (PP, a ductile polymer). He found the fracture toughness decreased approximately 3.3% for every 10C for highly oriented PP and approximately 5.4% for every 10C for unoriented oriented PP over the temperature range of -20C to 65

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66 20C. The decrease in fracture toughness means the ability of the polymers in sustaining stress decreases as the temperature increases. At higher temperatures, the molecules in polymers have more kinetic energy and the strength of mechanical interlocking between the polymeric chains decreases. These factors cause the fracture toughness of the polymers to decrease as the temperature increases. The results of the fracture toughness measurements of the bovine femur and manatee rib specimens have similar trends as the polymers do. Average fracture toughness values of bovine femur decreased from 7.0.7 MPam1/2 at 0C, to 4.3.5 MPam1/2 at 50C. Overall, the fracture toughness of bovine bone decreased 0.54 MPam1/2 (approximately 7.7%) for every 10C from 0 to 50C. The fracture toughness of manatee rib was shown to decrease from 5.5.2 MPam1/2 at 0C, to 4.1.1 MPam1/2 at 50C. Overall, the fracture toughness of manatee bone decreased 0.28 MPam1/2 (approximately 5.1%) for every 10C from 0 to 50C. The decrement in fracture toughness means the ability of bone in sustaining stress decreases as the temperature of the material increases. While the apatite minerals in bone are anticipated to exhibit a similar mechanical behavior over the testing temperature range, the reasons the fracture toughness of compact bone decreases as the temperature increases could be the ability of the organics in bone in sustaining stress decreases. Similar to polymers, at higher temperatures, the molecules in collagen have more kinetic energy and the strength of mechanical interlocking between the polymeric chains decreases. These factors cause bone to withstand lower stresses as the temperature increases. Carter and Hayes (1976) studied the effect of temperature on the fatigue life of adult bovine bone over the temperature range of 21 to 45C. They found the fatigue life of bone to decrease by

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67 approximately a factor of three as the temperature increased from 21 to 45C. The reason the fatigue life is shorter at higher temperatures could be because of the viscoelastic behavior of collagen. At higher temperatures, the viscoelastic effect in collagen is greater and the residual strains in bone require more time to recover after loads are removed. Under cyclic loading, the residual strains in bone may not recover in time and the total strains accumulate faster at higher temperatures than at lower ones. Throughout the testing temperature range, the fracture toughness of bovine femur was greater than that of manatee rib. This is consistent with our previous study (Yan et al., 2005a). We used a chevron-notched beam (CNB) method and found the fracture toughness of bovine femur, and manatee rib at 23C to be 5.8 MPam1/2, and 4.5 MPam1/2, respectively. Bovine femur has greater fracture toughness than manatee rib for several reasons. First, the bovine specimens had a greater apparent density (2.05 g/cm3) than the manatee specimens (1.98 g/cm3). Wright and Hayes (1977) studied the effect of apparent density and found a five percent increase in bone density (from 1.92 to 2.02 g/cm3) that resulted in a 30 percent increase in the fracture toughness. The second reason could be that the bovine specimens had a lower porosity (4.1%) than the manatee specimens (7.4%). Yeni et al. (1997) studied the influence of bone morphology and found the critical strain energy release rate (Gc) of human femur decreased as porosity of the bone increased. I studied the effect of porosity on the fracture toughness of manatee rib and found an increase of porosity from 2.6% to 11.6% to cause a 30 percent decrease in the fracture toughness (Yan, 2002). The above two factors are not isolated from each other. For same type of bone, lower porosity corresponds to higher density since the densities of bone constituents (mostly hydroxyapatite and collagen) are

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68 greater than the densities of the fluids (mostly water) in pores. The third reason could be that the bovine femur has a more organized microstructure than the manatee rib. The fracture surfaces of the bovine specimens show that many lamellae with a similar thickness of plexiform bone align parallel to each other (Figures 4-4A, 4-10A and 4-10B) or that secondary osteons having a similar size align parallel to the long axis of bovine femur (Figures 4-4B and 4-10C). On the other hand, the fracture surfaces of the manatee specimens (Figures 4-4C, 4-4D and 4-10D), though most osteons run parallel to the long axis of the rib, show that more vascular channels run horizontally than does the secondary osteonal bone in bovine femur. In general, the manatee specimens have a less defined microstructure than the bovine specimens do. Figure 5-1 shows the load-extension curves for 3 selected bovine specimens. Curve I is a specimen fractured at 0C. In general, for the specimens fractured at 0 and 10C, Figure 5-1. Load-extension curves of 3 bovine specimens tested at different temperature. Curve I is a specimen fractured at 0C, curve II is a specimen fractured at 23C, and curve III is a specimen fractured at 50C.

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69 most specimens (10 out 12) showed a similar curve as curve I with relatively little plastic deformation. Curve II is a specimen fractured at 23C. As the temperature increased, the maximum loads of the specimens decreased, the slopes of the curves decreased, and the amount of plastic deformation increased. Curve III is a specimen fractured at 50C. Of the 30 bovine specimens, 29 of them had catastrophic failure (fast, unstable fracture) at the end. The only bovine specimen fractured in a slow, stable manner (such as the curve VI in Figure 5-2) was a specimen with predominantly secondary osteonal bone fractured at 50C. Figure 5-2 shows the load-extension curves of 3 selected manatee specimens. Curve IV is a specimen that fractured at 0C. All 6 manatee specimens that fractured at 0C had load-extension curves similar to curve IV. Curve V is a specimen that fractured at 23C. In general, most manatee specimens (8 out of 12) fractured at 10 and 23C had load-extension curves such as curve V. The other 4 (1 at 10C and 3 at 23C) had curves such as curve VI. Curve VI is a specimen that fractured at 50C. All 12 specimens fractured at 37 and 50C had load-extension curves similar to curve VI, which had a slow, stable crack growth before the specimen fractured. Figures 5-1 and 5-2 show that for bovine and manatee specimens, the amount of plastic deformation increased as the temperature increased. Reilly and Burstein (1975) provided evidence that while the mineral phase is an important determinant of bone elasticity, the organics phase is responsible for the bone post-yield behavior. One important implication from this is that though the fracture toughness of compact bone decreases as the temperature increases, the energy spent in fracturing bone may not decrease. To verify this, the J integral measurements of bovine and manatee bone were carried out at 23 and 37C. Discussions of the J integral measurements are in section 5.2.

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70 Figure 5-2. Load-extension curves of 3 manatee specimens tested at different temperature. Curve IV is a specimen fractured at 0C, curve V is a specimen fractured at 23C, and curve VI is a specimen fractured at 50C. The results of the transverse fracture bovine specimens were compared with the results of five previous studies. Table 5-1 lists the fracture toughness values, testing method, temperature, and fracture rate of five previous studies on the fracture toughness of transverse fracture bovine femur. The results of current study on bovine femur at 23 and 37C are also listed in the table. Besides the possible variability in the testing methods, testing temperature and fracture rate were shown to affect the measured fracture toughness. Melvin and Evans (1973) used a SENB method and found the mean fracture toughness of slow crack growth and fast crack propagation specimens to be 5.58 MPam1/2 and 7.69 MPam1/2, respectively. Lucksanasombool et al. (2001) also used

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71 Table 5-1. Results of 5 previous studies on the fracture toughness of bovine femur. All specimens were fractured along transverse fracture direction. Investigators Testing Method Temperature Fracture Rate Fracture Toughness (MPam1/2) Melvin and Evans (1973) Single-edge notched beam Room temperature Slow Fast 5.58* 7.69* Wright and Hayes (1977) Compact tension Room temperature Slow 3.62* Robertson et al. (1978) Single-edge notched beam 37C Slow 5.71.4 Feng et al. (2000) Compact tension Room temperature Not available 6.00.4 Lucksanasombool et al. (2001) Single-edge notched beam Not available Slow Fast 3.480.33 (stored in saline) 5.050.96 (stored in alcohol) Current study Single-edge V-notched beam 23C 37C Fast Fast 5.50.6 4.80.4 For specimens having a catastrophic failure, the fracture rate is defined as fast. For specimens having a slow, stable crack growth throughout the test, the fracture is defined as slow. * Average values of the tested specimens. a SENB method to study the effect of storage media. They found the specimens from the bovine femur stored in saline fractured in a slow, stable manner (as curve VI in Figure 5-2) with an average fracture toughness 3.48 MPam1/2, whereas the specimens from the bovine femur stored in alcohol had catastrophic fracture (as curve I in Figure 5-1) with an average fracture toughness 5.05 MPam1/2. They claimed the alcohol could reduce the water in bone and inhibit viscoelastic behavior of the organics, which resulted in an increase in the fracture toughness. The results from the present study show that as the temperature increased from 23 to 37C, the fracture toughness of bovine femur decreased from 5.5 MPam1/2 to 4.8 MPam1/2 with a decrease of 12.7%. Robertson et al. (1978) used a SENB method and tested the specimens in Ringer’s solution at 37C. They found

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72 an average fracture toughness of bovine femur to be 5.7 MPam1/2 with a relatively large standard deviation of 1.4 MPam1/2 (standard deviations of other studies in Table 5-1 range from 0.33-0.96 MPam1/2). The average fracture toughness they estimated is 18.8% greater than the result of this study (4.8 MPam1/2). One of the reasons for the difference between the two studies and for the large standard deviation in their data is that the final sharp notches of their specimens were introduced by a wire saw, with the thickness of the wire approximately 50 m. The final sharp notches of the specimens in the present study were introduced using a razor blade with the radii of the tips approximately 1 m (Figure 3-1). Mukhopadhyay et al. (1999) studied the relationship between notch radius and Kc values of alumina (Al2O3) and silicon carbide (SiC) and found that the larger the notch radius, the larger the Kc value. A blunt pre-crack may result in a larger apparent Kc value since extra stress is needed to break the specimen. Presumably, a blunt notch needs to re-sharpen before further propagation resulting in a greater required stress to produce the sharp crack. The average fracture toughness value of the manatee specimens at 23C was 4.7 MPam1/2, which is close to what we reported (4.5 MPam1/2, Yan et al., 2005a). As shown in Figures 4-4 and 4-10, some bovine specimens had a predominantly plexiform bone structure, some had a predominantly secondary osteonal bone structure, and others had a combination of the two. It was found at the same temperature, the fracture toughness values of the specimens with a predominantly plexiform bone were consistently greater than the ones with a predominantly secondary osteonal bone or a combination of the two. The differences between the two groups are from 0.5-1.6 MPam1/2, which corresponds to 12-27%. This may explain why the standard deviations of the fracture toughness of the bovine specimens are consistently greater than

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73 the standard deviations of the fracture toughness of the manatee specimens (Table 4-2). The mean standard deviation of the bovine specimens at 5 testing temperatures is 9.8%, compared with 5.4% for the manatee specimens. The fracture surfaces of the manatee specimens (Figures 4-4C, 4-4D and 4-10D) all have a similar microstructure. Many studies have shown plexiform bone and secondary osteonal bone behave differently. Walmsley and Smith (1957) suggested that primary bone is stronger than secondary osteonal bone. Currey (1959) used quasi-static tensile tests to fracture the specimens from bovine femurs and confirmed Walmsley and Smith’s suggestion. Carter and Hayes (1976) studied the effect of temperature on the fatigue life of adult bovine bone and suggested that primary bone has greater fatigue resistance than secondary Haversian bone. Rimnac et al. (1993) studied the effect of microstructure on the creep of bovine compact bone and estimated the steady-state creep rate would be nearly one hundred times faster for a specimen with fully secondary osteonal bone than one with a fully plexiform bone. The reason for this could be the gaps (cement lines) between the osteons and bone matrix helped cracks in bone grew faster under tension. Wang and Agrawal (1996) studied the effect of sampling site on the fracture toughness of bovine femur. They found the samples from the outer sites had predominantly plexiform bone and showed a greater difference in the fracture toughness values of radial crack and circumferential crack specimens than the samples from the inner sites, which had more secondary osteons. Outer layers of bovine femur in young adult cows were newly laid down and were mostly plexiform bone. On the other hand, inner layers of bone existed since the animals were born and could have experienced bone remodeling.

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74 While many studies suggested that secondary osteonal bone is mechanically weaker than primary bone, secondary osteonal bone has its important functions in animal. Many suggested (Frost, 1973; Burr et al., 1988) the microcracks in bone can be repaired through bone remodeling, which creates secondary osteons. Burr et al. (1988) also suggested that although the morphology of the cement line in secondary osteons permits relatively easy crack initiation, it could prevent or slow any significant crack propagation. Their suggestion may explain why the specimens in this study with a predominantly secondary osteonal bone tended to have a lower strength but a greater amount of plastic deformation than the specimens with a predominantly plexiform bone. 5.2 Measuring the J Integral of Compact Bone J integral was used to evaluate the energy consumed in the linear-elastic and plastic deformations in the fracture process of bone. Before the results of the J integral are discussed, comparisons between the measured fracture toughness using the SEVNB method and the method suggested by ASTM E1820 were made to determine whether the two methods gave consistent results. Table 5-2 lists the results of the average fracture toughness values and their corresponding standard deviations of bovine femur and manatee rib using the SEVNB method and the method suggested by ASTM E1820 at 23 and 37C. The mean fracture toughness of the transverse fracture bovine and manatee specimens using the two methods is close to each other in all groups. Using a one-way ANOVA with = 0.05, the only group that shows a significant difference between the two methods is the manatee specimens tested at 37C. Thus, the results of the two methods are considered consistent. Table 4-4 lists the fracture toughness values, elastic moduli, Jel, Jpl, and the total J integral of the six studied groups. The results of the fracture toughness measurements

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75 Table 5-2. Fracture toughness of bovine femur and manatee rib at 23 and 37C. Both the bovine femur and manatee rib were studied using the SEVNB method and the method suggested by ASTM E1820 [unit: MPam1/2]. 23C 37C SEVNB S/W* = 10 Bovine Femur ASTM E1820 S/W* = 5 5.5.6 5.4.5 Not tested 4.8.4 5.1.5 4.9.7 SEVNB Manatee Rib ASTM E1820 4.7.3 4.4.3 4.5.3 4.1.2 * S/W is the ratio between the support span of the 3-point flexure, S, and the specimen width, W. show a great difference between the longitudinal fracture and the transverse fracture bovine specimens. Average fracture toughness of the longitudinal fracture bovine specimens was estimated to be 2.6 MPam1/2 with a standard deviation 0.3 MPam1/2. When compared to the transverse fracture bovine specimens with S/W=5, the average fracture toughness of the longitudinal fracture specimens is 53.1% of that of the transverse fracture specimens. This result is close to several previous studies. The ratios between the average fracture toughness of the longitudinal and transverse fracture bovine femur of some previous studies are: 0.65 (of specimens with fast fracture (Melvin and Evans, 1973)), 0.50 (Feng et al., 2000), and 0.66 (of specimens stored in saline (Lucksanasombool et al., 2001)). Elastic moduli of the transverse fracture bovine specimens at 37 and 23C were measured to be 17.5 GPa and 18.4 GPa, respectively. The difference between the two elastic moduli was 4.9%. Elastic modulus of cortical bone along the long axis of the bone is generally between 15-30 GPa (Table 1-1). Martin and Boardman (1993) measured the bending elastic modulus of bovine compact bone and found it to be 19.9 GPa at room

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76 temperature, which is 8.1% higher than the value (23C) in this study. However, the loading rate they used is 10 mm/min, which is 10 times faster than the rate used in this study. In general, the measured elastic modulus of bone is less if the loading rate is lower (Currey, 2002). The change of elastic modulus at different temperatures has been reported. Bonfield and Li (1968) used tensile specimens to study the effect of temperature on the deformation of bovine femur and found the elastic modulus decreased 24.4% from 0 to 50C, which corresponds to 6.8% for every 14C. They suggested the mechanical behaviors of the inorganic phases in this temperature range should be similar. The decrease of elastic modulus in bone as the temperature increases is primarily due to the decrease in the elastic modulus of collagen. This is because at higher temperatures, the molecules in collagen have more kinetic energy and the strength of mechanical interlocking between the polymeric chains decreases. The organic phases in bone then become easier to be deformed and the elastic modulus of bone decreases. The elastic modulus of the longitudinal fracture bovine specimen at 37C was measured 12.6 GPa, which is 38.9% lower than that of the transverse fracture bovine specimens. Compact bone has been shown to be anisotropic (or, close to transversely isotropic) with the elastic modulus of the longitudinal direction much greater than that of the radial and circumferential directions (Currey and Butler, 1975; Ashman et al., 1984). For manatee rib, the elastic moduli of the transverse fracture specimens at 37 and 23C are 15.5 GPa and 16.2 GPa with the difference of the two being 4.3%. The elastic moduli of manatee rib are 11.4% (37C) and 12.0% (23C) lower than those of bovine femur. The reasons the elastic modulus of manatee rib is less than that of bovine femur could be that the manatee rib has a greater porosity and a less density.

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77 As shown in Table 4-4, at 37C, the average Jpl values were shown to be 3.8 times (transverse fracture bovine femur with S/W=10), 4.7 times (transverse fracture bovine femur with S/W=5), 4.0 times (longitudinal fracture bovine femur with S/W=5), and 5.1 times (transverse fracture manatee rib with S/W=10) greater than the corresponding average Jel values. On average, the energy spent in advancing the crack beyond the linear-elastic deformation is 4.3 times that of bovine specimens and 4.6 times that of manatee specimens greater than the energy spent within the linear-elastic deformation. The reasons the average Jpl values are much greater than the average Jel values could be that bone has at least four toughening mechanisms and a high amount of organics. These toughening mechanisms, such as microcracking (Burstein et al. 1975; Zioupos et al., 1995), crack deflection (Yeni and Norman, 2000), and fiber bridging (Nalla et al., 2003; Yan et al., 2005a), have been observed in the fracture of compact bone. These mechanisms could effectively stop, slow, or deflect crack propagation. Therefore, extra energy has to be applied in order to fracture the specimens. The organic matrix, which controls the growth of the apatite minerals, is suggested to have strong bonding with the mineral phases (Eliana Lucchinetti, 2001). This strong bonding could force the crack to deflect, rather than go through the interface. Beck and Brodsky (1998) suggested collagen provides the post-translational modifications from cross-link formation and glycosylation. Compare the results of the transverse fracture bovine specimens (S/W=10) at 23C with those at 37C, the average Jel at 23C is greater than that at 37C, but the average Jpl at 23C is less than that at 37C. The reason for this is that at 37C the maximum stress decreased, but the amount of plastic deformation increased. As stated in section 5.1, the

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78 reasons the maximum stresses in bone specimens decreased at higher temperatures were the molecules in collagen had more kinetic energy and the strength of mechanical interlocking between the polymeric chains decreased. These factors cause bone to be deformed easier as the temperature increases. Although the fracture toughness of bone decreases as the temperature increases, the J integral results show a contrary trend (i.e., the J integral increased from 6.3 KJ/mm2 at 23C to 6.7 KJ/mm2 at 37C with the difference not statistically significant). This result is similar to what Grellmann (2001) found about the effect of temperature on the fracture toughness and J integral of polypropylene. They found the fracture toughness decreased as the temperature increased, while the J integral increased in the same temperature range (-30 to 20C). Grellmann suggested that in evaluating the temperature dependence of toughness, J integral is more appropriate than fracture toughness. The reason for this is that fracture toughness does not account for plastic deformation of materials. Figure 5-3 shows two SEM images of the fracture surfaces of bovine specimens. Before the onset of catastrophic failure, the surfaces are mostly relatively rough. A boundary between the fracture surfaces of slow, stable crack growth and catastrophic failure can usually be drawn by observing the difference. Figure 5-3A shows a specimen fractured at 23C with slow, stable crack growth approximately 5-10% of the uncracked ligament. Figure 5-3B shows a specimen fractured at 37C with slow, stable crack growth approximately 15-30% of the uncracked ligament. The cracks in all specimens of the 6 studied groups were guided to propagate along the grooves except for the group of transverse fracture bovine specimens at 37C with S/W=5. Figure 5-4 shows an angle view of the fracture surfaces of 3 specimens. Figure 5

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79 Figure 5-3. Two bovine specimens with different amounts of slow, stable crack growth. A boundary between the fracture surfaces of slow, stable crack growth and catastrophic failure can be drawn. A) A specimen fractured at 23C with slow, stable crack growth approximately 5-10% of the uncracked ligament. B) A specimen fractured at 37C with slow, stable crack growth approximately 15-30% of the uncracked ligament. 4A is a transverse fracture bovine specimen at 37C with S/W=10. Figure 5-4B is a longitudinal fracture bovine specimen at 37C with S/W=5. Fracture surface of the longitudinal fracture specimen is shown to be relatively smooth than that of the

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80 transverse fracture bovine specimen. The reason the fracture surfaces of the longitudinal fracture specimens are relatively smooth is because the crack propagation plane was parallel to fiber-reinforced direction. The advancing cracks were able to propagate without much retardation from the fibers. On the other hand, the cracks in the transverse fracture specimens had to break through many reinforced fibers. Figure 5-4C is a transverse fracture bovine specimen at 37C with S/W=5. The crack is shown to propagate in the plane of side grooves at first but ran obliquely afterwards. Of the ten specimens tested in this group, only 3 of them had cracks propagate along the grooves. The other seven specimens had a fracture surface similar to the one in Figure 5-4C. The reason the crack ran obliquely could be the S/W ratio of 5 is too low and the shear stress along the long axes of the specimens was high enough to drive the crack to propagate along the interface between fibers, instead of breaking through them. The Jpl values of this group were calculated without correcting the fracture surface area (Bbo in Equation 3.4) due to the tilting angles of the fracture surfaces, and where the tilts started varied a lot. Therefore, the results of this group were overestimated. The cracks in the longitudinal fracture bovine specimens did not go obliquely, even though the S/W ratio of the specimens is also 5, is because the cracks went through a plane parallel to the reinforced fibers. When compared to the transverse fracture bovine specimens at 37C with S/W=10, the average total J integral of the longitudinal fracture specimens is 37.3% of that of the transverse fracture specimens. The average Jel values of the transverse fracture manatee specimens at 23 and 37C were shown to be significantly less than those of the transverse fracture bovine specimens. However, the mean Jpl values of the transverse manatee specimens are

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81 Figure 5-4. Angle view of the fracture surfaces of 3 bovine specimens. A) A transverse fracture bovine specimen at 37C with S/W=10. C) A longitudinal fracture bovine specimen at 37C with S/W=5. C) A transverse fracture bovine specimen at 37 C with S/W=5.

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82 comparable to those of bovine specimens. Compare the results of the transverse manatee specimens (S/W=10) at 23 C with those at 37C, the average Jel and Jpl values at 23C are both greater than those at 37C. The load-extension curves of the manatee specimens, at both 23 and 37C, were mostly similar to curve IV in Figure 4-9. They had substantial slow, stable crack growth before fracture. Similar to Kc and KR (R-curve) measurement, a Jc and JR measurement may be more appropriate for materials with slow, stable crack growth (Saxena, 1998).Zioupos and Currey (1998) used a graphical method to estimate the J integral of human (35 year old) femurs and found it to be 1.2 KJ/m2. This value is less than many reported Gc values (Wang and Puram, 2004). Unless the bone specimens they tested fractured in a brittle manner without any plastic deformation, they might have underestimated the J integral. Zioupos (1998) derived the J integral of the equine hoof wall from the data reported by Kasapi and Gosline (1996) and found it to be between 10 and 15 KJ/m2. They also found the J integral to be constant over a loading rate spanning 5 orders of magnitude and suggested the J integral is a material property. Although several groups (Wright and Hayes, 1977; Bonfield et al., 1978) claimed that LEFM is applicable to the fracture of bone, the fact that often a moderate amount of yielding in bone is observed during bone mechanical tests suggests that LEFM may underestimate how tough bone is. Currey (2002) pointed out the basic assumption of LEFM, i.e., any part of the material behaves linear-elastically until fracture, is certainly not true for bone. The J-integral, a parameter that quantifies the energy consumed in the specimen in order to advance the crack, is shown in the present study to better describe the fracture process of bone.

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83 The statement made by Currey in the last paragraph is correct for bone in most in vivo and in vitro conditions. However, one should not assume LEFM can not be used to quantify the fracture behavior of bone in any condition. For example, Figures 5-1 and 5-2 show at 0C, the SEVN specimens fractured with little plastic deformation. In these cases, Kc or KR (R-curve) measurement may be sufficient to describe the fracture behavior. Another example is that, for most fracture mechanics tests (except for true dynamic fracture mechanics), the strain rate of the specimens is considerably less than certain physiological bone strain rate. The strain rate of the specimens tested in this dissertation is approximate 10-4/s, which is only about 1/100 of some measured strain rates in human bone at daily activities (such as walking and running) (Burr et al., 1996). If the specimens were fractured under a loading rate 100 times of the current test, the specimens may fracture with less plastic deformation. 5.3 Effect of Water and Organics on Fracture Toughness of Bone TGA curves in Figure 4-6 show two major weight loss stages. The first stage is related to the evaporation of water and the second stage is related to the burnout of the organics. The curves started to level off around 120C after the first stage with a weight loss around 10-11%. After the second stage, the curves started to level off around 450C with a weight loss around 33-35%. As shown in Table 4-5, the two water-removing processes removed a similar amount of water in both bovine femur and manatee rib. Comparing the results with the TGA curves, the majority of the water in bone is believed to be removed. It was anticipated that the fracture toughness of the specimens using the vacuum oven would be greater than that of the specimens heated in the furnace since the organics could be less altered. However, the average fracture toughness values of the two groups are not

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84 significantly different from each other. The reason for this could be that both processes did not significantly alter the nature of the collagen. Yamashita et al. (2001) used loss factor (tan ) and storage modulus to study the change in the viscoelasticity of bone at temperatures of 100 and 200C and found that significant collagen denaturation occurred only when bone was heated at 200C. The average fracture toughness values of the water-free bovine and manatee specimens are 44.7% and 32.4%, respectively, less than those of fresh bovine and manatee bone at 23C. Melvin and Evans (1973) used a SENB method to study the fracture toughness of bovine femur and found that the Kc value for the wet specimen was 60% greater than that for the dry companion specimen. Comparing the results in Table 4-6 and the TGA curves, the majority of the organics in bone is believed to be removed by the three organics-removing processes. When compared to fresh bone at 23C, fracture toughness values of the organics-free specimens are only 7.3% (bovine femur) and 8.5 % (manatee rib) of the fracture toughness of fresh bone specimens. The average fracture toughness values of both the bovine and manatee specimens heated at 600C for 24 h are significantly less than those of specimens under the other two heating processes. The reason for the difference between the specimens heated at 500C and heated at 600C could be that more organics were still in the specimens heated at 500C. Figure 3-5 shows the difference between the color of the specimens after heating at 500 and 600C. The reason for the difference between the specimens heated at 600C and heated at 800C could be that after heating at 800C, a sintering effect caused the specimens become much denser. It was found that the specimens shrunk linearly by about 5% after heating at 600C for 24 h and by about 12% after heating at 800C for 3 h. In terms of volume percentage, the specimens shrunk

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85 Figure 5-5. Fracture surfaces of two bovine specimens after different heating temperatures. A) A specimen after heating at 600C for 24 h. The surface shows many tiny mineral particles with diameters roughly 0.1-0.2 m. B) A specimen after heating at 800C for 3 h. The surface shows agglomerates with diameters mostly between 0.5 and 1 m.

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86 about 12-15 vol.% after heating at 600C and about 30-35 vol.% after heating at 800C. Figure 5-5 shows the SEM images of the fracture surfaces of two bovine specimens. Figure 5-5A is a specimen after heating at 600C for 24 h. Since the organics were mostly burned out, what is left in the specimen were tiny mineral particles with diameters roughly 0.1-0.2 m. Figure 5-5B is a specimen after heating at 800C for 3 h. The mineral particles are shown to sinter into larger agglomerates. The diameters of the particles were mostly between 0.5 and 1 m. Average fracture toughness of the specimens heated at 600C is 27.3% less than that of the specimens heated at 800C. Fracture toughness values of dense hydroxyapatite were found to be 0.9-1.2 MPam1/2 (Thomas and Doremus, 1981;Gross and Bhadang, 2004). The average fracture toughness values of the organics-free specimens were about 0.32 to 0.44 MPam1/2 (Table 4-6). Using the densities, 3.2 g/cm3 for minerals, 1.1 g/cm3 for organics, and 1.0 g/cm3 for water, suggested by Currey (1990), the volume percentage of the minerals, organics, and water in bovine femur are about 37, 42, and 21%, respectively. Assume the density of the minerals did not change and account the volume shrinkage after heating; the porosity of the specimens heated at 600C for 24 h was about 56-58% and the porosity of the specimens heated at 800C for 3 h was about 43-47%. This high porosity could be the primary reason the fracture toughness of the organics-free specimens were only about 30-50% of dense hydroxyapatite. The J integral of water-free and organics-free bovine femur can be estimated using Equation 3.3 since they were fractured with virtually no plastic deformation. E KGJccc2 . (3.3)

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87 Elastic moduli of water-free bone and organics-free bone were estimated using the slope ratios of the load-extension curves between them and fresh bone (Figure 4-12). Elastic moduli of water-free and organics-free femur were estimated 32 and 13 GPa, respectively. Using the fracture toughness values listed in Tables 4-5 and 4-6, the J integral of water-free (heated at 110C for 2 h) and organics-free (heated at 600C for 24 h) bovine femur were estimated 0.4 and 0.01 KJ/mm2, respectively. The J integral of fresh bovine bone at 23C was estimated 6.3 KJ/mm2, which is 15.7 times of that of water-free bone and 630 times of that of organics-free bone.

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CHAPTER 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 6.1 Conclusions Based on the data and images presented in this dissertation, the following conclusions can be drawn. (1) The fracture toughness of bovine femur and manatee rib decreases as the temperature increases. The mean fracture toughness of bovine femur decreased from 7.0 MPam1/2 at 0C, to 4.3 MPam1/2 at 50C. Overall, fracture toughness of bovine bone decreased 0.54 MPam1/2 (approximately 7.7%) for every 10C from 0C to 50C. The fracture toughness of manatee rib was shown to decrease from 5.5 MPam1/2 at 0C, to 4.1 MPam1/2 at 50C. Overall, fracture toughness of manatee bone decreased 0.28 MPam1/2 (approximately 5.1%) for every 10 C from 0C to 50C. (2) Although the fracture toughness of bovine bone decreases as the temperature increases, the J integral results show a contrary trend. The J integral of bovine femur increased from 6.3 KJ/mm2 at 23C to 6.7 KJ/mm2 at 37C. (3) LEFM is widely used to study the fracture toughness of bone. However, it may underestimate how tough bone is since it does not take plastic deformation into account. The J integral, a parameter that quantifies the energy consumed in the specimen in order to advance the crack, is shown to better describe the fracture process of bovine femur and manatee rib. 88

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89 (4) The J integral of transverse and longitudinal fracture bovine femur at 37C were estimated 6.7 and 2.5 KJ/m2, respectively. The J integral of transverse fracture bovine femur was 168% greater than that of longitudinal fracture bovine femur. (5) Fracture toughness of bovine femur is shown to be greater than that of manatee rib. The reasons that bovine femur has a greater fracture toughness are attributed to the facts that bovine femur has a greater apparent density, a lower porosity, and a more organized microstructure. (6) Fracture toughness of primary plexiform bone in bovine femur is greater than that of secondary osteonal bone by 12-27%. (7) The average Jpl values of bovine femur and manatee rib were shown to be 3.8-5.1 times greater than the average Jel values. On average, the energy spent in advancing the crack beyond the linear-elastic deformation regime is 4.4 times greater for bovine femur and manatee rib than the energy spent for the linear-elastic deformation. (8) The reasons the average Jpl values are much greater than the average Jel values could be that bone has at least four toughening mechanisms that leads to R-curve behavior and also because of bone has a high amount of organics. (9) TGA and DTA curves demonstrated that the two major weight loss stages in both bovine femur and manatee rib below 500C are the evaporation of water and the burnout of organics. (10) Both water and organics have a significant effect on the fracture toughness of bone. Fracture toughness of the water-free specimens was 44.7% (bovine femur) and 32.4% (manatee rib) less than that of fresh-bone specimens. Fracture toughness of the organics-free specimens was 92.7% (bovine femur) and 91.5 % (manatee rib)

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90 less than that of fresh bone specimens. The J integrals of water-free and organics-free bovine bone were found to be 0.4 and 0.01 KJ/mm2, respectively. The J integral of fresh bovine bone at 23C was estimated 6.3 KJ/mm2 (which is 15.7 times of that of water-free bone and 630 times of that of organics-free bone). (11) The organics-free specimens have an estimated porosity 43-58%. This large amount of porosity could be the primary reason that the fracture toughness values of the organics-free specimens were only about 30-50% of the dense hydroxyapatite. (12) Bone is a complicated composite that has at least four toughening mechanisms. To describe the mechanical property and fracture behavior of bone, a single parameter (such as strength and fracture toughness) is not sufficient. The mechanical properties of bone can be affected by many factors (such as age of the animals, type of the bone, bone composition, and bone morphology). It is important to understand the bone source and know what type of bone is being studied. When reporting data, other information (such as which direction the measurement is made, bone density, and porosity) should also be reported. It is also important to report technical conditions (such as loading rate, notching procedure, testing temperature, and how the bone is kept wet). 6.2 Suggestions for Future Work The strain rate of the specimens in this dissertation is only about 1/100 of some measured strain rates in human bone. It is anticipated the fracture toughness of bone at higher temperature (such as 37 and 50C), will be different since the viscoelastic nature of the organics will be less effective at faster strain rates. Also, in the J integral measurement, the ratio between Jpl and Jel may be different from the data in this

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91 dissertation. In the future, increase in the loading rate to create a strain rate that matches the high physiological strain rate will be important. In the J integral measurement, the load-extension curves of the manatee specimens, at both 23 and 37 C, were mostly similar to curve IV in Figure 4-9. For specimens having substantial slow, stable crack growth before fracture, Jc and JR measurements may be more appropriate in describing the fracture process of the material. More work is needed in this area.

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BIOGRAPHICAL SKETCH Jiahau Yan was born on July 18, 1976, in Taichung, Taiwan. After living in Taichung for 4 years, he moved to Kaohsiung, the city where he grew up. He entered National Taiwan University in 1994 and earned his B.S. from the Department of Mechanical Engineering in 1998. In his junior year, he became interested in materials science, and made up his mind to study in the USA. In 2000, he began graduate study in materials science and engineering at the University of Florida. He earned his master’s degree in 2002 under the supervision of Dr. John J. Mecholsky. His thesis title was “Biomechanical Properties of Manatee Rib Bone and Analytical Study Using Finite Element Analysis.” He enjoyed learning from Dr. Mecholsky so much that he decided to stay for his Ph.D. degree in the same group. Jiahau’s research interests include mechanical properties of bone, fracture and fractal analysis of materials, and mechanical properties of ceramics and dental biomaterials. He loves sports and enjoys playing basketball the most. He is also interested in volunteering and has learned a great deal through his volunteer services. In 2003, Jiahau married Paoyun Tao, a lady he met in his freshman year. 100