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Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.

Permanent Link: http://ufdc.ufl.edu/UFE0019617/00001

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Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Leismer, Jeffrey M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Jeffrey M Leismer.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Arakere, Nagaraj K.
Local: Co-adviser: Mecholsky, John J.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019617:00001

Permanent Link: http://ufdc.ufl.edu/UFE0019617/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Leismer, Jeffrey M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Jeffrey M Leismer.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Arakere, Nagaraj K.
Local: Co-adviser: Mecholsky, John J.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019617:00001


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1 NUMERICAL INVESTIGATION OF THE FR ACTURE PROPERTIES OF MANATEE RIB BONE USING EXPERIMENTALLY DETERMINED ANISOTROPIC ELASTIC CONSTANTS By JEFFREY MICHAEL LEISMER 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 2007

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2 2007 Jeffrey Michael Leismer

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3 To my sons, Finn and Jake, who make me laugh and always keep life exciting. To my amazingly beautiful wife, Renee, I am indebted to you for your patience and for doing such a wonderful job with our boys duri ng this multifaceted por tion of our lives. To my very supportive family and friends, who have believ ed in me every step of the way.

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4 ACKNOWLEDGEMENTS I thank my supervisory committee for their me ntoring and/or for providing me access to their labs, computers, equipment, supplies, and graduate students. I also thank Dr. Anusavice (Chair of the Dental Biomaterials Department ) and Dr. Daegling (Associate Professor in the Department of Anthropology) for extending the same courtesy with their labs and students. I thank my (current and former) lab mates in the Computational Solid Mechanics lab (Erik Knudsen, George Levesque, Linus Norlander, Matt Williams, Mika Lautaoja, Nathan Branch, Niraj Bidkar, Sangeet Srivastava, Shadab Siddi qui, Srikant Ranjan, Y ogen Utturkar, and Tae Joong (TJ) Yu) for their friendship and participat ion in many great discussions and outings as well as for assisting with various aspects of my research. I thank Kari Clifton and Jiahau (Bratt) Yan for inspiring me to work on this project and for helping to pave the path for my work. Maggie Stoll helped me set-up equipment for ha rvesting the rib bones used in my study and was very generous with her time. Mike Braddock he lped out tremendously by allowing for me to use his machine shop to prepare my bone specimens a nd fixtures. Brian Robinson machined a fixture for use in my experiments. Allyson Barrett pr ovided assistance locati ng persons and equipment to help with specimen preparation. Ben Lee as sisted with shear specimen preparation. Bret Stanford demonstrated how to use the VIC equi pment and answered many questions I had on the subject. Nancy Strickland spent many days wri ting custom Labview code for my experiments and showed me how to operate the universal tes ting machine used in my studies. Robert Smith spent many hours powder coating my fracture speci mens in preparation for the SEM analysis. The Major Analytical Instrumentation Center (M AIC) at the University of Florida performed SEM analyses for my quantitative fractography study. I thank those who discussed my research with me in Dr. Ifjus, Dr. Mecholskys, and Dr. Reeps labs. Drs. Kathy Howe and Laurie Gower

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5 facilitated motivating discussions by foundi ng/organizing the Bone Research Group, an interdisciplinary group of research ers performing research on bone a nd collagen at the University of Florida. Thanks to Dr. Walsh for being a gr eat conversationalist and for co-presenting on the topic of Bone (Bio) Mechanics with me at a UF Bone Research Group meeting. Mark Tillman and John Chow provided much guida nce while I was searching for a suitable dissertation topic. I could not have gotten where I am without the aid of my parents; I thank them for always being there and for providing me with life-shaping ad vice, knowledge, and skills. My nieces, Auriana and Kaiya, taught me much about nature, space, science, and technology, which helped me to gain a better perspective on my research. My brot her, Matt, enriched my learning experience by adding positive energy to my life. My sister-in-laws, Charlene a nd Melissa, my brother-in-law, Andrew, my in-laws, my extended family, and frie nds were always there to recharge me during my trips to Michigan. My wife Renee, provided much support dur ing my career as a graduate student, helped motivate the move to Florida, sh ared many great times with me, and has been a wonderful mother to our two boys. Thank you, Finn and Jake, for keeping my spirit light and for showing me how fun it is to lear n new things; you guys are so cool. Lastly, I would like to thank all of those who believed in my work and who c ontributed to my success at the University of Florida.

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6 TABLE OF CONTENTS page ACKNOWLEDGEMENTS.............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES................................................................................................................ .......10 ABSTRACT....................................................................................................................... ............13 CHAPTER 1 INTRODUCTION..................................................................................................................15 Motivation and Introduction...................................................................................................15 Florida Manatee (Trichechus manatus latirostris)..................................................................16 Hierarchy of Bone.............................................................................................................. .....19 Level 1: Major Components............................................................................................19 Level 2: The Mineralized Colla gen Fibril Building Block.............................................20 Level 3: Fibril Arrays......................................................................................................21 Level 4: Diversity in Fibril Array Organizational Patterns.............................................21 Level 5: Cylindrical Motifs Osteons............................................................................23 Level 6: Solid versus Spongy Bone.................................................................................24 Level 7: Whole Bones.....................................................................................................25 Manatee Bone Tissue............................................................................................................ ..25 2 EXPERIMENTAL METHODS IN BONE MECHANICS....................................................28 Specimen Preparation........................................................................................................... ..28 Cutting and Machining of Bone......................................................................................28 Preservation................................................................................................................... ..29 Potting........................................................................................................................ ......29 Standardized Tests............................................................................................................. .....29 Tensile Testing................................................................................................................30 Shear Testing.................................................................................................................. .31 Ultrasonic Testing...........................................................................................................32 Visual Image Correlation (Speckle Image Photogrammetry).........................................33 Deformation accuracy..............................................................................................34 Mirrored image correlation......................................................................................35 Rotated specimen method........................................................................................35 Fracture Tests................................................................................................................. .36 Single Edge V-Notched Beam (SEVNB) Tests..............................................................36 Chevron-Notched Beam Tests.........................................................................................37 Three Point Bend Tests...................................................................................................37

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7 Quantitative Fractography...............................................................................................37 3 BONE FRACTURE MECHANICS.......................................................................................43 Fracture in Engineering Materials..........................................................................................43 Fracture Testing of Bone....................................................................................................... .48 Validity of using LEFM to Model Bone Fracture...........................................................48 Rising R-curve Approach to LEFM................................................................................49 Elastic-Plastic Fracture Mechanics (EPFM)...................................................................49 Testing Procedures..........................................................................................................50 Anisotropic Fracture Analysis of Bone...........................................................................50 4 OVERVIEW AND ORGANIZATION OF THE EXPERIMENTS......................................54 Overview of the Experiments.................................................................................................54 Pilot Tensile Tests...........................................................................................................54 Tensile Tests.................................................................................................................. ..55 Shear Tests.................................................................................................................... ...56 Off-Axis Tensile Tests....................................................................................................57 Variable Strain Rate Tensile Tests..................................................................................57 Notched Tensile Tests.....................................................................................................58 Numerical Fracture Tests................................................................................................59 5 TENSILE PROPERTIES OF CORTICAL BONE USING MIRRORED IMAGE CORRELATION (MIC).........................................................................................................61 Short Summary.................................................................................................................. .....61 Introduction................................................................................................................... ..........62 Materials and Methods.......................................................................................................... .65 Results........................................................................................................................ .............69 Discussion..................................................................................................................... ..........71 Conclusions.................................................................................................................... .........72 6 ANISOTROPIC ELASTIC TENSILE PROPERTIES...........................................................81 Short Summary.................................................................................................................. .....81 Introduction................................................................................................................... ..........82 Methods........................................................................................................................ ..........84 Results........................................................................................................................ .............87 Discussion..................................................................................................................... ..........87 Conclusions.................................................................................................................... .........90 7 ANISOTROPIC ELASTIC SHEAR PROPERTIES..............................................................94 Short Summary.................................................................................................................. .....94 Introduction................................................................................................................... ..........95 Methods........................................................................................................................ ..........97 Results........................................................................................................................ ...........100

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8 Discussion..................................................................................................................... ........101 Conclusions.................................................................................................................... .......103 8 MATERIAL SYMMETRY ANALYSIS.............................................................................113 Short Summary.................................................................................................................. ...113 Introduction................................................................................................................... ........114 Material Symmetry.............................................................................................................. .116 Hookes Law.................................................................................................................... .....117 Isotropic Materials.........................................................................................................117 Transverse Isotropy.......................................................................................................118 Orthotropy.....................................................................................................................119 Anisotropy.....................................................................................................................120 Transformations in Anisotropic Materials.....................................................................121 Thermodynamic Restrictions on Elastic Constants.......................................................121 Tensile Testing................................................................................................................ ......122 Material Symmetry Analysis................................................................................................123 Results........................................................................................................................ ...........124 Discussion..................................................................................................................... ........125 Example 8-1...................................................................................................................130 Example 8-2...................................................................................................................133 Example 8-3...................................................................................................................135 9 ANISOTROPIC FRACTURE PROPERTIES OF MANATEE RIB (CORTICAL) BONE: A NUMERICAL AND EXPERI MENTAL INVESTIGATION............................137 Short Summary.................................................................................................................. ...137 Introduction................................................................................................................... ........138 Materials and Methods.........................................................................................................139 Results........................................................................................................................ ...........143 Discussion..................................................................................................................... ........144 Conclusions.................................................................................................................... .......147 10 CONCLUDING REMARKS................................................................................................154 APPENDIX A TENSILE TESTING AT VARI OUS CROSSHEAD SPEEDS...........................................157 B MATERIAL PROPERTY CALCULATIONS.....................................................................159 C VISUAL IMAGE CORRELATION (VIC) EQUIPMENT TUTORIAL.............................168 LIST OF REFERENCES.............................................................................................................179 BIOGRAPHICAL SKETCH.......................................................................................................186

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9 LIST OF TABLES Table page 4-1 Tensile tests.............................................................................................................. ..........60 4-2 Shear tests................................................................................................................ ..........60 4-3 Numerical tests............................................................................................................ .......60 5-1 Elastic moduli from MIC analysis. Mean va lues are presented standard deviations along with the number of specimens used to compute the mean values ( n )......................74 5-2 Poissons ratios from MIC analysis. I ndex 1 denotes the [100] direction, index 2 denotes the [010] direction, and inde x 3 denotes the [001] direction................................74 5-3 Summary of known manatee rib material properties.........................................................74 6-1 Tensile properties of the Florida manat ee rib bone. Mean values are presented as a function of specimen orientation.......................................................................................90 6-2 Summary of known manatee rib material properties.........................................................91 7-1 Shear modulus and coefficient of variation ( CV ) from cyclic tests. Mean values are presented standard deviations.......................................................................................104 7-2 Shear modulus of three orthogonal orienta tions of manatee rib bone. Mean values are presented standard deviations along with the number of specimens used ( n )..............104 8-1 Mean tensile and shear prope rties of manatee rib bone obtained from parts I and II of this paper, respectively.....................................................................................................127 8-2 Average load, number of specimens tested, average strains calculated by VIC (experimental)................................................................................................................. .127 9-1 Orientation dependent fr acture toughness and fracture parameters of manatee rib (cortical) bone................................................................................................................ ..148 9-2 Selected studies measuring fracture t oughness of compact bone (modified from Yan (2005))........................................................................................................................ ......149 A-1 Viscoelastic properties of manatee rib bone Three cross-head speeds were tested for their effect on elastic modulus for...................................................................................157

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10 LIST OF FIGURES Figure page 1-1 Florida manatees........................................................................................................... .....26 1-2 Hierarchical levels of bone architecture............................................................................27 1-3 Three-dimensional, sliced view of an osteon.....................................................................27 1-4 Cortical and trabecular bone within a section of long bone...............................................27 2-1 ASTM tensile specimen recommendations.......................................................................38 2-2 Tensile specimen with some bending present....................................................................39 2-3 Iosipescu shear specimen. This specimen is loaded uniaxially in order to generate nearly pure shear in the gage section.................................................................................39 2-4 Iosipescu shear specimens fitted in grips on a uniaxial testing machine...........................40 2-5 Ultrasonic test specimen geometries (A) cube used to obtain 18 velocities; B) shape used to obtain 9 velocities)................................................................................................40 2-7 A) Close-up of the notch in an SEVNB sp ecimen. B) Close-up of the v-notch in an SEVNB specimen..............................................................................................................42 2-8 Chevron-notched (CN) specimen......................................................................................42 2-9 Three-point bend specimen and loading constraints..........................................................42 3-1 Compact tension (CT) specimen dimensions....................................................................52 3-2 Single edge notched bend (SENB) specimen dimensions.................................................52 3-3 Definition of the coordinate syst em associated with a crack tip........................................53 3-4 Three modes of failure..................................................................................................... ..53 5-1 Three-dimensional surface as seen by tw o imaging sensors (courtesy of Correlated Solutions, Inc, West Columbia, SC)..................................................................................75 5-2 Movement of a sample square subset us ed for cross-correlatio n function estimation......75 5-3 Manatee skeleton........................................................................................................... ....76 5-4 Stereographic set-up used in the MIC analysis..................................................................76

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11 5-5 Example image and target region used by VIC3D to generate a dataset and the associated surface plot created by VIC3D.........................................................................77 5-6 Typical dV/dY curve showing all data points on th e gage section of a manatee rib bone specimen.................................................................................................................. ..77 5-7 Nomenclature used for the Poissons ratio analysis. Shaded faces represent the faces on which each respective Poissons ratio was evaluated...................................................78 5-8 Stress-strain curve for a [010] specimen............................................................................78 5-9 Schematic of Proximodistal [100] sp ecimens prepared from the middle 1/3rd of an adult manatee rib bone.......................................................................................................79 5-10 Proximodistal [100] stress-strain curve for three dry manatee rib bone specimens (A, B, and C) loaded to failure.................................................................................................79 5-11 Superficial-Deep [010] st ress-strain curve for three dr y manatee rib bone specimens (A, B, and C) loaded to failure...........................................................................................80 5-12 Craniocaudal [001] stress-strain curve fo r three dry manatee rib bone specimens (A, B, and C) loaded to failure.................................................................................................80 6-1 Articulated manatee skeleton.............................................................................................91 6-2 Visual image correlation performed on the gage section of a manatee rib bone tensile specimen....................................................................................................................... .....92 6-3 Plot of the deformation versus position data ( dv vs. dy ) in the gage region of a tensile specimen....................................................................................................................... .....92 6-4 Stress-strain plot for a given load step during a VIC analysis...........................................93 7-1 Three-dimensional surface as seen by two imaging sensors...........................................104 7-2 Movement of a sample square subset us ed for cross-correlat ion function estimation in the VIC analysis...........................................................................................................105 7-3 Manatee skeleton........................................................................................................... ..105 7-4 The middle third of each rib was used fo r the analysis to re duce potential sitespecific variability between specimens............................................................................106 7-5 Orientation and sectioning of the shear specimens used in this study.............................107 7-6 Iosipescu specimen dimensions. The length of the G23 specimens was limited by the physical dimensions of the manatee rib cross section.....................................................107 7-7 Iosipescu specimen prepared for simu ltaneous shear gage and VIC testing...................108

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12 7-8 Image showing the test section and load ing of an Iosipescu specimen prepared for simultaneous VIC and shear gage testing........................................................................108 7-9 Plots showing the high correlation of the dU/dY and dV/dX data points across the entire test section of an Iosipescu VIC specimen............................................................109 8-1 Specimen orientations used in the present analysis.........................................................128 8-2 Typical vertical position versus vertical deformation ( dV/dY ) curve..............................128 8-3 Surface plots showing the variation in material properties with material axes...............130 8-5 Rotation of the material coordinates to specimen coordinates fo r a specimen lying in the [110] material direction..............................................................................................133 9-1 Manatee skeleton. Center ribs from adu lt manatees were used in this study..................150 9-2 Mode I (opening mode) stress in tensity factor (SIF) nomenclature................................151 9-3 Notch tip geometry showing an illustration of the finite elements and variables used in the fracture toughness models......................................................................................151 9-4 Numerically-determined mode I SIF versus normalized crack front...............................152 9-5 Scanning electron micrographs of typical fracture surfaces from specimens loaded in each of the three principal orthogonal directions.............................................................152 9-6 Schematic (left) demonstrating the pee ling of fibrous layers during the fracture experiments.................................................................................................................... ..153 9-7 R-curve behavior of manatee rib bone de monstrating a trend of increasing fracture toughness with increasing critical crack length...............................................................153 A-1 Viscoelastic properties of manatee rib bone. Results are shown for one proximodistal specimen loaded at three different strain rates.................................................................158

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13 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NUMERICAL INVESTIGATION OF THE FR ACTURE PROPERTIES OF MANATEE RIB BONE USING EXPERIMENTALLY DETERMINED ANISOTROPIC ELASTIC CONSTANTS By Jeffrey Michael Leismer August 2007 Chair: Nagaraj Arakere Cochair: John Mecholsky, Jr. Major: Mechanical Engineering Manatees are docile marine mammals that are inhabitants of the Florida coastline as well as some of the natural springs. They are listed as an endangered species by the U.S. Department of the Interior. The coexistence of humans and manatees in Florid a waterways has contributed to the depletion of the manatee population: collisions between wa tercrafts and manatees are responsible for 25% of all manatee deaths. Previous research at the University of Florida aimed to reduce watercraft-related deaths by relating the work-t o-fracture of whole manatee ribs to the kinetic energy of a small boat traveling at various speeds. The intention was to provide scientific evidence for the reduction of speed limits in manatee populated wa terways. Data collected by the researchers was limited by the scope of their resear ch, and there still remains significant work to be done to understand the response of ma natee rib bones to mechanical loading. The purpose of the present study was to fully characterize the elastic anisotropy of the Florida manatee rib bone in terms of mechanical and fracture properties. Previous research conducted on anisotropic materials shows that neglecting anisotropy (dir ectional dependence of material properties) by modeling a material as isotropic (non-directionally dependent) reduces the accuracy of the analysis. Therefore, it was nece ssary to first establish an appropriate material

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14 model for manatee rib bone (specific aim 1) that could be used to accurately predict the response of the material to failure loading. Stereographi c imaging techniques utilizing visual (digital) image correlation were developed to measure the elastic anisotropic cons titutive constants of manatee rib bone (specific aim 2). These techniqu es were verified by comparison with traditional mechanical testing techniques. Once fully char acterized, experimental fracture tests were performed on the material (specific aim 3). Nu merical models of the fracture specimens were then developed that incorporated experimentally determined anisot ropic elastic constants, failure loads, and critical crack lengths (specific aim 4) to elucidate the fracture properties of manatee rib bone. It was found that manatee rib bone was more anisotropic and less tough than similar classifications of bone from other animals. Fu rther contributing to th e uniqueness of this material, the tissue was previously shown to have a greater mineral density than bone from most other animals, manatee ribs are mostly cort ical bone, and the ribs do not possess a marrow cavity. Thus manatee ribs are brittle, solid struct ures that exhibit a high degree of anisotropy and they have potential to frac ture with relative ease. From an engineering perspective, my study in troduces modeling and analysis tools never before used to study bone fracture. From a c onservation standpoint, the information gathered from this study (with proper interpretation) can be used to help protect the already depleted manatee population by providing additional scientific evidence for restrict ing speed limits of watercraft in manatee populated waterways.

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15 CHAPTER 1 INTRODUCTION Motivation and Introduction The issue of a 3D anisotropic stress intensity factor ( K ) solution was previously addressed by other students in Dr. Nagara j Arakeres research group for va rious materials (single crystal nickel base superalloys (Ranjan, 2005) and foam insulation for the space shuttle external fuel tank (Knudsen, 2006a)), but this is th e first time the 3D anisotropic K solution has been studied for a material as complex as bone. Bone is a compos ite material with a hier archical structure that exhibits a wide range of prope rties at each length scale (Wei ner and Wagner, 1998; Rho et al., 1998). Very little is understood about how bone fails or the means by which hierarchical structures affect failure (Akkus 2004). My analysis generates mo re information regarding the material and fracture properties of bone than what is currently available in the literature and therefore provides a significant contribut ion to the field of bone mechanics. Though a significant body of literature exists on the material properties of human and animal bone, reports of all 9 ort hotropic or 5 transver sely isotropic elastic constants (depending on how the bone is classified) come few and fa r between. My study explores the variation in elastic constants as a function of material orientat ion so that the elastic anisotropy of manatee rib bone can be fully characterized. The methods by which specimens are harveste d, prepared, preserved, and tested greatly affect the results of mechanical tests (A n and Draughn, 2000), while the influences of microstructure on fracture properties found from m echanical testing of bone have only recently been investigated (Malik et al., 2003; Nalla et al., 2003, 2005 a; Vash ishth, 2004). Yan et al. (2006b) presented discussion on the differences in fracture toughness values found from different

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16 species and attributed the differe nces to the microstructure of the bones. The microstructure of manatee rib bone is discussed along with the pr eparation, preservation, an d testing procedures used in my study. The intent is to provide sufficient information for a full and accurate comparison of the results found here to other work reported in the literature. This is a first study of its kind, in that no other researchers have employed the methods used here for the formulation of a 3D anisotropi c stress intensity factor solution for bone. In fact, current techniques used to compute the fracture properties of bone almost always neglect the anisotropy of the material. Therefore, one purpose of this investigation was to determine whether or not the anisotropic assumpti on plays a significant role in the fracture properties of this material. Arakere et al. (2005) f ound that the anisotropic assumpti on plays a significant role in the stress state about a notch in si ngle-crystal materials. It is ther efore hypothesized that fracture properties will likely be affected by the anisotropic assumption as well. An experimental component to this work is provided as a preface to the computational analysis. Experiments are used to determine the elastic constants of compact bone from the ribs of adult Florida manatees. The elastic constants ar e then used as input to finite element (FE) models of fracture specimens. The computationa l analysis provides a predictive framework for the determination of the fracture properties of compact bone based on input of the correct material properties to the numer ical model. Thus the methods re ported in this document have the potential to mitigate the need for experimental analyses to determine fracture properties and stress states in bone specimens and whole bone. Florida Manatee (Trichechus manatus latirostris) The Florida manatee ( Trichechus manatus latirostris ) (Figure 1-1) is listed by the U.S. Department of the Interior as an endangered specie s. This is due in large part to human presence in the manatees habitat. Each year, 25-30% of al l manatee deaths are a di rect result of collision

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17 with watercraft. Of these collisions, 66% of th e deaths are attributed to impact, while the remaining deaths are attributed to wounds caused by propellers (C lifton, 2005). A clear understanding of the mechanical and fracture proper ties of the manatee ri b bone is necessary to determine proper ways of protecting this enda ngered species from impact-related injuries. Researchers at the University of Florid a (Clifton, Golden, Koob, Mecholsky, Reep, and Yan) have been performing experiments with bo ne from the Florida manatee since 1999. Their work has resulted in the publicati on of several peer-reviewed articl es (Clifton et al., 2003; Yan et al., 2006a, b, 2007). Material and fracture prope rty data collected by these researchers was limited by the scope of their research, so a full pict ure of the material and fracture properties of manatee bone had yet to be developed. A significant percentage of manatees die as a result of collisions with watercraft. Therefore, one of the objectives of Dr. Kari Cliftons work was to propose a scientific basis for enforcing watercraft speed limits in manatee-populat ed waters. She was able to relate the energy of impact from a collision between a manatee and a small boat traveling at various speeds to the energy required to fracture a manatee rib. One of Dr. Cliftons co-wor kers, Dr. Jia-Hau Yan, developed a finite element model of manatee bone that could be used to predict stresses at critical locations for a variety of impact loading conditions (Yan, 2002). The techniques used in my study for modeli ng manatee bone further elaborate on the work conducted by Yan (2002). The models include incr eased material property data compared to Yans 2002 model by assuming that the material exhi bits some degree of anisotropy, rather than being isotropic. In the fracture study, notched tensile specimens are loaded to failure and the failure load and critical flaw sizes in the failed specimens are documented. The critical flaw is modeled in a finite element model, and failure loads from the experi mental fracture study are

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18 used as input to these models. Input of experimentally determin ed material properties, failure load, and critical flaw sizes allow for the first time ever the determination of the fracture toughness of manatee rib bone from a numerical model. The predictiv e capacity of the model is analyzed by comparison of the numerical results with experimental results found here and by Clifton (2005) and Yan (2002, 2005). This is the first study to analy ze the anisotropic fracture proper ties of manatee rib bone. To date, manatee rib bone fracture properties have on ly been tested for cracks propagated in the transverse direction of the ri b (Clifton, 2005; Yan, 2005). The transv erse orientation represents the most likely orientation for cracks resulting from a side impact from a watercraft. However, impact can occur at any angle, and it is sti ll not known if other crack orientations are as detrimental as the orientation already tested. The literature supports the idea that cracks propagated in the longitudinal direction of the bone result in a lower value of fracture toughness. Fracture toughness for cracks propagated in the transverse direction of long bones is ~50-100% greater than for longitudinally pr opagated cracks in cortical bone (Nalla et al., 2005b)). Thus the work by Yan et al. (2006 a, b, 2007a) may underest imate the severity of collisions between manatee and watercraft. The 3D anisotropic stress intensity factor formulations implemented by the finite element software is used to determin e which of the tested orientations is the least fracture resistant. In addition to further supporting the idea of enforcing speed limits in manatee zones (Clifton, 2005), the curr ent study lends itself towards the development of more realistic computational models of bone with improved pr edictive capabilities over current models. In a broader scope, it is anticipated th at the methods developed for this study can be extrapolated to analyses of fracture in the skeletal systems of other animal species including humans. The

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19 present work aims at providing a universally acc epted model of bone that uses known material properties to predict the fract ure toughness of any cortical bone specimen for any material orientation. This study aims to build on the work of other University of Florida researchers in several ways including: 1) fully characterizing manat ee bone material properties, 2) evaluating the feasibility of using image-based techniques to measure the material properties of manatee bone, 3) examining the microstructure and critical cr ack lengths of failed tensile specimens, and 4) numerically predicting the fract ure properties of manatee bone. Hierarchy of Bone Bone is a highly complex material, characteri zed by its structural hierarchy. At the very basic level, bone is predominantly comprised of collagen fibrils, hydroxyapatite crystals, and water. The apatite represents the inorganic component of bone, while the collagen is the dominant organic component. The structural arrangement of thes e materials varies depending on the length scale being observed (Figure 1-2), hence the hierar chy of bone architecture. The following sections will outline the seven hierarchi cal levels of bone identified by Weiner and Wagner (1998). Level 1: Major Components Hydroxyapatite crystals are plate-shaped and are on the order of 50 x 25 x 2.5 nm (length by width by thickness). The crysta ls exhibit hexagonal symmetry, a nd thus exhibit transversely isotropic material properties. No studies have yet looked at the mechani cal properties of these crystals when extracted from bone. However, th e elastic modulus of la rge single crystals of hydroxyapatite is in the vicinity of 114 GPa. Mineral content of bone and elastic modulus are positively correlated to one another, although the energy to fracture decreases with increasing

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20 mineral content (Currey, 1990). The dry weight pe rcentage of hydroxyapatite in bone is typically 66%, while the crystal accounts for roughl y of the volume (An and Draughn, 2000). Type I collagen fibrils comprise the la rgest portion of organic components in bone, although there exist over 200 other ty pes of proteins within bone. Each fibril is between 80 and 100 nm in diameter. The collagen fibrils are co mprised of three polypept ide chains that are wound into a triple helix. The helix is cylindrical in shape with a diameter of 1.5 nm and length of 300 nm. The fibril structure is orthotropic. Mechanical properties of individual fibers are difficult to obtain, since the fibrils exist in gr oups (called fibers) in bi ological materials. The properties of fibers vary dependi ng on the organization of fibrils within the fiber, which is different from one tissue to the next. Collagen accounts for ~1/2 of the volume of dry bone, weighing in at ~1/3rd the dry weight of the tissue (An and Draughn, 2000). Water exists within and between fibrils as well as between fibers. Studies have shown dramatic differences in the properties of hydrat ed versus dehydrated bone (Currey, 1990; Yan, 2005). Generally, the percentage of water in bone fo r various species is inversely proportional to the mineral content, while the co llagen content is fair ly constant for most species. Evans (1973) references reports of water cont ent ranging from a low of 9.49% in the tibia of cattle to a high of 73 volume % in the tibia of young dogs. Neuman and Neuman (1957) found that newly forming bone has a water content of 60%, while sen ile bone has a water content of only 10%. Level 2: The Mineralized Collagen Fibril Building Block At the mineralized collagen fibril building bloc k length scale, the stru cture is a plateletreinforced fibril. Here, hydroxyapati te crystals are arranged in laye rs inside of collagen fibrils. The layers align with the long axis of the elliptical cross-section of the crystals. The arrangement of crystals within the fibrils makes the fibrils orthotropic. As the crystals grow, they compress the triple-helical polypeptide chains, which provide an organic matrix in which the crystals are

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21 embedded. Hydroxyapatite also exists outside of th e fibrils. The mineral crystals that exist outside the fibrils exhibi t little order in their st ructural arrangement. Level 3: Fibril Arrays Mineralized collagen fibrils usually exist in bundles. The nature of how individual fibrils are arranged relative to one another in a bundle is poorly understood. One method of determining the structural arrangements of these bundles w ould be to assess their material properties. Orthotropic properties would coincide with an or dered arrangement of fi brils within the bundle, while transversely isotropic pr operties would imply a random ali gnment of fibrils within the bundles. Fibril bundles can be measured at the millim eter length scale, although level 3 generally pertains to bone structures at or below the micron length scale. Tendons are examples of fibril bundles. Mineralized tendons have modulus va lues ranging from 162 to 825 MPa, while demineralized tendons have a much lower modulus (67 to 103 MPa). Ten dons generally are less than 50 wt% mineral. Parallel-fibered bone is another structure comp rised of mineralized collagen fibrils. Their mineral weight percent is on the order of 65. Hardne ss tests indicate that th is bone is orthotropic in nature. Parallel-fibered bone has an elastic modulus of ~26 GPa along its long axis, while the transverse orientations have a modulus of only ~11 GPa. Level 4: Diversity in Fibril Array Organizational Patterns Several patterns exist for fibril arrays at le vel 4 of bone structural hierarchy. Pattern 1 is categorized as an array of para llel fibrils. These parallel fibers exist in tendons, and they are especially prevalent in minera lized form at tendon attachment points to bone. Parallel-fibered bone grows rapidly, leaving spaces between each layer of newly formed bone, where lamellar

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22 bone will be subsequently laid. The combination of parallel-fibered and lamellar bone is called fibrolamellar (plexiform) bone. Poorly arranged, loosely packed fibril bundles comprise the woven fiber structure of pattern 2. Both the matrix and the mineral are disorganized in woven bone. The disorganization of woven bone yields anisotropi c material properties rather th an isotropic properties. Woven bone is rapid in growth, and is suitably found at fracture sites where rapid bone growth is necessary. Fibril bundles and fiber bundles can exist in plywood-like layers, hence the name for pattern 3: plywood-like structures. The simplest structure has alternat ing orthogonal parallel fibril arrays. However, the thickness of each laye r is not always uniform, although layers usually repeat in a pattern of thick th en thin, creating a ba sic repeating unit. A more complex plywoodlike structure has five layers of parallel fibrils oriented at ~30 degrees from one another. This structural unit is not symmetrical since there are five, rather than six, layers. Within each layer exists an internal crystalline structure. Most of th e layers have a crystal layer that is parallel to the boundary of the fibril array, while a couple of the layers have a rota ted axis relative the boundary. The rotations of these laye rs is in one direction, thus th e lamellar unit is asymmetrical. Lamellar bone also exists in the form of cylinders, called osteons. Lamellar bone is less anisotropic than plexiform bone (anisotropi c ratio of 1.13-1.21 for lamellar bone versus 1.181.47 for plexiform bone). Pattern 4 represents radial fibril arrays. Thes e arrays are generally formed in dentin, the material at the inner layer of the teeth. Here, the fi brils lie in the plane parallel to the pulp cavity surface where dentin formation takes place. There is a poor organization of fibrils within this plane. Dentin is highly anisotropic in structur e due to the organization of collagen fibrils,

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23 although material properties reveal th at dentin is nearly isotropic. This reveals that the material properties are more dependent on the poor organi zation of crystals than on the structural organization of the collagen fibrils. Level 5: Cylindrical Motifs Osteons The long cylindrical lamellar bone structures th at are preferentially aligned along the axes of bones are called osteons (Figure 1-3). At the center of osteons are hollow tunnels that function as blood vessels. Capillary-like tunnels called canalicu li stem from the hollow center of the osteons. The canaliculi house bone cells called oste ocytes. Osteocytes can remain as osteocytes for the duration of their life, or they can turn into osteoclasts or osteoblasts to meet the remodeling needs of the surrounding tissue. During the process of remodeling, bone eating cells called osteoclasts eat old or damaged bone, creating a tunnel for the bone forming cells (osteoblasts) to lay new bone. The new bone is laid in layers, creating th e lamellar structure of osteons. As the newly formed osteonal bone ag es, its mineral content increases. Osteons involved in the process of remode ling are called secondary osteons to differentiate them from the similarly-shaped structures f ound in the spaces between the pa rallel-fibered bone in primary plexiform bone. Discussion on the stasis of bone becomes important at this level. Homeostasis occurs when the rate of bone formation equals that of bone absorption. If the rate of absorp tion exceeds that of formation, bone becomes structurally deteriorat ed. Osteoporosis, a disease characterized by low bone mass and structural deterioration of bone tissu e, is the result of inadequate bone formation in response to bone absorption. Various drugs are av ailable to prevent or slow the rate of bone absorption, resulting in a net gain of bone tissu e. However, there is still some debate over whether bones structural integrit y is compromised when newly fo rmed tissue is laid in voids without the repair of old, damaged tissue.

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24 Mechanical influences on the skeleton are also observed at this level of bone hierarchy. Mechanical loading governs where bone remodels while hormones determine when it remodels. Osteons tend to align themselves with the pr incipal stress direction, which has implications pertaining to the strength and dama ge resistance of osteonal bone. Transversely oriented cracks in bone tend to kink at the cement line interface between an osteon and surrounding bone tissue, while longitudinal cracks are able to propagate more freely. The cem ent line interface also acts as a barrier to crack propagation by blunting the tip of an approaching crack. The reduced mineral content of newly laid osteonal bone relative the surroundi ng mineralized matrix makes it difficult for cracks to ini tiate on the osteon. Thus, os teonal bone is both resi stant to fracture and fatigue damage. Level 6: Solid versus Spongy Bone Cortical bone comprises the outer surface of bones, while trabecular bone tends to be found internal the cortical outer layer of whole bones (Figure 14). Trabecular bone, also known as spongy or cancellous bone, is named for its por ous appearance. Trabecular bone porosity is generally in the 75 to 95% range (Yan, 2005). Trabeculae are rod or plate shaped structures that form the structure of cancellous bone. The streng th of trabecular bone is attributed to both the mineral content (~60%) and the architecture of the tissue (~ 40%) (Kleerekope r, 2006). Solid bone is referred to as compact, or cortical, bone. Cortical bone e xhibits very few pores. From its appearance one would not anticipate a great deal of anisotropy in cortical bone. However, the orientation of cortical bone constituents resu lts in macroscopic anisot ropy of its material properties. The elastic modulus of trabecular bone is on the order of 1 GPa, while that of cortical bone is ~20 GPa in the longit udinal direction (Cowin, 2002).

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25 Level 7: Whole Bones At the largest length scale, whole bone is characterized in terms of its shape and function. Long bones, such as the femur and humerus, ar e used primarily for mobility and for support. Their internal structure is g eared for a high strength-to-weig ht ratio. Flat bones are used primarily for protection of organs. The skull is an example of a flat bone. Irregular bones such as the scapula (shoulder bone), vertebra, and rib can be used for support and/or protection (e.g., the vertebrae protect the spinal cord from injury while at the same time providing support for our upper body). Manatee Bone Tissue Manatee rib bones are mostly primary pl exiform bone (Clifton, 2005), which is the predominant type of bone found in large land ma mmals. The Florida manatee is unique in the marine mammal kingdom in that it exhibits pach yostosis, characterized by thickening of bone tissue, replacement of cancellous bone with compact bone, and lack of a marrow cavity, while other marine mammals tend towards lower bone ma ss and density (Yan et al., 2006a). In addition to a more solid, dense bone structure, the minera l content of manatee rib bone is quite high (69% +/2%) (Clifton, 2005) compared to other mamm als. The high density may help the manatee overcome buoyancy (Ray and Chinsamy, 2004) and it is responsible for the material response to loading being highly linear up to failure. The highly linear response to loading provides a reasonable argument for the use of linear elastic fracture mechanics (LEFM) to model fracture in manatee rib bone. The large size of manatee bone make it ideal for testing the fracture behavior of the material, since specimen sizes can be la rge enough to satisfy standard fracture testing protocols that require plane stra in conditions to be met (e.g., ASTM E399). Also, the large size and the lack of a marrow cavity a llow for the preparation of larg e specimens oriented about all material axes required to capture orthotropic mate rial response to loading. The large size of the

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26 bone also allows for more specimens to be cu t from a single bone, thus reducing the natural variability between animals and between anat omical sites when aiming for statistically significant values of bone properties. De Buffrenil and Schoevaert (1989) found that the dugong, a close rela tive of the manatee in the marine mammal world, has a higher bone mineral density and mineral content at the anterior (cranial) ribs than at th e caudal ribs. It is expected that this trait is also present in the Florida manatee. The variation in mineral density by location means that spec ific subsets of ribs must be selected in order to reduce variability in mechanical test results. This will be discussed in more detail in later chapters. Figure 1-1. Florida manatees.

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27 Figure 1-2. Hierarchical leve ls of bone architecture. Figure 1-3. Three-dimensional, sliced view of an osteon. Figure 1-4. Cortical and trabecular bone within a section of long bone. Trabecular bone Cortical bone Macrostructure Microstructure Sub-microstructure Nanostructure Sub-nanostructure Osteon Haversian canal Lamella Collagen fiber Collagen fibril Collagen molecule 10-500 m 3-7 m 0.5 m 1 nm Bone crystal

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28 CHAPTER 2 EXPERIMENTAL METHODS IN BONE MECHANICS Specimen Preparation An and Draughn (2000) provide a detailed desc ription of specimen preparation techniques and considerations. Their work can be referenced for additional details not reported here. The discussion in this chapter is limited to specimen preparation procedures re levant to the current study. The techniques provided below refer to macr oscopic bone specimens in level 6 of the hierarchy of bone discussed in Chapter 1. Cutting and Machining of Bone Once bone is excised from an animal, it can be further cut into manageable sections. Rough cuts can be made using a band saw, hacksaw, or jigsaw. Efforts should be made to hydrate and cool the specimen while making any cu t, since each of thes e factors can greatly affect material properties of th e specimen. Water or saline can be used to cool and hydrate the bone. If the tools used to cut or machine the sp ecimen are not resistant to water damage or corrosion from saline, then 1 to 2 mm of bone tissue should be removed from the cutting/working surface using wet sand pape r (An and Draughn, 2000). Diamond-impregnated wire saws or low speed diamond-impregnated wa fering saws reduce specimen damage compared to other types of cutting tools, and they can be used for finer cuts that do not need to be sanded or polished. A diamond coring tool can be used for cylindrical specimens. Vertical milling machines and lathes can be used to mill and tu rn or face specimens, respectively. The specimen should be irrigated with water or saline when using a mill or a lathe.

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29 Preservation Drying, heating, freezing, embalming, or stori ng bone in saline or alcohol solutions can alter the material properties of bone. It is best to use freshly harvested bone when possible. If it is not possible to harvest fresh bone, it is recommen ded that the bone be wr apped in plastic wrap and stored in airtight ba gs in a freezer at -20 Celsius (An and Draughn, 2000) Properties will be best maintained with the muscles and soft tissues attached, although hydrating with saline solution after wrapping the bone in gauze is a reasonable alternative if storing the bone for shorter periods of time. Potting Bone specimens can be potted in polymet hylmethacrylate (PMMA) or epoxy to provide a better gripping surface for the specimen grips. The gripping surface should first be made free of fat and bone marrow using a water jet and soaking in trichloroeth ylene or a solution containing 10% bleach and 90% water. The surface should be allowed to dry before potting. Cyanoacrylate cement should be applied in several layers to improve the bond between the bone and the PMMA when specimens are subjected to large loads. Alternatives are availa ble that make use of specimen geometry to remove the need fo r potting materials (e.g., tapered ends, milled grooves/slots, etc.). Standardized Tests Mechanical test results can be greatly influenced by the specimen preparation techniques and testing methods. Therefore, st andardized protocols should be closely adhered to in order to improve the accuracy of results. Some limitations exist for bone specimens prepared according to standardized tests for engineer ing materials, since size, grip ping methods, and relatively low failure load levels often prevent standardized procedures from being adhered to (An and Draughn, 2000). The American Society for Testi ng and Materials (ASTM) has prescribed

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30 standardized tensile testing procedures (ASTM C565, D1623, D3039, D3044, E8, and E132), a shear testing procedure (ASTM D143), and fr acture testing procedures (ASTM E399, E561, E1820, and C1421, etc.), amongst other tests. Thes e procedures are always being updated and new standards are being put in pl ace in order to meet the demands of materials testing research. For example, although ASTM E399 is the standard for fracture testing of engineering materials, recent evidence suggests that ASTM E399 may not be appropriate to fully characterize the fracture properties of some materials with natural toughening mechanisms (Wallin, 2005). Tensile Testing Tensile tests are used to find the elastic modulus and Poisso ns ratio of a material. Dogbone shaped specimens are recommended by ASTM for tensile tests. These testing procedures mandate the following dimensional relations: 1) d/ D (gage width to grip width) ratio of 0.5, 2) gage length A ~3d, 4) grip length M = L/4, and 5) large fillet radius, R (Figure 2-1) (An and Draughn, 2000). A reduced gage section is introduced to guarantee that the largest values of strain occur at the central portion of the specimen. Strain gage s or extensometers are attached to the gage section to measure strain and de formation in the region, respectivel y. Strain gages are attached to a specimen using an adhesive and must be orient ed appropriately to capture the desired strain reading. There are several types of strain gages, which can be classified by their foil pattern and other parameters. The most common foil patter n is found on the uniaxial strain gage, which provides an average value of stra in in one direction over the foil region on a strain gage. The foil is oriented appropriately to capture strain in a particular direction. The elastic modulus of a specimen can be found when a foil gage is oriented along the loading axis and the load acting on the specimen is measured (e.g., w ith a load cell). When a uniaxia l force is applied, the load can

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31 be divided by the specimen cross-sectional area at the gage section and divided by the strain in the loading direction to obtain the elastic modulus (Equation 2-1). F A E (Eq. 2-1) where is the stress, F is the applied load, A is the cross-sectional ar ea of the specimen gage section, and is the strain measured by the strain gage A biaxial strain gage can be used to determine the Poissons ratio of a material. Biax ial gages measure strain in two (perpendicular) directions, thus the ra tio of the strains can be found (Equation 2-2). 2 12 1 (Eq. 2-2) It is good practice to place a strain gage on both the front and the back of a specimen to account for any bending that may be present in the specimen during testi ng. The test fixture can often induce bending if precautions are not made to insure proper specimen alignment with the loading axis of the testing machine. Figure 22 shows a tensile specim en with some bending present due to specimen misalignment. Averag ing the strains from gages mounted to two opposite faces of a bend specimen will remove th e contribution of bending stresses to the average strain. Thus, the average measured strain is that due only to tensile loads and not to bending loads. Shear Testing Shear specimens are generally rectangular in shape and contain a slot or groove on either side of the gage section (Figur e 2-3). Nearly-pure shear can be obtained in anisotropic materials when loading Iosipescu specimens on a uniaxial m achine. Graphical depictions of two Iosipescu test fixtures are provided in Figure 2-4. These fixtures remove the translational and twisting

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32 motion of the upper and lower mounts caused by 3D stresses generated in anisotropic materials during testing, and thus Iosipescu shear testing fi xtures like these are more useful for generating pure shear than when using other sh ear test specimens and fixtures. As with tensile tests, strain gages are used to measure the strain in a shear specimen. Specialized gages called shear gages were developed by Ifju (1992) to measure shear across the gage section of a shear specimen. The distribution of shear stress is nonuniform across the gage section. Th erefore, the shear strain is obtained by integrating over the gage region. Shear modulus can also be obtained using cy lindrical (torsion) specimens although shear specimens will be used in this analysis. Equation 2-3 can be used to find the shear modulus when using a torsion test. JG TL (radians), J TL G (N/m2) (Eq. 2-3) where is the angle of twist, T is the applied torque, L is the length, J is the polar moment of inertia, and G is the shear modulus. Ultrasonic Testing Ultrasonic testing of material properties is a non-destructive technique that is capable of providing elastic properties. A si ngle specimen can be used to determine several anisotropic properties, and therefore the technique may be advantageous over mechanical testing techniques. Elastic properties are found by m easuring wave velocity and i nputting known specimen densities into simple formulas. These formulas relate the sp eed of the wave and the apparent density of the material to the elastic constant s of Hookes law. Various specimen geometries can be used to determine material properties. Cube specimens w ith chamfered edges (Figure 2-5) are capable of providing measurement of 18 velocities (three in each of six directions) when ultrasonic waves

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33 are passed through the specimen at various an gles. These velocities provide sufficient information to determine all 9 independent elastic constants of an orthotropic material. However, thin, square-shaped specimens with a chamfer on two opposite corners (Fig ure 2-5) can provide only 9 velocities, and 2 of the 9 independent el astic constants cannot be measured with this geometry (Rho, 1996). Yan (2002) used thin, square specimens to determine a single elastic modulus, Poissons ratio, and sh ear modulus for manatee bone. Visual Image Correlation (Sp eckle Image Photogrammetry) Traditional measures of material properties involve the use of extensometers or strain gages to measure the response of a specime n to loading. These measures provide a representative value of deformation or strain over the gage section of a specimen, thus much of the data across the gage secti on is lost. For some heterogene ous materials, there may be considerable variation in deform ation within the gage section. A full-field deformation analysis is necessary to describe the constitutive rela tions for these non-homogeneous materials. In addition, understanding how every point in the gage section is behaving during a full-field deformation analysis increases the amount of da ta available for material property calculations over traditional measurement techniques. One method to obtain full-field data over a specimen gage section is to use photogrammetry, which provides 3dimensional coordinates of points on an object from 2 images. Photogrammetry uses a technique called spectroscopy, whereby tw o cameras are placed at some distance from one another and focused on the same object. Images from these cameras are taken simultaneously in order to create a ster eoscopic view of the object. Points on the object are then correlated between the images to provi de object feature representation in 3D space. A specialized form of photogrammetry is speckle image photogrammetry (SIP), also known as visual (or digital) image correlation (V IC). Visual image correlation requires that an

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34 array of speckles be applied to an object such that the speckles cont rast the color(s) of the object. In general, gray-scale images are used for th e correlation analysis. Cr oss-correlation (CC) and normalized cross-correlation (NCC) are commonly used methods to determine object features (e.g., points, edges, etc.). Once the images have been correlated for a reference image of an unloaded specimen, the specimen is loaded and more stereoscopic images are taken. The position of the points in the deformed specimen images are compared to thos e from the reference image, and the full-field deformation of the specimen gage section ca n be found. In addition, commercial software packages are available that apply algorithms to further reduce the data to strains, deformation rates, strain rates, etc. Deformation accuracy The accuracy of the full-field deformation analysis is in excess of 1/100th of a pixel when using VIC3D v3.1 (Correlated Solutions, Inc.). For out-of-plane deformation, accuracies vary as a function of the angle between the specimen and the cameras. Out-of-plane accuracy decreases with reduced camera spacing, while accuracy (in general) decreases as the camera/specimen positions deviate from equilateral. It is estimate d that the out-of-plane acc uracy is about the inplane accuracy for angles approaching equila teral (Simonsen, 2006). Strains are considered accurate from 500 to 500% when using VIC3D software. Visual image correlation is a relatively new technique that can be used to determine deformation and strain rate of any point on a spec imen. It is generally ac cepted that cross-head displacement be used for the calculation of deform ation in a specimen as well as to calculate the strain rate of a test. However, specimen slip a nd compliance of the test rig reduce the accuracy of these measures, making VIC a more attractive approach to this measure.

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35 Mirrored image correlation Mirrored image correlation (MIC) is a techni que that was developed for this study to provide full-field information on multiple spec imen faces simultaneously, thus reducing the number of tests required to capture the full constitutive res ponse of an anisotropic body. Mirrored image correlation involves the placem ent of mirrors in strategic locations to simultaneously view the desired specimen surfaces with both camer as. The simplest case of MIC involves the placement of a single mirror behind a specimen. The mirror is placed close enough to the specimen that the speckle patterns on both the reflected and non-reflected specimen images remain in focus. At the same time, the mirror must be located far enough from the specimen in order to view the reflected specime n image with both cameras. A camera lens with a long focal length is recommended when performing MIC. More specimen surfaces must be in view of the cameras than when using SIP alone, causing a need to move the cameras further from the specimen. Also, mirror placement may require special fixtures and careful positioni ng. Thus, the drawbacks of using MIC during SIP include increased set-up time and reduced image resolution. Rotated specimen method Another method that can be used to increase th e amount of data availa ble for calculation of material properties is the rotated specimen me thod (RSM), whereby a specimen with a polygonal cross-section is oriented relative to the cameras to maximize the viewi ng area of each specimen face. For an orthotropic material, a square cr oss-section provides the simplest case whereby material properties can be calculated. Therefor e, discussion in this document is limited to a specimen with a square cross-section oriented at a 45 angle relative to the midline between the cameras.

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36 Similar to MIC, a drawback of RSM is a re duction in image resolu tion compared to nonRSM MIC or VIC, since fewer pixels are associ ated with each face. Thus the combination of RSM and MIC significantly reduces the resolution of a full-field deformation analysis. However, the benefits of viewing all specimen surfaces during the analysis may outweigh the decrease in resolution for certain tests, depe nding on the resolution needs of the particul ar analysis. Should reduced resolution be an issue, higher resolution cameras can be incorporated in the analysis fairly inexpensively to make up for the loss of resolution associated with the combination of RSM and MIC. Fracture Tests The failure of a material is governed by the type of material being an alyzed (e.g., brittle vs. ductile), the microstructure of the material, a nd environmental factors su ch as the temperature and humidity levels at which the fracture test is performed. Several tests have been standardized for the determination of the fracture propert ies of a material (e.g., ASTM E399, E561, E1820, and C1421). Several of these tests and others are described below since results from these tests have been conducted on manatee rib bone and their results will be used for comparison to fracture toughness values f ound during the current study. Single Edge V-Notched Beam (SEVNB) Tests Yan et al. (2006b) used single edge V-notch ed beams (SEVNB) to analyze fracture in manatee rib bone. Although this method is not currently standardized, other authors have successfully applied the method to alumina, silicon nitride, a nd dental ceramics (Kubler, 1997 and 2002; Fischer and Marx, 2002). Figure 2-6 shows the dimensions and orientation of SEVNB specimens used in Yan et al.s study. Figure 2-7 shows a close-up of the Vnotch introduced to a notched bend specimen.

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37 Chevron-Notched Beam Tests Chevron-notched (CN) specimens have seve ral advantages over other methods (Yan, 2006c). Plane strain conditions at the crack tip ar e reached more easily than with other fracture specimens. This allows for smaller specimens to be prepared, thus more specimens can be obtained from a single bone. Also, cortical bone is generally not ve ry thick and testing of three orthogonal orientations of bone is not always possible. Therefore, thicker specimens may not be practical for fracture testing of cortical bone. Pre-cracking a speci men is not necessary with CN specimens, since the shape of the notch allows for the specimen to pre-crack easily during loading. The notches also help guide the crack to prevent crack turni ng from occurring, which often happens in transverse specimens. Also, a sharp crack is formed early during loading, and therefore special techniques are not required to introduce a sharp crack. Failure in CN specimens is stable, whereas crack growth is unstable for most fracture specimen geometries. Figure 2-8 shows a CN specimen along with the notch and appropriate dimensions. Three Point Bend Tests Yan et al. (2006a) studied fract ure in manatee rib bone using three point bend tests with no pre-crack or notch introduced pr ior to loading (Figur e 2-9). Fractography was used to identify the origin of the flaw which caused catastrophic failu re as well as to determine the critical flaw size that led to failure. Correction factors were im plemented to determine the critical stress at the flaw if failure initiated anywhere ot her than at the center of the specimen. Quantitative Fractography Quantitative fractography is a method that locat es the source of failure in a specimen and relates the fracture f eatures to the failure stress (Yan et al ., 2006a). Using this technique, fracture surfaces can be analyzed to determine artifacts su ch as failure mode, crack origin, and critical flaw size and to determine fract ure toughness without the need to introduce pre-cracks into a

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38 specimen. Scanning electron microscopy (SEM), a t ool used in quantitative fractography, uses a focused electron beam to scan the surface of a specimen. There is backscatter of primary electrons and ejection of secondary electr ons during SEM scanning. A positively charged collector attracts the ejec ted secondary electrons and the sign al is amplified and displayed on a cathode ray tube. A conductive coati ng is generally applied to a sp ecimen prior to SEM scanning to allow for sufficient electron ejection and back scatter. However, low-vacuum SEM and field emission (FE) SEM allow for the specimen to be observed without chemical fixation, dehydration, drying and coati ng (An and Draughn, 2000). These te chniques use backscattered electron images that provide both compositi on contrast and contrast caused by specimen topography over a long focal length compared to microscopy techniques. Image analysis software can be used to determine microstructu ral artifacts present in the images produced by SEM, including porosity. Microstr uctural artifacts have been s hown to have a strong influence on fracture parameters in bone (Akkus et al., 2005 ). Therefore it is good pr actice to compare the microstructure of specimens if fractur e properties are going to be compared. Figure 2-1. ASTM tensile specimen recommendations. ASTM Guidelines d/D ~ 0.5 A ~ 3d M=L/4 R = large L A M D d R

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39 Figure 2-2. Tensile specimen with some bending present. Averaging the total measured strain at points A and B of the above specimen rem oves the bending component of strain and we are left with only the tensile strain. Figure 2-3. Iosipescu shear specimen This specimen is loaded uniaxially in order to generate nearly pure shear in the gage section.

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40 Figure 2-4. Iosipescu shear specimens fitted in grips on a uniaxial te sting machine. Image courtesy of Nancy Strickland (Universit y of Florida, Mechanical and Aerospace Engineering). Figure 2-5. Ultrasonic test specimen geometries (A) cube used to obtain 18 velocities; B) shape used to obtain 9 velocities). A B

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41 Figure 2-6. Single edge V-notched beam (SEVNB ) specimen. Reproduced with permission from Yan, J., 2005. Elastic-plastic fracture mechanics of compact bone. Ph.D. thesis, University of Florida, Gainesville, Figure 3-2, p. 30. A

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42 B Figure 2-7. A) Close-up of the notch in an SEVN B specimen. B) Close-up of the v-notch in an SEVNB specimen. Reproduced with permi ssion from Yan, J., 2005. Elastic-plastic fracture mechanics of compact bone. Ph.D. thes is, University of Florida, Gainesville, Figure 3-1, p. 28. Figure 2-8. Chevron-notched (CN) specimen. Figure 2-9. Three-point bend speci men and loading constraints. Load B L W ao

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43 CHAPTER 3 BONE FRACTURE MECHANICS Fracture in Engineering Materials Stress fields have been analyzed in the vi cinity of a crack-tip by many authors for the isotropic case (Westergaard, 1939; Irwin, 1939; Irwin 1949; Williams, 1957; Gross and Srawley, 1964; Sih, 1973). According to Gri ffith, the critical stre ss that causes the crac k to propagate is C = [2E/(a)]0.5 (Eq. 3-1) where E is the elastic modulus and is the fracture surface energy. Orowan (1950) extended Griffiths theory to include ductile materi als by accounting for the energy dissipated by local plastic flow: C = [2E(+p)/(a)]0.5 (Eq. 3-2) where p is the energy due to plastic deformation. Irwin built on Orowans work by in troducing the fracture parameter, G, the strain energy release rate, in 1956. G = -dP/da = dW/da dU/da (Eq. 3-3) where P is the potential energy of the system, a is the crack half length, W is the work done by external forces per unit thickness, and U is the strain energy per unit thickness. Equation 3-3 says that the energy due to plastic deformation mu st be added to the energy associated with generation of new crack surfaces. The strain energy release rate is a measure of the change of elastic energy per change of crack length. Alternatively, G can be expressed as

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44 G = 2(+p). (Eq. 3-4) Irwin next introduced the stress intensity factor, a measure of the intensity of the stresses in the vicinity of a crack (Irwi n, 1957). According to ASTM E-399, K can be determined from the following equation: K = P*Y(a/W)/(B*W0.5) (Eq. 3-5) where P is the load, B is the specimen thickness, W is the width, and Y is the shape function, which is also known as the geometry factor The shape function can take several forms depending on the specimen and loading scenario. YCT = (2+a/W)/(1-a/W)3/2{.866 + 4.64(a/W) 13.36(a/W)2 + 14.72(a/W)3 5.6(a/W)4} for the compact tension (CT) sp ecimen (Figure 3-1), and YSENB = 3*S/W*(a/W)0.5/[2*(1 + 2(a/W))*(1 (a/W))3/2]*{1.99 (a/W)*(1 (a/W))*[2.15 3.93(a/W) + 2.7(a/W)2]} where S is the spacing between reaction loads in a single edge notched bend (SENB) specimen (Figure 3-2). The elastic stresses near the tip of a crack can be found as a function of K using the following relation: ij() = K*fij()/(r)0.5 + higher order terms (Eq. 3-6) where ij are the component stresses, and r are defined in Figure 3-3, fij is a dimensionless function of and the higher order terms reduce to zero or remain finite as r approaches zero, while the leading term appro aches infinity (Anderson, 1991). The critical stress intensity factor of a mate rial is also known as th e fracture toughness. The fracture toughness, Kc, of a material is defined as the critic al SIF at or above which stable crack

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45 propagation occurs. The fracture toughness of a ma terial depends on load, crack geometry, and specimen thickness, but it is independent of mate rial properties. Fracture toughness is calculated as KC = YC(aC)0.5, (Eq. 3-7) where Y is the yield strength and C is a geometry factor and aC is the critical crack length leading to failure. Opening mode fracture toughness is denoted as KIC. Here, the subscript I tells us that the crack is subjected to mode I loading. The thr ee types of loading a crack can experience are summarized in Figure 3-4. Mode I loading signifies that loads are applied normal to the crack plane, thus causing a crack to open. Mode II lo ading is the shearing mode, whereby in-plane shear loading slides one crack face relative the op posite face. Mode III loading results in tearing due to out-of-plane shearing. The stress intensity factor approach to fract ure mechanics requires an analysis of the stresses about a crack-tip, which presents some di fficulty for anisotropic materials. To date, there are very few analysis tools capable of computing stresses about a crack tip in an anisotropic material such as bone. Therefore, other approach es are generally used to analyze fracture in bone. The strain energy release rate approach is equa lly valid for isoand aniso-tropic materials. The strain energy release rate approach is based on the crack tip ope ning displacement method, whereby the crack tip displacements are related to the strain energy release rate (Gibson, R., 1994). The strain energy release rate of a material is not influen ced directly by microstructural features of a material, and thus provides an adequate representa tion of the global behavior of a

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46 material with complex microstructure such as bone (Norman, 1995). For an isotropic material, the strain energy release rate, G, as it relates to K is given as G = K2/E (Eq. 3-8) where E = E/(1-2) for plane strain, E = E for plane stress, and is Poissons ratio. Since G is a scalar, this relationship between G and K can be further delineated in to individual loading modes with the following equation (Anderson, 1991): G = KI 2/E + KII 2/E + KIII 2/(2 (Eq. 3-9) where is the shear modulus. Note that later discussions refer to shear modulus as G, while it is reported as in this chapter to differentiate it from strain energy release rate. For an anisotropic material the relationship between K and G is given by Liebowitz (1968) and Sih (1965) as G = K2(b11b22/2)0.5[(b22/b11)0.5+(2b12+b66)/(2b11)]0.5 (Eq. 3-10) where b11 = (a11a33-a2 13)/a33, b12 = (a12a33-a13a23)/a33, b22 = (a22a33-a2 23)/a33, b66 = (a66a33a2 36)/a33 for plane strain, and b11 = a11, b12 = a12, b22 = a22, b66 = a66 for plane stress. a11 = 1/E1, a13 = -13/E1, = -31/E3, a22 = 1/E2, a23 = -23/E1, = -32/E3, a33 = 1/E3, a12 = -12/E1 = -21/E2, a36 = 0 (due to elastic symmetry), a66 = 1/G12.

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47 Another method used to evaluate fracture is the J-integral approach (Rice, 1968a, b), which uses a path-independent line integral that measur es the strength of the singular stresses and strains near a crack tip. For the 2D case, J is defined as J = Wdy (Txu/x + Tyv/y)ds (Eq. 3-11) where is any path enclosing th e crack tip (Figure 3-5), W is the strain energy density (W=ijij/2), Tx is the traction vector along the x-axis (Tx=xnx+xyny), Ty is the traction vector along the y-axis (Ty=yny+xynx), is the component stress, n is the unit outer normal vector to path u is the x-direction displacement vector, v is the y-direction displacement vector, and s is the distance along the path The J-integral is related to G, where for small scale yielding, J = K2/E (Eq. 3-12) where E is the same as in Equation 3-8. Furtherm ore, by comparison with Equation 3-12, it can be noted that J = G for the linear range of a material, and that JC = GC for materials that exhibit linear elastic behavior up to failure. A limitation with the J-integral approach is that the individual Ks cannot be delineated for the mixed mode crack problem (Knudsen, 2006). Th erefore, another approa ch is necessary to find individual Ks. The present study will overcome the inhe rent hurdles associated with finding an anisotropic crack-tip stress solution by implementing a numerical technique developed specifically for this type of problem by the Co rnell Fracture Group (FRANC 3D, Cornell Fracture Group, Ithaca, N.Y.). The Cornell Fracture Gr oup developed a software program called FRANC3D, which is an acronym for Fracture Anal ysis Code 3D, to simplify the analysis of fractures. The mathematical formulations used by FRANC3D are omitted from this document

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48 due to their lengthy derivations. The reader is encouraged to review Knudsen (2006) for more information on this topic. Fracture Testing of Bone There has been a significant am ount of work done in the area of fracture testing of bone dating back to the early 1970s. Mo st often reported in the literature are values of fracture toughness (KC) and critical strain energy release rate (GC). However, there is much debate in recent times on the validity of using KC to describe bone fracture, since recent work by Nalla et al. (2005) demonstrated the importance of repor ting fracture resistance as a function of crack length (resistance-curves or R-curv es for short) as opposed to re porting a single value of fracture toughness. In addition, many of the models us ed to describe bone fracture are limited by isotropic assumptions, which is a major source of error when it comes to modeling anisotropic materials such as bone. Another major problem in the literature is th e lack of orientation dependent properties reported. Mo st authors report fracture to ughness for a single orientation of cortical bone, because the animal model chosen does not support large specimen sizes to be harvested for all orientations. Bone fracture literature can be segmented into several categories based on the animal species studied, the age of the animal from wh ich the bone was harveste d, the type of bone analyzed (cortical vs. trabecular), the level of bone hierarchy selected for analysis, and the location from which the bone was harvested. Thes e categorizations are necessary due to the extreme variation in material and fractu re properties between each category. Validity of using LEFM to Model Bone Fracture Fracture testing of bone is most often done according to ASTM E399 (Standard test method for plain strain fracture toughness tes ting of metallic materi als) and ASTM E561 (Standard practice for R-curve determination), wh ich were designed for linear elastic materials.

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49 The linear elastic assumption is important, since K and G methods are based on LEFM. There is some debate over the validity of the linear el astic assumption for bone, since the viscous nature of the material results in some degree of non-linearity prior to fracture. However, LEFM remains the standard in the literature for the de termination of bone fracture properties. Rising R-curve Approach to LEFM The rising resistance curve (R-curve) appr oach to LEFM accounts for materials which display microstructural tougheni ng mechanisms. The toughening mech anisms are responsible for increasing fracture resistance, KR, with increasing crack length. Note that KR is equal to KC, but it is differentiated when referring to R-curve ma terials. Toughening mechanisms include, but are not limited to crack turning at th e cement line interface between an osteon and the interstitial matrix which surrounds the osteon, crack tip blun ting at the weaker cement line interface, microcracking, crack bridging by uncracked ligamen ts, crack bridging by collagen fibrils, osteon pull out, and osteon micro-rotation. Several m odels of these micros tructural toughening mechanisms have been implemented in bone fr acture mechanics studies to account for their relative contribution to the fr acture toughness of human cortical bone (Nalla et al., 2005). Elastic-Plastic Fracture Mechanics (EPFM) Elastic-plastic fracture mechanics (EPFM) are equally valid for both elastic and inelastic materials. The J-integral approach is widely used for the determination of elastic-plastic fracture properties, and it was recently implemented by Yan (2005) to model fracture in manatee and bovine cortical bone. Yan (2005) found that the en ergy spent propagating a crack in the plastic regime was greater than the energy expended in the linear elastic range of bovine bones. This calls to question the validity of the linear elastic assumption of LEFM models of bone.

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50 Testing Procedures In brittle materials, such as bone, it is difficult to cause failure in Modes II and III, thus Mode I is the main mode of failure generally reported (e.g., KIC). Testing procedures used to determine KIC generally involve monotonic loading of a notched or pre-cracked specimen under constant load or displacement rate. The resul ting load-displacement curve is used in fracture property calculations. The curve tends to be line ar at start and deviates from linear prior to catastrophic crack propagation. The st ress intensity factor or strain energy release rate can be calculated from the load-displacement curve and a geometry factor which is a function of crack length and specimen geometry (Equation 3-5). Compact tension (CT) specimen geometries predominate the wealth of K and G experiments since the specimens are in accordance with standard test tech niques and there exists much control over the loading ra te (Akkus et al., 2004). However, other fracture tests are often employed as well. Several researchers have used three-point bend tests to determine the fracture toughness of cortical bone (Nalla et al., 2005b; Clifton, 2005; Yan, 2005). Yan (2005) used single-notched four-point bend test s, while Nalla et al., (2005b) used double-notched four-point bend tests to analyze fracture properties of cortical bone. Compact sandwich specimens (CSS) are sometimes used for smaller specimens that can not be analyzed using CT tests (Wang et al., 1996). Anisotropic Fracture Analysis of Bone A substantial body of literature exists on the fracture mechanics of bone. The bulk of the literature is confined to reports of fracture properties for a si ngle mode of fracture for one orientation of bone (Cooke et al., 1973; Pope and Murphy, 1974; Wright and Hayes, 1976; Wright and Hayes, 1977; Alto and Pope, 1979; Behi ri and Bonfield, 1980; Behiri and Bonfield, 1984; Moyle and Gavens, 1986; Bonfield, 1987; Norman et al, 1991; Norman et al., 1995;

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51 Zioupos and Currey, 1998; Yeni and Norman, 2000). Mo st of the existing literature neglects the anisotropic nature of bone, a nd hence, valuable information on orientation-dependent fracture properties is not always presented. There exists only a small body of literature that investigates anisotropic fracture properties of cortical bone (Bonfield, et al ., 1985; Bonfield, 1987; Behiri and Bonfield, 1989; Hoffmeister et al., 2000; Lucksa nambool et al., 2001; Nalla et al., 2005b). Nalla et al. (2005b) found KC in human cortical bone to be between 51-140% greater for cracks propagating in the transverse di rection relative long bones than for the cracks propagating in the longitudinal direction. The increase in transverse orientation toughness is the result of a ~90 kink, or deflection, in the cracks path. Yeni and Norman (2000) found that cracks can deflect from a transverse direction of propagation to grow along the cement lin e interface between an osteon and the surrounding mineral matrix. The cem ent line acts as a path of least resistance, causing the crack to redirect away from the nominal path of maximum stress. Akkus and Rimnac (2001) excluded from fr acture toughness calculations fatigue cracks that delineated from initial crack growth di rection by more than 20% since the fracture toughness formulation they used (Zeng and Dai, 1994) assumes that cracks remain straight. Crack deflection can lead to e rroneous calculations of fracture toughness if the crack deviates substantially from its initial growth direct ion (ASTM E-399), therefore other measures of fracture are generally obtained fo r the transverse orientation (N alla et al., 2005b). Nalla et al. (2005b) used work-of-fracture, Wf, as an alternative measure of toughness. Work-of-fracture is obtained by dividing the area under the load-displ acement curve by two times the nominal crack surface area. This fracture evaluation method has its limitations, since Wf is size and geometrydependent.

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52 Alternatively, several authors have placed gr ooves along the gage section of transverse fracture specimens to remove th e crack kinking effect (Behiri a nd Bonfield, 1989; Norman et al., 1992; Malik, 2003). The components of the fract ure toughness equation are modified according to Malik (2002) in order to account for the effects of the groove (Equation 3-13) KR, grooved = PY/[(BBn)0.5W0.5] (Eq. 3-13) where Bn is the thickness measured in the groove s and the other components correspond with those of Equation 3-5. In summary, bone is a highly complex materi al that requires special treatment when analyzing its properties. Standard test procedures for engineeri ng materials can be modified to accommodate the unusual characteristics of bone. Figure 3-1. Compact tension (CT) specimen dimensions. Figure 3-2. Single edge notched be nd (SENB) specimen dimensions. a W 1.25 W P P W a P S P/2 P/2

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53 Figure 3-3. Definition of th e coordinate system associated with a crack tip. Figure 3-4. Three modes of failure: Mode I is the opening mode, whereby loading occurs normal to the crack plane; Mode II is the shea ring mode, whereby in-plane shear loading slides one crack face with respect to the other; Mode III is the out-of-plane shearing mode, also known as the tearing mode. Figure 3-5. Path, enclosing the crack tip used for the J-integral formulation. r x yCrack tip yy xx yx xy

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54 CHAPTER 4 OVERVIEW AND ORGANIZATION OF THE EXPERIMENTS This chapter concisely describes the experime nts which are presented at greater length in the remaining chapters in order to put the experiments into pers pective. Tables 4-1 through 4-3 provide information on each of the experiments pe rformed in this study such as the number of specimens analyzed and the output variables from each experiment. The remaining chapters are organized as papers for publica tion in refereed journals with the exception of Chapter 10 which provides the concluding remarks of this dissertation. As such, some of the material from the introductory chapters (1 through 3) and the remain ing chapters (4 through 9) will be repeated in the remaining chapters to allow for each chapter to serve as a stand-alone article. An appendix appears at the end of this documen t to present work not included in the remaining chapters, since the material falls outside the sc ope of the articles or exceeds page limitations of the journals to which the remaining chapters will be submitted. Overview of the Experiments Pilot Tensile Tests The initial experiment was a tensile test conducted on 9 specimens oriented about 3 orthogonal axes (n=3 for each orientation). It was unknown how well the speckle pattern could be applied to wet bone specimens, so the pilot te nsile test specimens were air-dried at room temperature in an air-conditioned room (for 2 weeks) prior to testing. The specimens were loaded to failure at a constant load rate (1 mm/min). Mirrored image correlation (MIC) was used to capture deformations on the front and back speci men gage sections while a load cell was used to capture load data during the experiment. Outputs from these tests included three elastic moduli (for the proximodistal, superficia l-deep, and craniocaudal orientat ions, respectively) as well as six Poissons ratios. This was possible because for a given orientation, two specimen faces (90

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55 from one another) were analyzed. Because of th e limited number of specimens used in this experiment, only one or two specimens were used to determine each Poissons ratio. This pilot study aimed to: Determine the elastic modulus for 3-orient ations of manatee rib bone using MIC. Determine the Poissons ratio for several or ientations of manatee rib bone using MIC. Compare MIC results with those from three-point bend and ultrasound studies. Characterize the anisotropy of ma natee rib bone elastic modulus. Evaluate the ability of the test fixtures to prevent specimen bending by determining if results significantly differ on th e front and back of a specimen. Investigate the linearity of th e material for each orientation. Provide a list of the bene fits and draw backs of VIC relati ve to traditional deformation and strain measurement techniques. Tensile Tests This experiment consisted of loading and unloading 39 wet bone specimens (13 from each orthogonal orientation) within th eir elastic range at a rate of 1 mm/min, then rotating the specimens 90 degrees and repeating the loading a nd unloading cycle. Mirrored image correlation (MIC) was used to determine elastic modulus and Poissons ratios associated with each load cycle. These tests provided elastic modulus in 3 presumably primary or thogonal directions of manatee rib bone as well as Poissons ratios on both pairs of faces for specimens oriented in each of 3 orthogonal directions. The te nsile test study aims were to: Determine the elastic modulus for 3 orient ations of manatee rib bone using MIC. Determine the Poissons ratio for 3 orient ations of manatee rib bone using MIC. Characterize the anisotropy of mana tee rib bone tensile properties.

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56 Shear Tests Shear testing was done in order to determine the shear modulus of three orientations of manatee rib bone. Iosipescu (shear) specimens were used in order to produce nearly pure-shear in the specimen gage sections. Iosipescu specim ens have been evaluated many times over in the investigation of shear properties. Shear gages were thought to be the most accurate way to measure strain in the Iosipescu specimens, and were placed on the front and back of 9 wet bone specimens. Three specimens were oriented to obtain each of G12, G23, and G32, shear moduli for two orthogonal orientations of manatee rib bone (note that G23=G32). Visual image correlation (VIC) was performed over the top of shear gages in order to evalua te the feasibility of using VIC in the determination of shear modulus. Shear stra in from the strain gages was averaged over the front and back of the specimens, while strains were monitored by the VIC system on only one specimen face. Two additional wet bone specimens (G13) each had a shear gage applied to only the front face. VIC and shear gage acquisi tion were performed simultaneously on these specimens. The specimens were loaded in a cyclic manner to determine the repeatability of loading (i.e., was there property degradation with loading and how much variability exists in shear modulus when loading the same specimen several times?). Seven additional (wet) G13 specimens had VIC performed on them in the absence of shear gages. All loading was done at a rate of 1 mm/min. Output from these tests incl uded shear modulus for three presumably primary orthogonal orientations of manatee rib bone. The shear study aimed to: Determine the shear modulus for 3-orientati ons of manatee rib bone using both VIC and shear gages. Determine the repeatability of each technique by performing cyclic tests and looking at the coefficient of variation (CV) for all cycles. Determine if VIC is a viable alternative to shear gages in the calculation of shear modulus.

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57 Off-Axis Tensile Tests Twenty-two wet bone specimens oriented along three off-axis direc tions, [110], [101], and [001], were loaded and unloade d within the elastic range of the material at a rate of 1 mm/min. Specimens were rotated 90 about thei r longitudinal axis and the load cycle was repeated in order to asses Poi sson (lateral) strains on all specime n surfaces. A third loading was done to failure. Mirrored image correlation was us ed to sequester Poisson and normal strains on the specimen faces during testing. Several failed specimen fracture surfaces were scanned with a scanning electron microscope (SEM) to allow for critical crack lengths to be measured. Failed specimens were scanned on a high resolution scanner in order to measure the crack angle of each specimen. The aims of the off-axis study were to: Determine Poisson and normal strains in the gage section of off-axis specimens. Determine failure load of off-axis specimens of various orientations. Use fractography to determine critical crack lengths in failed specimens. Measure crack angles from scanne d images of failed specimens. Variable Strain Rate Tensile Tests Visual image correlation was performed on the crosshead of the universal testing machine operating in the absence of a load in order to de termine velocity of the crosshead at three speed settings. The speed settings were adjusted by manipulating the position of a rheostat on the universal testing machine. The position of the rh eostat was noted for three crosshead speeds, 1 mm/min, 2 mm/min, and 5 mm/min and the positi ons were marked on the universal testing machine in order to achieve the same crosshead speeds for each test. The same three rheostat settings were used while loading four medial-lat eral and four anteriorposterior specimens fixed with strain gages. The specimens were loaded cy clically (3-5 times each at 1 mm/min) within the linear range of the material to check for hyste resis. Several specimens were also loaded

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58 cyclically into the nonlinear range to assess the vi scoelasticity of the material. The loading cycles were repeated at each of the remaining rheostat settings in order to determine the effect of increased strain rate (calculated from crosshead speed and the specimen gage length) on elastic modulus. It took several weeks to modify the Labview code used to read in strain values from the specimens (after the specimens were already fitte d with strain gages) and modifying the code required that specimens be attached to the National Instruments equipment that received the strain readings, thus the specimens were air dried pr ior to testing. Unfortunately, equipment malfunction at the tim e of this experiment prevented crosshead speeds from varying between rheostat settings ex cept for during the testing of a single specimen (see data in Appendix A). This limited data provides a more thorough understanding of the viscoelastic behavior of manatee rib bone when put together with th e results of the cyclic tests. The aims of this study were to: Determine whether or not manatee rib bone is viscoelastic based on the presence or absence of hysteresis Determine the dependence of elastic modulus on strain rate (a nother indicator of viscoelasticity). Notched Tensile Tests The same wet bone specimens used in the tens ile test experiment along with 8 additional wet bone specimens were prepared for fracture testing. Specimens were wrapped in gauze, soaked in saline solution, and refrigerated until bringing them back to room temperature before testing. Notches were inserted ~1-1.5 mm deep on 2 opposite specimen faces using a diamondtipped wafering saw followed by inserting v-notches at the end of the la rger notches by pressing a razor blade into the notch tips. The v-notches were inserted in li eu of fatigue pre-cracks. Six to twelve specimens were prepared for each orient ation in order to later obtain each of 6 fracture toughness values (K12, K13, K23, K21, K31, and K32). Specimens were loaded to failure at a rate of 1

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59 mm/min. Failure load was recorded for each spec imen. MIC was performed on each specimen in order to have images available for the measurem ent of notch (crack) le ngths. This study aimed to: Determine the failure load for each specimen and crack orientation. Determine crack lengths for each specimen. Numerical Fracture Tests A 3-dimensional, orthotropic finite elemen t model of the notched tensile specimen was prepared in order to assess th e fracture toughness of manatee ri b bone. Material properties found from the previous studies were inserted into the model. Finite elements were oriented to match the three primary orthogonal orientat ions of manatee rib bone in or der to properly align material properties. One analysis was performed for each or ientation. Specimens were fitted with cracks equal in length to those measured by Yan (2005) when using the same razor blade crack insertion technique, and elements were oriented to ma tch those of the off-axis specimens. Boundary conditions were applied to clos ely approximate loading in th e test fixture used in the experimental studies. Loads were applied a ccording to the failure loads found during the experimental component of this analysis. Fracture analysis software was used to determine stress intensity factors (SIF or K) I, II, and III relative the crack growth direction. Hence KI is the opening mode, KII is the shear mode, and KIII is the tearing mode SIF. Because failure load and critical crack lengths were inse rted into the model along with appropriate material properties for the orientations being analyzed, KI values reported by the software are actually the mode I fracture toughness values (KIC) for manatee rib bone specimens of these particular geometries and orientations. The aim of this study was to determine the fracture toughness of various orientations of manatee rib bone for two different geometries.

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60 Table 4-1. Tensile tests. Analysis technique # of specimens per orientation # of orientations Total # of specimens used Output from tests MIC (pilot study) 3 3 9 Elastic moduli Poissons ratio MIC + strain gage 3-4 2 7 Elastic moduli at 3 strain rates Poissons ratio at 3 strain rates MIC 13 3 39 Elastic moduli Poissons ratio Off-axis MIC 1-11 3 24 Strains Critical crack length V-notch MIC 6-12 6 47 Failure loads Critical crack length Table 4-2. Shear tests. Analysis technique Number of specimens per orientation Number of orientations Total number of specimens used Output from tests VIC + shear gage 3 2 6 Shear modulus VIC 9 1 9 Shear modulus Table 4-3. Numerical tests. Specimen model # of orientations Output from tests V-notch 6 Fracture toughness

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61 CHAPTER 5 TENSILE PROPERTIES OF CORTICAL BONE USING MIRRORED IMAGE CORRELATION (MIC) Short Summary Manatee rib bone was the subj ect of a relatively new fullfield deformation technique called visual image correlation (VIC) which was us ed to determine the tensile properties of the material. Manatee rib is predominantly primary pl exiform bone and consists of very few osteons, no medullary cavity, and is almost completely cortical bone. Tension tests were performed on specimens oriented about three orthogonal directions assumed to be the principal material property directions for manatee rib bone under orthotropic assumptions. Nine bone specimens were analyzed in total (n=3 for each orientation). The specimens were loaded to failure at a rate of 1 mm/min under uniaxial tension. Mirrored im age correlation (MIC), derived for this study from VIC, was used to determine full-field defo rmations simultaneously on the front and back of loaded specimens. Strains were calculated from the slope of the position versus deformation curves on the specimen gage sections, where x=dU/dX and y=dV/dY. Elastic moduli and Poissons ratios were found from th e initial slope of the stress-s train curves a nd the ratios of lateral (Poisson) to longit udinal (normal) strain (yx=-x/y), respectively. Poissons ratios ranged from 0.10 for 21 to 0.28 for 12 and elastic moduli found for the three orientations were E1=14.2.6 GPa, E2=6.8.4 GPa, and E3=7.4.4 GPa, where subs cript 1 denotes the proximodistal orientation, subscr ipt 2 denotes the supe rficial-deep orienta tion, and subscript 3 denotes the craniocaudal orientati on of the specimens relative to th e anatomical dire ctions of the animal.

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62 Introduction Cortical bone mechanical testing presents many difficulties due to the nonlinearity, heterogeneity, and anisotropy of the material. Qu antifying material response to loading for the purpose of characterizing material properties be comes a challenging task. Nondestructive testing techniques such as ultrasound do not capture the nonlinearity of the material and require prior knowledge of the material density. Traditional mechanical testing pr ocedures make use of strain gages and extensometers to measure material response to loading. These measurement tools average strains or deformations across a specime n gage section, thus they can miss critical factors that influence material response such as stress risers and hetero geneity within a specimen. An alternative, non-contact method of measur ing deformations and calculating material properties is presented that overcomes difficu lties presented with current techniques. Visual (or digital) image correlation (VIC) is a non-contact full-field measurement technology, originally developed by researchers at the University of South Carolina (Sutton et al., 1986, 1991; Lichtenberger a nd Schreier, 2002). It is us ed to measure geometry, displacements, and plane strains. The underlying principle of the technology is to calculate the displacement field of a test sp ecimen by tracking the deformati on of a random speckling pattern applied to the specimen surface. The random pattern is digitally acquired by the cameras before and during loading. The VIC system then tries to find the region (i n the image of the deformed specimen) that maximizes a normalized cross-correlation functi on corresponding to a small subset of the reference image (taken when no load is applied to the structure) (Sutt on et al., 1986). The image space is iteratively swept by the parameters of the cross-correlation function, to transform the coordinates from the original reference frame to coordinates within the deformed image. An originally square subset in the un-deformed image can then be mapped to a subset in the

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63 deformed image, as can be seen in Figure 5-1. As it is unlikely that the deformed coordinates will directly fall onto the sampling grid of the image, accurate grey-value interpolation schemes (Schreier et al., 2000) are implemented to achieve optimal sub-pixel accuracy without bias. This procedure is repeated for a large numbe r of subsets to obtain full-field data. For two-dimensional plane stress problems, a si ngle camera is sufficient to measure the inplane displacements, directed orthogonally to the flat speckled surface. For the more general case of three-dimensional problems (where both in-p lane and out-of-plane di splacements are required over an arbitrarily non-flat surface), two cameras are needed. The acquisition of images is based on a stereo-triangulation t echnique, as well as the computing of the intersection of two optical rays. Referencing Figure 5-2, the stereo-cor relation matches the two 2-D frames taken simultaneously by the two cameras to reconstruc t the 3-D geometry, and then uses a tracking technique called temporal matching to follow the speckle deformations (Sutton et al., 1991). The calibration of the two cameras (to account for lens distortion and determin e pixel spacing in the model coordinates) is the initial fundamental step, which permits the determination of the corresponding image locations from views in the different cameras. Calibration is done by taking images (with both cameras) of a known fi xed grid of black and white dots. The twin cameras are connected with a PC via an IEEE 1394 (Firewire) cable, and a specialized unit is used to synchronize the camera triggers for instantaneous shots. A standard acquisition board installed in the computer carries out the digitalization of the images, and the image processing is carried out by specialized software (VIC3D v.3.1, Correlated Solutions, Inc., West Columbia, SC). Typical data results that can be obtained from the VIC system consist of the geometry of the surface in discrete X, Y, and Z coordinates (where the or igin is located at the

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64 centroid of the speckled area of interest, and th e outward normal points towards the cameras, by default), and the corresponding displacements on the surface (U, V, and W). A new technique was developed for this st udy that builds on the foundation of VIC in order to accurately capture th e response of a specimen to lo ading. Mirrored image correlation (MIC) is a technique that was developed by the au thors in order to provide an accurate account of deformations occurring simultaneously on the fr ont and back of a specimen loaded in tension. The intent of this methodology was to mitigate the effects of bending if the specimens are misaligned in the custom tensile grips. Bendi ng strains are removed by averaging strains found on the front and back of the specimen, leaving only tensile strains for the calculation of material properties. Visual image correlation alone ca nnot account for strain gr adients throughout the thickness of a specimen, thus MIC is superior to VIC when there is a chance of specimen misalignment. Furthermore, MIC re duces error compared to VIC because more data is available and averaged for the analysis. Visual image correlation also has advantages over traditional measurement techniques. For instance, the syst emic error of testing systems when using compliance measurement techniques (An and Frie dman, 1999) is removed by using VIC, since VIC is a non-contact method of measuring defo rmations on a specimen surface rather than measuring the relative movement of testing equi pment cross-heads, which introduces errors in measurement due to the natural compliance of testing equipment. The analysis technique presented here offers significant improvements towards the assessment of material properties. Our study aims to describe the role of MIC for measuring full-fiel d deformations during mechanical testing and use of these deformations in the determination of material properties. Manatee rib is ideally suited for mechanical testin g due to its large size, its lack of a medullary (hollow) cavity, and its large proportion of cortical bone to trabecular bone. All of these

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65 attributes make it possible to ha rvest large cortical bone specime ns oriented in any direction relative to the rib bone. The Florida manatee is an endangered species th at far too often is killed as a result of collisions with watercraft. Computer simulati ons that utilize the an isotropic properties of manatee bone tissue can aid in th e understanding of how manatee rib bone responds to impact from a watercraft. To this point, material properties are known fo r only the longitudinal orientation of manatee rib bone and computer simulations cannot accurately depict the true response of manatee rib to loadi ng with this limited input. This study will improve on the ability of computers to predict impact response of wa tercraft and manatees by identifying material properties for three orth ogonal orientations of rib bone under the presumption that manatee rib bone is orthotropic. The presen t study builds on work complete d by other researchers at the University of Florida that aims to reduce watercraft-related injury to this docile marine mammal (Clifton et al., 2003; Clifton, 2005; Yan, 2002, 2005; Yan, 2006a, b, 2007). The aims of this study are to: 1) determine the elastic modulus for 3-orientations of manatee rib bone using MIC, 2) determine the Poi ssons ratio for several or ientations of manatee rib bone using MIC, 3) compare MIC results w ith those from threepoint bend and ultrasound studies, 4) characterize the anisotropy of mana tee rib bone elastic modulus, 5) evaluate the ability of the custom test fi xtures to prevent specimen bendi ng by determining if results significantly differ on the front and back of a spec imen, and 6) investigate the linearity of the material for each orientation. Materials and Methods Manatee rib bones were obtained under US Fish and Wildlife permit #MA067116-0 issued to the Florida Fish and Wildlife Conservati on Commissions (FWC) Marine Mammal Pathology Lab (MMPL). Use of the tissue was governed by University of Florida Institutional Animal Care

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66 and Use Committee (IACUC) protocol #E407. One cen ter rib (#9 of 17-19 ribs) was selected for this study from a single adult male manatee (Figur e 5-3). The manatee rib was stored with flesh intact to preserve material properties (An a nd Draughn, 2000) at -20 Celsius. The middle third of the selected rib was used for testing, since Yan (2002) found that the fracture properties of manatee rib bone vary from the middle of each rib to the ends of the bone and it is assumed that material properties (in addition to fracture properties) vary as a function of position in the rib. The specimens were rough-cut with a wet ba nd saw: Close attention was paid to the orientation in which the specimens were harvested. Specimens were milled to their final dimensions (total length = 38 mm, width at grip = 12.7 mm, gage length = 19 mm, gage width = 6.4 mm, radius at gage section = 3.2 mm, and a s quare cross-section) und er constant irrigation with water to prevent over-heating. The specimens were allowed to air dry at room temperature for 2 weeks prior to testing to provide a dry surf ace for application of th e speckle pattern used during the VIC analysis. Specimens were spray pa inted white and dusted with black spray paint to generate the random speckle pattern used by the VIC stereo system during image correlation. During testing, specimens were fixed in a uni axial loading machine (MTI 30K, Measurement Technology, Inc., Roswell, GA) in custom holders that were designed and fabricated for this study in order to reduce shadows cast on the spec imen by halogen lights used to improve image quality during testing. A flat mirror was placed as close to the specimens as possible while still allowing for both the front and back of the specimens to be in view of the VIC cameras, since the focal length of the camera lenses (Schne ider-Kreuznach Cinegon CM120, Bad Kreuznach, Germany) was limited. Optimal performance of th e VIC stereo system was obtained when the cameras were placed ~13 mm behind the specimen s with the cameras (Retiga 1300, Quantitative Imaging Corporation, Burnaby, British Columbia, Canada) arranged to form a near equilateral

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67 triangle with the specimens (Figure 5-4). The cameras were calibrated with a dot matrix calibration grid generated using custom softwa re provided by Correlate d Solutions, Inc. and image correlation software (VIC3D v3.1, Correla ted Solutions, Inc., West Columbia, SC). Specimens were loaded to failure at a rate of 1 mm/min. Load data was monitored using a 1000lb load cell (Interface SM-100038, Scottsdale, AZ) accurate to 20 grams, received and amplified by a terminal block and signal amplifier (SCXI-1321 and SCXI-1121, National Instruments Corporation, Austin, TX), and logged using a custom Labview program (Labview v.7.0, National Instruments Corporation, Austin, TX). Correlation analyses were performed on the gage sections of loaded specimen images (Figure 5-5) us ing VIC3D v3.1. Output from the VIC analyses included position, deformation, and velocity for each data point on the specimen gage sections. Output files were exported from VIC3D and Labvi ew and input into a custom Mathcad program (Mathcad v.12.1, Mathsoft Engineering and Edu cation, Inc., Needham, MA) for further data reduction and analysis. Strains were calculated fo r each load step from the slope of the position versus deformation curves across the specimen gage sections, where x=dU/dX (Poisson strain) and y=dV/dY (normal strain) (Figure 5-6). Strains calcu lated for each load step were averaged over the linear range of the stress-strain curves for each specimen tested. The strains on the front and back of the specimen were averaged to remove bending effects on tensile property calculations. Poissons ratio was ta ken as the negative ratio of Po isson strain to normal strain (xy=-y/x) (Figure 5-7), while elastic modulus was calculated as the initial slope of the stressstrain curve (Figure 5-8). The linearity of the stress-strain curves was assessed by visual inspection. The analysis required several operations to be performed on the data before Poissons ratios and elastic moduli could be reported: 1) Poorly-correlated data points (as determined by

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68 VIC3D software) (correlation 0) were removed from the data set; 2) Data associated with negative or invalid Po issons ratios (0 > xy > 0.5) were removed from the analysis; 3) Data associated with deformation rates sufficiently di fferent than the prescrib ed rate (0.5 mm/min < velocity < 1.5 mm/min) were removed, since these points may be indicative of faulty data; 4) Several data points in the lower lo ading range were filtered out of the analysis because these data points did not meet minimum strain le vel requirements (strains less than 500 are considered to be too small for the VIC system to accurately de tect); and 5) Data points in the non-linear range of the stress-strain curve were removed from the elastic tensile property calculations. The anisotropic elastic consta nts of manatee bone were analyzed using VIC along three directions assumed to be principal material ax es of manatee rib (Figur e 5-9). Three specimens were prepared in each of the three orientations se lected for this analysis (labeled A, B, and C, respectively based on the order they were tested ). Miller indices (Mur ray, 1993) were used in addition to anatomical labeling of material orie ntations to accommodate those readers whom are more familiar with labeling of crystalline structures and to keep specimen labels concise. The Miller indices represent the loading axis of the specimen. Specimens labeled as [100] were oriented (loaded) in the proxim odistal direction. Superficial-deep specimens are labeled as [010]. The craniocaudal specimens are labeled as [001]. Statistics One-way ANOVA was performed on the means of all specimen orientations to determine if any modulus means were significantly different between the tested orientations. Independent group t-tests were performed on modulus means for each possible pair of specimen orientations to determine which means significantly diffe red. Not enough specimens were assessed to perform ANOVA or group t-tests on Poisson's ratio. Matched pair s t-tests were performed on

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69 results from the front and back of specimens for a given orientation to dete rmine if the effect of bending was significant on tensile properties for any orientation. Paired t-tests were also performed on elastic modulus and Poisson's rati o where all values were used regardless of orientation to see if bending had any effect on elastic modulus or Poisson's ratio as a whole. The null hypothesis for all tests was that no statisti cal difference existed between the variables. Significance was assumed if p<0.05. Statistical analyses were performed using Statistics Calculator v.3.0 (StatPac, Inc., Bloomington, MN). Results Results from each of the 9 specimens are prov ided in Tables 5-1 and 5-2 and Figures 5-10 to 5-12. Mean values and standard deviations (SD) are presented for elastic moduli along with the number of specimens used to calculate these values (n). Experimental values of several material properties of manatee rib bone are provided in Tabl e 5-3 for comparison to values obtained during this study. The proximodistal [100] orient ation represents the longitudi nal axis of the manatee rib bone. The strength for this orientation is much great er than for the other tw o orientations (Figure 5-10). The mean elastic modulus and standard de viation for the 3 proxim odistal specimens was 14.2.6 GPa. The superficial-deep [010] orie ntation represents the transver se plane minor-axis. Failure strength was the most uniform between specimens for this orientation. The strength was slightly greater than that for the cran iocaudal orientation, but much le ss than for the proximodistal orientation. Failure strain was qu ite variable for this orienta tion. The stress-strain curve was more nonlinear for this orientation than for the other two orientations (Figure 5-11). The mean elastic modulus and standard deviation for the 3 superficial-deep specimens was 6.8.4 GPa.

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70 The craniocaudal [001] orientati on represents the major-axis in the transverse plane of the manatee rib bone. The mean failure strain for th is orientation is lower than for the other orientations. The strength is nearly identical to that of the superficial-deep specimen orientation, but lesser than the prox imodistal orientation. The slope of th e craniocaudal stress-strain curve is (on average) more linear than the slope for the other two or ientations (Figure 5-12). The mean elastic modulus and standard deviation for the 3 craniocaudal specimens was 7.4.4 GPa. Statistical Results A one-way ANOVA test was performed on the tensile property means of three specimen orientations to determine if the difference betw een any of the orientati ons was significant. The pvalue (<0.001 for both modulus and Po isson's ratio) was significant (p<0.05). Therefore, the null hypothesis is rejected and it is c oncluded that at least one mean was significantly different from the others for both modulus and Poisson's ratio results. For the paired t-tests performed on elastic modu li for three orientations, the t-statistic was significant at the 0.05 critical alpha level (t(4)=3.442 with p=0.0262) for E1 and E2. Therefore, the null hypothesis is rejected a nd it is concluded that there is a significant difference between elastic moduli E1 and E2. The t-statistic was not significant (t(4)=0.226 with p=0.8322) for E2 and E3. Therefore, the null hypothesis is accepted and it is concluded that there is no significant difference between elastic moduli E2 and E3. The t-statistic was significant (t(4)=3.097 with p=0.0363) for E1 and E3. Therefore, the null hypothesis is rej ected and it is concluded that there is a significant difference between elastic moduli E1 and E3. In comparing tensile properties on the front a nd back of the specimens using matched pairs t-tests, the null hypothesis wa s accepted for all tests (p>0.05) and it is concluded that no significant difference exists between values found on the front and back of the specimens used in this study.

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71 Discussion Results from this study agree well with those from Clifton (2005) for a 3-point bend specimen taken from a center rib of the same manatee used in this study (14.3.0 GPa for the [100] direction here compared to 15.7 GPa in th e Clifton study). Poissons ratios from various directions in this study ranged from =0.10 to 0.28, and are on the same order as those from Yans ultrasound study (2002) (=0.25), although the re sults are not directly comparable since Poissons ratio from Yans study was cal culated using isotropic assumptions ( 1 2E G ). Results from this study provide an idea of the anisotropic nature of manatee rib bone elastic modulus, although more specimens shoul d be assessed prior to drawing any final conclusions due to the relatively large coefficient of variation (CV=SD/) for elastic modulus. In this study, elastic modulus from one of the thr ee orientations differed significantly from the others while the other two did not significantly differ from one another, implying that manatee rib bone elastic modulus is transv ersely isotropic (i.e., there is one plane of material symmetry). Although the anisotropic nature of manatee ri b bone has been found for a limited number of specimens, a complete characteri zation of manatee rib bone materi al properties would require the knowledge of the materials shear properties in addition to knowing the elastic moduli and Poissons ratios. It is recomme nded that additional tensile tests as well as shear tests be conducted in order to fully characterize the anisot ropy of manatee rib bone material properties. Manatee rib bone is significantly more complia nt in the transverse plane than in the longitudinal plane (E1 2E2 and E3). It is hypothesized that meas ures of fracture toughness will follow suit with one orientation be ing significantly tougher than th e other two orientations. Thus previous analyses assessing ma natee rib bone fracture toughness in the low-compliance direction (Clifton et al., 2003, Clifton, 2005; Yan et al ., 2006a, b, 2007) may over-estimate the minimum

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72 toughness of manatee bone. The re sults presented here may therefore imply that manatee rib bone is far more fragile than or iginally thought. It ha s already been shown that manatee rib bone is less tough than other animal bones consisti ng of primary plexiform bone (Yan et al., 2006b). The lack of significant difference between te nsile properties calcul ated on the front and back of a specimen show that MIC is not required to obtain material proper ties (i.e., VIC may be good enough given that there is no be nding in the experimental set up). It is cautioned that any experimental set-up used for assessi ng tensile properties other than the one used here should also be analyzed for bending effects if reporting VI C results for only one specimen face. Mirrored image correlation reduces error compared to VIC by removing the unwanted effects of bending induced by a tensile test fixture or misaligned sp ecimen and by allowing for more data points to be averaged over the specimen gage sections. An and Draughn (2000) assume that although bo ne is a viscoelastic material, it only exhibits a slight degree of viscoe lasticity and can therefore be trea ted as a linear-elastic material. This assumption was assessed by visual inspectio n of the experimentally determined stress versus strain curves found from the tensile tests. It was found that signifi cant nonlinearity existed for all orientations tested here (Figures 5-10 to 5-12). The cause for this nonlinear behavior remains unknown, since this study did not invest igate whether the nonlinearity was due to microcracking, other forms of plasticity, or visc oelasticity. Testing of the cyclic behavior and strain-rate dependent behavior of manatee rib bone can shed more light on this subject. Also, Xray diffraction, fractography, or other imaging tec hniques can be used to assess the existence of microcracks in the specimens preand post-testing. Conclusions Elastic moduli and Poissons ratios were found for three orientations of manatee rib bone. Elastic modulus was found to be transversely isot ropic. To the authors knowledge, this is the

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73 first study to use MIC in the determination of mate rial properties. The technique is superior to VIC if unwanted specimen bending is present. Mirro red image correlation is a viable alternative to 3-point bending for the determina tion of cortical bone modulus. Several benefits exist from the use of VIC ove r other measures of deformation and strain (e.g., compliance-based techniques, strain gages, and extensometers), including: 1) Deformations are measured on the specimen surface(s) and thus do not reflect machine compliance in the test rig; 2) Material heterogeneity can be accounted for by looking for changes in slope of the position versus deformation curves (indicating diffe rences in strain magnitude across the gage section), while most other techniques average stra ins over the entire specimen gage section; 3) Visual image correlation measurements are valid to 500% strain, while strain gages usually have a much lower top-out limit, thus VIC may be a better tool for capturi ng failure strain of a material than strain gages; 4) Localized strain (around notches or material imperfections) can be assessed using VIC, while they cannot w ith strain gages or extensometers. The drawbacks of VIC include: 1) The mini mum strain magnitude captured by the VIC system is limited by image correlation algorithms (currently, strains are not considered accurate below 500 ); 2) Set-up time can be significant when considering the time to position and focus the cameras and calibrate the stereo system; alth ough once the set-up is in place, testing can take place very rapidly.

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74 Table 5-1. Elastic moduli from MIC analysis. Mean values are presented standard deviations along with the number of specimens used to compute the mean values (n). Elastic modulus (GPa) Proximodistal [100] (E1) Superficial-Deep [010] (E2) Craniocaudal [001] (E3) Specimen A 16.0 9.8 7.8 Specimen B 13.9 7.6 7.3 Specimen C 12.8 3.1 7.0 Mean 14.2.6 (n=3) 6.8.4 (n=3) 7.4.4 (n=3) Table 5-2. Poissons ratios from MIC analysis. Index 1 denotes the [100] direction, index 2 denotes the [010] direction, and i ndex 3 denotes the [001] direction. Poissons ratio Proximodistal [100] (x Superficial-Deep [010] (x Craniocaudal [001] (x Specimen A 13=0.28 21=0.21 31=0.14 Specimen B 12=0.25 21=0.29 32=0.24 Specimen C 13=0.29 23=0.10 32=0.23 Table 5-3. Summary of known mana tee rib material properties. Investigators Testing method Bone source Load or wave direction Elastic modulus (GPa) Poissons ratio Shear modulus (GPa) Yan (2002) Ultrasonic Manatee rib Transverse 18.13.77 0.253.021 7.24.711 Clifton (2005) 3-point bend Manatee rib Transverse E1=4 to 18 Clifton (2005) 3-point bend MSW0253* Transverse E1=15.7 *MSW0253 is the ID for the same manatee as was used in the present study

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75 Figure 5-1. Three-dimensional surface as seen by two imaging sensors (courtesy of Correlated Solutions, Inc, West Columbia, SC). Figure 5-2. Movement of a sample square subs et used for cross-correl ation function estimation (courtesy of Correlated Solutions, Inc., West Columbia, SC). un-deformed image deformed image

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76 Figure 5-3. Manatee skeleton. Mo dified with permission from Yan, J., 2002. Biomechanical properties of manatee rib bone and analyti cal study using finite element analysis. M.S. Thesis, University of Florid a, Gainesville, Figure 3-2, p. 19. Figure 5-4. Stereographic setup used in the MIC analysis.

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77 Figure 5-5. Example image and target region us ed by VIC3D to generate a dataset and the associated surface plot created by VIC3 D showing the exaggerated XYZ position data. Figure 5-6. Typical dV/dY curve showing all data points on th e gage section of a manatee rib bone specimen. The normal strain (slope of the V vs. Y curve) is 3000 for this particular load step. The white line is the li near trend line used to fit the thousands of data points depicted in the plot.

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78 Figure 5-7. Nomenclature used for the Poissons ratio analysis. Shaded faces represent the faces on which each respective Poissons ratio was evaluated. Figure 5-8. Stress-strain curve for a [010] specime n. The initial slope of the stress-strain curve (depicted in light green) is the elastic modulus for th is specimen. The red dots represent the experimental data points obtai ned from the MIC analysis for each load step. 23 1 3 21 31 12 3 2 [001] Craniocaudal [010] Superficial-Deep [100] Proximodistal

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79 Figure 5-9. Schematic of Proxi modistal [100] specimens prepared from the middle 1/3rd of an adult manatee rib bone. Figure 5-10. Proximodistal [100] stress-strain curve for three dry manatee rib bone specimens (A, B, and C) loaded to failure.

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80 Figure 5-11. Superficial-Deep [ 010] stress-strain curve for thr ee dry manatee rib bone specimens (A, B, and C) loaded to failure. Figure 5-12. Craniocaudal [001] st ress-strain curve for three dry manatee rib bone specimens (A, B, and C) loaded to failure.

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81 CHAPTER 6 ANISOTROPIC ELASTIC TENSILE PROPERTIES Short Summary This chapter conveys the first study to analyze the nature of the anisotropy of manatee rib bone tensile properties. Three elastic moduli and three Poissons ratios were determined using a non-contact, full-field deformati on analysis technique called vi sual image correlation (VIC). Previously, tensile data existed for only one orientation of mana tee rib bone. A complete set of anisotropic material data is required in order to develop constitutive models that are capable of predicting the materials response to loading. This chapter is the first part of a three part series of studies aimed at characterizing ma natee rib bone material properti es. The specific aim of Part 1 (Chapter 6) is to determine the tensile propertie s of the material, while Part 2 (Chapter 7) will identify the shear properties of manatee ri b bone. Parts 1 and 2 will provide sufficient information for the completion of the anisotropi c elastic constants matrix under the assumption of material orthotropy. Part 3 (Chapter 8) will verify the or thotropic assumption by means of comparing experimental and predic ted strains in off-axis specimens using the constitutive model built for the material in Parts 1 and 2. Visual image correlation is an ideal tool fo r assessing the strain field in heterogeneous materials such as bone, since the technique is capable of discerni ng localized eff ects within the gage section. The output from the VIC anal ysis is the three-dimensional position and deformation data in the gage region of each sp ecimen. The derivative of the deformation with respect to the position is taken as the strain in each specimen (x=dU/dX and y=dV/dY). Poissons ratio is taken as the negative of th e ratio between Poisson and normal strain (xy=-x/y). Elastic modulus is taken as the init ial slope of the stress-strain curve (E=/y). Elastic modulus was found to be 16.9.5 GPa in the proxi modistal (longitudinal) direction of the bone,

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82 8.4.2 GPa in the superficial-deep (in-plane tr ansverse) direction, and 5.9.2 GPa in the craniocaudal (in-plane transverse) dir ection. Major Poissons ratios were 12=0.27.04, 13=0.25.04, and 23=0.25.04, while the minor Poissons Poissons ratios were 21=0.16.06, 31=0.15.07, and 32=0.28.08. The significant difference between elastic moduli in three orthogonal directions implies that the material is orthotropic, although material models that can prove this assumption still require input of shear modulus data. Introduction To date, no experiments have demonstrated th e anisotropic nature of manatee rib bone. Preliminary models of the material have been limited to isotropic assumptions due to the limited amount of material data availa ble (Yan, 2002). Without the proper material property input (e.g., treating an anisotropic material as isotropic), constitutive models for anisotropic materials cannot be used to accurately predict material response to loading (A rakere and Swanson, 2002; Arakere et al., 2005; Ranjan and Arakere, 2007). It is important to unders tand the response of this brittle tissue to mechanical loading in order to promote efforts to reduce the high casualty rate in manatees resulting from collisions with watercraft: Twenty-five percent of all manatees are killed by collision with watercraft with more than half of these deaths attributed to rib bone fracture (Clifton, 2005). Crack propa gation in manatee rib bone is facilitated by a high mineral density and a shortage of osteons that bridge and blunt cracks and a ssist in the healing process of bone in most mammals. The implication of these f eatures is that manatee rib fractures are often fatal. This study aims to provide the initial da ta required to complete the orthotropic elastic constants matrix for manatee rib bone so that predictive modeling of bone fracture will be possible in the future.

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83 Testing of cortical bone materi al properties presents severa l challenges due to material hierarchy, nonlinearity, anisotropy, and limitation s in specimen thicknes s. Several levels of hierarchy have been identified for bone, each co ntaining a unique set of properties and material characteristics (Weiner and Wagner, 1998; Rho et al., 1998). Tissue level properties were analyzed in the present study for the identifica tion of direction depende nt properties to help achieve the end goal of developing models of whole bone. In many mammalians, cortical bone would be too thin at most anatomical sites in order to assess tissue leve l properties in the three orthogonal directions required for orthotropic materials. However, manatee rib bone is unique in that it possesses no marrow cavity and it is pred ominantly cortical bone, thus it is quite conducive towards obtaining material properties in a ny direction. The analysis technique used to measure deformations in the tensile specimens is called visual image correlation (VIC). Visual image correlation is a full-field deformation analysis technique capable of providing threedimensional position and deformation information for the surface of a loaded specimen using stereoscopic imagery (Sutton et al., 1986; 1991; Schreier et al., 2000; Lichtenberger and Shcreier, 2002). The full-field deformation analysis is capable of identify ing localized effects in a specimen that could be indicative of material or structural inhomogene ities within the gage section that would go un-noticed using other defo rmation or strain analysis techniques (i.e., strain gages and extensometers average strain an d deformation data, respectively, across the gage section of a material and localized effects cannot be identified). The technique is considered accurate from 500 to 500% strain, thus it lends itself wonderfully towards testing biological materials undergoing large amounts of strain. Fu rthermore, commercial modules are available that allow for VIC to be performed at high speeds thus impact and vibrat ion testing is possible using the technique. Zhang et al (2007) used a two-dimensiona l form of VIC (using a single

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84 camera) to investigate th e tensile properties of bovine hoof horn. The technique has also been used to measure microstructural strains in corti cal bone (Nicolella et al ., 2001). To the authors knowledge, this is the first study of the anisotro pic material properties of cortical bone using VIC. Bone is generally considered to be a transv ersely isotropic (Yoon et al., 1976; Gotzen et al., 2003; Dong and Guo, 2004) or or thotropic material (Ashman et al., 1984; Hoffmeister et al., 2000). Material property data for one specimen or ientation is not sufficient for modeling the constitutive behavior of an an isotropic material. Therefore, th is study will analyze manatee rib (cortical) bone tensile speci mens along three orthogonal direc tions in order to assess the directional dependence (anisotropy) of the material. The nomencl ature used to describe the directional dependence of the material is as foll ows: the proximodistal di rection of the manatee was designated as the 1-directi on (i.e., the longitudinal directi on of the rib bone), superficialdeep as 2 (i.e., the in-plane transverse directi on of the rib bone), and craniocaudal as 3 (i.e., the out-of-plane transverse direction of the rib). The numbers appear as subscripts in the material properties analysis to describe the direct ion(s) of measured strain. For example, E1 is the elastic modulus in the 1-direction, and 12 is the ratio of Poisson strain in the 2-direction to normal strain in the 1-direction. Methods Manatee rib bones were obtained under US Fish and Wildlife permit #MA067116-0 issued to the Florida Fish and Wildlife C onservation Commissions (FWC) Marine Mammal Pathology Lab (MMPL). Use of the tissue was gove rned by University of Florida Institutional Animal Care and Use Committee (IACUC) protocol #E407. Ribs from the cen ter third of several adult manatees (#s 9-11 of 17-19 ribs) were sele cted for this study (Figure 6-1), and specimens

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85 were prepared from the middle third of the ribs since their properties were previously shown to vary as a function of anatomic position (Yan, 2006a). The manatee rib was stored with flesh intact to preserve material properties (An and Draughn, 2000) at -20 Celsius. Thirteen specimens were prepared for each orientat ion (proximodistal, superficial-deep, and craniocaudal). Specimens were rough-cut with a we t band saw, with close attention being paid to the orientation in which the specimens were harvested. Specimens were milled to their final dimensions (total length = 38 mm, width at grip = 12.7 mm, gage length = 19 mm, gage width = 6.4 mm, radius at gage section = 3.2 mm, and a square cross-section) und er constant irrigation with water to prevent over-heating. Specimens were wrapped in gauze and soaked in saline solution while being stored in a re frigerator for up to a week pr ior to testing. Specimens were wiped dry, spray painted white and dusted with black spray paint to generate a random speckle pattern used by the VIC stereo system during image correlation. Specimens were then rehydrated in saline solution for 3 hours before testing at room temperature. During testing, specimens were fixed in a uniaxial loading m achine (MTI 30K, Measurement Technology, Inc., Roswell, GA) in custom holders that were desi gned and fabricated for this study in order to reduce shadows cast on the specim en by halogen lights used to improve image quality during testing. Optimal performance of the VIC stereo system was obtained when the cameras were placed ~13 mm behind the specimens with the cameras (Retiga 1300, Quantitative Imaging Corporation, Burnaby, British Columbia, Canada) arranged to form a near equilateral triangle with the specimens. The cameras were calibrate d with a dot matrix calibration grid generated using custom software provided by Correlated So lutions, Inc. and image correlation software (VIC-3D Digital Image Correlati on v2006.0.0, Correlated Solutions, Inc., West Columbia, SC). Specimens oriented in the medial -lateral direction were loaded to 670 N, those oriented in the

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86 anterior-posterior direction were loaded to 1100 N, and specimens oriented in the cranial-caudal direction were loaded to 670 N at a rate of 1 mm/min (~5%/min). Loads were selected for each orientation based on results from a pilot study in which dry bone specimens of identical dimensions were loaded to failure at the same load rate. The maximum load in this study corresponds with the yield load fo r each orientation tested in th e pilot study. Load data was monitored using a 1000-lb load cell (Interface SM-1000-38, Scottsdale, AZ), received and amplified by a terminal block and signal amplifier (SCXI-1321 a nd SCXI-1121, National Instruments Corporation, Austin, TX), and l ogged using a custom Labview program (Labview v.7.0, National Instruments Corporation, Austin TX). Correlation analyses were performed on the gage sections of loaded specimen images (F igure 6-2) using VIC3D software. Output from the VIC analyses included position, deformation, and velocity for each data point on the specimen gage sections. Output files were expor ted from VIC3D and Labview and input into a custom Mathcad program (Mathcad v.12.1, Mathso ft Engineering and Edu cation, Inc., Needham, MA) for further data reduction and analysis. Strains were calculated for each load step from the slope of the position versus deformation curves across the specimen gage sections, where x = dU/dX (Poisson strain) and y = dV/dY (normal strain) (Figure 6-3) Strains calculated for each load step were averaged over the linear range of the stress-strain curves for each specimen tested. Poissons ratio was taken as negative the ratio of Poisson strain to normal strain (yx = -x/y). Major Poissons ratios are defined as 12, 13, and 23, while minor Poissons ratios are defined as 21, 31, and 32. Differentiation of major and minor Po issons ratios is necessary to support conventions used in the literatu re. Elastic modulus was calculate d as the initial slope of the stress-strain curve (Figure 6-4). Anisotropic ratios (rxy) were taken as the ratio of the modulus in the x-direction to that in the y-direction.

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87 One-way ANOVA was used to determine if el astic modulus or Poissons ratio means varied significantly between any of the orientat ions. Independent group ttests were performed on all possible pairs of specimen orientations to determine which means significantly differed, if any. Significance was taken if p<0.05. Statistical analyses we re performed using Statistics Calculator v.3.0 (StatPac, Inc., Bloomington, MN). Results Table 6-1 shows the elastic modulus and Poisso ns ratio of manatee ri b bone as a function of orientation. One-way ANOVA rev ealed that at least one mean wa s significantly different from the others for modulus (p<0.001) and minor Poisson's ratio (p=0.015), while the major Poissons ratios did not vary signifi cantly from one another (p=0.360). Independent group t-test results demonstrated that elastic modulus varied significantly between a ll combinations of orientations (p<0.001 for E1 and E2, p<0.001 for E1 and E3, and p=0.0014 for E2 and E3). Independent group t-test results for minors Poissons ratios were: 21 and 31 did not significantly vary (p=0.699), 21 and 32 varied significantly (p=0.021), and 31 and 32 varied significantly (p=0.014). The anisotropic ratio was largest between E1 and E3 (r13=2.9) and was smallest between E2 and E3 (r23=1.4). Discussion Results from this study agree with those fr om Clifton (2005) for a 3-point bend specimen from a center rib of the same manatee used in this study (E1=16.9.5 GPa in this study compared to 15.7 and 16.4 GPa in the Clifton study (Table 6-2)). Poissons ratios from this study are on the same order as those from Yans ultrasound study (2002) (Table 6-2), although the results are not directly comparable since Poisso ns ratio from Yans st udy was calculated using isotropic assumptions ( 1 2E G ).

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88 This study provides sufficient information to su ggest the anisotropic na ture of manatee rib bone. In this study, each of the three elastic moduli differed si gnificantly from one another, implying that manatee rib bone elastic modulus is orthotropic. Although the anisotropic nature of manatee rib bone elastic modulus has been found, a complete characterization of manatee rib bone material properties would require the knowle dge of the materials shear properties in addition to the tensile properties reported in this study. Part 2 of this series will assess the shear properties of manatee rib bone in order to provid e sufficient information to fully characterize the anisotropy of manatee rib bone material properties (Part 3). Part 3 will use the major Poissons ratios reported in this chapter in order to co mply with the most commonly used convention for this variable. It should be noted that the minor Poissons ratio can be calculated from the major Poissons ratio and the elastic modu li according to the following relation: y yxxy xE E for 1, 2, 3 xy Manatee rib bone is significantly more complia nt in the transverse plane than in the longitudinal plane (E1 2E2 and 3E3). It is hypothesized that measures of toughness will follow suit with one orientation be ing significantly tougher than th e other two orientations, thus previous analyses assessing ma natee rib bone fracture toughness in the low-compliance direction (Clifton et al., 2003; Clifton, 2005; Yan et al ., 2006a, b, and 2007) may over-estimate the toughness for the minimum toughness orientation of manatee bone. It has already been shown that manatee rib bone is less tough than other mammalian bone (Yan et al. 2006a). The results presented here imply that manatee rib bone may be far less tough than or iginally thought. Aside from being less tough than other bone, manatee ri b bone is also unique in that the anisotropic ratio (r12=2.9) is much higher than what is f ound in plexiform bone of other animals (rLT=1.18-

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89 1.47 (Weiner and Wagner, 1998), where L is the longitudinal direction of bone and T is the transverse direction). Manatee rib is predominantly primary plexifor m bone and has very few pores or osteons. Plexiform bone is a rapidly forming bone with a brick-like structure that is generally found in large mammals (Martin and Burr, 1989). The orga nization of the microstructure is likely responsible for the high anisotropic ratio in this material. Gotzen et al. (2003) used histology to identify the organization of microstructural feat ures around nutrient forame n, which are naturally forming holes in bone. They used the data they collected to build a fi nite element model to understand strain and modulus dist ribution in proximity to the foramen. A similar technique could be used to identify the effect of micros tructural organization on the anisotropic ratios in manatee rib bone. However, the scope of this st udy was to quantify mana tee rib tensile property anisotropy at the tissue level of the material, and investigation of microstructural effects on the elastic anisotropy of this material remains as future work. Several benefits exist from the use of VIC ove r other measures of deformation and strain (e.g., compliance-based techniques, strain gages, and extensometers), including: Deformations are measured directly on th e specimen and thus do not reflect machine compliance in the test rig. Reinforcement effects of strain gages and adhesives on low modulus specimens are not induced using VIC. Material heterogeneity can be accounted for using VIC by looking for changes in slope of the position versus deformation curves, while most other techniques average strains over the entire specimen gage section. The top-out limit in VIC measurements is on th e order of 500% strain while strain gages have a much lower top-out limit. Thus the failure strain of ma ny materials cannot be recorded with strain gages while VIC is well-suited for capturing large, and even nonlinear, strains up to the fa ilure of most materials. Localized strain (around notches or material imperfections) can be assessed using VIC, while they cannot with strain gages or extensometers.

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90 The drawbacks of VIC include: The minimum strain magnitude captured by th e VIC system is limited by image correlation algorithms (currently, strains are not considered accurate below 500 ). Set-up time can be significant when consid ering time to position and focus the cameras and calibrate the stereo system, although once th e set-up is in place, testing can take place very rapidly. Conclusions It is concluded that VIC is not only a viable alternative to st rain gages and extensometers, but it is also more versatile due to its ability to capture full-field deform ation and strain data up to high strain magnitudes on the order of 500%. Fu rthermore, it was shown in this study that VIC is capable of providing material property data c onsistent with other testing techniques and in sufficient detail to determine the nature of the anisotropy of manatee rib bone tensile properties. The results from this study suggest the need to assess shear propertie s about three orthogonal axes in order to assess the anisotropic na ture of manatee rib bone shear modulus under orthotropic assumptions. Part 2 of this three part series (Chapt er 7) will identify three shear moduli for manatee rib bone, while Part 3 (Chapt er 8) will demonstrat e a material symmetry characterization analysis for anisotropi c materials such as manatee rib bone. Table 6-1. Tensile properties of the Florida mana tee rib bone. Mean values are presented as a function of specimen orientation along with the standard deviat ion and number of specimens used to calculate each. Elastic modulus (GPa) Major Poissons ratio Minor Poissons ratio Component E1 E2 E3 12 13 23 21 31 32 Mean 16.9 8.4 5.9 0.27 0.25 0.25 0.16 0.15 0.23 SD 1.5 1.2 2.2 0.04 0.04 0.04 0.06 0.07 0.08 n 13 13 13 13 13 11 13 13 12

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91 Table 6-2. Summary of known ma natee rib material properties Investigators Testing method Bone source Load or wave direction Elastic modulus (GPa) Poissons ratio Shear modulus (GPa) Yan (2002) Ultrasonic Manatee rib Transverse 18.13.77 0.253.021 7.24.711 Clifton (2005) 3-point bend MSW02531 MSW02392 Transverse Transverse E1=15.7 E1=16.4 1,2MSW0253 and MSW0239 are the IDs of two manatees used in the present study as well as the Clifton study Figure 6-1. Articulated manatee skeleton. Specimens were pulled from the middle third of each rib in the center section of the manatees to reduce vari ability between specimens. Image courtesy of Roger Reep (department of physiological scien ces, University of Florida).

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92 Figure 6-2. Visual image correlation performed on the gage section of a manatee rib bone tensile specimen. The highlighted section on the sp ecimen is the area used for the visual image correlation (VIC) analysis. Figure 6-3. Plot of the defo rmation versus position data (dv vs. dy) in the gage region of a tensile specimen. The strain at the lo ad step plotted above was 900 Strain is calculated as the slope of the curve (i.e., the change in deformation with respect to the change in position is the strain (y=dv/dy)). The open circles repres ent all data points on the gage section while the line passing through th e data points is a linear trend line used to fit the data in order to calculate strain.

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93 Figure 6-4. Stress-strain plot for a given load step during a VI C analysis. The initial slope of the stress-strain curve is the elastic modulus of the material. The elastic modulus for this particular specimen was 10.1 GPa. The +s ar e representative of experimental data points from each load step, while the solid line represents the initial slope of the stress-strain curve.

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94 CHAPTER 7 ANISOTROPIC ELASTIC SHEAR PROPERTIES Short Summary Shear testing of cortical bone presents many difficulties due to the limited thickness of cortical bone from most animals, the orthotropy of the material, and the hierarchical structure of bone. This study aims to assess a new method fo r the measurement of cortical bone shear properties that overcomes many of the challenges a ssociated with shear testing of the material. Visual image correlation (VIC) is used to a ssess three orthogonal shea r moduli of manatee rib (cortical) bone from Iosipescu specimens. Ma natee rib bones are large, posses no marrow cavity, and are mostly cortical bone, so they are id eal for preparing large specimens (such as the Iosipescu specimen) of any materi al orientation. Shear strain from VIC is calculated as the sum of the slopes of the horizontal position versus vertical deform ation and the vertical position versus horizontal deformation curves (xy=dV/dX+dU/dY). Shear modulus is taken as Gxy=xy/xy, and is reported for both VIC a nd strain gage calculations. Visual image correlation was performed directly on top of shear gages in order to provide a direct comparison of the results. Specimens were loaded on a uniaxial test ing machine at a rate of 1 mm/min. For the VIC tests, G12=4.0.8 GPa (n=3), G13=4.1.4 GPa (n=9), and G23=2.7.4 (n=5), where the 1 directi on corresponds with the proxi modistal (longitudinal) direction of the rib, the 2 dire ction corresponds with the superfi cial-deep (in-plane transverse) direction of the rib, and the 3 direction corresponds with the craniocaudal (out-of-plane transverse) direction of the ri b. For the shear gage tests, G12=4.9.4 GPa (n=3), G13=4.7.5 GPa (n=2), and G23=3.4.3 (n=6).

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95 Introduction The anisotropic shear propertie s of manatee rib (cortical) bone have not yet been investigated. The shear properties of manatee rib bone are needed in order to fully characterize manatee rib bone material properties, which once characterized, can be useful in identifying critical loading scenarios that lead to failure of entire manat ee ribs. The elastic constitutive matrix used to describe materi al behavior in response to loading is near complete with contributions from this study as well as a tens ile properties study that aimed to determine the elastic moduli and Poissons ratio s of manatee rib bone under ort hotropic assumptions (Chapter 6). Once manatee rib bone has been fully char acterized and all of the elastic constitutive constants have been identified, accurate co mputer simulations that account for material anisotropy can be performed using finite element analysis to predict stre ss states and fracture properties of manatee rib in response to loading. To this point, material properties are known for only one orientation of manatee rib bone (Clifton et al 2003; Clifton, 2005; Yan et al., 2006a, b, 2007) and computer simulations woul d not accurately depict the tr ue response of manatee rib to loading with this limited input. This study will improve on the ability of computers to predict impact response of watercraft and manatees by identifying shear properti es for three orthogonal orientations of rib bone. Shear testing was done using Iosipescu specimens to determine the sh ear modulus of three orthogonal orientations of manat ee rib bone. Iosipescu specimens we re used in order to produce nearly pure-shear in the specimen gage s ections (Iosipescu, 1967). This specimen is advantageous because test fixtur es remove the translational an d twisting motion of the upper and lower mounts caused by three-dimensional stress es generated by anisotropic materials during testing. Although size limitations of cortical bone from most an imals prevent them from being prepared into Iosipescu specimens, manatee rib bone is large relative to ot her animals, it lacks a

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96 medullary (hollow) cavity, and it has a large prop ortion of cortical bone to trabecular bone, so these specimens are appropriate for shear testin g of manatee rib bone. Shear gages have been found to be an extremely accurate means of measuring shear stra in in the test section of Iosipescu specimens (Ifju, 1994), and were used in the assessment of manatee rib bone shear modulus in this study. A non-contact deformation analysis technique called visual image correlation (VIC) (Sutton et al., 1986 and 1991; Lichte nberger and Schreier, 2002) was also used in the calculation of manatee rib bone shear modulus. Visual image correlation has been us ed to assess biological tissue response to loading with applications incl uding the investigation of tensile properties of bovine hoof horn, the mechanical behavior of arte rial tissue, and the loosening of hip implants (Zhang and Arola, 2004). To the au thors knowledge, this is the fi rst study to use VIC for the determination of shear properties of cortical bone Visual image correla tion discretizes a surface in order to provide 3-dimensional coordinates of points on the surface from images taken by two cameras placed at some distance from one another (Figure 7-1). Points on the object are then correlated between the images to provide object f eature representation in 3D space. The analysis used here required that a random sp eckle pattern be applied to the test section of the specimen in order to provide sufficient cont rast and texture to the specim en images for the correlation analysis to perform properly. Cross-correlation (CC) and normali zed cross-correlation (NCC) are commonly used methods to determine object feat ures (e.g., points, edges, etc.) for image correlation analyses. Once images have been correlated for a reference image of an unloaded specimen, the specimen is deformed and more st ereoscopic images are taken. The position of the points in the deformed specimen images are compar ed to those from the reference image (Figure 7-2), and the full-field deformation of th e specimen gage section can be found.

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97 The aims of this study were to: Determine the shear modulus for 3-orientati ons of manatee rib bone using VIC and shear gages simultaneously Characterize the anisotropy of manatee rib bone shear modulus Determine if VIC is a viable alternative to shear gages in the calculation of shear modulus by comparison of the results Compare shear modulus results w ith those from ultrasound studies Determine the repeatability of each technique by performing cyclic tests and looking at the coefficient of variation (CV) for all cycles Methods Manatee rib bones were obtained under US Fish and Wildlife permit #MA067116-0 issued to the Florida Fish and Wildlife Conservati on Commissions (FWC) Marine Mammal Pathology Lab (MMPL). Use of the tissue was governed by University of Florida Institutional Animal Care and Use Committee (IACUC) protocol #E407. Two cen ter ribs (#s 9 and 10 of 17-19 ribs) were selected for this study from two adult male manat ees (Figure 7-3). The mana tee ribs were stored with flesh intact to preserve material properties (An and Dr aughn, 2000) at -20 Celsius, a temperature chosen to avoid freezer burn and, hence, tissue property degradation. Shear specimens were taken from the middle third of the selected ribs to meet specimen size requirements and to prevent site specific variatio n in material properties (Figures 7-4 and 7-5). Iosipescu specimen dimensions are shown in Figure 7-6. The length dimension of one Iosipescu specimen orientation (ori ented to obtain shear modulus G13) was limited by the crosssectional width of manatee rib bon e, thus the specimen length for this orientation was less than that used for the other two orientations. The cros s-section of the middle portion of the ribs used in this study was elliptical in shape, with a minor axis dimension (i n the superficial-deep direction) of ~40 mm and a major axis dimens ion (in the craniocaudal direction) of ~65 mm.

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98 The middle sections of the ribs were cut into large, rectangular bl ocks using a wet band saw. The blocks were then milled to their dime nsions, while keeping the thickness dimension as large as possible. Notches were milled into the specimens by mounting the specimens at a 45 angle relative a 1.59-mm radius end-mill. The specimens were then sliced to their appropriate thicknesses using a 102-mm wafering blade attached to an IsoMet low speed saw (Buehler Ltd., Lake Bluff, IL). Specimens were kept under cons tant irrigation with wa ter during all machining operations. Specimens were wrapped in gauze, soak ed in saline solution, and refrigerated until being brought to room temperature for shear gage and VIC preparation ju st before testing. Shear gages (Vishay Micro-Measurements, Rale igh, NC) were mounted to the front and back test section of nine specimens (n=3 for G12 and n=6 for G23 specimens) and to the front test section of two G13 specimens. The test section of these and seven additional G13 specimens were then spray painted white (for increased contrast) and speckled with black sp ray paint to generate a random speckle pattern used by the VIC stereo system during image correlation (Figures 7-7 and 7-8). Specimens were fixed in mounts design ed to prevent transla tion and twisting motion caused by 3-D stresses generated by anisotropic materials during test ing. Specimens were loaded in tension on an MTI uniaxial loading machine (Measurement T echnology, Inc., Roswell, GA), resulting in the generation of sh ear strains in the test section of the Iosipescu specimen. The G12 and G23 specimens were loaded to 650 N and unl oaded, rotated 180 without removing the specimens from the mounts, and the test was run again. This was done to allow for VIC to be performed on both the front and back specimen test sections in case if results were to differ due to specimen preparation or to the test fixture. Results were averaged for both loading-unloading cycles. The two G13 specimens with a shear gage adhered to a single test section were loaded cyclically (n=11 and n=6) from 0 to 650 N while VIC was performed simultaneously with shear

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99 gage testing. The seven G13 specimens prepared for VIC alone were tested to failure. All specimens were loaded at a rate of 1 mm/min. The VIC system used to capture test secti on deformation information consisted of Cinegon CM120 lenses (Schneider-Kreuzna ch, Bad Kreuznach, Germany) mounted to a pair of Retiga 1300 digital cameras (Quantitative Imaging Corpor ation, Burnaby, British Columbia, Canada), which were arranged to form a near equilatera l triangle with the specimens. The cameras were calibrated with a dot matrix calibration grid generated using custom software provided by Correlated Solutions, Inc. and image correlation software (VIC-3D Digital Image Correlation v.2006.0.0, Correlated Solutions, Inc., West Colu mbia, SC). Load data was monitored using a 1000-lb load cell (In terface SM-1000-38, Scottsdale, AZ), received by a National Instruments SCXI-1321 terminal block designed for use w ith the SCXI-1121 signal amplifier (National Instruments Corporation, Austin, TX), and l ogged using a custom Labview program (Labview v.7.0, National Instruments Corporation, Austin TX). Correlation analyses were performed on the gage sections of loaded specimen images (F igure 7-3) using VIC3D software. Output from the VIC analyses included position and deformation for each data point on the specimen gage sections. Output files were e xported from VIC3D and Labview a nd input into a custom Mathcad program (Mathcad v.12.1, Mathsoft Engineering and Education, Inc.) for further data reduction and analysis. Shear strain was calculated from VI C data by adding the hori zontal position versus vertical deformation and the vertical positi on versus horizontal deformation curves (xy = dV/dX+dU/dY) (Figure 7-9). Shear strains calculated for each load st ep were averaged over the linear range of the stress-strain curves for each specimen tested. The strains from the frontand back-mounted shear gages were averaged to remove bending and twisting effects. Shear modulus was taken as Gxy=xy/xy for both VIC and strain gage calculations.

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100 Statistics One-way ANOVA was performed on the sh ear modulus means for all specimen orientations to determine if any means were sign ificantly different from any other orientation as determined by both VIC and shear gages. Inde pendent group t-tests we re performed on all possible pairs of specimen orientations for both VIC and shear gage results to determine which means differed (if any). An inde pendent group t-test was performed to determine if there was a significant difference between VIC and shear ga ge shear modulus means for any specimen orientation. Significan ce was taken when p<0.05. Results Results are provided in Tables 7-1 and 7-2. Mean values and standard deviations (SD) are presented for all shear moduli. Cyclic testing resulted in G13=4.6.6 (VIC) and G13=5.3.2 (shear gage) for the 11 cycles performed on specimen 1, and G13=3.6.2 (VIC) and G13=4.2.1 (shear gage) for the 6 cycles performe d on specimen 2. The coefficient of variation (CV) for specimens 1 and 2 were 9% and 6% for VIC and 4% and 2% for shear gages. Shear modulus G12 was found to be 4.0.8 (n=3) for VIC and 4.9.4 (n=3) for shear gages, an 18% difference. Shear modulus G13 was found to be 4.1.4 (n=9) for VIC and 4.7.5 (n=2) for shear gages, a 13% difference. Shear modulus G23 was found to be 2.7.4 (n=5) for VIC and 3.4.3 (n=6) for shear gages with a 21% difference. Statistical Results One-way ANOVA revealed that at least one of the specimen orientation means was different for both VIC and shear gage calculations of shear modulus (p<0.001). Independent group t-tests showed that G12 and G13 did not significantly differ fo r either VIC or shear gage calculations (p=0.9139 and p=0.5933, respectively), while G12 differed from G23 (p=0.0226 for VIC; p<0.001 for gage), and G13 differed from G23 (p<0.001 for VIC; p=0.0039 for gage).

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101 Independent group t-tests demonstrated a lack of significant difference between VIC and shear gage means for both the G12 and G13 moduli (p=0.1516 and p=0.0991, respectively), although results for G23 differed significantly (p=0.0083). Discussion This was the first study to use VIC for the determination of cort ical bone material properties and to characteri ze the shear modulus of mana tee rib bone. Shear modulus G23 was found to differ from G12 and G13, while G12 and G13 did not significantly differ from one another, implying that manatee rib bone shear modulus is transversely isotropic. Visual image correlation results agreed reas onably well with those from shear gages, which were adhered to the specimen test secti ons for simultaneous comparison with VIC. It appears that VIC is a viable alternative to sh ear gages for the calculation of shear modulus as noted from the lack of significant diffe rence between VIC and gage values for G12 and G13. The significant difference between VIC and shear gage values for G23 specimens may be in part due the fact that G23 specimens were shorter than specimens from the other two orientations. Perhaps the stress state on the front and back of the Iosi pescu specimen is influenced by specimen length, since it is possible that specimen slip can occur if not enough materi al is gripped in the mounts. The averaging of shear gage values from the front and back of the specimen would account for differences in shear stress magnitudes on the fr ont and back of a specimen. However, the VIC results would not capture this difference sinc e VIC was performed on only the front-facing test section of the specimens. Visual image correlation position and deformat ion data were put into a shear strain algorithm that averages data acro ss the entire test section. Figu re 7-11 shows the sensitivity of VIC calculated shear modulus to shear strain. Shear modulus appears rather scattered until it collapses to a single value above ~700 Shear gages directly measure shear strain on a

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102 specimen test section using a foil pattern. Shear gages are si mple to align since they are designed specifically for the Iosipescu specimen (Ifju, 1994). Shear gage re sults have proven to be quite reliable for assessing composite material shear properties, because they account for the gradient in shear strain across the specimen test section by integrating over the entire test section (Ifju, 1995). Results from this study do not entirely agr ee with those from an ultrasound study on the material properties of manatee rib bone (Yan 2002). Yans study reported a shear modulus of 7.24.71 GPa for manatee rib bone (with no men tion of which shear modulus was measured), while the largest shear modulus measured in this study was from strain gage readings for the G12 orientation (4.9.4 GPa). The differences betw een these studies may be at least partially explained by the common observation of ultras onic methods yielding higher results than mechanical testing methods. Material differences could also have influenced the difference in results. For instance, age, porosity, mineral dens ity, and location within a manatee or within a specific rib all have potential for influencing the results of a shear modulus test. Differences in values were also found between shear gage a nd VIC results in the cu rrent study: The shear modulus from the gages was always slightly higher than that from VIC. Higher shear modulus calculated from strain gages are the result of lowe r strain readings being output from the shear gages than from the VIC system. This discrepanc y can be explained by the size differences of the effective gage region for each analysis method. The gage length used for the VIC analysis is slightly less than the effective gage length for the shear gages. A shorter effective gage length results in a higher shear strain measurement, because the near zero component of the nonuniform shear strain distribution is removed wh en averaging shear strain calculation (Figure 714). It is impossible to avoid a partial reduction in size of the selected VIC gage region, because

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103 of the algorithms used by the VIC software to map localized deformations to the gage region. Therefore, measured shear strains will likely alwa ys be slightly smaller using VIC than when using shear gages. It should be noted that sh ear modulus differences we re only significant for one of the three tested orienta tions, thus the influence of the reduced gage length from the VIC analysis compared to that from the shear gages is not substantial. Conclusions In conclusion, shear modulus was found for 3 presumably primary orthogonal orientations of manatee rib bone using both VIC and strain gages. Manatee rib bone shear modulus was found to be transversely isotropic. Visual imag e correlation results cons istently over-predicted strain gage shear strain due to smaller effective gage lengths being used in the VIC analyses than the shear gage analyses, causing a slight under-prediction of shear modulus in all cases (although this finding was not statistically significant for two of the three orientations tested). Visual image correlation is slightly less repeatable than shea r gages when cyclically testing a single specimen (CV 9% for VIC and 4% for shear gages). Results from th is study did not entirely agree with those from an ultrasound study conducted on manatee rib bone, alt hough the higher ultrasound values compared to mechanical test values were consistent with findings from other researchers. Visual image correlation and shear gage results from this study agreed reasonably well, and the slight differences in values can be explained by the difference in effectiv e gage lengths between the two methods. It may therefore be possible to develop a correction fact or to account for the slight under-prediction of shear modulus by VIC. The findings from this study demonstrate that VIC is a viable alternative to shear gages in the calculation of cortical bone shear modulus.

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104 Table 7-1. Shear modulus and coefficient of variation (CV) from cyclic tests. Mean values are presented standard deviations. Specimen 1 (11 cycles) Specimen 2 (6 cycles) Cyclic tests G13 (GPa) CV (%) G13 (GPa) CV (%) VIC 4.6.4 9 3.6.2 6 Gage 5.3.2 4 4.2.1 2 Table 7-2. Shear modulus of three orthogonal orie ntations of manatee rib bone. Mean values are presented standard deviations along with the number of specimens used (n). Shear modulus (GPa) Shear tests G12 G13 G23 VIC 4.0.8 (n=3) 4.1.4 (n=9) 2.7.4 (n=5) Gage 4.9.4 (n=3) 4.7.5 (n=2) 3.4.3 (n=6) Difference (%) 18 13 21 Figure 7-1. Three-dimensional surface as seen by two imaging sensors (courtesy of Correlated Solutions, Inc, West Columbia, SC). The im age demonstrates the stereographic set-up used in VIC analyses. The depicted imagi ng planes represent the 2-D viewing area for each camera.

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105 Figure 7-2. Movement of a sample square subset used for cross-correlatio n function estimation in the VIC analysis. Image courtesy of Corre lated Solutions, Inc., West Columbia, SC. Figure 7-3. Manatee skeleton. Mo dified with permission from Yan, J., 2002. Biomechanical properties of manatee rib bone and analyti cal study using finite element analysis. M.S. Thesis, University of Florid a, Gainesville, Figure 3-2, p. 19. un-deformed image deformed image Proximodistal (PD) Superficial-deep (SD) Craniocaudal (CC)

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106 Figure 7-4. The middle third of each rib was used fo r the analysis to reduce potential site-specific variability between specimens. The coordinate system used in the analysis is defined in terms of both anatomical directions a nd Miller indices. Th e proximodistal (PD) direction corresponds to the [100] directi on in Miller indices, wh ile the superficialdeep (SD) direction corresponds to the [ 010] direction and the craniocaudal (CC) direction corresponds to the [001] direction. PD [100] SD [010] CC [001]

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107 Figure 7-5. Orientation and secti oning of the shear specimens used in this study. The number of specimens available from each section depended on the dimensions of that section. Six to eight specimens were obtained for the G31 (reported herein as G13) orientation, while eight to ten could be prepared for the G21 (reported herein as G12) orientation. The number of specimens available from the G31 and G13 orientations depended on the length of the section cut for those orientations, rather than being limited by the ribs cross-sectional dimensions. Figure 7-6. Iosipescu specimen dimensions. The length of the G23 specimens was limited by the physical dimensions of the manatee rib cross section. t 3.18 mm L=34.9 mm ( G 2 3) or 76.2 mm ( G1 2 and G13) W=19.05 mm H=11.43 mm R=1.59 mm (90) PD SD CC G31 G21 G32 G23 Note that G12=G21 G13=G31 G23=G32

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108 Figure 7-7. Iosipescu specimen prepared for simu ltaneous shear gage and VIC testing. The test section first had a shear gage applied followed by being spray painted white and speckled with black spray paint for the VIC analysis. Figure 7-8. Image showing the test section and loading of an Iosipescu specimen prepared for simultaneous VIC and shear gage testing. A uniaxial load is applied to the specimen (left), while a shear gage and the VIC system measure deformations in the test section (left and middle images). An actual image us ed in the VIC analysis is shown in the image to the right. The image is partiall y obstructed by the large Iosipescu specimen holders, although the stereographic set-up was ar ranged to allow for the test section to be in view of both cameras. Image taken by VIC camera Test section F F

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109 Figure 7-9. Plots showing the high correlation of the dU/dY and dV/dX data points across the entire test section of an Iosipescu VIC sp ecimen. The blue dots represent VIC data points captured from the test section at a given load level, while the slope of the linear trend line was used in the shear strain algorithm (xy=dU/dY+dV/dX). Figure 7-10. Typical stress-strain plot obtained from VIC and sh ear gages. The peak strain measured by VIC (dots) is larger than that measured by shear gages (line).

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110 Figure 7-11. Sensitivity of VIC analysis to shear strain magnitude. Note the convergence of shear strain values at this load step at strains approaching and above 1000 This plot demonstrates the inability of the VIC syst em and shear strain algorithm to accurately capture strains below this cut-off value.

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111 Figure 7-12. Plot showing how VICcalculated shear strain accurately follows cyclic load curves. Note that this plot was not used in the cyclic analysis and is presented only to demonstrate how well VIC can track load even at low load levels. Figure 7-13. (Left) U and (right) V deformation contour plots for th e test region of an Iosipescu shear specimen during a VIC analysis. For the pictured load step, U deformations ranged from 0.15 (purple) to 0.41 mm (red), while V deformations ranged from -0.59 (red) to 0.71 mm (purple).

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112 Figure 7-14. Shear strain distribut ion and effective shear gage a nd VIC gage lengths in manatee rib bone Iosipescu specimens. Note that shea r strain can vary th rough the thickness of a specimen (dark blue region in the left image) and that shear strain is non-uniform across the test section. Also note the difference in average shear strain measured by VIC and shear gages resulting from the reduced effec tive VIC gage length (ri ght). The blue region represents shear strain measured by VIC, while the orange region represents the additional shear strain measured by shear gages.

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113 CHAPTER 8 MATERIAL SYMMETRY ANALYSIS Short Summary Traditionally, bone is presumed to be either transversely isotropic or orthotropic with no verification of the materials symmetry. Mode ls that utilize incorrectly defined material properties (often assuming the mate rial is isotropic) are prevalen t in the literature, although the accuracy of such models must be called in to question. A proper material properties characterization analysis requires verification of the material sy mmetry. In Chapters 6 and 7, experiments were performed on tensile and shear specimens, respectively, oriented about three axes presumed to be principal material axes. These tests were used to determine the nine independent elastic constants re quired to define the constitutiv e behavior of an orthotropic material. Miller indices were used to define material axes, with the [100] direction defining the proximodistal axis, [010] defining the superficia l-deep axis, and [001] defining the craniocaudal axis in anatomic coordinates. This study exam ines the symmetry of the material by comparing predicted and experimentally determined strains in off-axis specimens. Tensile tests were performed on specimens oriented along three off-a xis directions ([110], [101], and [011]). Visual image correlation was used as the experimental meas ure to determine the strain in each off-axis specimen, while three material symmetry mode ls (isotropic, transversely isotropic, and orthotropic Hookes law) were each used to pr edict strain the off-axis directions from the previously determined nine independent el astic constants. The material symmetry was determined by quantifying the difference between th e predicted and experimental strain values and choosing the model that best fit the experime ntal data. The agreement between predicted and

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114 experimental strains in all three off-axis direct ions was the greatest for the orthotropic model, thus it was concluded that mana tee rib bone is orthotropic. Introduction Cortical bone is generally assumed to be eith er transversely isotr opic (Yoon et al., 1976; Gotzen et al., 2003; Dong and Guo, 2004) or or thotropic (Ashman et al ., 1984; Hoffmeister et al., 2000). However, very little effort has gone into verifying the symmetry of the material. Part of the difficulty in characterizing the symmetr y of bone comes from specimen size limitations. Cortical bone is generally quite thin depending on the location and the animal from which the specimens are harvested (An and Draughn, 2000). In cancellous bone, size limitations have been overcome by the use of micro-finite-element (FE) models generated from high resolution images of the trabecular struct ure (van Rietbergen et al., 199 6; van Rietbergen et al., 1998). Testing can take place along any number of direc tions using this FE-based direct mechanics approach, and the elastic anisotr opic constitutive behavior of th e bone can be found. In cortical bone, microand ultra-structural constituents are responsible for the anisotropy of the material (Turner et al., 1995) and a similar FE approach can be used to model the ma terial (Gotzen et al., 2003). However, the technique has not yet been used to fully reveal cortical bone elastic anisotropic constitutive properties. The method that Gotzen et al. (2003) us ed to reveal localized properties is quite involved because it requir es the histological sectioning, imaging, and modeling of many slices of bone, thus the analysis is rather time consuming. Yang et al. (1999) made use of a previously developed methodology for averaging anisotropic elastic constants called spectral decomposition (M ehrabadi and Cowin, 1990) in order to demonstrate the compositional dependence of the elastic constant s of cancellous bone and wood. While the wood data was assumed to be orthotropic from the st art, the cancellous bone data was collected in a

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115 sufficient manner to fully populate the anis otropic elastic constants matrix using FE. Yang et al. demonstrated with 95% certainty that can cellous bone is orthot ropic. Due to the a priori assumption for the material symmetry of wood, no such conclusion could be made to verify the orthotropy of the material (i.e., not enough data was available to statis tically reduce the wood dataset as was done for cancellous bone). The pres ent study demonstrates that with the additional testing of an off-axis (non-principal) sp ecimen orientation, th at testing under an a priori assumption of material symmetry is sufficient fo r concluding the symmetry of a material. The specific objective of the present investigation is to determine the symmetry model that best fits experimental data collected from off-axis specimens. This objective is accomplished by comparing experimentally determined strain s in off-axis specimens with those found by transforming the isotropic, transversely isotropi c, and orthotropic compliance matrix in Hookes law. An essential component of the present analys is is to identify th e principal material directions of manatee rib bone. Th e longitudinal and two transverse axes are chosen according to the geometry of the bones cross-section. A caveat here is that principal material axes of micro and ultrastructural bone structures may not necessa rily align with external geometry (Turner et al., 1995), but given the structur e-function relationship of bone (Cowin, 2002), a good initial guess for the principal material axes of an ellip tically-shaped bone such as the rib places the principal axes along the major and minor axes of the cross-section. An alternative approach for determining the principal axes of bone would be to examine the microstructure (Wirtz et al., 2000) and orientation of constitu ents of the tissue (Gotzen et al., 2003). In our study, it is assumed that principal material directions correspond to macroscopic bone geometry.

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116 The nomenclature used in our analysis crosse s several disciplines in order to simplify mathematics involved with coordinate system tr ansformations as well as to make clear the specimen orientations relative the anatomic coor dinate system. Miller indices, traditionally used in the analysis of crystalline materials, were used to define material axes with the [100] direction defining the proximodistal axis (a lso referred to as the 1-axis), [010] defining the superficialdeep (2) axis, and [001] defining the cranial-caudal (3) axis in an atomic (material) coordinates. Chapters 6 and 7 provide the results require d to fully populate the orthotropic compliance matrix. Manatee rib bone tensile modulus was f ound to differ in all thr ee orthogonal directions (Chapter 6), while shear modulus was found to vary in one direction but not the other two (Chapter 7). Although it would appe ar that manatee rib bone is or thotropic at the macroscopic length scale due to findings from the tensile tests, the symmetry must be verified to ensure that the orthotropic material model best predicts material response to loading in non-principal directions. Our analysis is used to discern which material model best predicts strains in off-axis specimens. The method utilized in our study fo r examining the material symmetry of a previously uncharacterized material minimizes the number of tests required to verify the symmetry of the material, since only one additional specimen axis is required to be tested above and beyond the number of tests required for the material model chosen for the analysis. Material Symmetry There are a total of eight type s of linear elastic material symmetry: monoclinic, triclinic, orthotropic, cubic, tetragonal, trigonal, hexagonal (transvers ely isotropic), and isotropic (Mehrabadi and Cowin, 1990). The case of no material symmetry is called anisotropy. Our analysis will discern which material characte rization best matches th e observed constitutive response of manatee rib bone. In order to achi eve this goal while minimizing the number of required tests, manatee rib is initially assumed to be orthotropic, which, other than monoand tri-

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117 clinic, exhibits the least symmetry. Our analysis is not capable of discerning which lesser symmetry scenario best fits the material or if anisotropy is a better fit, but it will distinguish between greater symmetries. Hookes Law Hookes law relates the stresses to the strains in a material Various forms of Hookes law exist depending on the number of elastic constants required to define the constitutive matrix. Several forms of Hookes law are presented to describe the elastic constitutive response of materials with different material symmetries. The general form of Hookes law is provided in Equation 8-1. jijis (Eq. 8-1) where i and j are the components of the stress and strain tensors, respectively, and ijs is the compliance matrix for i,j=1. Equation 8-1 can be rewritten in terms of the stiffness matrix, ijc, which can be used to solve for stresses from strains. iijjc (Eq. 8-2) Material properties can be cal culated from Hookes law given that the stresses and strains are known. Example 8-1 demonstrates the calcu lation of material properties from strain measurements and applied loads. Isotropic Materials The simplest form of Hookes law is used to describe isotropic materials. Isotropic materials generally have randomly distributed constituents, which causes these materials to have properties that are inva riant to rotation (Bores i and Schmidt, 2003). This means that isotropic

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118 materials respond identically to lo ading in all directions. The comp liance matrix for an isotropic material is written as follows: 11 000 1 000 1 000 1 00000 1 00000 1 00000 EEE EEE EEE cs G G G (Eq. 8-3) where E is the elastic modulus, G is the shear modulus, and is the Poissons ratio of the material. There are two independent elastic cons tants in an isotropic material (any two of: E, and G, where G=E/[2(1+)]). Transverse Isotropy Transverse isotropy exists when a material exhibits isotropy in one material plane. An example of a transversely isotropic material is a fiber reinforced composite with uni-directional fibers. The plane that cuts across the fibers is the transverse plane. The compliance matrix for a transversely isotropic material is as follows:

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119 11 000 1 000 1 000 2(1) 00000 1 00000 1 00000LL LTT LT LTT LT LTT T T L LEEE EEE EEE cs E G G (Eq. 8-4) where ET is the elastic modulus in the transverse plane, EL is the elastic modulus normal to the transverse plane, GL is the shear modulus in the directi on normal to the transverse plane, T is the Poissons ratio in the transverse plane, and L is the Poissons ratio normal to the transverse plane of the material. There are 5 independent elastic constants in a transversely isotropic material (ET, EL, T, L, and GL). Orthotropy Orthotropy is generally the l east symmetry used to descri be bone in the literature. Orthotropy implies that material properties vary in three or thogonal directions (i.e., three orthogonal planes of material symmetry exist in an orthotropic material). The compliance matrix for orthotropic materials is as follows:

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120 31 21 123 32 12 123 1323 1 123 23 31 121 000 1 000 1 000 1 00000 1 00000 1 00000 EEE EEE EEE cs G G G (Eq. 8-5) where indices 1, 2, and 3 represent three primar y orthogonal directions. There are 9 independent elastic constants in an orthotropic material ( E1, E2, E3, 12, 13, 23, G12, G13, G23). Note that ijjiGG and i ijE = j jiE for ij=1, 2, 3 due to symmetry of the ma terial and the elastic constants matrix. Anisotropy Anisotropic materials respond differently to loads applied in all directions. A fully anisotropic material has a fully populated el astic constants matrix (Equation 8-6). The 36 coefficients reduce to 21 independent elastic co nstants due to symmetry of the elastic constants matrix. 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566ssssss ssssss ssssss s ssssss ssssss ssssss (Eq. 8-6)

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121 Transformations in Anisotropic Materials Lekhnitskii (1968) is well referenced for his wo rk in the area of transforming stresses and strains from material to specimen coordinates a nd vice versa when the two coordinate systems do not align with one another. An analysis of a speci men with its loading axis rotated away from its principal material direction requires a transf ormation of the elastic constants matrix (see Example 8-2). Given the compliance matrix for the principal material direc tions and strains in a specimen loaded in a non-princi pal direction, the co mpliance matrix can be transformed to predict the experimental st rains if an appropriate material mode l is selected for Hookes law. For example, if a material is ort hotropic, we should be able to apply orthotropic Hookes law to predict strains in off-axis specimens (i.e., spec imens not aligned along principal material axes). Our analysis uses this technique to determ ine the material symmetry of manatee rib bone. Example 8-3 demonstrates the use of the transf ormed compliance matrix for the calculation of strains in a specimen loaded along a non-principal direction. Thermodynamic Restrictions on Elastic Constants There are two requirements that must be met in order to ensure that thermodynamic restrictions imposed on elastic materials are satisfied (Cowin, 2002). The physical significance of these requirements is that the work done on an elastic material must be positive (e.g., tensile loads produce positive extension and hydrostatic co mpression will not result in expansion of the material) (Bertholet, J., 1999). The requirements are presented below in equation form. These equations will be used to verify the validity of experimentally determined elastic constants. Requirement 1 The stiffness and compliance matrices must be positive: 0 , ,66 55 44 33 22 11s s s s s s,

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122 0 0 033 13 13 11 33 23 23 22 22 12 12 11 s s s s s s s s s s s s, and 033 23 13 23 22 12 13 12 11 s s s s s s s s s. (Eq. 8-7) Requirement 2 The following conditions for the components of the stiffness matrix must be satisfied: 0 , ,12 31 23 3 2 1G G G E E E, 233213311221(1),(1),(1)0 and 0 2 113 32 21 13 31 32 23 21 12 (Eq. 8-8) Tensile Testing Manatee rib bones were obtained under US Fish and Wildlife permit #MA067116-0 issued to the Florida Fish and Wildlife Conservati on Commissions (FWC) Marine Mammal Pathology Lab (MMPL). Use of the tissue was governed by University of Florida Institutional Animal Care and Use Committee (IACUC) protocol #E407. Ribs from the center of se veral adult manatees (numbers 9-10 of 17-19 ribs) were selected for th is study. The manatee rib was stored with flesh intact to preserve material properties (An a nd Draughn, 2000) at -20 Celsius. Specimens were prepared from the middle third of the selected ri bs to conform to testing procedures outlined in Chapters 6 and 7, thus reducing variability associat ed with site-specific ma terial properties. Dogbone shaped specimens with a square cross-sect ion were machined to the following dimensions using a vertical end mill: length = 38 mm, widt h = 12.7 mm, gage length = 19 mm, gage width =

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123 6.4 mm, radius at gage section = 3.2 mm. Four to ten specimens were prepared in each of the [110], [101], and [011] directions (Figure 8-1). The harves t angles for off-axis specimens were measured and found to be within 15 of their targ et angle of 45 off-axis from the principal axes in each principal plane. All machining was done under constant irrigation with water to prevent over-heating. Specimens were wrapped in gauze, so aked in saline solution, and refrigerated for up to 1 week prior to testi ng. Specimens were wet when they were loaded to ~1100 at a rate of 1 mm/min at room temperature with a uniaxial loading machine (MTI 30K, Measurement Technology, Inc., Roswell, GA). The load level was selected from within th e linear range of the material based on results from a pilot study in which several air-dried specimens from each presumed primary orientation were loaded to fa ilure (Chapter 5). Loads were recorded for each specimen from readings taken by a 1000-lb load cell (Interface SM-1000 -38, Scottsdale, AZ). Strains were acquired using visual (digital) image correlation (VIC) (Figure 8-2). Strains measured by VIC are considered valid between 500 and 500% when using VIC3D software (VIC-3D Digital Image Correla tion v.2006.0.0, Correlated Solutions, In c., West Columbia, SC). In the current analysis, Poisson (lateral) strains fell below the 500 level, so normal strains (ranging from three to ten times Poisson strains) were used for comparison in the orthotropic verification analysis. Material Symmetry Analysis In the prerequisite studies (C hapters 6 and 7), manatee rib sh ear and tensile properties were measured along the major and minor axes of the transverse cross section and along the longitudinal direction of the rib bone (Table 8-1) thus completing the constitutive matrix relating stresses to strains for this pres umably orthotropic material. The constitutive matrix can now be transformed into any coordinate system desired. In our analysis, three sets of coordinate system

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124 transformations were performed, consisting of a 45 rotation about each of the presumed principal axes within the principal planes to correspond to the experimental specimens. The rotation was done for each of three material mo dels (isotropic, transv ersely isotropic, and orthotropic) in order to demonstr ate which model best predicts experimentally measured strains. A 500-N load was selected for implementation into the Hookes law models, since most of the experimental data contained stra in data for this load level. The material model yielding the lowest error when comparing predicted strains to experimental strains is the material model of choice for future analyses involving the manat ee rib bone. If none of the models reasonably predict experimental strains, tw o potential conclusions could be drawn: 1) the material is of lesser symmetry than orthotropic, or 2) the axes selected for the original material property collection studies (Chapters 6 and 7) are not the principal materi al axes. To test the whether manatee rib bone material properties exhibit less than orthotropic symmetry, additional experiments would be required in a number sufficient to complete the elastic constants matrix for whichever material model is being tested. To dete rmine if additional testing is necessary, an error tolerance was chosen for our pred icted strains to be plus or minus one standard deviation (SD) from the mean experimental strain. Predicted strains within one SD of the mean suggest that the material model yielding this result is a pr obable material symmetry for manatee rib bone. Results Table 8-1 shows the results from Chapters 6 and 7 including nine independent elastic constants, their standard deviat ions, and the number of specimens used to calculate each. The values found in Chapters 6 and 7 satisfy thermodyna mic restrictions listed in Equations 8-7 and 8-8. Table 8-2 shows the results from the off-axis Hookes law tests for various material models. Table 8-3 shows the measured normal strain values (mean SD) for specimens loaded in each off-axis direction. Table 8-4 shows the percent e rror associated with each material model when

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125 comparing results to those obtained experiment ally. It was found that the orthotropic model produced strains within one standard deviati on from the mean for each orientation tested experimentally, while the transversely isotropic model was within one standard deviation from the mean for two of the three orientations, and the isotropic model only produced strains within one standard deviation of the mean for one or ientation. Although the tr ansversely isotropic model predicted strains quite well, it did not meet the criteria defined in this paper for selection as the material symmetry model for manatee ri b bone. The only model to meet the criteria was the orthotropic model. It is th erefore concluded that manatee rib bone material properties are orthotropic. Material property plots for this orthotropic ma terial take the form of an ellipsoid with vertices equal in magnitude to the material pr operties measured about each respective principal axis (Figure 8-3). The amount of deviation from a spherical shape is propor tional to the value of the anisotropic ratios between material axes. Discussion Manatee rib material properties were previously measured onl y about the longitudinal axis of the bone (Yan, 2002; Clifton, 2005; Yan 2006a, b, 2007). Chapters 6 and 7 were used to expand the dataset for this material by obtaining nine independent elasti c constants of manatee rib bone under the assumption that the material is ort hotropic. The analysis presented in this study was used to quantify the er ror in modeling bone using orthot ropic Hookes law for off-axis specimens (i.e., specimens aligned in non-princi pal directions). Relatively few tests were required to determine which symmetry (orthotropy or greater) exists or if symmetry is less than orthotropic: Three pres umed principal directions are test ed to generate the orthotropic compliance matrix and at least one off-axis direc tion is experimentally tested for comparison to strain predictions from Hookes law. A limitation of the present me thod is that it is incapable of

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126 distinguishing between lesser symmetries than assu med in the experimental analysis. Therefore, the least symmetry expected for the material sh ould be assumed for the experimental component of the analysis. The distinction between greate r symmetries is only po ssible if testing those particular symmetry models using Hookes law. In Chapter 6, tensile properties were found to significantly vary in all three directions, thus orthotropy is the least symmetry possible for manatee rib bone. Our present analysis demonstrat ed that models with greater symmetry than orthotropic did not produce strain s within allowable tolerances as defined in this chapter for selection as the material symmetry model for manatee rib bone. Furthe rmore, our analysis confirms the finding that manatee rib bone is orth otropic by showing that error criteria were met and this model is valid for the material. Although we chose to apply our analysis technique to verify the material symmetry of manatee rib bone, the technique c ould be applied to any material exhibiting linear-elastic behavior.

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127 Table 8-1. Mean tensile and sh ear properties of manatee rib bone (from Chapters 6 and 7, respectively) used in the Hookes law anal yses. Components selected for each of the material models (e.g., orthotropic, transver sely isotropic, and isotropic) are also provided. Standard deviations (SD) are presented along with the number of specimens used to obtain each constant (n). Property Elastic moduli (GPa) Major Poissons ratios Shear moduli (GPa) Component E1, EL, E E2 E3, ET 12 13, L 23, T, G12 G13, GL G23, GT, G Mean 16.9 8.4 5.9 0.27 0.25 0.25 4.0 4.1 2.7 SD 1.5 1.2 2.2 0.04 0.04 0.04 0.8 0.4 0.4 n 13 13 13 13 13 11 3 9 5 Table 8-2. Predicted strain as a function of orie ntation and material model. The strains were obtained by rotating the compliance matrix fo r various material models (isotropic, transversely isotropic, and or thotropic) to conform to off-axis specimen orientations ([110], [101], and [011]) and applyi ng a 500 N load using Hookes law. Predicted strain (Hookes law) Orientation Isotropic Transver sely isotropic Orthotropic [110] 0.00142 0.00129 0.00122 [101] 0.00142 0.00129 0.00137 [011] 0.00142 0.00210 0.00186 Table 8-3. Strain measured by visual image corre lation. Values are provid ed as a function of specimen orientation for off-axis specimens ([110], [101], and [011]) with an applied load level of 500 N. Measured strain (visual image correlation) Orientation Mean SD n [110] 0.00126 0.00019 6* [101] 0.00121 0.00018 5** [011] 0.00187 0.00020 6 *Results from four outlier specimen s were removed from the mean and SD calculations **Results from three outlier specimens were removed from the mean and SD calculations Table 8-4. Error in Hookes law pr edicted strains as a function of orientation and material model. Percent error was calculated as (measured-predicted)/measured*100%. Percent error in Hookes law predicted strain Orientation Isotropic Transver sely isotropic Orthotropic [110] -13% -2% 3% [101] -17% -7% -13% [011] 24% -12% 1%

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128 Figure 8-1. Specimen orientations used in the present analysis. Figure 8-2. Typical vertical position versus vertical deformation (dV/dY) curve (right) showing data points from the gage section of a ma natee rib bone tensile specimen (upper left). The data points were generated using vi sual image correlation. The normal strain (slope of the curve) is 1100 for this particular specimen and load step. A linear trend line fits the data reasonably well (R2=0.821), hence the strain distribution along the gage length of the specimen is fairly uniform. Inhomogeneities, if present, would cause there to be a non-uniform strain dist ribution noted by a change in slope of the dV/dY curve. The 3D plot (lower left) maps the contour of the vertical deformations (V) onto the specimen surface. Values of V vary from -0.129 (red) to -0.135 mm (purple). [011] [101] [110] [001] [100] [010]

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129 A B

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130 C D Figure 8-3. Surface plots showing th e variation in material propert ies with material axes: A) elastic modulus, B) shear modulus, C) ma jor Poissons ratio, D) minor Poissons ratio. The values used to generate the minor Poissons ratio plot are listed in Chapter 6. The surface plot for an orthotropic material is an ellipsoid, whereas an isotropic material forms a sphere. Note that the ma terial property axes coincide with the principal material directions. That is, for example, E1, E2, and E3 coincide with the [100], [010], and [001] material directions, respectively. Example 8-1. Calculate the elastic modulus and Poiss ons ratio for the following scenario:

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131 Strains have been recorded on the surface of a specimen using a biaxial strain gage. Foil orientations correspond to the load ing axis direction [010] and the direction perpendicular to the loading axis [100]. Assume the material is orthotropic. Prov ide your response in variable form. Solution Orthotropic Hookes law relates the stresses to the strains using the following relationship: 31 121 23 123 32 122 13 1 123 2 13233 12 123 3 23 23 23 31 31 12 31 12 12000 000 000 00000 00000 00000 EEE EEE EEE G G G (Eq. 8-9) where 2ijijjiij for 1,2,3 ij y [010] x [100] z [001]

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132 The load is applied in the [010] material direction, thus the only component of stress acting on the system is 2 which is calculated by dividing applie d load by the crosssectional area of the specimen gage section. Break Equation 8-9 into 6 equations. 31 121 123 123 32 122 213 123 13233 312 123 23 23 23 31 31 31 12 12 12EEE EEE EEE G G G (Eq. 8-10) Because 2 is the only applied stress, set all othe r stresses equal to 0. Also note that 3 was not measured and should therefore be excluded from further calculations. 21 12 2 2 2 2E E (Eq. 8-11) Solve for E2 and 12. 222/E 1212/

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133 Example 8-2 Calculate the compliance matrix, [s], for a tensile specimen orie nted in the [110] material direction assuming the specimen underwent in-plane rotation about the [001] material coordinate system axis. Figure 8-5. Rotation of the materi al coordinates to specimen coor dinates for a specimen lying in the [110] material direction. Given: 10.0630.0160.015000 0.0160.0420.012000 0.0150.0120.059000 0000.14300 00000.1520 000000.144sGPa Solution Define the specimen coordinates in the material coordinate system. y [010] x [100] z, z [001] y [-110] x [110]

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134 111213 212223 313233['][110]1,1,0 ['][110]1,1,0 ['][001]0,0,1XaxisLLL YaxisLLL ZaxisLLL Solve for the direction cosines matrix ija. 111111 111 222222222 111213111213111213 23 2122 222 222222222 212223212223212223 313233 333 222222222 313233313233313233,, ,, ,, LLL LLLLLLLLL L LL LLLLLLLLL LLL LLLLLLLLL (Eq. 8-12) 111 222 3330.7070.7070 0.7070.7070 001 a Input the components of th e direction cosines matrix into the Q-matrix. 222 123233112 222 123233112 222 123233112 112233233213311221 112233233213311221 11223222 222 222 Q 3233213311221 (Eq. 8-13) Compute the compliance matrix in terms of the specimen orientation 'TsQsQ (Eq. 8-14)

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135 10.0540.0180.014000.01 0.0180.0540.014000.01 0.0140.0140.059000.003 0000.1470.00450 0000.00450.1470 0.0110.0110.003000.137 sGPa Example 8-3 Calculate the stresses and strains in a specimen when it is loaded in the [110] orientation (Figure 8-5), reporting values in both specimen and material coordinate systems. Use the compliance matrix from Example 8-2. Note that the values from the compliance matrix correspond to units of GigaPascals (GPa). Assume an applied stress along the specimen [100] axis (which corresponds to the material [110] ax is) of 0.1 GPa. Note that an apostrophe after a variable signifies that the values for that variable are reported in specimen rather than material coordinates. Given: 1 2 3 23 31 12' 0.1 0 0 0 0 0 GPa Solve for the stresses in the material coordinate system. 1' Q

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136 1 2 3 23 31 120.05 0.05 0 0 0 0.05 GPa Solve for the strains in the material coordinate system. s 1 2 3 23 31 120.00235 0.00130 0.00135 0 0 0.00720 Solve for the strains in the specimen coordinate system. 'TQQ 1 2 3 23 31 12' 0.00543 0.00178 0.00135 0 0 0.00105

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137 CHAPTER 9 ANISOTROPIC FRACTURE PROPERTIES OF MANATEE RIB (CORTICAL) BONE: A NUMERICAL AND EXPERIMENTAL INVESTIGATION Short Summary The fracture toughness of manatee rib (cortica l) bone was found using a finite element (FE) model incorporating an anisotropic stress in tensity factor formulation. Experiments were performed on six to twelve notched double-edge crack tensile (wet) bone specimens for each of six crack orientations in order to obtain fracture parameters used in the FE model. The specimens were loaded to failure at room temperature at 1 mm/min. Tensile loading in the longitudinal direction with cracks propagated in the out-of-plan e transverse direction re sulted in the greatest fracture toughness ( K13 IC), while specimens loaded in the out-of-plane transverse direction with cracks propagated in the in-plane transverse direction had the lowest fracture toughness ( K32 IC), where superscript 1 represents the proximodistal (longitudinal) axis, 2 denotes the superficialdeep (in-plane transverse) axis, and 3 denotes th e craniocaudal (out-of-plane transverse) axis in anatomic coordinates. Anisotropy was extremely significant in this st udy given the ~4-fold difference between the maximum and minimu m fracture toughness values of various orientations. The numerical tests resu lted in fracture toughness values of [ K12 IC K13 IC K23 IC K21 IC K31 IC K32 IC]=[3.2 3.4 1.0 1.1 1.0 0.8] MPa m. Results from traditional, isotropic-based experimental calculations ([ K12 IC K13 IC K23 IC K21 IC K31 IC K32 IC]=[3.2 3.2 0.9 1.1 0.9 0.7] MPa m) agreed well with those from the numerical analysis The procedures used in the present analysis can be used as a template for further cortical bo ne fracture studies that aim to discern a broader spectrum of fracture resistance values by combin ing numerical simulation with experimental fracture parameters.

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138 Introduction A substantial body of literature exists on the fracture mechanics of human and bovine bone. The bulk of the literature is confined to re ports of fracture properties for a single mode of fracture for one orientation of bone (Cooke et al ., 1973; Pope and Murphy, 1974; Wright and Hayes, 1976; Wright and Hayes, 1977; Alto an d Pope, 1979; Behiri and Bonfield, 1980; 1984; Moyle and Gavens, 1986; Norman et al., 1995; Zioupos and Currey, 1998; Yeni and Norman, 2000). Most of the existing literatur e neglects the anisotropic nature of bone, and hence, valuable information on orientation-dependent fracture prop erties is not always presented. There exists only a small body of literature that investigates the fracture properties of cortical bone as a function of material orientatio n (Bonfield, et al., 1985; Bonf ield, 1987; Behiri and Bonfield, 1989; Wang and Agrawal, 1996; Hoffmeister et al ., 2000; Lucksanambool et al., 2001; Nalla et al., 2005b). None of these studies, however, re port the six values of fracture toughness necessary to fully describe the fracture properties of an or thotropic material (Schach ner et al., 2000) such as bone. In addition, all of the studies to date have used experimental calculations valid only for isotropic materials (Stanzl-Tschegg et al., 1995 ). Yan (2002) modeled manatee rib bone with isotropic material properties using 3-D finite element analysis (FEA) in order to determine how an entire rib behaves in response to transverse loading. Paruchuru et al. (2002) modeled bone as a 2-D, isotropic material using FEA to determ ine the fracture toughness of small cortical bone specimens. Norman et al. (1992) modeled cortic al bone fracture using 2-D FEA to compare several fracture property formulations. Groteenbo er and Weersink (1982) used a 2-D model for single-edge notched specimens of human cortical bone. None of these models accounted for the effects of anisotropy in 3-dimensions, and hen ce the models are unable to cope with the 3-D stresses generated by material anisotropy (Arakere and Swans on, 2002; Arakere et al., 2005; Ranjan and Arakere, 2007; Knudsen and Arakere, 2006). In this study, an anisotropic stress

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139 intensity factor (SIF) formulation was used in combination with experimentally determined anisotropic elastic constants and fracture parame ters to determine the mode I (opening mode) critical SIF, or fracture toughness ( KIC), for each of six crack orient ations. To the best of the authors knowledge, this is the first study to combine 3-D numerical simulation of cortical bone fracture using anisotropic stress intensity factor formulations and experiment to analyze the fracture behavior of manatee rib bone as a function of its orientation. Materials and Methods Manatee rib bone is primary plexiform bone and does not contain a medullar canal. As such, larger specimens can be obtained from ma natee rib bones than from bones of most other animals. Manatee rib bones were obtained under US Fish and Wildlife permit #MA067116-0 issued to the Florida Fish and Wildlife Co nservation Commissions (FWC) Marine Mammal Pathology Lab (MMPL). Use of the tissue was gove rned by University of Florida Institutional Animal Care and Use Committee (IACUC) protocol #E407. Ribs from the cen ter of several adult manatees (#s 9-10 of 17-19 ribs) were selected for this study (Figure 9-1). The manatee rib was stored with flesh intact to preserve material properties (An and Draughn, 2000) at -20 Celsius. The middle third of the selected rib was used for testing, since Yan (2002) found that the fracture properties of manatee rib bone vary from the midd le of each rib to the ends of the bone. Close attention was paid to the orientation in which the specimens were cut from the middle third of the rib bone (Figure 9-2). Dog-bone shaped sp ecimens with a square cross-section were machined to the following dimensions using a vertical end mill: length = 38 mm, width = 12.7 mm, gage length = 19 mm, gage width = 6.4 mm, radius at gage section = 3.2 mm. Notches were introduced on opposite sides of the specimens us ing a low speed wafering saw (Isomet low speed saw, Buehler Ltd., Lake Bluff, IL). All machin ing was done under constant irrigation with water to prevent over-heating. A v-notch (used in place of a pre-crack) was introduced at the notch tips

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140 by pressing a sharp razor blade into the specime n (Yan, 2006b). Yan estimated the radius of the v-notches in his manatee rib bone sp ecimens to be on the order of 1 m when using this same technique as applied in the pr esent study. The resulting crack tip radius is governed by the material rather than the sharpness of the razor bl ade, since the razor blade effectively acts as a wedge that forces the material to crack to a char acteristic notch radius specific to the material. Specimens were wrapped in gauze, soaked in sa line solution, and refrigerated for up to 1 week prior to testing. Notch lengths were measured from digital photographs taken of the specimens prior to loading. The wet bone specimens were load ed to failure at a rate of 1 mm/min at room temperature with a uniaxial loading machin e (MTI 30K, Measurement Technology, Inc., Roswell, GA). Failure load was recorded for each specimen from readings taken by a 1000-lb load cell (Interface SM-1000-38, Scottsdale, AZ). Failure loads and notch lengths were averaged for each specimen orientation and the coefficient of variation ( CV ) was noted for each orientation ( SD CV where SD is the standard deviation and is the mean). A three-dimensional model of the notched, double edge crack tensile specimen was generated using CAD software (Pro-Engineer Wild fire 3.0, Parametric Technology Corporation, Needham, MA). Notch lengths were the only varying parameter between the models for each specimen orientation. A v-notch (0.2 mm (Yan 2006b)) was introduced at the tip of the right notch before exporting the file in IGES format The CAD models were imported into finite element software (ANSYS 10.0, ANSYS, Inc., Ca nonsburg, PA) for further analysis. Orthotropic material properties found during a prerequisite experimental study (chapter s 6 and 7) were input into the finite element (FE) model ( E1=16.9 GPa, E2=8.4 GPa, E3=5.9 GPa, 12=0.27, 13=0.25, 23=0.25, G12=4.0 GPa, G13=4.1 GPa, G23=2.7 GPa). A crucial step of the analysis was to implement direction cosines that transform th e element coordinate system into material

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141 coordinates. The material coordinates were ve rified by comparing numerical stress analysis results from an un-notched specimen of each or ientation with analytic al results. Next, a 2dimensional finite element mesh was applied to an area of the specimen associated with the right notch, where concentrated key points were applie d at the crack tip to generate quarter-point (singular) elements at that lo cation. The 2-D mesh was swept through the thickness of the specimen using 20-node brick elements (solid 95 ). A free mesh (also constructed using 20-node brick elements) was generated for the rest of the model. Boundary conditions that closely resembled the physical constraints of the test fixt ure were applied to the appropriate nodes of the models along with experimentally-determined failure loads. Swenson and Ingraffea (1987; 1988a; 1988b) at the Cornell Fracture Group ha ve developed FEA based fracture software: FRANC2D and FRANC3D. With an adaptive mesh technique, this fracture software can simulate crack growth without prescribing the cr ack path. Their work has provided fundamental understandings for the simulation of dynamic crack propagation based on FEA. We have adapted FRANC3D to compute anisotropic stress intensity factors for cortical bone specimens as a function of material orientation. The volume of the model associated with the left notch was exported to FRANC3D where an ed ge crack (0.2 mm (Yan 2006b)) was introduced at the tip of the notch. The volume was re-meshed by FRAN C3D with 3-D 10-node tetrahedral elements (solid 92) and solid 95 elements. The re-mes hed volume was re-inserted into ANSYS and the model was solved (Figure 9-3). The resulting no dal displacements were sent to FRANC3D for stress intensity factor (SIF) calculations. The SIFs were computed about the left crack tip according to the procedure outlined in the FR ANC3D documentation (FRANC3D Concepts and Users Guide, 2003), which are el aborated upon in Knudsen (2006).

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142 Experimental calculations were performed us ing theory for a double-edge crack plate in tension (Hertzberg, 1989) as a check for the numerical results. For plane strain, ICcKCa where c is the measured failure stress, a is the measured crack length, and C is the geometry factor ( 1 1 2 22 tan0.1sin 2 waa C aww where w is the specimen (far field) width). The effect of stress concentrations ( kt) near the notch on crac k-tip stresses were checked by comparing the following approxi mation to the v-notch (crack) length ( a ): 0.13 dDr (Eq. 9-1) where D is the notch length and r is the notch radius (Figure 9-3) The stress concentration factor should be included in the KIC calculations ( ICtcKkCa ) if a > d The yield stress of bovine (plexiform) cortical bone is 141 MPa (Mar tin et al., 1998), and was used to determine whether plain strain or plain stress conditions pr evailed in the notched tensile specimens, since yield strength, Y, of manatee bone was not investigated in our study. Plain strain prevails if the specimen thickness, B is greater than or equal to the right side of Equation 9-2. 22.5IC YK B (Eq. 9-2) A plastic zone is generated ahead of the crack tip in most materials. The plastic zone, if sufficiently large, affects the calculation of KIC. The plastic zone radius is found according to Equation 9-3 for plain strain and pl ane stress conditions, respectively. 2 21 2 1 6IC Y Y IC Y YK r K r (Eq. 9-3)

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143 In our study, the plastic zone is added to the crack length in the calculation of KIC if the plastic zone radius exceeds 10% of the crack length. Experimental results were obtained by pe rforming calculations using two different averaging techniques. First, the average failure load and average critical crack length for each crack orientation were used in the experimental fracture toughness calculati on in order to provide a direct comparison to computa tional results, which us ed the same input. Second, failure load and critical crack lengths were used to determine fracture to ughness for each specimen and averaged for each crack orientation. The latter results provide a standard deviation for fracture toughness. Results Fracture toughness values, crack lengths, and failure loads can be found in Table 9-1, while graphs of KIC versus normalized crack front are provi ded in Figure 9-4. As can be seen, mode I SIFs are non-uniform across the crack fron t with lesser values being found at the surfaces than in the middle of the specimen. Fractur e toughness was found to vary with specimen orientation with numerically evaluate d values ranging from 0.8 to 3.4 MPa m. Cracks propagated in the transverse direction relative to the rib were most resistant to fracture with fracture toughness values on the order of 4-times that of specimens with longitudinally propagated cracks. In particular, specimens load ed in the proximodistal direction with cracks propagated in the craniocaudal direction resu lted in the highest fracture toughness, while specimens loaded in the proximodistal direction with cracks propagated in the superficial-deep direction resulted in the lowest fracture toughness. Experimental calculation results were very similar to those from the numerical a ssessment, ranging from 0.9 to 3.2 MPa m. According to Equation 9-1, the effect of the stress concentration at the notch was deemed negligible to the mode I SIF computed at the cr ack tip. According to Equation 9-2, plane strain

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144 conditions prevailed in the notched specimens. Fr om Equation 9-3, it was determined that the plastic zone radius should not be included in the KIC calculations because the value was small relative to the crack lengths (<10%). Computation time varied from fifteen to forty minutes for each model using a Dell Optiplex GX280 work station with a 3.4 GHz Pen tium 4 processor and 1 GB of RAM. Models each contained approximately 120K nodes. Discussion According to Akkus et al. (2004), anisotr opy and inhomogeneity of bone tissue are the primary complications in modeli ng fracture behavior of bone. Both of these complications were accounted for in the present analysis. First, the SI F formulations utilized by the fracture analysis software are generalized for anisotropic materi als. Next, the inhomogeneities of the specimens were assessed during the material property eval uation that preceded th is study. A non-contact, full-field deformation analysis technique (Sutton et al., 1986; Sutton et al., 1991; Schreier et al., 2000; Lichtenberger and Schreier, 2002) was used to decipher pos ition and deformation of every discretized point on the gage se ctions of tensile and shear sp ecimens. Sudden changes in the strain magnitude are indicative of inhomogeneities in the specimen gage section. Strains were calculated as the slope of the change in pos ition versus change in deformation curves ( y; ; xxyuvuv x yyx ), where x, y, and xy are the Poisson, normal, and shear strains, respectively, x y and z are the positions and u v and w are the corresponding deformations in Cartesian coordinates. None of the specimens exhibited deviation from a linear slope (i.e., there were no changes in strain magnitude across the specimen gage sections). Therefore it was concluded that at the length scale of bone be ing studied here, inhomogeneities do not influence the apparent fr acture toughness of the specimens.

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145 Interpretation of results from experimental fracture toughness calcula tions is complicated by the small thickness available for most cortical bone (Akkus et al., 2004). This is because traditional fracture toughness calcula tions require an assumption of plane stress or strain, which is determined from Equation 9-2. Replacing th e unknown yield stress of manatee bone with that of bovine cortical bone shows that plane strain conditions prevailed in the notched specimens. The anisotropic stress intensity factor formulation used in the numerical analysis did not require an assumption of plane stress or strain, thus so me of the guess work for border line materials with a thickness that approaches the above inequa lity is removed when interpreting the results. Furthermore, yield stresses are not required as input to the numerical model. Because of the generalization of the SIF formulation used in th is study to anisotropic materials of any thickness and the fact that experimental calculations use a geometry factor determined from isotropic materials, the numerical KIC values were previously thought to be more accurate than those found from experimental calculations. To furthe r support this perception, Stanzl-Tschegg (2006) commented on the inappropriateness of utilizing the experimental KIC formulae for orthotropic materials. This is because the geometry factors used in KIC formulae are the result of fitting numerical or photoelastic data to experimental data from an isotropic material (Raju and Newman, 1979). However, it was found in this study that the numerical fracture model results agreed quite well with expe rimental calculations of KIC, thus suggesting the negligible effects of utilizing an isotropic-base d geometry factor in the KIC calculation. The difference in KIC between transverse and longitudinal orientations in this study is far larger than documented for other animals (Table 92). It appears that this difference is related to microstructural differences between specimen orie ntations (Figure 9-5). Manatee rib bone is mostly primary plexiform bone and contains ve ry few osteons (Yan et al., 2006a). During

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146 testing, sheets (or planes) of fibers bridged cracks in transversely loaded specimens as illustrated in Figure 9-6. Essentially, fract ure in transversely loaded sp ecimens occurred by peeling of fibrous sheets from one another. This would suggest that manat ee rib bone tissu e consists of layers of fibers oriented along th e longitudinal axis of the bone. Nalla et al. (2005b) found KIC in human cortical bone to be between 51-140% greater for cracks propagated in the transverse direction relative to the rib th an for cracks propagated in the longitudinal direction. The increase in toughne ss for cracks propagated in the transverse direction is the result of a ~90 kink, or deflection, in the crac ks path. Yeni and Norman (2000) found that cracks in osteonal bone ca n deflect from a transverse di rection of propagation to grow along the cement line interface between an os teon and the surrounding mineral matrix. The cement line acts as a path of least resistance, causi ng the crack to redirect away from the nominal path of maximum stress. Evidence suggests that failure of manatee rib cortical bone is governed by mode I loading even in the presence of mixed mode of loading. A characteristic of mode I failure that is not present during mode II or mode III failure is the ability to identify the fracture origin and the geometry of the flaw: These features were found during our study. Fracture toughness for all orientations of manatee rib bone is therefore reported as mode I fracture toughness. The toughness values found here were ~28% le ss for transversely propagated cracks than in Yan et al.s manatee rib single edge v-notch beam (SEVNB) study (2006b) ( KIC=4.7.3 for transversely propagated cracks in Yans st udy). Factors such as specimen size and microstructural variation within a single animal or between animals may be responsible for the difference in KIC values between these two studies.

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147 It should also be noted that bone is considered an R-curve material, meaning that fracture resistance increases with crack length. In bone, the R-curve beha vior is due predominantly to microstructural toughening mechan isms such as fiber bridging (Nalla et al., 2005a). Other studies in manatee bone fractur e support the idea of R-curve behavior. Yan (2006a) found a KIC value of 2 MPa m for a crack size of 0.2 mm; in another study Yan determined a KIC value of 3 MPa m for a crack size range of 0.30.6 mm; and in the 2006b study, a KIC of 4.7 MPa m for a crack size of 1 mm. Clifton et al (2007) found a toughness of ~8 MPa m for a crack size of 23.5 mm. Thus, all of the different studies are consistent with the concept of R-curve behavior in manatee bone (Figure 9-7) and with previous studies in other bone materials. Future studies on manatee rib should address the issue of R-curve behavior in a more sy stematic manner. The procedures used in the present analysis can be used as a template for further cortical bone fracture studies that aim to disc ern a broader spectrum of fractur e resistance values by combining numerical simulation with experi mental fracture parameters. Conclusions Fracture toughness values from th e computational analysis were nearly identical to results found using traditional experimental calculations fo r all crack orientations. The small difference between experimental and computational results reveals that the isotropic assumption placed on the geometry factor for the experimental calc ulation does not significan tly influence fracture toughness results, and it shows that computational analyses may not be necessary to determine fracture properties in co rtical bone. The largest anisotropic ra tio for fracture toughness values in manatee rib was found to be 4.3 in our study, wh ich is a much larger ratio than what is traditionally found in cortical bone (1.3-2.0). Further investigations should look into the mechanisms responsible for su ch a large ratio of maximum to minimum fracture toughness.

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148 Table 9-1. Orientation depende nt fracture toughness and fractur e parameters of manatee rib (cortical) bone. Mean valu es are presented for crack length and failure load along with the coefficient of variation, CV (in parentheses) for each fracture toughness orientation. The number of specimens ( n ) used during the experimental component of the test are also presented. Stress intens ity factors (SIFs) are denoted for varying orientations as Kxy, where x is the loading direction and y is the crack growth direction. The critical stress intensity factor s (fracture toughness, KIC) are obtained by implementing the mean crack lengths and fa ilure loads into the models. Fracture toughness values calculated by finite element analysis are reported as KIC FEA, while experimental plane strain fr acture toughness values with iden tical parameters to those used in the numerical model are reported as KIC exp1. Mean and standard deviations for plain strain fracture toughness calculated us ing traditional equa tions and averaging methods are reported as KIC exp2. SIF Orientation K12 ( n =7) K13 ( n =6) K23 ( n =8) K21 ( n =12) K31 ( n =8) K32 ( n =8) Crack Length (mm) 1.27 (31%) 1.30 (18%) 1.63 (19%) 1.14 (19%) 1.09 (14%) 1.17 (24%) Failure Load (N) 1856 (15%) 1860 (18%) 448 (26%) 641 (17%) 592 (25%) 448 (16%) KIC FEA (MPa m) 3.2 3.4 1.0 1.1 1.0 0. 8 KIC exp1 (MPa m) 3.2 3.2 0.9 1.1 0.9 0.7 KICexp2 (MPa m) 3.40.5 3.50.7 0.90.2 1.10.2 1.10.3 0.80.2

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149 Table 9-2. Selected studies measuring fractur e toughness of compact bone (modified from Yan (2005)). Investigators Testing Method Bone Source Fracture Direction Fracture Toughness (MPa m) Melvin and Evans (1973) Single-edge notched beam Bovine femur Longitudinal Transverse 3.21.43 5.58.52 Behiri and Bonfield (1984) Compact tension Bovine tibia Longitudinal 2.8-6.3 Norman et al. (1992) Compact tension Bovine tib ia Transverse 4.93-12.64 Vashishth et al. (1997) Compact tension Human tibia Longitudinal 1.6-2.5 Wang et al (1996) Compact sandwich Baboon femur Longitudinal 2.25.18 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 4.8.5 Nalla et al. (2003) Fatigue precracked flexure Human humerus Longitudinal Transverse 3.53.13 5.33.41 Yan et al. (2006a) 3-point bend Manatee rib Transverse 2.3.4 Yan et al. (2007) Chevron-notched beam Manatee rib Bovine femur Transverse Transverse 4.5.5 5.8.5 Yan et al. (2006b) Single-edge Vnotched beam Manatee rib Transverse 4.7.3

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150 Figure 9-1. Manatee skeleton. Cent er ribs from adult manatees we re used in this study. All specimens were harvested from the middle th ird of the ribs to reduce variability in fracture toughness results. Image courte sy of Roger Reep (department of physiological sciences, University of Florida).

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151 Figure 9-2. Mode I (opening mode) stress intensity factor (SIF) nomenclature (left). Mode I SIF is presented for each specimen orientation as Kij, where superscript i denotes the loading direction and j denotes the crack growth directi on. Note that the 1, 2, and 3 directions are denoted as [100], [010], and [001] in M iller index notation and as medial-lateral, anterior-posterior, and cr anial-caudal, respectively, in anatomic coordinates relative the manatee. The photogr aph (right) shows a failed cranial-caudal specimen ( K31). Figure 9-3. Notch tip geometry sh owing an illustration of the finite elements and variables used in the fracture toughness models (left), a nd vertical (opening mode) component stress contour at the crack tip (right).

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152 Figure 9-4. Numerically-determine d mode I SIF versus normalized crack front. The SIF values presented in this figure can be taken as the fracture toughness ( KIC) of the specimens because experimentally-measured crack length s and failure loads were inserted into the numerical model. The legend shows th e fracture toughness values in descending order. Figure 9-5. Scanning electron micrographs of typi cal fracture surfaces from specimens loaded in each of the three principal or thogonal directions: superficialdeep (left), proximodistal (middle), and craniocaudal (right). Notice the variation in microstructure with specimen orientation. The smooth regions in the images are the notches.

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153 Figure 9-6. Schematic (left) demonstrating the peeling of fibrous la yers during the fracture experiments. Transverse ( K12) fracture specimen (right) showing the overlap of two oppositely facing cracks growing in the longitudi nal direction, separated by a sheet of fibrous plexiform bone tissue (highlighted region). 0 1 2 3 4 5 6 7 8 9 00.511.522.53 a_critical (mm)KIC (MPa*m^0.5) Figure 9-7. R-curve behavior of manatee rib bone demonstrating a trend of increasing fracture toughness with increasing critical crack length. The data in this plot is presented from several experiments involving manatee ri b bone with various crack lengths (Yan, 2006a, b; Clifton et al., 2007; and the present study).

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154 CHAPTER 10 CONCLUDING REMARKS The objectives of the present work were to de velop an understanding of fracture in cortical bone and to characterize and document the material and fracture propertie s of manatee rib bone as a function of orientation. The former aspect is achieved via a unique synthesis of experimental and computational analyses that have never b een performed on cortical bone. This document provides the framework for bone mechanics rese archers to obtain more information on the orientation dependence of bone fracture than was previously available. The efficacy of this combined experimental and numerical te chnique is tested on manatee rib. The manatee rib bone work presented here s upports previous work aimed at reducing the impact of watercraft on the manatee populatio n. Prior to this work, fracture toughness was only known for cracks propagating along the tran sverse direction of the rib bone. When considering the potential for watercraft to impact a manatee at various angles, the next step is to identify the fragility of manatee rib bone as a f unction of material orientation. In our study, we analyzed fracture toughness for cracks traversing the longitudinal directio n of the rib bones and determined that this never before analyzed crack orientation produced fracture toughness values ~4 times smaller than for transversely propagati ng cracks, which were also analyzed in our study. The anisotropic ratio for fr acture toughness in this study is higher than that reported for bones from other animals (4.3 here compared to 1.3-2.0 elsewhere). Furthermore, findings from this study suggest that manatee ri b material properties are more anisotropic than plexiform bone from other animals (2.9 here compared to 1.18-1.47 in other animals). In conclusion, manatee rib bone material a nd fracture properties were found to vary strongly as a function of material orientation. The symmetry of manatee rib bone was determined

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155 to be orthotropic. Testing of specimens that had been air-dried for 2 weeks demonstrated the nonlinear behavior of the material. Complete re moval of the water content should result in a linear stress strain curve, suggesting that the water content was not comp letely removed from the specimens in our study. However, the sources of the nonlinearity were not fully investigated in this study, although an attempt was made to de termine whether or not manatee rib bone is viscoelastic. Unfortunately, malfunction of the universal testing machine used to load the specimens in this study caused the strain rates to not vary sign ificantly enough to provide strong evidence for the conclusion that manatee rib bone el astic modulus is strain rate dependent (i.e., viscoelastic). However, strain rate data varied significantly for one specimen, and that specimen demonstrated viscoelastic behavior (i.e., ther e was an increase in elastic modulus with an increased applied strain rate) (s ee Appendix A). Additional testing at largely differing strain rates on a plurality of specimens coul d be conducted to confirm the vi scoelasticity of the material. This would further support the evidence of viscoelasticity foun d by cyclically loading several specimens into the non-linear range of the material that demonstrated a lack of hysteresis. Other sources of nonlinearity could be investigated using histology. Histological st aining could be performed during cyclic te sting to confirm the presence of microcracking. Chapter 9 discusses the R-curve behavior of bone and provides evidence that manatee rib bone is an R-curve material. However, additional testing could further elucidate the effects of differing crack lengths on the fr acture toughness of the material. Th e effects of crack turning for transversely oriented cracks c ould also be further investigated Transverse cracks were found to deviate from their initial traject ory as the cracks grew in the manatee rib bone specimens. Several crack-turning models are available in the litera ture, and each could be used to determine which model best fits the manatee rib bone experimental fract ure test data. Furthermore, grooves could

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156 be machined into these specimens to keep the cr acks from deviating from their initial trajectory so that there is no mixed mode loading in th ese specimens. This approach has been used by several authors to retain opening mode failure in cortical bone specimens. Lastly, it is not known to what degree soft tissues surrounding the rib reduce localized trauma on the rib during impact with watercraft. This is an area of research that should be conducted to provide a direct link between fracture in manatee rib and collisions with watercraft. However, the scientific evidence presented in this work as well as that presented by Drs. Kari Clifton, Jiahau Yan, Roger Reep, John Mechol sky, Jr., and Laurie Go wer on skeletal tissue mechanics of the Florida manatee strongly suggest a need to modify boater activities in manatee populated waterways to help prot ect this endangered species. S uggested modifications to boater activities include: enforcement of reduced watercraft speeds, mandatory polarized sunglasses for all watercraft operators, technol ogies that alert manatees to approaching watercraft by making use of their unique sensory orga ns, and enforcement of safe wa tercraft operation practices.

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157 APPENDIX A TENSILE TESTING AT VARIOUS CROSSHEAD SPEEDS Various strain rates (calculated from crossh ead speeds and specimen gage lengths) were applied to tensile specimens fixed with strain gages. Four dog-bone shaped specimens oriented about each of two material dir ections (superficial-deep and proximodistal) were loaded and unloaded for three to five cycles at presumably different strain ra tes. A rheostat on the universal testing machine (MTI 30K, Materials Testing, Inc ., Roswell, GA) was adjusted to three different positions (one for each load cycle). The same three rheostat settings were used for all specimens. However, issues with one of the control modules in the testing machine resulted in crosshead speeds varying less with a load present than duri ng the zero-load crosshead speed calibration. This caused the crosshead speeds to not vary significantly between each test, and thus the modulus values for all but one specimen did not vary significantly betwee n tests. Results from the only specimen yielding significantly different elastic moduli between tests are provided in the table below. The cyclic tests at each rheost at setting did not result in any hysteresis although several specimens were loaded into their nonlinear range. This demonstrates the viscoelasticity of manatee rib bone. The viscoelast ic behavior of manatee rib bone can be further investigated by testing additional specimens at strain rates significantly different from one another to determine the strain rate dependence of the materials elastic modulus. Table A-1. Viscoelastic propertie s of manatee rib bone. Three cros s-head speeds were tested for their effect on elastic modulus for one specimen loaded in the proximodistal direction. Manatee rib bone viscoelasticity Crosshead speeds (mm/min) 0.58 0.80 0.91 Elastic modulus (GPa) 16.1 18.8 22.3

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158 Figure A-1. Viscoelastic propert ies of manatee rib bone. Results are shown for one proximodistal specimen loaded at three different strain rates.

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159 APPENDIX B MATERIAL PROPERTY CALCULATIONS The following pages show the custom Math cad code (Mathcad v.12.1, Mathsoft Engineering and Education, Inc.) used to aver age deformation rates and calculate strains, Poissons ratios, and elastic moduli from VIC data for three orientations of tensile specimens. Files generated by VIC3D (VIC-3D Digital Imag e Correlation v.2006.0.0, Correlated Solutions, Inc., West Columbia, SC) were input into Ma thcad for manipulation. The input data files included position, deformation, and deformation ra te information for the data points on the front and back gage sections of MIC tensile specimens. After performing filtering operations to remove potentially erroneous data, the deformation rate was calculated for data points within the gage section of the specimen. Next, the normal a nd Poisson strains are calculated as the slope of the vertical and horizontal position versus deformation data, respectively. Strains and deformation rates are averaged from the front an d back specimen gage sections to increase the data available for the analysis thereby reducing th e error of the analysis compared to analyzing only one specimen face. Also, averaging the data removes the effects of unwanted bending in the tensile specimens so the strains are those due only to tensile (not bending) loads. Poissons ratio is calculated as negative the ratio of Poisson to normal strain. Poisson strain, normal strain, deformation rate, and Poissons ra tio were next exported from the custom program for use in a data compilation file that links data from each lo ad step (including Labview generated load data) in order to calculate elastic modulus for each sp ecimen. A third program (not shown) was written to link data from each specimen in order to de termine mean Poissons ratios and elastic moduli and to determine the standard deviations for eac h specimen orientation. Similar programs were also developed for the shear tests.

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160 Mathcad File Used to Find the Strains and Di splacement Rate for a Single Pair of Images MSW 0253 (Manatee I.D.) 1ML (Specimen I.D.) Image 23 (Image I.D.) Begin the origin of arrays and matrices w ith subscript 1, not 0 as is the default ORIGIN1 Read input data from Excel file (import raw data from .csv file) Data file: 1ML_023_0.csv front 1 1 0 back 1 1 0 Column I.D. 1 2 3 4 ... 7 8 9 ... 16 17 18 X Y Z correlation ... U V W ... dU/dt dV/dt dW/dt ... Sort data (ascending order for column 4) frontcsortfront4 () backcsortback4 () rows (front) i_mfrowsfront () i_mf5.872103 if12 i_m f columns (front) j_mfcolsfront () j_mf21 jf12 j_m f rows (back) i_mbrowsback () i_mb3.286103 ib12 i_m b columns (back) j_mbcolsback () j_mb21 jb12 j_m b Extract vector elements to determine how many data points must be filtered from the dataset corrfsubmatrixfront1 i_mf 4 4 () corrbsubmatrixback1 i_mb 4 4 () Only allow data points with correlation values greater than zero cfxf ()0x f

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161 cbxb ()0x b ExtractfQfcf ()if1 Nf1 NfifQfjf ifif1 cfQfjf1 if jf1rowsQf ()1 for Nf ExtractbQbcb ()ib1 Nb1 NbibQbjb ibib1 cbQbjb1 if jb1rowsQb ()1 for Nb corrfExtractfcorrfcf () corrbExtractbcorrbcb () Remaining rows after removing poorly correlated data i_mfrowscorrf () i_mf4.687103 if12 i_m f i_mbrowscorrb () i_mb2.7103 ib12 i_m b Reverse the order of the matrix so the poorly correlated data poi nts appear at the bottom and crop the matrix to remove poorly correlated data frontreversefront () frontsubmatrixfront1 i_mf 1 j_mf () backreverseback () backsubmatrixback1 i_mb 1 j_mb () Re-order the matrix in ascending order of Y-axis position frontcsortfront2 () backcsortback2 () Filter out W values less than or greater than 2*stdev from the mean Wfsubmatrixfront1 i_mf 9 9 () Wbsubmatrixback1 i_mb 9 9 ()

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162 cfxf ()2 stdevWf ()meanWf () xf 2stdevWf ()meanWf () cbxb ()2 stdevWb ()meanWb () xb 2stdevWb ()meanWb () ExtractfQfcf ()if1 Nf1 NfifQfjf ifif1 cfQfjf1 if jf1rowsQf ()1 for Nf ExtractbQbcb ()ib1 Nb1 NbibQbjb ibib1 cbQbjb1 if jb1rowsQb ()1 for Nb WfExtractfWfcf () WbExtractbWbcb () Redefine rows after removing data with excessive out-of-plane displacement noise i_mfrowsWf () i_mf4.515103 if12 i_m f i_mbrowsWb () i_mb2.583103 ib12 i_m b Reverse the order of the matrix so the data poin ts with excessive noise appear at the bottom and crop the matrix to remove the noisy data frontreversefront () frontsubmatrixfront1 i_mf 1 j_mf () backreverseback () backsubmatrixback1 i_mb 1 j_mb () Re-order the matrix in ascending order of Y-axis position frontcsortfront2 () backcsortback2 ()

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163 Extract individual component s from the input matrix Xfsubmatrixfront1 i_mf 1 1 () Xbsubmatrixback1 i_mb 1 1 () Yfsubmatrixfront1 i_mf 2 2 () Ybsubmatrixback1 i_mb 2 2 () Zfsubmatrixfront1 i_mf 3 3 () Zbsubmatrixback1 i_mb 3 3 () Ufsubmatrixfront1 i_mf 7 7 () Ubsubmatrixback1 i_mb 7 7 () Vfsubmatrixfront1 i_mf 8 8 () Vbsubmatrixback1 i_mb 8 8 () Wfsubmatrixfront1 i_mf 9 9 () Wbsubmatrixback1 i_mb 9 9 () Find the average rate of deformation for the data set Noteuse the last 4/5ths of the data (bottom fifth of target region) since velocity in this region is representative of the cross-head speed dV_dt_fsubmatrixfront 4 5 i_mf i_mf 17 17 dV_dt_fmeandV_dt_f ()6 0 dV_dt_bsubmatrixback 4 5 i_mb i_mb 17 17 dV_dt_bmeandV_dt_b ()6 0 Average rate of deformation (mm/min) dV_dt_f0.988 dV_dt_b0.836 Strain in the X-direction (X and U are presented in mm) 4 2024 0.88 0.87 0.86 Uf Xf 4 20246 0.84 0.83 0.82 0.81 Ub Xb exfslopeXfUf () 8.371 104 exbslopeXbUb () 9.334 104 Strain in the Y-direction (Y and V are presented in mm)

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164 50510 0.3 0.25 Vf Yf 505 0.26 0.24 0.22 0.2 Vb Yb eyfslopeYfVf () eyf4.183103 eybslopeYbVb () eyb2.062103 Average the strains and stra in rates from the front and the back of the specimen x exfexb 2 x8.852 104 y eyfeyb 2 y3.122103 dV_dt dV_dt_fdV_dt_b 2 dV_dt0.912 mm/min Calculate Poisson's Ratio xy x y xy0.284 Write the results to a file using the prn format Results x y xy dV_dt WRITEPRN(C:\Jeff\School\...\MSW 0253\1ML\ML 1 023 .prn)=Results Mathcad File Used to Compile Datasets From a Single Specimen and to Calculate Elastic Modulus MSW 0253 (Manatee I.D.) 1ML (Specimen I.D.) Begin the origin of arrays and matrices w ith subscript 1, not 0 as is the default ORIGIN1 Read data sets (manually i nput each results data file) Results_001=READPRN(C:\Jeff\Sc hool\...\1ML_001.prn)=Results Results_002=READPRN(C:\Jeff\Sc hool\...\1ML_002.prn)=Results

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165 Results_024=READPRN(C:\Jeff\Sc hool\...\1ML_024.prn)=Results Compile data sets Results=augment(Results_001,Results_002,,Results_024) Extract individual components of the compiled data set xsubmatrixResults1 1 1 colsResults () () Results x xT x ysubmatrixResults2 2 1 colsResults () () Results y yT y zsubmatrixResults3 3 1 colsResults () () Results z zT z dV_dtsubmatrixResults4 4 1 colsResults () () Results dV_dtdV_dtT dV_dt mean(dV_dt) = -0.589 mm mi n xysubmatrixResults5 5 1 colsResults () () Results xy xyT xy i12 rows xy xy Filter out all invalid values of Poissons ratio c()=0<<0.5 ExtractQc ()i1 N1 NiQj ii1 cQj1 if j1rowsQ ()1 for N xyExtract xyc xy Read input data from Excel f ile (raw data from .xls file) Data file: 1ML Front Face.xls

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166 loadfile 1 1 0 Column I.D. 1 2 3 ... count load (lbs) time (s) ... rows i_mfrowsloadfile () i_mf48 if12 i_m f columns j_mfcolsloadfile () j_mf3 jf12 j_m f Extract the load and time vectors from the loadfile matrix loadsubmatrixloadfile24 i_mf 2 2 () timesubmatrixloadfile24 i_mf 3 3 () Calculate stresses area = 0.05806 mm2 stress load area psi Calculate the elastic modulus for each dataset modulus_psiistressi yi y psi Convert stresses to GPa for co mparison with Bratt and Karis reporting of elastic modulus psi2GPa6.8947610 6 Finn",.,.,,m,,,,,mmm" Jake"Gfff 0020" modulus_GPamodulus_psi p si2GP a modulus_psi

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167 Plot or tabulate the data to check for reasonableness of the values

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168 APPENDIX C VISUAL IMAGE CORRELATION (V IC) EQUIPMENT TUTORIAL University of Florida Mechanical and Aerospace Engineering Location (Responsible Faculty): 118 NEB (Dr. Peter Ifju) In This Document Specimen Preparation Equipment Setup Computer Setup Pre-Photo Adjustments Camera Calibration VIC Testing Pre-VIC Analysis VIC Analysis Example Analysis Specimen Preparation Begin by painting your specimen white (if it is not already light in coloration). Slightly enlarge the nozzle of a black spray paint can using a pin (to increase droplet size). Speckle the specimen with the black spray paint (do not overspray). Equipment Setup Place specimen in test fixture.

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169 Position the cameras to form an equilateral triangle with the specimen. However, if imaging 2 perpendicular surfaces at a time, you may need to place the cameras closer together. Note th at any deviation from an equilateral triangle degrades the calibration of the equipment. Connect wires to appropriately labeled components. Place light so that it illuminates the specimen. Remove lens covers from cameras.

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170 Computer Setup Turn on the computer and log on (user name = X ; password = X ). Open Windows Explorer and create a new directory for your images (named pics/date/ or some other identifiable name) and one for your calibration images (named cals/date/ or some other identifiable name). On the Windows Start Menu click on the shortcut to VicSnap Enter the calibration image directory name you provided earlier at the Image Directory prompt and choose a prefix name for you images (e.g., cal_date_name ). Your prefix will be followed by a numeric sequence (000, 001, 002, etc.) as images are stored to the co mputer. Images are stored in tiff format ( .tif ). Pre-photo Adjustments Turn on camera power (turn switches to 1394 PWR not Aux PWR). Flip the transformer power switch on (located on the surge protector).

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171 Turn on the light. If you have not already opened VicSnap do so now. You should be able to see the specimen in the VicSnap image viewer window on the computer screen. Adjust the position of the light to optimize the view of the specimen on the computer screen. An optimal image shows good visibility of the specimen without overor under-exposure. If viewing tw o perpendicular surfaces at once, try to position the camera to illuminate both surfaces equally. Lighting exposure can be adjusted using the slider on the camera lens. Exposure can also be adjusted by moving a slider bar in the VicSnap window, but it is recommended that the slider bar remain in the middle of the range. If the image is out of focus use an allen wrench to adjust the lens focal length. Be careful not to loosen the set screw too much or the lens will detach completely from the camera. Camera Calibration Change the directory to the cals/date/ directory if you did not specify this directory when you started VicSnap. This will require closing VicSnap and reopening the progra m (start menu => VicSnap) if you are in an alternate directory. If you did not yet adjust the camera focus and light exposure with your specimen in place, do that now. Otherwise, move on to the next step. Select a calibration dot-matrix image (black paper with white dots) according to your image resolution needs. Dot spaces range from 1.5 to 3mm. The closer your specimen is to the camera, the smaller dot spacing you should use (and the higher the image resolution will be).

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172 Glue the calibration dot-matrix paper to a hard, flat surface that is close to the size of the paper. Remove the specimen and place the calibration paper in front of the camera Take at least 20 photos while adjusting the position and angle of the paper, being sure to keep all dots in view of both cameras for each photo ( press the space bar to take photos ). Two images will be stored to your calibration image directory each time you press the space bar (one from the left camera and one from the right). Turn the light off Click shortcut to VIC3D on the start menu Click on the icon that looks like the calibration paper. In the pop-up window, locate the calibration image directory ( cals/cal_date/ ). Select all of the photos in the directory (to select all images at once, click on the first image, scroll to the last image, hold down shift and click on the last image) and press open Click on the calibration icon (it resembles a set of calipers). Select QImaging Retiga 1300 for the type of camera Under the type of target select the correct pixel (dot) dimension and spacing for the calibration paper in the images. Turn on adaptive thresholding by checking the box.

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173 Extract the images, checking to see that all of the dots appear red except for 3 dots which are different colors. If you see any images with white dots during the extraction process, right click on the images at the left of the screen and click remove Once erroneous pictures have been removed, click Calibrate and let the software calculate the standard deviations for each camera. Once standard deviations (SDs) for each image are displayed on screen, remove any image that resulted in an SD of 0.1 or higher and repeat the extraction. Once all SDs are below 0.1 (preferably below ~0.07), click Accept Go back to VicSnap and close the program. Re-open VicSnap and enter the pics image directory name you provided earlier at the Image Directory prompt and choose a prefix name for you images (e.g., pics_date_name ). Remember, your prefix will be followed by a numeric sequen ce (000, 001, 002, etc.) as images are stored to the computer. Save your work into your Pics directory (or some other identifiable directory) as analysis_date_name.v3d or some other identifiable name. VIC Testing

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174 Turn the light back on and place the specimen back in position ( in view of both cameras ). Make sure the on-screen lighting is still of good quality. R eadjust the light position or exposure slider(s) if there is too much glare or if the image is not bright enough. DO NOT adjust the camera positions or camera focus unless if you plan to recalibrate. Take a reference photo of the specimen with no load by pressing the space bar Two images will be stored to your Pics image directory with the prefix you chose followed by a 000_0.tif and 000_1. These are the name of your reference images. Note that one of the images (pref ix_000_0.tif) is from the left camera and the other (prefix_000_1.tif) is fro m the right camera. Make sure to either program the cameras to take photos in intervals during your test or be prepared to press the space bar to take photos of your specimen at key load or displacement intervals Begin loading the specimen and taking photos in appropriate intervals for your analysis. Once you are done loading the specimen, turn off the cameras light and transformer power switch (on the surge protector). Place the lens covers back on the cameras Pre-VIC Analysis Go back to Vic3D Click the reference image icon Select either the prefix_000_0.tif or prefix_000_1.tif image (either should work fine) and click Open Click the deformed image icon Select all of the images aside from the reference images and click Open Click an appropriate target region selection icon etc. Highlight areas of interest ( AOIs ) on the reference image. You can select multiple AOIs should your analysis require you to do so. Note that the red region is the active AOI.

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175 Cut out regions of disinterest such as areas of high reflectivity by clicking the cut region from aoi icon Use the left mouse button to draw a polygon around the region that you w ill cut from the selected region. Use the middle mouse button to cancel Use the right mouse button to confirm your cut Use the cycle active AOI icon to toggle between AOIs if your analysis requires cu tting operations to be performed on AOIs other than the active one (which is highlighted in red). Locate a seed point within an aoi which will be used to verify that both cameras are looking at the same point for your analysis. Click the seed point icon Click an easily identifiable feature (e.g., a distinguishable speckle) on the active aoi. Click the select initial guess icon Two images will appear on a split screen. If the image on the right does not show the same point as on the left, move the slider bar around to locate the point on the right imag e. You can also move the X marker on the bottom screen to locate the seed point If the point cannot be found, move the seed point by clicking the seed point icon again and place it on another identifiable feature. Note that the points may appear distorted when being viewed from the other camera. Once you locate the point on the bottom and right images click the point on both the left and right screens that matches the seed point as seen in the left image.

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176 Click Add Point then click Calculate The question mark over the deformed image name will change to a green check mark. Do this fo r all of the images. Click Close once a green check mark appears by ALL images Click the start correlation analysis icon Under the file tab make sure that all of the images you care to use in your analysis have a check next to their name. Under the option tab make sure to place a check by cubic B-spline interpolation and uncheck binomial lowpass filter Also, change the contrast to zero Under the thresholding tab uncheck use epipolar thresholding Click Run VIC Analysis

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177 Under the I mage Tab right click the Deformed Image you would like to analyze. Click Contour Plot Adjust the Data File and Variable fields according to your analysis needs. Use the Inspector Tools to Extract key information pertinent to your analysis. Turn off cameras, light, and transformer power switch a nd place the lens covers back on the cameras if you have not done so already. Example Analysis 1) Objective: Plot the U-deformation vs. load interval for data collected from a targ et region on the specimen. Use the circular (or similar) inspection tool to select a target region on the specimen aoi. Click Extract Change the Data File field to the next image and click Extract Repeat for all deformed image files (for each load interval). Use data extracted from the target region to plot U vs. File Number 2) Objective: Plot the U-deformation vs. Point Index for data collected from a verti cal line sketched on the fully loaded specimen and the V-deformation vs. Point Index for data collected from a ho rizontal line sketched on the loaded specimen. Select the desired image from the Data File field. Select U, V, and any other variable of interest in the Variable field. Use the line inspection tool to sketch a vertical and horizontal line on areas of interest on the image of the fully loaded specimen.

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178 Click the arrow icon in the inspection toolkit. Click on the vertical line you sketched. Click Extract to extract data from the vertical line. Plot U vs. Point Index Click the arrow icon in the inspection toolkit. Click on the horizontal line you sketched. Click Extract to extract data from the horizontal line. Plot V vs. Point Index This data can be transformed, filtered, or used in its raw form for your analysis. One final note is that you will want to check your analysis every step of the way. For example, your pre-analysis check should include a visualization of the VIC-generate d contour plots that represent the surfaces of interest (AOI's). To do this, right click on each output file under the Data Tab and click "3D Plot". VIC3D will display a 3D plot of your AOI's. Look for plots that don't appear to be repr esentative of the actual surfaces or that don't appear similar to other images in the group. You may want to consider removing these output files before continuing with your analysis. Good luck with your analyses! ***Special thanks to Dr. Peter Ifju for use of the VIC equipment and to Bret Stanford for providing the information necessary to compile this tutorial.

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179 LIST OF REFERENCES Akkus, O. Yeni, Y. Wasserman, N., 2004. Fract ure mechanics of cor tical bone tissue: A hierarchical perspective. Critical Reviews in Biomedi cal Engineering 32, 379-425. Alto, A. Pope, M., 1979. On the fracture toughness of equine metacarpi. Journal of Biomechanics 12, 415-422. An, Y., Friedman, R., 1999. Animal models in orthopaedic research. Boca Raton: CRC Press. An, Y., Draughn, R., 2000. Mechanical testing of bone and the bone-implant interface. Boca Raton: CRC Press. Arakere, N. Swanson, G., 2002. Effect of crystal orientation on fatigue fa ilure of single crystal nickel base turbine blade superalloys. Jour nal of Gas Turbines and Power 124, 161-176. Arakere, N. Siddiqui, S. Magnan, S. Ebrahimi, F. Forero, F., 2005. Investigation of threedimensional stress fields and slip system s for FCC single-crystal notched specimens, Journal of Engineering for Gas Turbines and Power 127, 629-637. Ashman, R., Cowin, S., Van Buskirk, W., and Rice, J., 1984. A continuous wave technique for the measurement of the elastic properties of cortical bone. Journal of Biomechanics 17, 349-361. Behiri, J. Bonfield, W., 1980. Crack velocity dependence of longitudi nal fracture in bone. Journal of Materials Science 15, 1841-1849. Behiri, J. Bonfield, W., 1984. Fracture mechanic s of bone the effects of density, specimen thickness and crack velocity on longitudinal fracture. Journal of Biomechanics 17, 25-34. Behiri, H. Bonfield, W., 1989. Orientation depe ndence of the fracture mechanics of cortical bone. Journal of Biomechanics 22, 863-872. Bertholet, 1999. Composite Materials Mechanical Behavior and Structur al Analysis. Springer Verlag, New York. Bonfield, W. Behiri, J. Charalambides, B., 1985. Orientation and agerelated dependence of fracture toughness of cortical bone. In: Perren, S. Schneider E. (Eds.), Biomechanics: Current Interdisciplinary Research, pp. 185-189. Bonfield, W., 1987. Advances in the fracture mechanics of cortical bone. Journal of Biomechanics 20, 1071-1081. Boresi, A.P., Schmidt, R.J., Sidebottom, O.M ., 2003, Advanced Mechanics of Materials (6th Ed.). John Wiley and Sons, Hoboken. Clifton, K., Golden, J., Reep, R., 2003. Material properties of subadult ma natee bone. Integrative and Comparative Biology 43, 951.

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180 Clifton, K., 2005. Skeletal biomechanics of the Fl orida manatee. Ph.D. di ssertation, Gainesville, University of Florida. Clifton, K., Reep, R. Mecholsky, J., 2007. Structural properties of manatee ribs tested in impact. Journal of Materials Science (submitted). Cooke, F. Zeidman, H. Scheifele, S., 1973. Th e fracture mechanics of bone another look at composite modeling. Journal of Biomed ical Materials Research 7, 383-399. Cowin, S., 2002. Bone Mechanics Handbook (2nd Ed.). CRC Press, Boca Raton. Currey, J.D., 1990. Physical characteristics aff ecting the tensile failure properties of compact bone. Journal of Biomechanics 23, 837-844. De Buffrenil, V., Schoevaert, D., 1989. Donnes qua ntitatives et observations histologiques sur la pachyostose du squelette du dugong, D ugong dugon (Muller) (Serenia, Dugongidae). Canadian Journal of Zoology 67, 2107-2119. De Santis, R. Anderson, P. Tanner, K. Ambrosio L. Nicolais, L. Bonfield, W. Davis, G., 2000. Bone fracture analysis on the short rod chevron-notch specimens using the X-ray computer micro-tomography. Journal of Materials Science: Materials in Medicine 11, 629-636. Dong, X., Guo, X. 2004. The dependence of transv ersely isotropic elasti city of human femoral cortical bone on porosity. Journal of Biomechanics 37, 1281-1287. Evans, F.G., 1973. Mechanical Propert ies of Bone. Thomas, Springfield. FRANC3D Concepts and Users Guide, 2003. Availa ble from the Cornell Fracture Group website (www.cfg.cornell.edu). Ithaca. Freund, L., 1978. Stress-intensity factor calc ulations based on a conservation integral. International Journal of Solids and Structures 14, 241-250. Gibson, R.F., 1994. Principles of Composite Ma terial Mechanics. McGraw Hill, New York. Gotzen, N., Cross, A., Ifju, P., Rapoff, A., 2003. Understanding stress co ncentrations about a nutrient foramen. Journal of Biomechanics 36, 1511-1521. Griffith, A., 1920. The phenomena of rupture and fl ow in solids. Philosophical Transactions of the Royal Society of London A221, 163. Grootenboer, H. Weersink, A., 1982. A composite model of cortical bone for the prediction of crack propagation. In: Huiskes, R. Van Camp en, D. De Wijn J. (Eds.), Biomechanics: Principles and Applications. The Hague: Martinus Nijhoff. Gross, B., Srawley, J., 1964. Stress intensit y factors for a single notch specimen by boundary collocation of a stress fu nction. NASA TN D-2395.

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186 BIOGRAPHICAL SKETCH Jeff Leismer was born in southeast Michigan in 1978. He moved to Houghton, Michigan in 1996 to begin working towards his bachelors degr ee in biomedical engine ering from Michigan Technological University. Jeff took an intere st in space physiology and supportive technology for astronauts under the tutelage of Dr. William Cooke during his junior year. He developed a particular interest in bone mechanics and went on to receive a master of engineering degree from the department of mechanical engineering at Michigan Tech, where he developed a vibration therapy device capable of stimulating muscle a nd bone in order to reduce disuse atrophy in astronauts. After completing his masters degr ee in 2002, Jeff was accepted to the PhD program in the department of mechanical and aerospace e ngineering at the University of Florida. His initial project involved the stress analysis of singl e crystal nickel base superalloys used in the space shuttle main engine, aircraft, and industria l gas turbines. The many similarities between modeling the orthotropic turbine blade material and bone (another orthotropic material) landed him on his current project of studying the anisotr opic elastic material and linear elastic fracture properties of the Florida manatee rib bone. Before moving to Gainesville in the fall of 2002, Jeff married his wife, Renee Morrow, of Alpena, Michigan. The couple now has two ra mbunctious boys born in the spring of 2006.