ABSTRACT EXPERIMENTAL TESTING AND NUMERICAL MODELING TO CAPTURE DEFORMATION PHENOMENON IN MEDICAL GRADE POLYMERS. by Colin Patrick Yeakle

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1 ABSTRACT EXPERIMENTAL TESTING AND NUMERICAL MODELING TO CAPTURE DEFORMATION PHENOMENON IN MEDICAL GRADE POLYMERS by Colin Patrick Yeakle Rate-sensitivity, creep and relaxation behavior of medical grade polymers has been investigated experimentally along with an assessment of different constitutive models. Two types of modified common biomedical material A and B were tested. All materials exhibited rate-sensitivity and ratereversal behavior during creep and relaxation testing with prior loading and unloading histories for tensile and compressive tests. Numerical modeling was performed through a modified version of the Viscoplasticity Based on Overstress (VBO) model. Simulation and prediction results with good quantitative and qualitative agreement were produced for all materials in tension and compression loading. Parameter fitting using the Hybrid and Three Network Models in PolyUMod software generated material constants for ABAQUS, and the finite element results for a knee joint were verified against experimental data. The parameter fitting was unable to produce acceptable results for MATERIAL C. Implementation of VBO into the PolyUMod library is recommended to enhance modeling capability.

2 EXPERIMENTAL TESTING AND NUMERICAL MODELING TO CAPTURE DEFORMATION PHENOMENON IN MEDICAL GRADE POLYMERS A Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science Department of Engineering and Applied Science by Colin P. Yeakle Miami University Oxford, OH 211 Advisor: Dr. Fazeel Khan Reader: Dr. James Moller Reader: Dr. Gregory Reese

3 TABLE OF CONTENTS List of Tables... v List of Figures... vi Acknowledgements... xi 1 Polymer Background... 1 Polymer Introduction Hip and Knee Implant Applications for Polymers Polymer Modifications Common biomedical material B Common biomedical material A Computational Modeling Background... 6 Introduction to Polymer Computational Modeling Modeling Polymer Viscoelastic Deformation Constitutive Modeling of Polymer Deformation Behavior Medical Grade Polymers and Modeling Scope of Research Experimental Testing Methods Equipment Materials and Samples MATERIAL C Samples Common Biomedical Material A Samples Experimental Testing Protocols Experimental Testing Protocol General Information Tensile Test to Failure Experimental Protocol Tensile and Compressive Loading Test Experimental Protocols Recovery Test Experimental Protocol Creep Test Experimental Protocols Relaxation Test Experimental Protocols Digital Scanning Calorimeter Experimental Protocols Dynamic Mechanical Analyzer Experimental Protocols Human Gait Profile Construction Human Gait Profile Construction Human Gait Control Verification Human Gait Profile Experimental Procedure Human Gait Profile Overload Experimental Procedure Data Collection Data Collection Introduction Instron Testing Frame Data Collection MTS Testing Frame Data Collection Digital Scanning Calorimeter Data Collection Dynamic Mechanical Analyzer Data Collection Experimental Data and Results ii

4 Common Biomedical MATERIAL A Experimental Data and Results Common Biomedical Material A DMA Experimental Results Common Biomedical Material A Tensile Loading and Unloading Experimental Results Common Biomedical Material A Tensile Loading Recovery Test Results Common Biomedical MATERIAL A A Compressive Loading and Unloading Experimental Results Common Biomedical Material A Human Gait and Overload Profile Common Biomedical MATERIAL A A Creep Tests in Compression Results Common Biomedical MATERIAL A A Relaxation Tests in Compression Results Common Biomedical Material A DSC Experimental Results MATERIAL C Material Experimental Data and Results MATERIAL C DMA Experimental Results MATERIAL C Tensile Loading and Unloading Experimental Results MATERIAL C Tensile Loading Recovery Test Results MATERIAL C Creep Tests in Tension Results MATERIAL C Relaxation Tests in Tension Results Computational Modeling Modifications and Results Introduction to the VBO Model Deriving the VBO Model and adjusting Parameters for modeling polymers The Evolution of the VBO Model for Modeling Polymers Parameter Value Assignment Recent enhancements of the VBO Model The Introduction of an Unloading Parameter The Introduction of a Double Element VBO Model Modeling COMMON BIOMEDICAL MATERIAL A Deformation Behavior VBO Modeling Tension Loaded MATERIAL B Samples VBO Modeling Tension Loaded MATERIAL A Samples VBO Modeling Compression Loaded MATERIAL A Samples Modeling MATERIAL C Deformation Behavior VBO Modeling Tension Loaded MATERIAL C Samples VBO Modeling Tension Loaded MATERIAL D Samples ABAQUS and PolyUMod Three Dimensional Modeling Introduction to The PolyUMod Software The PolyUMod Hybrid Model The PolyUMod Three Network Model PolyUMod Modeling and Parameter Fitting Process Hybrid Model Parameter Fitting Three Network Model Parameter Fitting ABAQUS Computational Modeling PolyUMod Parameter Value Implementation into Abaqus Abaqus Modeling Results Conclusions and Future Recommendations Experimental Conclusions Common Biomedical Material A Experimental Conclusions MATERIAL C Experimental Conclusions VBO Model Conclusions Enhancements and Upgrades to the VBO Model Modeling Medical Grade Polymer with the Modified VBO Model iii

5 ABAQUS and PolyUMod Conclusions PolyUMod Parameter Fitting Conclusions ABAQUS Three Dimensional Modeling Conclusions Future Work References iv

6 Tables Table 4.1: MATERIAL A recovery table for loading histories of 2%, 6% and 14% at loading rates of 1x1-2 s -1 and 1x1-3 s Table 4.2: MATERIAL B recovery table for loading histories of 2%, 6%, 14% and 2% at loading rates of 1x1-2 s -1 and 1x1-3 s Table 4.3: % Crystallinity obtained from DSC tests for two samples each of virgin, MATERIAL B, and MATERIAL A Table 4.4: % Crystallinity for non-deformed MATERIAL B and samples MATERIAL B samples with small deformation at several different strains Table 5.1: Modified VBO parameter values for HDPE and PC at tension deformation amounts of 1% and 6% Table 5.2: Modified double element VBO parameter values for HDPE and PC at tension deformation amounts of 1% and 6% Table 5.3: VBO parameter values for MATERIAL B at tension deformation amounts of 6% and 14% Table 5.4: VBO parameter values for MATERIAL A at tension deformation amounts of 6% and 14% Table 5.5: VBO parameter values for MATERIAL A at a compressive strain deformation amount of 6%. 64 Table 5.6: VBO parameter values for MATERIAL C at a tensile strain deformation amount of 4.5% Table 5.7: VBO parameter values for MATERIAL D loaded to a tensile strain deformation amount of 4.5% Table 6.1: Parameter List for the Hybrid Model in the PolyUMod Library Table 6.2: Parameter List for Three Network Model v

7 Figures Figure 1.1. An example of a Semi-crystalline polymer s molecular chain arrangement from [1]... 1 Figure 1.2: An example of an amorphous polymer s molecular chain arrangement from [1]... 1 Figure 2.1: Maxwell Model made up of a spring and dashpot in series with one another. This model represents Eqns. 1 and 2 summed together Figure 2.2: Kelvin-Voigt Model... 8 Figure 2.3: Standard Linear Solid Model... 9 Figure 3.1: MTS Servo-Hydraulic Testing Frame setup Figure 3.2: Instron Servo-Mechanical Testing Frame setup Figure 3.3: Human gait profile plot of force versus time for two cycles Figure 3.4: Initial test of the human gait profile plot of force versus time for two cycles Figure 3.5: Human gait force plot versus control output plot for calibration verification of the testing Figure 4.1: DMA testing of two Common Biomedical Material A samples for the glass transition temperature Figure 4.2: MATERIAL B sample tensile loaded to large strains to find an acceptable maximum testing strain Figure 4.3: MATERIAL B tensile loading at 1x1-2 s -1 and 1x1-3 s -1 strain rate control. The faster loading rate results in an increase in the stresses along the stress-strain curve... 3 Figure 4.4: MATERIAL A tensile loading at 1x1-2 s -1 and 1x1-3 s -1 strain rate control. The faster loading rate results in an increase in the stresses along the stress-strain curve... 3 Figure 4.5: MATERIAL B versus MATERIAL A tensile loading at 1x1-2 s -1 strain rate control. The modifications in the MATERIAL A samples result in higher stresses for deformation compared to MATERIAL B samples Figure 4.6: MATERIAL B versus MATERIAL A tensile loading at 1x1-3 s -1 strain rate control. The modifications in the MATERIAL A samples result in higher stresses for deformation compared to MATERIAL B samples Figure 4.7: MATERIAL B load and unload at 1x1-3 s -1 strain rate control to 2% maximum strain Figure 4.8: MATERIAL B versus MATERIAL A tensile loading at 1x1-2 s -1 strain rate control for 6% and 14% maximum strain Figure 4.9: MATERIAL B versus MATERIAL A tensile loading at 1x1-3 s -1 strain rate control for2%, 6%, and 14% maximum strain Figure 4.1: MATERIAL B tensile loading recovery test with a 1x1-2 s -1 loading rate vs a loading rate of 1x1-3 s -1 for 6% and 12% maximum strain Figure 4.11: MATERIAL A tensile loading recovery test with a 1x1-2 s -1 loading rate vs a loading rate of 1x1-3 s -1 for 6% and 12% maximum strain Figure 4.12: MATERIAL A compressive loading to 6% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Stress values of 2, 12, and 5 MPa where creep tests were performed are marked by dashed circles Figure 4.13: MATERIAL A compression loaded recovery test with a 1x1-3 s -1 loading rate versus a loading rate of 1x1-4 s -1 for a 6% maximum strain Figure 4.14: MATERIAL A human gait profile for two stress cycles Figure 4.15: MATERIAL A human gait profile 5 cycle maximum and minimum strains vs time vi

8 Figure 4.16: One hour recovery test performed at the conclusion of the MATERIAL A human gait profile 5 cycle test Figure 4.17: MATERIAL A human gait overload stress cycle with one normal stress cycle on each side.. 37 Figure 4.18: MATERIAL A human gait profile 5 cycle with one additional 4% overload cycle maximum and minimum strains vs time Figure 4.19: One hour recovery test performed at the conclusion of the MATERIAL A human gait overload profile test 5 cycle compressive test Figure 4.2: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 strain rate. Creep tests performed for 5 MPa on loading and unloading curves Figure 4.21: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 strain rate. Creep tests performed for 2 MPa on loading and unloading curves Figure 4.22: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 strain rate. Creep test was performed at a 12 MPa stress value on the unloading curve Figure 4.23: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Creep tests were performed at a 5 MPa stress values on the unloading curve Figure 4.24: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Creep tests were performed at a 2 MPa stress values on the loading curve Figure 4.25: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Creep tests were performed at a 12 MPa stress values on the loading curve Figure 4.26: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Relaxation tests were performed at a 12 MPa stress values on the unloading curve Figure 4.27: DMA testing for the glass transition temperature of a MATERIAL C sample Figure 4.28: DMA testing for the glass transition temperature of a MATERIAL D sample Figure 4.29: Two MATERIAL C samples tensile loaded to failure with the safe testing range circled Figure 4.3: Two MATERIAL D samples tensile loaded to failure with the safe testing range circled Figure 4.31: MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Stress values of 85, 6, 5 and 2 MPa where creep tests were performed are marked by dashed circles.. 46 Figure 4.32: MATERIAL D samples loaded to 1.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Stress values of 12, 8, 4 MPa where creep tests were performed are marked by dashed circles Figure 4.33: Recovery tests for MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 4.34: Recovery tests for MATERIAL D samples loaded to 1.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 4.35: MATERIAL C sample loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1. The creep test was performed at 85 MPa on the loading curve Figure 4.36: MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1. Creep test performed at 6 MPa on loading and unloading curves Figure 4.37: MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1. Creep test performed at 2 MPa on loading and unloading curves Figure 4.38: MATERIAL C sample loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Creep tests performed at 5 MPa on the unloading curve Figure 4.39: MATERIAL D sample loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1, and then unloaded to 4 MPa for a 1.5 hr creep test vii

9 Figure 4.4: MATERIAL D sample loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1, and then unloaded to 12 MPa for a 1.5 hr creep test Figure 4.41: MATERIAL D samples loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1, and then unloaded to 8 MPa for a 1.5 hr creep test Figure 4.42: MATERIAL C samples loaded to 4.5% (9 MPa) maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1, and then unloaded to 5 MPa for a 1.5 hr relaxation test Figure 4.43: MATERIAL D samples loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1, and then unloaded to 8 MPa for a 1.5 hr relaxation test Figure 5.1: VBO example plot of stress, equilibrium stress and the kinematic stress Figure 5.2: Original VBO example stress strain plot for HDPE with linear unloading simulation Figure 5.3: Modified VBO stress strain plot for HDPE with non-linear unloading function Figure 5.4: Modified VBO stress strain plot for PC with non-linear unloading function Figure 5.5: HDPE stress strain curve with high, middle, and low creep test locations to demonstrate rate reversal trend Figure 5.6: HDPE creep strain rate reversal exhibited at stress values of 8, 11 and 14 MPa Figure 5.7: General description of double element VBO and how rate reversal is reproduced on the unloading curve for creep Figure 5.8: Spring and dashpot representation of double element modeling for polymer deformation behavior Figure 5.9: Double element VBO simulation with unloading function for HDPE loaded at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.1: Double element VBO simulation with unloading function for PC loaded at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.11: Double element VBO creep simulations performed at 14, 11 and 8 MPas on the unloading curve for HDPE loaded at a strain rate of 1x1-3 s Figure 5.12: Double element VBO creep simulations performed at 14, 11 and 8 MPas on the unloading curve for HDPE loaded at a strain rate of 1x1-4 s Figure 5.13: Double element VBO load and unload simulations performed at 14% and 6% maximum strain deformation for MATERIAL B samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.14: Double element VBO load and unload simulations performed at 14% and 6% maximum strain deformation for MATERIAL A samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.15: Double element VBO compressive load and unload simulations performed at 6% maximum strain deformation for MATERIAL A samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.16: Double element VBO creep simulation for stresses of 5, 14 and 2% located on the unloading curve MATERIAL A samples at a strain rate of 1x1-3 s Figure 5.17: Double element VBO creep simulation for stresses of 5, 14 and 2% located on the unloading curve MATERIAL A samples at a strain rate of 1x1-4 s Figure 5.18: Double element VBO Relaxation simulations for a strain of 4.6% and 4.75% located on the unloading curve MATERIAL A samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.19: Double element VBO tension load and unload simulations performed at 4.5% maximum strain deformation for MATERIAL C samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.2: Double element VBO creep simulation for stresses of 2, 5 and 6 MPa located on the unloading curve for MATERIAL C samples at a strain rate of 1x1-3 s viii

10 Figure 5.21: Double element VBO creep simulation for stresses of 2, 5 and 6 MPa located on the unloading curve for MATERIAL C samples at a strain rate of 1x1-4 s Figure 5.22: Double element VBO Relaxation simulation for a strain of 3.5% located on the unloading curve for MATERIAL C samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.23: Double element VBO tension load and unload simulations performed at 1.5% maximum strain deformation for MATERIAL D samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 5.24: Double element VBO creep simulation for stresses of 4, 8 and 12 MPa located on the unloading curve for MATERIAL D samples at a strain rate of 1x1-3 s Figure 5.25: Double element VBO creep simulation for a stress of 8 MPa located on the unloading curve for MATERIAL D samples at a strain rate of 1x1-4 s Figure 5.26: Double element VBO Relaxation simulation for a strain of.94% located on the unloading curve for MATERIAL D samples at strain rates of 1x1-3 s -1 and 1x1-4 s Figure 6.1: Modified standard linear solid configuration used in the Hybrid Model Figure 6.2: Three Network Model element Figure 6.3: PolyUMod Hybrid Model simulation versus MATERIAL A compression loading to 6% final strain at a rate of 1x1-3 s Figure 6.4: PolyUMod Hybrid Model simulation versus MATERIAL C tension loaded to 4.5% final strain at a rate of 1x1-3 s Figure 6.5: PolyUMod Hybrid Model simulation versus MATERIAL D tension loaded to 1.5% final strain at a rate of 1x1-3 s Figure 6.6: PolyUMod Three Network Model simulation versus MATERIAL A compression loaded to 6.% final Figure 6.7: PolyUMod Three Network Model simulation versus MATERIAL C tension loaded to 4.5% final strain at a rate of 1x1-3 s Figure 6.8: PolyUMod Three Network Model simulation versus MATERIAL D tension loaded to 1.5% final strain at a rate of 1x1-3 s Figure 6.9: Company A three dimensional model of an Oxford Knee Design used for Abaqus three dimensional modeling polymer materials Figure 6.1: Company A three dimensional model of an Oxford Knee Design using parameter values for a three network model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The strain values are shown on the polymer insert for a 26 N applied force Figure 6.11: Company A three dimensional model of an Oxford Knee Design using parameter values for a three network model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The Mises stress values are shown on the polymer insert for a 26 N applied force Figure 6.12: Company A three dimensional model of an Oxford Knee Design using parameter values for the Hybrid model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The strain values are shown on the polymer insert for a 26 N applied force Figure 6.13: Company A three dimensional model of an Oxford Knee Design using parameter values for the Hybrid model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The Mises stress values are shown on the polymer insert for a 26 N applied force Figure 6.14: Company A three dimensional model of an Oxford Knee Design using parameter values for a Hybrid model within Abaqus for MATERIAL D. The strain values are shown on the polymer insert for a 26 N applied force ix

11 Figure 6.15: Company A three dimensional model of an Oxford Knee Design using parameter values for a three network model within Abaqus for MATERIAL D. The Mises stress values are shown on the polymer insert for a 26 N applied force x

12 Acknowledgements They say that success breeds success and this was evident in road to completion of this degree. I am very lucky to have been surrounded by bright and supportive people the entire time. First and foremost, I would like to express my gratitude and appreciation to my advisor Dr. Fazeel Khan. Without his knowledge and guidance for the past two years the advancement and depth taken in this research project would not have been possible. He could not have been a better advisor to me and his patience and trust helped to instill my confidence in my work and ability. I am truly grateful. Thanks to my wonderful friends and family who have helped to fuel my drive and believed in me along the way. They offered nothing but support and confidence in my decision making and capabilities. I thoroughly enjoyed the time spent with my classmates, who provided assistance and entertainment throughout the entire two year journey. A variety perspectives and personalities along with much needed distractions from work, helped the past two years go faster and smoother than I ever thought possible. Lastly, to the companies, Company A and Company B, for providing the materials needed to conduct all of the experiments and providing quick feedback when it was needed. The entire research venture was dependent on the materials and could have never taken place without the willingness to reach out to Miami University and myself. xi

13 1 Polymer Background 1.1 Polymer Introduction Several polymer deformation characteristics can be related to material molecular structure and chain arrangement. Polymers are characterized structurally as being either amorphous or semi-crystalline. A polymer with a semi-crystalline chain orientation has some its molecular chains arranged in a pattern that reflects some order or repetition. Regular folding of the chains can form lamellar plates, and localized, radial orientation of lamellar plates can result in the formation of spherulites. Conversely, an amorphous polymer s chains are tangled and intertwined with one another, exhibiting no order or regular pattern. Polymers can also be characterized as thermoplastics or thermosets based on the nature of their thermal response. Thermoplastics soften beyond the glass transition to have chain mobility. This gives the polymer the ability to be remolded and reused many different times throughout its useful life. The polymer s glass transition temperature is separate from its melting temperature. The glass transition temperature is the temperature at which the onset of chain mobility occurs and the polymers can be easily plastically deformed. Polymers such as COMMON BIOMEDICAL MATERIAL A and COMMON BIOMATERIAL B are classified as thermoplastic polymers. Thermosets are different from thermoplastics in that their individual molecular chains are chemically linked to one another by covalent bonds during their polymerization process. These chemically cross-linked networks of chains resist thermal softening, creep and chemical attack, but cannot be thermally processed again and again like thermoplastics. Two part epoxy resin adhesives are good examples of thermoset polymers in which a catalyst is combined with a resin to initiate a polymerization reaction. This results in the formation of a solid polymer. A general illustration of type of arrangement can be seen in Figures 1.1 and 1.2. Figure 1.1: An example of a Semi-crystalline polymer s molecular chain arrangement from [1] Figure 1.2: An example of an amorphous polymer s molecular chain arrangement from [1] The mechanical properties of each type of polymer, semi-crystalline and amorphous, can be quite different from one another simply based on the arrangement of the molecular chains. Amorphous polymers, due to the lack of order in their molecular chains, often have better properties than semicrystalline materials [2]. Longer chains and higher amounts of entanglement result in higher molecular 1

14 weights. Molecular weight; along with degree of polymerization and chain configuration, is a major factor in a polymer s mechanical behavior. High molecular weights produce higher density and superior strength for amorphous type polymers [3]. Semi-crystalline polymers typically have lower molecular weights and can be deformed easier than amorphous polymers. 1.2 Hip and Knee Implant Applications for Polymers The research methodology that is used when designing hip and knee implants can be explained by focusing on two major components; replacement kinematic design and material implementation. The main objective with the design enhancement of medical implants is the restoration of natural functionality. If the implant is unable to function properly like an organic body part, then restricted motion or even discomfort and pain can all negatively affect the patient. The forces incurred in human movement and everyday use on bone joints are measured and used for the development of each new knee joint model and material selection [4]. This data is used especially in the static and dynamic analysis of polymeric parts which form the wear surfaces and contribute to the range of motion provided by the joint. Early knee and hip joint designs featured metal on metal interfaces at the location of motion. These prosthesis joints had high failure rates and extremely limited motion due to material choice and design constraints [5+. As prosthesis joint s technology evolved and range of motion in the joints improved, COMMON BIOMEDICAL MATERIAL A was widely adopted for use in artificial joints. By increasing the range of motion, the degrees of freedom and the surface area affected by applied and frictional forces acting on the prosthesis parts are also increased. With more of the part being utilized while in service, the areas for deformation and chance of degradation of each part increased. Polymers were introduced to knee and hip prosthesis by adding COMMON BIOMEDICAL MATERIAL A as a separating layer between the metal part surfaces [5]. The material change greatly increased the life of the parts and reduced the amount of wear particles in the body because of the COMMON BIOMEDICAL MATERIAL A s highly favorable mechanical and frictional properties. Polymers, such as COMMON BIOMEDICAL MATERIAL A exhibit high strength, ductility, and wear resistance, all of which are desirable attributes for prosthesis joints and not all of which are always exhibited with metal parts. Also, due to the long term biocompatibility of some polymers, specifically COMMON BIOMEDICAL MATERIAL A became a necessity in the manufacturing of prosthesis joints. While polymers show a drastic improvement over their metallic counterparts, they have some disadvantages. No matter what material is implemented into the joint it is reasonable to say that it will incur wear. Polymer wear, while greatly reduced from the metal on metal surface wear that occurred in older models of knee joints, does still occur and tiny particles are released into the body and joints. These particles can cause inflammation and bone density loss. As bone density loss increases, the joint a can be loosened and require subsequent surgeries. The human body can only endure approximately two to four total joint replacement surgeries for a specific joint depending on the patient. This limitation becomes a major factor in the need for increased arthroscopic joint life. Creep and relaxation behavior can also put the polymer part at risk. Long periods of constant loading, such as standing, can cause creep behavior to occur. This additional strain increases the amount of plastic deformation within the part resulting in a shorter service life. Holding a constant strain on the polymer material can cause the stress within the polymer to decrease. This can be detrimental to inserts that are clipped into place and held. Over time the stresses that hold the part in place can relax causing loosening. Finally, fatigue is another issue for polymer parts. High amounts of cycling which occur in human joints, due to movement, can cause the materials to soften or harden, depending on the polymer, overtime. High fatigue life is a characteristic that polymers must exhibit for the longevity of a prosthetic joint. Among MATERIAL 2

15 Deformation and failure; wear, creep, relaxation and fatigue are at the forefront of material characteristics that drive research and demands for even better suited materials. 1.3 Polymer Modifications The increased demand for wear resistant and stronger materials for medical implants has spurred a constant search how to modify existing medical grade polymers and produce new variants. Because a medical device will end up implanted within the human body, many more factors than just basic mechanical properties need to be considered. Factors such as design, material selection, structural requirements, biocompatibility, biodegradability, processing and any clinical issues all have a hand in the production of an implant [3]. Certain daily human activities can cause high stresses within bones and joints. Therefore, if the load bearing polymer is not appropriately selected and suitable for the application, the polymer is put at a high risk of premature yield, wear, fatigue, or creep based failure. To combat these behaviors specifically some polymers undergo treatments such as radiation cross-linking [5] or are enhanced with other materials to become composites [2], such as carbon, to increase their mechanical behavior performance. Cross-linking radiation actually alters the molecular chains by creating bonds between them which results in improved surface properties (hardness) but can reduce other mechanical behaviors (strength and ductility) [6]. The bombardment of radiation breaks existing molecular bonds creating free radicals branching off the chain backbone. These free radicals have the ability to combine with one another and create secondary bonds. Due to the limited mobility of chains in the solid state, a significant fraction of the free radicals cannot recombine to form cross-links. This problem is addressed by applying additional thermal processes. For instance, annealing or melting the polymer allows chain mobility to increase and in turn, increases the cross-link bond density within the polymer. By reducing the amount of free radicals left over from gamma radiation, the polymer is also more resistant to oxidation, which severely degrades the mechanical properties of the part over time. Oxidation occurs when oxygen in the air, fluids, or surrounding medium bonds with the free radicals. Oxidation results in the degradation of mechanical properties such as strength, ductility, toughness and fatigue crack propagation resistance in COMMON BIOMEDICAL MATERIAL A from the modification treatments on the material [7]. Enhancements by the way of introducing fillers to the polymer matrix are also made. This is different from cross-linking because carbon reinforcement does not bond with the polymer chains but is mixed throughout the polymer matrix. These fillers cause improved wear performance compared to virgin materials under the same conditions [8]. The filler is often added to the matrix of the polymer in the form of fibers. The added filler rests within the voids of the polymer chains and the polymer matrix transfers the applied stress acting on the part to the fibers instead of directly on the molecular chains. Hence, the main role of the added fibers is to distribute the applied loads and reduce deformation of the polymer. The addition of the filler creates a new material modulus of elasticity for the part that is directly affected by the volume fraction of the fiber and the modulus of elasticity of the fiber. 1.4 Common Biomedical Material B MATERIAL C is a semi-crystalline, thermoplastic polymer that is part of a high stiffness polymer s family of polymers. As a semi-crystalline polymer, MATERIAL C s molecular chains exhibit both random order in areas and some order or pattern in others (approximately 2% - 3% crystalline) [9]. MATERIAL C s backbone molecular chain is made up of oxygen atoms and benzene rings giving it high strength and creep resistance. MATERIAL C has a glass transition temperature around 15 o C and a molecular weight around 1, g/mol [2]. MATERIAL C is also known for its high abrasion resistance, low water 3

16 absorption, high wear and radiation resistance. Many of these strengths are due to degradation mechanisms that occur with polymer parts in-vivo; therefore MATERIAL C is an excellent candidate for biomedical applications; more specifically implant parts and bearings. Two of the major medical applications in which MATERIAL C is utilized are hip and knee joint implants, specifically in joints where the polymer can be substituted to create polymer to metal bonded interfaces or in regions where there is a high propensity for wear. Despite its high wear resistance in natural form, this property can be enhanced through the use of fillers. MATERIAL C s high solvent resistance, impact strength and thermal stability make it suitable for high performance composites which are being developed for use in biomedical applications, for example MATERIAL D. The resistance to water absorption in particular mitigates the chances of filler leaching into the surrounding tissue/fluids. Fluids found in the body can also diffuse into the implant part and can weaken the strength of the interfacial adhesion, reducing the strength of the part and possibly leading to premature or catastrophic failure [2]. Due to MATERIAL C s radiation resistant characteristics, an enhancement to the polymer attributes comes by the way of fillers. Carbon, because of its own high strength and wear properties, is the predominate filler added to MATERIAL C. One of the leading producers of MATERIAL C for biomedical applications, Company B, has several lines of MATERIAL C that have been enhanced through the addition of fillers and specifically tailored for prosthesis applications [8] [1]. The two lines of MATERIAL C have been enhanced with the addition of carbon fibers. The carbon fibers significantly decrease wear in metal-to-polymer and ceramic-to-polymer interfaces such as hip and knee joints. MATERIAL C has shown in hip simulator wear tests up to a 7% decrease in polymer wear reduction when compared to industry standard COMMON BIOMEDICAL MATERIAL A [8]. The MATERIAL C line also has been shown to exhibit a high resistance to creep deformation, which can be another major concern in prosthesis applications. Due to the enhanced strength and wear properties, the polymer parts can be machined thinner and lighter, decreasing the weight of the prosthesis joints and increasing joint design flexibility [1]. Another research study with the aim to find bone replacement materials used ceramic, hydroxyapatite (HA) filler within MATERIAL C [11]. The theory behind this is that that because mammal bones are made up of almost 56% calcium hydroxyapatite that the composite is a suitable biomaterial along with its beneficial mechanical properties, notably in compression. The new composite was found to be suitable in high load bearing applications. 1.5 COMMON BIOMEDICAL MATERIAL A COMMON BIOMEDICAL MATERIAL A is a semi-crystalline thermoplastic polymer which has excellent physical and mechanical properties. Common biomedical material A is part of a group of thermoplastic polymers called polyolefins along with the polymer polypropylene. The common biomedical material A molecular chain is comprised of methylene monomers connecting to one another assembling the backbone of the molecular makeup. The basic and first common biomedical material A formulation is known as low density common biomedical material A (LDBM A) and was used for many everyday plastic applications such as food containers and bottles. LDBM A chain architecture is branched and has a low molecular weight around 5, g/mol [5]. It was discovered that common biomedical material A could be polymerized at a lower temperature than what had previously been accomplished for LDBM A and thus creating a linear backbone with fewer branches and having a higher crystallinity and a molecular weight of around 2, g/mol [5]. Obtaining a higher crystallinity leads to higher polymer density and strength, thus it is known as high density common biomedical material A (HDBM A). HDBM A can achieve molecular weights of up to 6 million g/mol, at which point it is labeled as COMMON BIOMEDICAL MATERIAL A. Common biomedical material A, in general, has a very low glass transition 4

17 temperature leaving the polymer chains in a state that makes them easily mobile and can give the polymer a high amount of deformation before fracture. That being said, the material still exhibits high strength and wear properties which are directly related to the crystalline content. The listed values for the glass transition and melting temperatures for HDBM A are -12 o C and 135 o C, respectively [5]. Common biomedical material A has been used extensively in medical device applications for over 45 years. Modifications to the polymer have been done to enhance properties compared to the standard untreated material. Modifications have been done extensively at different levels and doses to obtain high wear resistant types of COMMON BIOMEDICAL MATERIAL A. Other treatments to the polymer such as sample surface enhancements have also been implemented to try to obtain these characteristics without altering the entire part volume. For example, argon plasma treatments cannot penetrate completely through the polymer part and create a modified surface with increased wear resistance but leave the original properties unchanged within the majority of the part [12]. Similarly, carbon ion bombardment has also be utilized to create a layer of carbon onto the part surface in which wear properties of the carbon are used instead of the COMMON BIOMEDICAL MATERIAL A [13]. All three methods of enhancement have shown drastic increases in wear resistance for COMMON BIOMEDICAL MATERIAL A compared to the virgin material. Company A, a subsidiary of Johnson & Johnson, specializes in knee and hip prosthesis joints and utilizes COMMON BIOMEDICAL MATERIAL A in their products along with developing new versions with enhanced properties [14]. A previously-used type of COMMON BIOMEDICAL MATERIAL A, called MATERIAL B, underwent new modifications from MATERIAL A. A newer material in the experimental stages is the MATERIAL A. However; there is a company classified added filler within the polymer matrix to enhance the mechanical properties of the MATERIAL B. 5

18 2.1 Intro to Polymer Computational Modeling 2 Computational Modeling Background Due to the wide range of mechanical property values stemming from the differences in molecular structure, polymers can be used in myriad applications, and can even be substituted into applications where metals and alloys have previously dominated the market. Due to the large number of applications and multiple benefits derived from the use of polymers over other materials, there is a continuing desire to know more about polymers in order to enhance their properties and explore new potential applications. However, extensive experimental programs, which are required to assess the feasibility of polymers in new applications, can incur high operating costs and large amounts of time. To overcome these difficulties, different types of mathematical models have been developed to model the viscoelastic and viscoplastic deformation behavior of polymers under a variety of loading conditions and environmental temperatures [15], [16], and [17]. While there have been many significant accomplishments in the area of MATERIAL Deformation modeling, several shortcomings still remain within current computational models. While a large amount of studies and tests have been performed on the initial loading of polymers, with generally accepted results, capturing qualitative and quantitative data on the unloading portion of an experimental (stress-strain) loading test still poses some challenges for some models. Typically, complex loading histories and especially those involving loading and partial unloading of a sample before a creep or relaxation test, result in responses that lie beyond the capabilities of most models. One of these responses is rate-reversal behavior in which stress (relaxation) and strain (creep) begin with either a positive or negative rate and then switch to the opposite in the early stages of the test. As a result of their favorable mechanical characteristics and low cost, the demand for new and innovative polymers used in medical applications, specifically in implants and prosthesis, is increasing rapidly. Because of this rising demand, efficient and easily accessible methods for simulating or reproducing computationally the mechanical behavior of the medical grade polymers, such as COMMON BIOMEDICAL MATERIAL A and MATERIAL C, under diverse types of loading conditions are in high demand. Total hip and knee replacement joints are among two of the major products produced in biomedical applications in which medical grade polymers are implemented. Initially, hip and knee implants were designed with metal on metal components and, for a while, were a competing design [5]. Wear particles from the prosthetic joints -can cause severe complications with the human body. Therefore, the common biomedical material A-based designs are proving more robust and desirable moving forward. That being said, once implanted into the human body, these materials are subjected to different types of strenuous loading conditions in which can still cause wear, strain, and fatigue of the polymeric components. The degradation caused by these mechanisms on the materials in vivo provides incentive for continuing the task of creating new polymer matrix combinations and molecular enhancements that boast higher mechanical tolerance to these adverse conditions. With new fillerpolymer matrix combinations and molecular alterations that boost material mechanical behaviors, it is uncertain whether the changes resulting from these modifications can still be modeled effectively with the existing formulation. Therefore, current accepted computational models that can model deformation behavior need verification for their ability to model deformation behavior of medical grade polymers. A large quantity of the current experimental methods for testing polymers is not always accurate in depicting the complete deformation behavior experienced in-vivo for a prosthetic joint. While relevant and useful data results are obtained from these tests on the material s general deformation behavior 6

19 and characteristics, simple monotonic loading, whether in tension or compression, does not always capture the complete complex load profile experienced by a polymeric part in service. Also, traditional experimental loading programs need to be modified to produce more complex loading profiles such as one that mimics the human gait, which is a multi-peak compressive cycle test. Hence, computational models can be utilized to obtain accurate three dimensional simulations on part deformation with a wide array of geometries as well as more complex loading profiles in order to optimize design. It is important to touch on the interplay between modeling and simulation. The model serves as the basis of all future loading simulations. By obtaining a model that will accurately produce polymer behavior for any type of load condition, future simulations of knee joints and loading profiles will then produce accurate depictions of how the polymeric materials will react. It is of great importance, then, that the model can reproduce all types of deformation behavior and not have a glaring weakness in one or more areas. 2.2 Modeling Polymer Viscoelastic Deformation It is widely accepted that polymers can be modeled by various combinations of spring and dashpot configurations due to their viscoelastic nature. The need to exhibit both viscous and elastic properties requires the model to have elements to capture the strain rate effects of both. A linear spring element models the elastic behavior of the polymer while a dashpot is used to capture the viscous nature of the polymer. Both can be seen in Eqns. 1 and 2. where is the strain (subscript spring or dashpot denotes strain in given element), is the strain rate, is the stress, is Young s modulus (tangent modulus over small strain range) and is the viscosity of the dashpot component. Similarly to metals, polymer deformation can exhibit an elastic response over a small strain range. This elastic region is caused by the stretching and distorting of covalent bonds within the polymer chains allowing the material to deform. At higher stresses, the material molecular chains begin to slide, rotate and disentangle relative to one another. At these higher stresses, the material is beginning to plastically deform. Polymers are highly rate dependent materials meaning that the resistance to deformation is dependent on the rate of deformation. This nature is captured by the dashpot element and is representative of the aforementioned viscous nature of the material. Steric hindrance between the chains, cross-link bonds and filler materials all can impede the motion of the chains and also have an effect on the material Deformation behavior. Among some of the simplest models utilizing a spring and dashpot element arrangement are the Kelvin- Voigt, Maxwell, and Standard Linear Solid models. The Maxwell model is a configuration of spring and dashpot elements in series. The total strain is represented by the sum of the two individual strain behaviors of the elements. (1) (2) Figure 2.1: Maxwell Model made up of a spring and dashpot in series with one another. This model represents Eqns. 1 and 2 summed together. 7

20 While the Maxwell model correctly captures the rate dependence exhibited in polymers, the model also has its setbacks. The Maxwell Model lacks the ability to model creep mechanical behavior in polymers accurately. Equation 3 shows the basic equation for the Maxwell Model, where is the viscosity coefficient, is Young s modulus for the material, and denotes the time derivative for the stress and strain. It can be seen in Equation 3 that, when modeling a polymer s creep behavior, setting the stress rate of change to zero will produce a linear response. The stress and the viscosity coefficient are constants and therefore the only variable is time. However, the Maxwell Model can capture polymer relaxation behavior. Imposing a constant strain rate of zero and integrating with respect to time; an expression for stress as a function of time can be obtained to simulate the viscoelastic response in relaxation. The Kelvin-Voigt (KV) model has a parallel configuration of a spring and dashpot seen in Fig The KV model like the Maxwell model can accurately model polymer rate dependence but unlike the Maxwell model this is done by representing the total stress as the sum of the stresses in each component. (3) Figure 2.2: Kelvin-Voigt Model The KV model excels at modeling polymer creep behavior but lacks in its ability to model stress relaxation behavior. The fundamental relation for the KV model can be seen in Equation 4. where is the strain, is the stress, is Young s modulus, is the viscosity of the dashpot component, and is the time derivative for the strain. When using Equation 4 to model creep behavior, it can be seen that the KV Model has the ability to model viscoelastic creep behavior. Much like the Maxwell model s ability to model stress relaxation with a first order equation, integrating the equation with respect to time produces an equation for creep strain. However, when attempting to model stress relaxation by setting strain change to zero, again a linear equation appears. The KV model therefore cannot model stress relaxation qualitatively for polymers. There exists any number of additional configurations of the Maxwell and KV models with one another and with any number of the spring or dashpot elements in the makeup. To achieve an acceptable combination of characteristics exhibited in the Maxwell and KV models, namely creep and relaxation behavior, while keeping the simplicity of the two models, the Standard Linear Solid Model (SLS) was derived. This model is a modification of the previous two mentioned models in that it is a spring in series with the KV parallel configuration seen in Figure 2.3. (4) 8

21 Figure 2.3: Standard Linear Solid Model The SLS Model is assembled by using a purely elastic (spring) and an inelastic (spring dashpot parallel configuration) component with one another. This helps to capture the full viscoelastic response that is exhibited during deformation. The SLS Equation can be derived through a summation of the two component s representative equations, Equations 5 and 6. (5) (6) where and are the Young s modulus for the springs within the elastic and inelastic components, is the viscosity of the dashpot component, is the total stress, is the total stress rate, is the total strain and is defined as by Krempl [17]. When performing a creep test the elastic component becomes zero but the inelastic (KV) component provides the deformation behavior. Likewise, for a stress relaxation test, the elastic component is there to model this type of deformation behavior. 2.3 Constitutive Modeling of Polymer Deformation Behavior The general concept behind constitutive modeling for polymers, at its simplest form, is describing the relationship between the applied stresses to the resulting strain and deformation behaviors as a product of the former. This has been attempted many different ways in polymer research in an attempt to capture the complex nature of polymer deformation behavior. The main goal is to gain deformation behavior predictions that accurately capture as many, if not all, of the following; rate dependence, creep, relaxation, recovery, and temperature dependence. Most models can predict results for some of the above tests very well, while falling short of accurate results for the others. While some models are derived specifically for polymers, several have been designed to capture deformation behavior in metals. These models designed for metals are then modified to capture attributes such as curved unloading segments and higher recovery percentages at zero stress that aren t seen in metal deformation behavior. A wide variety of different models have been developed over the past 2 years attempting to model polymer deformation behavior. Several different models utilize different variations of the modified SLS for their constitutive equations. One model proposed by Khan et al. [19], uses a configuration made up of a Maxwell model placed in parallel with a KV Model. In an attempt to capture polymer temperature dependence along with other behaviors, one of the springs within the model exhibits elastic properties that are solely temperature based, unlike many other models that use the spring to represent only the elastic behavior of the polymer. A series of 15 material constants are adjusted and determined through a least squares optimization process in order to produce its output. This model was used by Khan et al. [19] during their research and is generally accurate in only polymer relaxation behavior and demonstrating strain rate sensitivity. It should be noted that this model among many others focused on prediction of relaxation tests with only prior loading rate histories at strain values located on the loading curve. 9

22 A state variable-based model known as the Viscoplasticity Based on Overstress (VBO) has been used for polymer loading deformation behavior that is also based on the modified SLS. This model focuses on an overstress term which is the difference between the stress value derived from an experimental loading rate and the stress value derived from a theoretical equilibrium stress (stress from loading at infinitesimally slow loading rates). VBO was originally conceived for the modeling of metal and alloy deformation behavior but has been modified and improved several times from its original formulation, to model polymers. VBO has extensively been used to model HDPE [2], [21], [22] and also shown the ability to model several other semi-crystalline and amorphous polymers as well [21], [23], [24]. Shortcomings in the unloading portion of a profile for a stress-strain curve have been previously reported where the model cannot qualitatively reproduce the non-linear unloading viscoelastic behavior exhibited by the polymer samples even with some newer modifications to the model [21] [23]. One such modification, implemented by McClung and Ruggles-Wrenn, was the addition of an isotropic stress rate of change function instead of considering the isotropic stress value constant and predicting deformation behavior of polymers at high temperatures, not just room temperature [24]. McClung and Ruggles- Wrenn s model, aimed at capturing softening during and after cyclic testing in the polymer, proved to produce satisfactory results. Using thermodynamic properties is another approach in constitutive modeling as opposed to using only a spring dashpot configuration methodology [16]. The model proposed by Ghorbel [16], was designed with a set of constitutive equations that introduced a new function focusing on the yielding behavior and the flow region of deformation that directly precedes the yielding of polymers. The yield criterion takes into account the material properties of the average molecular weight, molecular weight between cross-links, and the degree of crystallinity but is dependant of the first stress invariant, I 1, and the second invariant of the deviatoric stress, J 2 [16]. The model proved to be a valid way of modeling monotonic tensile and compressive loading but quantitatively did not produce matching values against experimental data when trying to predict polymer behavior during cyclic and shear loading profiles. While some models produce a set of equations that predict polymer behavior in a continuous fashion for a full load cycle (loading to a determined stress and unloading the sample back to zero force under one time set) [16], [17], [19], [2], other models attempt to piece together the elastic and plastic flow regions separately from one another using separate sets of equations for each portion of polymer deformation profile [25]. This type of model is assembled by piecing together an elastic region along with a high and low yielding/flow stress region depending on the strain rate that is used for the given test. The conception behind this type of modeling is motivated by a microstructure methodology. The computational trials were completed, by Wang and Arruda, only on loading to failure under tension and loading to high stresses under compression. No evidence was presented by Wang and Arruda, that the model could produce results for the non-linear shape of the unloading curve or any other type of deformation behavior such as creep, recovery, or relaxation. Results produced for loading curves proved to be quantitatively accurate but lacked qualitative results in modeling the viscoelastic unloading curvature that is produced by polymers when unloading a sample to zero force for experimental data under tension. With the increasing amount of polymer blending, composites, and molecular modifications being produced for higher stress and strain applications, several models have focused solely on these types of materials [26] [27]. Goldberg and Stouffer [26] utilized a micromechanics approach to model small loading rates and strains for polymer composites. In their model the effective properties and deformation behavior are calculated based on the properties of the individual constituents [26]. A unit cell of the material is analyzed with its response and is representative of the entire material response. While results were accurate, the experiments focused solely on loading and loading to failure. The 1

23 model did not model and viscoelastic behavior on the unloading segment or for faster loading rates. The second model, the Hybrid Model developed by Bergstrom et al [27] was specifically developed for modified COMMON BIOMEDICAL MATERIAL A only. This model focused on specific behavior in deformation zones that were represented by set strain ranges. While the Hybrid Model produced very accurate results for loading and unloading in COMMON BIOMEDICAL MATERIAL A it lacks universal polymer simulation. The Hybrid Model does produce a non-linear unloading curve but some qualitative shortcomings still arise on the unloading curve for the model predictions as have been previously reported and discussed in other models such as the VBO and Ghorbel models [15] [16]. As has been demonstrated through the discussion alone, several approaches can be adopted when developing constitutive models for polymer behavior. Initial loading paths of polymers have been successfully modeled by many different types of constitutive models. Polymer nonlinear unloading behavior still presents a problem for most models. And while reproduction of this nonlinear behavior has been improved, much progress is still needed in producing accepted qualitative and quantitative results. It should be also noted that, similar to the case of the model produced by Khan et al. [19], many models do not focus on creep and relaxation tests with a wide variety of prior loading histories leading up to the start of the creep or relaxation test. Creep and relaxation tests have been extensively studied by simply loading to a stress or strain value on the initial loading curve and performing the test. Stress and strain rate reversal behavioral results have been found in experimental data for polymers when performing creep and relaxation tests on the unloading segment of the loading profile [28] [29]. This behavior exhibits initial negative creep strain or stress relaxation and then changes to a positive rate. This behavior only occurs in a finite area along the unloading curve and can have negative implications on the polymer part if the behavior is not accounted for. For example inaccurate creep data can lead to miscalculations of service life if the correct final amount of strain obtained from a creep test is not documented. The behavior often is not documented due to a heavy research focus of creep and relaxation on the loading curve. Moreover, models have not accounted for or produced this type of behavior when modeling creep or relaxation deformation behavior. These findings along with successful computational loading results suggest that experimental and computational testing need to focus more extensively on the unloading segment of a load profile. Further development and focus into successfully modeling this more extensive prior loading history is suggested. Investigation and proper modification into the computational model s ability to capture rate reversal also needs to be incorporated into the already difficult task of modeling polymer deformation behaviors along with producing the correct profile curvature during loading and unloading as a result of the viscoelastic nature of the materials. The previously discussed models including the original version of VBO can produce some but not all of; accurate deformation behavior for multiple types of polymers, strain rate sensitivity at all ranges of strains, non-linear unloading profiles for stress-strain plots for tension and compression. Also, rate reversal in creep and relaxation tests in both tension and compression have not been documented in results from the any of the previously discussed models. These shortcomings present the need for a model that can capture the entire set of above mentioned polymer behaviors for a wide range of materials in both tension and compression. 2.4 Medical Grade Polymers and Modeling The VBO model has been extensively used to model the mechanical response of the semi-crystalline polymer, HDPE. Limited amounts of investigations have been completed in modeling the mechanical behaviors of medical grade polymers, more specifically medical grade COMMON BIOMEDICAL MATERIAL A, unfilled MATERIAL C and MATERIAL D, using this type of a constitutive modeling approach [29]. As previously mentioned, nonlinear deformation, at low strain rates only, was also successfully modeled for 11

24 an AS4/MATERIAL C composite using Ramaswamy Stouffer s method of implementing constitutive equations utilizing micromechanics [26]. It has yet to be determined whether VBO can still model the same mechanical behaviors that were successfully modeled in HDPE and to the same accuracy given the differences in the mechanical properties in HDPE and medical grade polymers. Perhaps more important is the question of whether the changed molecular bonds or the added composite particles yield deformation characteristics beyond the VBO model s capabilities. The current VBO formulation, along with many other models has been based on a two-dimensional formulation, which can be extended to a three-dimensional tensor based version. Experiments thus far have largely been done in uniaxial tension loading for stress, strain, creep and relaxation. The human knee and implants do experience some of this type of loading but more often they are placed mainly in compression loading instances. The Oxford Knee is the current prosthetic design used in most knee transplants [5]. A three dimensional part model of this knee transplant design can be used for finite element analysis for validation against experimental results. Polymers implanted into a human body are expected to experience complex loading conditions. There is, hence, a need to be able to apply a three dimensional model to the different types of biomedical materials to investigate how they react to the diverse loading profiles experienced by their in-vitro counterparts. Finite element analysis has become a favored method for testing and predicting the applied loads on a part. The program ABAQUS is used extensively to model polymers in a more realistic three-dimensional space [14]. Computational models can be loaded into ABAQUS to give three dimensional deformation results for material samples and parts. An extensive library of general purpose polymer material models has been developed as an add-on to ABAQUS called PolyUMod [3]. The Hybrid Model otherwise known as a modified KV model is used specifically for COMMON BIOMEDICAL MATERIAL A three-dimensional modeling. This model and several other models, mostly modifications to the elastomer based Bergstrom-Boyce model [3] are used to model various thermoplastic s deformation behavior. Part files with accurate dimensions of knee and hip implant polymer parts can be inserted into the Hybrid Model to give actual lifelike loading instances and predictions of deformation behavior that the implants will undergo when made of COMMON BIOMEDICAL MATERIAL A. The model can be calibrated with new parameters specific to different materials and their individual characteristics to reproduce a vast array of part and their deformation behaviors for each material, such as MATERIAL C. The PolyUMod model library was developed by Veryst Engineering [3]. Much of the library s models are developed with the use of the Bergstrom-Boyce model for elastomers with appropriate modifications. Bergstrom and Boyce originally designed their model for recreating elastomer and filled elastomer deformation behavior and is based on micromechanics of deformation for macromolecular networks [31] [32]. It was known that biological tissue exhibited characteristics similar to elastomers and thus the model later was modified to capture time-dependence and hysteresis in both elastomers and soft biological tissue, bringing the model into the biomedical area [29]. This modification of the Bergstrom-Boyce model began the assembly of many different types of models that are tailored for a broad range of types of polymers; thermosets, thermoplastics, foams, and biomaterials, and even some specifically for COMMON BIOMEDICAL MATERIAL A. 2.5 Scope of Research The aim of this research is threefold: 1. To investigate the mechanical behavior of medical grade polymers, namely MATERIAL C and COMMON BIOMEDICAL MATERIAL A, with and without different modificationsthrough a wide 12

25 array of uniaxial tension and compression tests such as, but not limited to; strain rate sensitivity, creep, relaxation, recovery and human gait cyclic loading (compressive only). 2. To determine the ability, or lack thereof, of the modified VBO two element model with new modifications (see Section 5.3) to produce more accurate simulations, than the single element model, of the mechanical behaviors of these medical grade polymers obtained in the experimental tests listed above and to recreate rate-reversal in creep strain and stress relaxation tests. 3. To use the PolyUMod model library and specifically the Hybrid Model for COMMON BIOMEDICAL MATERIAL A, in ABAQUS software to three-dimensionally simulate the deformation characteristics of the medical grade polymers listed above in knee prosthesis joint part files with the material characteristic parameters of MATERIAL C and COMMON BIOMEDICAL MATERIAL A. Extensive experimental testing has been conducted on both the COMMON BIOMEDICAL MATERIAL A and MATERIAL C samples. Tension and compression tests, when applicable, were completed along with a variety of other types of deformation analyses of tension and compression data for experimental tests such as; creep, recovery, and relaxation experiments. An emphasis on rate dependence was taken by loading the samples at several different strain rates along with experiments at a variety of maximum stress values. Results between virgin and modified materials were completed to discover the direct effects that these modifications have on the sample deformation behavior. There is also a need for a more complex loading profile, similar to that of a human gait or one that would more accurately produce stresses and strains that a knee implant would incur. This was developed and experimentally tested on the COMMON BIOMEDICAL MATERIAL A polymer samples per the request of Company A. VBO has shown promise in the modeling of polymer deformation behavior. The introduction of a new parameter to a VBO model, introduced by Dr. Ozgen Colak [23] (see Section 5.3), that has shown an increase in accuracy for modeling the shape of the unloading curve in polymers without affecting the already desirable modeling capability of the loading curve [18]. This parameter was implemented into the VBO model version used by Krempl and Khan to investigate its affect on the VBO model output compared to previous results [2] [21]. An alteration to the modified SLS model used in Krempl and Khan s VBO by using two modified SLS elements in series is to be investigated. This modification is implemented with the idea that the model, which currently cannot recreate rate reversal (see Section 5.3) observed in data obtained from creep and relaxation experimental tests, will be able to do so. Increases in the mechanical behavior occur from modifications to the material through radiation crosslinking the molecular chains and by the addition of fillers to the polymer matrix in medical grade polymers. This makes medical grade polymers unique from other non-medical grade polymers in that the modifications are introduced to mitigate the negative effects on the polymer parts that can be caused by everyday loading conditions within the knee and hip joints. Therefore, there is a need to investigate further the ability of the VBO model to produce similar qualitative and quantitative results for medical grade modified polymers as those found when modeling HDPE. Specifically, types of COMMON BIOMEDICAL MATERIAL A and MATERIAL C will be investigated. As previously mentioned, it is unknown whether the characteristics that enhance the mechanical behavior of polymers will have a negative effect on the ability to model these specific types of polymers. Resulting VBO outputs were compared with the data from the experiments done on the MATERIAL C and COMMON BIOMEDICAL MATERIAL A specimens in a multitude of tests in both tension and compression loading. VBO model parameter values with the best qualitative and quantitative output for each type of polymer were determined. 13

26 An introduction of three dimensional simulation tests run on the PolyUMod software was completed. These tests were fulfilled with the intent to give way to further research into the three dimensional deformation behaviors of the new formulation of COMMON BIOMEDICAL MATERIAL A (MATERIAL A) and MATERIAL C materials. Examining each material s response to the three-dimensional loading was completed to give insight as to how the old and new grades of polymers will perform under these conditions comparatively. Eventually, implementation of the three dimensional tensor adaptation of VBO into the ABAQUS software is desired. The quality of the predictions from the three-dimensional VBO model will then later be able to be verified by comparing VBO outputs to PolyUMod outputs. 14

27 3 Experimental Testing Methods 3.1 Equipment A variety of experimental tests have been conducted on COMMON BIOMEDICAL MATERIAL A and MATERIAL C samples. The tests include; loading and unloading in tensile and compression at multiple strain rates, strain recovery, creep, relaxation, percentage crystallinity, and three point bending. To be able to accommodate the vast array of tests and different polymer sample geometries, several different testing machines were utilized in completing the experimental program. A MTS servo-hydraulic testing frame, with a digital controller and Flex Test 4 software for data capture, was used for performing tensile and compression loading and unloading experiments on cylindrical polymer samples (Fig 3.1). The hydraulic collets installed on the MTS testing frame can only accommodate cylindrically shaped samples of various diameters for experimental testing. Samples are prevented from slipping by frictional force created by the gripping action of the collets. Low yield stresses in the instance of polymers result in excessive damage of the samples when the collet closes on the sample. Loading the samples into the machine required that the cylindrical ends of the samples be covered by stainless steel, ¾ diameter collars that bore the compression load of the hydraulic grips. These collars prevented the sample ends from unwanted deformation or even being crushed within the grips. The MTS machine applied load and gripping force to the samples through hydraulic fluid pressure delivered by an external pump and computer controlled external manifold. The grips have a maximum gripping pressure of 9 psi, but 1 psi was the maximum pressure used in the experiments that resulted in an axial force of 3663 lbf. The upper crosshead of the testing frame was adjusted for sample spacing in between the two grips and then secured in place. The lower grip which is attached to the actuator, applied all loads to the samples. A 22, lb capacity load cell installed in the crosshead, measured axial forces and enabled the machine to be controlled in load control mode. The samples could also be deformed under displacement control measured by a displacement (LVDT) sensor positioned on the actuator. A MTS axial extensometer, model B-2, was used to measure axial strain and control the experimental tests in strain rate control. The MTS machine has a maximum loading strain rate of 1 x 1-1 s -1. The extensometer has a range of 15% strain. A MTS, diametral extensometer, model E-2 having a 4 mm deflection range, was placed on the samples to measure diametral strain. The machine could have been controlled through the diametral extensometer but, this capability was never utilized in any of the experimental tests performed in this research study. An Instron servo mechanical model 1587 testing frame (Fig 3.2) running Bluehill-2 software was utilized for tensile loading and unloading of rectangular shaped dog bone samples provided by Company B. Company A s COMMON BIOMEDICAL MATERIAL A samples were also machined down into small cylindrical samples for loading and unloading compression tests on the Instron testing machine as well. By machining the samples into smaller cylinders, two samples could be created out of the original single cylindrical samples. The smaller machined sample s geometry was not susceptible to buckling where the original sample geometry had a higher probability of buckling due to its length and thinner diameter within the testing area. Sample geometry dictated the choice of which machine was to be used for which experimental test. The Instron testing frame applied loads through an external electrical motor located on the outer base of the testing frame. Installation and gripping of the samples was done manually through mechanical grips. An Instron axial extensometer, model with a 1 mm range was employed for measuring axial strain on the polymer samples along with controlling the testing frame in strain rate control. The Instron testing frame had a maximum loading strain rate of 1 x 1-2 s -1. An Instron diametral extensometer, model I M-ST with a five mm range was used to collect 15

28 diametral strain data. Similar to the MTS testing frame experiments, the diametral extensometer was not used in active control, rather only for data collection. The Instron testing frame could also be controlled in displacement or load control depending on the particular requirements of the test. Figure 3.1: MTS Servo-Hydraulic Frame setup. Figure 3.2: Instron Servo-Mechanical Testing Frame setup. Both testing frames used feedback control to carry out and complete each experimental profile. Feedback control is a method of measuring what is happening to the samples at any given time during each test and having the data relayed back to the testing frame so that it can adjust the load accordingly with the specifications of each experimental test routine. The three types of feedback control utilized in the testing programs for this research project were strain rate, load, and displacement control. Strainrate control was monitored through the axial extensometers. The extensometer measures the deformation rate and communicates with the testing frame as to whether it needed to apply more or less load to deform the material at the correct rate. Displacement control is measured by the sensor in the crosshead of the testing frame. The total displacement value and displacement rate could be controlled through this method. Load control is determined by the load cell on the crosshead of the testing frame. Like displacement, load values or load rate can be used as a control method. The maximum loading rate obtained during the experimental tests was 16 lb/s and was obtained in the human gait test. Tuning the testing frames is necessary for running more intricate programs and is sometimes required after long periods of nonuse. Determining whether the machine needs tuning can be done by watching the real-time plots of the data and the control output. A large discrepancy between the magnitude and phase between the control output and the data collection plot is the sign that the testing frame needs tuning. The command and feedback between the controller and the testing frame works as a closed loop system. By adjusting the three gain parameters, the control and feedback can be adjusted so that the error between the control and feedback plots can be minimized. The three gain parameters are proportional, integrate and derivative. Within the tuning mode, the testing frame is asked to produce a continuously varying force in the form of a sine wave or square wave. Typically, the sine wave is tuned 16

29 first because it is easier for the frame to produce a feedback plot that is close to the control plot. The gain parameter values are then tested against the square wave control, which is a more difficult control profile for the machine to produce an accurate feedback. The square wave tuning usually involves finer tuning of the parameter values. Once the control and feedback output plots are lined up and overlaying one another, the control gain parameters are saved in the controller interface. A Perkin-Elmer Jade Digital Scanning Calorimeter (DSC) was used to perform percent crystallinity analysis through time and temperature data from numerous pre-worked and as-received samples of COMMON BIOMEDICAL MATERIAL A. The DSC machine uses tap water and argon gas as a cooling medium for the inner chamber where the samples were located. The DSC machine was controlled by Pyris software. Small samples were placed into sealed aluminum containers to contain samples even after melting. A second empty container was also placed inside the heating chamber along with the polymer sample, and was used as a control sample for measurements taken by the machine during the experiments. The resulting temperature versus heat flow plot produces two peaks. One of which is the peak at the latent heat of melting and the other at the latent heat of fusion. The latent heat of fusion peak is then integrated by the Pyris software and the change in endotherm is obtained. The change in endotherm is then divided by the heat of fusion of pure COMMON BIOMEDICAL MATERIAL A and the percent crystallinity is calculated. This can be seen in Equation 7. (7 A dynamic mechanical analyzer (DMA), RSA3 made by TA Instruments, equipped with an environmental chamber was used to complete three-point bending tests. Samples were required to be machined down to thin rectangular strips due to the 36 N maximum load capacity of the DMA. In the 3-point bending tests, a small U-shaped holder supports the rectangular sample at each end and load is applied to the center of the sample by the actuator capable of quasi-static and dynamic loading up to 8 Hz. The DMA can record the modulus of elasticity as the temperature increases and the glass transition temperature for the material can be determined from the resulting plots. The DSC can be used on both thermoplastic and thermoset types of polymers. 3.2 Materials and Samples MATERIAL C Samples Two different types of MATERIAL C samples have been provided by Company B, a polymer manufacturing company. Twenty five regular MATERIAL C samples and 25 MATERIAL D samples were provided for various experimental testing procedures. The samples geometry is a rectangular dog bone shape with a gage section measuring in.385 in.157 in (approximately 67 mm 9.75 mm 4 mm). Samples run in tension were left in their given geometry. Samples tested in the DMA were machined down into thin strips with dimensions of in.295 in.75 in (approximately mm 5.32 mm 1.79 mm) for normal MATERIAL C and in.2 in.495 in (approximately mm 5.7 mm 1.25 mm) for MATERIAL D. DMA sample dimensions were dictated by length of the testing fitting and the stiffness of the material. For high stiffness values, thick samples would result in test errors and incomplete data. Type 1 ASTM D 638 samples (dog bone samples) only permit tensile testing and thus were not compression loaded. 17

30 3.2.2 COMMON BIOMEDICAL MATERIAL A Samples Three types of COMMON BIOMEDICAL MATERIAL A have been provided by Company A, a prosthesis manufacturing subsidiary of a well known company, for various experimental procedures. Normal untreated COMMON BIOMEDICAL MATERIAL A, MATERIAL A, and MATERIAL B with an undisclosed filler material added to the polymer matrix (MATERIAL A) were all provided for experimental testing. All three of the COMMON BIOMEDICAL MATERIAL A samples have the same cylindrical sample geometry. The experimental gage section has measurements of approximately 1.57 in ( mm) length and a.5 in (12.75 mm) diameter. For compression testing there was a fear of buckling under high compression loads for the COMMON BIOMEDICAL MATERIAL A samples due to their cylindrical geometry. Tests run in compression, on the Instron testing frame, were therefore machined down into cylindrical samples with dimensions of approximately 1.15 in (29.3 mm) length and a.6 in (15.5 mm) diameter. The new uniform cylindrical shape gave rise to a more stable geometry where buckling was no longer a concern. All samples that were tested in either tension or compression loading conditions on the MTS testing frame were left in their original form. Samples tested in the DMA were machined down into thin strips with dimensions of 1.8 in X.25 in X.8825 in (approximately mm X 6.35 mm X mm). 3.3 Experimental Testing Protocols Experimental Testing Protocol General Information Seven separate deformation tests, all run at room temperature, constituted the experimental program for this research project; tensile tests to failure, tensile tests to maximum strain values, compression tests to maximum strain values, creep tests, relaxation tests, recovery tests and a human gait profile test. Each experimental test required its own program and protocol for the given testing frame and software. Different external and internal variables needed to be addressed during each test setup in an attempt to keep equipment safe while acquiring complete and accurate data. Specific aspects of the different types of tests required changes in control methods, different loading schemes, important stress and strain target values, equipment selection, and engaging the proper safety interlocks. During startup for a round of testing, the MTS testing frame required a frame warm-up routine. This program ran the machine through a set of axial and rotational cycles to stabilize the oil temperature and exercise the servo valves to ensure proper functioning of the frame for initial experiments. Both frames also required that the force sensor readouts be initially zeroed before the first experiment. After every sample is loaded into the testing frame but before a test is begun both axial and diametral extensometers are also zeroed to ensure accurate deformation is reached for each experiment. The DSC machine required its own calibration before the initial test for the day could be run. With proper water and argon flow, the DSC machine required an initial heating and cooling of the sample chamber. This procedure was done to purge the chamber of any lingering moisture that could have been introduced from the surrounding environment while sitting idle. The chamber temperature was raised to 5 o C and held for several minutes before dropping the temperature back down to the experimental starting point of 3 o C Tensile Test to Failure Experimental Protocol In an attempt to ascertain the shape of the full stress-strain curve, which is necessary for planning all other tests, it was necessary to complete a tensile test to failure for each type of polymer. When selecting a polymer for a hip or knee joint application, the polymer must be able to withstand the 18

31 multitude of loading schemes without permanently deforming (reaching the plastic deformation zone) or exhibiting as little plastic deformation for each load cycle as possible. Thus any load applied to the material must lie within or close to the elastic zone of the polymer so that when the load is no longer being applied the polymer will return to or close to its original state. A continual process of plastically deforming the material will lead to extremely quick failure rates and place human joints in danger of health complications. A full tensile test to failure at rate and temperature, similar to those used in the simulations, will produce a complete loading stress strain curve in which the elastic deformation zone, plastic deformation zone, flow stresses and fracture stress will all be easily identified and obtained. The procedure for loading the polymers to failure required loading the polymer samples in displacement control. For a test where known failure was going to occur, the extensometer was left off the sample to prevent damage and thus the test cannot be controlled under strain-rate control. A conservative loading displacement rate of.1 in/s was chosen due to the unknown nature of a new sample s deformation behavior. Choosing a displacement rate as the control for this test enables the user to deform the material to capture the full stress strain curve without specifically tuning the machine for the material being tested. The displacement data is converted into strain in order to produce the stress-strain plot. A load ramp function within the software package was used to apply the loading rate. Conservative limit interlocks were set in place for the duration of the test to stop the testing frame immediately upon fracture for the MATERIAL C samples. Due to the stiffness and low ductility of MATERIAL C, the fracture would occur much more abruptly than in COMMON BIOMEDICAL MATERIAL A which has much higher ductility. An abrupt fracture actually causes the load sensor on the crosshead to recoil upward and act as a compressive (negative) force rather than the specified tensile (positive) loading force. Thus, a limit was placed on any negative force experienced during the test as well. The ductility of the COMMON BIOMEDICAL MATERIAL A allowed the limits to be set much more liberally. Only limits such as compressive forces and differences in the displacement rate were placed on the testing frame. These types of limits are set for the case of an incorrectly written testing procedure or a malfunction in the testing frame. Due to the high strength and probability of fracture from a flaw on the outer surface or within the sample, the procedure was run twice for each type of MATERIAL C material to validate that the initial test produced a suitable stress strain curve in reference to the materials deformation behavior Tensile and Compressive Loading Test Experimental Protocols Tensile and compressive tests for all materials were run under strain rate control. Strain rate control is performed through the axial extensometer and can be used because the sample is not expected to fracture in the pre-yield zone determined from the experimental tests to fracture. The samples are loaded to a maximum strain value and then unloaded at the same strain rate magnitude. Much like the test to failure, the sample is loaded up through a positive ramp function to a strain value and followed by unloading with a second ramp function with an opposite sign on the value for the rate. For compression, the same program is used except that the signs for the loading strain rates are changed to apply compressive forces when loading and unloading the sample. The unloading is stopped when the polymer reaches zero force. Zero force is chosen as the stopping point because the polymers often exhibit some residual strain. In this case, if the test would have been stopped at zero strain, the initial starting point, the machine would actually go into compressive (negative) forces for a tensile test or tensile (positive) force for a compressive test. This is undesired because recovery tests must be run at zero force. Safety limits for these tests are set for strains higher than the maximum desired strain. This action keeps the extensometer safe from over-extension and ensures the test only loads the material to the desired amount. Force limits are set for extremely high forces that would indicate a faulty test 19

32 procedure and for forces past the zero force point on the unloading curve to ensure the sample stays only in the desired tensile or compression mode. Tests of this nature were run at various strain rates to examine the rate dependence of the various materials. Samples experienced rates varying from 1x1-2 s - 1 to 1x1-4 s -1, depending on testing frame. The MTS testing frame can employ higher rates safely whereas the Instron testing frame was limited to 1x1-3 s -1 to conserve the electrical motor and to keep from having to calibrate the control gains for every change in rate Recovery Test Experimental Protocols Recovery tests were only run after the conclusion of tensile and compressive loading and unloading tests and at the conclusion of the human gait profile test. The purpose for recovery tests is to again test the rate dependence of the material by analyzing the effect on the polymer chain s ability to recover a percentage of the residual strain left at the conclusion of the test for two different prior loading rates. This can be examined thoroughly from the tensile and compressive loading and unloading tests and, therefore, is not necessary on the creep and relaxation tests. Recovery tests begin once the polymer has been unloaded to zero force. While loading to zero force, the testing frame often slightly over or under shoots the zero force value. While limit switches are placed to keep this from happening, they are set to larger force values relative to those experienced at the test conclusion. It is reasonable to expect this to happen based on the control and feedback ability of the machine and the over and under shoot values are so small that they do not compromise the test or sample. To ensure the recovery test begins at zero force, a two second force control command is implemented to bring the machine back to a value of zero. At the conclusion of this command the recovery test begins and a hold command is implemented under force control that holds the machine steadily at zero force. The recovery tests were initially run for a one hour time period to gauge the material s behavior. It was concluded after an initial test that the material sufficiently settles at a final strain and any additional strain recovery after an hour is minute comparatively, therefore; validating a runtime of an hour for recovery tests Creep Test Experimental Protocol Several sets of creep tests were performed on each type of material for two different prior loading strain rates, 1x1-3 s -1 and 1x1-4 s -1. The purpose of creep tests is to discover how the material behaves under constant stress for long periods of time and to also watch for previously noticed strain rate reversal behavior from the material. The creep test protocol requires the material to have either just a loading or a loading and unloading prior history to reach a desired set force value at which the creep test will then occur. The prior loading history is run exactly the same as the tensile and compressive tests with similar safety limits engaged. Once the target stress with the appropriate loading history is reached, the program switches into force control and holds a constant force corresponding to the target stress. The constant stress is held for 1.5 hours before the material is unloaded back down to zero force under strain control. The 1.5 hour long test duration allows a sufficient amount of time for the material to undergo strain rate reversal if it is going to do so, it allows the material to switch from initial primary creep to secondary creep, and lets the material either level off at a final strain value during secondary creep or to switch into tertiary creep at which time the safety limits will stop the testing frame. Safety limits are set on the amount of strain 1%-2% below the allowable maximum strain value (15%) for the extensometer arms. All creep tests that were completed began the constant force at strains below 6%. This value along with the extensometer maximum total strain limit ensures that enough strain data can be obtained to fully capture the nature of the MATERIAL Deformation behavior while also keeping the extensometer safe from over extension. 2

33 3.3.6 Relaxation Test Experimental Protocol Stress relaxation tests follow much the same protocol as the creep tests, as in they too need to have a either a loading or a loading and unloading prior history. However, the target for the relaxation test is a final strain value instead of a final stress value. Also, due to the limited number of samples provided, only one relaxation test per loading rate was performed instead of a high, middle and low value as was done in the creep tests. The strain value found on the x-axis that corresponded to the stress value in which the middle creep test was performed at on the stress-strain plot was the strain value selected for this single relaxation test for both loading rates. By choosing this strain value, the relaxation tests can be used to examine each material comparatively, investigate rate dependence, and stress rate reversal. Loading the material up to a maximum strain and then down to the targeted relaxation test strain value, the test is controlled under strain control in the same manner as a standard tensile or compressive test. Once at a target strain, the test holds at that strain (under a strain rate control) at a level of zero for 1.5 hours. Again, the hour and a half time frame allows the material enough time for the stress to relax asymptotically to a final stress value much like the creep test and the corresponding strain deformation behavior. Because the strain rate is held at zero for this test there will be no extensometer arm movement. There is also an expectation that the stress will ultimately decrease due to the nature of stress relaxation. Due to these two assumptions no new safety limit settings are introduced into this particular experimental procedure Digital Scanning Calorimeter Experimental Protocol More sample preparation was needed for DSC tests than most of the other testing. Tiny samples of COMMON BIOMEDICAL MATERIAL A were taken from previously-deformed and undeformed samples for percentage crystallinity comparison. The goal of this experiment was to determine whether crystallinity increased due to small percentages of permanent strain deformation. Pieces of COMMON BIOMEDICAL MATERIAL A were cut from the previously-deformed samples. The pieces were cut from sections near the center of the gage section as far away from the surface as possible. The samples were cut with the intent to have as little surface area of the samples being sliced. This was done to leave as little induced surface deformation from the blade as possible that may possibly have an effect on the crystallinity values. This practice was implemented because pieces closer to the center deform more uniformly and will represent the deformation from experimental tests more accurately. Each sample s mass (in grams) was obtained and input into the Pyris software as a parameter before each test. The samples were then placed into a small aluminum container and sealed to conserve mass during each test. It was important to make sure each container was centered on the heating element to ensure complete melting for accurate data and results. During a full test cycle, the latent heat of melting and fusion are both obtained. Due to the physical properties of cooling, the latent heat of fusion data is not always completely accurate and, therefore, was not of great interest in this experiment. Each test was heated at a slower rate of 5 o C/min to obtain a complete and accurate heating curve. Cooling was performed at 1 o C/min to expedite the remaining portion of the test since the data from this part of the experiment was not going to be used. After some initial data analysis, it was determined that complete melting could occur and finish with a maximum heating temperature of 18 o C. Multiple tests were run on unmodified MATERIAL B, MATERIAL B, and MATERIAL A to ensure accurate results. 21

34 3.3.8 Dynamic Mechanical Analyzer Experimental Protocol Thin samples of COMMON BIOMEDICAL MATERIAL A and both types of MATERIAL C were machined from undeformed samples for the DMA testing. The goal of this experiment was to determine the glasstransition temperature and to verify value of Young s modulus for all of the materials. Full samples were milled down into the thin strips needed for the test. The length of the samples had to be longer than the length of the U-shaped holder and the thickness was determined by the stiffness of the samples. The MATERIAL C samples were milled down much thinner than the COMMON BIOMEDICAL MATERIAL A samples. The appropriate thickness was found by running a test on the sample. If the sample was too thick the test would stop and give an error message. The samples were then placed onto the holder and centered underneath the probe. The probe was lowered to the surface of the sample until it made contact. The probe, sample, and holder were then sealed within the environmental chamber. The probe was touched to the surface of the material prior to the beginning of the tests. A small preload of around 2 N was placed onto the MATERIAL C samples due to their high stiffness. This would ensure the probe properly oscillated the samples and remained touching the surface past the glass transition temperature and provided a clear data curve. During a full test cycle, the probe would oscillate at the defined frequency of two Hz. At the point that the glass-transition temperature was reached, the sample would become soft and begin to sag below the point of contact with the probe. The output plot data of Young s modulus versus temperature would trail off to zero and the glass-transition temperature could be determined as the point at which the slope begins to drastically drop off. Each sample was heated at a rate of 5 o C/min to obtain a complete and accurate modulus versus temperature curve. The test starting temperature was set to 35 o C and increased until 17 o C was reached. This final temperature was set above the MATERIAL C and MATERIAL D glass transition temperature so that a definitive transition could be obtained from the data. The environmental chamber was then cooled with air before removing the sample. 3.4 Human Gait Profile Experimental Procedure and Protocol Human Gait Profile Construction The human gait profile test was the only experimental test run on the MTS machine that involved compressive loading forces. The MTS machine was chosen for this particular test because it could handle the control needed for quick load transitions and the collet grips were more ideally equipped to hold the sample under the complex loading scheme. The collars placed onto the sample ends were designed to hold the sample under tensile loads. They have a notched edge that fits under the rim at the sample top to prevent the sample from slipping out under tension. There is no stop on the top of the collars to prevent the sample from slipping when placed under compressive loads. The collars do come with a pin that can be slid through the samples horizontally to hold them into place under compression. Each sample run under the human gait test therefore, had to be prepped by drilling a hole through each end in order to secure them tightly in the testing frame. The gait profile is constructed based on experimental data taken from the forces experienced by a knee joint from a walking motion on a level surface by elderly subjects [4]. The forces enacted by the motion of a step are initialized through a heel strike. As the body weight acting on the heel transfers to the toes, there is a slight dip in the compressive forces applied by the weight of the subject while stepping and experienced by the knee before a final spike while the toes are experiencing pressure from the entire body weight. Two cycles of the human gait profile can be seen in Figure

35 Force (lbf) -1 Airborne Heel Strike Heel -5 Toe Strike Time (s) Figure 3.3: Human gait profile plot of force versus time for two cycles With the possibility of sample buckling in the test region under compression again arose for the human gait profile test and verification that the sample would not fail under buckling was needed. Equation 7 was used to determine if the sample could withstand the maximum compressive forces enacted by the human gait profile without buckling. with and being the compressive modulus and the polar moment of inertia from the material characteristics and sample geometry respectively, and and are the effective-length factor and the sample length. It was determined that the samples could withstand 146 lbf before buckling would occur. During the human gait profile test the maximum compressive load the samples experiences, under the assumption that the test is representing a 185 lb person, is 498 lbf. This value is well below the maximum limit of 146 lbf with a safety factor of almost three. The human gait test protocol is comprised of several different program element functions, such as load ramps and sinusoidal waves, and is much more complex than the previous experimental tests that were performed. An initial load ramp under force control of 8 lbf/s compresses the material. A half sinusoidal wave with a frequency of four Hz/cycle and amplitude of 1 lbf is then implemented to reach the first peak of compression force when the heel strikes the ground. Another half sinusoidal wave with frequency of 8 Hz/cycle and amplitude of 1 lbf finishes the transfer from the heel to standing on the toe. The profile is then finished off with another force ramp of the same 8 lbf/s magnitude as the initial ramp but with a negative rate. The set of program commands were combined within a group that could be manually set by the user to cycle any given amount of times. The overall time for one cycle, pulled from the experimental data plots, is approximately seconds for one step. Loading rates were determined from this overall cycle time and by deconstructing the force-time data plots from the given experimental study data into the separate steps detailed above [4] Human Gait Control Verification It was initially unknown once the human gait profile test program was constructed if the MTS machine could indeed handle the quick changes in amplitude and rate for a 1.5 second cycle. To determine this, a systematic method was implemented to test the ability and accuracy of the MTS testing frame. To begin, the human gait was assembled using much slower loading rates and sinusoidal frequencies along with smaller compressive force values to determine the machine s capabilities in quick positive to negative 23 (7)

36 Force (lbf) Force (lbf) changes in rate. The MTS testing frame was given a human gait profile with a 15 second gait cycle with maximum loading and unloading rates of 5 lbs/s and maximum compressive forces of 25 lbf, seen in Figure Time (s) Figure 3.4: Initial test of the human gait profile plot of force versus time for two cycles Once the test was deemed a success the sample staying intact and visual verification of the profile shape, force and time values, the loading rates were doubled along with the data sampling rates to maintain the same level of accuracy on the plots. The rates were doubled until the second cycle range was obtained at which time the compressive forces were then increased. To validate the testing frame s output, a new control data output was added to the output data file. The control output was then plotted against the force output to determine the control feedback accuracy of the MTS testing frame, seen in Figure MATERIAL B 1 Cycle Gait Control Output -6 Time (s) Figure 3.5: Human gait force plot versus control output plot for calibration verification of the testing The plot of the control output and the force data output shows very little discrepancy between the control and feedback output of the MTS testing frame. There is also good agreement in the phase for the time at which the each output cycle occurs and the force s value output seen mainly at peak 24

37 compression times. This plot suggests that no further calibration was needed for the MTS control gains within the controller Human Gait Profile Experimental Procedure It was initially unknown how many cycles a sample would need to undergo to obtain a useful amount of strain data. The methodology was to proceed systematically much like the testing of the MTS s capability to perform the test. Initially 1 cycles were completed to verify that the program worked properly. After it was determined that the test could reproduce a couple cycles worth of data similar to that which was seen in the experimental studies, the focus shifted towards identifying if the strain of the material would increase indefinitely or asymptotically settle at a maximum strain value and if the difference between the maximum and minimum strain for each cycle would continue to grow showing cyclic softening. Cycles were completed in numbers of 5, 1, 2 and 5 before it was determined that enough data was obtained to make definitive statements on the cyclic deformation behavior of the material. A recovery test was placed at the end of the cyclic test to also compare the materials recovery under cyclic loading profiles to monotonic tensile and compression loading Human Gait Profile Overload Experimental Procedure The human gait overload procedure was performed to investigate the effects of strain behavior when an isolated overload cycle was introduced into the 5 cycle procedure. This type of overload procedure is often performed in fatigue testing and is relevant to knee loading from activities such as jumping. The overload procedure used the same human gait procedure with one overload cycle introduced at the midway point (25 th cycle) of the test. This overload profile had an increased total force and amplitude within the cyclic portion of the cycle of 4%. This amount was chosen to provide the overload cycle but also to stay somewhat conservative because it was unknown how the testing frame would respond to a quick increase in load rate and magnitude and then back down to the original gait profile cycle. With an increase of force also came a 4% increase in load rate, with the attempt to keep the 1.5 second cycle time uniform with the other 5 cycles. A recovery test was placed at the end of the cyclic test to compare the materials recovery behavior with the regular gait profile. 3.5 Data Collection Data Collection Introduction Data collection rates varied among the different types of experimental tests. The aim was to collect enough data to accurately depict the deformation behavior of the material, but to not inundate the plots with needless amounts of data points. When deciding how many data points and at what rate they should be collected, having prior knowledge from similar experimental tests assisted in proper data collection selection. Stress strain plots for tensile and compression experimental tests were among the easiest to choose data collection rates for. Key aspects of the plots require more points than others. Transition from the elastic into the plastic flow region produces smooth curves different from the more linear type behavior seen in the flow region. More data points are needed in these transition curves and in places where there is more of a continuous change in a deformation profile. For creep tests, primary creep has more curvature as it proceeds into secondary creep. There is also the added effect of possible rate reversal in primary creep that has a large amount of change in its curvature. As secondary creep sets in there is large drop in the change in strain as the strain asymptotically settles out. Therefore, this secondary creep region does not require a high data collection 25

38 rate to record the strain time behavior. Should the material begin to exhibit tertiary creep, any data rate previously set for primary creep should also be able to match the increase in the rate of deformation. Creep tests at low stresses can show very little change in strain. Taking too many data points in this case, can cause plots to have stair step like attributes. These attributes can give a false sense of what is actually occurring and hide the correct behavioral response of the material. Relaxation tests require similar types of data collection as creep tests. Relaxation has been observed to have the same type of rate reversal characteristic as has been documented in creep tests. This early change in slope along with a general trend of more stress rate changes in the early stages of published data from relaxation tests requires a much faster data collection to accurately depict its behavior. Relaxation stress eventually settles out much like secondary creep and requires much less data points as longer amounts of time pass to capture its behavior. Recovery tests work the same way as creep and relaxation tests in that all of the quick changes happen in the early stages. Most of the percentage of strain that is recovered is done so in the first five minutes of the test. Strain recovery values level off similarly to secondary creep after long periods of time and very little change if any can be seen. The human gait profile requires many data points per cycle to capture its complete deformation characteristics. Between fast cycle rates and many changes and reversals in slope there is the need for a very quick data collection rate. A high data collection rate of about one data point per every.1 seconds along with 5 cycles produces higher amounts of data points than has been previously seen in other experimental tests. For this particular test, Excel could not handle the large amount of data points (>6,), and thus Matlab was used to plot the data results for this specific test Instron Testing Frame Data Collection The Bluehill-2 software provided a user friendly option of setting different cases and a specific collection rate for each case. The interface is similar to an if-then statement used in programming languages. For tension and compression tests on the Instron testing frame, a two case data collection rate was implemented. Data was collected every change of 1 lbf or one data point every 1 seconds. For tension and compression loading tests with creep data, two collection rates were used to capture both types of deformation. The software recorded data for every change in.1% strain or one point every 6 seconds. Due to the similar time based behavior between creep and recovery tests, the same data collection rate was used for recovery tests as was used for creep tests. In all cases, data points were output for time, force, axial strain, diametral strain, and displacement values MTS Testing Frame Data Collection The MTS testing frame has a much different method for data collection than that of the Instron testing frame. Data collection is done by assigning a drag and drop data collection element into the program. Each element can only be assigned one rate for data collection but can be assigned a start and stop time during the test unlike the Instron machine where the data rate is continuous for the duration of the experiment. For tension tests performed on the MTS machine, four points were taken for every.1% of strain. The Flex Test 4 program converts and displays the actual decimal rate as being one point every third of a second. Since loading rates for the tests were varied by factors of ten, obtaining equally accurate collection rates is done by moving the decimal point one spot to the right or left depending on whether there was an increase or decrease in rate change. Recovery tests performed on the MTS machine required two data elements. As mentioned previously, the majority of change in strain occurs 26

39 around the first five minutes of the recovery test. Therefore, one data element was set to take data at one point a second for the first five minutes of the recovery test. A second element took over at the end of five minutes and took one data points every minute for the duration of the test. The human gait profile only required one data element to capture the polymer s deformation behavior. Initially it was unknown, but through successive testing, it was determined that approximately 15 data points could accurately capture the complete profile of one cycle. Therefore, a data collection rate of one point per every.1 seconds was implemented for the human gait profile. For the addition of the overload cycle, the data collection rate was increased by a factor of five, to ensure that the monotonic and cyclic loading schemes could be captured accurately with the faster loading rates Digital Scanning Calorimeter Data Collection Data collection for the DSC can be determined through the user defined parameter, heating rate, and the thermal characteristics of the material. The data collection for the set of DSC tests completed in this research study was approximated and set to 36 data points per test. Due to the simple nature of the test, the program only one allows a single collection rate. Therefore; a single rate that could capture the latent heat of melting and fusion transition peaks was needed, but not one that would produce an excessive amount data points to sort through. The heating rate for each test was five degrees Celsius per minute. The peaks were approximated to have a start and stop range around 5 o C, which would produce a 1 minute window for complete melting of the solid. It was determined through this knowledge, that one point per second (6 points for a peak) would accurately capture the latent heat peak s profile while producing a manageable amount of excess data points Dynamic Mechanical Analyzer Data Collection Data collection for the DMA can be determined by the user prior to the test. The data collection for the set of DMA tests was uniform for all of the materials tested because the testing parameters such as the rate of temperature increase and frequency of the probe were all uniform as well. With the slow heating rate, the materials had enough time to heat throughout the entire volume of the sample allowing for a slow transition between states. The collection was set to one data point per every five seconds. This provided approximately 325 data points per temperature increase. No data was captured for the cooling of the material as no relevant information was needed from this part of the procedure. 325 points accurately captured the heating curve profile and a provided clear depiction of the transition temperatures for the different materials. 27

40 Modulus of Elasticity (Pa) 4 Experimental Data and Results 4.1 COMMON BIOMEDICAL MATERIAL A Experimental Data and Results COMMON BIOMEDICAL MATERIAL A DMA Experimental Results To identify any material characteristics that may differ between batches as a result of minor variations in processing parameters, a three point bending test was performed on a DMA. Also depending on the material, the glass transition temperature or melting temperature can be located from this type of testing. Plots constructed from the data obtained from the three point bending tests for two samples of COMMON BIOMEDICAL MATERIAL A can be seen in Figure E+9 9.E+8 8.E+8 7.E+8 6.E+8 5.E+8 4.E+8 MATERIAL B Sample 1 MATERIAL B Sample 2 3.E+8 2.E+8 1.E+8.E Temperature ( o C) Figure 4.1: DMA testing of two COMMON BIOMEDICAL MATERIAL A samples for the glass transition The listed value for the elastic modulus for a piece of general COMMON BIOMEDICAL MATERIAL A, at room temperature, is typically around.5 GPa [5]. The elastic modulus is influenced by the molecular weight of the material and can vary greatly even in one material family such as COMMON BIOMEDICAL MATERIAL A. The data plots for MATERIAL B samples produce results with some variation from the listed values for the tensile modulus at a temperature of 32 o C. For samples of MATERIAL B, the DMA produced a value for the tensile modulus of.8 GPa. The slight difference between experimental and published values is deemed acceptable because the experimental values were expected to be higher than the listed values for general COMMON BIOMEDICAL MATERIAL A. This is due to the fact that the experimental samples had undergone modification procedures. The modifications will make the stiffness higher for equal amounts of deformation because some of the molecular chains are linked together impeding normal sliding or rotating under the corresponding stress amounts. The listed glass transition temperature for COMMON BIOMEDICAL MATERIAL A lies at -12 o C [5] and therefore cannot be verified under the experimental test conditions in the DMA test. By looking at the resulting plot in Figure 4.1, the melting temperature can be estimated. The listed value for the melting temperature of COMMON BIOMEDICAL MATERIAL A was 137 o C [5]. Around 145 o C the data line on the plot changes slope noticeably and is also extremely low in value. At this point the sample began to sag noticeably between the two end supports. With the material entering a rubbery state, data readings are no longer accurate. The material is approaching the melting temperature and the probe is no longer touching the sample s surface and applying any force. The temperature at the point of sagging is representative of the melting temperature or very close to it. The melting temperature found from the test is slightly higher than the listed value, like the tensile modulus, and can also be said to be a result of 28

41 Stress (MPa) the modifications. At this temperature, the chains are moving freely in a viscous state, but the modifications still create some resistance at the normal melting temperature causing the value for the temperature to be higher than the listed value COMMON BIOMEDICAL MATERIAL A Tensile Loading and Unloading Experimental Results Initially, one sample of MATERIAL B COMMON BIOMEDICAL MATERIAL A was placed into the MTS machine and loaded in tensile mode to very large strains in an attempt to locate the material s pre-yield strain range for future testing. The resulting stress strain curve from this loading procedure can be seen in Figure MATERIAL B Strain (%) Figure 4.2: MATERIAL B sample tensile loaded to very large strains to find an acceptable maximum testing strain. The region that was determined suitable for testing is circled on the stress strain plot in Figure 4.2. The stress range represents the upper bound of stress values experienced by polymeric parts in implants and it lies sufficiently below the yield point that the strain can safely be assumed to be uniform. There is a large strain range that can be tested within this region and thus makes it safe to test and compare deformation behavior between several strain values. It is important to know if the material behaves similarly during loading and unloading at these different values to pinpoint any differences or singularities in the material s deformation behavior. Anomalous behaviors that go undetected can lead to unexpected failures after the polymer part is implemented into service if they occur within the operating stress range. Samples of both MATERIAL B and MATERIAL A have undergone tension loading tests, at a minimum of three different maximum strain values per material, on the MTS servo hydraulic testing frame. The maximum strain values selected were 2%, 6%, and 14%. Each tensile test run to one of these strain values was completed by using strain rate control at a rate of 1x1-3 s -1. Limitations in the distance the extensometer arms can travel safely restricted the maximum values of strain to 14%. For the maximum strain values of 6% and 14%, a second round of tensile tests using a higher loading strain rate of 1x1-2 s - 1 was conducted. This grouping of identical tensile profiles with different loading rates was performed with the intent to highlight the material s rate dependence and to uncover what the actual effects on deformation behavior that the modifications have for COMMON BIOMEDICAL MATERIAL A. It was unknown prior to testing whether the material would exhibit different amounts of rate dependence for a range of final strain values or whether there would be no difference within the deformation behavior for different loading histories. 29

42 Stress (MPa) Stress (Mpa) MAT B LUL 6% 1E-3 MAT B LUL 14% 1E-3 MAT B LUL 6% 1E-2 MAT B LUL 14% 1E Strain (%) Figure 4.3: MATERIAL B tensile loading at 1x1-2 s -1 and 1x1-3 s -1 strain rate control. The faster loading rate results in an increase in the stresses along the stress-strain curve Strain (%) MAT A LUL 6% 1E-3 MAT A LUL 14% 1E-3 MAT A LUL 6% 1E-2 MAT A LUL 14% 1E-2 Figure 4.4: MATERIAL A tensile loading at 1x1-2 s -1 and 1x1-3 s -1 strain rate control. The faster loading rate results in an increase in the stresses along the stress-strain curve As seen in Figures 4.3 and 4.4, the MATERIAL B and MATERIAL A samples show rate dependence when loaded at two different tensile loading rates. At the point of maximum difference, MATERIAL B shows approximately a 5 MPa difference in stress to reach the maximum strain target and MATERIAL A sample data recorded a stress difference of about 3 MPa in stress values. There also appears to be about a.5% difference in the final leftover residual strain when the test is stopped at force for the separate loading rates used on MATERIAL B as opposed to a difference of.2% for MATERIAL A. This implies that there was slightly more plastic deformation exhibited for the MATERIAL B samples than MATERIAL A when increasing the loading rate. The filler in the MATERIAL A appears to decrease the difference in amount of final stress at the maximum strain experienced within the material when comparing data between samples loaded to the same amount of strain at different strain rates. The maximum stresses reached in the two types of COMMON BIOMEDICAL MATERIAL A samples were higher than those found in regular COMMON BIOMEDICAL MATERIAL A. These higher stresses can be directly attributed to the modification process for the MATERIAL B samples and the modifications in the MATERIAL A samples. 3

43 Stress (Mpa) Stress (Mpa) MAT A LUL 6% 1E-2 MAT A LUL 14% 1E-2 MAT B LUL 6% 1E-2 MAT B LUL 14% 1E Strain (%) MAT A LUL 6% 1E-3 5 MAT A LUL 14% 1E-3 MAT B LUL 6% 1E-3 MAT B LUL 14% 1E Strain (%) Figure 4.5: MATERIAL B versus MATERIAL A tensile loading at 1x1-2 s -1 strain rate control. The modifications in the MATERIAL A samples result in higher stresses for deformation compared to MATERIAL B samples. Figure 4.6: MATERIAL B versus MATERIAL A tensile loading at 1x1-3 s -1 strain rate control. The modifications in the MATERIAL A samples result in higher stresses for deformation compared to MATERIAL B samples. Figures 4.5 and 4.6 show the comparison of MATERIAL A and MATERIAL B samples to one another loaded at the same two strain rates. Both plots highlight higher loading stress values for the MATERIAL A samples than those of MATERIAL B samples. Overall there is about a 3 MPa stress increase at the maximum strain point for MATERIAL A at both the 6% and 14% final strain tests. The plots show that the deformation profile in the flow region of the curve produces the same slope and shape independent from the rate. Quantitatively the values are higher for rate differences not in the overall profile shape. Also of importance is that both materials show the same, or very close to the same, amount of residual strain when unloaded back down to zero stress. It can be concluded from these two plots that the filler agent within the MATERIAL A samples increases the strength of the MATERIAL During tensile loading. The differences in stress values between the two materials stay consistent with typical rate dependant deformation behavior and with what would be expected by adding a reinforcing filler agent to enhance the material s deformation behavior and properties. The tensile loading program written in the Flextest 4 software that collected the data shown in Figures was modified in an attempt to circumvent the maximum limit of 15% strain measurement in the extensometer so that a larger deformation test could be achieved. The new program achieved this by introducing a temporary pause function at a user defined strain value to allow the extensometer to be manually reset and balanced out. By balancing the extensometer, the final recorded strain at the pause could be summed with each subsequent data value to achieve accurate measurements of strain for the duration of the data. This modification to the tensile program allowed the samples to be tested up to a maximum strain of 2%. With the sample held in tension during the program for a small amount of time at a higher stress, there became a higher probability that the sample could exhibit small amounts of deformation caused by creep behavior that would leave a noticeable bump in the stress strain profile. The new modified test proved to be cumbersome to accurately complete and, therefore, was not reproduced for more samples. This creep behavior in the loading curve can be seen in Figure 4.7 at the 11% strain point. The purpose of completing this experimental test was to note any possible differences in the deformation behavior at what is considered a higher level of strain deformation. Secondly, the sample was examined to see if higher strain deformation results in a higher amount of deformation- 31

44 Stress (MPa) induced crystallinity in the molecular chain arrangement. The tensile test to 2% strain can be seen in Figure MATERIAL B LUL 2% Strain (%) Figure 4.7: MATERIAL B load and unload at 1x1-3 s -1 strain rate control to 2% maximum strain COMMON BIOMEDICAL MATERIAL A Tensile Loading Recovery Test Results At the conclusion of each tensile loading test, once the sample was unloaded down to zero stress, a one hour recovery test was then performed. The purpose was to examine any effects that all of the modifications would have on the strain recovery each material would exhibit. Recovery occurs when internal stresses, within the molecular chains left over after unloading the sample, are relieved and the polymer recovers some of its residual strain and is left with a lower amount, which is different for each material, than when the sample is first unloaded to zero force. It is necessary to see how the samples recover at different load rates and deformation amounts to gage the overall final plastic deformation after loading and unloading. This is essential to determining the overall ability of the polymer to withstand loading schemes within the knee joint and its lifespan. Figures 4.8 and 4.9 compare each material s recovery behavior when prior loading was performed at two different loading rates to one another. The actual recovery values that can be seen in Tables 4.1 and 4.2 show that the MATERIAL B samples have a higher percentage of strain recovery than the MATERIAL A samples ranging from 7% to almost 1% more recovery per test. However, it should be noted that at 2% maximum strain and a loading rate of 1x1-3 s -1, the MATERIAL A sample exhibits a higher percentage of strain recovery by just over 8%. A trend between for the data emerges such that the difference in the amount of recover between MATERIAL A and MATERIAL B increases as the maximum loading strain.material A appears to recover more strain at small strain deformations, whereas MATERIAL B records high and consistent recovery results at higher maximum strain loading values. 32

45 Strain (%) Strain (%) Strain (%) Strain (%) 6 7 MATERIAL A LUL 2% MATERIAL A LUL 6% MATERIAL A LUL 14% MATERIAL B LUL 6% MATERIAL B LUL 14% MATERIAL A LUL 6% MATERIAL A LUL 14% MATERIAL B LUL 2% MATERIAL B LUL 6% MATERIAL B LUL 14% Time (s) Figure 4.8: MATERIAL B versus MATERIAL A tensile loading at 1x1-2 s -1 strain rate control for 6% and 14% maximum strain Time (s) Figure 4.9: MATERIAL B versus MATERIAL A tensile loading at 1x1-3 s -1 strain rate control for2%, 6%, and 14% maximum strain. Figures 4.1 and 4.11 display the differences in the percentage strain recovered at a given maximum strain value for two different loading rates for each material separately. The final recovered strain value for MATERIAL A samples that were loaded at different deformation rates have final recovery values that are much closer to one another than those seen in the MATERIAL B sample trials. It would appear that prior loading rate has a smaller affect on the final strain value for MATERIAL A samples than that of the MATERIAL B samples. The overall total percentage of strain recovery for MATERIAL A samples, however; is lower than the MATERIAL B as seen in previous comparison. It seems to be apparent in the MATERIAL A samples that for the given amounts of residual strain left over after a tensile loading test and at the start of the recovery test, there is a much less rate dependence, resulting in a tighter limit on the final amount of residual strain at the recovery conclusion MAT B LUL 2% Rcv 1E-3 MAT B LUL 6% Rcv 1E-3 MAT B LUL 14% Rcv 1E-3 MAT B LUL 2% Rcv 1E-3 MAT B LUL 6% Rcv 1E-2 MAT B LUL 14% Rcv 1E MAT A LUL 2% 1E-3 MAT A LUL 6% 1E-3 MAT A LUL 14% 1E-3 MAT A LUL 6% 1E-2 MAT A LUL 14% 1E Time(s) Figure 4.1: MATERIAL B tensile loading recovery test with a 1x1-2 s -1 loading rate vs a loading rate of 1x1-3 s -1 for 6% and 12% maximum strain Time (s) Figure 4.11: MATERIAL A tensile loading recovery test with a 1x1-2 s -1 loading rate vs a loading rate of 1x1-3 s -1 for 6% and 12% maximum strain.. 33

46 Stress (MPa) Table 4.1: MATERIAL A recovery table for loading histories of 2%, 6% and 14% at loading rates of 1x1-2 s -1 and 1x1-3 s -1. Material Rate = 1E-2 1/s Rate = 1E-3 1/s MATERIAL A 2% 6% 14% 2% 6% 14% Initial Strain Final Strain Total Recovery Table 4.2: MATERIAL B recovery table for loading histories of 2%, 6%, 14% and 2% at loading rates of 1x1-2 s -1 and 1x1-3 s -1 Material Rate = 1E-2 1/s Rate = 1E-3 1/s MATERIAL B 2% 6% 14% 2% 6% 14% 2% Initial Strain Final Strain Total Recovery MATERIAL A Compressive Loading and Unloading Results With the scarcity of published data, due to MATERIAL A being a new material, and the compressive nature that is observed in knee implants, the MATERIAL A samples were tested by loading and unloading to a maximum strain value in compression as well as tension. Due to the nature of compression tests and material behavior, it was not expected that the material could be loaded up to the maximum values of strain (14% or 2%) that were previously achieved when the materials were loaded in tension. To simulate service conditions in which maximum compressive stresses approach 2 MPa (discussion with Company A staff), an upper strain value of 6% was selected. Experiments were conducted with loading the sample to 6% maximum strain at a strain rate of 1x1-3 s -1. Similarly to tensile tests, a second sample was loaded to 6% maximum strain but at a slower rate of 1x1-4 s -1 to document compressive rate dependence and to note differences from tension deformation behavior. Figure 4.12 shows the resulting stress strain plots for the two compressive loading rates MAT A LUL 6% w/ 1 hr Rcv 1E-3 MAT A LUL 6% w/ 1 hr Rcv 1E-4-25 Strain (%) -3 Figure 4.12: MATERIAL A compressive loading to 6% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Stress values of 2, 12, and 5 MPa where creep tests were performed are marked by dashed circles. The observed rate dependence produces a maximum stress difference of around 4.5 MPa as seen at the final strain points where maximum stresses of approximately 24.5 MPa and 2 MPa are recorded. Although higher stress differences were incurred at the maximum strain, the samples were unloaded down to zero stress with very little difference in the amounts of residual strain. This would suggest that 34

47 Δ Strain (%) there is less viscoelastic rate dependence on the unloading profile than in the loading profile for compression testing. This is a new behavior separate from the tensile loading profiles and behavior discussed earlier. Tensile tests produced different loading and unloading profiles at different loading rates. The phenomenon of compression rate dependence (in compression) can also be seen in recovery tests. Each of the samples loaded at the two different rates underwent an hour recovery test at the conclusion of the loading program. The two recovery tests are compared in Figure MAT A LUL 6% 1 hr Recovery 1E-3 MAT A LUL 6% 1 hr Recovery 1E Recovery Calculation MAT A LUL 6% 1E-3 (Compression) Strain Initial = -2.1% Strain Final = -1.1% %Recovery - 1hour = 47.62% Recovery Calculation MAT A LUL 6% 1E-4 (Compression) Strain Initial = -1.9% Strain Final = -1.% %Recovery - 1hour = 47.37% Time (s).4.2 Figure 4.13: MATERIAL A compression loaded recovery test with a 1x1-3 s -1 loading rate versus a loading rate of 1x1-4 s -1 for a 6% maximum strain From the recovery plots seen in Figure 4.13, the final amounts of recovery show no significant difference for the two prior loading rates. This behavior is unlike that which was observed when MATERIAL A samples were loaded in tension. It would seem that for low amounts of compressive deformation it can be said that prior loading rate does not affect the strain recovery. With the limited range of compressive strains tested for the prior loading histories, more experimental testing must be completed before this can be concluded for higher or all amounts of compressive deformation COMMON BIOMEDICAL MATERIAL A Human Gait Profile After the human gait profile was verified as an accurate depiction of the experimental data taken from human subjects, a new MATERIAL A sample was placed into the MTS testing frame for experimental testing. The final amount of cycles decided upon to capture an adequate amount of data to determine the strain deformation behavior was 5. The data plot for the stresses exhibited during two of the gait cycles can be seen in Figure The general stress profile observed for each cycle does not change throughout the duration of the cycles, since true stress is being measured and the loading profile does not change from cycle to cycle either. A plot with the separate lines depicting the values of the minimum and maximum strain exhibited within each cycle subsequently throughout the entire 5 cycle test can be seen in Figure

48 Δ Strain (%) Strain (%) -1-2 AOX MAT A 5 5 Cycle Minimum Strains AOX MAT A 5 Cycle Maximum Minimum Strains Cycle Force Profile Time (s) Figure 4.14: MATERIAL A human gait profile for two stress cycles. Figure 4.15: MATERIAL A human gait profile 5 cycle maximum and minimum strains vs time. From examining the data trend from the 5 cycle strain gait profile it can be seen that after 5 cycles the strain has not completely leveled out to a steady value and is still somewhat increasing, albeit at a slower rate. The trend seen from the plot, however, suggests that the maximum and minimum strain plots are indeed converging to a final strain values in time. It is important to also point out the difference between the maximum and minimum strain values of each plot stops diverging from one another and reaches a constant difference value around the 2 cycle mark. The amplitude for the minimum to maximum strains after 2 cycles remains steady at approximately 1.5%. This is an important attribute in that deformation behavior known as cyclic softening ceases and the material is not continually weakening as the test progresses. At the conclusion of the 5 cycles of the human gait profile, the MATERIAL A sample was subjected to a one hour recovery test. The resulting data plot for this recovery test can be seen in Figure MAT A 5x Gait Recovery Recovery Calculation 5x Gait Profile Strain Initial = % Strain Final = % % Recovety - 1hour % Time (s) Figure 4.16: One hour recovery test performed at the conclusion of the MATERIAL A human gait profile 5 cycle test. At the conclusion of the one hour recovery test for the human gait profile, the sample shows a total strain recovery of 42.55% relative to the strain at the start of the test. This resulting data plot shows similar results to those seen in the recovery tests run in compression and tension test samples, loaded to various strains that were previously discussed. The recovery values for all three types of tests fall within the 4%-5% recovery range. This comparison validates that the materials does not have unique differences in recovery deformation characteristics when deformed under the various loading schemes. 36

49 These conclusions, specifically from the gait profile experiments, are favorable when determining that the new MATERIAL A can withstand typical loading stresses and strains experienced in human knee joint. The data implies that the material should not exhibit unexpected failure or plastic deformation under normal knee loading forces. The material also has not shown a final maximum strain change of zero after 5 cycles and will continue to increase the amount of strain deformation per cycle for cycle numbers above 5. The change in difference between the maximum and minimum strains stops at approximately the 2 cycle mark and holds constant for the remainder of the test. It was concluded that at higher amounts of cycles, the difference of the maximum and minimum strains should still remain constant and not experience any effects of cyclic softening. For the gait overload profile a new MATERIAL A sample was placed into the MTS testing frame for experimental testing. A single 4% increased overload cycle was placed into the procedure after the 25 th cycle. The data plot for the stresses exhibited the overload cycle with one normal cycle on each side for comparison can be in seen in Figure The general stress profile observed for each cycle again does not change throughout the duration of the test, except for the overload, much like the normal gait test. A minimum and maximum strain plot was produced with the overload cycle very noticeable in the middle of the test, which can be seen in Figure Cycle Force Profile MAT A 5 Cycle Minimum Strains MAT A 5 Cycle Minimum Strains Figure 4.17: MATERIAL A human gait overload stress cycle with one normal stress cycle on each side. Figure 4.18: MATERIAL A human gait profile 5 cycle with one additional 4% overload cycle maximum and minimum strains vs time. From examining the data trend from the 5 cycle strain gait overload profile it can be seen that after the overload shock the material does acquire some extra deformation. It should also be noted that the difference in the maximum and minimum strain values does not change for the duration of the test after the overload cycle. The material appears to be able to withstand the shock without changing the deformation behavior between the difference of the maximum and minimum amounts of strain per cycle. For some materials the maximum and minimum strains may actually converge or diverge after the overload, but in the MATERIAL A sample the difference between the two remain steady. This is an important attribute to observe because it shows that real life overload events will not affect the deformation behavior resulting in the occurrence unexpected actions by the polymer inserts. There appears to be a lower amount of final strain in this sample compared to the original gait sample. Taking a closer at the two plots reveals that the overload sample exhibiting smaller strain values throughout the first half of the test than those exhibited in the original gait test. After the overload cycle the strain values are still less than those seen in the original gait test. This implies that the difference in strain 37

50 Δ Strain (%) exhibited between the original test and the overload test is most likely due to sample variation and not the overload cycle itself. Had the overload strain values been the same as the original test and then deviated after the overload cycle or the strain values shown an overall change in behavior in the form of a change in the difference between the maximum and minimum strains then the overload cycle might have been the cause. The consistency in the strain behavior of the material suggests otherwise. At the conclusion of the 5 cycles of the human gait overload profile, the MATERIAL A sample was subjected to a one hour recovery test. The resulting data plot for this recovery test can be seen in Figure Recovery Calculation 5x Gait Overload Profile Strain Initial = % Strain Final = % % Recovety - 1hour % MAT A 5 Gait Overload Recovery Time (s) Figure 4.19: One hour recovery test performed at the conclusion of the MATERIAL A human gait overload profile test 5 cycle compressive test. At the conclusion of the one hour recovery test for the human gait profile with and overload cycle, the sample shows a total strain recovery of 49.72% relative to the strain at the start of the test. This resulting data plot shows similar results to those seen in the normal gait profile recovery test run in compression along with the other compressive and tensile tests that all fell between 4-5% recovery. The recovery value is slightly higher than the original gait test; however, the final residual strain left from the overload test was also different from the original gait test. The difference in the maximum and minimum strain values also stayed constant before and after the overload event. This suggests that the difference between the two recovery values may differ due to sample variation and not behavior spurred by the overload event since the only noticeable effect from the overload was a slight increase in residual strain. These conclusions, specifically from the gait profile experiments, are favorable when determining that the new MATERIAL A can withstand an overload event randomly while the material is experiencing normal stresses and strains in human knee joint. The data implies that the material should not exhibit unexpected failure or plastic deformation under these types of knee loading forces MATERIAL A Creep Tests in Compression Results Five separate creep tests were performed at unique stress values along the stress-strain curve produced from loading an MATERIAL A sample to 6% strain at a rate of 1x1-3 s -1. This grouping of data sets was then duplicated for a loading rate of 1x1-4 s -1. The main focus of the analysis and comparison for these creep tests performed, will be on the stresses located on the unloading portion of the stress strain curve and their possible rate reversal behavior. The three unloading curve creep tests were conducted at stress values of 5, 12 and 2 MPa. The two tests performed on the loading curve were conducted at 5 38

51 Δ Strain (%) Δ Strain (%) and 2 MPa to coincide with the max and minimum stresses used in the creep tests performed on the unloading curve. The creep test data plots at a 1x1-3 s -1 strain rate can be seen in Figures MAT A L 5 MPa Creep Test Strain Initial = -.1% Strain Final = -.7% Total Creep = -.6% MAT A LUL 5 MPa Creep Test Strain Initial = -3.4% Strain Final = -2.7% Total Creep =.7% Loading and Unloading MAT A L 6% to 5 MPa MAT A LUL 6% to 5 MPa MAT A L 2 MPa Creep Test Strain Initial = -2.8% Strain Final = -9.9% Total Creep = -7.1% Loading and Unloading MAT A LUL 2 MPa Creep Test Strain Initial = -5.7% Strain Final = -11.3% Total Creep = -5.6% MAT A L 6% to 2 Mpa MAT A LUL 6% to 2 Mpa Loading -6-7 Loading -.8 Time (s) -8 Time (s) Figure 4.2: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 strain rate. Creep tests performed for 5 MPa on loading and unloading curves. Figure 4.21: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 strain rate. Creep tests performed for 2 MPa on loading and unloading curves. For the tests performed at the high and low stress values, some interesting behavior characteristics of the polymer samples can be pointed out. For the 5 MPa creep tests there is a negative (compressive) creep occurring from a prior loading history of just loading while on the unloading curve there is a positive (tensile) creep occurring. The positive creep is occurring from forces enacted by the material s recovery behavior while being uncompressed. The recovery energy within the polymer chains is slightly overpowering the low 5 MPa compressive stress being applied to the specimen until an equilibrium strain is reached. Both tests only exhibit fractions of a percent of creep strain and only differ by.1% total strain. The creep tests performed at higher stress values exhibit behavior similar to one another than that at the lower values. Final amounts of creep measured differed by 1.5% between the two sets of data with the loading curve creep test producing the greater amount of strain. In both test cases there is not a clear asymptotic limit reached and at longer time periods the material strain behavior could move into tertiary creep. The difference in the two data plots seems to reach the 1.5 % maximum amplitude difference around the 2 second mark and then stay constant for the duration of the experimental test. The difference can be explained by the prior loading history differences in the two samples. The general trends in the curves behave similarly at after longer periods of time, which suggests that qualitatively after longer time periods loading history has less of an effect than during the initial moments of the test. 39

52 Δ Strain (%) Δ Strain (%) Δ Strain (%) MAT A LUL 12 MPa 1-3 Creep Test Strain Initial = -4.6% Strain Final = -5.% Total Creep = -.4% MAT A LUL 6% to 12 MPa Time (s) Figure 4.22: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 strain rate. Creep test was performed at a 12 MPa stress value on the unloading curve. Rate reversal was detected when analyzing the creep test data plot performed at 12 MPa on the unloading curve for a sample loaded to a maximum strain of 6% at 1x1-3 s -1, shown in Figure The sample initially exhibits a positive rate of strain that increases to a maximum strain value of.2% and then quickly reverses its rate and finishes with a minimum final strain of -.4%. This rate reversal behavior has been documented before in other polymers such as HDPE and PPO [28]. To highlight the rate-dependent properties of the MATERIAL A samples during creep deformation, the creep tests on the unloading curves for the slower rate of 1x1-4 s -1 were plotted with their 1x1-3 s -1 rate counterparts. The plots for creep tests performed at 5, 12 and 2 MPa for both rates can be seen in Figures MAT A L 5 MPa 1E-4 Creep Test Strain Initial = -.6% Strain Final = -1.3% Total Creep = -.7% MAT A L 5 MPa 1E-3 Mat A L 5 Mpa 1E-4 MAT A L 5 MPa 1E-3 Creep Test Strain Initial = -.1% Strain Final = -.7% Total Creep = -.6% MAT A LUL 2 MPa 1E-4 Creep Test Strain Initial = -6.% Strain Final = -11.3% Total Creep = -5.% MAT A LUL 2 MPa 1E-3 Creep Test Strain Initial = -5.7% Strain Final = -11.3% Total Creep = -5.6% MAT A LUL 2 MPa 1E-4 MAT A LUL 2 MPa 1E Time (s) -6 Time (s) Figure 4.23: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Creep tests were performed at a 5 MPa stress values on the unloading curve. curves. Figure 4.24: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Creep tests were performed at a 2 MPa stress values on the loading curve. curves. 4

53 Δ Strain (%) While the MATERIAL A samples show some effect of prior rate dependence in creep, the creep tests exhibit much smaller magnitudes than that seen in the tensile or compression loading tests for MATERIAL A. The differences in strain values for the two plots are again on the fraction of a percent. It should be noted that the length of time that the experiment for the 2 MPa creep test at a rate of 1x1-4 s -1 was less than that of the 1x1-3 s -1 rate test due to an interruption from a limit trip. Based on the data obtained and the observed deformation trend there is no evidence to believe that any type of drastic change from the exhibited deformation behavior will occur within an additional 15 seconds. Thus prior loading rate seems to have very little effect on the creep deformation behavior for the MATERIAL A samples MAT A LUL 12 MPa1E-4 Creep Test Strain Initial = -5.1% Strain Final = -5.3% MAT A LUL 12 MPa 1E-3 Creep Test Strain Initial = -4.6% Strain Final = -5.% Total Creep = -.4% MAT A LUL 12 Mpa 1E-3 MAT A LUL 12 Mpa 1E Figure 4.25: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Creep tests were performed at a 12 MPa stress values on the loading curve. Time (s) The data shown in Figure 4.25 continues to reinforce the findings found in the 5 and 2 MPa creep experiments for the effects of rate dependence for a stress value within the middle of the unloading curve. However; it should also be noted that changing the rate of loading does not seem to inhibit or induce the rate reversal behavior of the MATERIAL A samples. While the quantitative values of creep are slightly affected by loading rate, the qualitative nature of the material is not. The rate reversal occurs for both samples and at the same maximum amount of strain before reversing rate, but the time at which the reversal occurs is later for the slower prior loading rate than that of the faster rate MATERIAL A Relaxation Tests in Compression Results Relaxation tests were completed on MATERIAL A samples for strain values coinciding with the middle stress values at which the creep tests were completed. This area of the unloading was chosen due to the high amount of rate reversal behavior relative to the other test points used in the creep tests. Rate reversal in experimentation and modeling studies in relaxation (as well as creep) are rare, and Company A is currently not able to model this in their material analysis. At this middle stress value, the goal is to capture both rate dependence and rate reversal within one experiment. This was done in order to conserve samples due to the low number available. Two separate tests were conducted, again using two separate rates of 1x1-3 s -1 and 1x1-4 s -1. The relaxation plots can be seen in Figure

54 Δ Stress (MPa) MAT A LUL 12MPa Relaxation Test 1E-4 Stress Initial = MPa Stress Final = MPa Total Relaxation = MPa MAT A LUL 12 MPa Relaxation Test 1E-3 Stress Initial = 12.1 MPa Stress Final = MPa Total Relaxation = MPa MAT A LUL 1.5 hr Relaxation 1-3 MAT A LUL 1.5 hr Relaxation Time (s) Figure 4.26: MATERIAL A compression loading to 6% strain at 1x1-3 s -1 and 1x1-4 s -1 strain rate. Relaxation tests were performed at a 12 MPa stress values on the unloading curve. Both rates for the relaxation tests produced rate reversal behavior. In both cases the stress values rose positively before changing and then decreased with a final negative amount of stress change. A high amount of rate dependence was exhibited within the profile of the relaxation tests. The slower rate exhibited.7 MPa less relaxation than the faster rate. Rate dependence was also seen in the rate reversal behavior unlike in the creep tests. Not only was the rate of the reversal slower for the slower prior loading rate, but the magnitude of stress change in the positive direction was noticeably lower also. From comparisons made from the results of the creep and relaxation plots, the rate reversal appears to have a greater effect on behaviors induced by stress than those induced by strain COMMON BIOMEDICAL MATERIAL A DSC Experimental Results Samples of each type of COMMON BIOMEDICAL MATERIAL A were initially placed in a DSC and heated to find their latent heat of melting. From this value, the crystallinity of their molecular chains could be determined using Eqn. 7. Two samples of each material were tested in order to establish an average. This method was used because crystallinity values could possibly vary depending on sample and the calibration of the DSC. The data results for the six samples can be seen in Table 4.3. Table 4.3: Percent Crystallinity obtained from DSC tests for two samples each of virgin, MATERIAL B, and MATERIAL A. COMMON BIOMEDICAL MATERIAL A curves. Sample Crystallinity % MATERIAL A MATERIAL A Virgin Virgin MATERIAL B MATERIAL B The two MATERIAL A samples produced the highest values in crystallinity compared to the virgin COMMON BIOMEDICAL MATERIAL A and the MATERIAL B samples. The values seen here are in agreement with what ultimately may be expected with these types of materials. While the MATERIAL B samples have been modified and will show higher strength than the virgin samples, the modification process that MATERIAL B is subjected to, in order to eliminate free radicals, actually causes its crystallinity to decrease because of large amounts of chain movement occurring at this high 42

55 temperature. The recombination of free radicals during the remelting process pulls chains away from their current ordered state and reduces the crystallinity. The MATERIAL A is not submitted to the same remelting process as MATERIAL B but uses the filler material to reduce free radicals and thus not requiring any movement in the chains that could reduce crystallinity. There should be some error expected within the tests here also with a slight range of varying crystallinity differences among similar samples. Multiple crystallinity experimental tests have also been completed on previously loaded and fresh samples of MATERIAL B. The purpose of this second set of experiments was to determine if small amounts of strain deformation have the ability to increase the crystallinity of the material by aligning the chains. This can be expected for high amounts of deformation when the material is plastically deformed and chains are rotating and sliding relative to one another and lining up into a more uniform order. Table 4.4, shows the resulting values for the crystallinity of several samples and the differences in the crystallinity from the average of the non-deformed MATERIAL B. Table 4.4: % Crystallinity for non-deformed MATERIAL B samples and MATERIAL B samples with small deformation at several different strains. Non-Deformed MATERIAL B vs Deformed MATERIAL B Sample Number Loading Value Crystallinity % Crystallinity % +/- 1 % % % % % % % Note: Sample 3 is from a non-deformed area of the tensile tested specimen loaded to 6% The change in crystallinity was determined by taking the difference of the average value of the nondeformed samples and subtracting the values from the deformed samples. There was a clear variation between the four deformed samples and their difference from the non-deformed samples. However, no clear pattern emerged from the obtained values. The only conclusion that can be made from the data obtained is that there is not a noticeable change in crystallinity value attributed to any particular amount of deformation between strains of 2% and 2%. At this point any change in crystallinity is attributed to slight differences in crystallinity between each small piece taken from the larger deformed samples and measurement accuracy of the DSC. For example, there is almost a 2% difference in crystallinity for trials 1 and 2 which were taken from the same MATERIAL B sample. There is almost a 3% change in crystallinity from the pieces taken from two non-deformed samples of MATERIAL B that were produced from the same irradiated batch of COMMON BIOMEDICAL MATERIAL A. So it is reasonable to assume that the changes in crystallinity found in the deformed samples can largely be attributed to material and equipment variation, seen in the non-deformed samples data, instead of being caused by amount of strain deformation. 43

56 E' (Pa) E' (Pa) 4.2 MATERIAL C Material Experimental Data and Results MATERIAL C DMA Experimental Results Following the same initial experimental process as the COMMON BIOMEDICAL MATERIAL A, the glass transition temperature and Young s modulus were verified for the provided batch of MATERIAL C and MATERIAL D samples, through three point bending tests on the DMA. Data plots from the three point bending tests for these samples can be seen in Figures 4.27 and E+9 1.E+9 1.E+8 1.E+8 MATERIAL C 1.E+7 MATERIAL D 1.E+6 1.E Temperature ( o C) Figure 4.27: DMA testing for the glass transition temperature of a MATERIAL C sample. 1.E Temperature ( o C) Figure 4.28: DMA testing for the glass transition temperature of a Material D sample The listed values for the tensile modulus at room temperature for normal unfilled MATERIAL C is 3.6 GPa and for MATERIAL D the tensile modulus is listed as GPa depending on the amount of modifications within the polymer matrix [34]. The data plots from the three point bending tests produce values with some variation for the tensile modulus at a temperature of 4 o C from the listed values. For the unfilled MATERIAL C samples the modulus of elasticity is about.8 GPa and for the MATERIAL D the experiment produced a value of.43 GPa. These results show the importance of verifying the values the elastic modulus for a given polymer batch before its implementation as it can have some variation between batches of polymer samples. The glass transition temperatures were also verified from the DMA data plots. The listed values for the glass transition temperature for normal unfilled MATERIAL C is 149 o C and for MATERIAL D the glass transition is listed as 145 o C [34]. The temperatures can be taken off of the data plots within the circled areas just after the values for elastic modulus in the plot begins decrease rapidly from its previous constant flat slope line [2]. The temperature taken off of the plots for the unfilled MATERIAL C samples was 146 o C. For the MATERIAL D the temperature from the plot was determined to be 14 o C. Again, like the tensile modulus, there is some variation between listed and experimental values. This variation is likely a cause of the amount of crystallinity within the samples, variations in molecular weight along with the amount of modifications in the MATERIAL D samples, as was previously discussed for the MATERIAL A samples. 44

57 Stress (Mpa) Stress (Mpa) MATERIAL C Tensile Loading and Unloading Experimental Results Both MATERIAL C and MATERIAL D samples were initially loaded to failure in tension to obtain the full stress-strain curves. From this region, a maximum testing strain that would introduce minimal plastic deformation within the material samples could be determined. The tests were run twice for each material type to validate any findings. Stiff materials such as MATERIAL C require high stresses for small deformations. Since the onset of fracture can be strongly influenced by surface cracks and/or internal voids, it is prudent to perform at least two tests to verify the experimental findings. The pre-yield strain ranges in MATERIAL C and MATERIAL D, for future testing, are circled in Figures 4.29 and 4.3. The maximum strain values determined for elastic testing for MATERIAL C and MATERIAL D were 4.5% and 1.5% strain respectively but this nevertheless, amply covered the in-service stress range the materials will experience. Unlike the COMMON BIOMEDICAL MATERIAL A samples, the ranges of safe deformation are small therefore limiting the ability to perform tensile tests on a wide range of strain values like COMMON BIOMEDICAL MATERIAL A. The samples of MATERIAL C and MATERIAL D were only tested on the one strain value for each type mentioned before MATERIAL C Specimen 1 4 MATERIAL C Specimen Strain % MATERIAL D Specimen 1 4 MATERIAL D Specimen Strain (%) Figure 4.29: Two MATERIAL C samples tensile loaded to failure with the safe testing range circled. Figure 4.3: Two Material D samples tensile loaded to failure with the safe testing range circled. To examine the rate dependence of MATERIALS C and D, the samples were loaded to values found in the tensile failure tests at two separate loading rates under strain rate control; 1x1-3 s -1 and 1x1-4 s -1. As seen in Figures 4.31 and 4.32, rate dependence in the MATERIAL C does exist but it is not as prominent as it is in COMMON BIOMEDICAL MATERIAL A. MATERIAL C exhibits higher stress levels at certain strain points located on the loading segment of the stress strain plot for the higher loading rate, but the final stress reached at 4.5% strain is almost identical to that of the slower rate. On the unloading curve it can be seen that the faster rate shows slightly less deformation, on the order of fractions of a percent, at a stress level of zero than the slower rate. The MATERIAL D exhibits slightly higher rate dependence than the MATERIAL C. The higher loading rate reveals noticeably higher stresses almost throughout the entire loading and unloading segments compared to the slower rate data plot. Both loading rates seem to produce the same amount of final residual strain at the end of the unloading process, showing that the rate dependence doesn t affect the residual strain at these two different loading rates. Also noticeable within the plots is that the MATERIAL D samples exhibit stress values that are as much as a factor of 2.5 times more than those seen in the 45

58 Δ Strain (%) Δ Strain (%) Stress (Mpa) Stress (Mpa) MATERIAL C samples at similar strain points. This is expected due to the addition of modifications in the polymer matrix. These fibers bear much of the applied load on the polymer samples and, therefore, increase the amount of stress it takes to deform the samples MATERIAL C LUL 1E-3 MATERIAL C LUL 1E Strain (%) MATERIAL D LUL 1E-3 MATERIAL D LUL 1E Strain (%) Figure 4.31: MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s - 1 and 1x1-4 s -1. Stress values of 85, 6, 5 and 2 MPa where creep tests were performed are marked by dashed circles. Figure 4.32: Material D samples loaded to 1.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Stress values of 12, 8, 4 MPa where creep tests were performed are marked by dashed circles MATERIAL C Tensile Loading Recovery Test Results At the conclusion of the tensile loading tests, 3 minute strain recovery tests were performed on the samples to examine the material s ability to recover strain deformation and to highlight the rate dependence of this type of behavior. The results from these experimental tests can be seen in Figures 4.33 and MATERIAL C LUL Rate of 1E-3 MATERIAL C LUL Rate of 1E-4 Recovery Calculation LUL 4.5% at 1E-4 Initial Strain = % Final Strain = % % Recovery = 49.1 % Recovey - 3 min Recovery Calculation LUL 1.5% at 1E-3 Initial Strain =.166 % Final Strain =.111 % % Recovery = 33.13% Recovery - 3 min Recovery Calulation LUL 1.5% at 1E-4 Initial Strain =.175 % Final Strain =.123 % % Recovery = 29.71% Recovery - 3 min -.4 Recovery Calculation LUL 4.5% at 1E-3 Initial Strain = % Final Strain =.5375 % % Recovery = 9.96% Recovery - 3 min -.4 MATERIAL D LUL Rate of 1E-3 MATERIAL D LUL Rate of 1E Time (s) -.6 Time(s) Figure 4.33: Recovery tests for MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Figure 4.34: Recovery tests for MATERIAL D samples loaded to 1.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s

59 Δ Strain (%) The regular MATERIAL C samples showed a much higher propensity for strain recovery than the MATERIAL D samples. Regular MATERIAL C recorded recovery values between 2.5 and 3.5 times larger than that of MATERIAL D. It also exhibited a recovery two times the amount that MATERIAL A and MATERIAL B samples did by recovering 9.95% of its deformation for a rate of 1x1-3 s -1. MATERIAL D had much smaller recovery amounts with its largest being 33.13% for the faster rate. Though, the 33.13% is still around the range of recovery experienced in COMMON BIOMEDICAL MATERIAL A samples For rate dependence, the behavior of the MATERIAL C samples was slightly reversed from the tensile loading findings. While MATERIAL C did not show high rate dependence for tensile loading, there was a high dependence on rate for recovery behavior. MATERIAL C samples recovered twice as much of the residual strain at a faster prior loading rate. MATERIAL D in this case also showed some dependence on rate for a difference in the amount of recovery of about 4% more for the faster prior loading rate. In general, MATERIAL C has exhibited ability to recovery much of its residual strain at the conclusion of an applied load MATERIAL C Creep Tests in Tension Results Five separate creep tests were performed at five unique stress values (for each material) along the stress strain curve produced from loading a MATERIAL C sample to 4.5% strain at a rate of 1x1-3 s -1. This process was then duplicated for a rate of 1x1-4 s -1. The main focus of analysis and comparison in the creep tests performed was on the unloading portion of the stress strain curve and its rate reversal behavior, much like that of the COMMON BIOMEDICAL MATERIAL A samples. The three creep tests (on the unloading segment) for MATERIAL C were conducted at stress values of 2, 5, 6, and 85 MPa and can be seen in Fig Initially 85 MPa was arbitrarily picked as an upper limit creep stress. Tertiary creep quickly set in on the sample after 1 seconds of constant 85 MPa stress on the sample and the test was aborted. The data for this can be seen in Figure The upper limit for the creep tests was then decreased to 6 MPa and the experimental procedure was continued. Two tests performed on the loading curve were conducted at 2 and 6 MPa to compare with the unloading curve creep data obtained. The five creep tests performed at a 1x1-3 s -1 strain rate can be seen in Figures MATERIAL C L 85 MPa Creep Test Strain Initial = 3.931% Strain Final = % MAT C L 85 Mpa Creep 1.5 hrs Time (s) Figure 4.35: MATERIAL C sample loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1. The creep test was performed at 85 MPa on the loading curve. 47

60 Δ Strain (%) Δ Strain (%) Δ Strain (%) MATERIAL C L 6 MPa Creep Test Strain Initial = 1.73% Strain Final = 1.975% MATERIAL C LUL 6 MPa Creep Test Strain Initial = 3.42% Time (s) Loading MATERIAL C L 6 MPa 1.5 hr Creep MATERIAL C LUL 6 MPa 1.5 hr Creep Loading and Unloading MATERIAL C LUL 2 MPa Creep Test Strain Initial = 1.718% Strain Final = 1.33% Total Creep = -.415% MATERIAL C L 2 MPa Creep Test Strain Initial =.4635% Strain Final =.4772% Total Creep =.137% Time (s) Loading MATERIAL C LUL 2 MPa 1.5 hr Creep MATERIAL C L 2 MPa 1.5 hr Creep Loading and Unloading Figure 4.36: MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1. Creep test performed at 6 MPa on loading and unloading curves. Figure 4.37: MATERIAL C samples loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1. Creep test performed at 2 MPa on loading and unloading curves. For the tests performed at high and low stress values some interesting deformation characteristics of the polymer samples can be pointed out. For the 6 MPa creep tests performed on the loading curve there is a negative (compressive) creep occurring while on the unloading curve there is a positive (tensile) creep occurring similar to the behavior seen in COMMON BIOMEDICAL MATERIAL A. For the creep test performed on the unloading curve a small amount of rate reversal appears to be present in which the amount of negative strain is decreasing and heading back towards positive values. It doesn t appear that the strain will cross over into the positive strain region, but the strain rate does become positive. Both tests performed at 6 MPa only exhibit fractions of a percent of total creep strain and only differ by approximately.15%. The creep tests performed at the lower stress values are also different from one another. The creep performed on the loading curve shows a low value of strain and thus a high resistance to creep for a low stress value. Final amounts of creep measured for the loading curve were.14% and differed from the amount of strain seen in the creep test performed on the unloading curve by.4%. Thus it is safe to say that MATERIAL C does have a high resistance to creep behavior relative to COMMON BIOMEDICAL MATERIAL A MATERIAL C LUL 5 MPa Creep Test 1E-3 Strain Initial = 3.335% Strain Final = 3.157% Total Creep = -.178% MATERIAL C LUL 5 MPa Creep Test 1E-4 Strain Initial = 3.399% Strain Final = 3.224% Total Creep = -.175% MATERIAL C LUL 5 MPa 1E hr Creep MATERIAL C LUL 5 MPa 1E hr Creep Time (s) Figure 4.38: MATERIAL C sample loaded to 4.5% maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Creep tests performed at 5 MPa on the unloading curve. 48

61 Δ Strain (%) Δ Strain (%) Rate reversal was again detected within the sample materials when analyzing the data plot for the creep test performed at 5 MPa on the unloading curve for a sample loaded to a maximum strain of 4.5% for the two experimental rates, in Figure Rate reversal was expected since the beginnings of this deformation behavior were seen at a creep test performed at 6 MPa. The sample initially exhibits a negative rate of strain that decreases to a minimum strain value of -.19% and then reverses its negative rate and finishes with a final strain just above -.18%. The results from a 5 MPa creep test for both rates have produced minimal amounts of rate reversal compared to COMMON BIOMEDICAL MATERIAL A. Further testing at stress values between 5 MPa and 2 MPa may produce results that exhibit a more pronounced reversal in the creep plot. It is not expected that the rate reversal behavior will reach the high amounts seen in UHMPWE due to the general stiffness of MATERIAL C and the high amount of stress that is necessary to deform the material. Much like the testing methods for MATERIAL C and COMMON BIOMEDICAL MATERIAL A, samples of MATERIAL D were put through the same testing regimen. The stress values on the unloading curve that were selected for creep tests, based off the tensile stress strain curve, were 4, 8, and 12 MPa and can be seen on Fig As seen previously, MATERIAL D can withstand much higher stresses than any of the previously tested materials. These deformation behaviors resulting from high stresses are therefore expected to produce similar results to that of the MATERIAL C samples MATERIAL D L 4 MPa Creep Test Strain Initial =.285% Strain Final =.31% Total Creep =.16% MATERIAL D LUL 4 MPa Creep Test Strain Initial =.62% Strain Final =.53% Total Creep = -.72% Time (s) Loading MATERIAL D LUL 4 MPa 1.5 hr Creep MATERIAL D L 4 MPa 1.5 hr Creep Loading and Unloading Figure 4.39: MATERIAL D sample loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1, and then unloaded to 4 MPa for a 1.5 hr creep test Loading Loading and Unloading MATERIAL D L 12 MPa Creep Test Strain Initial =.966% Strain Final = 1.198% Total Creep =.232% MATERIAL D LUL 12 MPa 1.5 hr Creep MATERIAL D L 12 MPa 1.5 hr Creep MATERIAL D LUL 12 MPa Creep Test Strain Initial = 1.32% Strain Final = 1.389% Total Creep =.69% Time (s) Figure 4.4: MATERIAL D sample loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1, and then unloaded to 12 MPa for a 1.5 hr creep test. Data plots from the creep tests at the higher and lower stress limits can be seen in Figures 4.39 and 4.4. Again, tests at low stresses produced positive and negative creep for a prior loading history with partial unloading to the target stress. Only hundredths of a percent of total creep occurred on both loading and unloading. The higher stress creep tests produced all positive creep values. Behavior trends for the MATERIAL D followed those of the MATERIAL C samples. Higher creep totals were found on the loading curve than on the unloading curve. After 1 seconds though the profile slopes are almost identical as seen in Figure 4.4. For even high creep stresses the total amount of creep exhibited is on the order of tenths and hundredths of a percent. MATERIAL D is showing an even higher resistance to 49

62 Δ Strain (%) creep strain than that of MATERIAL C samples. The modifications have a very positive effect on the ability of the material to experience high stresses without exhibiting high deformations MATERIAL D LUL 8 MPa Creep Test 1E-3 Strain Initial = 1.26% Strain Final = 1.1% Total Creep = -.25% MATERIAL D LUL 8 MPa Creep Test 1E-4 Strain Initial = 1.12% Strain Final =.982% Total Creep = -.3% -.2 MATERIAL D LUL 8 MPa 1.5 hr Creep 1E-3 MATERIAL D LUL 8 MPa 1.5 hr Creep 1E Time (s) Figure 4.41: MATERIAL D samples loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1 s and 1x1-4 s -1, and then unloaded to 8 MPa for a 1.5 hr creep test. Figure 4.41 shows the middle stress value for the creep tests performed on the unloading curve for the two different rates. The effects of prior rate can be seen between the two specimens through the rate reversal behavior. The sample loaded at a higher rate exhibits rate reversal behavior unlike the slower rate and actually has a lower final strain amount than the slower rate. This is the first time that a faster rate has produced lower amounts of strain in a creep test, exhibiting an interesting characteristic for faster loading. For such small amounts of total creep strain seen in Fig. 4.41, this behavior most likely doesn t play a major role. If this type of behavior is exhibited in other polymer materials at higher deformations and less resistance to creep, follow up research on this creep behavior may be worthwhile MATERIAL C Relaxation Tests in Tension Results Relaxation tests were conducted at strain values that coincided with the middle stress values on the stress strain curve that were used for rate reversal behavior in the creep experiments. The two stress values used to pick the holding strains were 5 MPa for the regular MATERIAL C samples and 8 MPa for MATERIAL D samples. For MATERIAL C the strains at which the relaxation tests were performed were 3.5 % and 3.8% for loading rates of 1x1-3 s -1 s and 1x1-4 s -1. Similarly for MATERIAL D the strain values at which the tests were performed were.97% and 1.1% respectively. Only one set of relaxation tests were performed at two rates for each material. The resulting relaxation data plots can be seen in Figures 4.42 and

63 Δ Stress (MPa) Δ Stress (MPa) MATERIAL C LUL 5 MPa Relaxation Test 1E-4 Stress Initial = 49.9 MPa Stress Final = 51.71% Total Relaxation = 1.81 MPa MATERIAL C LUL 5 MPa Relaxation Test 1E-3 Stress Initial = 56.2MPa Stress Final = 55.99% Total Relaxation = -.21 MPa Time (s) MATERIAL C 1.5 hr Relaxation 1E-3 MATERIAL C 1.5 hr Relaxation 1E MATERIAL D 1.5 hr Relaxation 1E-3 MATERIAL D 1.5 hr Relaxation 1E-4 MATERIAL D LUL 8 MPa Relaxation Test 1E-3 Stress Initial = 8.155MPa Stress Final = MPa Total Relaxation = MPa MATERIAL D LUL 8 MPa Relaxation Test 1E-4 Stress Initial = MPa Stress Final = MPa Total Relaxation = 3.18MPa Time (s) Figure 4.42: MATERIAL C samples loaded to 4.5% (9 MPa) maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1, and then unloaded to 5 MPa for a 1.5 hr relaxation test. Figure 4.43: MATERIAL D samples loaded to 1.5% (155 MPa) maximum strain at strain rates of 1x1-3 s -1 and 1x1-4 s -1, and then unloaded to 8 MPa for a 1.5 hr relaxation test. The occurrence of rate reversal behavior is similar to that found in the MATERIAL C and MATERIAL D creep. Rate reversal is only exhibited in the faster loading rate for both materialsandd thus shows that the rate reversal is therefore rate-dependent itself much like several other derformation behaviors. The rate reversal in the MATERIAL C sample is the only example where the amount of stress (or strain) begins in the positive total stress region and then crosses completely into the negative total stress region (or vice versa). MATERIAL C exhibits a maximum change in stress of almost 2 MPa while MATERIAL D exhibits a maximum stress change of 3.2 MPa. These values are very small compared to the amount of stresses that are being applied within the tensile loading. Therefore, MATERIAL C and MATERIAL D also show a high resistance to stress relaxation as well as creep. 51

64 5.1 Introduction to the VBO Model 5 Computational Modeling Modifications and Results Among the current computational models, the Viscoplasticity Theory Based on Overstress (VBO) has been demonstrated to be highly capable at modeling the mechanical behaviors of polymers [15]. An initial shortcoming of the model was that it produced a linear unloading curve, which is very much the case for metals, but fails to capture the non-linearity of polymers during unloading [15], [2], and [21]. Recent advances to the model have shown promise in capturing this non-linear unloading curve profile producing a more accurate simulation of the entire stress strain curve [35]. High density polyethylene (HDPE) and polycarbonate (PC) have been the two materials in which non-linear unloading curves have been produced. Rate reversal behaviors that polymers can exhibit in creep and relaxation tests performed at different stress and strain levels during the unloading phase of the stress strain curve had previously been beyond the scope of the model. Rate reversal has been observed only on the unloading portion of the stress strain curve and is, therefore, highly dependent on the materials prior loading history and stress level. Enhancements to the model undertaken as a part of this research project have produced good quantitative and qualitative results through VBO modeling. There are no research ventures in the literature on the application of VBO to commonly-used medical grade polymers such as medical grade COMMON BIOMEDICAL MATERIAL A, MATERIAL C and MATERIAL D. Polymers such as these have different material characteristics, caused by enhancements at the molecular level, compared to nonmedical grade polymers. Radiation treatments and the addition of fillers are two of the most common enhancements to medical grade polymers. Therefore, it is important to ascertain whether the increase in the strength of the materials for high stress applications and enhanced wear properties has an effect on the VBO model s ability to successfully capture the deformation behavior of these biocompatible polymers. The central focus of the computational aspect of this research project is aimed at investigating the use of the modified VBO formulation for modeling the loading histories encountered by polymer implants. 5.2 Deriving the VBO Model and Adjusting Parameters for Modeling Polymers The Evolution of the VBO Model for Modeling Polymers VBO, a state variable model, was originally conceived for the prediction of the mechanical response seen in different types of metals and alloys [15]. VBO is a constitutive model able to reproduce strain rate sensitivity, creep, recovery, and relaxation using a single set of parameters. The model is comprised of three coupled first order differential equations; the true stress rate, equilibrium stress rate, and the kinematic stress rate. A polymer s behavior is considered to be a combination of an elastic solid and a viscous liquid or viscoelastic [2]. The elastic range on polymers lies within a very small percentage of strain deformation compared to metals where it is visible up until yielding occurs. The mechanical behavior of polymers is very time dependent. At a rapid deformation rate a more elastic response will occur, but at slow deformations the polymer will deform viscously [2]. As mentioned in Chapter 2, typically a spring is used to represent the elastic behavior, while a dashpot is used to represent the viscous fluid nature of the polymer. There are several combinations of these two elements used to try and capture the polymer s complete deformation behavior. A modified Kelvin-Voigt model is used to represent this type of 52

65 mechanical behavior using the flow law. The KV model is modified by adding a spring in series with the damper seen in Fig 2.2. The flow law is described as the sum of the elastic and inelastic strain rates and is given by where and are the true and equilibrium stresses and represents the overstress. and denote the elastic modulus and the viscosity function. The symbol is the change in stress over time or stress time derivative. The overstress term is the difference between the stress and the equilibrium stress. The equilibrium stress is known as the stress the material can sustain at infinitesimally slow loading rates. The kinematic stress sets the tangent modulus,, for the true stress and equilibrium stress in the flow stress region. Both parameters can be seen in Equations 9 and 1 and are given by (8) (9) where A is denoted as the isotropic stress, ψ as the shape function, k as the viscosity function, and Γ is an invariant of the overstress tensor of -g for uniaxial loading conditions. A general schematic of outputs from VBO and the relationships between the true stress, rate dependent overstress term, rate independent isotropic hardening, and kinematic stress terms can be seen in Equation 11 and Figure 5.1 (1) (11) Figure 5.1: VBO example plot of stress, equilibrium stress and the kinematic stress. 53

66 The overstress term then makes it possible to model deformation characteristics common to polymers and metals, such as rate dependent inelastic behavior. Due to this capability, VBO was modified to model the mechanical behaviors of polymers along with metals [2] [21]. The final two functions that help make up the VBO model and which are located in the equilibrium and kinematic stress are known as the shape and viscosity functions Eqns. 12 and 13, respectively. Each function, seen in Equations 12 and 13, utilize a few additional parameters that, along with parameters in some of the other previously discussed equations, are adjusted by the user for each specific polymer that is being modeled. (12) VBO was initially formulated on the basis of a modified parallel spring and dashpot arrangement (Kelvin- Voigt model) in series with a spring representing both the elastic and inelastic regions of the polymer seen in Figure 2.3. Multiple variations, with some added modifications, of the VBO model have been developed over time. Three selectively modified representations of VBO, with some minor differences within the governing constitutive equations, are presented by Krempl and Khan [2] [21], Colak [22] [23], and McClung and Ruggles-Wrenn [24]. Krempl and Khan first established similarities within the inelastic deformation behavior between metallic and polymeric materials. This study gave validity to using VBO to model deformation behavior in polymers as well as metallic materials. Colak [23] introduced a new unloading function into a variation of the Krempl and Khan model [2] that produced a more qualitative nonlinear unloading curve. The VBO model was predominately used to capture deformation behavior of polymers at room temperature in the previous two models. McClung and Ruggles-Wrenn [24] modified these models by introducing an isotropic stress parameter that helped capture temperature dependence for polymers at high temperatures along with trying to establish a direct methodology for tuning the parameter value assignments Parameter Value Assignment There are eight parameters to adjust per material for the original VBO model. The VBO model parameter values are currently determined through a systematic, iterative approach. McClung and Ruggles-Wrenn [24] attempted to quantify a direct and efficient method for solving the parameter values for any given polymer. To do this, the user must have a general understanding of the effect that changing each parameter or function has on the model s output. The starting value for the parameters of and, can be determined from the experimental stressstrain data. The elastic modulus,, is the slope of the stress strain plot at small strains. This value is typically reported in MATERIAL Data tables. E is much more pronounced in a full load to failure plot where loading the material produces several distinct zones of deformation such as, the elastic region, plastic deformation, necking and fracture. For a tensile test of loading the material within the elastic region and then unloading back to zero force, the elastic modulus can sometimes be difficult to obtain due to the polymer s viscoelastic behavior. E is often the slope of the plot for small amounts of deformation at the start of a test. The tangent modulus, E t, appears as the slope of the flow region. The value for this parameter is easily obtainable from plotting experimental results and measuring this slope. (13) 54

67 The isotropic stress, A, is the difference between the equilibrium stress and the kinematic stress as the equilibrium stress has reached an asymptotic limit. McClung and Ruggles-Wrenn have determined that a value for this parameter may be determined experimentally [24]. They performed relaxation tests intermittently along the loading curve for a given polymer at four different constant loading strain rates. Each relaxation test was performed at a stress corresponding to two strain values located at points along the loading curve. The relaxation tests all asymptotically reached the same minimum stress value at each of the corresponding strains for which the tests were performed. These final stresses formed a curve in which they have denoted the equilibrium stress curve. From this value, the isotropic stress could be determined from the difference in the equilibrium stress and the kinematic stress. The number of parameters needed to be adjusted can be dropped from nine to either six or seven depending on whether the isotropic stress is determined experimentally or not through the previously discussed methods of determination. Either way, the values assigned for the experimentally-found parameters still may need fine adjustments from their experimental assignments to produce the most reliable simulations. The remaining six or seven parameters have distinct effects on the shape and magnitude of values that are represented in the stress-strain plot. It is from knowing these effects that a systematic approach can be taken to assign the parameter values instead of a trial and error approach. The parameter values for,, and are all located within the viscosity function,. Adjusting the viscosity function parameters changes the magnitude of the strain rate sensitivity and the overall final stress values for the stress-strain plot. The shape and behavior of relaxation and creep profiles are directly affected by these parameter values. By knowing that is a coefficient multiplied through the function, is a divisor of the overstress invariant, and that is a negative power of the quotient; the magnitude of change in the stress-strain plot can be intuitively estimated by which parameter is to be adjusted and the magnitude of change that will occur. The parameter values for and are located within the shape function, ψ. The shape function models and determines the shape of the transition from quasi-elastic deformation behavior to the flow region. This section can also be described at the rounded top of the loading curve. The shape function also determines the profile of the unloading curve and what final value for residual strain is obtained. The magnitude of change in the stress strain plot from tuning the shape function parameters can again be intuitively estimated by which parameter is to be adjusted and knowing the how each parameter effects the magnitude of the plot. 5.3 Recent Enhancements of the VBO Model The Introduction of an Unloading Function Although VBO could produce very accurate results in the loading history of both amorphous and crystalline types of polymers, the major shortcoming of the original model was its inability to accurately reproduce the curvature in the unloading profile of the stress-strain curve [18]. This can be seen in Figure 5.2, where the unloading curves produced by the VBO model simulation, at loading strain rates of 1x1-3 s -1 and 1x1-4 s -1, are linear and not curved like those of the matching experimental data plots. 55

68 Stress (MPa) 2 HDPE LUL 1E-3 1/s HDPE LUL 1E-4 1/s VBO Sim HDPE 1E-3 1/s 16 VBO Sim HDPE 1E-4 1/s Strain (%) Figure 5.2: Original VBO example stress strain plot for HDPE with linear unloading simulation. A new rate independent unloading function was introduced by Colak to capture the curvature of the unloading curve [18].The new function proved beneficial and improved the accuracy in the profile of the unloading curve produced from the VBO model prediction without disturbing the accuracy in the profile of the loading segment. Quantitative discrepancies still remained and recent enhancements of the unloading function by Colak [18], implemented into the original polymer VBO model by Krempl and Khan [2] [21], have produced more acceptable results both quantitatively and qualitatively than previously achieved. The unloading function, C, is defined as where and are two new additional parameters added to the model. Equation 14 is introduced into the VBO model within the flow law, specifically the elastic strain rate given as The addition of the unloading function, C, is done to vary the value of the elastic modulus on the unloading segment to produce the non-linear profile shape. Without the unloading function the elastic modulus is just a constant value resulting in a constant, linear slope. The new model has produced acceptable quantitative and qualitative results for HDPE and PC samples at two different loading rates of 1x1-3 s -1 and 1x1-3 s -1 and can be seen in Figures 5.3 and 5.4. (14) (15) 56

69 Stress (MPa) Stress (MPa) 2 16 HDPE LUL 1E-3 1/s HDPE LUL 1E-4 1/s VBO Sim HDPE 1E-3 1/s VBO Sim HDPE 1E-4 1/s PC LUL 1E-3 1/s PC LUL 1E-4 1/s VBO Sim PC 1E-3 1/s VBO Sim PC 1E-4 1/s Strain (%) Figure 5.3: Modified VBO stress strain plot for HDPE with non-linear unloading function. The parameters used in the production of the stress strain curves in Figures 5.3 and 5.4 are shown in Table Strain (%) Figure 5.4: Modified VBO stress strain plot for PC with non-linear unloading function. Table 5.1: Modified VBO parameter values for HDPE and PC at tension deformation amounts of 1% and 6%. Material Element # Loading E Et A k1 k2 k3 C1 C4 Lambda Omega HDPE 1 Tension 1% PC 1 Tension 6% Introduction of a Double Element VBO Model Experimental data from creep and relaxation tests performed at two different strain rates, differing by an order of magnitude, have brought to light unusual behavior in polymers [28]. Creep tests are predominately performed and modeled on the loading segment of a stress-strain plot. The polymer is typically loaded up to a desired stress level at which the stress is held constant for a specific time. The strain is then recorded as a function of time and increases, implying a positive rate of change, during the test. Significant differences arise when the sample is loaded up to the desired maximum stress or strain value and then unloaded down to a desired stress level before the creep test is performed. Three different test points located on the unloading curve at high, low and intermediate stress values relative to the unloading point produced three different resulting output plots seen in Figures 5.5 and 5.6. An increasing positive strain rate was observed when the creep tests were performed at high stress levels on the unloading curve. Alternatively, increasingly negative strain values were seen when the creep test was performed at the bottom of the unloading curve where the stress/strain values were comparatively much lower than those at the top of the unloading curve. A third test conducted at a stress in-between the high and low stress values at which the first two creep tests were conducted, the creep test showed a rate reversal in which the strain would start off moving in the negative rate direction before switching and moving in the positive rate direction. 57

70 Stress (MPa) Δ Strain (%) 2 HDPE LUL 1E-3 1/s Intermediate High 4 3 HDPE Creep 14 MPa 1E-3 1/s (High) HDPE Creep 11 Mpa 1E-3 1/s (Int.) HDPE Creep 8 Mpa 1E-3 1/s (Low) 8 Low Strain (%) Figure 5.5: HDPE stress strain curve with high, middle, and low creep test locations to demonstrate rate reversal trend Time (s) Figure 5.6: HDPE creep strain rate reversal exhibited at stress values of 8, 11 and 14 MPa. From the two polymers previously tested, HDPE and PC, this rate reversal has been observed in the experimental data at different values for each polymer. The only commonality seen thus far between the two materials is that at high stress levels on the unloading segment the rate is only positive, at low stresses the rate is only negative, while the rate reversal range is located somewhere in-between these two regions on the unloading curve. The significance of this finding is that previously creep strain in polymers was thought to travel only in a positive or a negative direction for tensile and compressive stress states, respectively. The rate reversal occurs during the early stages of the creep and relaxation tests and produce negative and positive stresses, depending on the loading conditions, in instances where they were thought not to occur. This rate reversal needs to be accounted for within any type of constitutive model used to predict deformation behavior as it may have a strong impact on the accuracy of simulation results. The original single modified Kelvin-Voigt element version of VBO could not accurately reproduce this rate reversal seen in the experimental data from the central region of the unloading curve. By holding stress constant, the sign of the strain in a creep test is determined directly by the sign of the overstress term because the elastic strain rate drops out of the equation. With a single spring dashpot model only one overstress term is used in determining the inelastic portion of the strain rate. The difference between the stress and the equilibrium stress on their unloading curves at a corresponding strain rate determines the sign of the creep and relaxation values. If the stress is larger than the equilibrium stress, the sign of the creep and relaxation terms is positive and vice versa. By adding a second set of constitutive equations, two modified Kelvin-Voigt elements in series, two overstress terms and plots are produced that when summed together produce the value considered to be the total overstress value. The two element model form splits the original one equilibrium stress curve into two equilibrium stress curves. The region, in which rate reversal occurs, lies within the area in which the true stress is lower than one equilibrium stress and higher than the other. A depiction of the differences between stress and the two equilibrium stresses is presented in Figure

71 Figure 5.7: General description of double element VBO and how rate reversal is reproduced on the unloading curve for creep. The Khan and Krempl VBO model, mentioned above, has therefore also been modified to include the two spring damper models in series with one another to reproduce this unusual rate reversal seen in the mechanical behavior in polymers previously observed within the experimental data from creep and relaxation tests. The spring dashpot representation of the double element model can be seen in Figure 5.8. Figure 5.8: Spring and dashpot representation of double element modeling for polymer deformation behavior. The addition of another modified standard linear solid inherently means that the number of parameters within the model will double from the eight original to 16 with also the two additional unloading parameters. It is important to find out before creep tests are modeled what the double element model does to the qualitative and quantitative nature of the stress strain curve. The objective is to continue to make progress in capability without losing the accuracy that has already been obtained. The stress strain curve for HDPE and PC can be seen in Figures 5.9 and 5.1 for a double element model. 59

72 Δ Strain (%) Δ Strain (%) Stress (MPa) Stress (MPa) Strain rate: 1E-3 1/s Strain rate: 1E-4 1/s VBO Sim 2 HDPE 1E-3 1/s VBO Sim 2 HDPE 1E-4 1/s PC LUL 1E-3 1/s PC LUL 1E-4 1/s VBO Sim PC 1E-3 1/s VBO Sim PC 1E-4 1/s Strain (%) Strain (%) Figure 5.9: Double element VBO simulation with unloading function for HDPE loaded at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Figure 5.1: Double element VBO simulation with unloading function for PC loaded at strain rates of 1x1-3 s -1 and 1x1-4 s -1. After verification against an experimental data stress-strain curve, the parameter must be tuned further. Through parameter adjustment, acceptable reproductions of stress strain curves can be obtained through multiple sets of parameter values. Analyzing how the parameter values reproduce creep or relaxation behavior along with the stress strain curve modeling is how the parameter values can be verified as acceptable. The reproduction of creep tests for HDPE at the two previous loading rates can be seen in Figures 5.11 and VBO Sim Creep 14 MPa 1E-3 VBO Sim Creep 11 MPa 1E-3 VBO Sim Creep 8 MPa 1E-3 HDPE Creep 14 MPa 1E-3 HDPE Creep 11 MPa 1E-3 HDPE Creep 8 MPa 1E HDPE VBO Sim 14 MPa 1E-4 HDPE VBO 11 MPa 1E-4 HDPE VBO 8 MPa 1E-4 HDPE Exp Creep 14 MPa 1E-4 HDPE Exp Creep 11 MPa 1E-4 HDPE Exp Creep 8 MPa 1E Time (s) Figure 5.11: Double element VBO creep simulations performed at 14, 11 and 8 MPas on the unloading curve for HDPE loaded at a strain rate of 1x1-3 s Time (s) Figure 5.12: Double element VBO creep simulations performed at 14, 11 and 8 MPas on the unloading curve for HDPE loaded at a strain rate of 1x1-4 s -1. Parameter values for HDPE and MATERIAL C materials modeled in the double element VBO model with the modified unloading function can be seen in Table 5.2. It should be noted that the PC model parameters were used without creep modeling and experimental data to verify completely. Therefore, further investigation may require adjusting of the parameter values. 6

73 Stress (MPa) Table 5.2: Modified double element VBO parameter values for HDPE and PC at tension deformation amounts of 1% and 6%. Material Loading E Et A k1 k2 k3 C1 C4 Lambda Omega g HDPE Tension 1% h g PC Tension 6% h Modeling COMMON BIOMEDICAL MATERIAL A Deformation Behavior VBO Modeling Tension Loaded MATERIAL B Samples Medical grade polymers have previously not been modeled using VBO or, more specifically, the two element VBO model with an additional unloading parameter modification. Tension tests at two loading rates were performed on the COMMON BIOMEDICAL MATERIAL A specimens. Verifying that the model can reproduce deformation behavior in MATERIAL B samples is the first step to test the ability of VBO to model medical grade polymers because the modifications in MATERIAL B are the only new material modification and produce the closest representation to normal COMMON BIOMEDICAL MATERIAL A. The VBO output can be seen superimposed over experimental data for tension loading and unloading to 6 and 14% strain at rates of 1x1-3 s -1 and 1x1-4 s -1 in Figure MAT B LUL 6% 1-2 MAT B LUL 6% 1-3 VBO Sim LUL 6% 1E-2 VBO Sim LUL 6% 1E-3 MAT B LUL 14% 1-2 MAT B LUL 14% 1-3 VBO Sim LUL 14% 1E-2 VBO Sim LUL 14% 1E Strain (%) Figure 5.13: Double element VBO load and unload simulations performed at 14% and 6% maximum strain deformation for MATERIAL B samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. The results produced by the VBO model show good qualitative and quantitative agreement with the experimental data for the MATERIAL B. The model can accurately capture the loading profile for small as well as larger amounts of deformation. The unloading curve profiles are non-linear much like the results produced from the HDPE model stress strain plots. It should be noted that a small amount of quality is lost in the area of transition between elastic behavior and the flow region compared to the HDPE VBO results but the deviation is not significant. The model predicts the stress amounts slightly over the experimental data at small strains. This appears to occur more so in the faster loading rate. Also of note, because the unloading curve for 6% strain deformation and 14% are not parallel to one another, separate values for the unloading function parameters ( and ) were needed. All other original VBO parameters were identical for the two separate deformation amounts. The values for each parameter can be seen in Table

74 Stress (MPa) Table 5.3: VBO parameter values for MATERIAL B at tension deformation amounts of 6% and 14%. Material Loading E Et A k1 k2 k3 C1 C4 Lambda Omega g Tension 14% h MATERIAL B g Tension 6% h The model s main benefit is using a single set of parameter values for deformation rate and type. Further testing of amounts of deformation versus the values for the two unloading parameters and the resulting profiles is needed, but the indication is that a separate set of values will be needed for different amounts of deformation to maintain the accuracy seen in Figure VBO Modeling Tension Loaded MATERIAL A Samples The double element VBO model produced acceptable results for MATERIAL B samples in tension and, thus, was also used to model MATERIAL A samples that have new modifications in the polymer matrix. VBO was used to reproduce tension tests that were performed on MATERIAL A samples loading them to maximums of 6% and 14% similar to the MATERIAL B samples. The results for the VBO modeling can be seen in Figure 5.14 superimposed over the experimental data for these amounts of deformation Strain (%) AOX LUL 6% 1E-2 AOX LUL 6% 1E-3 VBO Sim LUL 6% 1E-2 VBO Sim LUL 6% 1E-3 AOX LUL 14% 1E-2 AOX LUL 14% 1E-3 VBO Sim LUL 14% 1E-2 VBO Sim LUL 14% 1E-3 Figure 5.14: Double element VBO load and unload simulations performed at 14% and 6% maximum strain deformation for MATERIAL A samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. Much like the results seen for the MATERIAL B samples, the VBO model produces simulates the experimental data very well. The model accurately produces a similar loading profile for both strain ranges. The VBO plot shows less deviation off of the loading profile in the MATERIAL A output than in the MATERIAL B output. Again the unloading behavior of the MATERIAL A samples did not produce parallel or like profiles for the unloading segments, thus causing different parameter values for the unloading function parameters. The parameter values for the VBO model producing MATERIAL A tension deformation behavior can be seen in Table 5.4. Table 5.4: VBO parameter values for MATERIAL A at tension deformation amounts of 6% and 14%. Material Loading E Et A k1 k2 k3 C1 C4 Lambda Omega g Tension 14% h UHMWPE AOX g Tension 6% h

75 Stress (MPa) The general shape of the stress-strain curve is near identical for HDPE, MATERIAL B, and MATERIAL A VBO Modeling Compression Loaded MATERIAL A Samples The experimental tests and procedures completed on the MATERIAL A samples for compression along with the data obtained had never been produced before because of the newness of the material. Therefore, it would be reasonable to say that VBO modeling on compression deformation has also never been completed and verified for the MATERIAL A. The VBO model was used to try and reproduce two loading rates, 1x1-3 s -1 and 1x1-4 s -1, of the material compressively loaded to 6% strain deformation. VBO for both rates can be seen with the experimental data overlaid in Figure MAT A LUL 6% 1-E3 MAT A LUL 6% 1E-4 VBO Sim LUL 6% 1E-3 VBO Sim LUL 6% 1E Strain (%) Figure 5.15: Double element VBO compressive load and unload simulations performed at 6% maximum strain deformation for MATERIAL A samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. The output produced shows approximately the same level of accuracy as the output for the MATERIAL A tension modeling. For the parameter set used in this output, there seems to be slightly less agreement for the faster loading rate. While the final maximum compressive stress is slightly lower than that of the experimental data, adjusting it slightly higher would not drastically fix the model output and data differences seen in Figure The parameter values used for the compressive model output are different from those used in the tension modeling. When the MATERIAL A samples were placed in compression it required a reasonably higher amount of stress to obtain the same 6% of strain deformation compared to tension loading. Before fully exploring the capability of the model to produce compression outputs, it was assumed that compression and tension modeling could be done using the same set of parameter values. After inputting in the tension parameter values it became apparent that the model compressive output did not fully represent the experimental data. The model reproduces lower stress values for compression compared to those seen in tension. The model was therefore adjusted as if it was a new material and a unique set of parameters were given to the compression loading seen in Table 5.5. The creep and relaxation simulations under compression loading should match the experimental data with the same set of parameters as used for the stress-strain curve in compression

76 Δ Strain (%) Δ Strain (%) Table 5.5: VBO parameter values for MATERIAL A at a compressive strain deformation amount of 6%. Material Loading E Et A k1 k2 k3 C1 C4 Lambda Omega g MATERIAL A Compression h Creep deformation modeling for stress values located on the unloading segment produced by compressive loading MATERIAL A samples, were performed for values of 5, 12, and 2 MPa for the two previously mentioned loading rates. The intent is to verify that the model continues to model ratereversal in creep tests with compressive prior loading histories. The creep outputs were produced using the same set of parameters as those seen in Table 5.5. The model outputs and overlaid experimental data can be seen in Figures 5.16 and Time (s) MAT A LUL 5 MPa 1-3 MAT A LUL 12 MPa 1-3 MAT A LUL 2 MPa 1-3 VBO Sim 5 MPa 1 VBO Sim 12 MPa 1 VBO Sim 2 MPa 1 Figure 5.16: Double element VBO creep simulation for stresses of 5, 14 and 2% located on the unloading curve MATERIAL A samples at a strain rate of 1x1-3 s Time (s) MAT A LUL Creep 5 MPa 1E-4 MAT A LUL Creep 12 MPa 1E-4 MAT A LUL Creep 2 MPa 1E-4 VBO Sim Creep 5 MPa 1E-4 VBO Sim Creep 12 MPa 1E-4 VBO Sim Creep 2 MPa 1E-4 Figure 5.17: Double element VBO creep simulation for stresses of 5, 14 and 2% located on the unloading curve MATERIAL A samples at a strain rate of 1x1-4 s -1. The creep modeling produced a good match to the experimental data. The VBO output for high stress creep tests exhibited rate reversal in the 5 and 6 MPa creep stress values and almost identical long term secondary creep output. There was slight deviation between the model output and primary creep results in the 6 MPa creep stress plots, but, as mentioned, the plots both reached similar asymptotic values. The VBO model had a much higher initial slope in the primary creep region than the experimental data. The major quantitative difference between the model and experimental data occurred in the 2 MPa test. The model exhibits a similar profile shape as that of the experimental data plot but exhibits less total strain for most of the duration of the test. The VBO model also trends towards leveling out at its asymptotic values well before the experimental data appears to. The model produced almost identical characteristics for both the fast and slow rates, this implies the accuracy may be able to be increased by tuning and there is not a unique issue with the overall simulation. Stress relaxation behavior for strain values located on the unloading segment produced by compressive loading MATERIAL A samples at the two previously mentioned loading rates were modeled using the VBO model. The strains at which the tests were conducted were at approximately 4.6% and 4.75% strain. The target was to pick a strain value that correlated with the stress values at which rate reversal was observed in creep tests. The relaxation outputs were produced using the same set of parameters as those seen in Table 5.5. The model outputs for both rates and overlaid experimental data can be seen in Figure

77 Stress (Mpa) Δ Stress (MPa) MAT A LUL 1.5 hr Relaxation 1E-3 MAT A LUL 1.5 hr Relaxation 1E-4 VBO Sim Relaxation 4.6% 1E-3 VBO Sim Relaxation 4.75% 1E Time (s) Figure 5.18: Double element VBO Relaxation simulations for a strain of 4.6% and 4.75% located on the unloading curve MATERIAL A samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. The output for the VBO model shows a very good ability for modeling the rate reversal in stress relaxation. The rate reversal is quite prominent in this plot. The VBO plots show quantitative and qualitatively acceptable results for the initial part of the relaxation test. The slope of the output then begins to deviate a bit from the experimental data plots in that it does not model the curvature of the experimental data profile as well. Quantitatively, the final asymptotic stress values do not exhibit a great amount of difference from the experimental data especially in the faster loading rate. 5.5 Modeling MATERIAL C Deformation Behavior VBO Modeling Tension Loaded MATERIAL C Samples The VBO model was used to try and reproduce two rates, 1x1-3 s -1 and 1x1-4 s -1, for MATERIAL C samples loaded in tension to 4.5% strain. VBO outputs for both rates can be seen with the experimental data overlaid in Figure MAT C LUL 1-3 MAT C LUL1-4 VBO Sim LUL 1E-3 VBO Sim LUL 1E Strain (%) Figure 5.19: Double element VBO tension load and unload simulations performed at 4.5% maximum strain deformation for MATERIAL C samples at strain rates of 1x1-3 s -1 and 1x1-4 s

78 Δ Strain (%) Δ Strain (%) For a material with low rate dependence, as evidenced by the small difference in the final stress values for two different loading rates, the VBO model does a satisfactory job of capturing this type of behavior. The unloading curve again shows good non-linear unloading and deviates slightly from the experimental data by small values comparatively to the other VBO output stress strain curves. The model does follow almost perfectly the last half of the loading curve, ending at the same place as the experimental data. The parameter values used for the VBO MATERIAL C modeling can be seen in Table 5.6. Table 5.6: VBO parameter values for MATERIAL C at a tensile strain deformation amount of 4.5%. Material Loading E Et A k1 k2 k3 C1 C4 Lambda Omega g MATERIAL C Tension h Creep deformation modeling for stress values located on the unloading segment produced by tension loading MATERIAL C samples, were performed for values of 2, 5, and 6 MPa for the two previously mentioned loading rates of 1x1-3 s -1 and 1x1-4 s -1. The creep outputs were produced using the same set of parameters as those seen in Table 5.6. The model outputs and overlaid experimental data can be seen in Figures 5.2 and MAT C 2 MPa Creep 1E-3 MAT C 5 MPa Creep 1E-3 MAT C 6 MPa Creep 1E-3 VBO Sim Creep 2 MPa 1E-3 VBO Sim Creep 5 MPa 1E-3 VBO Sim Creep 6 MPa 1E MAT C 2 MPa Creep 1E-4 MAT C 5 MPa Creep 1E-4 MAT C 6 MPa Creep 1E-4 VBO Sim Creep 2 MPa 1E-4 VBO Sim Creep 5 MPa 1E-4 VBO Sim Creep 6 MPa 1E Time (s) -.45 Time (s) Figure 5.2: Double element VBO creep simulation for stresses of 2, 5 and 6 MPa located on the unloading curve for MATERIAL C samples at a strain rate of 1x1-3 s -1. Figure 5.21: Double element VBO creep simulation for stresses of 2, 5 and 6 MPa located on the unloading curve for MATERIAL C samples at a strain rate of 1x1-4 s -1. The VBO model fails to reproduce the same high level of accuracy for the creep plots at these parameter values as previously seen in the HDPE and MATERIAL A creep plots. While the scale on the plots is in tenths of a percent, the overall differences are mainly in the profile. For the faster rate the model produces a positive rate and amount of creep strain while the experimental data is a negative rate and amount for a 6 MPa creep test. Yet the total difference between the model and experimental data here is about a tenth of a percent compared to the lower stress creep test of 2 MPa where the plot is qualitatively better but differs by a little over three tenths of a percent. The VBO model comes closer qualitatively to the profile shape at the slower loading rate than at the higher rate by at least capturing the correct profile shape but still differs by around the same amount of strain quantitatively. Overall this is the first really noticeable deviation from the experimental data. However; again the amounts of difference are still relatively small. 66

79 Δ Stress (MPa) Stress relaxation behavior for strain values located on the unloading segment produced by tension loading MATERIAL C samples at the two previously mentioned loading rates were modeled using the VBO double element model. The tests were conducted at approximately 3.5% strain. The target was again to pick a strain value that correlated with the stress values at which the most or any rate reversal was observed in creep tests. The relaxation outputs were produced using the same set of parameters as those seen in Table 5.6. The model outputs for both rates and overlaid experimental data can be seen in Figure MAT C Relaxation 1E-3 MAT C Relaxation 1E-4 VBO Sim Relaxation at 3.5% 1E-3 VBO Sim Relaxation at 3.5% 1E Time (s) Figure 5.22: Double element VBO Relaxation simulation for a strain of 3.5% located on the unloading curve for MATERIAL C samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. The output for the VBO model again shows a very good ability for modeling the rate reversal in stress relaxation but of a smaller magnitude than that seen in the MATERIAL A samples. The rate reversal is much more gradual for the MATERIAL C material than the quick transitions seen in the MATERIAL A samples. The VBO plots show quantitative and qualitative results for the entire relaxation plot for the faster rate of 1x1-3 s -1. For the slower rate of 1x1-4 s -1 the initial part of the relaxation test is modeled well but the values for later stages of the relaxation test are overshot a little bit. The slope of the output is steeper than the experimental data plot but levels out to a constant stress value around the same time as the experimental data does. Qualitatively there are some differences as mentioned but the overall profile of the relaxation curve is captured. Quantitatively the final stress values do not exhibit a substantial difference from the experimental data, about 22% difference VBO Modeling Tension Loaded MATERIAL D Samples The VBO model was used to try and reproduce two rates, 1x1-3 s -1 and 1x1-4 s -1, for MATERIAL D samples loaded in tension to 1.5% strain. VBO outputs for both loading rates can be seen with the experimental data overlaid in Figure

80 Stress (Mpa) MATERIAL D LUL 1E-3 MATERIAL D LUL 1E-4 VBO Sim LUL 1E-3 VBO Sim LUL 1E Strain (%) Figure 5.23: Double element VBO tension load and unload simulations performed at 1.5% maximum strain deformation for Material D samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. The MATERIAL D samples provided a much higher stress plot for very little strain. Slightly more rate dependence is exhibited by the material making it easier to compare the accuracy of the model output for each rate separately. Some variance between model and experimental data can be seen in the figure. The chosen model parameters capture the high final stresses in the two loading rates but provide some values along the loading curve that are lower than the experimental data much like what was observed for the MATERIAL C VBO output stress strain plot. Qualitatively though the model captures much of the profile shape. Again the places where the model under predicts the stress values, an area where the profile bows out a bit, is the only qualitative deviation. The unloading curve again shows good non-linear unloading and doesn t deviate much from the experimental data. Where it does, the difference is by small values relative to some of the other VBO output stress strain curves. Much like the MATERIAL C VBO output the unloading curve produces nearly the exact amount of final residual strain as the experimental data plot. The parameter values used for the VBO MATERIAL C modeling can be seen in Table 5.7. Table 5.7: VBO parameter values for Material D loaded to a tensile strain deformation amount of 4.5% Material Loading E Et A k1 k2 k3 C1 C4 Lambda Omega g MATERIAL D Tension h The parameter values chosen for the MATERIAL D material behavior plots are at much greater values than previously observed in the other material parameter values. This is a direct correlation to the high stress values needed to produce small amounts of deformation. Previous parameter values for MATERIAL C samples were thought to be large compared to the other materials tested. The MATERIAL D uses,, and values that are 2.5 to 4 times larger than the MATERIAL C material parameters in Table 5.7. Creep deformation modeling for stress values located on the unloading segment produced by tension loading MATERIAL D samples, were performed for values of 4, 8, and 12 MPa for the two previously mentioned loading rates of 1x1-3 s -1 and 1x1-4 s -1. The creep VBO output plots were produced using the same set of parameters as those seen in Table 5.7. The model outputs and overlaid experimental data can be seen in Figures 5.24 and

81 Δ Strain (%) Δ Strain (%) Time (s) MAT D Creep 12 MPa 1E-3 MAT D Creep 8 Mpa 1E-3 MAT D Creep 4 MPa 1E-3 VBO Sim Creep 12 Mpa 1E-3 VBO Sim Creep 8 MPa 1E-3 VBO Sim Creeo 4 MPa 1E-3 Figure 5.24: Double element VBO creep simulation for stresses of 4, 8 and 12 MPa located on the unloading curve for Material D samples at a strain rate of 1x1-3 s MAT D LUL 8 MPa Creep 1E-4 VBO Sim Creep 8 MPa 1E-4 Time (s) Figure 5.25: Double element VBO creep simulation for a stress of 8 MPa located on the unloading curve for Material D samples at a strain rate of 1x1-4 s -1. Based on the VBO model outputs seen for MATERIAL C samples, the creep outputs for MATERIAL D show better agreement with the experimental data. The model produces the correct profile shape and strain values down to a couple hundredths of a percent. As described in Chapter 3 for the experimental procedure, only one set of data was collected for the slower rate of MATERIAL C in the creep tests due to a limited number of samples. The VBO model does well capturing the slower rate profile and strain values for the single set of data. The ability of the model to produce plots of this quality for MATERIAL D along with the similar high stress for low strain deformation behavior suggests that further parameter adjustments to the MATERIAL C parameters may produce better qualitative and quantitative results for that material also. Stress relaxation behavior for strain values located on the unloading segment produced by tension loading MATERIAL D samples at the two previously mentioned loading rates were modeled using the VBO double element model. The strains at which the tests were conducted were at approximately.94% strain. Once again, the target was to pick a strain value that correlated with the stress values at which the most or any rate reversal was observed in creep tests. The relaxation outputs were produced using the same set of parameters as those seen in Table 5.7. The model outputs for both rates superposed on the experimental data are shown in Figure

82 Δ Stress (MPa) MAT D Relaxation.94% 1E-3 MAT D Relaxation.95% 1E-4 VBO Sim Relaxation.94% 1E-3 VBO Sim Relaxation.95% 1E Time (s) Figure 5.26: Double element VBO Relaxation simulation for a strain of.94% located on the unloading curve for Material D samples at strain rates of 1x1-3 s -1 and 1x1-4 s -1. The output for the VBO model again shows a very good ability for modeling the rate reversal in stress relaxation. In the case of the MATERIAL D samples the rate reversal is captured correctly in the exact time frame in which it occurred during the test. The downside to the model output is that it over predicts the magnitude of the rate reversal for the faster loading rate. That being said, the final values of stress at later stages of the relaxation test are nearly the same. Like creep tests, this is important because it shows where the material stress values will settle at longer periods of time and ultimately where they will remain while in service. For the slower rate of 1x1-4 s -1 the model again accurately predicts the profile shape for the relaxation plot and gives good qualitative results. However; for this loading rate the model over predicts the entire curve a bit. The later stages of the test and the final stress is over predicted by about 2 MPa producing less accurate final values than the faster loading rate. Overall, though, the model output plots for MATERIAL D were among the best model predictions for the set of tested materials. 7

83 6 ABAQUS and PolyUMod Three Dimensional Modeling 6.1 Introduction to the PolyUMod Calibration Software The PolyUMod Hybrid Model The Hybrid Model in the PolyUMod model library was developed specifically to model the deformation behavior exhibited by modified COMMON BIOMEDICAL MATERIAL A [27] [36]. The Hybrid Model has also been used to model other types of thermoplastics. The model utilizes 13 parameters. These parameters are split into four separate type categories; the elastic, hyperelastic, backstress network flow, and the yield and viscoplastic flow constants. A list of the parameter symbols and names can be seen in Table 6.1. Table 6.1: Parameter List for the Hybrid Model in the PolyUMod Library. Category Symbol Description Elastic Young s Modulus Poisson s Ratio Hyperelastic Locking Stretch Shear Modulus Bulk Modulus Backstress Network Flow Yield and Viscoplastic Flow Constants Initial Stiffness Final Stiffness Transition Rate Stiffness Flow Resistance Stress Exponent Pressure Dependence of Flow Flow Resistance Stress Exponent The Hybrid model is represented by a modified standard linear solid. The configuration comprises of different arrangement and types of springs and dampers compared to the VBO modified standard linear solid and can be seen in Figure 6.1. Figure 6.1: Modified standard linear solid configuration used in the Hybrid Model. The way the model is mathematically designed; there are specific simulated deformations that occur for general ranges of strains. For small amounts of deformation, the model produces a purely elastic response. For middle strain ranges, viscoelastic behavior is exhibited. With increasing amounts of 71

84 deformation the model then exhibits larger and larger amounts of viscous deformation [3]. Since the model is primarily designed around COMMON BIOMEDICAL MATERIAL A s behavior, the ranges of strain deformation can be directly compared to the strain ranges seen in the experimental data found for MATERIAL B samples in chapter PolyUMod Three Network Model Very similar to the Hybrid Model, the Three Network Model (TNM) is used to model thermoplastics but is not tailored to a specific material. The TNM is considered to be computationally more efficient than the Hybrid Model [3].The TNM is a configuration of three different elements in parallel with one another. Two of the elements are Maxwell Models with the third element being just a single spring. The three networks act as molecular networks and produce a total polymer deformation comprised of mechanical deformation and thermal expansion [32] [37] [38]. The model configuration is shown in Figure 6.2. Figure 6.2: Three Network Model element The TNM uses 17 different parameters all together. Three of the parameters are temperature dependent. This adds a new deformation characteristic to the model that has not been previously seen in the models discussed previously in this document. Since all of the experiments performed in this research study have been completed at room temperature, the temperature dependence does not provide an additional benefit over the other model types here. In other types of tests and applications where temperature is a factor, the temperature dependence could prove to be a unique characteristic of the model that propels it ahead of the others. The list of parameters for the TNM can be seen in Table 6.2 below. 72

85 Table 6.2: Parameter List for Three Network Model. Symbol Description Shear Modulus of Network A Temperature Factor Locking Stretch Bulk Modulus Flow Resistance of Newtwork A Pressure Dependence of Flow Stress Exponential of Network A Temperature Exponential Initial Shear Modulus of Network B Final Shear Modulus of Network B Evolution Rate of Flow Resistance of Network B Stress Exponential of Network B Shear Modulus of Network C Relative Contribution of I2 to Network C Thermal Expansion Coefficient Thermal Expansion Reference Temperature 6.2 PolyUMod Modeling and Parameter Fitting Process Hybrid Model Parameter Fitting The first material used for the Hybrid Model parameter fitting was MATERIAL A. Since this is the type of material that this model is designed for, it would essentially be the best material with which to gage its modeling capability. The model requires that the experimental data used for the parameter fitting be formatted such that columns of time, strain and stress data, obtained from the experiments described in Chapter 3, be inserted into a text file for importing in the named order. The program gives two different outputs to the user. The first output is in the form of material parameter values as an Abaqus input file, used for specifying a user material and its deformation characteristics in Abaqus. The second output is a plot of experimental data versus the model simulation. This plot output provides the user with an idea of how well the material properties will be represented within Abaqus and how accurate the resulting deformation simulation will likely be. The resulting output plot for the MATERIAL A compression loading to 6% strain and a loading rate of 1x1-3 s -1 can be seen in Figure

86 MAT A (experimental) MAT A (prediction) Figure 6.3: PolyUMod Hybrid Model simulation versus MATERIAL A compression loading to 6% final strain at a rate of 1x1-3 s -1. The Hybrid model exhibits non-linear loading and unloading curves, thereby modeling the viscoelastic nature of the polymer fairly well. Compared to the VBO model outputs, different issues arise in the accuracy of the resulting unloading curve profile. While VBO s original problem was producing an unloading curve that was too linear, the Hybrid model appears to produce an exaggerated bend around 5.5% strain deformation. There also appears to be one location on the loading curve and on the unloading curve in which a sudden negative spike in stress occurs unexpectedly. After the resulting stress spike, the curves continue on producing the same profile shape as was previously being exhibited. It is unknown as to why the model would produce this strange behavior and then continue on as it previously had been. The Hybrid model was also used to fit parameter values for MATERIAL C and MATERIAL D for use in Abaqus. Both MATERIAL C materials exhibit very high stiffness. With the Hybrid model tailored to produce elastic results at small deformation strain values, it was unclear how precise of a fit to the experimental data the model could produce. It was thought that even though both MATERIAL C materials were viscoelastic, their high stresses for low strain deformation is more similar to linear elastic behavior compared to COMMON BIOMEDICAL MATERIAL A and thus the Hybrid model may have been able to fit parameters to these two data plots. The two resulting plots for MATERIAL C and MATERIAL D loaded at 1x1-3 s -1 to 4.5% strain and 1.5% strain can be seen in Figures 6.4 and

87 MAT C no rlx (experimental) MAT C no rlx (prediction) MAT D no rlx (experimental) MAT D no rlx (prediction) Figure 6.4: PolyUMod Hybrid Model simulation versus MATERIAL C tension loaded to 4.5% final strain at a rate of 1x1-3 s -1. Figure 6.5: PolyUMod Hybrid Model simulation versus Material D tension loaded to 1.5% final strain at a rate of 1x1-3 s -1. The two resulting predicted plots show vastly different final modeling results between the MATERIAL C and MATERIAL D materials. At 4.5% final strain, the fitted parameter values are showing a purely elastic behavior and do not capture the experimental data s profile during loading or unloading. The model also misses the correct slope denoted by the Young s modulus value, as it is not as steep as the MATERIAL C loading curve. There is also approximately an excess of 35 MPa in the difference in the final stress given from the simulation plot compared to the experimental data. The Hybrid model falls short of expectations on multiple levels for modeling MATERIAL C monotonic loading and unloading deformation behavior. The model has been found to perform better at simulating the deformation of MATERIAL D compared to the MATERIAL C samples. While qualitatively the simulation lacks accuracy on the second half of the loading curve, the final stress and unloading curve agree well with the experimental data. This is a completely opposite result from the MATERIAL C simulation plot. The small hysteresis of the simulation plot profile is likely due to the model attempting to produce elastic behavior. Since MATERIAL D has a higher Young s modulus, the steepness of the experimental data curve is more in line with the linear elastic trend seen in the MATERIAL C simulation plot and would explain the more accurate final profile. However; changes to the model must be made to improve the simulation accuracy for the MATERIAL D experimental data loading and unloading curve Three-Network Model Parameter Fitting The TNM has previously been used by Company A to fit parameters to their MATERIAL B samples. Thus it seemed reasonable to explore the output from modeling the newer COMMON MATERIAL A samples as well. The resulting output can be compared with that of the Hybrid Model to see determine the better fit for the material. The resulting output plot for the MATERIAL A compression loading to 6% strain and a loading rate of 1x1-3 s -1 can be seen in Figure

88 MAT A (experimental) MAT A (prediction) Figure 6.6: PolyUMod Three Network Model simulation versus MATERIAL A compression loaded to 6.% Analyzing the parameter fitted predicted plot profile against the experimental plot shows that the TNM produces a very accurate representation of the MATERIAL A deformation behavior. So much so, that it appears to present a slightly better representation than the profile produced by the Hybrid model. There appears to be another distinct bend on the unloading curve much like that seen in the Hybrid model. The magnitude of the bend and deviation from the experimental data profile is much less than the Hybrids deviation from the curve. The TNM also produces the strange stress spikes that were produced by the Hybrid model in the exact same places on both the loading and unloading curves. The magnitude of the quick spike in stress appears to be smaller than that of the Hybrid model, but is still present nonetheless. Searching for a better representation of the deformation behavior of MATERIAL C and MATERIAL D the TNM was used to fit parameter values for as well. With the Hybrid model tailored to reproducing COMMON BIOMEDICAL MATERIAL A deformation the thought was that since the TNM is typically employed for thermoplastics and considered more efficient than the Hybrid model, that it would give a better representation than the Hybrid model s results. The two resulting plots produced by the TNM for MATERIAL C and MATERIAL D can be seen in Figures 6.7 and 6.8. MAT C no rlx (experimental) MAT C no rlx (prediction) MAT D no rlx (experimental) MAT D no rlx (prediction) Figure 6.7: PolyUMod Three Network Model simulation versus MATERIAL C tension loaded to 4.5% final strain at a rate of 1x1-3 s -1. Figure 6.8: PolyUMod Three Network Model simulation versus Material D tension loaded to 1.5% final strain at a rate of 1x1-3 s

89 Once again, MATERIAL C and MATERIAL D simulated loading and unloading plots left a lot to be desired from the parameter fitting. The TNM clearly cannot handle the high stress per strain loading profile that is produced by either material. The loading profile produces a concave loading curve while the two MATERIAL C materials produce a clear convex curve. The unloading curve stresses are severely under predicted from those seen in the experimental data plots. Out of the two models, the Hybrid Model produces an possibly acceptable representation for the MATERIAL D only. Final validation from the Abaqus results is needed before any definitive statement on the acceptability of the MATERIAL D materials parameter fit curve can be made. 6.3 Abaqus Computational Modeling PolyUMod Parameter Value Implementation into Abaqus The obtained parameter values from the PolyUMod model library are exported and saved as an Abaqus input file. Company A has provided a sample three dimensional Abaqus assembly of a knee implant that the parameters are implemented within to model the polymer insert s deformation behavior. The sample model is an assembly consisting of three separate input files. Two of the files are three dimensional part files, one being the upper metal femoral insert of the knee joint and the second being only the polymer insert that rests on the top of the metal insert of the fibula. The third file is the assembly file containing all of the material characteristics, specifications of the assembly and references to the two parts. The parameter values obtained from the MC Calibration program within PolyUMod are copied and pasted into the assembly file while the material type specified as a user material within the Abaqus program. While performing any type of deformation modeling the Abaqus program must be able to call upon the PolyUMod software and the specified model in which the parameter fitting was performed. The three dimensional knee insert is designed with the polymer insert being constrained at the bottom surface where there is contact with the fibula. The constraint acts as the lower force counteracting the weight of the body on the lower half of the insert in which it is placed in compression. Since there is no specific interactions that are dependent on material behavior and the insert lays uniformly flat against the polymer surface the lower metal part s force against the polymer insert can be left out of the model. This simplifies the model and reduces the computational runtime of the program. The model was received with the orientation of the metal and the polymer insert in the position of the knee in a bent state and thus needed to be rotated and adjusted in order to accurately depict the position the knee would be in for a compression loaded state. The upper metal insert was rotated about its central point 9 degrees, above the polymer insert. Two translations were then made to line the metal insert s runners up with the two channels on the polymer insert s surface in which they rotate. The same exact steps were taken for modeling all three of the sets of parameter fit values obtained from PolyUMod to create uniformity within the computational modeling procedures. The metal part was lowered to almost the point of contact with the polymer insert making sure the two parts were not overlapping with one another. The polymer was deformed by using a compressive ramp load of -26 N, normal to the plane of the polymer s upper surface where metal insert is pushes against the polymer part. Any little space left between the metal and polymer from the alignment performed before the test would be negated from the load, as Abaqus translates the upper metal insert until it made contact with the polymer. The non-deformed three dimensional assembly of the metal and polymer inserts can be seen in Figure

90 Figure 6.9: Company A three dimensional model of an Oxford Knee Design used for Abaqus three dimensional modeling polymer materials Abaqus Modeling Results Three dimensional plots for COMMON BIOMEDICAL MATERIAL A implementing the Hybrid model and TNM along with MATERIAL D implementing the Hybrid model were all completed using Abaqus and the PolyUMod library. The strain and stress results for COMMON BIOMEDICAL MATERIAL A modeled with the TNW model can be seen in Figures 6.1 and EE, Max. Principle (Avg: 75%) e e e e e e e e e e e e e-6 S, Mises (Avg: 75%) e e e e e e e e e e e e e-2 Figure 6.1: Company A three dimensional model of an Oxford Knee Design using parameter values for a three network model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The strain values are shown on the polymer insert for a 26 N applied force. Figure 6.11: Company A three dimensional model of an Oxford Knee Design using parameter values for a three network model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The Mises stress values are shown on the polymer insert for a 26 N applied force. The applied compressive force of 26 N is nearly equivalent to the forces that would be experienced by a 46 lb person, or double the weight of a 23 lb person. From this applied force it can be seen that the maximum values of strains within the material are at 3.12 % and maximum stresses at these same high 78

91 strain areas are on the order of 22.7 MPa. The Abaqus output lists the strains in Figure 6.1 as elastic strains on the material. From the experimental data obtained in this study deforming MATERIAL A to 2% strain deformation, it can be said that the deformation at this strain value is not perfectly elastic. The stresses exhibited on the material, as seen in Figure 6.11, are very similar to the amounts seen in the compression tests performed as part of the experimental testing protocols. This verifies that the fitted material parameters within the model can accurately reproduce the three dimensional deformation behavior of MATERIAL A for a knee insert design compared to the experimental data from monotonic compression tests. Implementation of the Hybrid model for the modeling of MATERIAL A can be seen in Figures 6.12 and The same load conditions and knee joint configuration were used in this modeling as was used in the COMMON BIOMEDICAL MATERIAL A modeling. EE, Max. Principle (Avg: 75%) e e e e e e e e e e e e e-6 S, Mises (Avg: 75%) +2.2e e e e e e e e e e e e e-2 Figure 6.12: Company A three dimensional model of an Oxford Knee Design using parameter values for the Hybrid model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The strain values are shown on the polymer insert for a 26 N applied force. Figure 6.13: Company A three dimensional model of an Oxford Knee Design using parameter values for the Hybrid model within Abaqus for COMMON BIOMEDICAL MATERIAL A. The Mises stress values are shown on the polymer insert for a 26 N applied force. From the 26 N applied force, it can be seen that the maximum values for strains within the MATERIAL D material are at 1.16 % and maximum stresses at these same high strain areas are on the order of 22.2 MPa. The stress output from Abaqus in both Fig is very similar to that of COMMON BIOMEDICAL MATERIAL A modeled by the TNM and the experimental results. The strain output by Fig is about half of that seen in the TNM. The stress fields are different from those seen in the TNM also. The alignment process was the same for the Hybrid model and the TNM so it is not suspected to be the culprit. The parameter fitted output plot for the hybrid model was slightly worse than the TNM and it is suspected as the reason behind the strain and stress field differences. The model did produce the correct range of stresses from the applied force. Implementation of the Hybrid model for the modeling of MATERIAL D can be seen in Figures 6.14 and The same load conditions and knee joint configuration were used in this modeling as was used in the COMMON BIOMEDICAL MATERIAL A modeling. 79