Multiscale mechanics and multiobjective optimization of cellular hip implants with variable stiffness

Transcription

1 Multiscale mechanics and multiobjective optimization of cellular hip implants with variable stiffness Written by: Sajad Arabnejad Khanoki Doctor of Philosophy Department of Mechanical Engineering McGill University Montreal, Quebec April 13 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Sajad Arabnejad Khanoki, 13

2 I lovingly dedicate this thesis to my wife Zahra for her love, support, and encouragement. ii

3 Acknowledgements I would like to take this chance to express my sincerest thanks to my advisor, Professor Damiano Pasini, for his technical support, scientific guidance, and encouragement during my PhD. His kindness, enthusiasm, and dedication to this work have always motivated me for research. He is a great teacher and a true mentor and has always been generous with his time for me. Without his support, the completion of this work would be impossible. I would like to thank the nice friends from The Pasini Lab whom I was pleased to meet and know. Particularly, thanks go to Dr. Ehsan Masoumi Khalil for his assistance, advice, and kindness. I would like to express my special thanks to my parents and my parents-in-law for their encouragement, unconditional love, and continuous support. I spent many hours thinking how to thanks my dear wife Zahra. Words fall short in describing your kindness, love, and support. I will always be indebted and grateful to you for all sacrifices you made for me and for the many ways in which you influenced my life. This thesis and the rest of my life are dedicated to you. I would like to thank the Faculty of Engineering of McGill University, the National Sciences and Engineering Research Council of Canada (NSERC), and the McGill Engineering Doctoral Award (MEDA) for their collaborative financial support for this project. iii

4 Abstract Bone resorption and bone-implant interface instability are two bottlenecks of current orthopaedic hip implant designs. Bone resorption is often triggered by mechanical bio-incompatibility of the implant with the surrounding bone. It has serious clinical consequences in both primary and revision surgery of hip replacements. After primary surgery, bone resorption can cause periprosthetic fractures, leading to implant loosening. For the revision surgery, the loss of bone stock compromises the ability of bone to adequately fix the implant. Interface instability, on the other hand, occurs as a result of excessive micromotion and stress at the bone-implant interface, which prevents implant fixation. As a result, the implant fails, and revision surgery is required. Many studies have been performed to design an implant minimizing both bone resorption and interface instability. However, the results have not been effective since minimizing one objective would penalize the other. As a result, among all designs available in the market, there is no implant that can concurrently minimize these two conflicting objectives. The goal of this thesis is to design an orthopaedic hip replacement implant that can simultaneously minimize bone resorption and implant instability. We propose a novel concept of a variable stiffness implant that is implemented through the use of graded lattice materials. A design methodology based on multiscale mechanics and multiobjective optimization is developed for the analysis and design of a fully porous implant with a lattice microstructure. The mechanical properties of the implant are locally optimized to minimize bone resorption and interface instability. Asymptotic homogenization (AH) theory is used to capture stress distribution for failure analysis throughout the implant and its lattice microstructure. For the implant lattice microstructure, a library of D cell topologies is developed, and their effective mechanical properties, including elastic moduli and yield strength, are computed using AH. Since orthopaedic hip implants are generally expected to support dynamic forces generated by human activities, they should be also iv

5 designed against fatigue fracture to avoid progressive damage. A methodology for fatigue design of cellular materials is proposed and applied to a two dimensional implant, with Kagome and square cell topologies. A lattice implant with an optimum distribution of material properties is proved to considerably reduce the amount of bone resorption and interface shear stress compared to a fully dense titanium implant. The manufacturability of the lattice implants is demonstrated by fabricating a set of D proof-of-concept prototypes using Electron Beam Melting (EBM) with Ti6Al4V powder. Optical microscopy is used to measure the morphological parameters of the cellular microstructure. The numerical analysis and the manufacturability tests performed in this preliminary study suggest that the developed methodology can be used for the design and manufacturing of novel orthopaedic implants that can significantly contribute to reducing some clinical consequences of current implants. v

6 Résumé La résorption osseuse et l'instabilité de l'interface os-implant sont deux goulots d'étranglement de modèles actuels d'implants orthopédiques de hanche. La résorption osseuse est souvent déclenchée par une bio-incompatibilité mécanique de l'implant avec l'os environnant. Il en résulte de graves conséquences cliniques à la fois en chirurgie primaire et en chirurgie de révision des arthroplasties de la hanche. Après la chirurgie primaire, la résorption osseuse peut entraîner des fractures périprothétiques, conduisant au descellement de l'implant. Pour la chirurgie de révision, la perte de substance osseuse compromet la capacité de l'os à bien fixer l'implant. L'instabilité de l'interface, d'autre part, se produit à la suite d'un stress excessif et de micromouvements à l'interface os-implant, ce qui empêche la fixation des implants. De ce fait, l'implant échoue, et la chirurgie de révision est nécessaire. De nombreuses études ont été réalisées pour concevoir un implant qui minimise la résorption osseuse et l'instabilité de l'interface. Cependant, les résultats n'ont pas été efficaces, car minimiser un objectif pénaliserait l'autre. En conséquence, parmi tous les modèles disponibles sur le marché, il n'y a pas d'implant qui puisse en même temps réduire ces deux objectifs contradictoires. L'objectif de cette thèse est de concevoir une prothèse orthopédique de la hanche qui puisse simultanément réduire la résorption osseuse et l'instabilité de l implant. Nous proposons un nouveau concept d'implant à raideur variable qui est mis en œuvre grâce à l'utilisation de matériaux assemblés en treillis. Une méthodologie de conception basée sur la mécanique multi-échelle et l'optimisation multiobjectif est développé pour l'analyse et la conception d'un implant totalement poreux avec une microstructure en treillis. Les propriétés mécaniques de l'implant sont localement optimisés pour minimiser la résorption osseuse et l'instabilité d'interface. La théorie de l'homogénéisation asymptotique (HA) est utilisée pour capturer la distribution des contraintes pour l'analyse des défaillances tout le long de l'implant et de sa microstructure en treillis. Concernant vi

7 cette microstructure en treillis, une bibliothèque de topologies de cellules D est développée, et leurs propriétés mécaniques efficaces, y compris les modules d'élasticité et la limite d'élasticité, sont calculées en utilisant le théorie HA. Puisque les prothèses orthopédiques de hanche sont généralement censées soutenir les forces dynamiques générées par les activités humaines, elles doivent être également conçues contre les fractures de fatigue pour éviter des dommages progressifs. Une méthodologie pour la conception en fatigue des matériaux cellulaires est proposée et appliquée à un implant en deux dimensions, et aux topologies de cellules carrées et de Kagome. Il est prouvé qu un implant en treillis avec une répartition optimale des propriétés des matériaux réduit considérablement la quantité de la résorption osseuse et la contrainte de cisaillement de l'interface par rapport à un implant en titane totalement dense. La fabricabilité des implants en treillis est démontrée par la fabrication d'un ensemble de concepts de prototypes utilisant la fusion par faisceau d'électronsde poudre Ti6Al4V. La microscopie optique est utilisée pour mesurer les paramètres morphologiques de la microstructure cellulaire. L'analyse numérique et les tests de fabricabilité effectués dans cette étude préliminaire suggèrent que la méthodologie développée peut être utilisée pour la conception et la fabrication d'implants orthopédiques innovants qui peuvent contribuer de manière significative à la réduction des conséquences cliniques des implants actuels. vii

8 Table of Contents Acknowledgements... iii Abstract... iv Résumé... vi List of Tables... xi List of Figures... xii CHAPTER 1: Orthopaedic hip replacement implant Background and motivation Cemented and cementless implants Failure of prosthesis Implant design requirements Biocompatibility High wear resistance Implant stability Preservation of bone tissue Conflict between implant stability and bone resorption Thesis rationale Practical feasibility of the implant concept: lattice materials via additive manufacturing Objectives Structure of the thesis... CHAPTER : Multiscale mechanics of cellular structures Introduction Asymptotic homogenization (AH) method Effective mechanical properties of cellular structures by AH Comparative analysis of homogenization schemes and discussion Conclusion CHAPTER 3: Design of lattice materials against fatigue failure using asymptotic homogenization Introduction viii

9 3. Fatigue design methodology Ultimate strength of lattice materials Cell geometries under investigation Mathematical formulation of the optimization problem Numerical modeling Results Stress distribution in the unit cells Failure surfaces of the unit cells and experimental validation Modified Goodman diagrams Conclusion CHAPTER 4: Multiscale Design and Multiobjective Optimization of Orthopaedic Hip Implants with Functionally Graded Cellular Material Introduction Methodology Formulation of the multiobjective optimization problem Application of the methodology to design a D femoral implant with a graded cellular material FEM model of the femur at the macroscale FEM model of the cellular implant at the microscale Design of the graded cellular material of the implant Analysis of microstructure failure Results and discussion Conclusion... 9 CHAPTER 5: Fatigue Design of a Mechanically Biocompatible Lattice for a Proofof-concept Femoral Stem Introduction Fatigue analysis of cellular implants Fatigue design of a hip implant with controlled lattice microarchitecture Design of a D femoral implant with a graded cellular material D FEM model of the femur FEM model of the cellular implant ix

10 5.5 Results Verification of the numerical results Manufacturability of designed graded cellular implants Discussion Conclusion CHAPTER 6: Conclusions and future work Summary Original contribution Future work Creating a library of three-dimensional lattice cell topologies suitable for bone replacement implants Generalization of AH theory Fatigue failure analysis of lattice materials based on the generalized AH Experimental validation of AH results Design and optimization of a 3D femoral stem Design and optimization of acetabular cup...11 Appendix: The matrix operators of the shape optimization procedure presented in Chapter References...14 x

11 List of Tables Table 1 1: Relationship between cause of failure a and time to failure (time interval to revision) of hip implant [55]. (Adapted with permission from Springer)... 6 Table 1: Yield surfaces as a function of relative density for square, hexagonal, and Kagome unit cells... 4 Table : The expressions of the effective mechanical properties obtained in literature for various cell topologies Table 3 1: Material properties of bulk solid materials Table 3 : Yield and ultimate strength of the G 1 and G unit cells Table 3 3: Fatigue to monotonic performance ratio, / e ult, of G 1 and G lattices made of titanium (Ti) and aluminum (Al) 661T6 alloy at given relative densities Table 4 1: Effective mechanical properties of the square unit cell as a function of relative density Table 5 1: Comparison of microscopic stress distribution obtained by detailed FEA and AH for the unit cells located at the proximal region and closed to the implant border xi

12 List of Figures Figure 1 1: total hip replacement with cemented and uncemented hip implants [19]. (Reprinted with permission from Medical Multimedia Group)... Figure 1 : Alternative porous surfaces in cementless implants: a) TaperLoc hip stem coated with titanium porous plasma spray (BioMet Inc.) [9, 3] b) Epoch stem with titanium fiber mesh coating (Zimmer Inc.) [31-34], c) Trabecular metal stem with tantalum foam coating (Zimmer Inc.) [35-38], and d) VerSys stem with sintered beaded coating (Zimmer Inc.) [34, 39-43]. (Reprinted with the permission from BioMet Inc and Zimmer Inc)... 5 Figure 1 3: Theories proposed over the years regarding aseptic loosening [54]. (Adapted with permission from Springer)... 7 Figure 1 4: The flutes and splines applied to a stem, b) the clothespin design and its lateral radiograph. (Reprinted with the permission from Current Orthopaedic Practice [4]) Figure 1 5: Multiple orthopaedic applications for porous tantalum. Top row: monoblock acetabulum, and a revision acetabular augment. Middle row: monoblock tibia, revision TKR augments, and salvage patella button. Bottom row: osteonecrosis implant, and spine arthrodesis implants (Courtesy of Zimmer, Warsaw, IN) [117]. (Reprinted with permission from Elsevier) Figure 1 6: Rapid prototyping manufacturing technique, (a) SLA technique, (b) SLS or SLM technique, (c) EBM technique Figure 1: Common cell topologies... 4 Figure : Homogenization concept of a cellular structure... 8 Figure 3: Periodic boundary conditions for a pair of nodes located on the opposite surfaces, A and A, of the RVE Figure 4: Effective elastic constants as a function of relative density for the cell topologies: (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B xii

13 Figure 5: Yield strength as a function of relative density for: (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B cells Figure 6: Yield surface of an hexagonal cell honeycomb under combined inplane stress state ( x, and xy ) for a relative density 5% y Figure 7: Yield surface of a square cell topology under combined multiaxial macroscopic stress state ( xx, and yy xy ) for a relative density 5% Figure 8: Yield surface of a Kagome cell honeycomb under combined in-plane stress state ( x, and xy ) for a relative density 3% y Figure 9: Effective elastic constants for (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B, for relative density below.3 (. 3). The closed-form expressions of the effective elastic constants obtained by Gibson and Ashby [144], Wang and McDowell [151], Elsayed [166], and Vigliotti and Pasini [167] are plotted Figure 1: Relative error between the effective elastic constants obtained via the closed-form expressions given in [144, 151, 166, 167] and those obtained by asymptotic homogenization for different cell topologies, (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B, for the range of relative density lower than.3 (.3 ) Figure 11: Yield strength as a function of relative density lower than.3 (.3 ) for: (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B. The closed-form expressions plotted in figure are those obtained by Gibson and Ashby [144], Wang and McDowell [151], and Vigliotti and Pasini [167] Figure 1: Relative error between the yield strength obtained via the closed-form expressions given in [144, 151, 167] and those obtained by asymptotic homogenization for (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) xiii

14 mixed square/triangular A, and (f) mixed square/triangular B, at density lower than.3 (. 3) Figure 3 1: (a) SEM micrograph of the notch root region of an Al-foam obtained during an in situ crack propagation test, (b) stress concentrated at the vicinity of the sphere-sphere bonding [9]. (Reprinted with permission from Elsevier)... 5 Figure 3 : (a) SEM photograph of a section of a undamaged joint showing the fatigue crack growth path starting at the toe of the braze joint of struts carrying tensile load [191], (b) optical photograph of a braze joint of a diamond lattice core of a sandwich structure [19]. (Reprinted with permission from Elsevier) Figure 3 3: Schematic view of (a) a stress-life curve and (b) a cyclic load with constant stress amplitude Figure 3 4: Fatigue design diagrams showing the relation between alternating stress and mean stress for a given number of cycles Figure 3 5: Flowchart of the asymptotic homogenization theory to obtain the ultimate strength of a lattice material Figure 3 6: Flowchart of the design methodology. For a given cell geometry, shape synthesis is coupled with computational analysis followed by size optimization. The goal of the first step is to smooth the transitions between the geometric primitives defining the cell inner boundaries, thereby reducing stress localization. In the second module, the effective strength properties of the lattice are determined through asymptotic homogenization theory. The third step involves the cell size optimization to reduce at minimum the maximum von Mises stress in the cell wall Figure 3 7: Schematic views of: (a) G 1 square unit cell; (b) G 1 hexagonal unit cell... 6 Figure 3 8: Schematic views of: (a) G continuous square unit cell; (b) G continuous hexagonal unit cell; (c) Parameterization of the inner profile of a unit cell xiv

15 Figure 3 9: Mesh sensitivity showing the independency of the results from the mesh size Figure 3 1: Von Mises stress (MPa) distribution in hexagonal and square unit cells made out of Ti 6Al 4V. Lattices under fully reversed uni-axial loading, defined by: G 1 cell with small arc (left), optimum G 1 cell (middle), and optimum G cell (right) Figure 3 11: Von Mises stress (MPa) distribution in hexagonal and square unit cells made out of Ti 6Al 4V. Lattices under fully reversed pure shear, defined by: G 1 cell with small arc (left), optimum G 1 cell (middle), and optimum G cell (right) Figure 3 1: Yield and ultimate surfaces of Ti-6Al-4V square and hexagonal unit cells for relative density of 1%. Projection of yield and ultimate surfaces of the G 1 and G square unit cells in the planes: (a) xx yy, (b) xx xy, (c) xx yy, (d) xx xy Figure 3 13: Effective yield strength of the square and hexagonal unit cells under uni-axial and shear loading as a function of relative density. Yield strength for square unit cell under (a) uni-axial and (b) shear loading respectively; (c) and (d) for hexagon Figure 3 14: Modified Goodman diagram for Ti-6Al-4V square and hexagonal unit cells at given relative densities. G 1 and G square under (a) uni-axial loading and (b) shear loading condition; G 1 and G hexagon under (c) uni-axial loading and (d) shear loading Figure 4 1: Flow chart illustrating the design of a graded cellular hip implant minimizing bone resorption and implant interface failure Figure 4 : D Finite element models of the femur (left) and the prosthesis implanted into the femur (right) Figure 4 3: D hollow square unit cell for given values of relative density Figure 4 4: a) 3 3 Gauss points in the RVE; b) superposition of the RVE on the macroscopic mesh of the homogenized model xv

16 Figure 4 5: Trade-off distributions of relative density for the optimized cellular implant Figure 4 6: Distribution of bone resorption around (a) fully dense titanium implant, (b) cellular implant with uniform relative density of 5%, (c) graded cellular implant (solution B in Figure 4 5) Figure 4 7: Distribution of local interface failure f ( ) around (a) fully dense titanium implant, (b) cellular implant with uniform relative density of 5%, (c) graded cellular implant (solution B in Figure 4 5) Figure 4 8: Polypropylene proof-of-concept of the optimal graded cellular implant (solution B in Figure 4 5) Figure 5 1: Flow chart illustrating the fatigue design methodology of a graded cellular hip implant Figure 5 : a) D Finite element models of the femur and b) the prosthesis implanted into the femur Figure 5 3: Trade-off distributions of relative density for the optimized cellular implant made of a) square and b) Kagome lattices Figure 5 4: Regions used to assess the accuracy and validity the AH model (left and middle) with respect to a detailed FE analysis of a 5 5 lattice microstructure (right) Figure 5 5: Macroscopic strain distribution (solution B in Figures 5 3a and b) as a result of load case 1 at (a) the proximal part and (b) the border of the square lattice implant, and (c) the proximal part and (d) the border of the kagome lattice implant Figure 5 6: Fabricated Ti6Al4V graded cellular implants and the corresponding microstructures Figure 5 7: Cell wall thickness and pore size of cellular implants fabricated with alternative cell size are compared to the nominal design parameters. Δ represents the difference between the average as-fabricated and the nominal values Figure 5 8: Distribution of bone resorption around (a) fully dense titanium implant, (b) graded cellular implant with square topology (solution B in Figure 5 xvi

17 3a), (c) graded cellular implant with Kagome topology (solution B in Figure 5 3b) Figure 5 9: Distribution of local shear interface failure f ( ) around (a) fully dense titanium implant, (b) graded cellular implant with square topology (solution B in Figure 5 3a), (c) graded cellular implant with Kagome topology (solution B in Figure 5 3b) xvii

18 1 CHAPTER 1: Orthopaedic hip replacement implant 1.1 Background and motivation A hip joint consists of two main components that fit together like a ball-andsocket: the femoral head at the top of the leg and the acetabulum in the pelvis. The surfaces of the femoral head and the acetabulum are covered with a layer of cartilage to protect the joint against wear and provide smooth movement. When this cartilage is damaged due to severe injury or disease, such as arthritis or rheumatism, the joint becomes stiff and makes the execution of basic daily activities problematic and painful [1-3]. A total hip replacement (THR) can be performed to restore normal joint motion by replacing the femoral head and the acetabulum with artificial prostheses. A hip prosthesis consists of a cup, namely, the acetabular component, and a metal stem, the femoral component. The acetabular cup is placed in the acetabulum socket, which hosts the head of the femoral component, as shown in Figure (1 1). Although significant improvements have been made on the design of orthopaedic hip implants, two main design bottlenecks are still recognized for current hip prostheses: bone resorption and implant stability [4-1]. The problem of bone resorption in bone-replacement implants lies in the mechanical mismatch between femoral bone and implant. Materials currently used in hip implants are 316L stainless steel, cobalt chromium alloys, and titanium-based alloys. The elastic modulus of these materials is considerably higher than that of bone. The elastic moduli of bone vary in magnitude from 4 to 3 GPa depending on the type of the bone and the direction of measurement [11]. The values of Young s modulus of chromium alloys and titanium alloys, for example, are and 11 GPa, respectively, which are almost one order of magnitude higher than that of bone. As known from basic concepts of solid mechanics, when two different materials are joined, the stiffer structure bears the majority of the load. Once a metal implant is 1

19 inserted inside bone, especially for applications where the primary function of the bone is to support load and body weight, such as hip and knee joints, the majority of load will be transferred to the implant due to its high stiffness. Therefore, bone will be under-loaded. Bone, like almost all biological tissue, has the ability to adjust to mechanical and physiological loads by adding tissue at locations of high stress and removing it from under-loaded areas [1-17]. Therefore, after the implantation, under-stimulated areas will start to lose mass by an adaptive process known as bone resorption. This phenomenon is usually referred to as stress shielding effect [4, 18]. Bone resorption will increase the porosity of bone and weakens the integrity of the implant fixation. This occurrence might result in periprosthetic fracture and implant failure, necessitating revision surgery. Cemented implant Cementless implant Bone Metal Metal Bone Cement MMG 1 Porous coating Figure 1 1: total hip replacement with cemented and uncemented hip implants [19]. (Reprinted with permission from Medical Multimedia Group) Bone loss has also an undesirable influence on revision surgery because it causes an increase in the risk of failure. The safe removal of the failed implant from severely stress shielded bone without fracture is not only technically challenging, but may not leave adequate bone for satisfactory support of a revision implant. Therefore, the surgeon is required to bypass or reconstruct the remaining bone by opting for a larger and longer implant, as well as bone grafting with morselized,

20 structural or synthetic bone grafts. As a result, revision surgery has a lower chance of achieving successful implant fixation than a primary implant surgery. Despite numerous studies performed for the design of orthopaedic hip implants, there is no existing hip implant that can simultaneously avoid bone resorption while providing implant stability. Any design to prevent bone resorption has led to a flexible and mechanically unstable implant with high amount of micromotion at bone-implant interface resulting in implant failure. On the other hand, any design for a stable implant has led to a stiff implant that cause bone resorption. Therefore, the motivation of this thesis is to develop a novel total hip replacement implant that is mechanically stable for long-term survival, and preserves the host femoral bone stock needed for both primary and successful revision surgery. This chapter first introduces the two main categories of orthopaedic hip implants: cemented and cementless implants. Then, the main causes of hip implant failure are discussed in order to define the design requirements for orthopaedic hip implants. Based on these requirements, the thesis underlying principle and objectives are defined. 1. Cemented and cementless implants Hip implants are commonly categorized with reference to their modes of fixation, specifically cemented and cementless implants (Figure 1 1). In cemented prosthesis, introduced by Dr. Charnley in 196 s [], the implant adheres mechanically to the bone by cement. This procedure was a major achievement in the treatment of hip arthritis that allowed the THR to become the most popular hip reconstruction surgery in the world. Initial results of Charnley s implant encouraged surgeons to use the cemented technique in large number of patients, including young and active patients. However, radiographic and clinical results showed a high incident of failure in young patients [1, ]. The debonding at the bone-cement interface and the progressive fatigue failure of the cement mantle were found as the major culprits for the limited longevity of a cemented implant [3]. As a result, this implant is commonly used in elderly patients [4]. In the early to mid 197 s, cementless implant were introduced as an alternative to 3

21 provide long-term stability [5]. As shown in Figure (1 ), they integrate at their surface a porous metal to create a bone ingrowth interface and to biologically fix the implant to the bone [6-8]. Sintered beaded coating, titanium fiber mesh coating, titanium porous plasma coating, and trabecular tantalum foam are various approaches to produce a porous structure on the implant surface. Porosity, pore size, interconnectivity between pores, and surface roughness are parameters of the coating surface that affect the bone-implant bonding. Increasing porosity with interconnected pores enhances vascularization and eases the transportation of nutrients for bone cells; therefore high porosity promotes bone ingrowth. Many studies have been performed to obtain the optimum range of porosity and pore size that provides the maximum strength for implant fixation. Bobyn et al. [44] reported that pore sizes from 5 to 4 micron can provide the maximum fixation strength, while Bragdon et al. [45] reported that a mean pore of P m and a porosity of 4% are the optimum ranges for bone ingrowth. In another study, it has been shown that increasing pore size from 175 to 4 μm has almost no effect on the implant fixation strength [46]. Several values, sometimes not aligned, can be found in the literature regarding the optimum pore size for bone ingrowth; a broad range is often specified with boundary values of 5 and 9 μm [47-5]. Surface roughness and microtexture also have a positive effect on bone formation at the implant surface [43, 51, 5]. Surface roughness increases friction coefficient against bone and therefore improves implant initial stability for bone ingrowth and mechanical fixation. Surface microtexture and microporosity (pore size of 1m) play an important role on bone formation as well. Microporosity results in larger surface area that is believed to contribute to higher bone inducing protein adsorption. Microporosity also encourages ion exchange and bone-like apatite formation by dissolution and reprecipitation [5]. 4

22 (a) (b) (c) Porosity: 3-6% Pore size: 5-1 µm Porous layer thickness: 15-5 µm Surface roughness: 1-5 µm (d) Reed sprunger Titanium fiber mesh PEEK polymer CoCr core Porosity: 4-6% Pore size: 8-56 µm Porous layer thickness: µm Surface roughness: 45-1 µm Porosity: 75-8% Pore size: 4-6 µm Porous layer thickness: 15-5 µm Surface roughness: 5-5 µm Porosity: 3-5% Pore size: 1-3 µm Porous layer thickness: 1-15 µm Surface roughness: 5-1 µm Figure 1 : Alternative porous surfaces in cementless implants: a) TaperLoc hip stem coated with titanium porous plasma spray (BioMet Inc.) [9, 3] b) Epoch stem with titanium fiber mesh coating (Zimmer Inc.) [31-34], c) Trabecular metal stem with tantalum foam coating (Zimmer Inc.) [35-38], and d) VerSys stem with sintered beaded coating (Zimmer Inc.) [34, 39-43]. (Reprinted with the permission from BioMet Inc and Zimmer Inc) 5

23 1.3 Failure of prosthesis Although in the last decades hip arthroplasty is considered one of the greatest achievements of orthopaedic surgery, revision surgery is still required as a result of several, sometimes concomitant, factors. Among these, the most common are: aseptic loosening, bone-implant interface instability and micromotion, bacterial infection at the joint, components failure, periprosthetic fracture, and pain [53-55]. Patient-related factors, such as sickle cell anaemia [56], poor bone quality [57], high body mass index [58], may also predispose the patient to infections, dislocation, or prosthetic failure. In a recent study, a relationship between the primary reason for failure and time of revision for patients who underwent revision THR over a 6-year time period has been obtained [55]. The results are summarized in Table 1 1. Table 1 1: Relationship between cause of failure a and time to failure (time interval to revision) of hip implant [55]. (Adapted with permission from Springer) Time interval to revision Number of hips requiring revision Aseptic loosening (%) Infection (%) Instability (%) Component failure (%) Periprosthetic fracture (%) < years years years >1 years <5 years >5 years Total a Cause of failure is given in the table as the percentage of the group Pain (%) It has been found that the major cause of failure after 5 years is aseptic loosening, while most failures occurring within the first years can be attributed to joint instability and infection [55]. Aseptic loosening, defined as an extensive localized bone resorption resulting in loosening without infection, was first described by Harris et al. [59]. The reasons for aseptic loosening are multifactorial, and several theories have been proposed regarding aseptic loosening, as shown in Figure (1 3). For many years, the dominant theory of aseptic loosening was the 6

24 dispersion of wear particles from the acetabular cup into the joint fluid. These particles activate macrophages, which can cause osteoclasts to initiate bone resorption. The resulting bone loss has the effect of enlarging the bone-implant interface, thereby exposing more mineral bone to the joint fluid and wear debris. This process leads to a gradual loosening of the implant [6]. Several studies have addressed the role of wear particles from cement, metal and polyethylene (PE) role in aseptic loosening [54, 61-63]. Metal-particle August (1986) [66] Stress shielding Engh and Bobyn (1988) [64] Sealed interface Schmalzried et al. (199) [6] Endotoxin Ragab et al. (1999) [67] Cement disease Harris et.al (1976) [59] and Willert (1977) [68] PE-particle Howie et al. (1988) [69] Micromotion Ryd and linder (1989) [7] 7 High fluid pressure Linder (1994) [71] and Aspenberg and Van der Vis (1998) [7] Individual variations Matthews et al. () Figure 1 3: Theories proposed over the years regarding aseptic loosening [54]. (Adapted with permission from Springer) Another theory that has been proposed for aseptic loosening is the remodeling of bone to adapt to new loading conditions emerging after surgery [54, 64]. When inserted into the femur, the implant carries the majority of the load, causing a reduction of stress in some regions of the femoral bone. Hence, the stress normally associated with controlling bone remodeling is not transferred to the adjacent bone, which leads to bone resorption [64]. Due to bone resorption, the bone mineral density (BMD) is reduced, and bone becomes more osteoporotic, which may open the bone-implant interface and ease the flow of joint fluid. This facilitates the ingress of wear particles, a process that contributes to induce implant loosening [65] [4, 18]. To prevent the progression of osteoporosis and aseptic loosening, it is thus important to preserve bone from resorption and maintain the natural bone mineral level in a patient femur as much as possible. Another factor contributing to implant loosening is the micromotion of the stem relative to the femur. Excess micromotion at the bone-implant interface may jeopardize implant stability. In addition, micromotion triggers the formation of an

25 undesired fibrous cortical tissue around the implant, thereby preventing osseointegration at the interface. The extent of micromotion can yield different outcomes on osseointegration. For example, for a low range of micromotion between -4 μm, osseointegration can still occur to stabilize the implant since both bone and the implant elastically deform at the interface [73, 74]. On the other hand, higher values of micromotion affect implant stability and may open the interface to joint fluid and wear particles, leading to the resorption of bone tissue and subsequent loosening of the implant. It has been demonstrated that an osseointegrated implant with a sealed interface may not be affected by wear particles, and the aseptic loosening may not occur [75]. The integrity of the bone/implant or bone/cement interface thus plays an important role on the survival of the implant. Sealing off the interface by creating porous surface, suitable for bone ingrowth, can effectively prevent wear debris from entering the interface and causing subsequent periprosthetic osteolysis [76, 77]. As a result, implants with smooth surface are more susceptible to wear debris, since these particles can be transported into the interface space by the pressure of joint fluid and can affect bone tissue around the prosthesis. The effect of high joint fluid pressure on implant loosening has also been a factor studied in literature [6, 71, 7]. It has been demonstrated that an oscillating fluid pressure can disturb the perfusion and the oxygenation of bone, and may result in osteolysis and osteocyte death. The resulting bone loss is a process that is believed to occur without the activation of macrophages, which usually takes place as a reaction of cells to foreign body. Other theories have been also proposed to explain the reasons of implant loosening, such as bacterial infection of an artificial joint, and individual and genetic variations. The presence of bacterial membrane on the implant surface containing endotoxin may trigger macrophage activation and initiate bone resorption [78]. It has been shown that removal of endotoxin from titanium particles decreases the particle-induced osteolysis by 5 7% [79]. Genetic factors may also play an important role in macrophages reaction to the wear particles [8]. It has been observed that the reaction of macrophages to the wear debris may vary 8

26 up to times in different patients [8]. Some patients may present barely noticeable wear particles but might experience rapid osteolysis with implant failure. Other patients, on the other hand, might have severe wear particles, but no signs of implant failure. The aforementioned studies of the causes of femoral implant failure can be used to identify the requirements for the design of orthopaedic hip implants, as shown next. 1.4 Implant design requirements The physical characteristics of an implant play a crucial role for the survival of both primary and revision THR. The materials used for bone replacement in shoulders, knees, and hips should meet important functional requirements [3]. These requirements include i) biocompatibility of the implant material with the body environment, ii) high wear and corrosion resistance of the articulating surfaces, iii) high implant stability in short and long term, and iv) preservation of bone tissue around the implant. A detailed description of each design requirement along with technologies currently available for the implant design is given here Biocompatibility The biocompatibility of the implant material and the surface conditions are two important factors for the acceptance of an implant in the body environment [3, 81]. Biocompatibility refers to the chemical, biological, and physical compatibility of the implant material with the host tissue. The material should possess superior corrosion resistance in the body environment. It should be non-toxic and should not cause any inflammatory or allergic reactions in the human body [3, 8, 83]. The material should also have high mechanical strength, fatigue, and wear resistance to prevent mechanical failure and perform for long-term under cyclic loading. For bone replacement applications, the implant surface should promote bone cell attachment, proliferation, migration, and differentiation. The surface should also have an interconnected porosity with suitable pore size to facilitate bone ingrowth and transport of nutrients and regulatory factors. Surface compatibility of an implant can be enhanced by applying a specific surface treatment, such as heat- 9

27 treating [3], coating a layer of apatite to increase bone cell attachment [84], and coating a porous surface [85]. Presently, the materials used for orthopaedic implant applications are 316L stainless steel, cobalt chromium alloys, and titanium-based alloys; however, titanium alloys are fast emerging as the first choice for majority of applications due to their excellent biocompatibility and elastic modulus closer to that of bone. Commercially pure Ti and Ti6Al4V are most commonly used titanium materials for implant applications. For long-term applications using Ti6Al4V, concerns on the toxic effect of vanadium and aluminum have led to the development of a second generation of titanium alloys with nontoxic alloying elements, such as Ta, Nb, Zr [3, 86]. The substitution of vanadium and aluminum by these alloying elements also changes the allotropic form of Ti6Al4V from to phase, which results in the decrease of elastic modulus of the material without compromising the strength [87]. This leads to the reduction of implant stiffness, and thereby has a positive influence on decreasing bone resorption. Extensive research has been pursued on alloys to understand the effect of alloying additions and heat treatment procedures on the evaluation of material microstructure, phase transformation, and elastic modulus. A large number of β-type titanium alloys with an elastic modulus range of 8-99 GPa have been developed, which can be used in different biomedical applications [86]. It should be mentioned that the improvement of biocompatibility or surface conditions of orthopaedic hip implants is not within the scope of this thesis High wear resistance As discussed in the previous section, the accumulation of volumetric wear debris resulting from the degradation of the cup surface often leads to osteolysis followed by aseptic loosening [54]. Consequently, there has been considerable effort to decrease the number of wear particles generated from the bearing surface to minimize the inflammatory response and increase the longevity of the arthroplasty. 1

28 Metal-on-polyethylene (MOP) bearings have been the most common bearings used in THR. In an attempt to improve the wear resistance of the polyethylene liner, highly cross-linked polyethylene (HXLPE) liners have been developed [88, 89]. Conventional ultra-high molecular weight polyethylene (UHMWPE) is exposed to gamma radiation to produce crosslinking polyethylene. The component is then re-melted and cooled slowly to extinguish free radicals while maintaining mechanical properties. An in-vitro study performed on a hip simulator showed a 9% wear reduction of the HXLPE compared with conventional UHMWPE [9, 91]. The issue regarding this type of cup is the relatively small size of the femoral head [9] which limits range of motion, especially of concern in young and active patients [93]. There has been renewed interest in metal-on-metal (MOM) bearings which have been shown to have 1 times lower volumetric wear than MOP articulations [94-96]. Subsequent reduction in osteolysis has also been reported in a number of published studies [63, 97]. The advantage of MOM is the large diameter of heads which creates greater head-neck ratio. This increases range of motion and reduces potential for impingement of the femoral neck on the acetabular component rim [98]. The main concerns with MOM bearings involve the elevation of serum and urine metal ion levels and the possible risk of carcinogenicity or teratogenic effects [99]. Many studies are currently being conducted to develop new materials or to obtain an optimal design for the acetabular cup [1-1]. The design of an acetabular cup to reduce wear is beyond the scope of this thesis Implant stability Initial implant stability is essential in ensuring surgery success and long-term implant performance. Failure to obtain initial implant stability leads to implant micromotion, a phenomenon that promotes fibrous attachment to the implant and thus prevents bone ingrowth. As a result, the implant fails or the patient may suffer hip pain, necessitating revision surgery. Implant stability can be quantified by the relative displacement and shear stress at the bone-implant interface. Although an exact threshold value of allowable interface micromotion is not known, it has been 11

29 suggested that in order to achieve bone ingrowth the designed implant should experience interface micromotion lower than 4 μm under daily activities [73, 74]. The control of interface stresses is also crucial for the implant performance. Excessive interface stresses may cause failure and permanent deformation of the supporting cancellous bone resulting in subsidence of the implant and failure of fixation [13-15]. Many studies have been performed to optimize the shape of implants for improved initial stability by minimizing interface stresses and micromotion [16-19]. Huiskes et al. [16] introduced a numerical procedure for the optimization of a cemented implant to determine the optimal prosthetic shape which minimize interface stresses. The results showed that 3-7% interface stress reduction can be obtained with an optimized stem shape. In another study, a three-dimensional shape optimization procedure was also used to obtain the optimal geometry of a hip implant that minimizes the relative displacement at the bone-implant interface [19]. All optimized shapes presented a collar component to avoid subsidence; however, these designs present surgical difficulties in terms of insertion into the femur. Despite being initially stabilized and osseointegrated, an implant might still fail as a result of mechanical failure, aseptic loosening, or patient related factors, as previously discussed. When joint revision is required, the success of the surgery strongly depends on the quality of the stock bone. Due to bone degradation around the primary implant, revision surgery requires the surgeon to bypass or reconstruct the remaining bone by opting for a larger and longer implant, as well as bone grafting with morselized, structural or synthetic bone grafts Preservation of bone tissue Bone loss is undesirable for both primary and revision surgery. For the former, stress shielding, severe or progressive, can result in serious clinical consequences such as periprosthetic fractures. In addition, increased intracortical porosity of the stress shielded bone can make it more susceptible to the ingress of particulate wear debris which can result in osteolytic bone loss and implant loosening [54, 77]. For the revision surgery, the loss of normal bone stock severely compromises the 1

30 bone s ability to adequately fix to the cementless revision implant and increases the risk of an intraoperative fracture. As a result, hip revision surgery has a lower chance of achieving successful implant fixation than a primary THR. a) b) Figure 1 4: The flutes and splines applied to a stem, b) the clothespin design and its lateral radiograph. (Reprinted with the permission from Current Orthopaedic Practice [4]) Over the last three decades, several alternative implant designs have been proposed to deal with the stress shielding effect and reduce its serious clinical consequences. Different parameters, including stem stiffness, stem material, porous coating level, ingrowth surface, and stem geometry, have been recognized to affect implant performance. One strategy for reducing stem stiffness is to decrease the second moment of area of the implant cross-section by a selective, focal removal of implant material from an existing stem [4]. Design features, such as slots or flutes, reduce the second moment of area and consequently the implant stiffness (Figure 1 4). Nevertheless, it has been reported that none of these designs has significantly reduced bone loss [4]. The reason is that in all of these designs, the material is removed from the middle or distal part of the stem, while the greatest stiffness mismatch between implant and host bone occurs proximally. Therefore, a stiffness reduction in the proximal stem is more likely to be effective in reducing stress shielding. To overcome the mismatch between the stiff stem and the more elastic bone, the concept of an isoelastic stem was introduced by Robert Mathys in 197s [11-11]. This concept was based on the assumption that bone and implant would deform as one unit to avoid stress shielding [113]. The implant was designed with a 13

31 layer of polyacetal resin with elastic modulus similar to that of bone, and with a stainless steel core to improve the mechanical strength. Although it was expected that the implant would perfectly match with surrounding bone, the results of a 15- year follow-up study revealed the poor performance of this prosthesis [111]. High micromotion with secondary debris production was demonstrated to be the main reason for the high failure rate. In another study, Harvey et al. [31] examined the effect of femoral stem flexibility on bone-remodeling and stem fixation in a canine total hip arthroplasty model. Two fully porous-coated stems of different stiffness were used in their study: a titanium-alloy stem and a composite stem. The composite stem was three to five times more flexible than the canine femur. The investigation demonstrated that despite the markedly greater flexibility of the composite stems, no significant difference could be detected in the overall degree of femoral stress-shielding between these stems and the stiffer ones. The reason was that the composite stems had more bone-implant interface motion than did the titanium-alloy stems, a factor that caused fibrous tissue ingrowth rather than osseointegration. Other advances in total hip replacements have used a porous material surrounding a fully dense material. Hip implants with a porous tantalum coating have been proposed in hip replacement surgeries [114]. The fabrication of tantalum foam begins with the pyrolysis of a thermosetting polymer foam to obtain a lowdensity carbon skeleton which has a repeating dodecahedron open cell with interconnected pores. Pure tantalum is then deposited on the carbon skeleton using chemical vapour deposition to create a porous metal construct. Tantalum foam is an excellent material for this application due to its biocompatibility, high volumetric porosity, and modulus of elasticity similar to that of bone. It has been used as a porous coating on the surface of acetabular cups (Figure 1 5) and femoral stems (Figure 1 c) for bone ingrowth. Completely porous tantalum augments are also designed to reconstruct bone stock in revision surgeries as shown in Figure 1 5. Additionally, porous tantalum implants have applications in revision and primary total knee replacement (TKR), the treatment of osteonecrosis of the femoral head, and spinal arthrodesis surgery. Due to the manufacturing process, the 14

32 microstructure of a tantalum foam implant has an almost uniform and random distribution of pore shape and size throughout the implant [6]. This material characteristic has been demonstrated incapable of solving the conflicting nature of the physiological phenomena occurring in an implant [115, 116]. Whereas the reduced stiffness of the foam decreases bone resorption, the uniform distribution of cells has the undesired effect of increasing the interface stresses. Figure 1 5: Multiple orthopaedic applications for porous tantalum. Top row: monoblock acetabulum, and a revision acetabular augment. Middle row: monoblock tibia, revision TKR augments, and salvage patella button. Bottom row: osteonecrosis implant, and spine arthrodesis implants (Courtesy of Zimmer, Warsaw, IN) [117]. (Reprinted with permission from Elsevier) 1.5 Conflict between implant stability and bone resorption It can be concluded from this survey of literature that current prosthesis with either i) fully-dense material [4, 64, 118, 119], ii) porous-coating [1-1], iii) 15

33 composite and isoelastic hip stems [11, 111, 13, 14] or iv) porous material with uniform pore distribution [115, 15], fail to simultaneously preserve bone stock, and minimize implant micromotion. The preservation of bone stock and the reduction of implant instability are two conflicting requirements, which current available implants are unable to optimally reconcile. A stiff implant stem shields the bone from necessary mechanical loading. On the other hand, decreasing the stem stiffness increases the interface shear stress between the implant and the bone, which increases the risk of interface motion and finally accelerates implant failure. The conflict existing between stress shielding and interface shear stress has also been identified in the seminal work of Kuiper and Huiskes [115, 116], who attempted to find a trade-off design of a bidimensional hip implant. Their numeric results showed that one solution to this issue is an implant whose material properties vary locally throughout the structure. Under a given loading condition, the strain energy of bone before implantation was considered as a reference value for the evaluation of bone resorption after implantation. If the strain energy of the implanted femur was lower than a threshold value, bone resorption was considered to occur. It was found that a non-homogeneous distribution of elastic properties within the hip stem could contribute to minimizing the probability of interface failure while concurrently limiting the amount of bone loss. In their approach, however, the solution of the multi-objective problem has been simplified, reformulated and solved with a single objective optimization strategy. As a result the whole set of trade-off designs could not have been captured. Hedia et. al. [16, 17] attempted to reconcile the conflicting nature of these objective functions by proposing to use three bioactive materials: hydroxyapatite, Bioglass, and collagen, to design a graded cementless hip stem. Although their implant design reduced bone resorption and bone-implant interface stresses, the use of such bioactive materials have limitations due to their brittleness and insufficient strength when applied to load-bearing applications [18-13]. In a more recent study, Fraldi et al. [131] applied a maximum stiffness topological optimization strategy to re-design a hip prostheses with the goal of reducing stress shielding in the femur. Following this method, elements with intermediate volume fraction (between and 1) are 16

34 penalized to limit their presence in the final solution. For regions with intermediate relative density, certain microstructures should be proposed to match those materials in terms of effective elastic properties. Laser micro-drilling is suggested to create the required micro-porosity, an option that can be used only on the implant surface, not throughout the implant. Moreover, the effect of these micropores on the mechanical integrity of the implant was not considered. Although many optimization procedures have been proposed for the design of an implant, none of these designs can simultaneously minimize bone resorption and implant instability. Furthermore, no systematic design methodology is available that takes into account the manufacturability and the mechanical strength of the implant under loading conditions. 1.6 Thesis rationale In the previous sections, all requirements for the design of orthopaedic hip implants, including i) biocompatibility of the implant material, ii) high wear resistance of the articulating surfaces, iii) high implant stability, and iv) preservation of bone tissue were discussed. In this thesis, we examine only two conflicting requirements: implant stability and preservation of bone tissue. We propose to design an orthopaedic hip replacement implant that can simultaneously minimize implant instability and bone resorption. To achieve this goal, we introduce a novel concept of a variable stiffness implant implemented through the use of a graded lattice material. The mechanical properties of the lattice implant are optimized to locally match the strain energy of the host femur to that of the femur before implantation in order to minimize bone resorption. In contrast to the design of current femoral implants, generally made of a fully solid material, this implant is completely porous with controlled periodic lattice microarchitecture displaying non-homogeneous distribution of material properties. Multiscale mechanics is applied for failure analysis of the implant, and additive manufacturing technique is considered for the fabrication of the design. 17

35 1.7 Practical feasibility of the implant concept: lattice materials via additive manufacturing While the idea proposed above to design a cellular implant with variable stiffness may improve the performance of current implants, the practical feasibility of manufacturing specifically designed cellular implants needs to be investigated. Here, additive manufacturing techniques, such as Electron-Beam Melting (EBM), Selective Laser Sintering/Melting (SLS/SLM), and Stereolithography Apparatus (SLA), are considered for the fabrication of cellular implants [13-136]. With SLA, the parts are created layer by layer through the solidification of photopolymeric resin exposed to an ultraviolet laser light tracing the pattern of each Figure 1 6: Rapid prototyping manufacturing technique, (a) SLA technique, (b) SLS or SLM technique, (c) EBM technique. 18

36 cross section. Then, the prototypes are used as master pattern for injection molding, blow molding, and other metal casting processes to obtain the final metallic part, while with EBM and SLM, parts are directly created by melting metal powder layer by layer. The manufacturing process in EBM and SLM is quite similar. The machine consists of an electron gun (for EBM) or a laser head (for SLM), a powder container, spreader, and a building table. At each step, a new layer of metal power spread on the building table, and the electron or the laser beam locally melt the material in accordance to the slice data generated from the input CAD file of the model. After melting of a layer, the building table is lowered by one layer thickness, and a new layer spread out and melted. This process is repeated till the part is completed. Parts are fabricated in a vacuum chamber to avoid chemical reaction between metal powder and oxygen and other chemical species available in the atmosphere. Figure 1 6 shows the schematic of these manufacturing processes. Various experimental studies have been performed to examine the material and mechanical properties of cellular titanium structures fabricated by rapid prototyping techniques [47, 13, ]. It has been found that these manufacturing processes allow for the manufacturing of cellular components with a high level of quality, accuracy and reliability. They are also capable of building graded cellular structures with an exceptional degree of control over the mechanical properties. 1.8 Objectives Multiscale mechanics is one of the main aspects required to design a cellular hip implant. Since the characteristic length of the implant microstructure is much smaller than the overall macro geometry of the component, modeling the entire geometry including each individual cell strut will result in a large model, which may be difficult and costly to solve. To overcome this challenge, the whole problem is decoupled into macro and micro problems using asymptotic homogenization (AH) theory [ ]. For a given lattice cell topology, we will obtain the effective mechanical properties, including effective elastic modulus and yield strength, to use for the analysis of macrostructure. That is, the cellular microstructure is substituted with a continuum material, which possesses equivalent material properties. To design a cellular hip implant, the structure should also resist 19

37 the repetitive loading originating from body movement. Therefore, the lattice microstructure should be designed against fatigue fracture. The objectives of the thesis are to: 1. Compute the effective moduli, yield and ultimate surfaces of a library of lattice cells for use in the design of cellular implants.. Perform a comparative study of different homogenization techniques to gain the insight into the level of accuracy that alternative homogenization techniques can attain and thus select the most appropriate for this work. 3. Develop a computational method for the fatigue design of lattice materials that can accurately capture the stress distribution within unit cells. 4. Optimize the shape of the unit cell by minimizing the curvature of its interior borders to smooth stress localization and improve its fatigue resistance. 5. Develop a methodology based on multiscale mechanics and multiobjective optimization for the design of cellular hip implants with variable stiffness. 6. Apply the computational method developed in step 3 for the design of cellular implant against fatigue failure. 7. Fabricate a set of proof-of-concepts of planar implants via EBM to demonstrate the manufacturability of the designs. 1.9 Structure of the thesis Several analytical and numerical schemes exist in literature to characterize the mechanics of cellular materials. Each one has its respective assumptions, advantages, and limitations that control the level of accuracy a method can provide. In this thesis, AH theory is selected among several numerical approaches for the multiscale analysis of the cellular implant due its rigorous mathematical foundation. In Chapter, the underlying assumption and the mathematical formulation of AH are discussed, and a comparative study on the accuracy of different multiscale mechanics approaches is performed, while AH is considered as the benchmark method. AH is first applied to determine the effective elastic moduli and yield strength of six lattice topologies for the whole range of relative density. With respect to the relative density, the results are then compared to those obtained with other methods available in literature. The results of this chapter on the

38 effective properties of lattice materials provide not only handy expressions of prompt use in multiscale design problems, but also insight into the level of accuracy that alternative homogenization techniques can attain. In Chapter 3, a numerical method based on AH theory is presented for the design of lattice materials against fatigue failure. The linear elastic material model used in chapter is extended to consider the effect of material nonlinearity as a result of plastic deformation. This elastic-plastic model is used to determine the endurance limit and the ultimate strength of lattice materials under multiaxial loading conditions. The method is applied to study the effect of unit cell shape on the fatigue strength of planar cell topologies, including square and hexagonal unit cells, under uniaxial tension or shear loading conditions. Optimum cell shapes are obtained by minimizing the curvature of the interior borders to smooth stress localization and improve their fatigue resistance. The results are also compared with experiments available in literature for the numerical validation. In Chapter 4, the methodology to synthesis a graded cellular implant is introduced. The two main requirements for the implant design, (i) implant stability in the short and long term, (ii) preservation of bone tissue around the implant, are translated into mathematical functions for the assessment of implant performance. The multiscale mechanics based on AH, discussed in Chapter, and a multiobjective optimization algorithm are integrated into the methodology to obtain the optimal material distribution throughout the implant to concurrently minimize bone resorption and implant interface failure. In Chapter 5, the fatigue design methodology, discussed in Chapter 3, is extended to model multiaxial loadings. The approach is then applied to the fatigue design of a graded cellular hip implant loaded under cycling forces of walking. The microstress distribution found with AH is also compared with that obtained from a detailed finite element analysis to verify the periodicity assumption of AH, and to assess the estimation of material yield in the lattice. Finally, a set of proof-ofconcepts of planar implants have been fabricated via Electron Beam Melting (EBM) to demonstrate the manufacturability of Ti6AL4V into graded lattices with alternative cell size. Optical microscopy has been used to measure the 1

39 morphological parameters of the cellular microstructure, including cell wall thickness and pore size, and compared them with the nominal values.

40 CHAPTER : Multiscale mechanics of cellular structures.1 Introduction Cellular materials are hybrid materials obtained by an interconnected network of solid struts or plates which form edges and faces and confine voids in cells [144]. Materials with cellular microstructure can be found very often in nature, such as wood, cork, human trabecular bone, honeycomb, and plant tissue. Taking inspiration from natural materials, artificial cellular materials have been developed with the aim of utilizing their outstanding properties, including high strength to weight ratio, and unique thermal, acoustic and energy-absorbing properties. They are widely used in many industrial applications where weight savings and multifunctional properties are critical. Aerospace sandwich panels, vibration and sound insulators, compact heat exchangers, and biomedical implants are a few examples [84, ]. Cellular materials can be loosely categorized with respect to the arrangement of the cells into either stochastic or ordered [147]. Foams and most natural structures fall in the first category, as their cellular structure has an intrinsic microscopic randomness. Lattice materials, on the other hand, have an ordered arrangement of cells that partition periodically a prescribed domain in one, two or three dimensions. They can be designed with superior mechanical properties compared to stochastic foams. The lattice cell topology performs the key role in its structural behaviour and can be tailored to construct materials with unprecedented mechanical properties. In cellular materials, the term material is usually used when the characteristic length of the component is several orders of magnitude higher than that of its constituent unit cell (e.g. individual cell wall length). To analyze these cellular components, a direct approach, involving the modeling of each cell strut individually, can be applied. However, this approach would result in considerably large models, which are computationally expensive and often impractical to 3

41 manage. A more successful approach is to substitute the cellular microstructure with a continuum material, which possesses equivalent material properties [148]. For this purpose, a representative volume element (RVE) is isolated and analysed to calculate its physical properties which represent those of the material at the macroscale. Several analytical and numerical approaches, as well as experimental investigations, have been proposed in the literature to determine the mechanical properties of cellular materials [ , ]. Noteworthy contributions in the area of cellular materials are those of Gibson and Ashby [144], Masters and Evans [149], Christensen [15], and Wang and McDowell [151], which provide closed-form expressions of the effective mechanical properties. These methods rely on certain premises. They generally assume the cell walls behave like Euler- Bernoulli beams, examine the individual cell wall and determine the effective mechanical properties of the cell by solving deformation and equilibrium problems. For simple cell topologies, macroscopic stresses are applied to the unit cell, and internal forces of cell struts are computed. The deformation and stress distribution throughout the cell walls are determined to compute the elastic stiffness and yielding strength of the material. Following this procedure, Gibson et al. [163] computed the in-plane elastic stiffness and strength of hexagonal unit cell, and Wang and McDowell [151] characterized six cell topologies shown in Figure 1. This procedure works well for topologies that have a simple arrangement of the cell members, but presents limitations if the geometry of the unit cell has a complex topology. Bending dominated Stretching dominated Mixed Square Hexagonal Kagome Triangular triangular A Figure 1: Common cell topologies Mixed triangular B 4

42 More recently, matrix-based techniques using the Bloch s theorem and the Cauchy Born hypothesis have been used to homogenize the properties of planar lattice materials [164, 165]. Hutchinson and Fleck [164] first formulated the microscopic nodal deformations of a lattice in terms of the macroscopic strain field, from which the material macroscopic stiffness properties are derived. A methodology was proposed to characterize cell topologies with a certain level of symmetry, e.g. the Kagome lattice and the Triangular-Triangular lattice. Elsayed and Pasini [165] and Elsayed [166] extended this method to deal with planar topologies that can possess any arbitrary geometry of the cell. Vigliotti and Pasini [167, 168] presented a more general matrix-based procedure for the analysis of arbitrary bidimensional and tridimensional cell topologies with open and closed cells. Other models have been proposed to model the cellular microstructure as an equivalent micropolar medium [ ]. In micropolar elasticity, in addition to the translational deformation, an independent microscopic rotational field is usually introduced [169, 17]. The displacement field at the cell joints is approximated by a Taylor series about the center of the unit cell, and the strain energy of the lattice is determined. Then, the parameters of the equivalent micropolar continuum are obtained by comparing the expression of the strain energy of the lattice with those of continuum model [15-155]. Because the Taylor expansion of displacement field is derived based on the displacement field of the unit cell center, the application of this approach is limited to cell topologies with a single internal joint and central symmetry. Among numerical approaches, asymptotic homogenization (AH) theory has been successfully applied to predict the effective mechanical properties of materials with a periodic microstructure [ , 171]. AH has been widely used not only for the analysis of composite materials and topology optimization of structures [17-175], but also for the characterization of porous materials, such as tissue scaffolds [158, ]. AH assumes that any field quantity, such as the displacement, can be described as an asymptotic expansion, which replaced in the governing equations of equilibrium allows to derive the effective properties of the 5

43 material [141, 14]. Verification with experimental results has shown that the effective mechanical properties predicted by AH are reliable and accurate [179-18]. Takano et al. [179] applied AH to analyze micro-macro coupled behaviour of the knitted fabric composite materials under large deformation conditions. The predicted deformed microstructures were compared with the experimental results, and a very good agreement was observed. In another study, to validate the accuracy of AH results, the predicted value of the effective elastic modulus of a porous alumina with 3.1% porosity was compared with the elastic constant measured from the experiment [18], and the relative error of 1% was found. Guinovart-Díaz et al. [181, 18] computed the thermoelastic effective coefficients of a two-phase fibrous composite using AH. The results were compared with different experimental results, and a good agreement was found. Despite being accurate, AH has not been applied for the analysis of lattice materials. Compared to other homogenization schemes, a noteworthy advantage of AH is that the stress distribution in the unit cell can be determined accurately and thus be used for an detailed analysis of the strength and damage of the materials [171, 183, 184]. Furthermore, AH has neither limitation on the cell topology nor on the range of relative density; essentially, AH can handle any lattice regardless of its relative density. Here, AH is applied for the characterization of lattice structures, shown in Figure 1, and the results are considered to serve as benchmark in comparative studies. As briefly described above, several methods exist in literature to model the mechanics of cellular materials. Each one has its own assumptions, advantages, and limitations. One central issue to the process of solving a given problem is the careful selection of the homogenization technique that is most effective in terms of accuracy and the most convenient as far as the computational cost is concerned. This task, however, presents often trade-offs, which are not always properly defined. For example, closed-form expressions, such as those in [144, 151, 166, 167], can be conveniently used to fast compute the effective properties of a lattice material; in addition, problems of accuracy might emerge if the microstructure does not respect the model assumptions. The hypothesis that cell walls behave like beam/rod elements ceases to provide reliable results for increasing values of 6

44 relative density. Moreover, the Euler-Bernoulli beams and rod elements cannot capture the deformation of the solid material at the cell joints, a problem that might affect the estimation of the yield strength of the material. Other techniques, on the other hand, have been proved to be accurate over the whole range of relative densities, but for certain problems they might have the drawback to require longer time of computation [184]. This represents a curb in large multiscale optimization problems of lattice materials, because the material properties must be iteratively evaluated several times. In this case, the trade-off solution would be to prefer a method which is computationally faster, as long as it is used in a range of relative density where the results are considered satisfactory within an acceptable range of error. It is thus essential to be able to contrast the validity of alternative homogenization schemes so as to select the most effective method to solve a given problem. This chapter is composed of three main parts. In the first part, the concept and fundamentals of AH is reviewed. The homogenization procedure for the characterization of cellular structures in terms of effective elastic constants and yield strength under multiaxial loading condition is described. For the second part, six lattice topologies (Figure 1), representative of either bending or stretching dominated behaviour, are selected and characterized under uniaxial and shear loading conditions. For three common cell topologies, square, hexagonal, and Kagome, AH is used to obtain yield surfaces under multiaxial loading condition. As part of the analysis, approximated closed-form expressions of the results are also given for a practical use in multiscale analysis and design problems of cellular materials [145, 185, 186]; in particular, they might serve for i) rapid calculation of the effective mechanical properties of lattice materials, ii) validation purposes of experimental data [16, 18], and iii) topology optimization problems, as described in the work of Hassani and Hinton [17], and Liu et. al [187]. In the last part, a comparative study on the accuracy of classical and more recent homogenization techniques [144, 151, 166, 167] is presented; for a given range of density, the relative error of the properties predicted by each technique is discussed with respect to those obtained with AH. 7

45 . Asymptotic homogenization (AH) method The deformation and failure mechanisms of a structure with heterogeneous material can occur at both macroscopic and microscopic scales. In a full scale simulation, each heterogeneity is explicitly modeled at the microscale to obtain a high level of accuracy. The computational effort, however, can be very lengthy and time-consuming. As an alternative, the microstructure can be replaced by a homogeneous medium, and the mathematical theory of homogenization can be used to characterize the mechanical behavior of heterogeneous media [184]. As shown in Figure, a body traction t at the traction boundary boundary with a periodic microstructure subjected to the t, a displacement d at the displacement d, and a body force f can be replaced by a homogenized body with the prescribed external and traction boundaries applied to details and voids of the local coordinate system., without geometrical Figure : Homogenization concept of a cellular structure In general, a representative volume element (RVE) is analyzed, and the effective mechanical properties of material at the macroscale are obtained. Here, AH is applied to determine the effective stiffness and the mechanical strength of the cell topologies under investigation [171]. The underlying assumption of AH is that each field quantity depends on two different scales: one on the macroscopic level x, and the other on the microscopic level, y=x/ε. ε is a magnification factor that scales the 8

46 dimensions of the unit cell to the dimensions of the material at the macroscale. In addition, field quantities, such as displacement, stress, and strain, assumed to vary smoothly at the macroscopic level, are periodic at the microscale [14, 188]. Based on AH, each physical field, such as the displacement field, u, in a porous elastic body, can be expanded into a power series with respect to ε: u ( x) u ( x, y) u ( x, y) u ( x, y) (.1) 1 where the functions u, u 1, u are Y-periodic with respect to the local coordinate y, which means they yield identical values on the opposing sides of the unit cell. u 1 and u are perturbations in the displacement field due to the microstructure. u can be shown to depend only on the macroscopic scale and to be the average value of the displacement field [14]. Taking the derivative of the asymptotic expansion of displacement field with respect to x and using the chain rule allows the small deformation strain tensor to be written as: 1 ( u ) ( ) u u u u O x y T T (.) where () x and () y are gradient of the field quantity with respect to the global and local coordinate systems, respectively. Neglecting terms of O( ) and higher, the following strain tensors can be defined: where ε( ) T ( u) u u * ( ) ( ) ( ), u u u * 1 T ( u) u1 u1 1 * u is the average or macroscopic strain, and ε ( ) x y (.3) u is the fluctuating strain varying periodically at the microscale level. Substituting the strain tensor into the standard weak form of the equilibrium equations for a cellular body with the pertinent geometrical details of voids and cell wall, the following equation is obtained [188]: ε ( ) ε 1 ( ) T * E ε( ) ε ( ) t T v v u u d v d (.4) t

47 where Eis the local elasticity tensor that depends on the position within the RVE, ε ( ) 1 v and ε ( v) are the virtual macroscopic and microscopic strains, respectively, and t is the traction at the traction boundary t. Being the virtual displacement, v may be chosen to vary only on the microscopic level and be constant on the macroscopic level. Based on this assumption, the microscopic equilibrium equation can be obtained as: T 1 * ε ( v) E ε( u) ε ( u) d (.5) Taking the integral over the RVE volume (V RVE ), equation (.5) may be rewritten as: ε 1 ( ) T E ε * ( ) 1 ε ( ) T v u dv E ε( ) RVE v u dvrve (.6) V RVE V RVE The above equation represents a local problem defined on the RVE. For a given applied macroscopic strain, the material can be characterized if the fluctuating * strain, ε ( u ), is known. The periodicity of the strain field is ensured by imposing periodic boundary conditions on the RVE edges (Figure 3); the nodal displacements on the opposite edges are set to be equal [188, 189]. Equation (.6) can be discretized and solved via finite element analysis as described in [141, 143, 183, 188]. For this purpose, equation (.6) can be simplified to obtain a relation between the microscopic displacement field D and the force vector f as: where K is the global stiffness matrix defined as: K m e k e1 KD f (.7) T e, k B EB e e dy (.8 a,b) Y 3

48 u v w u v w u 1 v 1 w1 A A Figure 3: Periodic boundary conditions for a pair of nodes located on the opposite surfaces, A and A, of the RVE. m being ( ) the finite element assembly operator, m the number of elements, B e1 the strain-displacement matrix, and in equation (.7) is expressed as: e Y the element volume. The force vector f m e e f f, f BEε( u) e e1 Y dy e (.9) Equation (.9) can be either used for the linear elastic analysis of microstructure or to model the effect of material nonlinearity as a result of elastoplasticity deformation of the unit cell. The material yield strength and the effective elastic modulus can be characterized by the linear analysis of the microstructure, while the ultimate strength of the material can be obtained through the elastoplasticity analysis. A detail discussion on elastoplasticity analysis of the microstructure is provided in Chapter 3. Considering the assumption of small deformation and elastic material behavior, the solution of equation (.7) leads to a linear relation between the macroscopic u and microscopic ε( u ) strain through the local structural tensor ε( ) ( u) M ( u) 31 M : (.1)

49 For a two-dimensional case, three independent unit strains are required to construct the M matrix. T 1, 1 T, 1 T (.11) 11 1 The macroscopic strains are applied to equation (.9) to obtain the force vector for the computation of microscopic displacements through equation (.7). Using the * strain-displacement matrix B, the fluctuating strain tensor ε ( u ) is determined and used to calculate the microscopic strain tensor ε( u ) through equation (.3). The local structural tensor M can then be obtained at the element centroid by solving three sets of matrix equations (for D) once ε( ) u and ε( ) u are known. Here, since three independent unit strains are considered, each column of the matrix M is the microscopic strain tensor ε( u ) obtained. The effective stiffness matrix can be simply derived by the taking the integral of the microscopic stress over the RVE and dividing by the RVE volume: 1 dv V RVE V (.1) RVE EM RVE from which the effective stiffness matrix E H can be defined as H 1 E E M dv V RVE V (.13) RVE RVE The homogenized stiffness matrix relates the macroscopic strains to the macroscopic stresses of the homogenized material. AH can also allow obtaining the macroscopic stresses that lead to the microscopic yield, or the endurance limit, as well as fracture of the lattice. To calculate the yield strength of the unit cells, the microscopic stress distribution can be obtained from the multiaxial macroscopic stress using the following equation: 3

50 H 1 EME (.14) The von Mises stress distribution at the microstructure can then be used to capture the yield surface of the unit cell expressed as follow: max ( ) (.15) y ys vm 33 where y is the yield surface of the unit cell, ys is the yield strength of the bulk material, and () vm is the von Mises stress corresponding to the applied macroscopic stress. To calculate the yield strength of the unit cells under uni-axial tension in x and y directions and the pure shear loading condition, the macroscopic unit stresses can be applied to equation (.15)..3 Effective mechanical properties of cellular structures by AH For the representative lattices in Figure 1, we apply AH to obtain the effective stiffness and yield strength as a function of the relative density. To capture the effective characteristic of each cell topology, regardless of the solid material, the properties are normalized with respect to the base material. A parametrized geometry of the unit cell is created assuming cell walls with uniform thickness, which in turn is varied at discrete values to span the whole range of relative density. ANSYS (Canonsburg, Pennsylvania, U.S.A) is used to build, mesh, and solve the D problem of the lattice material, which is modelled with planar eightnode elements (Plane 8). Figure 4 illustrates the homogenized elastic constants of the cell topologies as a function of relative density. As can be seen, the effective Young s modulus, shear modulus, and Poisson s ratios converge to the elastic constants of the base solid material as the relative density reaches one. Since hexagonal, Kagome, and triangular cell topologies are elastically isotropic in their plane, the Young s modulus is equal in both x and y directions. This applies also to the other cell topologies, which have orthotropic symmetry. As can be also seen in Figure 4, the square cell has a superior elastic stiffness due to the alignment of the cell walls in the loading direction, but it exhibits very low stiffness under shear loading as a

51 result of cell wall bending. While the hexagon has also low Young s and shear moduli, its Poisson s ratio is high, as shown in Figure 4b. The yield strength of the cell topologies is also obtained as a function of relative density, as shown in Figure 5. The results were obtained by applying three loading conditions (uniaxial load in x and y directions, and shear load) to determine the maximum von Mises stress in the cell walls and the yield strength of each lattice. For the hexagonal, Kagome, and triangular lattice, the yield strength is dissimilar in the x and y direction, unlike the other in-plane elastic properties. For the relative density equal to one, the material is fully dense with yield strength equal to that of its solid material. A common feature in the plots of Figure 5 is a sudden decrease of the effective yield strength for decreasing values of relative density. The reason is the presence of stress concentration at the cell joints, which locally increases the level of stress. As illustrated in Figure 5, the material yield strength decreases of about 6% at relative density of 99%. As discussed in Chapter 3, the mechanical strength of the lattice can be significantly improved by optimizing the cell shape and removing the curvature discontinuity at the joints. While the yield strength provided in Figure 5 is limited to the uniaxial and shear forces, many applications require the material to withstand multiaxial loadings. We thus obtain here the yield surfaces for alternative combinations of macroscopic stress. Being in linear elasticity, the location of the yield point on the yield surface of each lattice is obtained by multiplying the macroscopic stress with the ratio of the material yield strength and the maximum von Mises stress. 34

52 (a) Effective Young s and shear modulus, E x /E and s, E Poisson s y /E s, G/E ratio s, s (c) E x/e/ E s, se, y /EE y s / Es G/E / E s s / s s (b) Effective E * Young s /E, x s E* /E and, shear y s G* /E modulus s (d) E * /E / E, x s E*, /E E (AH) / E y s x s y s / s G * /EE s (AH) s (AH) / s Effective Poisson s / s s ratio Effective Young s and shear modulus, E x /E and s, E Poisson s y /E s, G/E ratio s, s (e) Effective E * /E Young s, x s E* /E and, shear modulus, and Poisson s y s G* /E ratio s E x/e/ E s, se, y /E Ey s / Es G/E / E s s / s s E x/e/ s E, se, y /EE y s / Es G/E / E s s / s s Effective E * /E Young s, x s E* /E and, shear modulus, and Poisson s y s G* ratio /E s Effective E * /E Young s, x s E* /E and, shear modulus, and Poisson s y s G* /E ratio s Figure 4: Effective elastic constants as a function of relative density for the cell topologies: (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B (f) E * /E / E, x s E*, /E E (AH) / E y s x s y s / s G * /EE s (AH) s (AH) / s E * /E, s E* /E x / Es, E / y sy Es G * /E / E s s / s s 35

53 (a) Effective y /, xx uniaxial ys y /, yy and ys y shear / xy ys strength (c) Effective y /, xx ys uniaxial y /, yy ys and y / xy shear ys strength (e) y y, xx ys y y xx / ys, / yy ysyy ys y y / xy xy ys ys / y y xxyy ys ys y y yyxx / ys ys y y / xy xy ys ys (b) Effective * / uniaxial, x ys * / and, y ys * shear / strength xy ys (d) Effective * / uniaxial, x ys * / and, y ys * shear / strength xy ys (f) 1. * y (AH) xyy / ys ys * y.8 y xx / (AH) ys ys * y / (AH) xy ys * y (AH) xyy / ys ys * y.8 y xx / (AH) ys ys * y / (AH) xy ys.6.4. xy xy ys ys Effective * / uniaxial, x ys * / and, y ys * shear / strength xy ys * y, x ys * (AH) y /, / y ys xx ys yy ys * y / (AH) xy ys xy ys Figure 5: Yield strength as a function of relative density for: (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B cells Figures 6 to 8 show the yield surfaces normalized with respect to the yield strength in the uniaxial and shear directions for the hexagonal, square, and Kagome lattices at given relative density. In particular, Figures 6 and 7 refer to the hexagonal and square lattice for the relative density of 5%, and Figure 8 pertains to the Kagome cell for the relative density of 3%. We selected 3% for 36 Effective * / uniaxial, x ys * / and, y ys * shear / strength xy ys * y, x ys * (AH) y /, / y ys xx ys yy ys * y / (AH) xy ys xy ys

54 the Kagome, because for a.5 relative density the base material fills almost completely the triangular voids, and thus the Kagome structure cannot be realized. (a) xx y xx xy y xy yy y yy (b) yy y yy m m xy y xy xx y xx (c) 1. xy y xy (d) 1. xy y xy xx y xx yy y yy Figure 6: Yield surface of an hexagonal cell honeycomb under combined inplane stress state ( x, and xy ) for a relative density 5%. y As can be seen in Figures 6 to 8, the yield surfaces are controlled by the shape of the unit cell. For example, the yield surface of the hexagonal cell (Figure 6b) resembles a parallelogram. For design purposes, it is often convenient to resort to closed-form expressions that can approximately describe the geometry of a yield surface. For this reason, Table 1 lists the functions along with relative fitting parameters of the yield surfaces for the unit cells here under investigation. For the hexagonal cell (Figure 6b), m 1 and m are the slopes of the parallelogram lines expressed as a function of relative density. For the Kagome cell, the surface can be also approximated as a parallelogram, while for the square cell, a pyramid with an elliptical base has been used to resemble the yield surface. 37

55 * F (Table 1) governs both the slenderness ratio and the inclination of the major xy y y y axis of the elliptical base. The parameters,, (Table 1) are the yield xx yy xy strength of the unit cell under uni-axial and shear stresses, which can be obtained from Figure 5 as a function of relative density. The results illustrated in Figures 4 to 8 and the expressions given in Table 1 can be handily used in multiscale design and topology optimization problems of cellular materials [145, 17, ], where the effective mechanical properties are required as a function of relative density. They can also be used to generate an efficient data base for the validation of experimental results, as recently suggested in [16, 18]. (a) xx y xx y (b) xy xy yy y yy yy y yy xy y xy xx y xx (c) 1. xy y xy / xy (d) y xx 1. xy y xy xx y xx yy y yy Figure 7: Yield surface of a square cell topology under combined multiaxial macroscopic stress state ( xx, and yy xy ) for a relative density 5%. 38

56 (a) xx y xx (c) xy y xy xy y xy 1 (b) yy y yy (d) 1..5 yy y yy xy y xy xy y xy xx y xx xx y xx yy y yy Figure 8: Yield surface of a Kagome cell honeycomb under combined in-plane stress state ( x, and xy ) for a relative density 3%. y.4 Comparative analysis of homogenization schemes and discussion Although AH has been proved to have no inherent limitation on the cell topology and to provide consistent results for the whole range of relative density, its main drawback is often considered to be the computational cost. This can be high if the problem at hand is complex and of multiscale nature, as well as if it contains a large number of variables [184]. In such instances, less precise but faster methods might be preferred. This section discusses the validity and compares the accuracy of classical and more recent homogenization techniques [144, 151, 166, 167], with respect to AH. 39

57 Table 1: Yield surfaces as a function of relative density for square, hexagonal, and Kagome unit cells Yield function a yy xx xy 1, y y y xx m1 yy xy max b yy xx xy m 1 y y y yy xx xy xx * xx yy yy xy F y xy y y y y xx xx yy yy xy.3 for.3 1 m1.8e 3.3 Constant a, b 4 a 1, b 8 m * ( ) ( )( ) ( ) (1 ) F xy a a for xx yy.6 1 a 1.5 xy 1, y y y xx m1 yy xy.6 b for max.6 1 b 3 b yy 6.9 xx xy m 1.6 m1 1.15e y y y for yy xx xy.6 1 m m.35 for m.3783 The closed-form expressions of the effective elastic modulus and yield strength obtained via alternative methods [144, 151, 166, 167] are given in Table. For the cell topologies shown in Figure 1, the effective mechanical properties are contrasted in Figures 9 and 11 for relative density lower than.3 (. 3 ), where the assumption of Euler-Bernoulli beams, considered by the selected methods, holds. Figure 9 shows that by decreasing the relative density, the effective elastic modulus converges to the values obtained with AH. The mathematical expressions generally underestimate the effective elastic constant, with the exception of the Poisson s ratio of the hexagonal cell obtained by Gibson and Ashby [144]. We note that in the study by Wang and McDowell [151], the cell walls of stretching dominated cells were modelled as rod elements, while Elsayed [166], and Vigliotti and Pasini [167] considered cell wall as beam elements. As can be seen, whereas for stretching dominated lattices the contribution of bending moments on the effective elastic constants is almost negligible, this is not the case for bending dominated cells, such as the hexagon. Figure 9b shows that if the 4

58 effect of axial deformation is modelled in the formulation, as in [167], lower values of the effective Young s and shear modulus can be observed. Figure 1 shows the relative error of each effective property normalized with its respective one obtained via AH. As can be seen, the assumption of modelling cell walls as beams or rod elements yield error less than 1% for stretching dominated lattices. In contrast for bending-dominated lattices, the error can be much higher. For example for the square cell, the error of the effective shear modulus calculated by Wang and McDowell [151] at. 15 can be as high as 4%. Figure 11 illustrates the yield strength of the cell topologies under uni-axial and pure shear stresses. The results obtained by asymptotic homogenization are compared with those obtained from the analytical expressions listed in Table [144, 151, 167]. As can be seen in Figure 11, the analytic expressions converge to the AH results at low range of relative density. The yield strength of the cell topologies is overestimated, and the deviation increases with increasing relative density. Similarly to Figure 1, Figure 1 shows the relative errors between the yield strengths obtained by AH and the analytical expressions. Compared to the range of error in Figure 1, we observe here that the assumption of beam/rod elements for the cell walls is less accurate for the yield strength. For example, the effective elastic constants of stretching dominated can be estimated with an error less that 1% for. 5. For the yield strength, on the other hand, the closedform expressions are overpredictive, since the assumption of rod element for the cell walls cannot capture any stress localization. This observation is critical and cannot be neglected if the lattice material is to be designed to withstand cyclic loading. In these cases, the accuracy can be improved if model refinements are implemented. For example, for the Kagome and triangle cells (Figures 1c and d), if cell walls are modelled as beam elements, as opposed to rod, the error considerably reduces; for instance, for 3% relative density an error decrease of more than % can be obtained. Furthermore, for the hexagonal cell (Figure 1b) the results of the yield strength at low relative density show that the analytic 41

59 expressions can be more accurate if axial forces, besides bending moments, are modelled in the cell walls. Yet for higher relative density, the relative error rises since the deformation of the solid material at the cell joints cannot be captured. In particular for the hexagon, the relative error of the yield strength is more than % for. 1. The results given in this section, therefore, provide insight into the accuracy of different homogenization techniques that can be used to predict the effective properties of lattice materials. The results show that the Euler-Bernoulli beam assumption of cell walls can capture the effective elastic moduli of lattice materials with an error less than 1% for.5. However, the effective mechanical strength can be estimated with an error less than % but only for low range of relative density (.1). Therefore, if the material is designed only for effective elastic moduli, analytical approaches, discussed above, can be used as a cost-effective technique for the analysis, design, and optimization of lattice materials. However, the Euler-Bernoulli beam assumption fails to accurately capture the stress and strain distribution within the lattice cell walls. AH, on the other hand, has the capability to evaluate both macroscopic constitutive behavior and microscopic distributions of stress and strain with high accurately. As a result, in this thesis, AH is selected for the analysis of lattice materials. 4

60 Effective E * /E Young s, x s E* /E and, shear y s G* /E modulus s (a) Effective E * /E Young s, x s E* /E, modulus and Poisson s y s ratio s (c) (e) Effective E * /E Young s, x s E* /E and, shear modulus y s G* /E s E * /E / E, x s E*, E /E / [6] E [6] y s x s y s G * /EE s [6] / s / s s [6] E * /E / E,, x s E* E /E / E (AH) (AH) y s x s y s G * /EE s (AH) / s s (AH) / s s [6] / s s [3, 4] / s / s (AH) E * /E / E, x s E*, E /E / E [6] [6] y s x s y s G * /EE s [6] / s E * /E, s E* /E [3, 4] x / Es, E / y y E [3, 4] s s G * /EE s [3, 4] / s E * /E / E, x s E*, E /E / E (AH) (AH) y s x s y s G * /EE s (AH) / s s [6] / s s [3] [3, 4] / s s (AH) / s E * /E / E, x s E*, E /E / E [6] [6] y s x s y s G * /EE s [6] / s E * /E / E, x s E*, E /E / E [3] [3, 4] y s x s y s G * /EE s [3] [3, 4] / s E * /E / E, x s E*, E /E / E (AH) (AH) y s x s y s G * /EE s (AH) / s Effective Poisson s ratio s Effective Poisson s ratio Effective G shear * /E s modulus Figure 9: Effective elastic constants for (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B, for relative density below.3 (. 3). The closed-form expressions of the effective elastic constants obtained by Gibson and Ashby [144], Wang and McDowell [151], Elsayed [166], and Vigliotti and Pasini [167] are plotted. (b) E * /E, x s E* /E, y s G* /E Effective Young s and shear s modulus (d) Effective E * /E Young s, x s E* /E and, y s G* shear /E modulus s (f) Effective E * /E Young s, x s E* /E and, y s G* shear /E modulus s.6 E * /E / E, x s E*, E /E / [3] E [3] y s x s y s G * /EE s [3] / s s [3] / s E * /E / E, x s E*, E /E / [4] E [4] y s x s y s G * /EE s [4] / s s [4] / s E * /E / E, x s E*, E /E / (AH) E (AH) y s x s y s G * /EE s (AH) / s s (AH) / s s [6] / s s [3, 4] / s s (AH) / s E * /E / E, x s E*, E /E / [6] E [6] y s x s y s G * /EE s [6] / s E * /E, s E* /E x / Es, E / [3, 4] y y E [3, 4] s s G * /EE s [3, 4] / s E * /E / E, x s E*, E /E / (AH) E (AH) y s x s y s G * /EE s (AH) / s s [6] / s s [3] [3, 4] / s s (AH) / s E * /E / E, x s E*, E /E / E [6] [6] y s x s y s G * /EE s [6] / s E * /E / E, x s E*, E /E / E [3] [3, 4] y s x s y s G * /EE s [3] [3, 4] / s E * /E / E, x s E*, E /E / E (AH) (AH) y s x s y s G * /EE s (AH) / s Effective Poisson s s ratio Effective Poisson s s ratio Effective Poisson s ratio 43

61 Error (%) Wang and McDowell [6] Elsayed [3] Error (%) Wang and McDowell [6] Elsayed [3] Error (%) Wang and McDowell [6] Elsayed [3], Vigliotti and Pasini [4] { { { { Error (%) Wang and McDowell [6] Elsayed [3], Vigliotti and Pasini [4] Error (%) { { { { Wang and McDowell [6] { Error (%) Gibson et al. [3] Vigliotti and Pasini [4] { { (a) (b) 3 E * /E, s E* /E x / Es, E y y / E s s G * //EE s / s s 3 E * /E, s E* /E x / Es, E y y / E s s G * //EE s / s s Error % 1 Error % 1 (c) (d) E * /E, s E* /E x / Es, E / y y E s s G * //EE s 3 E * /E, s E* /E x / Es, E / y y E s s G * //EE s / s s / s Error % Error % 1 1 (e) (f) E * /E, s E* /E x / Es, E / y y E s s G * //EE s 3 E * /E, s E* /E x / Es, E / y y E s s G * //EE s / s / s Error % Error % Figure 1: Relative error between the effective elastic constants obtained via the closed-form expressions given in [144, 151, 166, 167] and those obtained by asymptotic homogenization for different cell topologies, (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B, for the range of relative density lower than.3 (. 3 ). 44

62 Effective uniaxial strength Effective uniaxial and shear strength Effective uniaxial and shear strength (a) (c) (e) * y, x ys * y /, [6] / [6] y ys xx ys yy ys * y / [6] [6] xy ys xy ys * y, x ys * y /, (AH) / (AH) y ys xx ys yy ys * y / (AH) xy ys xy ys * y / [6] [6] x ys xx * y / [6] [6] y ys * y / [6] [6] xy ys * y / [4] [4] x ys xx * y / [4] [4] y ys yy * y / [4] xy ys xy yy xy ys ys ys ys ys ys * y /, x ys *, y, / y ys *, [6] y / [6] xy ys * y, x ys * y /, (AH) / (AH) y ys * y / (AH) xy ys * y / (AH) (AH) x ys * y / (AH) (AH) y ys * y / (AH) xy ys xx yy xy xx ys yy ys xy ys xx ys yy ys xy ys ys ys ys Effective shear strength Figure 11: Yield strength as a function of relative density lower than.3 (.3 ) for: (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B. The closed-form expressions plotted in figure are those obtained by Gibson and Ashby [144], Wang and McDowell [151], and Vigliotti and Pasini [167]. (b) Effective uniaxial and shear strength (d) Effective uniaxial and shear strength (f) Effective uniaxial and shear strength * y, x ys * y /, [3] / [3] y ys xx ys yy ys * y / [3] [3] xy ys xy ys * y / [4] [4] x ys xx ys * y / [4] [4] y ys yy ys * y / [4] xy ys xy ys * y / (AH) (AH) x ys xx ys * y / (AH) (AH) y ys yy ys * y / (AH) xy ys xy ys * y / [6] [6] x ys xx * y / [6] [6] y ys * y / [6] [6] xy ys * y / [4] [4] x ys xx ys * y / [4] [4] y ys yy ys * y / [4] xy ys xy yy xy ys ys ys ys * y, x ys * y /, [6] / [6] y ys xx ys yy ys * y / [6] [6] xy ys xy ys * y, x ys * y /, (AH) / (AH) y ys xx ys yy ys * y / (AH) xy ys xy ys * y / (AH) (AH) x ys xx * y / (AH) (AH) y ys yy * y / (AH) xy ys xy ys ys ys 45

63 Error (%) Wang and McDowell [6] { Error (%) Wang and McDowell [6] Error (%) Wang and McDowell [6] Vigliotti and Pasini [4] { { Error (%) Wang and McDowell [6] Vigliotti and Pasini [4] Error (%) Wang and McDowell [6] Error (%) Gibson and Ashby [3] Vigliotti and Pasini [4] { { { { { { (a) 1 * y / xxx ys ys (b) 1 * y / xxx ys ys 8 * y y yy / ys ys * y / xy ys ys 8 * y yyy / ys ys * y / xy ys ys 6 6 Error % 4 Error % 4 (c) * y / xxx ys ys * y / yyy ys ys * y / xy ys ys (d) * y xxx / ys ys * y yyy / ys ys * y / xy xy ys ys Error % 6 4 Error % 6 4 (e) * y xxx / ys ys * y yyy / ys ys * y / xy ys ys (f) * y xxx / ys ys * y yyy / ys ys * y / xy ys ys Error % 6 4 Error % Figure 1: Relative error between the yield strength obtained via the closed-form expressions given in [144, 151, 167] and those obtained by asymptotic homogenization for (a) square, (b) hexagonal, (c) Kagome, (d) triangle, (e) mixed square/triangular A, and (f) mixed square/triangular B, at density lower than.3 (.3 ). 46

64 Table : The expressions of the effective mechanical properties obtained in literature for various cell topologies Unit cell Study Wang and McDowe ll [151] Gibson and Ashby [144] Vigliotti and Pasini [167] Wang and McDowe ll [151] Elsayed [166] Relative density ( ) E * x E s E * y or E t l t 3l 3t l / s 1 9 / 4 3 (1 / 6) 3 / 6 G * E s 3 16 * s /8 1 3 / 4 s 1 3 / 4 (1 9 / 4) s * x s 3 (1 4 /3) 8 ys * y or 3 6 or 3 6 or ys * xy ys / s (3 / 6) Vigliotti and Pasini [167] Wang and McDowe ll [151] Elsayed [166] Vigliotti and Pasini [167] 3t l (1 / 6) 3 / 6 3 (1 /1) 3 /1 (1 /1) 3 /1 1 / 6 (1 4 /3) 8 (3 / 6) s s 3 (1 /1) 8 (1 / 6) or 3 / 6 (1 / 6) / 6 or (1 / 6) 3(1 ) 3 1 /1 - - s (3 /1) 1 /1 (1 /1) 8 (3 /1) s (1 /1) or 3(1 /36) (1 /1) 3 / /1 (1 /1) (1 /1) 3(1 ) ( / 3) Wang and McDowe ll [151] Elsayed [166] ( )t 4 l 7 (148 9 ) (73 11 ) s ( 1) ( ) s Wang and McDowe ll [151] Elsayed [166] ( )t l 1 4( 1) s ( 1) 7 4 (148 9 ) s ( ) 1 ( )

65 .5 Conclusion In the first part of this chapter, the basic concept and the mathematical formulation of asymptotic homogenization method is presented. In the second part, AH has been applied to characterize six cell topologies of either stretching or bending dominated behaviour. The effective elastic modulus and yield strength have been obtained for the whole range of relative density. The yield surfaces for multiaxial loading have been provided for three common cell topologies, i.e. square, hexagonal, and Kagome, and closed-form expressions approximating the geometry of the yield surfaces have been given for handy use in applications involving multiscale analyses and design problems of cellular materials. In the last part of the chapter, AH results have been compared with those obtained with the other homogenization schemes available in literature. For stiffness and strength properties, the relative error of each method with respect to AH has been also contrasted. The results have shown that closed-form expressions can capture the effective elastic constants with higher accuracy compared to the effective yield strength. Material deformation and stress localization at the cell joints cannot be captured by modelling cell walls as beam or elements; thus such mathematical expressions significantly overpredict the yield strength of a lattice material. For instance, the effective elastic constants of stretching dominated lattices can be estimated with an error below 1% for. 5, whereas the initial yield strength might yield relative errors less than % only for.1. These results can contribute to gain insight into the level of accuracy that a homogenization scheme might offer, and help to select the most accurate and costeffective technique for a given problem. In the next chapter, a numerical method based on AH theory is presented for the design of lattice materials against fatigue failure. The method is applied to study the effect of unit cell shape on the fatigue strength of hexagonal and square lattices. 48

66 3 CHAPTER 3: Design of lattice materials against fatigue failure using asymptotic homogenization 3.1 Introduction In several applications, the load applied to a lattice material is far from being static. The recurring waves on a ship hull, the aero-acoustic excitation of a turbine engine, and cyclic loadings acting on an orthopedic implant during walking are all classical examples of time-dependent loads [19-195]. A cyclic load has generally a detriment impact on the fatigue resistance of a lattice material. Factors that govern fatigue failure are not limited to the alternating and mean stresses, load frequency, environment conditions, and macroscopic form of the structure, but include also the geometry of the microstructure, i.e. the geometry in which the unit cell of the lattice is shaped. If geometric discontinuities are embedded at either the macro or/and the microscale, then a severe drop in the fatigue resistance is observed. From literature, it appears that the study of fatigue failure in cellular materials, with both random (foam) and periodic arrangement of cells (lattice), has received less attention than their monotonic quasi-static and dynamic properties [144, 146, 149, 151, ]. Among the investigations, both experimental and theoretical approaches have been developed for metallic and polymeric cellular materials under fatigue conditions. The experimental works are predominant and mainly stem from studies on foams [-7], which aim at determining the stress-life curves under shear and axial loadings. For example, Burman [6] experimentally characterized the fatigue behavior of sandwich panels with polymeric PVC and PMI foam cores. The effect of the alternating strain on the fatigue life of polymeric cellular materials was shown to be more significant than that of the mean stress. It was also observed that reversible cyclic loadings, with negative max to min stress ratio, significantly reduce the fatigue resistance of polymeric foams. 49

67 In another work, closed cell aluminum alloy foams, Alulight and Alporas, were tested under alternating loads of fracture mode I to measure their fatigue resistance to crack propagation [7]. Crack evolution was observed to be controlled by linear elastic fracture mechanics. On the other hand, due to the high sensitivity of the crack propagation rate to the alternating stress intensity, the conventional damagetolerant methodology was found to be inappropriate for the fatigue design of aluminum foams. McCullough et al. [8] observed that ratcheting in closed cell Alulight foam is the dominant cyclic deformation mode for a tension tension and compression compression load. In another study on crack propagation, Motz et al. [9] experimentally showed that a cellular solid with hollow spheres made of a stainless steel (316L) has fatigue strength 5% to 6% lower than that of a closed cell aluminum foam. Such strength reduction was attributed to the presence of stress concentration at the bonding points between spheres (Figure 3 1). (a) (b) Stress concentration Figure 3 1: (a) SEM micrograph of the notch root region of an Al-foam obtained during an in situ crack propagation test, (b) stress concentrated at the vicinity of the sphere-sphere bonding [9]. (Reprinted with permission from Elsevier) Studies on the fatigue of lattice materials are even less than those on foams. On the experimental side, Côté et al. [19], among few researchers, characterized the stress-life curves of sandwich panels with a lattice core made of stainless steel. Sandwich panels with a unit cell of diamond shape have been observed under alternating shear weaker than under axial load. A fatigue collapse map was 5

68 developed for stainless steel sandwich beams with pyramidal core [191] and crack nucleation sites leading to failure were identified in the cellular structure. It was shown that the weakness of brazed joints under cycling loading significantly reduces the shear fatigue strength of the panel core (Figure 3 ). In another study, Côté et al. [19] examined the available experimental data for metallic lattice and foam cores and showed that the fatigue to monotonic strength ratio, /, are close to.3 and., respectively, for R=.1 and R=.5. The load ratio (R) is defined as the ratio of minimum shear stress ( min ) to maximum shear stress ( max ). The data, gathered from a wide range of lattice cores, suggested that the fatigue to monotonic strength ratio generally does not depend on core topology, relative density, and material properties. e ult (a) (b) Figure 3 : (a) SEM photograph of a section of a undamaged joint showing the fatigue crack growth path starting at the toe of the braze joint of struts carrying tensile load [191], (b) optical photograph of a braze joint of a diamond lattice core of a sandwich structure [19]. (Reprinted with permission from Elsevier) On the theoretical side, empirical relationships of finite/infinite life as well fracture mechanics approaches have been employed to predict the fatigue life of a periodic cellular solid [1-1]. In these studies, the cell walls were assumed to flex like a beam. However, as it is shown in Chapter, the use of beam elements, rather than continuum elements, such as plane elements, is not appropriate to capture accurately the real stress distribution in the lattice cells. In fact, this assumption can lead to unrealistic results that may limit the use of these methods for fatigue design. 51

69 From the literature on lattice materials, it appears that most of the methods for fatigue design rely on experiments, which are tailored to handle selected lattice topologies and materials, besides being time consuming and often expensive. Theoretical approaches, on the other hand, seem to lack accuracy, since they might fail to capture the real stress distribution in the lattice cells. As described in Chapter, the AH theory is able to capture accurately the stress distribution throughout the unit cell of a periodic lattice. In this chapter, a methodology based on AH is proposed to design lattice materials against fatigue failure. The linear elastic material model used in Chapter is extended to consider the effect of material nonlinearity as a result of plastic deformation. This elastic-plastic model is used to determine the endurance limit and the ultimate strength of lattice materials under multiaxial loading conditions. Using this information, the constant fatigue life diagram of the material can be created and be used as a data base for the design of macroscopic component against fatigue failure. The method is also applied to study the role of the unit cell shape on the fatigue strength of hexagonal and square lattices. Cell shapes with regular and optimized geometry are examined. A unit cell is considered to possess a regular shape if the geometric primitives defining its inner boundaries are joined with an arc fillet. An optimized cell shape, on the other hand, is obtained by minimizing the curvature of its interior borders, which are conceived as continuous in curvature to smooth stress localization. For a given multi-axial cyclic loading, failure surfaces of metallic hexagonal and square lattices are provided along with their modified Goodman diagrams to assess the effect of mean and alternating stresses on the fatigue strength. The numeric simulations are also validated through experimental data available in the literature. 3. Fatigue design methodology Several methodologies and criteria have been developed for the design of components against fatigue fracture and degradation, including infinite-life, safelife, fail-safe, and damage-tolerant criteria. The selection of these methodologies depends on design objectives, constraints, and the location where the device is being used. In infinite-fatigue life, the component is designed to have stress levels 5

70 below the endurance limit of the material. This methodology is usually used for the fatigue design of components which should operate under high cyclic loading conditions and whose regular inspection is expensive or impractical. Cardiovascular stents, ventricular assist devices, and orthopaedic implants are some examples of medical implants that need to be designed with on infinite-life criterion. For other fatigue design methodologies, regular inspection is required to detect probable cracks and to indicate when the component should be replaced. These methodologies are applied for the design of components with high safety concerns usually used in aerospace and nuclear power industries. The fatigue strength of a material can be generally determined through traditional approaches, e.g. the stress (strain) life method. The stress (strain) life method is an empirical approach introduced by Wohler to find the number of cycles that a smooth test specimen can resist under alternating stresses before fatigue fracture. The experimental data are usually plotted as a stress-life curve where x- axis represents the number of cycles and y-axis is the alternating stress. A schematic view of a stress-life curve, also known as S/N curve, is shown in Figure 3 3a. At high cycles, the S/N curve attains a plateau value, which is referred to as the fatigue endurance limit. For alternating stress below the endurance limit, it is expected that the material have an infinite life, and the infinite-life fatigue criterion can be used to design the component. In cyclic loading condition, the alternating (S a ) and mean (S m ) stresses are obtained from the maximum (S max ) and minimum (S min ) stress levels (Figure 3 3b) as follow: (a) S a S S max min (3.1) (b) Stress (S) Endurance limit Stress (S) T S a S min S max S m Number of cycle (N) Time (t) Figure 3 3: Schematic view of (a) a stress-life curve and (b) a cyclic load with constant stress amplitude. 53

71 S S S max min m (3.) Mean stress can also affect the fatigue strength of the material. It has been shown that a positive mean stress decreases the fatigue resistance of the mechanical components [13], while a compressive mean stress prevents cracks to grow and thus increases the fatigue life [14, 15]. To study the effect of mean stress on the allowable alternating stress, experimental data are plotted as alternating stress versus mean stress for constant design life. Different formulas have been proposed to fit the experimental data; Soderberg, Goodman, Modified Goodman, and Gerber diagrams are common models, as shown in Figure 3 4. These graphs divide the stress space into a safe and an unsafe operating region. The left side is the safe cyclic stress space under which the component will operate without failure for the desired life cycle; the stress space on the right side, on the other hand, causes failure before the designed life cycle. The following expressions can be written for the constant life diagrams: Soderberg: Goodman: Modified Goodman: m a 1 (3.3) y m ult f a 1 (3.4) f f Soderberg Modified Goodman Goodman Gerber 45 Figure 3 4: Fatigue design diagrams showing the relation between alternating stress and mean stress for a given number of cycles. 54 y ult

72 Gerber: ( ) m y f a ult 1 for m ult f ( ult f ) 1 ( ) ( m a ) 1 for m y ( ult f ) y f ult (3.5) where m and m a 1 ult f a are the mean and alternating stresses respectively. are the yield and ultimate strength of the material respectively, and 55 and y (3.6) ult is the f allowable alternating stress for a given life cycle, which is obtained from the stresslife curve of the material (Figure 3 3a). For the infinite-life design, the allowable alternating stress is the endurance limit of the material e. To construct the constant life fatigue diagram, material properties, including allowable alternating stress, yield strength, and ultimate strength are required. For isotropic materials, these mechanical properties are usually obtained under unidirectional loading conditions, such as a fully reversible tension-compression loading and uniaxial tension. However, in many practical situations, various stresses which may include both normal and shear stresses can act simultaneously on the material. The effect of multiaxial loading conditions can be taken into account by using a criterion such the von Mises failure criterion. For orthotropic materials, such as composite and several lattice materials, the mechanical properties are a function of fiber or cell architecture orientation, and their mechanical strength varies with loading directions. Tsai-Wu is a well-known failure criterion for the design of composite materials, where a quadratic polynomial is defined as the yielding function to analyze the material under multiaxial loading conditions [16, 17]. For lattice materials, the yield strength under multiaxial loading condition can be obtained from equation.15 as described in Chapter. For the fatigue analysis of lattice materials, it is assumed here that the specimen is free of defects, and no type of damage, such as scratches, notches and nicks, has occurred in the lattice cell walls. As a result, the constant life diagram can be constructed for the design of the

73 material against fatigue failure [18]. The damage-free assumption of the microstructure would also ensure the validity of the periodicity assumption of the microstructure. Hence, AH theory can be used to capture the stress distribution within the unit cell. To account for the effect of micro-defects on the fatigue fracture of the material microstructure, computational techniques, such as the multilevel computational approach [19, ] or the mesh superposition method [1], can be included in the method to model local defects explicitly. It is also assumed that the cell wall material is a continuum media with properties comparable to those of its bulk material, which means that the material has one level of structural hierarchy. This hypothesis sets the limit of applicability of the proposed method to a minimum length-scale that depends on the material microstructure, slip system, grain and size [-4]. If the effect of nanostructure defects, grain and size of the material, has to be modeled, then two length scales in the lattice microstructure should be considered, as described by Zienkiewicz and Taylor [5]. Since for a high cycle fatigue failure, as it is considered in this work, the stress level is lower than the yield of the bulk material, the linear elasticity assumption holds. Hence, using AH results from Chapter, the endurance limit of the unit cell can be obtained through the product of the unit cell yield strength with the ratio of the endurance limit to yield strength of the bulk material as: e y es (3.7) where ys and es are respectively the yield strength and the endurance limit of the bulk material, and e is the endurance limit of the unit cell. To construct the constant-life diagram, the ultimate strength of the material is also required. The AH theory, presented in the previous chapter, is extended here to capture the ultimate strength of lattice materials. 3.3 Ultimate strength of lattice materials The analysis of a lattice material is often performed by isolating a RVE and calculating its properties. The AH theory can be also used to determine the ultimate 56 ys

74 strength of lattice materials. The effect of material non-linearity as a result of plastic deformation is modeled, and the macroscopic stresses that lead to the microscopic failure can also be obtained. The basic assumptions of the AH theory described in Chapter, including the dependence of field quantity on two different length scales and the periodicity of field quantities at the microscale level, are still valid for the nonlinear behaviour of the lattice material. Following the procedure, described in Chapter, the equation of local problem defined on the RVE, equation.6, can be derived. The only change is to replace the local elasticity tensor E with the tangent modulus of the elastic plastic bulk material.9 can be written as: K m e k e1 57 E t. Equations.7 to KD f (3.8) T e, k B E B e t e dy (3.9 a,b) m e e f f, f BE ε( ) e t u e1 Y e dy (3.1) Y It is interesting to note that ε( u ) in equation 3.1 describes the initial strain field applied to each element of the unit cell. As a result, the force vector f is a function not only of the applied strain but also of the properties of the solid material. Since the plastic deformation is required to determine the ultimate strength, equation 3.8 can be considered as a system of non-linear equations that can be solved with the Newton Raphson scheme. As shown in Figure 3 5, the computational analysis is accomplished by imposing increments of macroscopic strain, as described in plastic theory [6, 7]. While the macroscopic strain ε( u) is increased monotonically, the force vector increment is computed and replaced in equation 3.8 to evaluate the displacement field. During the procedure (Figure 3 5), the effective material coefficients are kept constant until the stress reaches the yield strength of the base material. The mechanical properties of the yielded elements are replaced, iteratively, with the tangent modulus of the elastic plastic bulk material [6, 7].

75 Procedure of finite element analysis for asymptotic homogenization V RVE ε 1 ( ) T E ε * ( ) 1 ε ( ) T E ε( ) RVE v u dv v u dv V RVE RVE KD f Updating force vector and fluctuating strain ε( u) i e i T Δf B Et Δε( ) e Y u dy e i i1 m i i e i K D f e1 NO * i i ε ( u) BD Save the results YES? ε( u) ε i Updating microscopic strains and stresses * ε( ) i i u ε( u) ε ( u) i i i1 i ε ε ε( u) i el i σ( u) Eε( u) i i1 i σ( u) σ( u) σ( u) 1 V i i σ σ( u) RVE V RVE dv RVE Figure 3 5: Flowchart of the asymptotic homogenization theory to obtain the ultimate strength of a lattice material. * Once the displacement field is obtained, the fluctuating strain ε ( u) is determined by the product of the strain-displacement matrix, B, and the nodal displacement vector. The increment of the microscopic strain is then determined through the equation: * ε( u) ε( u) ε ( u) (3.11) 58

76 The stress field in the unit cell is updated with respect to the elastic strain, and the stress average is computed over the unit cell to attain the effective macroscopic strength as: 1 dv V RVE V (3.1) RVE σ( u) RVE The macroscopic stress, equation 3.1, represents the ultimate strength of the unit cell when the stress level at any local point of the microstructure reaches ultimate strength of the solid material. The generation of the model and its mesh, as well as the nonlinear plasticity analysis, has been implemented into ANSYS (Canonsburg, Pennsylvania, U.S.A), where the von Mises yield criterion with the associated flow rule has been considered for plasticity analysis. The fatigue, yield, and ultimate strength are required properties to construct constant life diagram for any type of cell topology. 3.4 Cell geometries under investigation The topology and shape of the unit cell has a significant effect on the static and fatigue properties of the lattice material [8-3]. Here, we analyze square and hexagonal lattices with three different cell shapes: G 1 cells with sharp corners, G 1 cells with optimum fillet radius, and G cell with continuous optimum curvature of the fillet. Figure 3 6 shows the methodology to obtain the fatigue properties of the optimized cell geometry. The procedure consists of three integrated parts that combine notions of shape synthesis, asymptotic homogenization theory, and design optimization. First, the shape of the unit cell is generated for a given geometrical parameters; then its fatigue properties are computed to generate the constant life diagram at given relative densities. Here, the modified Goodman diagram is considered. The last step involves the minimization of the von Mises stress in the cell wall. For square and hexagonal lattices, we study cell shapes with the characteristics described below: ult 59

77 Initial guess [ t, t ] 1 Shape synthesis Minimize 1 n wk k n 1 n is the number of supporting points k is the curvature at supporting point is the weighting factors obtained iteratively by FEM w k Procedure of finite element analysis for asymptotic homogenization V RVE ε 1 ( ) T E ε * ( ) 1 ε ( ) T E ε( ) RVE v u dv v u dv V RVE RVE KD f Updated variables [ t, t ] NO 1 ε( u) i Updating force vector and fluctuating strain e i T Δf B Et Δε( ) e Y u dy e Save the results YES Optimum set of [ t? 1, t] NO i i1 m i i e i K D f e1 Gradient based optimization to find Minimize max{ ( t, t )} in RVE von Mises 1, [ t, t ] 1 YES YES? ε( u) ε i i i1 i ε ε ε( u) * i i ε ( u) BD Updating microscopic strains and stresses * ε( ) i i u ε( u) ε ( u) i i el i σ( u) Eε( u) i i1 i σ( u) σ( u) σ( u) 1 V i i σ σ( u) RVE VRVE dv RVE Figure 3 6: Flowchart of the design methodology. For a given cell geometry, shape synthesis is coupled with computational analysis followed by size optimization. The goal of the first step is to smooth the transitions between the geometric primitives defining the cell inner boundaries, thereby reducing stress localization. In the second module, the effective strength properties of the lattice are determined through asymptotic homogenization theory. The third step involves the cell size optimization to reduce at minimum the maximum von Mises stress in the cell wall. 1. G 1 cells with sharp corners. These cells represent a conventional lattice with regular cell geometry. The fillets at the cell joints are specified by an arc with radius equal to 1% of the RVE length, which is representative of a sharp fillet resulting from a given manufacturing process. The choice of the small-arc fillet 6

78 enables to obtain a realistic material distribution in a cell member [31], approximately similar to that measured through experiments [191, 19]. These cells are named here G 1, where G represents geometry and the superscript shows the degree of continuity of the geometric primitives defining the cell shape. As seen in Figure 3 7, G 1 cells have continuous tangent along their inner boundary profile, but the curvature at the blending points between the straight and arc primitives is discontinuous.. G 1 cells with optimum fillet radius. The G 1 cells with sharp corners are here optimized to obtain a value of the fillet radius that reduces the effect of stress concentration. A classical optimization problem is formulated with the objective of minimizing the maximum value of the microscopic von Mises stress in the lattice under a given loading. The fillet radius and the thickness at the middle of the struts (Figure 3 7) are considered as interdependent design variables. A design constraint is set on the relative density. The problem is solved by implementing a conjugate gradient method. For given relative density and design variables a unique solution is found. 3. G cell with continuous optimum curvature of the fillet. The inner profile of these unit cells are synthesized to obtain lattice structures free of stressconcentration. The procedure is based on a well-known engineering theory stating that the stress concentration due to the presence of curvature discontinuity in a component has the effect of localizing high peak of stress [3]. Thus, synthesizing the inner boundary of a cell with curves that are continuous in their curvature, i.e. G -continous curves [33], improves the fatigue and monotonic strengths of the lattice. Furthermore, to avoid high bending moments caused by curved cell members, the cell walls can be designed to be as straight as possible, i.e. with the smallest curvature. The top left quadrant of Figure 3 6 shows the optimization formulation used in this study, while the details of the design optimization problem are given in the next section. Figure 3 8a and b show the parametric view of square and hexagonal cells, whose G continuous profiles of their inner profiles are optimized with 61

79 minimum curvature. We note that as shown in Figure 3 6, once the yield strength of the G cell are obtained, the thickness in the middle and corner of the cell walls (Figure 3 8a and b) are optimized with the goal of minimizing the maximum von Mises stress. The optimum values of these variables are found through a conjugate gradient optimization method, where the density is set as a constraint. Blending points t r r t (a) Figure 3 7: Schematic views of: (a) G 1 square unit cell; (b) G 1 hexagonal unit cell (b) t t 1 t1 t y A ds Pk B (a) (b) (c) Figure 3 8: Schematic views of: (a) G continuous square unit cell; (b) G continuous hexagonal unit cell; (c) Parameterization of the inner profile of a unit cell. 3.5 Mathematical formulation of the optimization problem The design method used to find a novel lattice cell topology is based on the synthesis of structural members with G -continuous curves that minimize the rms value of their curvature [33]. The shape optimization of the lattice strut is stated as: under given end conditions, find a boundary-curve that connects two given 6 x

80 end points A and B of the cell strut as smoothly as possible and with a G - continuous curve. By parametrizing the cell strut boundary-curve as a function of the arc-length s along the strut, we can formulate the optimization problem as [33]: B 1 J ( ) ds L A min () s (3.13) where is the rms value of the curvature of the cell strut boundary-curve, L is the strut length, A and B are the two end-points of the boundary curve, and ds is the arc-length along the strut, starting from at point A, as shown in Figure 3 8c. Each segment of the strut boundary-curve is subjected to four constraints at each endpoint: two constraints are used to fix the end-point coordinates, while the other two constraints are used to determine the tangent and curvature of the curve at those points. Equation 3.13 can be solved as a mathematical programming problem by means of non-parametric cubic splines [34]. Hence, each boundary curve is 1 discretized by n supporting points { P } n that are defined by P (, ) in a polar coordinate system. As shown in Figure 3 8c, curve; P k k k k P is a generic point of the k A and P n1 B, where A (, ), and B(, ) are two end-points of A the boundary-curve of the cell strut. Moreover, if we assume that the discrete points are located at constant tangential intervals, the tangential increment will be [35]: A B B B A n 1 A cubic spline, ( ), between two consecutive supporting points be defined as: P k and k 1 (3.14) P can ( ) A ( ) B ( ) C ( ) D (3.15) 3 k k k k k k k The radial coordinates, the first and second derivatives of the cubic splines at the k th supporting point,, and,, respectively, are represented by the following three vectors: 63

81 [,,..., ] T 1 n n1 [,,..., ] T 1 n n1 [,,..., ] T 1 n n1 (3.16) Imposing the G -continuity condition results in the following linear relationships between and and between and [35]: A 6C and P Q (3.17) where A, C, P and Q are defined in appendix. Furthermore, and A n 1 are known from a given cell design parameter vector. Now, if x is the vector of the design variables, defined as: 1 n x [,... ] T (3.18) The discretized shape optimization problem can be written as [35]: B 1 z( ) w k k n n x 1 min x (3.19) where w k is the weighting coefficient of point th k, defined at each supporting point, and representing the contribution of each point on the curvature of the optimum curve. As an initial guess, we solve the optimization problem of the cell strut shape only as a function of its geometry by assuming equal weighting coefficients, i.e. 1. The required number of supporting point to represent the curve shape depends on the geometric boundary conditions. After performing a sensitivity analysis, 1 supporting points has been selected to capture the boundary curves. The curvature at each point Pk is given by: ( ) k k k k k 3/ ( k ( k) ) (3.) Discretizing the objective function (equation 3.19) and applying the constraints at the end point of the boundary curve, allows to solve the problem with mathematical programming [36]. A sequential-quadratic-programming algorithm using orthogonal decomposition is implemented to solve this problem [37]. 64

82 As mentioned above, the first stage of the geometrical optimization procedure assumes an equal weighting factor for the supporting points, and the effect of loading condition and material properties are not taken into account. Once the initial design is obtained, the weighting factors can be defined as a function of stress (strain) along the curve [38]. Higher weighting coefficients will be assigned to the supporting points with higher stress (strain). The curvature of each supporting point will be further optimized to design a boundary curve with uniform stress (strain) distribution for the given loading condition and material properties. The weighting coefficients are therefore not uniform along the curve, and they can be defined as follow [38]: w k k (3.1) T where k and T are, respectively, the rms value of von Mises stress at the k th supporting point of the curve and the rms value of von Mises stress over the whole cell, and they are defined as: T 1 m (3.) m i 1 i 1 m (3.3) 5 k k ki, k k i 1 where m is the total number of nodes in the FE model, i is the von Mises stress at the i th node of the FE model, and ki is the von Mises stress of the k nodes (% of total number of nodes of the FE model) which are relatively close to the k th supporting point. The structural optimization algorithm stops when the maximum reduction of stress is less than %. 3.6 Numerical modeling Both G 1 and G optimum cell geometries have been automatically obtained via in-house Matlab (MathWorks, Natick, Massachusetts) subroutines, which have been coupled with ANSYS (Canonsburg, Pennsylvania, U.S.A) to build, mesh, and solve the D model of the lattice material. Assuming in-plane loading conditions, a 65

83 D eight-node element type, Plane 8, with plane strain formulation was used, as it can model curved boundaries with high accuracy. Different combinations of axial and shear loadings are considered to obtain the data required to plot the failure surfaces. The effect of the material properties on the normalized fatigue to monotonic strength ratio, /, is studied by using aluminum and titanium alloys as bulk e ult properties of the solid materials. Aluminum was selected because of its broad application for lightweight foams; Titanium is of interest because it is widely used in the aerospace, automobile, sport equipment, and biomedical sectors. Table 3 1 lists the material properties used in this study [39-4]. A bilinear elasto-plastic constitutive law was considered to obtain the macroscopic ultimate stress of the cellular material. An analysis of the mesh sensitivity (Figure 3 9) was performed to ensure the independency of the results from the mesh size. During this test, the element size of the FE model was refined, especially around regions of high stress concentration, until the value of the maximum stress converged H exagonal Square xx ys Number of Elements Figure 3 9: Mesh sensitivity showing the independency of the results from the mesh size. 66

84 Table 3 1: Material properties of bulk solid materials. Ti-6Al-4V [39, 4] Aluminum 661 [41, 4] Ultimate Yield Fatigue Elongation Young s Modulus Poisson s tensile strength endurance limit to fracture (E s, GPa) ratio (ν) strength (MPa) (MPa) (%) (MPa) Results Stress distribution in the unit cells Figures 3 1 and 11 show the distribution of von Mises stress (MPa) in the titanium square and hexagon with G 1 and G cell boundaries under fully reversed uni-axial and shear loading conditions. As can be seen, the cell shape has a pivotal role on stress distribution within the microstructure which can affect yield, ultimate and fatigue strengths of the lattice. The stress level significantly changes at the blending points of the G 1 cells, while the stress in G cells is distributed uniformly along the cell struts. In addition, it can be seen that under uni-axial loading there is no stress concentration at the joints of the G square cell. For the G 1 unit cell, stress concentration occurs at each blending point between the arc and the straight segment, as shown in Figure 3 1. This shows that the curvature discontinuity of the strut can locally increase the stress level which reduces fatigue strength as opposed to what occurs with smooth G corners. Using the maximum value of stress in Figure 3 1 and 11, it can be found that optimum square G 1 or G cells have respectively 48.8% (6%) and 54% (119%) higher yield strength in comparison with G 1 cell with sharp edges under uniaxial tension and (shear) loadings. For the hexagonal unit cell, the yield strength of optimum G 1 or G cells has significantly improved by 173% (14%) and 89% (44%) in comparison with G 1 cell with sharp edges under uniaxial tension and (shear) loadings, respectively. These results suggest that the fatigue strength of a 67

85 lattice material can be substantially improved by optimizing the unit cell variables that reduce stress concentration within the walls of the unit cell. Figure 3 1: Von Mises stress (MPa) distribution in hexagonal and square unit cells made out of Ti 6Al 4V. Lattices under fully reversed uni-axial loading, defined by: G 1 cell with small arc (left), optimum G 1 cell (middle), and optimum G cell (right). Figure 3 11: Von Mises stress (MPa) distribution in hexagonal and square unit cells made out of Ti 6Al 4V. Lattices under fully reversed pure shear, defined by: G 1 cell with small arc (left), optimum G 1 cell (middle), and optimum G cell (right) Failure surfaces of the unit cells and experimental validation Figure 3 1 shows the yield (solid marks) and ultimate (open marks) surfaces for both G 1 and G cells, calculated at relative density of 1%. The yield and 68

86 ultimate surfaces are obtained for different biaxial loading conditions in xx yy and planes. Figure 3 1 shows that G cells have higher yield strength xx xy than that of G 1 cells, which confirms the detrimental effect of the sharp corner on the strength of G 1 cells. Figure 3 13 shows the effective yield strength of both cells under uni-axial and shear loadings as a function of the relative density. As mentioned previously, the values are extracted from the intersection of yield/ultimate surfaces with the xx and xy axes. Figures 3 13a and b show that although both optimum G 1 and G square lattices have comparable yield strengths under axial loading condition, the effect of stress concentration at the joints of the G 1 cell significantly reduces its yield strength under shear loading. Figures 3 13c and d show that at low relative densities, the hexagonal G cell has up to 5% higher yield strength in comparison to the optimum hexagonal G 1 cell; for higher relative density, on the other hand, the two unit cells have comparable yield strength. Table 3 lists the yield/ultimate stresses of the G 1 and G cells for different relative densities. Table 3 reports the fatigue to monotonic performance ratio, /, of the G 1 and G lattices made of titanium and aluminum 661T6 alloy [41, 4]. It can be observed that for conventional lattices with small arc, the numerical results are close to the experimental values (.3 and. respectively for R=.1 and R=.5) reported by Côté et al [19]. Furthermore, in agreement with [19], the fatigue to ultimate monotonic stress ratio does not depend on the geometry of the unit cell, its relative density, and its material. The conformity of the results with the experimental data obtained by Côté et al. [191, 19] contributes to validate the computationalmechanics approach presented in this work. e ult 69

87 (a).6 / yy ys (b). / xy ys.3.1 / xx / ys xx ys (c).4 / yy ys (d).4 / xy ys. 4 Sm all arc / xx ys O pt arc / xx ys G cell.1.1 Sm all arc Sm all arc -. O pt arc 4 -. O pt arc 15 G cell G cell -.1 G Sm 1 cell all arc with small arc ( 4 1 =1%) Sm G all cell arc with small arc ( =%) Sm G cell all 4arc with small arc ( =3%) G O pt cell arcwith optimum arc ( =1%) 1 OG pt 1 cell arc with optimum arc ( =%) -.4 O G 1 pt cell arc with optimum arc ( =3%) 4-4 G cell - Optimum G cell ( G 4 =1%) Optimum cell G cell ( =%) GOptimum cell G cell ( =3%) Small arc (Yield) Sm all arc -.1 Sm all arc Small arc (Ult) 1 O pt arc Small O pt arc arc(yield) Opt arc (Yield) G cell - Small G arc cell(ult) Opt arc (Ult) 1 1 Small G cell Smarc all with arc (Yield) small arc (Yield) -4 5 Opt G cell arc - with (Yield) optimum arc (Yield) 4 G Optimum (Yield) G cell (Yield) Small G cell arc with (Ult) small arc (Ultimate) Opt G cell arc with (Ult) optimum arc O pt arc -4 (Ultimate) G Optimum (Ult) G cell (Ultimate) Opt -4 arc (Yield) - G (Yield) 4 Opt G arc cell (Ult) G (Ult) Small arc (Yield) -4 Figure : G (Yield) Yield and ultimate surfaces of Ti-6Al-4V - square Small and arc (Ult) hexagonal unit G (Ult) Small arc (Yield) Opt arc (Yield) cells for relative density of 1%. Projection of yield and ultimate surfaces of the G 1 and G - Small arc (Ult) Opt arc (Ult) 1 1 square unit Small G cells arc with (Yield) small in the arc planes: (a) Opt G 5 cell arc with (Yield) optimum xx 1 yy, arc(b) 15 xx G Optimum (Yield) xy, 5 (c) G cell 3 xx yy, (d) - Small arc (Ult) Opt arc (Ult) -4 G (Ult). Opt arc (Yield) G (Yield) Modified Goodman diagrams xx xy Opt arc (Ult) -4 G (Ult) G (Yield) G (Ult) Figure 3 14 shows the modified Goodman diagrams of the G 1 and G cells at given relative densities, obtained with the values listed in Table 3. The diagrams are constructed from the intersection of (i) the line connecting the unit cell ultimate and fatigue strengths with (ii) a 45 sloped line, originated from the yield strength of the lattice material. Here, the values are normalized with respect to the yield strength of the balk material. The corresponding value on the vertical axis is the alternative macroscopic stress, which as described in Section 3. generates a 7 4.

88 stress level equal to the fatigue endurance limit of the solid material at the microstructure. As expected, the modified Goodman diagrams of G cells cover a wider range of applied alternating/mean stresses in comparison with their G 1 counterparts. It should be noted that Figure 3 14 shows the modified Goodman of lattices only for uni-axial and pure shear loadings. In practice, however, a mechanical component should resist complex load combinations, which produce multi-axial macroscopic stress states at each point. Thus for fatigue design, the modified Goodman diagrams should be obtained at critical regions subjected to a multi-axial stress state. Figure 3 1 illustrates the failure surfaces, which are given here for this purpose. For example, for a prescribed multi-axial stress state, such as.51 and, the distance between the origin and intersection points with the ultimate/yield surfaces can be chosen as yield and ultimate stresses to derive the corresponding modified Goodman diagram. Such a methodology can be of interest for the design of lattice materials under multiaxial cyclic loading conditions. 3.8 Conclusion A methodology based on computational mechanics has been presented in this chapter to design planar lattice materials for fatigue failure. Asymptotic homogenization has been used to determine the macroscopic yield strength, ultimate strength, and endurance limit, each required to generate the modified Goodman diagrams of the lattice. A comparison with the experimental data available in literature shows a good agreement of the results. As a case study, the methodology has been applied to investigate the effect of cell shape on the fatigue/monotonic strength of lattices with hexagonal and square cells. Failure surfaces of unit cells with their respective material distribution within the RVE and with no geometric stress concentration have been obtained. The results show that cells with continuous and minimum curvature geometry have superior fatigue performance than cells with shape boundaries defined by line and arc primitives. In addition, unit cells with G minimum curvature should be 71

89 preferred over cells with arc-rounded joints, especially if manufacturability is not a constraint..16 (a). (b).1.15 xx ys.8 xy ys (c) G cell.6 with small arc ( 4 1 =1%) G cell with small arc ( =%) G cell 4 with sm 1 G cell with optimum arc ( 1 =1%) G cell with optimum arc ( 1 =%) G cell with opt -4 - Optimum G cell ( 4 =1%) Optimum G cell ( =%) Optimum G ce Sm all arc (Y ield.4. - Sm all arc (U lt) xx xy Sm all arc (Y ield) O pt arc (Y ield) ys - Sm all arc (U lt) ys O pt arc (U lt) 1 1 G cell. with small arc (Yield) -4 OG ptcell arc - with (Y ield) optimum.1 arc (Yield) 4 GOptimum (Y ield) G cel 1 1 G cell with small arc (Ultimate) OG ptcell arc with (U lt) optimum arc -4 (Ultimate) GOptimum (U lt) G cel -4 - G (Y ield) 4-4 G (U lt) Sm all arc (Y ield) Sm all arc (U lt) Sm all arc (Y ield) O pt arc (Y ield) - Sm all arc (U lt) O pt arc (U lt) 1 Sm G cell all arc with (Ysmall ield) arc 1 OG ptcell arc with (Y ield) optimum arc GOptimum (Y ield) G cell Sm all arc (U lt) O pt arc (U lt) -4 G (U lt) O pt arc (Y ield) G (Y ield) Figure 3 13: Effective yield strength of the square and hexagonal unit cells under uni-axial and shear loading as a function of relative density. Yield strength for square unit cell under (a) uni-axial and (b) shear loading respectively; (c) and (d) for hexagon. 4 (d) 7

90 .75 (a).9 (b) a ys.5 a ys (c) m ys (d) m ys. a ys.1 a ys Sm all arc Figure 3 14: Modified Goodman Small diagram arc (Yield) for Ti-6Al-4V square Optand arc (Yield) hexagonal unit - Small cells at given relative densities. 1 arc (Ult) and G Opt arc (Ult) 1 square under (a) uni-axial loading and Small arc (Yield) 5 (b) shear Small loading arc (Ult) condition; G 1 1 Opt arc (Yield) and Opt arc G(Ult) G cell with small arc (Yield) -4 G cell - with optimum arc (Yield) 4 G Optimum (Yield) G cell (Yield) 1 1 G cell with small arc (Ultimate) G cell with hexagon optimum arc -4 (Ultimate) under (c) uni-axial G Optimum (Ult) loading G cell (Ultimate) and (d) Opt -4 arc (Yield) - G (Yield) 4 shear loading. Opt arc (Ult) G (Yield) 4-4 G (Ult) - 1 Small G cell arc with (Yield) small arc - Small arc (Ult) -4 4 O pt arc G cell..3 Sm all arc m m 4 O pt arc ys G cell ys 1 G cell with small arc ( 4 1 =1%) G cell with small arc ( =%) Sm G cell all 4arc with small arc ( =3%) 1 G cell with optimum arc ( =1%) 1 G 1 cell with optimum arc ( =%) O G 1 pt cell arc with optimum arc ( =3%) -4 - Optimum G cell ( 4 =1%) Optimum G cell ( =%) GOptimum cell G cell ( =3%) Small arc (Yield) - Small arc (Ult) Opt arc (Yield) Opt arc (Ult) -4 G (Yield) G (Ult) G (Ult) - Small arc (Yield) Small arc (Ult) Small arc (Yield) Opt arc (Yield) Small arc (Ult) Opt arc (Ult) 1 Opt G 5 cell arc with (Yield) optimum 1 arc15 G Optimum (Yield) 5G cell 3 Opt arc (Ult) -4 G (Ult) G (Yield) G (Ult) 73

91 Table 3 : Yield and ultimate strength of the G 1 and G unit cells Square unit cell 1% % 3% Uni-axial tension shear Uni-axial tension shear Uni-axial tension shear y (MPa) ult (MPa) y (MPa) ult (MPa) y (MPa) G 1 cell with Small arc Optimum G 1 unit cell Optimum G unit cell Hexagonal unit cell G 1 cell with Small arc Optimum G 1 unit cell Optimum G unit cell ult (MPa) y (MPa) ult (MPa) y (MPa) ult (MPa) y (MPa) ult (MPa) Table 3 3: Fatigue to monotonic performance ratio, (Al) 661T6 alloy at given relative densities. / e ult, of G 1 and G lattices made of titanium (Ti) and aluminum 1% % 3% ult e ( R. 1 ) ult e ( R. 5 ) ult e ( R. 1 ) ult e ( R. 5 ) ult e ( R. 1 ) ult e ( R. 5 ) Al Ti Al Ti Al Ti Al Ti Al Ti Al Ti Square unit cell G 1 cell with Small arc Optimum G 1 unit cell Optimum G unit cell Hexagonal unit cell G 1 cell with Small arc Optimum G 1 unit cell Optimum G unit cell

92 4 CHAPTER 4: Multiscale Design and Multiobjective Optimization of Orthopaedic Hip Implants with Functionally Graded Cellular Material 4.1 Introduction In this chapter, a methodology based on multiscale mechanics and design optimization is introduced to synthesize a graded cellular hip implant that can minimize concurrently bone resorption and implant interface failure. AH theory, described in Chapter, is used to capture the mechanics of the implant at the micro and macro scale, and multiobjective optimization is applied to find optimum gradients of material distribution that minimize concurrently bone resorption and bone-implant interface stresses. The procedure is applied to the design of a D implant with optimized gradients of relative density. To assess the manufacturability of the graded cellular microstructure, a proof-of-concept is fabricated by rapid prototyping. The results from the analysis are used to compare the optimized cellular implant with a fully dense titanium implant and a homogeneous foam implant with a relative density of 5%. 4. Methodology Kuiper and Huiskes [115, 116] showed that the use of a graded material in an orthopedic stem can lead to a reduction of both stress shielding and bone-implant interface stress. To this end, hierarchical computational procedure [43-46] can be implemented to design an optimum material distribution within the implant. This strategy might generally require a high computational cost besides yielding a microstructure which is difficult to fabricate. In this work, we suggest to design gradients of material properties through a tailored lattice microstructure, whose geometrical parameters are optimized in each region of the implant to achieve minimum bone loss and implant interface failure. 75

93 The mechanical properties of a cellular structure depend on the relative density and the geometric parameters of the unit cell, as described, for example, by the expression of the Young s modulus [144]: * E CE s s p (4.1) where * E is the effective Young s modulus of the unit cell, is the density of the unit cell, and E s and s are the Young s modulus and density of the constitutive material, respectively. p has a value varying from 1 to 3 as determined by the mechanical failure mode of the unit cell, and C is a geometric constant of the unit cell. By changing the relative density of the lattice microstructure, it is thus possible to obtain desired values of mechanical properties in any zone of the implant. b () t 3 t 1 a c t (1) CT-scan data & FEA model of hip (3) Characterization of cell topology H E y Homogenized elastic modulus Multiaxial yield surfaces (4) FEA, Macro-stresses/ strains (5) xx y xx Microstructure failure i1 j1 1 W ij xy y xx 1 ij yy y yy (6) Multiobjective optimization min F( b) { m ( b), F( b)} r (7) Update design variables No Converge Yes Optimal design SF 1 Safety factor 4% Mean porosity S.t. 5m P 9m Mean pore size t tmin Cell wall thickness withb {,..., } 1 m Vector of design variables Relative density at each for 1 sampling point i where i i m Figure 4 1: Flow chart illustrating the design of a graded cellular hip implant minimizing bone resorption and implant interface failure. Figure 4 1 summarizes the procedure proposed here to design a cellular implant with controlled gradients of mechanical properties. The method integrates a multiscale mechanics approach to deal with the scale-dependent material structure and a multiobjective optimization strategy to handle the conflicting nature of bone 76

94 resorption and implant interface failure. The main steps identified by the numbers reported in the flow chart of Figure 4 1 are described here: (1-) A finite element model of the bone is created by processing CT-scan data of a patient bone. The design domain of the prosthesis is assumed to possess a 3D lattice microstructure, where the unit cell, i.e. the building block, can be of any arbitrary topology (Figure 4 1). The microscopic parameters of the unit cell geometry and the macroscopic shape of the implant are the design variables of the vector b. The unit cell is assumed to be locally periodic, and its field quantities, such as stress and strain, to vary smoothly through the implant. (3) The characteristic length of the unit cell in the cellular implant is assumed to be much smaller than the characteristic length of the macro dimensions of the implant. Hence, the microstructure can be replaced with a homogeneous medium whose equivalent mechanical properties, in particular the homogenized stiffness tensor and the multiaxial yield surface of each unit cell, are calculated through the AH theory, as described in Chapter. (4) The homogenized stiffness tensors are then used to construct the stiffness matrix which will be the input to the Finite Element (FE) solver. As a result, the average strains and stresses throughout the bone and the structure of the prosthesis are calculated. (5) To analyze microstructure failure, the average of macroscopic stress over the unit cell is computed using a Gaussian quadrature integration [5] with 3 3 Gauss points, and then is compared with the yield surface of the unit cell to measure the safety factor. A more detailed description is given in section (6) If the microscopic stress level is below a predefined failure criterion, the macroscopic stresses and strains representing the mechanical behavior of the implant are used to evaluate bone loss ( m (b) ) and interface failure ( F (b) ). In the formulation of the multiobjective optimization problem, the constraints are set on the average porosity of the cellular implant (b), the mean pore size P, and the minimum thickness of cell walls t min. In particular, ( b) 4% and 77 r

95 5m P 9m are selected to ease bone ingrowth [45, 47]. The thickness of the cell walls is selected to be greater than the minimum resolution t min offered by a given manufacturing process. For example, t min is 1 m and 7 m respectively for SLM and SLA [135, 47]. (7) If the solutions of the optimization have not converged, then the vector b of the design variables is updated to find the set of non-dominated solutions of the Pareto front. Multiscale mechanics and multiobjective optimization are integrated aspects of the method proposed here. The methodology based on AH, described in Chapter, is used to compute the homogenized stiffness matrix and microscopic stresses within each unit cell of the cellular implant. The solutions of FE analysis are then postprocessed to evaluate the objective functions of the multiobjective optimization problem, whose formulation is described in the following section. 4.3 Formulation of the multiobjective optimization problem For the design of an optimum implant, we impose the simultaneous minimization of the amount of bone loss around the prosthesis, and the probability of mechanical failure at the bone-implant interface. As illustrated in Figure 4 1, the multiobjective optimization problem can be formulated as: Minimize: m r ( b) F( b) SF 1 (b) 4% Subject to 5m P 9m t tmin with b { 1,..., m } bone loss interface failure safety factor average porosity mean pore size cell wall thickness vector of design variables where i for 1im Relative density at each sampling point i (4.) The amount of bone loss around the stem is determined by assessing the amount of bone that is underloaded. Bone can be considered locally underloaded when its local strain energy ( U ) per unit of bone mass ( ), averaged over k loading cases ( j 78

96 1 k U j S k ), is beneath the local reference value S, which is the value of S ref j1 when no prosthesis is present. However, it has been observed that not all the underloading leads to resorption, and a certain fraction of underloading (the threshold level or dead zone s) is tolerated. Indeed, bone resorption starts when the local value of S is beneath the value of the resorbed bone mass fraction ( 1 s)s [115, 48]. Using this definition, ref m r can be obtained from: 1 mr ( b ) g( S( b)) dv (4.3) M V where M and V are the original bone mass and volume respectively, and g ( S( b)) is a resorptive function equal to unity if the local value of S is beneath the local value of ( 1 s)s and equal to if ( 1 s) S S. In this study, the value of dead zone s ref is assumed to be.5 [115]. ref The other objective is to minimize the probability of local interface failure, which is expressed by the following functional of the interface stress distribution: 1 F b f d (4.4) k b ( ) ( j ) k j1 b where F (b) is the global interface function index, is the interface stress at the loading case j, depending on the design variable b, is the interface area, and b ( j ) f is the local interface stress function, which is defined based on the multiaxial Hoffman failure criterion [49]. This function is used to determine where local debonding might occur along the bone-implant interface [115, 5]. The probability of local interface failure f ( ) is given by: f ( ) n n StS c St S (4.5) c Ss j where S t and S c are the uniaxial tensile and compressive strengths, respectively, S s is the shear strength, and n and are normal and shear stresses at the bone- 79

97 implant interface, respectively. For f ( ) 1, a high probability of failure is expected, whereas for f ( ) 1 the risk of interface failure is low. Tensile, compression, and shear strengths of the bone can be expressed as a function of bone density according to the power law relation obtained by Pal et al. [5]: S t 14.5, Sc 3.4, Ss 1.6 (4.6) The bone density distribution can be obtained through a CT-scan data of bone and then used in equation (4.6) to find the effective mechanical properties of the bone, from which the local interface failure is determined via equation (4.5). Finally, the interface failure index, F (b), is evaluated by means of equation (4.4). It should be noted that F (b) serves as a qualitative indicator only; lower values of F(b) indicate reduced probability of local interface failure in the implant. The expressions of bone resorption after implantation, and mechanical failure of bone-implant interface described in this section are used in the finite element analysis to evaluate the objective functions to be optimized. The following sections shows the application of the method for the design of a two dimensional cellular implant. 4.4 Application of the methodology to design a D femoral implant with a graded cellular material FEM model of the femur at the macroscale The left hand side of Figure 4 shows the geometry of the left femur considered in this work along with the appropriate loads and boundary conditions. These data have been obtained from the work of Kuiper and Huiskes [115]. The 3D geometry of the femur is simplified into a D model where the thickness of the stem and bone varies such that the second moment of area about the out-of-plane axis does not vary in both models [16]. Furthermore, the implant material is designed to be an open cell lattice to ease bone ingrowth in the implanted stem and obtain a full bond. Although bone ingrowth does not exist in a postoperative situation, it can appear later if local mechanical stability is guaranteed. The minimization of interface stress reduces the possibility of occurrence of interface 8

98 micromotion and instability [17]. Therefore, to decrease the computational cost required by a stability analysis based on a non-linear frictional contact model, the prosthesis and the surrounding bone are assumed fully bonded. Figure 4 : D Finite element models of the femur (left) and the prosthesis implanted into the femur (right). The load case represents daily static loading during stance phase of walking [51]. The distal end of the femur is fixed to avoid rigid body motion. For the material properties of the model, we consider GPa as the Young s modulus of the cortical bone and 1.5 GPa for the proximal bone. The Poisson s ratio is set to be.3 [115] FEM model of the cellular implant at the microscale The right hand side of Figure 4 illustrates the model of a cementless prosthesis implanted into the human femur. The grid depicts the domain of the implant to be designed with a lattice material of graded properties. Figure 4 3 shows the unit cell geometry at given relative densities of the sampling points used 81

99 for the tessellation of the whole implant. The gradients of material properties are governed by the lattice relative density, which is a variable controlled by the cell size and wall thickness of the hollow square. For the material property of the implant, we consider Ti6Al4V [13], which is a biocompatible material commonly used in EBM. Its mechanical properties are the following: 9 MPa for the yield strength of the solid material, 6 MPa for the fatigue strength at 1 7 cycles, 1 GPa for the Young s modulus, and.3 for the Poisson s ratio. These properties are experimental values obtained from mechanical testing of EBM samples after postprocess by hot-isostatic-pressing (HIP) [5]. Although micro defects and voids can be largely eliminated by the HIP process, remnants may still persist in built samples. Here, we assumed that the specimen is free of micro defects, and the cell wall material is a continuum medium with properties comparable to those of the bulk material. y x 3% 45% 65% 85% Figure 4 3: D hollow square unit cell for given values of relative density Design of the graded cellular material of the implant To design the graded cellular implant, the relative density of the unit cell is assigned to 13 sampling points (i.e. m 13 in equation 4.1), 6 rows along the prosthetic length and 5 columns along the radial direction, as shown in the right side of Figure 4. The number of sampling points has been chosen to be 13 to limit the computational time required for the analysis and optimization, while providing a reasonable resolution for the relative density distribution. For a more refined density distribution, the number of sample points can be increased. Their values have been constrained in the range.1 1 [151] to prevent elastic 8

100 buckling from occurring in the unit cell prior to yielding. The values of the relative density between the sampling points are obtained through linear interpolation. Although not considered in the current research, the shape of the implant could be included in the vector b as a design variable (Figure 4 1). Table 4 1: Effective mechanical properties of the square unit cell as a function of relative density E E x y ( ) s G E s s E E s ( ) 1.5.7( ) ( ).5( ) ( ) 1.33( ).51( ).65.68( ).8( ).6 Once the values of relative density for each element of the FE model is obtained, the effective elastic constants of the square unit cell, shown in Figure 4a, can be used to construct the stiffness matrix of each FE element. To have the continuous functions for the properties, we calculate through the least squares method the expressions (Table 4 1) for two ranges of relative density, i.e.. 3 and. 3. The functions allow the values of the stiffness for a given relative density assigned to each sample point to be found for the finite element model of the implant. We note that the expressions of Young moduli in the x and y change since the cell thickness is uniform. directions do not Once the stress and strain regimes of the cellular material have been calculated, the non-dominated sorting genetic (NSGA-II) algorithm [53] is employed to solve the multiobjective optimization problem, formulated in Section 4.3. The strain energy within the bone and the stress distribution at the bone-implant interface is then calculated and used in equations 4.3 and 4.4 to evaluate the objective functions. The initial population is then sorted based on the non-domination front criterion. A population of solutions, called parents, are selected from the current population, based on their rank and crowding distance. Then, genetic operators are

101 applied to the population of parents to create a population of off-springs. Finally, the next population is produced by taking the best solutions from the combined population of parents and off-springs. The optimization continues until the userdefined number of function evaluations reaches 5 [53]. The computational cost required to run the optimization process in a single.4 GHz Intel processor was about 3, CPU seconds. Using parallel computing with a PC cluster will considerably reduce the computational time, since each function evaluation can be performed independently. (a) y.6l (b).6l x Gauss point L=mm L L Figure 4 4: a) 3 3 Gauss points in the RVE; b) superposition of the RVE on the macroscopic mesh of the homogenized model Analysis of microstructure failure For each point in the objective function space, the stress distribution at lattice microstructure should be obtained to verify if the design does not fail. For this purpose, the average macroscopic stress inside each unit cell is found and compared with the yield strength of the unit cell. To this end, the position of each unit cell within the implant is obtained after imposing a proper cell tessellation, which in this work has been set to be uniform. The size of the unit cell is selected as small as possible to capture the relative distribution contour with higher resolution. For a relative density of.1, the square cell sizes for the cell wall thickness of either 7 or 1 m have been selected respectively as 1.36 and 1.8 mm. Once the position of the unit cells has been determined, 3 3 Gauss points are assigned to each cell, as shown in Figure 4 4. The values of relative density and 84 Macroscopic element

102 macroscopic stress at these points are obtained from the relative density distribution and macroscopic stress field. Using a Gaussian quadrature integration [5], the average of relative density and macroscopic stress of each cell are calculated: where 3 3 W (4.7) i1 j1 3 3 i1 j1 ij ij ij W (4.8) ij, ij, and W ij are the relative density, macroscopic stress, and weight factors at each Gauss point, respectively. The values at each Gauss point are obtained with respect to its local coordinates within the macroscopic element of the homogenized model (Figure 4 4). For the given average relative density, the yield surface of the square unit cell is obtained from the closed-form expression given in Table 1. The comparison of the average macroscopic stress (equation 4.8) and the yield surface provide the safety factor of the unit cell. This procedure is applied to all unit cells of the selected optimal design located on the Pareto frontier and the minimum local safety factor of a cell is specified as design safety factor. 4.5 Results and discussion The advantage of multiobjective optimization with a posteriori articulation of preference is that a set of optimum solutions are available without requiring the designer to choose in advance any weighting factors to the objective functions. Once the whole set of Pareto solutions has been determined, the designer has the freedom to select the desired solution based on the importance of bone mass preservation relative to the amount of interface stress. Figure 4 5 shows all the optimum solutions, i.e., the relative density distribution, for a hip stem implant with graded cellular material. The x axis represents the amount of bone resorption for the implanted hip; on the y axis is the interface failure index. Among the optimal solutions, we examine three representative relative density distributions: the extreme points, A and C, of the Pareto frontier, for which one objective function has importance factor and the other 1%, and a solution B characterized by a ij 85

103 5% weight factor. For each solution, Figure 4 5 gives the following performance metrics: bone resorption ( m ), interface failure index ( F (b) ), maximum interface failure ( f ( ) max r ), average porosity of each stem ( ), and design safety factor (SF) after implementing the stress recovery procedure. The maximum interface failure f ( ) max is included since F (b), which quantifies only the overall effect of the implant stiffness on the interface stresses, is not sufficient to provide information on the probability of failure. Figure 4 5: Trade-off distributions of relative density for the optimized cellular implant. As seen from the performance metrics in Figure 4 5, the porosity of solutions A, B, and C, is greater than 4%, which is satisfactory for bone ingrowth [45]. By comparing implants C and A, we observe that a raise of the implant porosity from point C to A results in an implant stiffness decrease, which, on one hand, lowers bone loss and, on the other hand, enhances the risk of interface failure. When solution B is compared to C, a reduction of 8% of bone resorption is noted with a slight increase of the peak value of the interface failure. On the other hand, by contrasting solution B to A, we note a significant increase (6%) of the peak value of interface failure, which is below the Hoffman failure strength, and a minor reduction (%) of the amount of bone resorption. The main benefit of solution A is 86

104 the maximum porosity of the microstructure that can promote bone ingrowth. Solution B, on the other hand, might be the preferred solution with respect to low bone resorption and interface failure. These observations emerge only by inspection of the objective functions selected in this work; we remark here that other parameters should be taken into account for the selection of the best implant. These include patient s bone characteristics, the range of activity, age, and desired level of bone mass preservation after implantation. For prescribed geometric loading and constraint conditions, we now compare b the metrics of resorbed bone mass ( m ) and distribution of interface stress ( f ( ) ) r of the optimal solution B with those of i) a currently-used fully dense titanium stem and ii) a cellular implant with a uniformly distributed relative density of 5%. Figure 4 6 and 4 7 illustrate the results of the comparison. For the solid titanium stem, the amount of bone resorption calculated through equation (4.3) is 67%, and the interface failure index F(b) obtained from equation (4.4) is Using the distribution of f ( ) generated around the titanium stem (Figure 4 7a), we observe that the maximum value of interface failure (.51) occurs at the distal end of the implant. As expected, this implant is much stiffer than the surrounding bone, thereby resulting in a higher amount of bone resorption. For the numerical validation, the interface shear stress of the titanium implant at the proximal region is also compared with the one obtained by Kowalczyk for a 3D model [17]. The mean and the maximum values of interface shear stress for the 3D titanium implant in the work by Kowalczyk [17] are.57 and.8 MPa, respectively. These values are.31 and.15 MPa respectively for the titanium implant in this paper. The contribution to the higher level of shear stress in the 3D model of Kowalczyk is the distribution of shear force on a smaller area. In Kowalczyk s study [17], the implant and bone are bonded only at the proximal region, while in our work the whole bone-implant interface is bonded, which results in a decrease of the mean and the maximum values of interface shear stress. 87

105 Non resorbed bone Resorbed bone Implant Figure 4 6: Distribution of bone resorption around (a) fully dense titanium implant, (b) cellular implant with uniform relative density of 5%, (c) graded cellular implant (solution B in Figure 4 5). Figure 4 7: Distribution of local interface failure f ( ) around (a) fully dense titanium implant, (b) cellular implant with uniform relative density of 5%, (c) graded cellular implant (solution B in Figure 4 5). The cellular implant with uniform relative density of 5% is approximately three times more flexible than the titanium stem. This implant can qualitatively simulate the behaviour of an implant made out of tantalum foam. For this stem, the amount of bone resorption and the interface failure index are about 34% and.87, respectively, and the interface failure is maximum (.71) at the edge of proximal 88

106 region. Compared to the solid titanium implant, the amount of bone resorption decreases by 5%, whereas the maximum interface failure increases about 4%. This shows that a decrease of the implant stiffness with uniform porosity distribution aiming at reducing bone resorption has the undesirable effect of increasing the risk of interface failure at the proximal region. This result confirms the findings of the previous work by Kuiper and Huiskes [115]. Figure 4 8: Polypropylene proof-of-concept of the optimal graded cellular implant (solution B in Figure 4 5). Figure 4 6c and 4 7c show the results for the graded cellular implant B. Its bone resorption and interface failure index are 16% and 1.15 respectively. The peak value of the local interface failure is.5. Compared to the titanium stem, both the amount of bone resorption and the interface failure peak decrease by 76% and 5%, respectively. With respect to the uniformly-distributed cellular implant, the 89

107 decrease of bone resorption and interface failure peak is of 53% and 65%, respectively. A graded cellular implant with optimized relative density distribution is thus capable of reducing concurrently both the conflicting objective functions. In particular, bone resorption reduces as a result of the cellular material which makes the implant more compliant; the interface stress, on the other hand, is minimized by the optimized gradients of cellular material. Figure 4 8 shows the polypropylene prototype of solution B, which was manufactured with the 3D printer Objet Connex5 [54]. A uniform tessellation and a square unit cell of 1.8 mm size were assumed to draw the model. The cell geometry was calculated from the average relative density using the Gaussian quadrature integration method described in section An STL file of the graded cellular implant, solution B, was finally used for rapid prototyping. 4.6 Conclusion The focus of this chapter has been on the methodology integrating multiscale analysis and design optimization to design a novel hip implant made of graded cellular material. The method can contribute to the development of a new generation of orthopedic implants with a graded cellular material that will reduce the clinical consequences of current implants. For the multiscale analysis, asymptotic homogenization has been used to capture the mechanics of the implant at the micro and macro scale. For the aspects of design optimization, NSGAII has been applied to find optimum gradients of material distribution that minimize concurrently bone resorption and bone-implant interface stresses. The results have shown that the optimized cellular implant exhibits a reduction of 76% of bone resorption and 5% of interface stress, with respect to a fully dense titanium implant. Although encouraging, these results have been obtained by applying a static loading regime to the implant, thus neglecting the impact of an applied cyclic loading that generally boosts the risk of fatigue failure. Bone has been also considered as consisting of a cancellous and a cortical part. In the next chapter, the methodology will be extended to design a microarchitectured material for a femoral implant against fatigue fracture caused by cyclic loadings on the hip joint as a result of walking. For the fatigue failure 9

108 analysis, the Soderberg diagram is developed for multiaxial loading conditions. Two cell topologies, square and Kagome, are chosen to obtain optimized property gradients for a two-dimensional implant. Bone is also modeled as a functionally graded material with the density distribution obtained from CT-scan data. 91

109 5 CHAPTER 5: Fatigue Design of a Mechanically Biocompatible Lattice for a Proof-of-concept Femoral Stem 5.1 Introduction An orthopaedic hip implant is expected to support dynamic forces generated by human activities. To avoid progressive and localized damage caused by daily cyclic loading, the prosthesis is to be designed for fatigue under high cycle regime. In the previous chapter, a static loading regime was applied for the design of the implant, thus neglecting the impact of cyclic loading that generally boosts the risk of fatigue failure. Here, the methodology is extended to design a spatially periodic microarchitectured material for orthopaedic hip implants under cyclic loading conditions. The material is composed of a graded lattice with controlled property distribution that minimizes concurrently bone resorption and interface failure. The implant microstructure is designed for fatigue fracture caused by cyclic loadings on the hip joint as a result of walking. The Soderberg diagram is used for the fatigue design under multiaxial loadings. Two cell topologies, square and Kagome, are chosen to obtain optimized property gradients for a two-dimensional implant. AH theory, described in Chapter, is used to address the multiscale mechanics of the implant as well as to capture the stress and strain distribution at both the macro and the microscale. The microstress distribution found with AH is also compared with that obtained from a detailed finite element analysis to verify the periodicity assumption of AH. Moreover, the metrics of bone resorption and interface shear stress are used to benchmark the graded cellular implant with existing prostheses made of fully dense titanium implant. To demonstrate the manufacturability, a set of proof-of-concepts of planar implants have been fabricated via Electron Beam Melting (EBM). The titanium alloy Ti6AL4V is used for the fabrication. Optical microscopy has been used to measure the morphological parameters of the cellular 9

110 microstructure, including cell wall thickness and pore size, and compared them with the nominal values. 5. Fatigue analysis of cellular implants In literature, there are several experimental and numerical studies focusing on the fatigue analysis of hip implants [55-6]. For example, fatigue loading conditions, ISO 76/3, have been applied to a hip stem to predict its elastic stress via large deflection finite element analysis [59]. It has been demonstrated via experiments that the high cycle fatigue-life of hip stems can be adequately predicted by using alternative fatigue theories, such as Morrow, Smith Watson Topper (SWT), and Goodman. The Soderberg theory has also been used to design a cemented implant for infinite life; the results have been proved to be accurate although more conservative than those obtained with Goodman and Gerber theories [55, 57]. Among the biocompatible materials used for reconstructive orthopaedics, porous tantalum has been lately the object of studies aiming at characterizing its fatigue fracture mechanisms [63, 64]. Similar to open cellular foams, porous tantalum has a random cellular microstructure which is typically imparted by the manufacturing process, involving a chemical deposition of pure tantalum on carbon skeleton [114, 136, 65]. Due to its pore structure, the fracture propagation of porous tantalum under fatigue occurs as a result of bending dominated failure mode of the unit cell [168, 63, 64, 66-68]. High level of stress at the cell joints nucleates cracks that propagate throughout a strut until the final break [63, 64, 69]. Several analytic methods have been proposed to study the fatigue life of cellular structures [19-19, 7, 1-1]; however, the majority fails to accurately capture the real stress distribution generated in the lattice cells, as shown in Chapter. This can lead to unrealistic results that may limit the use of these methods for fatigue design. Here, we resort to AH theory to determine the homogenized properties of the lattice material and capture the microscopic stress and strain distribution via the analysis of a RVE. It is also assumed the specimen to be free of defects, such as scratches, notches and nicks. As a result, the constant life 93

111 diagram can be constructed for the design of the material against fatigue failure [18]. Yield strength and endurance limit of the lattice material is used to construct the Soderberg diagram for the fatigue analysis of the structure under multiaxial loading conditions: m a 1 (5.1) y e SF where denotes the second norm operator to compute the vector length, y is the yield strength of the lattice for the given loading condition, and e is the endurance limit of the lattice computed from the yield strength through equation m 3.7. and a are, respectively, the mean and alternating macroscopic stresses and are calculated by the following relations: where m max min max min a, (5.a, b) max min and are the multiaxial macroscopic stresses that cause, respectively, the highest and the lowest values of the von Mises stress in the microstructure. In this study, the Soderberg diagram is applied to design a D graded cellular implant against fatigue failure. To generate the lattice, we select the square and kagome unit cells, as representative of bending and stretching dominated topologies, and their mechanical properties, calculate in Chapter, are used to construct the Soderberg diagram. For the material properties of the lattice, we consider Ti6Al4V [13] with mechanical properties described in Chapter Fatigue design of a hip implant with controlled lattice microarchitecture Figure 5 1 illustrates the methodological steps to design a graded cellular implant for infinite fatigue life. The approach combines multiscale mechanics and multiobjective optimization. The former deals with the scale-dependent material structure, where the local problem of the RVE is first solved, and then the effective 94

112 elastic moduli and yield strength are obtained and used as homogenized properties of the macroscopic model of the implant. The latter handles the conflicting nature of bone resorption and implant interface stress. A fatigue failure theory can thus be embedded in the procedure to design the implant for infinite fatigue life. A brief description of the main steps identified by the numbers in the flowchart is given in Figure 5 1. b () t 3 t 1 a c t (1) CT-scan data & FEA model of hip (3) Characterization of cell topology H E y e Homogenized elastic modulus Multiaxial yield surfaces Multiaxial fatigue surfaces (4) FEA, Macro-stresses/ strains (5) m e ij a ij Microstructure fatigue failure max min max min a, Soderberg Modified Goodman Goodman Gerber m ij y ij ult ij (6) Multiobjective optimization min F( b) { m ( b), F( b)} r (7) Update design variables No Converge Yes Optimal design SF 1 Safety factor 4% Mean porosity S.t. 5m P 9m Mean pore size t tmin Cell wall thickness withb { 1,..., m } Vector of design variables Relative density at each for 1 sampling point i where i i m Figure 5 1: Flow chart illustrating the fatigue design methodology of a graded cellular hip implant (1) A finite element model of the bone is created by processing CT-scan data of a patient bone. (, 3) A 3D lattice microstructure is considered as the building block of the implant, and its mechanical properties are predicted through AH. The homogenized elastic modulus, yield and fatigue surfaces of the cell topology under multiaxial loading conditions are obtained. (4, 5) From FEA, the mean and alternative macroscopic stresses are obtained, and used in the fatigue design diagram to determine the design safety factor (SF). In this study, the Soderberg s fatigue failure criterion is considered for the analysis. 95

113 (6) The two conflicting objective functions, bone resorption mr ( b ) and interface failure index F( b ), are minimized via a multiobjective optimization strategy subjected to a set of inequality constraints. The amount of bone resorption is determined by comparing the local strain energy per unit of bone mass between the preoperative and the postoperative situation, as described in detail in section 4.3. On the other hand, the interface failure index F( b ) is expressed by the following relation: b f ( j ) F( b ) min 1 (5.3) b f ( j ) da A A b where j is the loading case (1,, and 3), and A is the interface area. f ( ) is j defined as the interface failure caused by shear stress, and is expressed as, S s where is the local shear stress at the bone-implant interface, and S s is the b bone shear strength. In equation 5.3, the interface failure f ( ) is normalized with its average over the bone-implant interface area. The minimization of F( b ) will lead to a design with minimum and uniform shear stress distribution at the interface. The shear strengths of bone can be expressed as a function of bone apparent density according to the power law relation obtained by Pal et al. [5]: j Ss (5.4) During the optimization procedure, the values of mean porosity and pore size are selected to ensure bone ingrowth [45, 47], and the minimum thickness of the cell walls is determined by the resolution of the manufacturing process, i.e. the manufacturing limits. (7) The vector b of the design variables is updated until the set of nondominated solutions of the Pareto front are obtained. 96

114 The methodology described above is now applied for the design of a D graded cellular implant. Square and Kagome cell topologies, which are characterized in Chapter, are considered as the cell architecture of the implant. The lattice is designed to support the cyclic load of walking and is optimized to reduce bone resorption and interface stress. The FEA model of the femur and implant, loading and boundary conditions, and the results are described in the following sections. Frame 1 7 Mar 1 D Hip Relative Density Distribution ensity Distribution Bone density (a) (b) Figure 5 : a) D Finite element models of the femur and b) the prosthesis implanted into the femur. 5.4 Design of a D femoral implant with a graded cellular material D FEM model of the femur Figure 5 a shows the geometry of the femur considered in this work along with the applied loads and boundary conditions. CT scan data of a 38-year-old male, obtained through the visible human project (VHP) database of the national library of medicine (NLM, USA), is used to construct the 3D model of the femur. The stack of CT images are imported into ITK-SNAP [7] to create the STL file of the femur geometry by using the semi-automated segmentation process. The 3D 97

115 geometry is then created by using SolidWorks software package, and meshed with tetrahedron elements in ANSYS (Canonsburg, Pennsylvania, U.S.A). The apparent density for each element of the FE model is then determined from the Hounsfield value (HU) measured from CT data ranging from -14HU to 1567 HU. The CT data set represents a regular cubic grid where a HU value is assigned at each point. A linear relation between HU and apparent density is considered. The maximum value of HU corresponds to the densest region of the cortical bone with apparent density of. g/cm3, and HU value of water and its apparent density equal to zero. Once the apparent density of each CT grid point is obtained, elements of the FE model are superimposed on the CT grid points to evaluate the average of relative density for each element using the procedure described in [71]. From the apparent density distribution, the effective elastic moduli of bone are obtained through the relation [7-74]: E E 65.95< < ,.3 (5.5) An isotropic material model is considered for the bone, as this simplification does not lead to a noticeable difference from those results obtained by assigning to the bone orthotropic material properties [73, 74]. For the purpose of this exploratory study, the 3D geometry of the femur is simplified to a D model, which is assumed to have a side plate of variable thickness [75, 76]. The mid-frontal section of the femur is considered for the D model geometry, and the anterior and posterior parts of the femur are represented by the side plate. The D model and the side plate have variable thickness such that the second moment of area about the out-of-plane axis of the D model does not differ from that of the 3D model (Huiskes et al., 1987; Weinans et al., 199). The material properties of the front plate are extracted from the mid-frontal section of the 3D model, and the mechanical properties of the cortical bone are considered for the side plate. This simplification helps reduce the computational cost involved in the optimization process. Nevertheless, many of the essential features of the implant physics can still be captured with a D model. For mid-frontal loadings, 98

116 von Mises and interface stresses distribution can be calculated with an accuracy similar to that of a full 3D model [75]. As a result, the remodeling process in the metaphyseal and diaphyseal parts, and the failure at the bone-implant interface, can be approximated with a D geometry. The distal end of the femur is fixed to avoid rigid body motion, and three loading cases, 1,, and 3, representing the cyclic load during walking movements are applied to the hip joint and the abductor [48, 51, 77]. With respect to the load cases, magnitude and direction of the hip joint are given here together with the abductor forces in brackets: 1) 317 N at 4 from vertical (7 N at 8 from vertical), ) 1158 N at 15 from vertical (351 N at 8 from vertical), 3) 1548 N at 56 from vertical (468 N at 35 from vertical). ANSYS (Canonsburg, Pennsylvania, U.S.A) is used to build, mesh, and solve the D model. Assuming in-plane loading conditions, a D eight-node element type (Plane 8) is used since it can model curved boundaries with high accuracy FEM model of the cellular implant Figure 5 b illustrates the model of a cementless prosthesis implanted into the femur. The grid depicts the domain of the implant to be designed with a functionally graded lattice material. The variable of the lattice model is the relative density attributed to 115 sampling points, 3 rows along the prosthetic length and 5 columns along the radial direction. The values of relative density are constrained in the range.1 1 to prevent elastic buckling in the unit cell from occurring prior to yielding [151]. The relative density distribution throughout the implant is obtained by linear interpolation between the corresponding values at the sampling points. The homogenized stiffness matrix and the yield surfaces of each element are then computed from those values respectively illustrated in Figure 4 and Table 1. The former is employed to assemble the global stiffness matrix for the Finite Element (FE) solver, and the latter is used to construct the Soderberg diagram for fatigue analysis. Since the implant is designed to have a cellular microstructure with suitable pore size for bone ingrowth, it is assumed that the prosthesis and the surrounding bone are fully bonded [17]. This choice significantly decreases the computational 99

117 cost required for the stability analysis based on a non-linear frictional contact model [78]. 5.5 Results The procedure illustrated in section 5.3 is applied for the fatigue design of the implant after having calculated the yield and fatigue strengths of the microstructure, as described in Chapter and 3. To solve the multiobjective optimization problem, the non-dominated sorting genetic (NSGA-II) algorithm [53] is here used. The optimization continues until the user-defined number of function evaluations reaches 5 [53]. Figure 5 3a and b show the optimum relative density distributions for a D hip stem designed with square and Kagome cell topologies. The x axis represents the amount of bone resorption for the implanted hip; on the y axis is the interface failure index. Among the optimal solutions, we examine three representative relative density distributions: the extreme points, A and C, of the Pareto frontier, for which one objective function has importance factor and the other 1%, and solution B characterized by weight factors of 5%. For these solutions, the following characteristics are also illustrated in Figure 5 3: amount of bone resorption ( m ), interface failure index ( F (b) ), maximum shear interface failure ( f ( ) max r ), average porosity of each hip stem ( ), and design fatigue safety factor (SF) from the Soderberg diagram. Through a comparison of the results, we observe that an increase in implant porosity from point C to A results in a stiffness decrease of the implant. This increase, on one hand, lowers bone loss, and, on the other, enhances the risk of interface failure. The implant initial stability is the first objective in hip replacement surgery as it governs the long term performance and the success rate of the implant. Therefore, the implant with maximum stability, solutions C in Figures 5 3a and b, might be selected from the Pareto front. By contrasting solutions B and C, we note a significant reduction of bone resorption with only a slight increase of the interface failure index. From solution C to B (Figures 5 3a and b), the amount of bone resorption decreases by 6% and 51%, respectively, and the interface 1

118 failure index increases by 17% and 15%. Solutions B can thus be considered as preferred designs. It should be noted, however, that the selection of the best implant depends also on other factors, such as patient s bone characteristics, the range of activity, age, and the desired level of bone mass preservation after implantation. m r (a) m r F( b) f ( ) m r 14.1% (b) max 57% SF % F( b) 1 f ( ) max 66.% SF % F( b) 1 f ( ).43 F(b) max 66.%.63 F(b) 1 5 F(b) A B SF m r.3 m r 17.6% Fb ( ) 4.9 f ( ) % Fb ( ) 3.7 f ( ). max 58.3% SF.3 SF 1.58 B C A m r m r 58.3%.1..3 m r max 54% SF % F( b) 3.7 f ( ).3C max m r f ( ) m r m r 46.5% Fb ( ) 4.17 f ( ) % F( b) 3. f ( ) max 38.5% SF 1.73 SF % F( b) 3. max 45.5%.5 max 45.5% SF 1.73 Relative Density m r Figure 5 3: Trade-off distributions of relative density for the optimized cellular implant made of a) square and b) Kagome lattices. As can also be seen compared to the implant with square lattice, the implants designed with Kagome cells have better performance in terms of bone loss and interface shear stress. If solutions B in Figure 5 3a and b are compared, we note that the amount of bone loss decreases of about 4.% and the shear stress 11

119 concentration factor at the interface reduces by up to about 4.5%. While both implants have been designed for infinite fatigue life, the fatigue safety factor has improved approximately 81% for the implant designed by the graded Kagome cell topology. The reason for this is that Kagome is a stretching dominated cell with higher mechanical strength compared to the square cell for a given relative density. This provides a wider range of relative density for the optimization search to choose the design variable from, and control the stress distribution at the interface. Moreover, lower values of relative density can be selected to increase the implant flexibility and reduce bone resorption. We remark here that beside mechanical strength, other physical parameters, such as pore shape, interconnectivity, permeability and diffusivity of the unit cell, should be taken into account for the selection of a proper lattice cell for bone tissue scaffolding [178, 79-8]. Further research is required to address these aspects. 5.6 Verification of the numerical results During the optimization procedure, AH is applied for the multiscale analysis of the cellular implants. Although this method is quite effective in computing the stress and strain distribution at each scale, the results needs to be verified especially at regions where the underlying assumption, Y-periodicity of field quantities, is not satisfied. This can include regions with locally varying structure, areas with a high gradient of field quantities, or zones in the vicinity of borders [19-1, 83-85]. The multilevel computational method can be used for the analysis of these critical regions [19, ]. This method decomposes the computational domain into two levels of hierarchy: a) the detailed cellular microstructure and b) the homogenized medium. The region of interest, composed of a cellular microstructure, is modeled by a fully detailed FE analysis, and the results are compared with those obtained from the homogenization method to verify the periodicity assumption of AH. The following criterion can be defined to measure the departure from the periodicity conditions: 1

120 FEA F(, ) F(, ) ij ij ij ij RVE F( ij, ij ) RVE C (5.6) where the function F is a function of (, ) and can be defined, for example, as ij the average of the microscopic stress over the RVE. The superscript FEA refers to the evaluation of the function F via a detailed finite element analysis of a given microstructure. The macroscopic displacement solution, obtained from the homogenized model, is imposed on the unit cell boundary of the detailed FE model, and the stress and strain distribution within the microstructure is obtained. The superscript RVE, on the other hand, corresponds to the computation of F for each RVE through the imposition to the unit cell of a macroscopic strain with periodic boundary conditions. C is a user defined adaptation tolerance; C=.1 can be considered as an appropriate transition value to map the homogenized model to the detailed analysis of the local microstructure []. Here as functions, a) the average and b) the maximum value of von Mises stress over the unit cell, are considered respectively to verify the periodicity assumption of field quantities at the macroscale, and to assess the estimation of material yield in the lattice. We investigate two regions to verify the results of AH: one at the proximal part, where the Y-periodic assumption of field quantities is expected, and the other at the vicinity of the implant boundary, where this assumption does not hold. Figure 5 4 illustrates the macroscopic von Mises stress distribution throughout the square and kagome lattice implants associated with the loading condition number 1 applied to the hip joint. The mesh of the macroscopic elements at the vicinity of the implant border has been refined to capture the interface stresses with a higher resolution. The stress and relative density distribution, shown in Figure 5 4, corresponds to the solutions B in Figure 5 3. We can observe almost a uniform stress distribution in the proximal region of the implants; however, there is higher stress gradient at the vicinity of the implant boundary especially for the square lattice implant, which might affect the periodicity assumption of AH. ij 13

121 (a) von Mises stress distribution Relative density distribution Microstructure (b) SF 1.58 von Mises stress distribution 66.% f ( ) max.43 F ( b ) 1 m r 1.4% Relative density distribution Microstructure 1 F(b) 5 m r 13.4% F ( b ) 3.7 f ( ) max 58.3% SF m r SF % f ( ) max.5 F ( b ) 3. m r 7.4% MPa Relative Density Relative density Figure 5 4: Regions used to assess the accuracy and validity the AH model (left and middle) with respect to a detailed FE analysis of a 5 5 lattice microstructure (right). To perform the detailed FEA and verify the results of AH, the microstructures need to be constructed at the specified regions. For the square cell, a mm size is selected to satisfy the manufacturing constraint ( t min.1mm for.1) and to uniformly tessellate the regions with a 5 5 cells block. For the Kagome topology, the RVE has a rectangular shape with the same cell size as the square in the x direction. To produce the cell geometry from the relative density distribution,

122 Gauss points are assigned to each cell, as shown in Figure 4 4. Using a Gaussian quadrature integration [5], the average relative density of the RVE is obtained from equation 4.7. The relative density at each Gauss point is obtained with respect to its local coordinates within the macroscopic element of the homogenized model (Figure 4 4). Once the average relative density is obtained, the cell geometry can be constructed for both the square and Kagome lattices, as depicted in Figure 5 4. The displacement of the macroscopic solution is then imposed on the boundary of the cells block [], so as to calculate the stress distribution of the microstructure. The average and the maximum von Mises stress for the unit cells is then computed and used in equation 5.6 to verify the periodicity assumption and the results of AH. To recover the stress distribution throughout the microstructure via AH, the average macroscopic strain is needed over the RVE. Figure 5 5 illustrates the macroscopic strains distribution, xx, yy, xy, over the regions proximal and closed to the boundary of the square and Kagome lattice implants. As can be seen, there is a uniform variation of macroscopic strains in the proximal region, while there is high strain gradient close to the boundary which might affect the AH periodicity assumption. Therefore, the results obtained by the homogenization method needs to be verified. 3 3 Gauss points are assigned to each cell, and using the procedure described above, the average macroscopic strain for each unit cell is computed. The strain tensor is then used in equation.1 to obtain the microscopic strain distribution throughout the microstructure, from which the microscopic stresses are calculated via the constitutive equation of the base material. For the block at the proximal region, the microscopic stress distribution of the unit cell located at the center of the block is compared with those obtained from a detailed FEA. For the block at the implant border, the stress distribution within the cell in the middle of the first column of Figure 5 4 is considered. Based on the results of several analyses, we have observed that a change of the block position has a negligible effect on the unit cell stress distribution if the location of the selected unit cell is prescribed within the implant. 15

123 (a) -3 xx yy xy (b) -3 xx yy xy (c) -3 xx yy xy (d) -3 xx yy xy Figure 5 5: Macroscopic strain distribution (solution B in Figures 5 3a and b) as a result of load case 1 at (a) the proximal part and (b) the border of the square lattice implant, and (c) the proximal part and (d) the border of the kagome lattice implant. The von Mises stress distribution of the unit cells, obtained by AH and by detailed FE analysis, are given in Table 5 1. The average and the maximum value of von Mises stress over the unit cells obtained by AH are also compared with the detailed FE analysis, and the relative errors, defined by equation 5.6, are illustrated in Table

124 Table 5 1: Comparison of microscopic stress distribution obtained by detailed FEA and AH for the unit cells located at the proximal region and closed to the implant border. Proximal cell Boundary cell Proximal cell Boundary cell AH FEA Error ( ) 1.98% 1.8% 1.% 3.8% vm Error (max 7.1% 8.7% 8.% 18.6% { vm }) 1- The error corresponds to the average of von Mises stress over the unit cell - The error corresponds to the maximum von Mises stress over the unit cell For the square unit cell located in the proximal region, the average and the maximum value of von Mises stress can be estimated with an error of.98% and 7.1%, respectively. However, for the unit cells close to the boundary, a higher relative error for the microscopic stresses is observed as the Y-periodic assumption is not satisfied. For the Kagome lattice located in the proximal region, the relative error for the average and the maximum von Mises stress is 1.% and 8.%, respectively, percentages that increase to 3.8 and 18.6 when the Kagome unit cell is located at the implant boundary. Considering C=.1 as the criterion for creating the 17

125 transition from the homogenized model to the fully detail analysis, it can be seen that the periodicity assumption can capture the average of the macroscopic stress distribution throughout the implant with an error below 1%. The average macroscopic stress in Table 1 can be used to assess the material yield at the microscopic level of the lattice struts. For unit cells located at the implant boundary, where non-periodic local heterogeneity and non-uniform macroscopic field exist, the finite element mesh superposition method integrated with AH is better suited to capture the microscopic stress distribution with higher accuracy [1, 86-88]. This would significantly reduce the computational cost of the numerical simulations, since a fully detailed FEA of the implant might be unfeasible. The computational cost required to perform a single simulation of a fully detailed FE model of a cellular implant on a.4 GHz Intel processor is about 1,5 CPU seconds. Considering 5 function evaluations for the optimization procedure, the simulation time required for the fully detailed FE model would be seconds which is about 1 times higher than the simulation time needed for the analysis of a homogenized model. 5.7 Manufacturability of designed graded cellular implants To assess the manufacturability of the implant microarchitecture, D proof-ofconcepts implants were fabricated via EBM. Ti6Al4V powder supplied by ARCAM [5] with powder particle size between 45 and 1 micron was used for fabrication. The relative density distribution of solution C in Figure 5 3a was selected and mapped into a square lattice implant with the following cell size: 1 mm, mm, and 3 mm. A uniform cell tessellation was assumed to draw the geometric model. 33Gauss points were assigned to each cell, and the average of relative density was computed using Gaussian quadrature integration (equation 4.7). The geometry of each lattice cell was then calculated from the average relative density and the size of the unit cell. STL files of the cellular implants were created and finally processed by EBM. Figure 5 6 shows the implants with their representative microstructure. An optical microscope equipped with a digital camera was used to measure morphological parameters of the cell, such as cell wall thickness and pore size. No 18

126 sign of fracture or incomplete cell walls was inspected, an observation that shows good metallurgic bond between cell walls and structural integrity of the lattice. Figure 5 6: Fabricated Ti6Al4V graded cellular implants and the corresponding microstructures. Figure 5 7 provides the comparison of the average cell wall thickness and pore size between the nominal values and the fabricated parameters for each prototype. For the implant with unit cell size of 1mm, the average wall thickness and pore size exhibit a high relative error. The average wall thickness of the fabricated implant is 33.5% higher than the nominal value, while its average pore size is 53.6% lower than the nominal one. As can be seen in Figure 5 6, the pores are partially filled with non-fully melted powder particles, a side-effect that increases the wall thickness and decreases the effective pore size. Although the pores size range 19

127 Cell wall thickness (mm) Cell pore size (mm) between 6 and 56 μm is still an optimum range for bone ingrowth, the average pore size has decreased of 46%, from the nominal value of 66 μm to the real size of 36 μm Designed Fabricated 1.5%.%.5. Designed Fabricated.1% % % % Cell size (mm) Cell size (mm) Figure 5 7: Cell wall thickness and pore size of cellular implants fabricated with alternative cell size are compared to the nominal design parameters. Δ represents the difference between the average as-fabricated and the nominal values. On the other hand, for cell size of mm and 3 mm, the morphological parameters are in a good agreement with the design values. The difference between the average pore size decreases from 5.5% for mm unit cell size to a negligible value of.1% for 3mm cell size. While a cell size increase provides more control on the results, it may lead to a pore size greater than the suitable size for bone ingrowth. For the prototype with unit cell size of mm, the pore size varies between 33 and 135 μm with an average of 14 μm; this value is slightly higher than the optimum size for bone ingrowth. For the implant with unit cell size of 3mm, the pore size range is between 395 and 4 μm with an average of 166 μm, which is significantly higher than the optimum size prescribed for bone ingrowth. To provide a suitable environment for bone ingrowth, a periphery layer with either a smaller unit cell size or a conventional porous coating is suggested to be integrated on the implant surface. 11

128 5.8 Discussion In this section, we examine the results within the context of a performance comparison of other implants currently available in the market as well as on the manufacturability aspects. As a benchmark for the comparative study, a fully dense titanium implant is chosen. Its bone resorption and the distribution of local shear interface failure are determined, and then compared with those of the cellular implants represented by solutions B in Figure 5 3 for both the square and Kagome lattice. As expected, Figure 5 8a shows that for a fully dense implant, bone mass loss is about 71.4%. This initial postoperative configuration of bone loss is in good agreement with that in literature [5, 48]. A high amount of bone resorption is found throughout the medial and lateral part of the femur around the fully dense stem. Compared to the fully dense implant, the amount of initial postoperative bone loss of the square and kagome lattice implants decreases, respectively, by 53.8% and 58%. This shows that the design of a flexible implant through a graded cellular material has the beneficial effect of improving the load-sharing capacity of the implant with the surrounding bone, thereby reducing bone resorption. Figure 5 9 shows the distribution of the local shear interface failure, f ( ), around the fully dense titanium, square and kagome lattice implants. At each point, the maximum value of interface failure caused by any of three loading cases is shown. Since the function f ( ) is the interface shear stress normalized with respect to the local shear strength of the bone, high probability of interface failure is expected for f ( ) 1, whereas for f ( ) 1 the risk of interface failure is low. For the fully dense titanium implant, we observe that the maximum value of shear interface failure occurs at the distal end with magnitude of.96. This means that the shear stress is almost equal to the shear strength of the host bone, which may cause interface micromotion and prevent bone ingrowth. For the square and kagome lattice implants, the maximum shear interface failure reduces significantly of about 79% to.19 and., respectively. An optimized graded distribution of the cellular microarchitecture can reduce the stress distribution at the implant interface. 111

129 For the numerical verification, the interface shear stress of fully dense titanium implant is also compared with those obtained in literature [115, 15]. We have that the interface shear stress varies from 3.8 MPa at the proximal region to the maximum value of 4 MPa at the distal end, which is in good agreement with the stress regime available in [115, 15]. The fatigue analysis of the fully dense titanium implant shows that its safety factor is Although this value is about two times higher than the corresponding value of the kagome lattice implant, a safety factor of.3 for kagome lattice implant can be still considered as a reasonably safe margin for the design against fatigue fracture. To improve the implant fatigue strength, a lattice with smooth cell geometry could be considered. As it was shown in Chapter 3, the geometry of the cell joints, i.e. the locations where the struts converge, can be designed to level out any curvature discontinuity, thereby improving the fatigue strength of the cellular material. (a) (b) (c) Resorbed bone Non resorbed bone Implant Figure 5 8: Distribution of bone resorption around (a) fully dense titanium implant, (b) graded cellular implant with square topology (solution B in Figure 5 3a), (c) graded cellular implant with Kagome topology (solution B in Figure 5 3b). 11

130 (a) (b) (c).5 Figure 5 9: Distribution of local shear interface failure f ( ) around (a) fully dense titanium implant, (b) graded cellular implant with square topology (solution B in Figure 5 3a), (c) graded cellular implant with Kagome topology (solution B in Figure 5 3b). 5.9 Conclusion The focus of this chapter was to introduce a methodology for the design of the lattice microarchitecture of a hip implant against fatigue fracture as a result of cyclic loads in the hip joint. Asymptotic homogenization has been used for the multiscale analysis of the structure to obtain the stress distribution at the macro and micro scale, while the Soderberg fatigue criterion has been integrated in the procedure to design the implant for infinite fatigue life. The procedure was applied to design a D proof-of-concept implant with square and Kagome lattice topologies. The graded distribution of material properties throughout the implant was optimized using a multiobjective optimization procedure to simultaneously minimize bone resorption and interface failure. It has been found that for the square and kagome lattice implants the amount of bone loss is respectively 54% and 58% lower than that of a fully dense titanium implant. The maximum shear interface failure at the distal end of the implants decreases as well of about 79%. The numerical results obtained with AH have been verified via a detailed FE analysis. It was found that the periodicity assumption can capture the average of the macroscopic stress distribution throughout the implant with an error below 1%. 113