Hierarchical Analysis and Multiscale Modelling of Cellular Structures: from Meta Materials to Bone Structure

Hierarchical analysis and multiscale modelling of cellular structures: from meta materials to bone structure

A Dissertation Presented

Ramin Oftadeh

Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy in the field of Mechanical Engineering

Northeastern University Boston, Massachusetts December 2015

Abstract

Materials with structural hierarchy over nanometer to millimeter length scales are found throughout Kingdoms Plantae and Animalia. The idea of using structural hierarchy in engineering structures and materials goes back at least to Eiffel’s Garabit Viaduct and then Tower.

Incorporating hierarchy into honeycomb lattice structures has been the focus of a number of studies and has significance with regard to the application of honeycombs in impact energy absorption and structural protection, thermal isolation and as the structural core of sandwich panels. Here we explore the mechanical properties of two kind of cellular structures: hierarchical honeycombs and trabecular bone.

Hexagonal honeycomb structures are known for their high strength and low weight. We construct a new class of fractal-appearing cellular metamaterials by replacing each three-edge vertex of a base hexagonal network with a smaller hexagon and iterating this process. The mechanical properties of the structure after different orders of the iteration are optimized. We find that the optimal structure (with highest in-plane stiffness for a given weight ratio) is self-similar but requires higher order hierarchy as the density vanishes. These results offer insights into how incorporating hierarchy in the material structure can create low-density metamaterials with desired properties and function.

The second aim of this study was to explore the hierarchical arrangement of structural properties in cortical and trabecular bone and to determine a mathematical model that accurately predicts the tissue’s mechanical properties as a function of these indices. By using a variety of analytical techniques, we were able to characterize the structural and compositional properties of cortical and trabecular bones, as well as to determine the suitable mathematical model to predict the tissue’s mechanical properties using a continuum micromechanics approach. Our hierarchical analysis

2 demonstrated that the differences between cortical and trabecular bone reside mainly at the micro- and ultrastructural levels. By gaining a better appreciation of the similarities and differences between the two bone types, we would be able to provide a better assessment and understanding of their individual roles, as well as their contribution to bone health overall.

Keywords: structural hierarchy; anisotropy; honeycombs; cellular structures; cortical and trabecular bone; hierarchical analysis; continuum micromechanics; bone mechanical properties; bone composition and structure

Acknowledgement

I dedicate this current work to my family specially my wife who was very caring and understanding during my PhD study.

I would like to thank my advisors, Prof. Vaziri and Prof. Nazarian for their help and guidance during my PhD study. Also, I would like thank Prof. Hashemi for his help and comments during many projects I worked under his super vision.

Finally I would like to thank all of my lab-mates, Davood, Hamid and Ali who made these years joyful and filled with enthusiasm to work.

Table of contents

Abstract ...... 2 Acknowledgement ...... 4 Chapter 1 Introduction...... 15 Chapter 2 Biomechanics and Mechanobiology of Trabecular bone ...... 18 2.1. Introduction ...... 19 2.2. Trabecular bone biology...... 21 2.2.1. Cell populations ...... 21 2.2.2. Mechanosensation ...... 23 2.2.3. Mechanotransduction ...... 27 2.3. Tissue properties ...... 28 2.3.1. Tissue Composition ...... 28 2.3.2. Tissue Elastic Properties ...... 28 2.4. Elastic Behavior of Trabecular Bone ...... 35 2.5. Strength of trabecular bone ...... 42 2.6. Damage, fatigue and creep ...... 47 2.7. Conclusion ...... 50 Chapter 3 - Mechanics of Anisotropic Hierarchical Honeycombs...... 53 3.1. Introduction ...... 54 3.2. Analytical modeling of Elastic properties ...... 57 3.3. Fabrication using 3D printing ...... 62 3.4. Elastic properties of anisotropic hierarchical honeycombs: Analytical Modeling ...... 62 3.4.1. Elastic modulus in principal directions ...... 63 3.4.2. Effective shear modulus and Poisson’s ratio ...... 65 3.5. Elastic analysis: Numerical simulation ...... 66 3.6. Plastic collapse strength analysis ...... 67 3.7. Results and discussion ...... 68 3.7.1. Elastic modulus properties...... 68 3.7.2. Shear modulus and Poisson’s ratio ...... 73 3.7.3. Plastic collapse strength ...... 76 3.8. Concluding remarks ...... 79

Chapter 4 Optimal Fractal-like Hierarchical Honeycombs ...... 83 4.1. Introduction ...... 84 4.2. Optimal Elastic Modulus ...... 84 4.3. Concluding remarks ...... 94 Chapter 5 Hierarchical analysis and multi-scale modelling of rat cortical and trabecular bone . 96 5.1. Introduction ...... 97 5.2. Material and methods ...... 100 5.2.1. Specimen preparation ...... 100 5.2.2. Macrostructural properties ...... 102 5.2.2.1. Extrinsic structural properties ...... 102 5.2.2.2. Bone tissue density (흆풕) ...... 102 5.2.2.3. Mineral and matrix content ...... 103 5.2.3. Microstructural properties ...... 103 5.2.3.1. Morphometric indices ...... 103 5.2.3.2. Apparent material properties...... 103 5.2.4. Nanostructural properties ...... 104 5.2.4.1. Nanoindentation ...... 104 5.2.5. Compositional properties ...... 105 5.2.5.1. Total protein and collagen content ...... 105 5.2.5.2. Phosphate (PO4), hydrogen phosphate (HPO4), carbonate (CO3), carbonate/phosphate and protein/mineral content ...... 105 5.2.5.3. Calcium and phosphate (퐏퐎ퟒ) content ...... 106 5.2.6. Statistical analysis...... 107 5.2.7. Mathematical modeling ...... 107 5.2.7.1. Nanoscale ...... 110 5.2.7.1.1. Interaction of water and non-collagenous proteins with hydroxyapatite ..... 110 5.2.7.1.2. Interaction water and non-collagenous proteins with collagen ...... 111 5.2.7.2. Submicroscale ...... 112 5.2.7.3. Microscale ...... 113 5.2.7.4. Macroscale ...... 113 5.2.7.5. Elementary-phase stiffness values and modelling parameters ...... 115

5.2.8. Finite element analysis ...... 118 5.3. Results ...... 119 5.4. Discussion ...... 123 Chapter 6 Curved Beam Computed Tomography based Structural Rigidity Analysis of Bones with Simulated Lytic Defect: A Comparative Study with Finite Element Analysis ...... 128 6.1. Introduction ...... 129 6.2. Material and Methods...... 130 6.2.1. Specimen Preparation ...... 130 6.2.2. Imaging and Image Analysis ...... 131 6.2.3. Mechanical Testing ...... 131 6.2.4. Finite Element Analysis...... 131 6.2.5. Structural Rigidity Analysis ...... 132 6.3. An ideal case ...... 137 6.4. Results ...... 138 6.5. Discussion ...... 145 Chapter 7 Concluding remarks ...... 151 Chapter 8 Appendix...... 154 Chapter 9 References...... 157

List of Figures

Figure 2-1. An illustration of the hierarchical nature of trabecular bone ...... 20

Figure 2-2. An illustration of bone cell population...... 22

Figure 2-3. Strain-amplification model illustrating the osteocyte process in cross section and longitudinal section. Actin filaments span the process, which is attached to the canalicular wall via transverse elements. Applied loading results in interstitial fluid flow through the pericellular matrix, producing a drag force on the tethering fibers...... 25

Figure 2-4. Illustration of an integrin-based strain-amplification model ...... 26

Figure 2-5. (a) Scanning electronmicroscopy image of atrabeculum. (b) Indent locations across the width of a trabeculum. (c) Tissue Young modulus of trabecular bone using nanoindentation from skeletally mature sheep after undergoing overiectomy (OVX). (From Reference [134] with permission) ...... 33

Figure 2-6. (a) The layout of the cored specimens (S1–S7) demonstrated on a proximal femur image. Three-dimensional visualization of the average (b) Modulus (E) with the upper and lower limits of data at each site; and (c) bone volume fraction (BV/TV) distribution of human proximal femur. In (a), sites S1, S4, S6 and S7 form a loop or belt from the femoral head, through the neck and onto the trochanteric region, where the applied load (in a relatively uniform magnitude) traverses through the proximal femur and disburses into the cortical shaft. It is possible that the loads resultant from normal daily activities are mostly translated though this loop, whereas sites

S2 and S3 encounter the higher loads applied to the proximal femur for higher impact activities.

(From reference [157] with permission.) ...... 36

Figure 2-7. Schematic flowchart of computing multiscale material properties: (a) RVE homogenization for estimation of the effective material properties of the bone model at all

8 intermediate levels; (b) a correlation between the porosity of the geometrical models and their respective effective material properties (c) inverse local material properties model as a function of porosity and (d) computational model verification. (From reference [184] with permission.) 40

Figure 2-8. Spatial decomposition of trabecular bone. The initial binary image that served as input for our algorithm is shown in panel A. A skeletonization and optimization algorithm is applied to get a homotopic shape preserving skeleton as shown in panel B. This skeleton is then point-classified, thus arc-, surface-, border-, and intersection-points are shown in different colors.

(C) This point-classified skeleton is then spatially decomposed by removing the intersection points. (D) A two-way multicolor dilation algorithm was applied to find the volumetric extend of each element, yielding in the final spatially decomposed structure. (From reference [39] with permission.) ...... 41

Figure 2-9. Failure occurs at subregions with the lowest BV/TV values. Subregions number 1, 2,

3 and 4 with the lowest BV/TV values here coincide with the 4 regions that fail based on the visual data provided by the time-lapsed mechanical testing. (From reference [204] with permission) ...... 45

Figure 2-10. Reductions in secant modulus and accumulation of strain with increasing number of load cycles characterized the cyclic behavior of trabecular bone. Failure was defined as the cycle before which a specimen could no longer sustain the prescribed normalized stress, as indicated by a rapid increase in strain upon the subsequent loading cycle. Creep strain was defined by translation along the X-axis (ck), and damage strain was defined by the difference of the hysteresis loop strains (dk + d1). Total strain was the sum (ck + dk). (From reference [225] with permission.) ...... 49

Figure 3-1. Sections of the non-hierarchical honeycomb structure (left) and honeycomb structures with one (middle) and two orders (right) of hierarchy. In order for the intersections of newly generated hexagons lie on the edge of previous hexagons, ℎ푖 = 2푙푖 ∗ sin휃 for 푖 = 0,1,2, … should be satisfied. (b) Images of regular honeycombs with 푙0 = ℎ0 = 2 cm fabricated using three-dimensional printing (figure 3.1b is taken from [18])...... 56

Figure 3-2. a) Images of 1st order honeycombs of 휃 = 10o, 30o and 70o with 푙0 = 2 cm fabricated using three-dimensional printing. b,c) Free body diagrams of the subassembly of hierarchical honeycombs of 1st and 2nd order hierarchy used for finite element and theoretical analysis. 푁푖 and 푀푖 (푖 = 1 to 3) denote the reaction vertical forces and moments in the edges of the subassembly structures. Note that the solid lines represent the subassembly used for evaluating the elastic modulus/strength, while the whole structure (both dashed and solid lines) is used for shear modulus analyses...... 60

Figure 3-3. (a, b) The elastic modulus bandwidth of specific 1st to 4th order hierarchical honeycombs, normalized by the zero-order modulus of isotropic honeycomb (i.e., 30 degree wall angle) for 푦 and 푥 direction. (c) Surface plot of maximum 푥-direction and 푦-direction normalized elastic modulus of zero to fourth order hierarchical honeycomb...... 69

Figure 3-4. (a, b) Normalized elastic moduli of 1st order hierarchical honeycomb for 푦 and 푥 direction. (b,c) Normalized elastic moduli of 2nd order hierarchical honeycomb with 훾1 = 0.29, at 푦 and 푥 direction. The elastic moduli is normalized by that of zero order honeycomb with the same oblique wall angle 휃. Data markers are the numerical frame analysis results. Note that 퐸푥 and 퐸푦 are the same for 휃 = 30o...... 71

Figure 3-5. (a) The normalized shear modulus bandwidth of 1st to 4th order hierarchy. (b, c)

Normalized shear modulus of 1st and 2nd order hierarchy versus 훾1 and 훾2, respectively. For (a)

10 the shear modulus is normalized by that of zero order regular honeycomb (휃 = 30o), whereas for (b) and (c) it is normalized by that of zero order honeycomb with the same oblique wall angle

휃; Data markers are the MATLAB frame analysis results...... 74

Figure 3-6. a) Poisson’s ratio of 1st order hierarchical honeycomb in 푦-direction versus 훾1 for different values of oblique wall angle, 휃. b) Poisson’s ratio of 2nd order hierarchical honeycomb in 푦-direction with 훾1 = 0.29 versus 훾2 for different values of oblique wall angle, 휃. Star points are the numerical frame analysis results. Note that the results are normalized by non-hierarchical

Poisson’s ratio at the same angle, which varies significantly with anisotropy...... 76

Figure 3-7. (a, b) The normalized plastic collapse bandwidth of 1st to 4th order hierarchical honeycomb for 푦 and 푥 direction (c, d) The normalized plastic collapse of 1st order hierarchical honeycomb for 푦 and 푥 direction. Plastic collapse mechanisms for different values of 훾 are shown on the top. Note that for the plastic collapse of structure in y-direction with oblique angle of 휃 = 10o, only two mechanisms in left and right are dominant for the whole range of 훾. For (a) and (b) the plastic collapse strength is normalized by that of zero order regular honeycomb (휃 =

30o), whereas for (c) and (d) it is normalized by that of zero order honeycomb with same oblique wall angle, 휃...... 78

Figure 3-8. Plastic collapse strength versus elastic modulus for 1st, 2nd, 3rd, and 4th order hierarchical honeycombs under uniaxial loading for (a) 푦-direction and 휃 = 28oand (b) 푥- direction and 휃 = 45o, respectively. The plastic collapse strength and elastic modulus are normalized by those of regular honeycomb of same density. Note that the 1st order is shown by solid line, while higher orders are shown by areas of different shade...... 81

Figure 4-1 (a) Unit cell of regular to 4th order hierarchical honeycomb fabricated using 3D printing. (b) Unit cell of the hierarchical honeycombs with regular structure (left) and with 1st order hierarchy (right)...... 85

Figure 4-2. Maximum achievable stiffness (stiffness limit) of hierarchical honeycombs for different relative densities, 𝜌, and different hierarchical orders, n, normalized by the stiffness of regular honeycomb of the same mass. The limit for the maximum stiffness of hierarchical honeycombs are shown by dashed line. (∆) shows the stiffness limit found from scaling analysis

(equation 4.5). The inset shows comparison with the experimental results for hierarchical honeycombs with density of 𝜌 = 0.054...... 88

Figure 4-3. Topology of the stiffest hierarchical honeycombs at different relative densities. The results shows the values of 훾 coressponding to the optimim topology of the hierarchical honeycombs at different relative densities. Maximum achieavable hierarchical order and selected topologies of the stiffest hierarchical honeycombs on the specified relative density range are also shown on the top...... 89

Figure 4-4. (a) The order of hierarchy which yields stiffest hierachical honeycomb versus the honeycomb relative density. (numerical analysis (solid line) are shown together with scaling analysis results (equation 4.8) (dashed line) (b) Stiffness limit of the hierarchical honeycomb versus the relative density. (the numerical analysis (circle markers) are shown together with scaling analysis results (equation 4.9) (dashed curve)...... 91

Figure 4-5. Stiffness range for different order of hierarchy, 푛, versus relative density. Dashed line shows the stiffness limit of the hierarchical honeycomb for specified relative density

(equation 9) ...... 94

Figure 5-1. An illustration of the hierarchical nature of cortical and trabecular bone ...... 101

Figure 5-2. An illustration of the preparation process for cortical and trabecular specimens .... 102

Figure 5-3. Calculation of minimum cross-sectional area for (a) cortical and (b) trabecular specimens ...... 104

Figure 5-4. a) Infrared spectrum in the 휈4 PO4 domain of a synthetic nanocrystalline apatite

(maturation time 3 days, exempt of foreign ions). Curve-fitting (Lorentzian band shape) showing the different absorption bands and their attribution: b) Infrared spectrum in the 휈4CO3 domain of a synthetic nanocrystalline apatite (maturation time 3 days, prepared in the presence of carbonate ions). Curve-fitting (Lorentzian band shape) showing the different absorption bands and their attribution...... 106

Figure 5-5. Micromechanics representation of hierarchical structure of bone with four levels from nano to macro scale ...... 108

Figure 5-6. Tissue modulus frequency plots a) Cortical bone b) Trabecular bone...... 120

Figure 5-7. Comparison of axial elastic stiffness between micro mechanics modeling and finite element modelling with experiments for cortical and trabecular bone...... 121

Figure 5-8. Axial displacement contours of a) trabecular and b) cortical obtained from finite element analysis...... 122

Figure 5-9. Relation between a) Apparent mineral 𝜌푚푖푛 ∗ and organic 𝜌표푟푔 ∗ densities b)

Apparent water and noncollagenous proteins 𝜌푤푛 ∗ and mineral 𝜌푚푖푛 ∗ densities c) Apparent water and noncollagenous proteins 𝜌푤푛 ∗ and mineral 𝜌표푟푔 ∗ densities d) Apparent densities and e) Volume fractions of mineral (𝜌푚푖푛 ∗, 휙푚푖푛), organic (𝜌표푟푔 ∗, 휙표푟푔) and water and noncollagenous proteins (𝜌푤푛 ∗, 휙푤푛), versus extracellular bone density (𝜌푒푐)...... 126

Figure 6-1. The curve beam model representation. (ABCD section from right figure is shown on the left before and after deformation with corresponding strains) ...... 133

Figure 6-2. Ideal case A) schematic representation of hallow curved shaft used in ideal case example with radius of curvature 푅, inner diameter 퐷1 and outer diameter 퐷2. B) Percentage difference of critical strain of curved beam and straight beam model with Finite element based on changing 푅/퐷2 C, D, E) contour plot of vertical principal strain for sample 푅/퐷2 = 3 for straight beam, curved beam and finite element model...... 138

Figure 6-3. Linear regression between failure loads predicted by A) straight beam model B) curved beam model and C) Finite element analysis versus failure load from mechanical testing.

...... 139

Figure 6-4. Linear regression between failure load from mechanical testing versus straight beam model A) axial rigidity C) bending rigidity E) torsional rigidity and curved beam B) axial rigidity

D) bending rigidity F) torsional rigidity ...... 141

Figure 6-5. Bland–Altman plots for A) straight beam model (B) curved beam model and C)

Finite element failure load versus mechanical testing failure load...... 144

Figure 6-6. Fracture location as demonstrated by A) mechanical testing, B) Finite element analysis C) Curved beam model and D) straight beam model for a representative sample with defect. For straight and curved beam model the red areas shows the most critical locations which the fracture is imminent...... 146

Figure 6-7. Fracture locations for all specimens as predicted by FEA, curve beam and straight beam models and failure load from mechanical testing...... 147

Chapter 1 Introduction

Materials with structural hierarchy over nanometer to millimeter length scales are found throughout Kingdoms Plantae and Animalia. Examples include bones and teeth [1, 2], nacre

(mother-of-pearl) [3], gecko foot pads [4], Asteriscus (yellow sea daisy) [5], Euplectella sponge

[6], wood [2, 7] and water repellent biological systems [8]. The idea of using structural hierarchy in engineering structures and materials goes back at least to Eiffel’s Garabit Viaduct and then

Tower [9]. More modern examples include polymers [10], composite structures [11-13] and sandwich panel cores [14, 15]. The effect of structural hierarchy on mechanical and chemical properties of biological and biomimetic systems has been extensively documented [9-17]. The type and order of the hierarchy and the general organization of these structures play a significant role in their properties and functionality [16, 17].

Incorporating hierarchy into honeycomb lattice structures has been the focus of a number of studies [15, 18-21] and has significance with regard to the application of honeycombs in impact energy absorption and structural protection [22-26], thermal isolation [27] and as the structural core of sandwich panels [28-32]. Recently, a new generation of honeycombs with hierarchical organization was achieved by replacing nodes in the regular honeycomb with smaller hexagons

[18, 19].

Here we explore the mechanical properties of two kind of cellular structures: hierarchical honeycombs which is the designed meta-material and the other one is trabecular bone which is the biological material. In chapter 2, we first focus on the biology of trabecular bone and then on classical and new approaches for modeling and analyzing the trabecular micro- and macro- structure and their corresponding mechanical properties.

In chapter 3, anisotropic hierarchical honeycombs with various oblique-wall angles are compared to hierarchical conventional honeycombs (with 휃 = 30o). The stretches not only alter

16 the cell wall lengths, but it also changes the oblique wall angle, 휃 (which is equal to 30o in the conventional isotropic honeycomb). Note that uniform stretch leaves oblique cell walls still pointing at the centers of hexagons above and below. Since equal vertical and horizontal stretches would leave the hexagonal geometry undistorted, we ‘normalize’ the transformation: the length of an oblique cell wall was taken as fixed, while its angle is selected within the range 0 < θ < π/2.

In chapter 4, the optimal configuration of hierarchical honeycombs in terms of effective stiffness is determined for various structural densities using finite element simulation, scaling analysis and experiments. We investigate whether such structures can be systematically optimized, in particular by adjusting the number of hierarchical orders.

In chapter 5, the hierarchical arrangement of structural properties in cortical and trabecular bone will be explored and a mathematical model that accurately predicts the tissue’s mechanical properties as a function of these indices will be determined. By gaining a better appreciation of the similarities and differences between the two bone types, we will be able to provide a better assessment and understanding of their individual roles, as well as their contribution to bone health overall.

In chapter 6, a new method for Computed tomography-based structural rigidity analysis

(CTRA) derived from curved beam theory (curved CTRA) has been introduced. To that end, the failure load predictions from curved CTRA, traditional CTRA and Finite Element (FE) modeling has been compared to those of mechanical testing in an ex-vivo human model of femoral lytic defects. We hypothesize that CTRA will outperform traditional CTRA in terms of the accuracy of the predicated failure load and also the failure location and will correlate well with FEA and mechanical testing results. At the end, in chapter 7, concluding remarks and potential for future work will be mentioned.

Chapter 2 Biomechanics and Mechanobiology of Trabecular bone

2.1. Introduction

Trabecular bone tissue is a hierarchical, spongy, and porous material composed of hard and soft tissue components which can be found at the epiphyses and metaphyses of long bones and in the vertebral bodies (figure 2.1). At the macrostructural scale, the hard trabecular bone lattice, composed of trabecular struts and plates, forms a stiff and ductile structure that provides the framework for the soft, highly cellular bone marrow filling the intertrabecular spaces. At a microstructural scale, trabecular architecture is organized to optimize load transfer. Mineral and collagen content and architecture determine the mechanical properties of trabecular bone tissue

[33].

In the appendicular skeleton, trabecular bone transfers mechanical loads from the articular surface to cortical bone, whereas in the vertebral bodies it represents the main load bearing structure. Bone tissue mechanical properties and architecture of trabecular bone are two main factors which determine the mechanical properties of trabecular bone. Fragility fractures that arise in the context of metabolic bone diseases such as osteoporosis usually occur in regions of trabecular bone.

Several numerical tools such as micro finite element methods have been used to investigate the mechanical properties of trabecular bone from the compositional to organ levels [34-36].

Several new approaches relate the mechanical properties of trabecular bone to its compositional material properties [37], including decomposition of trabecular bone into its volumetric components (i.e. plates and rods) [36, 38-40]. In this review paper, we first focus on the biology of trabecular bone and then on classical and new approaches for modeling and analyzing the trabecular micro- and macro-structure and their corresponding mechanical properties.

Figure 2-1. An illustration of the hierarchical nature of trabecular bone

2.2. Trabecular bone biology

2.2.1. Cell populations

The integrity of the skeletal system is maintained by a continuous remodeling process that responds to mechanical forces and that results in the coordinated resorption and formation of skeletal tissue. This process occurs on a microscopically scale within bone tissue by basic multicellular units (BMUs) in which the cellular components are osteoclasts and osteoblasts [41].

Osteoclasts differentiate from hematopoietic progenitor cells of the monocyte/macrophage lineage, and it is hypothesized that they recognize and target skeletal sites of compromised mechanical integrity and initiate the bone remodeling process, although the exact signals and underlying mechanisms that target osteoclasts to specific sites remain unknown. [42] Osteoclastic bone resorption is followed by the recruitment of osteoblasts, which are derived from mesenchymal stem cells.[43, 44] Osteoblasts actively synthesize extracellular matrix on bone surfaces, which is subsequently mineralized.[45, 46]

Osteoblasts entrapped in matrix differentiate into osteocytes and compose 90% to 95% of the cells embedded in the mineralized matrix of bone.[47] Osteocytes residing in lacunae distributed within the matrix communicate through their interconnecting dendritic processes through a large lacuno-canalicular network which allows osteocyte communication with cells on the bone surface and access to the nutrients in the vasculature (figure 2.2). [48, 49] Osteocytes are ideally distributed to sense external mechanical loads[50-52] and to control the process of adaptive remodeling by regulating osteoblast and osteoclast function.[53]

Stem cells

Monocyte

Pre-osteoblast

Pre-osteoclast

Osteoblast Bone-lining cell Bone-lining cell

Osteoclast

Macrophage

Osteocyte

Figure 2-2. An illustration of bone cell population

2.2.2. Mechanosensation

A key regulator of osteoblast and osteoclast activity is mechanical strain. Bone has an intrinsic ability to adapt its morphology by adding new bone to withstand increased amounts of loading, and by removing bone in response to unloading or disuse. [54, 55] How the osteocytes sense the mechanical loads and coordinate adaptive alterations in bone mass and architecture is not yet completely understood.[56] However, it is accepted that mechanical loads placed on bones generate several stimuli that could be detected by the osteocyte. These include physical deformation of the bone matrix itself,[57-59] load-induced flow of canalicular fluid through the lacuno-canalicular network[60, 61], and electrical streaming potentials generated from ionic fluid flowing past the charged surfaces of the lacuno-canalicular channels. [62-66] In vivo, it is difficult to separate the 3 types of stimuli because mechanical loading will result in osteocyte exposure to bone matrix deformation, canalicular fluid flow shear stress, and associated streaming potentials.

[53] Weinbaum et al. [61] proposed a fluid flow shear stress hypothesis to explain how bone cells detect mechanical loading and developed a mathematical model for flow through the pericellular matrix surrounding an osteocyte process in its canaliculus. The model predicted that despite the small deformations of whole bone tissue and the small dimensions of the pericellular annulus

(typically 0.1 μm), the fluid flow shear stress on the membranes of the osteocyte processes was roughly the same as for the vascular endothelium in capillaries. This is relevant because of the hypothesis that fluid flow shear stress on osteocyte processes in the lacuno-canalicular porosity signaled cellular excitation for many actions such as detecting mechanical loading, activating ATP, cyclic AMP and prostaglandin E2 release, inhibition of osteocyte apoptosis and inhibition of osteoclast formation among others. [67]

Several studies demonstrated that bone cells are more responsive to fluid flow than to mechanical strain. [60, 68, 69] These studies strongly suggested that, in culture, direct mechanical strains appeared to be far less important than fluid flow shear stress in cellular excitation, as no biochemical responses were detected for cellular-level mechanical strains less than 0.5%.[60] This represents a fundamental paradox in bone tissue: cellular-level mechanical strains greater than

0.5% may cause bone tissue damage, yet tissue-level strains caused by locomotion seldom exceed

0.2%.[70, 71] This paradox suggests that whole tissue mechanical strains need to be amplified to elicit a cellular biochemical response.[67] A strain-amplification model for the mechanical stimulation of the osteocyte was developed based on the hypothesis that the dendritic process of the osteocyte behaves as a suspended cable by virtue of its attachment to adhesion proteins lining the canalicular wall.[72] The fibers mediating this attachment were proposed to be proteoglycans that spanned the fluid annulus and attached to the membrane of the osteocyte process. Accordingly, when the mineralized tissue is deformed, the fluid passing through the osteocyte pericellular matrix creates a hydrodynamic drag that would put the tethering fibers in tension, thereby producing a radial strain on both the process membrane and its underlying central actin filament bundle (figure

2.3). The predictions of this model demonstrated a cellular level strain amplification of 10- to 100- fold. According to the strain-amplification model, the activating mechanical signal was not fluid flow shear stress but the flow-induced drag on the tethering fibers.

Proteoglycans are not initiators of intracellular signaling, and therefore it has been suggested that the osteocyte processes might be attached directly to the canalicular wall by β3 integrins at the apex of canalicular projections (figure 2.4). [73] A theoretical model was developed that predicts that the tensile forces acting on these integrins can be as large as 15 pN, and thus provide stable attachment in the range of physiological loading. [74] The model also predicts that

axial strains caused by the sliding of actin microfilaments relative to the fixed attachments are two

orders of magnitude greater than whole-tissue strains thereby producing local membrane strains in

the cell process that can exceed 5%. In vitro experiments indicated that membrane strains of this

order are large enough to open stretch-activated cation channels.[60] It is likely that stretch-

activated ion channels play a role in the transduction of mechanical stimuli into a chemical

response in osteocytes. However, the involvement of specific ion channels in the mechanoresponse

of osteocytes has not been elucidated.

Force balance on transverse element

Transverse Velocity element Transverse Fm element Drag Cell Fd force Deformed membrane Cell membrane Actin filament . Membrane ...... Fimbrin Fi Fluid flow Deformed actin filament

Canalicular wall Deformed cell membrane Pericellular matrix

Force balance on cytoskeleton

Figure 2-3. Strain-amplification model illustrating the osteocyte process in cross section and

longitudinal section. Actin filaments span the process, which is attached to the canalicular wall via

transverse elements. Applied loading results in interstitial fluid flow through the pericellular

matrix, producing a drag force on the tethering fibers.

Y Transverse tethering

T6 element T5

T 4

T 3 X Intracellular anchor protein complex T2

T0 Canalicular Integrin protrusion

Figure 2-4. Illustration of an integrin-based strain-amplification model

Thus far, it has not been determined which cellular component of the osteocyte is the most important in sensing mechanical strain. [75] It has been proposed that the osteocyte only senses mechanical loads through its dendritic processes, and that the osteocyte cell body is relatively insensitive to mechanical strain. [50, 76, 77] Others have proposed that osteocytes sense strain through both the cell body and the dendritic processes, [78] or that the primary cilium, a single hair-like projection, is the primary strain-sensing mechanism in the osteocyte. [79, 80] There appears to be evidence for all three mechanisms, and it remains unclear whether the cell body, cell

26 processes, and cilia work separately or in conjunction to sense and transmit mechanical stimuli.

[81]

2.2.3. Mechanotransduction

An important step leading to adaptation of bone to mechanical loading is the transduction of physical stimuli into biochemical factors that can alter the activity of the osteoblasts and osteoclasts. In osteocytes, fluid flow shear stress induces the increase of intracellular calcium through ion channels and the release of intracellular stores. [63, 82, 83] The rise in intracellular calcium concentration is necessary for the activation of calcium/calmodulin-dependent proteins such as nitric oxide synthase (NOS). Additionally, the activation of phospholipase A2 results in the stimulation of arachidonic acid production and prostaglandin (PGE2) release mediated by the enzyme cyclooxygenase (COX). [84, 85] In bone, nitric oxide (NO) released from osteocytes and osteoblasts in response to loading inhibits resorption and promotes bone formation [86] and may also prevent osteocyte apoptosis. [87, 88] On the other hand, PGE2 released by mechanical loading acts through the β-catenin pathway to enhance connexin expression and gap junction function [89] and to protect osteocytes from glucocorticoid-induced apoptosis.[90]

Another family of molecules that very recently has been identified as mediator of the adaptive response of bone to mechanical loading is the Wnt family of proteins. Osteocytes use the canonical Wnt–β-catenin signaling pathway to transmit signals of mechanical loading to lining cells on the bone surface. [91] Wnt binds to specific receptors, called frizzled, and to low-density lipoprotein receptor–related proteins 5 and 6 (LRP5 and LRP6). These interactions lead to the stabilization of β-catenin, which translocates to the nucleus and regulates gene expression. [92]

Inactivating mutations in the LRP5 cause osteoporosis, [93] while gain-of function mutations in the LRP5 co-receptor increase Wnt signaling and result in higher bone mass. [94, 95]

Mechanostransduction involves many different pathways that includes fluid flow shear stress inducing the increase of intracellular calcium through ion channels and the activation of several molecules, such as arachidonic acid, prostaglandins, cyclooxygenase and NO, that will inhibit bone resorption and promote bone formation. At the gene expression level, there is the Wnt family of proteins that activate specific signaling pathways and interactions which will result in the translocation of the β-catenin protein to the nucleus and regulates gene expression.

2.3. Tissue properties

2.3.1. Tissue Composition

Trabecular bone, just like compact cortical bone consists of mainly hydroxyapatite, collagen, and water. However, trabecular bone has lower calcium content [96], tissue density

(1.874 g/mm3) and ash fraction (33.9%) [97] compared to cortical bone. Consequently, it has higher water content (27 % compared to 23% for cortical bone). These results are consistent with the fact that trabecular bone is more active in remodeling and, as a consequence, less mineralized. In other words, more recently formed bone has lower mineralization than older bone. Trabecular bone has high surface to volume ratio and the considerable bone remodeling compared to cortical bone (26% volume per year turnover rate for trabecular and 3% for cortical bone) [98].

2.3.2. Tissue Elastic Properties

In this section, unless noted otherwise, tissue properties imply the properties at the trabacular level. Collagen and mineral orientations and organizations are the most important factors to determine bone tissue properties since they are building blocks of bone at nano level structure [99]. At the microstructural scale, single trabeculae consist of groups of parallel lamellae bounded by cement lines primarily oriented parallel to trabecular surfaces. The lamellae are

28 composed of mineralized collagen fibrils with ellipsoidal shaped lacunae that house osteocytes distributed among the lamellae. The size and distribution of the lacunae are another important factor in bone microstructure since the elevated stress concentration located at the longitudinal direction of lacuna can cause microdamage in trabecular bone packets [100]. As of any other biological structure, bone tissue composition along with microstructural architecture determines the tissue mechanical properties of trabecular bone [99].

Characterizing the tissue-level mechanical properties of trabecular bone is relatively difficult due to the minuscule dimensions of the trabeculae. Various methods including buckling analysis [101, 102] and nanoindentation [33, 103-112] have been used to determine the tissue modulus of trabecular bone at trabecular level. Other methods include uniaxial tensile test [113,

114], bending test [115-117], ultrasonic measurements [107, 113, 118, 119], combinations of mechanical testing and finite element modeling [120-122], and microindentation [108, 123]. Other new methods have used macro-scale relationships between bone density and apparent elastic modulus [124] and digital volume correlation in conjunction with X-ray computed tomography

[125].

Studies of buckling analysis implement the Euler buckling formula [126] for elastic beam to find the maximum stress that a single trabeculae can bear. Townsend et al. [102] evaluated the buckling stress as a function of slenderness ratio (ratio of trabecular length to the minimum radius of gyration of the trabecular cross section) and then extrapolated it for an ideal slender ratio of a single trabeculae. They estimated the modulus as 14.13 GPa for dry and 11.38 GPa for wet tissue

(table 2.1). There are several drawbacks for this buckling test method which include difficulty in measuring the slenderness ratio of intact trabeculae and also the assumption of constant tissue modulus for the bone tissue.

Table 2.1-Tissue Elastic Modulus of Trabecular bone Tissue Testing Reference Bone type Modulus Technique (Gpa) 11.38 (wet), [102] Buckling Human proximal tibia 14.13 (dry) Human Proximal Tibia 23.6 ± 3.34 [127] Human Greater Trochanter 24.4 ± 2.0 Human Femoral Neck 21.4 ± 2.8 [33] Experiment- Human femoral neck 18.0 ± 2.8 [128] FEA Bovine proximal tibia 18.7 ± 3.4 [120] Human vertebra 6.6 ± 1.0 [121] Human vertebra 5.7 ± 1.6 [129] Human Proximal femur 10 ± 2.2 [130] Bovine proximal tibia 6.54 ± 1.11 [131] Human femur 13.0 ± 1.47 [107] Human femur 17.5 ± 1.12 [131] Ultrasonic Bovine femur 10.9 ±1.57 [118] technique Human tibia 14.8 ± 1.4 [118] Human vertebra 9.98 ± 1.31 [113] Human tibia 14.8 ± 1.4 [106] Human femoral neck 11.4 ± 5.6 [107] Human distal femur 18.1 ± 1.7 [109] Human vertebra 13.4 ± 2.0 Human femural head 21.8 ± 2.9 [132] Nanoindentation Human femur trochander 21.3 ± 2.1 Human distal radius 13.75 ± 1.67 [133] Human vertebrae 8.02 ± 1.31 [108] Porcine femur 21.5 ± 2.1 [111] Human tibia/vertebrae 19.4 ± 2.3 [134] Sheep proximal femur 20.78 ± 2.4

Ultrasonic technique, which mainly has been used to calculate the apparent elastic moduli of trabecular bone, [135] can also be used for determination of mechanical properties at tissue level.

The ultrasonic technique can be applied to either a whole specimen [118, 131] or a single trabecula

[113]. The reported tissue moduli for trabecular bone are different based on species and anatomical site: 13.0 ± 1.47 GPa [131] and 17.5 ± 1.12 GPa [107] for human femur, 10.9 ±1.57 GPa for bovine femur [131], 14.8 ± 1.4 GPa for human tibia, 9.98 ± 1.31 GPa for human vertebra [118] and 14.8 ± 1.4 GPa for human tibia [113] (table 2.1). In ultrasonic technique, the measurement is usually based on sample length and not ultrasound wave length which limits accuracy of the measurement.

In recent years, nanoindentation has been used to characterize the tissue properties of trabecular bone. Nanoindentation resolution can be as small as 0.05 μN in load and 0.01 nm in displacement [134]. The elastic modulus is calculated based on the unload portion of the displacement curve. Using nanoindentation, the mechanical properties of trabecular bone can be found at material level (table 2.1). Zysset et al. [106] reported 11.4 ± 5.6 GPa for average tissue elastic moduli of wet trabecular bone in the human femoral neck. 13.4 ± 2.0 GPa for tissue modulus is also reported using dry samples of vertebral trabeculae [109]. Turner et al.. [107] reported 18.1 ± 1.7 GPa for trabecular bone tissue from a distal femoral condyle which is higher compared to previously reported results. Using high resolution nanoindentation, Brennan et al.

[134] were able to measure elastic modulus across normal and ovariectomized sheep trabecular specimen. They reported that the modulus decreases as the distance from the trabecular core increases (figure 2.5). As seen in Figure 2.5c, elastic modulus ranges from 17.2 GPa in the superficial region to 23.4 GPa in trabeculae core. These results can be served as an input for finite element modeling of trabecular bone.

Table 2.2 - Mathematical relationships of Elastic modulus trabecular bone Reference Bone type range Elastic modulus (Gpa) R2

[136] Human vertebra ρapp: 0.11 – 0.27 E=2.1 ρapp ̶ 0.08 0.61 [137] Human proximal femur BV/TV: 0.15 – 0.40 E = 7.541(BV/TV) ̶ 0.637 0.88 αa: 0.174 – 0.662 [138] Human E = 84.37 (BV/TV)2.58α2.74 0.97 BV/TVb: 0.022 – 0.843 1.56 Human vertebrae ρapp: 0.11 – 0. 35 E = 4.730 (ρapp) 0.73 1.93 Human proximal tibia ρapp: 0.09 – 0. 41 E =15.520 (ρapp) 0.84 2.18 [127] Greater trochanter ρapp : 0.14 – 0. 28 E = 15.010 (ρapp) 0.82 Human femoral ρ : 0.26 – 0. 75 E = 6.850 (ρ )1.49 0.85 neck app app 1.83 Pooled ρapp: 0.09 – 0. 75 E = 8.920 (ρapp) 0.88 Human distal [139] ρ : 0.102 – 0.331 E = 10.88 (ρ )1.61 0.78 femur ash app [132] Human Femur BV/TV: 0.06 – 0.33 E = 10.89 (BV/TV)2.84 0.95 0.1063 (BV/TV) E1 = 0.02054 e 0.69 0.1499 (BV/TV) E2 = 0.006001 e 0.80 0.1753 (BV/TV) Human mandibular E3 = 0.001037 e 0.76 [140] BV/TV: 0.09 – 0.28 0.1332 (BV/TV) condyle G12 = 0.004812 e 0.81 0.1218 (BV/TV) G13 = 0.03215 e 0.82 0.1486 (BV/TV) G23 = 0.001458 e 0.73 b 1.87 [141] Rat femur ρapp : 0.301 – 1.553 E = 3.711 (ρapp) 0.74 Density in g/cm3. R2 is the determination coefficient a Ash fraction b Trabecular and cortical bone

Using finite element analysis in conjunction with experimental testing is another way of estimating tissue modulus of trabecular bone. First, apparent elastic modulus is determined based on a conventional method such as ultrasonic technique or mechanical testing and a 3D model of the sample is generated using μCT or μMRI imaging. By applying the same boundary conditions as in the experiments, the 3D model is solved assuming the tissue modulus as an arbitrary value

퐸푖. Assuming linear elasticity, the real tissue modulus 퐸푡 can be found as:

퐸푒푥푝 퐸푡 = 퐸푖 (2.1) 퐸퐹퐸푀

where 퐸푒푥푝 and 퐸퐹퐸푀 are the apparent elastic moduli of the bone based on experiment and finite element method, respectively. Niebur et al. [128] used high resolution finite element models and experiments to calibrate this linear model and reported 18.7 ± 3.4 GPa for bovine trabecular bone tissue modulus, which is in agreement with the results reported by Turner et al. [107] using nanoindentation. Using a similar method, 18.0 ± 2.8 GPa for human femur [33] and 6.54 ± 1.11

GPa for bovine tibia [130] also have been reported (table 2.1). The results for tissue elastic modulus show high variability across anatomical sites and species. Bayraktar et al. [33] argue that this discrepancy can be cause of end-artifacts or measuring of transverse modulus. Other factors include spatial sampling and anatomic site-dependence. Verhulp et al. [130] consider this variability as a result of variations in tissue density, sample size, strain rate and the way the strain is measured. Generally, the results based on the back calculation using finite element modeling shows higher variability than other methods which suggest that these methods find an “effective” tissue modulus to correlate the elastic modulus in the apparent level. On the other hand,

34 nanoindentation quantifies tissue modulus locally and can show heterogeneity along the trabecular bone tissue [134].

2.4. Elastic Behavior of Trabecular Bone

Studying the elastic behavior of trabecular bone is important as it is the main load bearing bone in vertebral bodies and also transfers the load from the joints to the cortical bone in long bones. Furthermore, it relates to the strength and affects fracture risk of the bone structure [142,

143]. The elastic properties of trabecular bone are showcased in its mechanical behavior during normal daily activity, and different experiments have shown it to have a linear behavior [144].

Therefore, linear elasticity can predict the elastic properties of trabecular bone. Based on the generalized Hook’s law, the elastic properties of the structure can be described by a fourth rank

tensor 퐶푖푗푘푙 , where it linearly relates stresses and strains in the structure as 𝜎푖푗 = 퐶푖푗푘푙휀푘푙. The elastic tensor in its most general form has 21 independent components. Trabecular bone generally is assumed to behave as an orthotropic structure with three planes of symmetry (9 independent components to fully describe the elastic behavior of the structure). However, it also can be described as a transversely isotropic structure which is rotationally symmetric around its axis of symmetry (5 independent components).

Predicting the mechanical properties of trabecular bone is challenging because of the heterogeneous [140, 145, 146] and anisotropic nature of bone [142, 147-151]. The elastic behavior, and in general the mechanical properties of the trabecular bone depend on loading direction [152-

155], anatomical site [127, 156, 157], size of the sample under consideration [150, 158, 159] and even cartilage damage adjacent subchondral trabecular bone [160]. Day et al. [160] have shown that volume fraction of subchondral trabecular bone increases to balance the loss of tissue modulus caused by cartilage damage. Nazarian et al. [157] have shown that the mechanical performance of

35 each region in human proximal femur is highly dependent on the corresponding trabecular microstructure (figure 2.6).

S2 and S3 encounter the higher loads applied to the proximal femur for higher impact activities.

(From reference [157] with permission.)

Many studies have shown that the elastic behavior of trabecular bone in compression and tension are the same [161, 162]. There are two different experimental setups to assess the elastic modulus of trabecular bone: mechanical testing [163, 164] and ultrasound techniques [135, 161,

165]. Mechanical testing can be performed in compression and tension [163] to evaluate the axial moduli or in torsion to evaluate the shear moduli [144]. To increase the reproducibility of mechanical testing, samples should first go through a number of conditioning cycles before reaching a steady state [163, 164]. Ultrasound technique is another mode to assess elastic modulus which can give nine orthotropic constants of bone specimens.

To generate the relationship between elastic properties and structural parameters of trabecular bone, mechanical and structural specific parameters are gathered from trabecular bone samples. Then, based on statistical analysis, they found the best fit between these parameters [138,

166]. Several single- and two parameter power law or linear functions have been proposed to predict the elastic modulus of trabecular bone (table 2.2)[137, 138, 141, 166]. Apparent density

(𝜌푎푝푝), which is the product of bone volume fraction and bone tissue density, is the primary component affecting the mechanical properties of trabecular bone [167, 168]. The general form of

푟 퐸 = 푎(𝜌푎푝푝) is proposed for this relationship, and several studies have found ‘r’ to be nearly two with high correlation rate between the mathematical relationship and experimental results.[167,

169]. However, as shown by Ulrich et al. [170], although 86% of the variation in elastic properties can be explained by bone volume fraction, the difference between elastic moduli can be up to 53% at certain volume fractions. Furthermore, due to the challenges mentioned at the beginning of this

37 section, these functions cannot individually predict the elastic properties of the trabecular bone in different anatomic sites and species (table 2.2).

Several studies have correlated elastic properties of trabecular bone to the fabric tensor, since fabric tensor is a descriptor of the anisotropy of trabecular bone, [165, 171]. The relation between elastic constants and fabric tensor was first introduced by Cowin [152]. Based on this model, Turner et al. [165] quantified it for trabecular bone using bovine samples. Later, Rietbergen et al. [171] found a reliable fit between components of fabric tensor and elastic constants in trabecular bone.

Numerical methods (mainly finite element models) based on nondestructive imaging such as μCT and μMR, have been introduced to determine the elastic moduli of trabecular bone from

3D generated models [172, 173]. Two methods have been employed to build the 3D finite element model based on the bone images: one method is to convert each voxel in the computer reconstructed representation of bone structure to a brick element with same size and coordinate

[174, 175]. Another method is to use marching cubes algorithm [172, 176] and divide the bone structure into tetrahedron elements which vary in size and shape based on their coordinates in the structure. Using this method the generated model is smoother; however, the computational effort increases as a result.

Many limitations and errors associated with mechanical testing, due to end-artifacts [149] and off- axis measurements, [155] can be eliminated using μFE analysis. Finite element models gather displacements and forces throughout the sample and not just the surfaces. Also, different boundary conditions can be applied to evaluate all independent components of the elastic tensor.

On the other hand, it is difficult to employ heterogeneity and anisotropy of bone tissue in the FE model, where simplifications are in order [177]. Micro Finite element analysis has been applied to

38 large sets of data to find the orthotropic components of trabecular bone.[178, 179] Both studies have shown that there are strong correlations between bone volume fraction and elastic and shear moduli, whereas this correlation is weak for Poisson’s ratio and bone volume fraction. For the FE analysis, many studies have shown that anisotropy of trabecular bone at tissue level has little impact on overall anisotropy of the trabecular bone, which means that anisotropy of bone architecture is dominant [120, 180]. To find the elastic tensor, many studies verified the use of orthotropic tensors to represent the elastic behavior of trabecular bone [181, 182]. Rietbergen et al.

[181] found that there is little error, assuming elastic tensor as orthotropic by comparing two whale vertebral samples. Odgaard et al. [182] observed that the orthotropic principal axes are nearly aligned with fabric tensor directions.

Expansion of computational resources have led to larger finite element models of up to 1 billion degrees of freedom [183]. However, these large finite element models require supercomputers for computation which are not available everywhere. On the other hand, homogenized macro scale models lag in accuracy. Therefore, multiscale finite element analysis of trabecular bone has been introduced [184]. Recently, Podshivaloy et al.. [184] have proposed a multiscale finite element model to fill the gap between homogenized macroscale and high resolution micro finite element models. Their model has several intermediate levels, in which bone material characteristics are updated based on change of porosity in different material scales (figure

2.7). In their model, effective mechanical properties vary at each intermediate level due to the changes in geometry. Using fourth order polynomial equation they further improve their model by correlating porosity and effective material properties [185].

In recent years, individual trabecular segmentation (ITS) of trabecular bone into rods and plates has been developed [36, 38-40] and has been used to identify and relate the separate plate-

39 rod configurations to mechanical properties of bone. One of the pioneering publications in this area was the work conducted by Stauber and Mü ller [39]. They decomposed the trabecular bone into its volumetric elements using skeletonization, optimization and multicolor dilation algorithms

(figure 2.8). The advantage of ITS is that it reduces the computational effort for finite element analysis and also examines the contribution of each component to the mechanical properties of trabecular bone. Hong et al. [186] examined the accuracy of this conversion by constructing idealized plate-rod and rod-rod microstructures at typical 휇CT resolution. They compared the ITS- based finite element model with a voxel-based finite element model and found that the ITS based

FE model significantly reduced the computational effort and yet preserved the accuracy of Young moduli and yield strength predictions.

Figure 2-7. Schematic flowchart of computing multiscale material properties: (a) RVE homogenization for estimation of the effective material properties of the bone model at all intermediate levels; (b) a correlation between the porosity of the geometrical models and their

40 respective effective material properties (c) inverse local material properties model as a function of porosity and (d) computational model verification. (From reference [184] with permission.)

, border-, and intersection-points are shown in different colors. (C) This point-classified skeleton is then spatially decomposed by removing the intersection points. (D) A two-way multicolor dilation algorithm was applied to find the volumetric extend of each element, yielding in the final spatially decomposed structure. (From reference [39] with permission.)

Helgason et al. [187] reviewed the several mathematical relationships between elastic modulus and apparent density and categorized the relationships based on specimen boundary

41 conditions, specimen geometry and anatomic site. Although they could not draw a definite conclusion from these relationships, they proposed a roadmap to standardize the mechanical testing and also set up indirect validation methodologies to find the most reliable mathematical relationships. As mentioned earlier, trabecular bone is a highly anisotropic and heterogeneous structure, whose mechanical properties are highly dependent upon anatomical site and species.

Therefore, the perfect mathematical model should be chosen based on these variables.

2.5. Strength of trabecular bone

Strength is defined as the ultimate stress which the structure can bear before failure, which is the maximum stress in the stress-strain curve. Studying the strength of trabecular bone is important, since it can be related to bone fracture, damage, which causes the bone remodeling, and failure of bone implants [188-190]. To understand the mechanisms of failure in trabecular bone, several models have been proposed. One of the earliest models is based on the cellular solid theory which uses the power law relationship between strength and bone apparent density [191]. In cellular solid theory, trabecular bone is assumed to be a structure with periodic boundary conditions, and a unit cell for trabecular bone is derived. Solving this unit-cell with basic analytical equations, cellular solid theory quantifies the effect of architecture and bone material properties on apparent mechanical properties.

Another model, considers trabecular bone as a lattice type structure, where the structure is solved using numerical methods such as finite element analysis [192, 193]. None of these models creates a realistic representation of trabecular bone. Recent improvements in high-resolution imaging and processing power make it possible to have realistic 3D representations of trabecular bone and then solve the model based on micro-finite element analysis [194-196]. The advantage of this method is that the sample under consideration can be tested multiple times with different

42 loading types and boundary conditions for failure analysis. In one of the earlier works in this area,

Fyhrie and Hou [194] used large scale nonlinear finite element analysis and found that the results depend on tissue mechanical properties. In another study, Reitbergen et al. [195] predicted the failure behavior of five human trabecular bone samples from tissue yield criteria, and found that the predicted strength is in the 15% range of measured strength from the experiments; however, ultimate strain was underestimated by 35%. In a later study, Niebur et al. [128] successfully predicted apparent ultimate compressive and tensile stresses and failure strains for seven bovine tibia samples using asymmetric tissue yield strains in tension (0.6 %) and compression (1.01 %).

When studying the failure behavior of trabecular bone, multi-axial analysis of trabecular bone has clinical importance, since multi-axial stresses can occur during fall, accidents and also in the bone implant interface [188-190]. For multi-axial strength analysis, bone volume fraction and architectural variation in specimens should be taken into consideration. There are several fracture criteria applied in material science which have been adopted for bone mechanics. Von Mises criterion is one of the first formulas to predict the bone fracture [138, 197]. This formula uses principal stresses 𝜎푖 and ultimate stress 𝜎푣 in compression (or tension) can be written as:

2 2 2 √(𝜎2 − 𝜎3) + (𝜎3 − 𝜎1) +(𝜎1 − 𝜎2) = 𝜎푣 (2.2)

It has been shown that this criterion may not be a good fracture predictor, since it does not account for asymmetry of strength in compression and tension [198]. The maximum principal stress criterion [199], maximum principal strain criterion [198] and maximum shear stress and strain criterion [199] have also been applied for predicting bone fracture. Mechanistic analysis using cellular solid criteria has also been used, since it accounts for different mechanisms of failure in

43 the analysis [200, 201]. This criterion has been applied to bovine tibia bone [198] with the percentage error between failure prediction and experimental failure as low as 7.7% for compression-shear and 5.2% for tension-shear. Among these failure criteria, Tsai-Wu criterion seems to be a very good candidate for trabecular bone failure analysis, since it accounts for anisotropy, loading direction and strength asymmetry of trabecular bone. The Tsai-Wu criterion which considers the existence of a failure surface in the stress space is in the form of:

푓(𝜎푘) = 퐹푖𝜎푖 + 퐹푖푗𝜎푖𝜎푗 = 1 푖, 푗 = 1,2 … 6 (2.3) here 퐹푖 푎푛푑 퐹푖푗 are the second and forth order tensors depending on tissue material properties,

2 and 𝜎푖 s are the principal stresses. The constraint 퐹푖푖퐹푗푗 − 퐹푖푗 ≥ 0 should also be satisfied for accurate analysis. One of the drawbacks of this criterion is the large number of constants which should be determined through experiments. Fenech and Keaveny [198] used this criterion and predicted fracture load for bovine femurs specimens within a 20% error. In another study, Keaveny et al. [202] found material dependent parameters for Tsai-Wu criterion as a function of apparent density using bovine tibial specimens. They found that failure surface depends on apparent density and is aligned with the principal material directions.

Several studies have shown that the axial strength of the bone structure better correlates with axial elastic modulus than structural density [142, 143]. In contrast to elastic modulus, which is the same in compression and tension, tensile strength is reported to be less than compressive strength for trabecular bone [142, 143]. Similar to elastic modulus, heterogeneity of trabecular bone makes it difficult to establish a general rule for strength. To overcome this issue, use of non- dimensional parameters such as strain has been proposed [203]. Nazarian et al. [204]showed that because of the heterogeneity of trabecular bone, subregions with minimal BV/TV values are better

44 predictors of trabecular failure than the average specimen BV/TV (figure 2.9). Similar to elastic analysis, studies show that the anisotropy of trabecular tissue material can be ignored, since in most cases trabecular bone elements (i.e. struts and plates) are loaded uniaxially [181]. This assumption forces the apparent principal axes of trabecular bone to coincide with the principal axes of microstructural anisotropy (i.e. principal axes of fabric tensor).

BV/TV [%]

12.0 8.0 4.0 0.0

1 2

3 Failure 4 Regions 5 6 7 8 9 10

sub-regions

Figure 2-9. Failure occurs at subregions with the lowest BV/TV values. Subregions number 1, 2,

3 and 4 with the lowest BV/TV values here coincide with the 4 regions that fail based on the visual data provided by the time-lapsed mechanical testing. (From reference [204] with permission)

Animal age, bone organ type, anatomic site and diseases such as osteoporosis have a significant impact on strength of trabecular bone structure by impacting bone apparent density, architecture and tissue mechanical properties. Regarding anatomic site, failure stresses for human bone can vary between 2 MPa for vertebral trabecular bone to 7 MPa for distal femoral bone [147,

205]. With regards to age, studies show that the ultimate strength decreases by almost 7% and 11%

45 for proximal femoral and vertebral bone, respectively between the ages of 20-100, mainly due to volume fraction decreases [147, 206]. Other studies have shown that strength variation may not be fully predicted by age. Maximum strength reported for human proximal tibial and vertebral bone occur in the age range of 40-50 years [207] and 30-40 years [147], respectively. Loading mode is another factor that impacts trabecular bone strength. Different studies on bovine tibial bone [196,

208] show that compression strength is higher than tensile and shear strength and shear strength is the lowest of all [208]. Tested bovine trabecular bone specimens are more plate like structures and so may not be plausible to generalize those findings for human trabecular bone, which its architecture varies by anatomic site. For human trabecular bone, Morgan and Keaveny [156] studied different anatomic sites including vertebra, proximal tibia, femoral greater trochanter and femoral neck and showed that yield strain is dependent upon anatomic site. They found that yield strain is higher in the femoral neck in compression than in vertebrae in tension. They also have shown that for all anatomic sites, yield strain for compression is higher than tension. Considering both anisotropy and heterogeneity of osteoarthritic trabecular bone, Tassani et al. [209] have shown that error in predicting compressive strength can be reduced by 17% in residual error.

The quest for having strength-density relation have led to various power law relationships, which most have reported that square relationship are more accurate [210]. Sanyal et al. [211] reported that compressive and shear strength depend on bone volume fraction with an exponent of

1.7 in human trabecular bone. Based on the shear to compressive strength ratio (0.44 ± 0.16), they concluded that shear strength is much weaker than compressive strength. For human trabecular bone, strength-density relations do not significantly change with anatomic site. However, Morgan and Keavery [156] argue that for yield criteria, the relationships predict more accurate yield strains when accounting anatomic site in the analysis. In addition, predicting the failure of trabecular bone

46 based on apparent density may not be accurate [121], since different microstructure failure mechanisms occur during apparent mechanical testing. However, experiments on human femoral head have shown that variations in ultimate strength correlate well with variations in bone volume fraction; and therefore, local BV/TV is a better strength predictor than overall BV/TV [212].

Regarding tensile and compressive strengths, Keaveny et al. [203] showed that the post yield load bearing capacity of trabecular bone for compression is higher than for tension. A study on bovine bone [203] and human bones [136] showed that differences between compressive and tensile strength increase linearly with elastic modulus. The interpretation of these results leads to the use of strains to describe the failure of trabecular bone, since they are independent of elastic modulus, nearly homogenized, and higher in compression. This interpretation is broadly supported by the experiments showing nearly no dependence of failure strain on apparent density [121, 136,

203]; however, they can depend on anatomic site [156]. Adopting strain failure criteria have shown to be accurate in finite element analysis of human vertebra bone [213] and rat tibia [214]. In this regard, there is a controversy about the isotropy of yield strength: Turner et al. [215] reported a weak relationship between failure strain and fabric tensor for bovine distal femur, suggesting that failure strains are isotropic; and Mosekilde et al. [147] found that for human vertebral body, the failure strains are anisotropic. In addition, experiments on bovine tibial bone have also shown slight anisotropy for shear strain [208] or no anisotropy at all [202]. It seems that although yield and ultimate strains varies across anatomic sites, since they are generally uniform within the particular site, they are the good predictors of trabecular bone failure.

2.6. Damage, fatigue and creep

Damage and repair of trabecular bone is a daily physiologic process [216]. Time dependent behavior and damage susceptibility behavior during cycling loading are the two main

47 characteristics of trabecular bone [217]. Damage has a direct effect on fracture risk in musculoskeletal diseases such as osteoporosis [216, 218] and bone remodeling [219] and can occur by implantation of prostheses [218] and bone joint diseases [220].

Experiments on trabecular bone specimens of bovine tibia [221] and human vertebra [218] show that, after the yield point, the structure unloads to a residual strain (1.05% for human vertebra with 3.0% compressive strain) with no stress [218]. This level of strain causes an 85% reduction in modulus and can be used as a quantitative measure of bone damage behavior. Several studies have shown that damage does not depend on bone organ type [222], density or anatomic site [221,

223], and that it occurs at the compositional level of collagen and hydroxyapatite [221, 224]. Along with that, Haddock et al. [225] showed that fatigue behavior of human vertebral and bovine tibial trabecular bone are similar by quantitative comparison of cycles to failure in cycling loading. They conclude that dominant failure mechanisms are in bone ultrastructural level for cyclic loading, regardless of anatomical site and species. Figure 2.10 shows the cyclic test data for human vertebral bone which shows the progressive loss of modulus and accumulation of strain similar to tibial bovine trabecular bone tests [226].

Creep is the tendency for bone to permanently deform under applied mechanical loads, and fatigue is the weakening of bone under repetitive or cyclic loading. Trabecular bone shows the classical creep characteristics with three phases: high elastic strain response, steady state response, and necking (which the strain rate exponentially increases) [227]. Creep and cyclic loading tests are implemented to model daily mechanical loadings on trabecular bone. Then, the standard stress- life and strain-time curves are plotted to understand the effects of these types of loadings on trabecular bone structure. Fatigue test results, conducted in-vitro and without considering bone

6 healing, show that cyclic loading reduces stiffness up to 70% after 10 cycles [226]. Therefore, the

48 results can serve as a lower bound for the lifetime of bone. Using specimens from human femur and bovine vertebra, Dendorfer et al. [228] showed that in cyclic loading, strain localizes even at very low load levels, and microcracks are induced just after load cycles. Consistent with this finding, it has been shown that microcracks and microdamage propagation are major failure mechanisms and result in large specimen modulus reduction [229].

49 along the X-axis (ck), and damage strain was defined by the difference of the hysteresis loop strains

(dk + d1). Total strain was the sum (ck + dk). (From reference [225] with permission.)

2.7. Conclusion

Volume fraction, trabecular tissue material properties and architecture determine the mechanical properties of trabecular bone. These features are of great interest for the study and understanding of biomechanics and mechanobiology of trabecular bone.

Bone cell population is comprised of osteoblasts, osteoclasts and osteocytes. Osteoblasts are derived from hematopoietic progenitor cells; recognize and target specific skeletal sites; and begin the bone remodeling process. Osteoblasts differentiate from mesenchymal stem cells; are recruited when remodeling process starts; and actively synthesize extracellular matrix on bone surface and will later differentiate into osteocytes. Osteocytes compose approximately 95% of the cells in the mineralized matrix of bone; sense mechanical loads; and control the process of adaptive remodeling by regulating osteoblast and osteoclast function.

The way by which osteocytes sense mechanical strain has yet to be determined. Many theories have been proposed to explain this. For instance, the osteocyte only senses mechanical loads through its dendritic processes, and that the osteocyte cell body is relatively insensitive to mechanical strain. Alternatively, other authors proposed that osteocytes senses mechanical loads through both, the cell body and dendritic processes, or that the primary cilium is the strain-sensing mechanism. This issue is still undetermined since evidence for all theories have been found.

Mechanotransduction is the mechanism by which these mechanical strains are transmitted to bone cell to maintain bone tissue. Several studies demonstrated that bone cells are more responsive to fluid flow than to mechanical strain. These studies strongly suggested that, in culture,

50 direct mechanical strains appeared to be far less important than fluid flow shear stress in cellular excitation as no biochemical responses were detected for cellular-level mechanical strains less than

0.5%. Different pathways as the induction of the increase of intracellular calcium through ion channels and the activation of several molecules are involved in mechanotransduction.

Additionally, the β-catenin protein, at the gene expression level, is theorized to be involved in this process.

The results for tissue elastic modulus show high variability across anatomical sites and species as shown in table 2.1. This discrepancy can be caused by several reasons including end- artifacts, measuring of transverse modulus, spatial sampling, anatomic site-dependence, variations in tissue density, sample size, strain rate and the way the strain is measured. [33, 130]. As seen in table 2.1, tissue modulus calculation using back calculation with finite element modeling shows higher variability than other methods such as nanoindentation, which calculate the tissue modulus locally [134].

With respect to apparent elastic behavior of trabecular bone, Helgason et al. [187] compared different mathematical relationships and have shown high discrepancy among these relationships. They have suggested a road map to standardize the mechanical testing and set up indirect validation methodologies to find the most reliable mathematical relationships. Based on the highly anisotropic and heterogeneous nature of trabecular bone, the perfect mathematical relationship should be chosen considering these variables.

In recent years, micro finite element analysis has provided a substantial tool for researchers to evaluate different aspects of mechanical properties of trabecular with high accuracy. The growth of these computational tools has led to finite element models with almost 1 billion degrees of

51 freedom. [183]. In addition, individual trabecular segmentation (ITS) of trabecular bone into its basic structures of rods and plates has been developed to relate the different rod-plate configurations to mechanical properties of trabecular bone [36, 38-40].

Regarding strength of trabecular bone, it appears that although yield and ultimate strains are good predictors of trabecular bone failure, and as they vary across anatomic sites, they are generally uniform within the particular site. With respect to damage, several studies have shown that damage behavior of trabecular bone does not depend on bone organ type [222], density or anatomic site [221, 223]; and therefore damage occurs at the compositional levels of collagen and hydroxyapatite [221, 224].

As discussed earlier, developing high resolution finite element modeling and sophisticated experimental tools and techniques have greatly improved our understanding of trabecular bone complexity and its behavior under different types of loading. In future, multidisciplinary approaches and multi-scale modeling of trabecular bone can address more complex behavior of this biological tissue, reduce the computational time, and maintain model accuracy.

Chapter 3 - Mechanics of Anisotropic Hierarchical Honeycombs

3.1. Introduction

Materials with structural hierarchy over nanometer to millimeter length scales are found throughout Kingdoms Plantae and Animalia. Examples include bones and teeth [1, 2], nacre

(mother-of-pearl) [3], gecko foot pads [4], Asteriscus (yellow sea daisy) [5], Euplectella sponge

[6], wood [2, 7] and water repellent biological systems [8]. The idea of using structural hierarchy in engineering structures and materials goes back at least to Eiffel’s Garabit Viaduct and then

the toughness but decreases the strength, suggesting that an optimal level of hierarchy could be defined.

[18, 19]. One or two orders of optimized hierarchical refinement offered up to 2 and 3.5 times the in-plane stiffness [18] and almost 2 times the plastic collapse strength [19, 20] of conventional honeycomb with the same mass. Majority of these works address the mechanical and thermal

54 properties of isotropic honeycomb structures. However, there are relatively little investigations on the mechanical properties of honeycomb structure with stretched cells resulting in anisotropic honeycomb structures. The present paper extends these previous works by horizontally/vertically

‘stretching’ or reformulating the underlying hexagonal network prior to the hierarchical refinement steps, so that the developed structure is no longer isotropic (Wall thickness is maintained uniform, while being adjusted to have fix overall average density as hierarchy is introduced).

In this work, anisotropic hierarchical honeycombs with various oblique-wall angles are compared to hierarchical conventional honeycombs (with 휃 = 30o). The stretches not only alter the cell wall lengths, but it also changes the oblique wall angle, 휃 (which is equal to 30o in the conventional isotropic honeycomb). Note that uniform stretch leaves oblique cell walls still pointing at the centers of hexagons above and below. Since equal vertical and horizontal stretches would leave the hexagonal geometry undistorted, we ‘normalize’ the transformation: the length of an oblique cell wall was taken as fixed, while its angle is selected within the range 0 < θ < π/2.

The distorted hexagons of the underlying network therefore have height 2푙0푐표푠(휃) and horizontal edge length 2푙0푠푖푛(휃) (figure 3.1a). In hierarchically refined structures of uniform thickness, the structural organization is uniquely defined by the ratio of the introduced oblique edge lengths (푙1 and 푙2, respectively, for first and second order of hierarchy) to the original hexagon’s oblique edge length, 푙0. These are denoted 훾1 = 푙1/푙0 and 훾2 = 푙2/푙0 , etc., where 푙0, 푙1, 푙2 are defined in figure

3.1a. At each order of hierarchy the introduced horizontal edge length conforms to ℎ푖 = 2푙푖푠푖푛(휃), where 푙푖 is the introduced oblique edge length, 휃 is the oblique wall angle and ℎ푖 is the introduced horizontal edge length.

Figure 3-1. Sections of the non-hierarchical honeycomb structure (left) and honeycomb structures with one (middle) and two orders (right) of hierarchy. In order for the intersections of newly generated hexagons lie on the edge of previous hexagons, ℎ푖 = 2푙푖 ∗ sin(휃) for 푖 = 0,1,2, … should be satisfied. (b) Images of regular honeycombs with 푙0 = ℎ0 = 2 cm fabricated using three- dimensional printing (figure 3.1b is taken from [18]).

Here we explore up to four orders of hierarchy. The elastic properties of first and second order hierarchy are evaluated theoretically by Castigliano’s method and compared to a matrix frame analysis carried out in MATLAB. The elastic moduli of third and fourth order hierarchy are therefore evaluated numerically only. In Section 2, the effective 푥 and 푦 elastic moduli are determined. In Section 3, the effective shear modulus and Poisson’s ratio of the structure are also examined. In Section 4, the effective plastic collapse strength for uniaxial in-plane loading in principal structural directions is determined using elastic-plastic beam elements in the finite element package ANSYS. In section 5, conclusions and potential for further performance improvement of hierarchical honeycombs are presented.

3.2. Analytical modeling of Elastic properties

Elastic modulus in principal directions

we use Castigliano’s second theorem [230] to determine analytically the elastic moduli of first and second order hierarchical stretched honeycombs in their principal directions. The cell walls of thickness 푡 consist of an isotropic elastic material with elastic modulus 퐸푠, Poisson’s ratio

휈푠 and yield stress 𝜎푌. Since a conventional hexagonal network extending to infinity has six-fold rotational symmetry, any linear second order tensorial operator (e.g., thermal conductivity) or linear fourth-order operator (e.g., elastic modulus) must be isotropic. However when anisotropy is present, the resulting ‘material’ has only two-fold rotational symmetry, and an orthotropic elasticity tensor. The macroscopic in-plane elastic behavior of an orthotropic material can be described by 5 constants (i.e., 퐸푥, 퐸푦, 휈푥푦, 휈푦푥 푎푛푑 퐺푥푦) , of which four are connected by the reciprocal relation [200]: 퐸푥휈푦푥 = 퐸푦휈푥푦. Here, 퐸푥 and 퐸푦 are the elastic moduli in the x and y directions normalized by material elastic modulus 퐸푠, and 휈푥푦 and 휈푦푥 are orthotropic Poisson’s

57 ratios in the x and y directions, defined as (− strain in second index direction)/(strain in first index direction) due to uniaxial stress in first index direction. Thus, for network angles other than 300, instead of two elastic constants governing in-plane deformation, the structure has four.

For the analytical investigation, each cell wall is treated as an Euler-Bernoulli beam, and the stored bending energy is evaluated based on the loading conditions. The modulus is determined from unit-cell boundary point displacements. Following [18], the far-field uniaxial stresses 𝜎푦푦 =

−퐹/ (3푙0sin (휃)) or 𝜎푥푥 = −푃/ (푙0cos (휃)) were imposed to determine the 푦- and 푥-direction elastic moduli, where 푙0 is the oblique wall length of the underlying zero-order honeycomb, and 휃 is the oblique wall angle. Vertical stress is equivalent to applying a vertical force 퐹 at every cut point of a horizontal line passing through the mid-points of oblique edges in a row of underlying

(i.e. no hierarchy) hexagons (such as points 퐻 and 퐻′ in figure 3.1a). Horizontal stress is equivalent to applying a horizontal force 푃 at every cut point of a vertical line passing through the midpoints of oblique edges in a row of underlying (i.e. no hierarchy) hexagons (such as points 푉,

푉′, 푉′′푎푛푑 푉′′′ in figure 3.1a). Due to 180o rotational symmetry, these cut points are moment-free for any choice of uniform remote stress components (i.e. 𝜎푦푦, 𝜎푥푥 and 휏).

st figure 3.2b shows a unit cell with 1 order hierarchy subjected to both vertical (𝜎푦푦) and horizontal loading (𝜎푥푥), however in this section they are analyzed separately. Due to the geometrical and force symmetry, we may apply Castigliano’s method to a reduced subassembly, namely the upper half of the unit cell. For loading in the 푥- and 푦-directions, point 4 is subjected only to force −푃 in the x direction or force −퐹 in the 푦 direction. (For shear loading, to be considered later, the bottom half of the unit cell is also needed, as shown with dotted lines, and non-symmetrical forces as shown are also required). Due to the structural symmetry, point 3 is

58 clamped for all the analysis [18]. At the subassembly centerline cut points 1 and 2, the 푦-direction forces acting on the upper subassembly are denoted by 푁1 and 푁2, and moments are denoted by

푀1 and 푀2. There can be no horizontal force acting on the upper half-assembly at point 1 because of reflection symmetry about the 푥 axis.

For first order hierarchy, the elastic bending energy stored in the statically indeterminate subassembly can be expressed as a function of the external force 퐹 or 푃, and the force and moment

2 푁1 and 푀1, applied to the structure at point 1: 푈(퐹 표푟 푃, 푀1, 푁1) = ∑푎푙푙 푏푒푎푚푠 ∫(푀 /(2퐸푠퐼))푑푠.

Here, 푀 is the bending moment at the location 푠 along a cell wall, and 퐼 is the beam’s area moment of inertia, which is constant because of the uniform-thickness refinement. To simplify the analysis, we have assumed the normalized structural density to be so low that all macroscopic deformations can be attributed solely to bending of cell walls. (Effectively, axial and shear stiffness are taken to be infinite.)

Assuming zero vertical displacement and zero rotation at points 1, 3, and also along the horizontal beam connecting points 2 and 3 (due to symmetry), 푁1 and 푀1 can be obtained from 휕푈/휕푁1 = 0 and 휕푈/휕푀1 = 0. These two relations allow 푁1 and 푀1 to be expressed algebraically in terms of

퐹 for vertical loading, or 푃 for horizontal loading. The displacements 훿푦 at the point 4 can then be

found from 훿푦 = 휕푈(퐹)/휕퐹|푁1,푀1 and 훿푥 = 휕푈(푃)/휕푃|푁1,푀1, which allows to obtain strains in the principal directions of structure (for evaluating Poisson’s ratio, both 퐹 and 푃 should be applied simultaneously).

st o o o Figure 3-2. a) Images of 1 order honeycombs of 휃 = 10 , 30 and 70 with 푙0 = 2 cm fabricated using three-dimensional printing. b,c) Free body diagrams of the subassembly of hierarchical

st nd honeycombs of 1 and 2 order hierarchy used for finite element and theoretical analysis. 푁푖 and

푀푖 (푖 = 1 to 3) denote the reaction vertical forces and moments in the edges of the subassembly structures. Note that the solid lines represent the subassembly used for evaluating the elastic modulus/strength, while the whole structure (both dashed and solid lines) is used for shear modulus analyses.

For second order of hierarchy, elastic energy in the section model of figure 3.2c is obtained as a function of the external force 퐹 or 푃, and unknown reaction forces and moments 푁1, 푀1, 푁2

2 and 푀2 at points 1 and 2: 푈 = 푈(퐹 표푟 푃, 푁1, 푀1, 푁2, 푀2) = ∑푎푙푙 푏푒푎푚푠 ∫(푀 /(2퐸푠퐼))푑푥 .(where

푁3 and 푀3 can be written as functions of 푁1, 푀1, 푁2, 푀2 and 퐹 or 푃 by means of the equilibrium equations). 푁1, 푀1, 푁2 and 푀2 are obtained by invoking the concept of the least work: 휕푈/휕푁1 =

0 , 휕푈/휕푀1 = 0 , 휕푈/휕푁2 = 0 , 휕푈/휕푀2 = 0. Similar to the first order hierarchy, point 4 is clamped for all the analysis [18].

Shear modulus

To obtain the shear modulus we imposed far-field shear stress 휏, which gave rise to forces

푃 = −휏(3푙0 sin 휃) below the horizontal cut HH’, and forces 퐹 = −휏(푙0cos 휃) to the left of vertical cut VV’ in figure 3.1a. In a state of pure shear stress, 푃 = 3퐹 ∗ 푡푎푛(휃). Since the loading is not symmetric with respect to the unit cell midline, the full subassembly of figure 3.2b and 3.2c must be analyzed to find the stored bending energy.

Using the same procedure as above, the bending energy stored in a first order hierarchical structure can be expressed as a function of the simultaneously applied external forces 퐹 and 푃:

푈 = 푈(퐹, 푃). The bending moments along each side of the complete hexagon are determined from a subsidiary analysis in which it is divided at nodes 5 and 6, and then three compatibility conditions are enforced at each of those nodes [18]. Displacements 훿푦 and 훿푥 at point 4 can be found from:

훿푦 = 휕푈(퐹, 푃)/휕퐹 and 훿푥 = 휕푈(퐹, 푃)/휕푃.

Poisson’s ratio

To obtain the dependence of the orthotropic Poisson’s ratios on the dimension ratio of the hierarchical structure, we again used Castigliano’s second theorem and analyze the upper half of the figure 3.2 subassembly. We determined 휈푦푥 only, since 퐸푥휈푦푥 = 퐸푦휈푥푦 allows to find 휈푥푦.

Applying far-field stress 𝜎푦푦 = −퐹/ (3푙0sin (휃)) in a vertical direction is equivalent to applying a vertical force 퐹 at point 4 of figure 3.2b for first order hierarchy and at point 5 of figure 3.2c for second order hierarchy. Here we apply the dummy horizontal Force P which allows using

Castigliano’s theorem to find the horizontal displacement of the node. When 푃 is zero, the 푥 and

푦 displacements of the upper node due to force 퐹 can be expressed as follows: 훿푥 =

(휕푈/휕푃)|푁1,푀1,푃=0 and 훿푦 = (휕푈/휕퐹)|푁1,푀1,푃=0, where 푈 is the bending energy stored in the structure due the forces 퐹 and 푃.

3.3. Fabrication using 3D printing

Figure 3.1b shows samples of both zero order and hierarchical regular-hexagon honeycombs with relative density of 𝜌 = 𝜌/𝜌푠 = 0.10 and 푙0 = 20 푚푚, where 𝜌 is the structural density and 푙0 is the oblique hexagon edge length [18]. These samples were fabricated using 3D printing (Dimensions 3D printer, Stratasys Inc., Eden Prairie, MN). The regular honeycomb has

푡 = 1.75 푚푚; the honeycomb with one level of hierarchy has 훾1 = 0.3 and 푡 = 1 푚푚; and that with two-level hierarchy has 훾1 = 0.3, 훾2 = 0.12, and 푡 = 0.75 푚푚. These were printed as three-dimensional extruded shells from an ABS polymer (acrylonitrile butadiene styrene, elastic modulus = 2.3 GPa). figure 3.2a shows the images of 1st order anisotropic honeycombs with 휃 =

o o o 10 , 30 and 70 and 푙0 = 2 푐푚 fabricated using three-dimensional printing. All three structures have 훾1 = 푙1/푙0 = 0.3.

3.4. Elastic properties of anisotropic hierarchical honeycombs: Analytical Modeling

Some geometric constraints on the hierarchically introduced edges must be imposed to avoid interfering with pre-existing members: in a honeycomb with first order hierarchy, 0 ≤ 푙1 ≤

푙0/2 (figure 3.1a) and thus, 0 ≤ 훾1 ≤ 0.5, where 훾1 = 0 denotes the regular honeycomb structure. For second order hierarchy, the two geometrical constraints are 0 ≤ 푙2 ≤ 푙1 and 푙2 ≤

푙0/2 − 푙1. For uniform wall thickness of 푡, the relative density of the structure (i.e. area fraction) can be given as a function of the length ratios and 푡/푙0:

푛 𝜌 1 + sin휃 푖−1 푡 = ∗ (1 + 2 ∑ 3 훾푖) ∗ (3.1) 𝜌푠 3sin휃cosθ 푙0 1

where 𝜌푠 is the material density, 휃 is the oblique wall angle, and 푛 is the hierarchical order.

This relation is used to adjust 푡, so that various choices of 휃 and the 훾푖 maintain a fixed relative density. A vertical stretch makes the oblique walls much longer than the horizontal walls, as can be seen by letting 휃 approach 0 in the above expressions (figure 3.2a). In following sections, we mainly focus on results and findings. For detailed description of analytical modeling please refer to Appendix A.

3.4.1. Elastic modulus in principal directions

The effective elastic modulus (to be normalized by the material elastic modulus, 퐸푠) is defined as the ratio of average stress (𝜎̅푦 = −퐹/(3푙0푠푖푛휃) , 𝜎̅푥 = −푃/(푙0푐표푠휃)) and average strain (휀푦̅ = −2훿푦/(푙0푐표푠휃), 휀푥̅ = −2훿푥/(3푙0푠푖푛휃)). 푙0 is the oblique wall length of the underlying zero-order honeycomb, and 휃 is the oblique wall angle. 푃 and 퐹 are unit-cell boundary points forces in 푥 and 푦 directions, respectively and 훿푥 and 훿푦 are the unit-cell boundary point displacements (figure 3.2b and 3.2c). The cell walls of thickness 푡 consist of an isotropic elastic material with elastic modulus 퐸푠 and Poisson’s ratio 휈푠. Using the notation 훾1 = 푙1/푙0 , the elastic

3 3 modulus in 푥 and 푦 directions are finally obtained as 퐸푥/퐸푠 = 푡 /푙0 ∗ 푓푥(훾1, 휃) and 퐸푦/퐸푠 =

3 3 푡 /푙0 ∗ 푓푦(훾1, 휃) where:

12 sinθ(sinθ+7)/(cosθ (1−sinθ)) 푓푥(훾1, 휃) = 2 3 2 2 2 (96sin θ+596sinθ−148)γ1 +(−48 sin θ−168sinθ+312)γ1 +(−93 sinθ−165)γ1+(4 sin θ+32 sinθ+28)

and

4 cosθ(sin2θ+8 sinθ+7)/(3sin3θ) 푓푦(훾1, 휃) = 2 3 2 2 2 (192sin θ+644sinθ−196) γ1 +(−96 sin θ −192 sinθ +336) γ1 +(−93 sinθ−165) γ1+(4 sin θ+32 sinθ+28)

These expressions, are valid for any allowable 훾1, in particular for the case when 훾1 = 0, namely zero order hierarchy. Note also that when 휃 = 30o, the resulting isotropy means these expressions should be equal for all values of 훾1.

To find the value of 훾1 = 푙1/푙0 which yields the maximum 푥 or 푦 elastic modulus for a given relative density 𝜌/𝜌푠, equation 3.1 must be used to express 푡/푙0 as a function of 𝜌/𝜌푠. For hierarchical order 푛 = 1 this results in 푡/푙0 = 3sin휃cosθ/(1 + sin휃) ∗ 𝜌/𝜌푠(1 + 2훾1). (Note that 푡 approaches zero when 휃 approaches either 0 or 90.) Substituting this into the modulus expressions yields:

3 3 3 퐸푥,푦 3sin휃cosθ 𝜌 1 = ( ) ( ) ∗ 푓푥,푦(훾1, 휃) ( ) (3.2) 퐸푠 1 + sin휃 𝜌푠 1 + 2훾1

where 퐸푥,푦 represents 퐸푥 or 퐸푦, and 푓푥,푦 correspondingly represents 푓푥 or 푓푦. The analytical tool of MATLAB was used to obtain 훾1which maximized equation 3.2. A similar procedure can be used to derive elastic moduli of anisotropic honeycomb structures with two orders of hierarchy

(figure 3.2c). The elastic moduli in the principal directions were obtained by finding displacements at location of applied forces, 퐹 and 푃. These moduli are function of 훾1, 훾2 and 휃 and can be

3 3 3 3 presented as : 퐸푥/퐸푠 = 푡 /푙0 ∗ 푓푥(훾1, 훾2, 휃) And 퐸푦/퐸푠 = 푡 /푙0 ∗ 푓푦(훾1, 훾2, 휃). To find the optimal values of 훾1 and 훾2 at any constant density ratio 𝜌/𝜌푠, the 푡/푙0 term is expressed as a function of density ratio 𝜌/𝜌푠. From (1): 푡/푙0 = 3sin휃cosθ/(1 + sin휃) ∗ 𝜌/𝜌푠 ∗ 1/(1 + 2훾1 +

6훾2). Substituting this into the elastic modulus expressions yields

3 3 3 퐸푥,푦 3sin휃cosθ 𝜌 1 = ( ) ( ) ∗ 푓푥,푦(훾1, 훾2, 휃) ( ) (3.3) 퐸푠 1 + sin휃 𝜌푠 1 + 2훾1 + 6훾2

For the sake of conciseness, both 퐸푥 and 퐸푦 are shown in one equation with 푓푥,푦 representing 푓푥 or

푓푦.

3.4.2. Effective shear modulus and Poisson’s ratio

To fully characterize the linear elastic behavior of horizontally or vertically stretched hierarchical honeycombs, the orthotropic shear modulus 퐺푥푦 and Poisson’s ratio 휈푦푥 should be obtained as a function of 휃 and the hierarchical dimension ratios. We applied Castigliano’s second theorem to the full subassemblies of figure 3.2b and 3.2c to find shear modulus퐺푥푦 and Poisson’s ratio 휈푦푥.

The effective shear modulus (normalized by the material shear modulus 퐺푠) is defined as the ratio of average shear stress 휏 = −푃/ (3푙0sin 휃)) to average shear strain (휀푥푦 =

2훿푦/ (3푙0푠푖푛휃) + 2훿푥/(푙0푐표푠휃)). The result can be presented as,

3sin휃cosθ 3 𝜌 3 1 3 퐺푥푦/퐺푠 = ( ) ( ) 푔1(훾1, 휃) ( ) (3.4) 1 + sin휃 𝜌푠 1 + 2훾1

where 푔1 is a complex function of 훾1and 휃. For a first-order hierarchical structure the maximum

shear modulus occurs when 훾1 = 0.34, virtually regardless of wall angle 휃. However its magnitude

depends on 휃.

Similarly, for the case of the structure with two hierarchical orders (figure 3.2c), the

effective shear modulus can be defined as,

3sin휃cosθ 3 𝜌 3 1 3 퐺푥푦/퐺푠 = ( ) ( ) 푔2(훾1, 훾2, 휃) ( ) (3.5) 1 + sin휃 𝜌푠 1 + 2훾1 + 6훾2

where 푔2 is a function of 훾1, 훾2and 휃. Differentiating (5) with respect to 훾1 = 푙1/푙0 and 훾2 =

푙2/푙0 at constant density shows that the maximum normalized shear modulus occurs at 훾1 =

푙1/푙0 = 0.3 and 훾2 = 푙2/푙0 = 0.14.

The Poisson’s ratio 휈푦푥can then be obtained from 휈푦푥 = 훿푥/(3tan (휃)훿푦), which results

in:

휈푦푥(훾1, 휃)

2 2 2 3 2 2 2 (3.6) (cos θ /3 sin θ) ((−112 sin θ − 564sinθ + 196) γ1 + (72 sin θ + 180sinθ − 324) γ1 + (93sinθ + 165γ1) γ1 − 4 sin θ − 32sinθ − 28) = − 2 3 2 2 2 ((192 sin θ + 644sinθ − 196) γ1 + (− 96 sin θ − 192sinθ + 336) γ1 + (−93sinθ − 165) γ1 + 4 sin θ + 32sinθ + 28)

Performing the same procedure for the second order hierarchy structure of figure 3.2b, we can

obtain 휈푦푥 as a function of oblique wall angle 휃, 훾1 and 훾2 (i.e. 휈푦푥 = 휈푦푥(휃, 훾1, 훾2)).

3.5. Elastic analysis: Numerical simulation

To explore the effect of hierarchical order greater than 2, a matrix frame analysis procedure

was implemented in MATLAB. Shear and stretching were included in the governing equations for

66 each beam [231] : 푑/푑푥(퐴퐸푠 ∗ 푑푢/푑푥) = 0 for stretching, 푑/푑푥[푘푠퐴퐺푠(푑푣/푑푥 − 휙)] = 0 for shearing, and 푑/푑푥(퐸퐼 ∗ 푑휙/푑푥) + 푘푠퐴퐺푠(푑푣/푑푥 − 휙) = 0 for bending, where 퐴 is the cross sectional area, 퐼 is the second moment of area, 퐸푠 and 퐺푠 are the elastic and shear moduli of the cell wall material, 푘푠 is the shear coefficient (equal to 5/6 for a rectangular cross section [231]),

푢, 푣 are the longitudinal and transverse displacements, and 휙 is the beam cross-section rotation about the 푧-axis. Transformation matrix is calculated for each beam to transform its stiffness matrix to global coordinates and Points 3 and 4 were clamped in figure 3.2b and 3.2c, respectively.

This program was highly efficient and allowed us to systematically change geometry of the hierarchical structure in order to find unique geometry which maximized desired mechanical properties. Compared to the commercial FEA software, the developed program was faster and more flexible. We explored the whole range of 훾 to determine the elastic modulus range at each hierarchical order.

3.6. Plastic collapse strength analysis

In this section, FE simulation with beam elements is performed to study the in-plane uniaxial plastic collapse strength of hierarchical honeycombs with up to four orders of hierarchy. Using beam element, a subassembly of the structure, as defined in figure 3.2b and 3.2c, was modeled in

ANSYS. The modeled structure was then meshed using plastic 2D cubic beam element (Beam 23) with plastic, creep, and swelling capabilities. Elastic-perfectly plastic behavior was assigned to the chosen material, and element length was taken as 푙0/500, where 푙0 is the length of an oblique wall

(short elements are needed to capture localized plastic hinges). The simulations were performed with uniaxial loading separately applied in 푥 and 푦 directions. The elastic modulus, Poisson’s ratio and yield strength of material used in the simulations were 퐸푆 = 70 퐺푃푎, 휈 = 0.3 and 𝜎푦 =

130 푀푃푎, respectively. The relative density of the structure used for plastic collapse strength analysis was 0.01.

The modeled subassembly of the structure was subjected to compressive displacement- controlled loading (i.e. on the points 4 and 5 in figure 3.2b and points 5 and 6 in figure 3.2c) with free transverse expansion. Strength was defined as the stress associated with the plateau level in the force-displacement curve. The values of collapse strength in each direction are denoted by

푆푦,휃 and 푆푥,휃 , where 푥 or 푦 denotes the strength direction and 휃 is the oblique wall angle.

3.7. Results and discussion

3.7.1. Elastic modulus properties

figure 3.3a and 3.3b show the achievable elastic modulus range in the 푦 and 푥 directions as a function of wall angle, for up to 4 hierarchical orders. The computed orthotropic elastic moduli

0 are normalized by the isotropic elastic modulus of zero order unstretched honeycomb (i.e. 퐸30).

As can be seen from these two figures, by increasing 휃, elastic modulus in the 푦-direction

0 decreases to zero, while normalized 푥-direction modulus (퐸푥/퐸30) increases to approximately 6.

(The opposite behavior is seen as 휃 decreases to zero.) These results are explained by the fact that as 휃 approaches either 0 or 90, beam thickness reduces and bending stiffness rapidly approaches zero. In contrast axial stiffness contribution increases resulting in an overall finite modulus.

figure 3.3a and 3.3b also show that as hierarchical order is increased, not only the maximum elastic modulus but also the elastic modulus range covered by each additional order increases.

Values of 훾 for maximum elastic modulus at each hierarchical order are provided in the figures.

These values are approximately independent of 휃 and the selection of 푥 or 푦. For third order hierarchy we found 훾1 = 0.27, 훾2 = 0.13, 훾3 = 0.06 and for fourth order we found 훾1 = 0.26, 68

훾2 = 0.13, 훾3 = 0.06, 훾4 = 0.03. The trend in 훾 values suggests that when increasing the hierarchical order, the inserted edge length which gives the stiffest structure tends to be half the edge length of the previous hexagons (i.e., 훾푖+1 = 훾푖/2).

Figure 3-3. (a, b) The elastic modulus bandwidth of specific 1st to 4th order hierarchical honeycombs, normalized by the zero-order modulus of isotropic honeycomb (i.e., 30 degree wall

69 angle) for 푦 and 푥 direction. (c) Surface plot of maximum 푥-direction and 푦-direction normalized elastic modulus of zero to fourth order hierarchical honeycomb.

0 figure 3.3c compares normalized elastic moduli in the 푥- and 푦-directions (퐸푦/퐸30 vs

0 퐸푥/퐸30). The 휃-constant contours in this figure are shown with solid lines. As can be seen from the figure, by increasing the order of hierarchy (푛) the covered area of the map increases. Also, for

o 0 0 o 0 휃 < 30 the value of 퐸푥/퐸30 is less than 퐸푦/퐸30 so that for 휃 < 10 the value of 퐸푥/퐸30 is

0 o negligible compared to 퐸푦/퐸30 and for 휃 > 60 the opposite is true. Therefore the design region for bi-axial loading should be within the range of 15o < 휃 < 50o.

In order to understand the effect of hierarchical structure on the elastic moduli of the structure, the elastic modulus for various hierarchical structure are normalized respect to elastic moduli of the non-hierarchical structure for various wall angle 휃. Figure 3.4a and 3.4b show the

o o 푥 and 푦 elastic moduli of a first-order hierarchical structure, now normalized by 퐸푥,휃 and 퐸푦,휃, the non-hierarchical structure of the same wall angle, using data shown by a dark line in figure 3.3a and 3.3b. Here we divided the stiffness by the stiffness of the zero order value at the same wall

0 angle θ, unlike figure 3.3 in which it is divided by a constant (퐸30). Surprisingly, the maximum normalized elastic moduli occur in a very narrow range of 훾1 = 0.3147 to 0.3232 as 휃 varies from

o 2 2 0 to 90. At 훾1 = 푙1/푙0 = 0.32, 퐸푦/퐸푦,휃 = 2(sin 휃 + 8 sin 휃 + 7)/(1.02 sin 휃 + 8.12 sin 휃 +

7.02) ≈ 1.98, which varies less than 1% over the entire range of wall angle 휃. The maximum elastic modulus in the 푦-direction for the first order honeycomb is thus about twice that of non- hierarchical honeycomb with the same density ratio, for all 휃 angles. In contrast, while the 푥- direction peak elastic modulus occurs for essentially the same hierarchical structure (same choice of 훾1), its value does depend significantly on anisotropy, i.e. 휃. Both 푥 and 푦 results clearly show

70 that hierarchical refinement within some range of 훾 values always increase elastic modulus (i.e., exceed 1.0), regardless of the wall angle.

Figure 3-4. (a, b) Normalized elastic moduli of 1st order hierarchical honeycomb for 푦 and 푥

nd direction. (b,c) Normalized elastic moduli of 2 order hierarchical honeycomb with 훾1 = 0.29, at

푦 and 푥 direction. The elastic moduli is normalized by that of zero order honeycomb with the same

71 oblique wall angle 휃. Data markers are the numerical frame analysis results. Note that 퐸푥 and 퐸푦 are the same for 휃 = 30o.

The data markers in figure 3.4 show the results of computational analyses on the upper half of the subassembly defined in figure 3.2b and 3.2c. They represent the MATLAB frame analysis results for four different oblique wall angles and five different 훾1. The results were obtained by clamping point 3 in figure 3.2b and allowing points 1 and 2 to move horizontally due to structural symmetry. Similarly in figure 3.2c, point 4 was clamped and points 1, 2 and 3 were confined to move horizontally. The frame computation results show good correlation with the theoretical results (maximum error < 1%). The analytical results were based on using bending energy only.

In contrast frame analysis was based on considering all deformation energies, (bending, shear and stretching). The agreement could be justified by considering that the bending energy is dominant energy for small relative density of less than 0.01.

Figure 3.4c and 3.4d show the normalized elastic modulus for second order hierarchy in the 푦- and 푥-directions. The results for the first order hierarchy can be inferred from the plots in figure 3.4a and 3.4b at 훾2 = 0. The derivatives of (3) with respect to 훾1 = 푙1/푙0 and 훾2 = 푙2/푙0 at constant density show that the maximum normalized elastic modulus in the 푥- and 푦-directions occur at 훾1 = 푙1/푙0 = 0.29 and 훾2 = 푙2/푙0 = 0.135, respectively (virtually independent of θ),

o 3 where 퐸푦/퐸푦,휃 ≈ 3.52(𝜌/𝜌푠) . As can be seen in figure 3.4c, the maximum normalized elastic modulus in 푦-direction for 훾1 = 0.29 and for all different oblique wall angles is the same. The maximum elastic modulus in the 푦-direction is almost four times greater than the elastic modulus of a regular honeycomb structure with the same density ratio. In contrast, the maximum normalized

72 elastic modulus in the 푥-direction depends on the wall angle 휃. There is also a good correlation between frame analysis results and theoretical results (maximum error < 1%).

3.7.2. Shear modulus and Poisson’s ratio

Based on the numerical frame analysis in MATLAB, figure 3.5a shows the shear modulus

0 bandwidth (퐺푥푦/퐺30) of the structure versus oblique wall angle, 휃 for up to 4 hierarchical orders,

0 o where 퐺30 is the shear modulus of zero order isotropic honeycomb(휃 = 30 ). Figure 3.5a shows that by increasing hierarchical order, the possible range of shear modulus increases and the maximum shear modulus is also increases. The oblique wall angle for the maximum shear modulus is approximately 휃 = 28o, for all hierarchical orders. Values of 훾 for maximum modulus at each hierarchical order are virtually independent of 휃. For third order hierarchy this occurs at

훾1 = 0.27, 훾2 = 0.13, 훾3 = 0.06 , and for the fourth order at 훾1 = 0.27, 훾2 = 0.13, 훾3 = 0.06,

훾4 = 0.03. The enhancement in the normalized shear modulus is noticeable as it increases from

2.59 and 5.04 for 1st and 2nd order hierarchical honeycombs to 7.87 and 10.78 for 3rd and 4th order.

Just as in the previous section on the effect of hierarchical order on the elastic modulus, the trend in 훾 values which maximized the shear modulus tends to be half the edge length of the previous hexagons (i.e. 훾푖+1 = 훾푖/2). Frame analysis results are also shown which were obtained by applying the same loading conditions as those in the analytical analysis.

figure 3.5b shows the shear modulus for first order hierarchical structures, normalized by the modulus of the zero order structure with that same angle. As the oblique wall angle increases,

o the normalized shear modulus (퐺푥푦/퐺휃 ) is increased slightly. When 훾1 = 0.34, the shear modulus of the first order hierarchical honeycombs increases two times of the non-hierarchical honeycombs regardless of wall angle.

Figure 3.5c shows the normalized shear modulus for second order hierarchical structure.

The maximum normalized shear modulus is moderately dependent on oblique wall angle 휃. Trends are the same as those for structures with first order hierarchy. Adding the second order of hierarchy makes the shear modulus almost 2 times greater than for first order hierarchy.

Figure 3-5. (a) The normalized shear modulus bandwidth of 1st to 4th order hierarchy. (b, c)

st nd Normalized shear modulus of 1 and 2 order hierarchy versus 훾1 and 훾2, respectively. For (a) the shear modulus is normalized by that of zero order regular honeycomb (휃 = 30o), whereas for (b)

74 and (c) it is normalized by that of zero order honeycomb with the same oblique wall angle 휃; Data markers are the MATLAB frame analysis results.

Figure 3.6a shows the Poisson’s ratio of 1st order hierarchical honeycomb normalized by the

o Poisson’s ratio of zero order honeycomb structure at the same wall angle, θ (i.e. 휈푦푥,휃 =

2 2 표 cos θ /(3 sin θ)). For isotropic honeycomb (휃 = 30 ) and 훾1 = 0 , 휈푦푥 is equal to one. This can be attributed to the deformation of these structures under hydrostatic pressure. Under hydrostatic loading, beams in these structures experience only axial loading. Since these beams are assumed to be inextensible, the Poisson’s ratio should be equal to one. However, for honeycomb with other values of , 2D hydrostatic loading results in a different loading in beams, and thus different values for Poisson’s ratio. Furthermore, any hierarchical replacements lead to

o beam bending (hence strain) so Poisson’s ratio must be reduced. It was found, 휈푦푥/휈푦푥,휃 ≅ 0.5 at

o o 훾1 = 0.5 for all values of 휃, with the minimum value of 0.39 for 휃 = 10 and 0.33 for 휃 = 70

nd at 훾1 = 0.4. Figure 3.6b shows the normalized Poisson’s ratio of 2 order hierarchical honeycomb

o (i.e. 휈푦푥/휈푦푥,휃) in 푦-direction with 훾1 = 0.29 versus 훾2 for different values of oblique wall

o angle, 휃. The value of 휈푦푥/휈푦푥,휃 is between 0.74 to 0.78 at 훾2 = 0 and between 0.35 to 0.38 at

훾2 = 0.21, with the minimum value between 0.29 to 0.34 at 훾2 = 0.17. As can be seen from these two figures, the curves intersect at a point which correspond to locations in which maximum elastic

st nd modulus occurs (i.e. 훾1 = 0.32 for the 1 order and 훾1 = 0.29, 훾2 = 0.135 for 2 order). Frame computations in MATLAB are also provided for comparison with the analytical solution. The results show a good agreement between analytical results and the frame analysis. Although the frame analysis considered all deformation modes in the structural components, analytical and frame analysis are in good agreement because bending energy is the dominant energy for low density structures.

st Figure 3-6. a) Poisson’s ratio of 1 order hierarchical honeycomb in 푦-direction versus 훾1 for different values of oblique wall angle, 휃. b) Poisson’s ratio of 2nd order hierarchical honeycomb in

푦-direction with 훾1 = 0.29 versus 훾2 for different values of oblique wall angle, 휃. Star points are the numerical frame analysis results. Note that the results are normalized by non-hierarchical

Poisson’s ratio at the same angle, which varies significantly with anisotropy.

3.7.3. Plastic collapse strength

figure 3.7a and 3.7b show the normalized plastic collapse bandwidth of 1st to 4th order hierarchical honeycomb versus oblique wall angle, 휃, for 푦- and 푥- directions respectively. Here, the plastic collapse strength is normalized by that of zero order regular honeycomb (i.e. 휃 = 30o).

These two figures show that the maximum achievable plastic collapse strength saturates at the 3rd order for both 푦- and 푥-direction. The maximum plastic collapse strength for different hierarchical orders occurs at approximately 휃 = 28o for 푦-direction and at 휃 = 45o for 푥-direction and is

76 nearly independent of hierarchical order. The values of 훾 at which maximum plastic collapse strength occur in each hierarchical order are virtually independent of 휃. For the second order hierarchical structure it was found the maximum 푥-direction collapse strength occurred when 훾1 =

0.29, 훾2 = 0.14 for 푦-direction and 훾1 = 0.29, 훾2 = 0.15. For the third order the maximum strength was achieved at 훾1 = 0.27, 훾2 = 0.14, 훾3 = 0.07 for both 푥- and 푦- direction strengths.

The trend in 훾 values show that by increasing the hierarchical order the edge length of newly generated hexagons tends to be half the edge length of the previous hexagons (i.e. 훾푖+1 = 1/2 ∗

훾푖). These results show that the collapse behavior of the hierarchical honeycomb can be improved up to the 3rd order by increasing the hierarchical order. However, further refinement does not make any improvement in collapse strength.

figure 3.7c and 3.7d show the normalized 푥 and 푦 direction plastic collapse strengths of a first-order hierarchical structure versus 훾1 for different values of oblique wall angle, 휃. The plastic collapse strength is normalized by that of a zero order honeycomb with same oblique wall angle

휃. The results suggest that the 훾1 values corresponding to the maximum normalized plastic collapse strength in 푥- or 푦-direction are not fixed for different oblique wall angles. For 푦 direction, the maximum occurs between 훾1 = 0.33 − 0.36 for the range of oblique beam angle varying between

o o 휃 = 70 and 10 , while for 푥 direction it occurs between 훾1 = 0.34 − 0.35 for the same range of angle. For each curve corresponding to a fixed oblique wall angle, 휃, there are two points with slope discontinuity for the plastic collapse in the 푦 direction figure 3.7c, and one point for plastic collapse in the 푥 direction figure 3.7d. These points represent the locations in which the mechanism of failure and the location of plastic hinge points change by increasing the value of 훾1.

Corresponding plastic collapse mechanisms and locations of plastic hinges for different range of

훾 are shown on top of figure 3.7c and 3.7d. For the plastic collapse in the 푦 direction as 휃 decreases

77 the normalized plastic collapse strength increases. The maximum normalized plastic collapse for

휃 = 10o is equal to 1.37 whereas for 휃 = 70o is equal to 1.08. Figure 3.7d shows that 휃 = 30o

o st gives the largest normalized plastic collapse 푆푥/푆푥,휃 which is equal to 1.2 for the 1 order hierarchy structure.

Figure 3-7. (a, b) The normalized plastic collapse bandwidth of 1st to 4th order hierarchical honeycomb for 푦 and 푥 direction (c, d) The normalized plastic collapse of 1st order hierarchical

78 honeycomb for 푦 and 푥 direction. Plastic collapse mechanisms for different values of 훾 are shown on the top. Note that for the plastic collapse of structure in y-direction with oblique angle of 휃 =

10o, only two mechanisms in left and right are dominant for the whole range of 훾. For (a) and (b) the plastic collapse strength is normalized by that of zero order regular honeycomb (휃 = 30o), whereas for (c) and (d) it is normalized by that of zero order honeycomb with same oblique wall angle, 휃.

Having the maximum plastic collapse strength at 휃 = 28o for the 푦 direction and at 휃 =

45o for the 푥 direction suggests a map comparing plastic collapse to elastic modulus. Figure 3.8a and (8b) show the maps of normalized collapse strength versus of normalized elastic modulus for zero to 4th order honeycombs for 휃 = 28o in 푦 direction and for 휃 = 45o in x direction, respectively, obtained from the finite element analysis. It should be noted that an nth order hierarchical honeycomb is a special configuration of honeycombs with higher order of hierarchy.

For example, a 1st order hierarchical honeycomb is a special configuration of 2nd order hierarchical honeycombs with, 훾2 = 0. Thus, the entire colored area in figure 3.8 shows the range of achievable elastic modulus and strength with four orders of hierarchy. Compared to previous data given in

[19] for isotropic hierarchical honeycombs, current graphs show a larger range of achievable elastic modulus and plastic collapse strength by stretching the structure in vertical/horizontal direction.

3.8. Concluding remarks

Mechanical properties of anisotropic hierarchical honeycombs were investigated, where the term ‘anisotropic’ represents the change of honeycomb’s oblique wall angle by uniform horizontal or vertical stretching of the underlying network, and the term ‘hierarchical’ represents

79 the replacement of each three-edge vertex of base hexagon structure with a smaller hexagon of same wall angle. The oblique wall angle of the honeycombs and the relative side lengths of newly generated hexagons are two key parameters which determine the stiffness and strength of the structure, at any fixed relative density.

The effective elastic modulus, Poisson’s ratio and plastic collapse strength of anisotropic hierarchical honeycombs were obtained as a function of dimension ratios and wall angles of the structure. The analytical results for the elastic part were based on Castigliano’s second theorem, and numerical frame analysis was used to validate the analytical approach, and explore higher orders of hierarchy. The results show that a wide range of elastic modulus and strength ratio can be obtained for anisotropic hierarchical honeycombs by varying geometrical parameters. Table 3.1 summarizes the elastic modulus/strength values of anisotropic hierarchical honeycombs at different hierarchical orders. Increasing the hierarchy order improves the structural performance in terms of its elastic modulus and plastic collapse strength.

Table 3.1

Summary of elastic modulus /strength of anisotropic hierarchical honeycombs

o o o o o order 퐸̅푥(휃 = 90 ) 퐸̅푦(휃 = 0 ) 퐺푥푦̅ (휃 = 28 ) 푆푥̅ (휃 = 45 ) 푆푦̅ (휃 = 28 )

zero value 6.75 6.02 1.02 1.21 1.01

property value 10.77 11.98 2.86 1.44 1.15 1st 훾 values [.32] [.32] [.34] [.35] [.34]

property value 17.48 21.20 5.04 1.67 1.27 2nd 훾 values [.29, .13] [.29, .13] [.30, .14] [.29, .15] [.29, .14]

property value 25.48 32.01 7.87 1.79 1.29 3rd 훾 values [.27, .13, .06] [.27, .13, .06] [.27, .13, .06] [.27, .14, .07] [.27, .13, .07]

property value 33.92 43.5 10.78 1.79 1.29 4th 훾 values [.26, .13, .06, .03] [.26, .13, .06, .03] [.26, .13, .06, .03] [.26, .13, .07, .03] [.26, .13, .07, .03]

o o Maximum elastic modulus/strength values of anisotropic hierarchical honeycombs, 퐸̅푥 for 퐸푥/퐸30, 퐸̅푦 for 퐸푦/퐸30,

̅ o ̅ o ̅ o 퐺푥푦 for 퐺푥푦/퐺30, 푆푥 for 푆푥/푆30 and 푆푦 for 푆푦/푆30.

Chapter 4 Optimal Fractal-like Hierarchical Honeycombs

4.1. Introduction

Hierarchically structured material systems are characterized by the existence of structure at different length scales and often exhibit superior mechanical properties such as enhanced stiffness [9, 18], strength [9, 232], toughness [233-235] and negative Poisson’s ratio [236-238].

They are used in many fields including polymers [10], composite structures [12, 13, 239], sandwich panel cores [15, 240] and biomimetic systems [9, 17, 235]. Along these lines, a new family of honeycomb structures was designed with a hierarchical refinement scheme, where the structural hexagonal lattice was replaced by smaller hexagons. This process can be repeated to create honeycombs of higher hierarchical order (figure 4.1a). These cellular solids are shown to have improved in-plane stiffness and strength compared to the counterpart regular honeycombs

[18, 241, 242]. However, it is still unknown whether such structures can be systematically optimized, in particular by adjusting the number of hierarchical orders. Here, the optimal configuration of such hierarchical honeycombs in terms of effective stiffness is determined for various structural densities using finite element simulation, scaling analysis and experiments. This chapter is derived from a paper published by the author [243].

4.2. Optimal Elastic Modulus

The structural organization (the set of 훾푖) is defined by the ratio of the newly introduced hexagonal edge length (푙푖) to previous hexagon edge length (푙푖−1), where 푖 varies from 2 to 푛

(hierarchical order) (i.e. 훾푖 = 푙푖/푙푖−1). For convenience, 훾1 is defined as 2푙1/푙0 (see below). Some geometric constraints on the hierarchically introduced edges must be imposed to avoid overlapping

th 푛−1 with pre-existing edges. For n hierarchical order (푛 > 1), 0 ≤ 푙푛 ≤ 푙푛−1 and 푙푛 ≤ 푙0/2 − ∑푖=1 푙푖 which can be written based on structural organization parameters as

0 ≤ 훾푛 ≤ 1 { 푛 푖 (4.1) ∑푖=1 ∏푗=1 훾푗 ≤ 1 which holds for all hierarchical orders (푛 ≥ 1). For uniform wall thickness of 푡, the relative density of the structure (i.e. area fraction) can be given as a function of the length ratios and 푡/푙0:

푛 푖−1 푖 𝜌̅ = 𝜌/𝜌푠 = 2/√3 ∙ (1 + ∑1 3 ∏푗=1 훾푗) ∙ 푡/푙0 ( 4. 2)

where 𝜌푠 is the material density. This relation is used to adjust 푡 to maintain a fixed relative density

𝜌̅ for various choice of hierarchical order and values of 훾푖.

Since a honeycomb network extending spatially to infinity has six-fold rotational symmetry, any linear second order tensorial operator (e.g., thermal conductivity) or linear fourth- order operator (e.g., elastic modulus) must be isotropic. Therefore, the macroscopic in-plane elastic behavior of honeycomb structure can be described by just two constants [244]. For numerical and analytical analysis, the far-field uniaxial stress in vertical direction 𝜎푦푦 = −2/3 ∙ 퐹/푙0 was imposed to determine the elastic modulus of the hierarchical honeycomb structure, where 퐹 is an arbitrary concentrated force. Vertical stress is equivalent to applying the force 퐹 in vertical direction at every mid-point of oblique edge in original (i.e. zero hierarchy) hexagons (refer to the supplementary material for detail). To carry out the analysis, the unit cell of the lattice (figure 4.1b) was selected to represent the loaded lattice structure. Each beam in the lattice can undergo stretching, shear, and bending. In the bending dominated regime, the elastic moduli of first order hierarchical honeycomb (to be normalized by material elastic modulus, 퐸푚) can be written as (The analytical modeling is detailed in supplementary material):

푚 3 퐸/퐸 = (푡/푙0) 푓(훾1) (4.3)

2 3 where 푓(훾1) = √3/(0.75 − 1.7625훾1 + 0.575훾1 + 0.3625훾1 ). 푡/푙0 can be eliminated using equation 4.2 to solve the effective elastic modulus for fixed relative density as:

푚 3 3 퐸/퐸 = 3√3/(8(1 + 훾1) ) ∙ 푓(훾1) ∙ 𝜌̅ (4.4)

The finite element analysis for higher orders was performed using MATLAB. The developed program allowed us to validate the theoretical results and systematically change the geometry of the hierarchical structure to find the unique geometry which maximizes the effective elastic modulus of the honeycomb. Figure 4.2 shows the maximum effective elastic modulus for different relative structural densities. As can be seen from this figure, the maximum effective elastic modulus saturates above certain number of hierarchical orders. For example, for the density of

0.018 to 0.026, the maximum modulus is achieved for hierarchical order of 6. The dashed line shows the behavior of bending dominated structure where shear and stretching energies are eliminated from the analysis. This curve is unbounded and the elastic modulus does not saturate.

Figure 4.3 shows the structural organizations (훾푖) of stiffest structures at different relative densities. As can be seen form this figure, the values of 훾푖 approach to 1/2 as the density decreases.

Also, as the density decreases, the maximum reachable hierarchical order for each density increases and the density range for each specific hierarchical order decreases.

In experiments, the unit cell of regular honeycomb and 1st to 4th order hierarchical honeycomb with constant relative density of 0.054 were fabricated using 3D printing (figure 4.1a).

The fabrication and mechanical testing are described in the supplementary material. The

87 experimental results show good agreement with the results obtained from the numerical simulations as seen in the inset of figure 4.2.

Figure 4-2. Maximum achievable stiffness (stiffness limit) of hierarchical honeycombs for different relative densities, 𝜌̅, and different hierarchical orders, n, normalized by the stiffness of regular honeycomb of the same mass. The limit for the maximum stiffness of hierarchical honeycombs are shown by dashed line. (∆) shows the stiffness limit found from scaling analysis

(equation 4.5). The inset shows comparison with the experimental results for hierarchical honeycombs with density of 𝜌̅ = 0.054.

89 topologies of the stiffest hierarchical honeycombs on the specified relative density range are also shown on the top.

For scaling analysis, we seek the maximum amplification of the effective bending elastic modulus, when replacing a Y-shaped structure by a hexagon. For 훾 ≤ 0.64, the effective elastic modulus increases with 훾; therefore based on figure 4.2, the optimal value of 훾 is assumed to be

1/2, which also yields a self-similar structure. In figure 4.1b, the effective elastic modulus of first order structure in right is 1.598 times the modulus of regular honeycomb in the left (equation 4.3 for 훾1 = 1/2). However, for the first hierarchical order, only three triangles out of four have their elastic modulus amplified by this factor. Consequently, the amplification of the elastic modulus from one generation to the next is 3/4 × 1.598 ≈ 1.2. Therefore we expect the elastic modulus, normalized by the regular honeycomb, to be 1.2푛 where 푛 is the hierarchical order. However, this result is based on the assumption of perfect self-similarity. Figure 4.3 suggests that this assumption holds for 푛 ≥ 5 where 훾푖approaches to 1/2 for all hierarchical orders. Therefore we take the value of elastic modulus for 푛 = 5 and propose:

푏 푛−5 퐸 /퐸0 = 9.2 × 1.2 (4.5)

Equation 4.5 gives an upper bond value for 푛 > 5 and agrees well with finite element simulations

푚 3 as shown in figure 4.2. Using the elastic modulus of regular honeycomb 퐸0/퐸 = 1.5 𝜌̅ [200], equation 4.5 can be written as:

퐸푏/퐸푚 = 5.58 × 1.2푛𝜌̅3 (4.6)

91 versus the relative density. (the numerical analysis (circle markers) are shown together with scaling analysis results (equation 4.9) (dashed curve).

where 퐸푚 is the elastic modulus of the bulk material. Note that equations 4.5 and 4.6 are valid as long as cell walls undergo only bending, which is relevant to the limit of thin beams, i.e. vanishing density. Therefore, to determine the maximum achievable stiffness for each order of hierarchy, we also need to compute the shear-based and stretching-based modulus of the structure. For this purpose, we seek the shear energy and stretching energy stored in the zero and first order hierarchical honeycomb. For zero order, the projection of 퐹 perpendicular to the beam is 퐹/2. The

2 푚 stored shear energy can be written as 1/(8√3) ∙ 푟퐹 /(퐸 𝜌̅), where 푟 = 2푘푠(1 + 휈). 휈 is the

Poisson’s ratio of bulk material and 푘푠 is the shear coefficient (equal to 5/6 for rectangular cross

푠ℎ 푚 sections). Therefore, the corresponding stiffness is 푘0 /퐸 = 4√3/푟 ∙ 𝜌̅. For the first order (훾1 =

1/2), all the beams have the length 푙0/4. The reaction forces at the midline are shown as 푁1 and

푚 2 2 2 푁2 (figure 4.1b). The shear energy is determined as √3/8 ∙ 푟/(퐸 𝜌̅) ⋅ (퐹 + 푁1 + 5푁2 ). As 푁1 +

푁2 = 퐹, minimizing the energy with respect to 푁1 yields 푁1 = 5퐹/6 and 푁2 = 퐹/6. Consequently

2 푚 푠ℎ 푚 the shear energy is 11√3/192 ⋅ 푟퐹 /(퐸 𝜌̅), while the corresponding stiffness is 푘1 /퐸 =

푠ℎ 푠ℎ 32√3/(11푟) ∙ 𝜌̅ . The shear modulus is multiplied by the ratio 푘1 /푘0 = 8/11. Similar to bending, only 3/4 of the structure has its modulus changed. So the shear-based modulus is multiplied by 3/4 ⋅ 8/11 = 6/11. Therefore the shear based modulus takes the form:

퐸푠ℎ/퐸푚 = 4√3푓 ⋅ 0.545푛𝜌̅ (4.7) where 푓 = 1/푟 and for 휈 = 0.3 and rectangular cross section is equal to 0.3205. Using the same

푛 procedure, the stretching energy is given in scaling terms by 푈푠푡~ (9/8) . Since the shear energy

푛 is given by 푈푠ℎ~ (11/8) , the stretching energy can be ignored compare to shear energy for large

푛. Therefore elastic modulus saturates when the shear modulus of the structure become comparable to bending modulus. The above equations show that if the order of hierarchy increases, the bending based-modulus increases while the shear based modulus decreases. The two expected to be optimal when 퐸푠ℎ~퐸푏. Therefore the optimum hierarchy order can be found as

푛 = ⌊−2.54 ln 𝜌̅ + 푐⌋ (4.8) where ⌊∙⌋ is the floor function (푛 is integer). 푐 can be found from numerical data as −3.03. Figure

4.4a shows optimum hierarchical order for different densities. The scaling analysis is in good agreement with the results of finite element simulations especially for small densities (푅2 = 0.93).

Replacing the value of 푛 from equation 4.8 in elastic modulus of equation 4.5 gives the maximum reachable elastic modulus at each density as

(−0.46) 퐸/퐸0 = 푐1 𝜌̅ + 푐2 (4.9)

where 푐1 and 푐2 can be found from numerical data as 2.15 and −3.19, respectively. Figure 4.4b shows the maximum achievable elastic modulus as a function of relative density. The results of scaling analysis are in good agreement with FE simulations, especially for small densities

(푅2 = 0.99).

In figure 4.5, we have plotted the achievable elastic modulus range as a function of density for up to 10 order of hierarchy. The upper bound of this range, which is depicted with a dashed line, shows the maximum achievable elastic modulus for different densities, same as the one shown in figure 4.4b. As can be seen from this figure, increasing the hierarchical order while preserving the structural density can significantly increase the effective elastic modulus of hierarchical

93 structure. Also, the maximum achievable hierarchical level has increased by reducing the structural density.

Figure 4-5. Stiffness range for different order of hierarchy, 푛, versus relative density. Dashed line shows the stiffness limit of the hierarchical honeycomb for specified relative density (equation 9)

4.3. Concluding remarks

In summary, new class of fractal-appearing cellular meta-materials is introduced. The results show that the effective elastic modulus of the developed cellular material can be increased significantly by increasing the hierarchical order while preserving the structural density. The

94 optimal hierarchical level has also shown to be increased by reducing the structural density. This particular case of hierarchical refinement can be served as a promising realization of performance enhancement due to hierarchy. Moreover, the current work provides insight into how incorporating hierarchy into the structural organization can play a substantial role in improving the properties and performance of the materials and structural systems and introduces new avenues for development of novel metamaterials with tailorable properties.

Chapter 5 Hierarchical analysis and multi-scale modelling of rat cortical and trabecular bone

5.1. Introduction

Cortical and trabecular bone are arranged within a hierarchical structure in the osseous tissue: this diversity of structures allows the skeleton to perform its mechanical and metabolic functions.

Different factors, such as bone mass, geometry, material properties, cortical to trabecular proportion, molecular composition, microstructure and architecture, contribute to the tissue’s strength and quality [245-247].

Collagen fibers, as an organic component, and carbonated apatite crystals, as a non-organic component, contribute to bone’s strength by resistance against loads applied to its structure [245].

The mineral phase is the main determinant of stiffness, whereas collagen content governs its post- yield ductility. The mechanical properties of bone are ultimately determined by the mineral content and its distribution pattern within the collagenous matrix, as well as the tissue’s structural, microstructural and nanostructural organization [248].

Composed primarily of osteons of concentric lamellae, cortical bone is remarkably stiff and contributes substantially to the tissue’s mechanical strength. On the other hand, trabecular bone is arranged in a mosaic of angular segments of parallel sheets of lamellae and shows a greater rate of metabolic activity, lower modulus and larger surface-area-to-volume ratio [117, 246, 249, 250].

Extensive research has been conducted to distinguish cortical from trabecular bone [33, 96, 97,

104, 107, 115, 117, 123, 248, 251-266]. Although a variety of animal models and analytical techniques have been employed to assess the differences between these two structural components—especially in the context of endocrine, dietary and stress variables—few studies have compared them in a comprehensive manner. Bigi et al. [256] have reported the CO3 and Ca/P

97 content of trabecular and cortical bone in mouse. Bagi et al. [258] analyzed the yield load and stiffness of cortical bone in mouse based on bone volume fraction. Toolan et al. [267] analyzed the effects of bisphosphonates on the mechanical behavior of rat bones. Hodgskinson et al. [123] and Kuhn et al. [254] reported the hardness and Ca, CO3, C/PO4 and Ca/P content for trabecular and cortical bone of bovine. Limited data have been reported on pig and dog [108, 263]. The majority of work on human contains mineral density, bone volume fraction, tissue modulus and module of elasticity [33, 96, 97, 104, 107, 115, 117, 264, 266]. These studies have produced inconsistent results and have only provided snapshots of the vast spectrum of data on which to base an exhaustive comparison (table 5.1). Given the complex nature of bone, a comparison between its cortical and trabecular components should consider the hierarchical arrangement of structural properties for these two distinct tissues. Recent technological advancements have allowed researchers to evaluate bone’s properties at ultra-, micro- and nanostructural levels, facilitating new insights into the tissue’s material properties. Moreover, by considering the relative influences of certain structural parameters on bone strength and modulus, the tissue’s mechanical properties can be predicted by mathematical modelling with single- and two-parameter power-law or linear functions [166, 268-271]. However, owing to the heterogeneity and anisotropic material properties of cortical and trabecular bone, these methods cannot fully predict the mechanical properties of bone. In recent years, several methods have been proposed to overcome this shortcoming [37, 99, 272-275]. One of the methods considered for this purpose is the continuum micromechanics approach [276, 277]. Continuum micromechanics is the analysis of heterogeneous or anisotropic materials at the level of the individual material elements forming these materials [276, 277]. It has been used in several applications including modelling of defects

98 in solids [278], mechanical properties of composites [279], electroelastic moduli of piezoelectric composites [280] and recently in modelling of mechanical properties of bone [37, 274, 275].

Table 5.1. A chronological snapshot of comparative hierarchical properties of cortical and trabecular bones.

Level Indices Ref. Bone Type Testing Technique Cortical Bone Cancellous Bone [255] Rat femur/tibia Thermogravimetry 66.4 (0.3) 62.0 (0.3) Ash (Inorganic) content % [97] Steer vertebra/tibia gravimetry 67.86 64.55 Protein content (% ) [97] Steer vertebra/tibia gravimetry 28 31.09 [259] Mouse femur/tibia μCT 1.089 (.017) 0.745 (.102) Bone Mineral Density Human (black) tibia 0.229 (.088) 1.188 (.043) (BMD) (g.cm-3) [264] Pripheral QCT Human (white) tibia 0.255 (0.053) 1.117 (3.6) [262] Rat vertebra/femur Gravimetry 2.066 (.005) 1.908 (.011) Bone Tissue Density (ρ ) t [97] Human vertebra/tibia Gravimetry 1.91 1.87 (g.cm-3)

Macro-structure Steer vertebra/tibia Gravimetry 1.995 (0.01) 1.93 (0.22) [257] Rat Femoral midshaft Three point bending 47.24 (6.59) -- Stiffness (N.mm-1) [266] Rat Femoral midshaft Three point bending 588 (75) -- Rat Vertebra Compression -- 1327 (336) Failure Load (N) [257] Rat femoral midshaft Three point banding 160.09 (30.80) -- [257] Rat femur μCT 0.46 0.11 (0.04) BV/TV (mm3.mm-3) [259] Mouse femur/tibia μCT -- 0.25 (0.06) [261] Fetal pig mandibular μCT -- 24.14 (4.14) [265] Human iliac crest (23 yr) Three point bending 3.76 (1.68) 3.03 (1.63) Human iliac crest (63 yr) Three point bending 5.26 (2.09) 4.16 (2.02) Mod. of Elasticity (GPa) [117] Human tibia Four point bending 6.75 (1.00) 5.72 (1.27) Micro-structure [115] Human proximal tibia Three point banding 5.44 (1.25) 4.59 (1.6)

Yield Strength (YS) (GPa) [33] Human femur Compression 0.109 0.089 [108] Porcine femur Microindentation 11.6 (9.5) 5.9 (4.3) Nanoindentation 16.4 (1.3) 21.5 (2.1) [104] Human vertebrae/tibia Nanoindentation 25.8 (0.7) 13.4 (2.0) Tissue Modulus (GPa) [107] Human femur Acoustic microscopy 17.73 (0.22) 17.5 (1.12) Nanoindentation 20.02 (0.27) 18.14 (1.7) [247] Human femur Nanoindentation 21.2 (5.3) 11.4 (5.3)

Nano-structure [104] Human vertebrae/tibia Nanoindentation 0.736 (.034) 0.468 (.079) Hardness (GPa) [247] Human femur Nanoindentation 0.234-0.76 0.234-0.76 [255] Rat femur/tibia FTIR 3.8 (0.2) 2.3 (0.2) CO3 (-) [253] Bovine femur/tibia Chemical analysis 5.33 (0.1) 5.33 (0.18) C/Po4 (-) [253] Bovine femur/tibia Chemical analysis 0.17 0.17 [123] Bovine femur Colorimetry 271 257 Ca (mg.g-1) [96] Child vertebrae/Femur Gravimetry 194 47.4 HPO4 (% ) [253] Bovine femur/tibia FTIR 20.3 (0.2) 20.7 (0.2) [253] Bovine femur/tibia FTIR (%) 9.6 (0.1) 8.7 (0.1)

Composition PO4 Chemical analysis [96] Child vertebrae/Femur (mg/g of bone) 24.1 90.3 [255] Rat femur/tibia Spectrophometry 1.63 (0.2) 1.5 (0.2) Ca/P (-) [253] Bovine femur/tibia Chemical analysis 1.64 (0.02) 1.58 (0.06)

The aim of this study is to explore the hierarchical arrangement of structural properties in cortical and trabecular bone and to determine a mathematical model that accurately predicts the tissue’s mechanical properties as a function of these indices (figure 5.1). By gaining a better appreciation of the similarities and differences between the two bone types, we will be able to provide a better assessment and understanding of their individual roles, as well as their contribution to bone health overall [254, 261].

5.2. Material and methods

5.2.1. Specimen preparation

The study protocol was approved by the Institutional Animal Care and Use Committee at Beth

Israel Deaconess Medical Center, Boston, MA. Thirty Sprague–Dawley female rats (20 weeks old) were obtained from Charles River Laboratories (Charlestown, MA, USA) and euthanized via CO2 inhalation. Cylindrical samples of diaphysis cortical bone (height 6.85 + 0.85 mm) and distal metaphyseal trabecular bone (height 5.17 + 0.65 mm) specimens were obtained from each femur

(figure 5.2). Additionally, secondary specimens for embedding were obtained by cutting 1-mm- thick diaphysis and distal metaphyseal sections from all femurs. The specimen preparation protocol has been published in detail elsewhere [141]. All specimens underwent cleaning via sonic agitation (Fisher Scientific International, Hampton, NH, USA) while suspended in distilled water for 20 min, followed by centrifugal removal of excess water and marrow at 9g for 15 min. The details of the analytical methods will be presented in hierarchical fashion as follows.

100

Figure 5-1. An illustration of the hierarchical nature of cortical and trabecular bone

101

Figure 5-2. An illustration of the preparation process for cortical and trabecular specimens

5.2.2. Macrostructural properties

5.2.2.1. Extrinsic structural properties

All cylindrical specimens underwent uniaxial compression (INSTRON 8511, Instron Corporation,

Norwood, MA, USA) for determination of properties through analysis of the load–displacement curve. Structural stiffness was defined as the slope of the linear portion of the curve, whereas yield load was represented at the point where the curve ceased to be linear. The point with the highest load value represents the ultimate load.\

5.2.2.2. Bone tissue density (흆풕)

Bone mass and tissue volume (TV) of cylindrical specimens were measured by a precision scale

(AnalyticalPlus, Ohaus, Pine Brook, NJ, USA) and gas psychometry (AccuPyc 1330,

102

Micromeritics, GA, USA). Bone tissue density was calculated by dividing bone mass by bone tissue volume.

5.2.2.3. Mineral and matrix content

In order to determine the mineral (ash mass/dry mass) and matrix (1 2 (ash mass/dry mass)) contents, the cylindrical specimens were dried at 708C for 24 h and ashed at 6008C for 96 h

(Furnace 48000, Thermolyne, Dubuque, IA, USA). It has been shown that some parts of mineral evaporate at 6008C and quantitatively contribute to 6.6% of mineral weight. Therefore, the measured mineral content should be multiplied by 1.066 to show the actual mineral content in the bone.

5.2.3. Microstructural properties

5.2.3.1. Morphometric indices

Bone volume fraction (BV/TV) and bone-surface-to-volume ratio (BS/BV) of the trabecular and cortical cylindrical specimens were assessed using micro-computed tomography (mCT40; Scanco Medical AG, Bru¨ttisellen, Switzerland—tube energy and current, 55 kVp and 145 mA, respectively; integration time, 250 ms; and isotropic voxel size, 20 휇m).

5.2.3.2. Apparent material properties

Apparent mechanical properties were calculated from the stress – strain curves obtained from uniaxial compression testing. The minimum cross-sectional areas for cancellous and cortical bone specimens were calculated from 휇CT images (figure 5.3). The modulus of

103 elasticity (E) was determined from the slope of the linear portion of the curve, while the point where the curve ceased to be linear was designated as the yield strength (YS). The point with the highest strength value represented the ultimate strength (US).

Figure 5-3. Calculation of minimum cross-sectional area for (a) cortical and (b) trabecular specimens

5.2.4. Nanostructural properties

5.2.4.1. Nanoindentation

The secondary specimens were dehydrated with ethyl alcohol, embedded in epoxy resin and polished. The midsections of the trabecular elements and the cortical shells were selected as indentation sites using a Berkovich indenter (Hysitron Tribo- indenter,

Minneapolis, MN, USA) to avoid boundary condition errors. Thirty-five indentations distributed across the cross section of each sample were done and the results were averaged per sample.

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5.2.5. Compositional properties

5.2.5.1. Total protein and collagen content

Following uniaxial compression testing, cylindrical specimens were subjected to amino acid analysis. For this purpose, specimens were powdered using a Spex mill (SPEX Freezer/Mill;

SPEX Industries Inc., NJ, USA) and lyophilized to recover cortical and trabecular bone powder. The matrix analysis was performed with an amino acid analyzer (Beckman System

7300; Beckman Coulter Inc., CA, USA). The amino acids were separated by ion-exchange chromatography followed by post-column derivatization using ninhydrin for detection.

Signals at 440 and 570 nm wavelengths were integrated, and the concentration of each ninhydrin-reactive component was recorded.

5.2.5.2. Phosphate (PO4), hydrogen phosphate (HPO4), carbonate (CO3), carbonate/phosphate and protein/mineral content

Fourier transform infrared (FT-IR) spectroscopy was performed on cylindrical specimens using a spectrometer (Perkin-Elmer, Waltham, MA, USA). The spectra were curve-fitted in the 휈4 PO4, 휈2CO3 and amide band domains (Galactic GRAMS Software, Salem, NH,

USA). The 휈4PO4 domain shows five main phosphate bands at 600, 575 and 560 cm-1 for

PO4 groups in an apatite lattice, and 617 and 534 cm-1 for non-apatitic environments corresponding to surface location (figure 5.4a) [281]. The 휈2 CO3 domain was decomposed into three bands at 879, 871 and 866 cm-1 related to types A and B carbonate, and carbonate ions in non-apatitic environments, respectively (figure 5.4b) [282]. The amide

–휈3 carbonate domain (1300 – 1800 cm-1) was decomposed into seven bands at 1750, 1670,

105

1640, 1550, 1510, 1450 and 1410 cm-1. The relative intensity of the mineral and protein bands has proven to be an accurate measure of the mineral-to-protein ratio [283].

Figure 5-4. a) Infrared spectrum in the 휈4 PO4 domain of a synthetic nanocrystalline apatite

5.2.5.3. Calcium and phosphate (퐏퐎ퟒ) content

Calcium content was determined us Calcium content was determined using an NBX electron microprobe (Camera Instrument, Nampa, ID, USA) on isolated mineral crystals pressed into a flat pellet (beam voltage, 10 keV; beam current, 30 nA; and a rastered beam, 64 × 64 휇m) [284, 285].

On the other hand, a modified Fiske and Subbarow colorimetric method at the peak absorption of

660 휇m was used to quantify the phosphate content [286-288].

106

5.2.6. Statistical analysis

Normality of continuous data was assessed by using the Kolmogorov–Smirnov test. Comparative analyses were performed by one-way analysis of variance (ANOVA), with bone type (cortical and trabecular) as independent variables, and outcome measures from different testing modalities as dependent variables. In addition, a regression analysis was conducted to determine the correlation between axial stiffness derived from experiments and that derived from the micromechanics model. Statistical analysis was performed using the PSAW software package (version 19.0; SPSS

Inc./IBM, Chicago, IL, USA). Two-tailed values of p < 0.05 were considered statistically significant.

5.2.7. Mathematical modeling

Bone has a hierarchical structure [289, 290]; therefore, each level of hierarchy plays a significant role in the mechanical properties of bone structure. Figure 5.5 shows the four levels of hierarchy inspired from material composition and structure of the bone: nanoscale (10–100 nm), submicroscale (1–10 휇m), microscale (10–100 휇m) and macroscale (0.5–10 mm). The objective of mathematical modelling is to relate macrostructural mechanical properties of bone to its elementary components, namely hydroxyapatite (HA) crystals, collagen, non-collagenous proteins

(NCPs) and water.

A micromechanics approach is ideal for modelling the mechanical properties of bone owing to the tissue’s heterogeneity and complex structure. The basic concept behind it is to hierarchically label the representative volume elements (RVEs) in the bone structure. Based on the micromechanics approach framework, the RVEs should have two main aspects: first, the characteristic dimension of these volume elements (푙) should be considerably larger than the

107 characteristic length (푑) of the elements constructing them and at the same time extremely smaller than the characteristic dimensions (퐿) of the architecture built by the RVEs (i.e. 푑 ≪ 푙 ≪ 퐿).

Second is the ability of RVEs to be divided into phases with constant material properties.

Figure 5-5. Micromechanics representation of hierarchical structure of bone with four levels from nano to macro scale

108

At each level of hierarchy, the phases and their properties are defined (volume fractions

휙 and elastic stiffnesses 퐜). Based on the linear elasticity estimate of continuum micromechanics and assuming constant elastic modulus for the phases in the RVEs, the homogenized stiffness of

0 RVEs, 푪푒푠푡, can be determined as [276, 277]

−1 0 푛 0 0 −1 푛 0 0 −1 푪est = ∑푟=1 휙푟퐜푟: (퐼 + 퐏푟 : (퐜푟 − 퐂 )) : [∑푠=1 휙푠(퐈 + 퐏푠 : (퐜푠 − 퐂 )) ] (5.1) where 푛 is the number of phases in the RVEs, 퐜 is the phase stiffness, 퐈 is the fourth-order unity tensor and 퐂0 is the homogeneous elastic matrix stiffness which is included in the phases. Tensor

푷0 is related to the Eshelby tensor [291] (푷0 = 푺0퐸푠ℎ: 퐶0,−1), which characterizes the interaction between phases in the RVEs.

In micromechanics modelling, the elastic stiffness of RVEs found in each level of hierarchy will be used as phase stiffness for the analysis of subsequent levels. There are several estimates in the literature for choosing 푪0. The Mori–Tanka scheme [292, 293] is the model which chooses 푪0 =

푪matrix, meaning that there is an inclusion phase consisting of small particles embedded in the continuous-matrix phase. This model, which is best suited for particle-reinforced composites, has

0 0 an explicit solution. The self-consistent scheme [294, 295] is the model which chooses 퐂 = 퐂푒푠푡, meaning that the phases are dispersed with the stiffness properties of homogenized RVEs. For the self-consistent scheme, equation (2.1) is reduced to a set of nonlinear equations. Here, the micromechanics representation of each phase and corresponding elastic modulus tensors are derived.

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5.2.7.1. Nanoscale

5.2.7.1.1. Interaction of water and non-collagenous proteins with hydroxyapatite

At the nanostructural level, HA crystals, water and NCPs (figure 5.5) interact with each other. At this level, phases are dispersed, thus warranting the use of a self-consistent scheme

−1 −1 −1 푛 ̂ 푛 ̂ 푪HA,w = ∑푟=1 휙푟퐜푟: (퐼 + 퐏푟 : (퐜푟 − 퐂퐻퐴,푤)) : [∑푠=1 휙푠 (퐈 + 퐏푠 : (퐜푠 − 퐂퐻퐴,푤)) ] (5.2)

HA minerals are platelet shaped [296-299], and water is considered to have a spherical shape.

Volume fractions of HA and water are 휙̂퐻퐴 and 휙̂푤푛1 , with a sum equal to 1,

휙̂퐻퐴 + 휙̂푤푛1 = 1 (5.3)

Platelet-shaped HA minerals make the RVE matrix anisotropic. Using Laws formula [300, 301] for determining the P tensor in a transversely isotropic matrix, the P tensor can be found for water and HA (appendix A.1). Assuming isotropic material properties for water and HA, the corresponding elastic matrices can be written as

푣표푙 푑푒푣 퐜퐻퐴 = 3퐾퐻퐴퐈 + 2퐺퐻퐴퐈 (5.4) and

푣표푙 푑푒푣 퐜푤푛1 = 3퐾푤퐈 + 2퐺푤퐈 (5.5)

푣표푙 where 퐾퐻퐴, 퐺퐻퐴, 퐾푤 and 퐺푤 are the isotropic stiffness properties of the HA and water. 퐈 and

푰푑푒푣 are, respectively, the volumetric and deviatoric part of the fourth-order unity tensor (푰푣표푙 =

푑푒푣 푣표푙 1/3훿푖푗훿푘푙 and 퐈 = 퐈 − 퐈 ). For highly mineralized tissues, radial stiffness was shown to be

110 equal to axial stiffness [302], which means that HA isotropically contributes to the bone tissue stiffness. For the exploration of the volume fraction–radial stiffness relation to the point where the volume fraction of HA is equal to 1, the isotropic elastic stiffness of HA is equal to 100 GPa. Based on the relation: 퐶11,퐻퐴 = 퐸퐻퐴(1 − 휈ℎ퐴)/((1 + 휈퐻퐴)(1 − 2휈퐻퐴)) and assuming 푣퐻퐴 = 0.27 [303], the elastic modulus, bulk modulus and shear modulus of HA can be found as 79.76, 57.8 and 31.4

GPa, respectively (table 5.2). Equation (5.2) leads to five coupled nonlinear equations, which should solve simultaneously to reach five constants of the transversely isotropic matrix 퐶퐻퐴,푤.

Table 5.2. Isotropic mechanical properties of bone elementary components.

Elastic Bulk Shear Poisson's Component Modulus* E Modulus K Modulus Reference Ratio (ν) (GPa) (GPa) G (GPa) Hydroxyapatite 79.76 0.27 57.8 31.4 [301] Collagen 5.4 0.28 4.1 2.1 [303] Water and Non-Collagenous Protein 0 0.49 2.3 0 *Only two of these parameters are independent. The other two can be found based on universal relations for isotropic material

5.2.7.1.2. Interaction water and non-collagenous proteins with collagen

At this level, fibrillar collagen molecules are attached to each other, and the space between them is filled with water and NCPs. Considering collagen molecules as a matrix and the interspace water and NCPs as an inclusion, from the Mori–Tanaka scheme, the corresponding stiffness can be written as

−1 ̂ ̂ ̂ ̂ 푪col,w = [휙푐표푙퐜푐표푙 + 휙푤푛2퐜푤푛2: (퐈 + 퐏푤푛2: (퐜푤 − 퐂푐표푙)) ] : [휙푐표푙퐈 + 휙푤푛2 (퐈 +

−1 −1 퐏푤푛2: (퐜푤 − 퐂푐표푙)) ] (5.6)

111

Assuming isotropic material properties for the collagen matrix, the corresponding elastic modulus matrix can be written as

푣표푙 푑푒푣 퐜푐표푙 = 3퐾푐표푙퐈 + 2퐺푐표푙퐈 (5.7)

where 퐾푐표푙 and 퐺푐표푙 are, respectively, the bulk modulus and shear modulus of the collagen stiffness matrix [304] (table 5.2). Because the matrix is isotropic, the 퐏푐표푙 tensor is defined based on the cylindrical inclusions embedded in an isotropic matrix (appendix A.2). Volume fractions of phases are 휙̂푐표푙 and 휙̂푤푛2 for collagen and water, respectively, where

휙̂푐표푙 + 휙̂푤푛2 = 1 (5.8)

5.2.7.2. Submicroscale

At this level, the organic and mineral phases interact with water and with each other. Stiffness matrices of mineral and organic phases come from RVEs at the nanolevel. Dispersion of the phases in the RVEs warrants the use of a self-consistent scheme:

−1 −1 −1 푛 푛 푪subm = ∑푟=1 휙푟퐜푟: (퐈 + 퐏푟 : (퐜푟 − 퐂푠푢푏푚)) : [∑푠=1 휙푠 (퐈 + 퐏푠 : (퐜푠 − 퐂subm)) ] (5.9)

where 푪subm is the stiffness of submicroscale. Volume fractions of phases occupying the RVEs are 휙퐻퐴,푤, 휙푐표푙,푤and 휙푤푛3, where

휙퐻퐴,푤 + 휙푐표푙,푤 + 휙푤푛3 = 1 (5.10)

Spherical phase inclusions are chosen for mineral and water phases (i.e. 퐏푚푖푛 = 퐏푤 = 퐏푠푝ℎ) and cylindrical phase inclusions are chosen for organic phases (i.e. 퐏푚푎푡 = 퐏푐푦푙; appendix A.1).

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5.2.7.3. Microscale

At a microstructural level, lacunae containing osteocytes are enclosed by the continuous bone matrix. From the Mori– Tanaka scheme, considering the bone material as a matrix and lacunae as spherical inclusions with volume fractions of 휙푠푢푏푚 and 휙푙푎푐, the stiffness of bone material at the microscale, 퐂푚푖푐, becomes

−1 푪mic = [휙푠푢푏푚퐜푠푢푏푚 + 휙푙푎푐퐜푙푎푐: (퐈 + 퐏푙푎푐: (퐜푙푎푐 − 퐂푠푢푏푚)) ] : [휙푠푢푏푚퐈 + 휙푙푎푐 (퐈 +

−1 −1 퐏푙푎푐: (퐜푙푎푐 − 퐂푠푢푏푚)) ] (5.11)

푣표푙 푑푒푣 where 퐜푙푎푐 = 3퐾푤퐈 + 2퐺푤퐈 , 퐏푙푎푐 = 퐏푠푝ℎ for a transversely isotropic matrix and

휙푠푢푏푚 + 휙푙푎푐 = 1 (5.12)

The non-zero terms of 퐏푙푎푐 are presented in appendix A.1.

5.2.7.4. Macroscale

The structure of cortical and trabecular bone becomes different at this level. Therefore, the modelling has been divided into cortical and trabecular bone (figure 5.5). The Haversian canals contain blood vessels and nerve cells; therefore, it is reasonable to assign water stiffness to them

푣표푙 푑푒푣 (퐜ℎ푎푣 = 3퐾푤퐈 + 2퐺푤퐈 ). The volume fractions of phases in the RVEs are 휙ℎ푎푣 and 휙푚푖푐, where

휙ℎ푎푣 + 휙푚푖푐 = 1 (5.13)

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For cortical bone, considering bone microstructure as a matrix and Haversian canals as inclusions and using the Mori–Tanaka scheme, the stiffness matrix can be written as

−1 푪mac = [휙푚푖푐퐜푚푖푐 + 휙ℎ푎푣퐜ℎ푎푣: (퐈 + 퐏ℎ푎푣: (퐜ℎ푎푣 − 퐂푚푖푐)) ] : [휙푚푖푐퐈 + 휙ℎ푎푣 (퐈 +

−1 −1 퐏ℎ푎푣: (퐜ℎ푎푣 − 퐂푚푖푐)) ] (5.14)

Finally, to model the stiffness of trabecular and cortical bone structure (퐂푏표푛푒), porosities in trabecular bone have considered as spherical inclusions, and medullary cavity and restoration cavities in cortical bone have been considered as cylindrical inclusions in a transversely isotropic matrix (퐏ℎ푎푣 = 퐏푠푝ℎ, 퐏푐푎푣 = 퐏푐푦푙). Because porosities and cavities are vacant, their stiffnesses are set to zero (퐜푝표푟 = 퐜푐푎푣 = 0). Using the Mori–Tanaka scheme, from equation (5.1) 퐂푏표푛푒 can be written as

−1 푪bone = [휙푀퐜푀 + 휙푁퐜푁: (퐈 + 퐏푁 : (퐜푁 − 퐂푀 )) ] : [휙푀퐈 + 휙푁 (퐈 + 퐏푁 : (퐜푁 −

−1 −1 퐂푀 )) ] (5.15) and

휙푀 + 휙푁 = 1 (5.16) where subscript M stands for mac and mic and subscript N stands for cav and por regarding cortical and trabecular bone, respectively.

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5.2.7.5. Elementary-phase stiffness values and modelling parameters

Having a micromechanics model in hand, the elementary-phase stiffness values of the bone structure should be determined (i.e. 퐾퐻퐴, 퐺퐻퐴, 퐾푐표푙, 퐺푐표푙, 퐾푤, 퐺푤,). Table 5.2 shows the values which are chosen for the model. The phase stiffness matrices can be built based on these properties

(풄퐻퐴, 풄푐표푙, 풄푤). Then, the tissuespecific composition data should be determined: 휙̂푐표푙, 휙̂푤푛1, 휙̂퐻퐴 and 휙̂푤푛2 for nanoscale; 휙퐻퐴,푤, 휙푐표푙,푤 and 휙푤푛3 for submicroscale; 휙푠푢푏푚 and 휙푙푎푐 for microscale; and 휙푚푖푐, 휙ℎ푎푣, 휙푚푎푐, 휙푐푎푣 (cortical bone), 휙푝표푟, 휙푚푖푐 (trabecular bone) for macroscale.

First, the volume fractions of bone elementary components 휙푐표푙, 휙퐻퐴 and 휙푤푛 are determined

∗ from experimental data. Having mineral density (𝜌푚푖푛 = 푚푚푖푛/푉푏표푛푒) as a ratio of mineral mass

푚푚푖푛 to bone volume 푉푏표푛푒, found from 휇CT analysis (table 5.3) and the mass density of HA as

−3 𝜌퐻퐴 = 3 푔 푐푚 [305] the volume fraction of HA can be obtained as

∗ 휌푚푖푛 휙퐻퐴 = 휙푚푖푛 = (5.17) 휌퐻퐴

Bone mineral content (BMC), which is the ratio of mineral mass (푚푚푖푛) over dry bone mass

(푚푑푟푦), is determined using ashing analysis). Taking the mass density of organic matrix as 𝜌표푟𝑔 ≈

−3 𝜌푐표푙 = 1.41 푔푐푚 [305, 306], 휙표푟𝑔 reads as

1−퐵푀퐶 휌퐻퐴 휙표푟𝑔 = ( ) 휙퐻퐴 (5.18) 퐵푀퐶 휌표푟푔

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Table 5.3. Composition and axial module of elasticity of cortical and trabecular bone.

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Here, the mass densities of protein and collagen are assumed to be the same.

Approximately 90% of the mass density of protein is collagen [290, 305], therefore 휙푐표푙 =

0.9 휙표푟𝑔 (table 5.3). Then, the volume fraction of water and non-collagenous protein can readily be found as

휙푤푛 = 1 − 휙푐표푙 − 휙퐻퐴 (5.19)

The total volume fraction of water and non-collagenous proteins can be written as

휙푤푛 = 휙̂푤푛1휙푐표푙,푤 + 휙̂푤푛2휙퐻퐴,푤 + 휙푤푛3 (5.20)

At nanoscale observation, volume fractions of HA and collagen (휙̂퐻퐴 and 휙̂푐표푙) can be found as

휙푐표푙 휙̂푐표푙 = (5.21) 휙푐표푙,푤 and

휙퐻퐴 휙̂퐻퐴 = (5.22) 휙퐻퐴,푤

Using equations (5.3), (5.8), (5.10), (5.20), (5.21) and (5.22) and assuming water contents in the two considered RVEs at the nano level are the same (휙̂푤푛1 =휙̂푤푛2), and also the water contents at the nano- and microlevels are equal (휙푤푛3 =휙푤푛/2), the volume fraction composition at the nano- and submicroscale can be found as

휙푤푛 휙̂푤푛1 = 휙̂푤푛2 = (5.23) 2−휙푤푛

2−휙푤푛 휙푐표푙,푤 = 휙푐표푙 (5.24) 2−2휙푤푛

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2−휙푤푛 휙퐻퐴,푤 = 휙퐻퐴 (5.25) 2−2휙푤푛

At the microlevel, the volume fraction of the lacunae is defined as the area of lacunae in the examined area divided by the total area. Here, 휙푙푎푐 is assumed to be 2% [307-309]. The volume fraction of Haversian canals is defined as the ratio of the area of Haversian canals to the total considered area. 휙ℎ푎푣 varies from 2% to 5% for healthy cortical bone [310], and here it is assumed to be 3% [311]. At last, for determining the volume fraction at the macrolevel (cortical and trabecular bone), 휙푚푎푐 (cortical) and 휙푚푖푐 (trabecular) are assumed to be equal to bone volume fraction (BV/TV). The experimental data for bone volume fraction can be found in table 5.3.

5.2.8. Finite element analysis

휇CT-based finite-element analysis was performed, for both trabecular and cortical bone, to evaluate the proposed micromechanics model. The models were meshed with eight-node linear hexahedral elements. The material was assumed to be linear elastic with the elastic modulus taken from experimental tissue modulus results for each sample (table 5.4). The number of elements ranged from 110 000 to 630 000 for trabecular samples and from 990 000 to 1 900 000 for cortical bone samples. To mimic the mechanical testing conditions, the lower surface of the models was fixed, whereas a linear displacement load was applied at the upper surface of the model. Then, the reaction forces at the superior surfaces were evaluated to determine the apparent elastic modulus of the samples.

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Table 5.4. Macro-, micro-, nano- and compositional-level properties of rat cortical and trabecular bone found by this study.

5.3. Results

At the macrostructural level, no differences were observed between cortical and trabecular bone regarding tissue density (𝜌푡), as measured by gravimetric methods (푝 = 0.78), and mineral and matrix contents, as assessed by ash content ( = 0.42 and 0.41, respectively). Nonetheless, stiffness and yield load values were significantly greater in cortical bone (푝 < 0.001 for both cases; table

5.4).

Cortical bone has a larger volume fraction (푝 < 0.001) and a smaller bone-surface-to volume ratio (BS/BV; 푝 < 0.001) than trabecular bone. Apparent mechanical properties showed

119 that the cortical bone modulus of elasticity (E) and yield strength (YS) values were approximately four times greater than those of trabecular bone ( 푝 < 0.001 for both cases). Both bone types failed at the segment of the smallest cross section (table 5.4).

At the nanostructural level, no significant differences in tissue modulus and hardness were observed between the two bone types (푝 = 0.46 and 0.72, respectively; table 5.4). Cortical bone demonstrated higher modulus variability than trabecular bone (standard deviation was 5.73 GPa for cortical bone and 3.77 GPa for trabecular bone; figure 5.6). Amino acid analysis indicated no differences in total protein and collagen levels between the two bone types (푝 = 0.59 and 0.90, respectively). However, collagen content in cortical specimens was on average 7% greater than that of trabecular bone specimens (푝 < 0.001, table 5.4).

Figure 5-6. Tissue modulus frequency plots a) Cortical bone b) Trabecular bone.

−3 Non-apatitic phosphate (PO4 ) content did not differ between groups (푝 = 0.25), neither did the HPO4 content in the hydrated surface layer (푝 = 0.44). Carbonate (CO3) content also showed no difference among trabecular and cortical specimens (푝 = 0.01).

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There were no differences between groups in carbonateto- phosphate (C/P) and protein-to-mineral ratios (푝 = 0.09 and 0.89, respectively). Additionally, there were no differences in calcium and phosphate contents of the two bone types (푝 = 0.29 and 0.76, respectively; table 5.4).

The mathematical model’s validation is based on elastic moduli values obtained during mechanical testing (table 5.3). The 푅2 between the experimental and modelling axial stiffness values is 0.82, which shows relatively high agreement between the results (figure 5.7). The average values of axial elastic modulus (C33) found from micromechanics modelling for cortical and trabecular bone are 8.40 and 3.02 GPa, respectively. Experimental results show these values to be 8.34 and 2.85

GPa for cortical and trabecular bone, respectively. Based on the continuum micromechanics approach, the elastic tensor for rat cortical and trabecular bone can be evaluated as

Figure 5-7. Comparison of axial elastic stiffness between micro mechanics modeling and finite element modelling with experiments for cortical and trabecular bone.

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5.78 3.12 3.31 0 0 0 3.12 5.78 3.31 0 0 0

3.31 3.31 8.40 0 0 0 푪푡푟푎푏푒푐푢푙푎푟 = (5.26) 0 0 0 4.04 0 0 0 0 0 0 4.04 0 [ 0 0 0 0 0 2.88] and

1.98 1.06 1.15 0 0 0 1.06 1.98 1.15 0 0 0

1.15 1.15 2.85 0 0 0 푪푡푟푎푏푒푐푢푙푎푟 = (5.27) 0 0 0 1.34 0 0 0 0 0 0 1.34 0 [ 0 0 0 0 0 0.94]

Axial displacement contours of cortical and trabecular bone samples were obtained using finite- element analysis, as shown in figure 5.8. The axial displacement contour distribution in cortical bone is more regular than that in trabecular bone. The elastic moduli derived from finite element analysis are shown in figure 5.7. There was a strong correlation between finite-element analysis results and mechanical testing results (푅2 = 0.84).

Figure 5-8. Axial displacement contours of a) trabecular and b) cortical obtained from finite element analysis.

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5.4. Discussion

The aim of this study was to explore the hierarchical nature of the two major bone types in rats.

By using a variety of analytical techniques, we were able to characterize the structural and compositional properties of cortical and trabecular bone, as well as to determine the best mathematical model to predict the tissue’s mechanical properties.

Our hierarchical analysis demonstrated that the differences between cortical and trabecular bone reside mainly at the micro- and macrostructural levels. Our findings are consistent with those of previous studies: modulus of elasticity and yield strength values were significantly lower in trabecular bone specimens [33, 107, 115, 117, 264, 266]. Although not evidenced in our study,

Choi & Goldstein [117] made the same asseveration, emphasizing the higher mineral density values seen in trabecular bone. These findings can be explained by the configuration of lamellar/collagen fibers within the tissue, along with other microstructural characteristics that altogether support the fact that tissue morphology, and not just mineral density, plays a major role in determining the mechanical properties of cortical bone. The wide range of apparent elastic moduli can be explained by the wide range of bone volume fraction in the samples. Another factor causing the high degree of dispersion could be the way of calculating the apparent modulus: the resultant force is divided by the minimum cross-section area to determine the apparent elastic modulus (figure 5.3). Therefore, some of the apparent elastic moduli for trabecular bone are larger than those of cortical bone.

As shown by previous studies that also used FT-IR, carbonate content is significantly greater in cortical bone [256]. This finding may be explained by the critical role played by this ion during mineralization, coupled to the fact that cortical bone undergoes less remodelling over time. Khun

123 et al. [254] described the differences in the mineral content and crystal maturation process in young and old animals, which mirror tho[312]se seen in cortical and trabecular bone, respectively. For this reason, the differences between the mineral crystals may be attributed to their age, as well as to contrasting extents of post-translational modifications in the collagen structure [123, 255, 256].

The higher protein-to-mineral ratio and collagen content in protein seen in cortical bone seems to be similarly linked to its mechanical properties. The intermolecular cross-linking of collagen strongly determines the way fibrils are arranged to ultimately provide matrices with tensile strength and viscoelasticity [313, 314]. Although a weak trend was evidenced in the carbonate-to-phosphate ratio between bone types, this finding further demonstrates the reigning similarities of cortical and trabecular bone at the compositional level.

Analysis performed at the nanostructural level yielded results that were consistent with previous reports in the literature, where hardness is basically considered to be similar between both bone types [104, 108, 123]. Hodgskinson et al. [123] described a strong relationship between calcium content and hardness, all equally similar across specimens compared in this study.

The purpose of mathematical modelling was to predict the bone’s mechanical properties

(i.e. anisotropic elastic moduli) as a function of the elementary components of the bone. For mathematical modelling, two approaches have been proposed in the literature for attributing the anisotropy to the bone structure, namely ‘mineral-reinforced collagen matrix’ [315-317] and

‘collagen-reinforced mineral matrix’ [302, 318, 319]. Both these approaches have been incorporated in the proposed model; the choice of platelet-shaped HA as an inclusion imposes anisotropy at the nanoscale and the choice of cylindrical-shaped collagen molecules as an inclusion imposes anisotropy at the submicroscale (figure 5.5). Regarding the shape of the elementary constituents of bone, it has been shown that HA crystals have plate-like shapes [320]. In this study,

124 the HA crystals are assumed to be plate-like, which affects the Eshelby tensor and eventually the micromechanics model. Other approaches have employed spherical shapes to model HA inclusions [99]. The results of mathematical modelling are highly dependent on the choice of the mechanical properties of bone elementary components (i.e. HA and collagen).Here, we used the data from Katz & Ukraincik [303] and Yang et al. [304] for HA and collagen, respectively. These values also have been used by Hellmich & Ulm [302] and Hamed et al. [321] for multi-scale modelling of bone, and the results have shown good agreement with experimental findings. As seen in figure 5.7, finite-element modelling results better correlate with the mechanical testing results, especially for trabecular bone samples (푅2 = 0.56 for finite-element modelling versus

푅2 = 0.2 for micromechanics modelling). The reason is that micromechanics modelling provides more crude results, as the bone structure become more disorganized.

For comparison, bone mineral, organic and water densities and bone volume fractions of the samples are plotted along with the results of other organs and species in the literature [97, 310,

322-328] (figure 5.9). These plots were first reported by Vuong & Hellmich [328] to verify the universal relation among the bone constituents. As outlined in Vuong & Hellmich [328], these figures can be divided into two regions and be presented by bilinear functions. Figure 5.9a–c is

∗ ∗ ∗ plotted based on mineral (𝜌푚푖푛), organic (𝜌표푟𝑔) and water and non-collagenous protein (𝜌푤푛) densities. Alternatively, extracellular bone density can be found as

푒푐 ∗ ∗ ∗ 𝜌 = 𝜌푚푖푛 + 𝜌표푟𝑔 + 𝜌푤푛 (5.28)

∗ ∗ ∗ Densities of bone composition elements (𝜌푚푖푛, 𝜌표푟𝑔 and 𝜌푤푛) are plotted versus extracellular bone density in figure 5.9d. The volume fractions of bone constituents (휙푚푖푛, 휙표푟𝑔 and 휙푤푛) versus extracellular bone density are shown in figure 5.9e. In figure 5.9a, in the region with positive

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∗ slope, the organic density (𝜌표푟𝑔) is increased by increasing extracellular bone mineral density

∗ (𝜌푚푖푛).

∗ ∗ Figure 5-9. Relation between a) Apparent mineral (𝜌푚푖푛) and organic (𝜌표푟𝑔) densities b) Apparent

∗ ∗ water and noncollagenous proteins (𝜌푤푛) and mineral (𝜌푚푖푛) densities c) Apparent water and

∗ ∗ noncollagenous proteins (𝜌푤푛) and mineral (𝜌표푟𝑔) densities d) Apparent densities and e) Volume

∗ ∗ fractions of mineral (𝜌푚푖푛, 휙푚푖푛), organic (𝜌표푟𝑔, 휙표푟𝑔) and water and noncollagenous proteins

∗ 푒푐 (𝜌푤푛, 휙푤푛), versus extracellular bone density (𝜌 ).

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This region is represented by growing organisms and species, whereas the region with negative slope represents the adult organisms [328]. Figure 5.9a–e shows that the reported results in this study for bone composition densities and volume fractions are comparable to previous studies in the literature. In addition, our results further validate the universal relation between different bone composition elements [328].

In this study, we did not consider the potential differences in geometry in the two bone types being compared. We strongly believe that it would be relevant to address the roles of lacunae and osteons in the structural properties of trabecular and cortical bone. In addition, the inferior resolution of FT-IR at very small scales introduces another limitation to our study [261]. In spite of these shortcomings, this study provides a comprehensive framework for trabecular and cortical bone properties that can be expanded upon in future studies.

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Chapter 6 Curved Beam Computed Tomography based Structural Rigidity Analysis of Bones with Simulated Lytic Defect: A Comparative Study with Finite Element Analysis

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6.1. Introduction

The skeleton is the third most common site of metastatic cancer and nearly half of all cancers metastasize to bone [329-331]. Approximately 30-50% of bone metastases lead to pathologic fractures [332], where the orthopaedic surgeon faces the dilemma of determining the probability of such an event, oftentimes based on subjective assessments of bone strength. Patients deemed to have a low risk of fracture are treated using nonsurgical approaches [333, 334], and operative treatment is reserved for cases of impending and pathological fractures in long bone and pelvic girdle metastases. Nevertheless, the scoring systems frequently used to evaluate fracture risk are now recognized to be inaccurate [335]. Therefore, there is a need for a reliable clinical tool to objectively assess fracture risk based on the material and geometric determinants of bone strength.

Computed tomography-based structural rigidity analysis (CTRA), which takes into account the material properties and structural organization of bone, can reliably predict failure load in rat and human bones with lytic defects [214, 336-341]. However, CTRA calculations are derived from straight beam theory, where the influence of bone curvature on strength has not been considered which has been shown to be significant in calculating bone fracture load [342]. Therefore, in current study, a new method for CTRA derived from curved beam theory (curved CTRA) has been introduced. To that end, the failure load predictions from curved CTRA, traditional CTRA and

Finite Element (FE) modeling has been compared to those of mechanical testing in an ex-vivo human model of femoral lytic defects. We hypothesize that CTRA will outperform traditional

CTRA in terms of the accuracy of the predicated failure load and also the failure location and will correlate well with FEA and mechanical testing results.

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6.2. Material and Methods

6.2.1. Specimen Preparation

Following Institutional Review Board approval, ten paired femurs from fresh frozen human cadavers (mean age 81.7±10.65 years) were obtained from the Department of Anatomy at Radboud

University Medical Center [343, 344]. One of the femurs in each pair was left intact and assigned to the control group. The contralateral femur was assigned to the simulated lytic defect group, where one or more defects were created. Size and location of these lesions resembled clinical appearance of lytic metastatic lesions, as discussed with orthopedic oncologists. Lesion sizes and locations on defect femurs are shown in table 6.1.

Table 6.1 Artificial Lesion sizes and locations on defect femurs Specimen Lesion characteristics Size Location (mm) 1 40 Med, prox 2 40 Med, shaft 3 22 Med, prox 4 40 Post, prox 5 45 Med, prox 6 40 Lat, prox 7 2×22 Med,prox & shaft 8 40 Ant, prox 9 22 Ant, prox 10 Ant, prox & 2×30 shaft

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6.2.2. Imaging and Image Analysis

Quantitative computed tomography (QCT) images were acquired with the following settings: 120 kVp, 220 mA, slice thickness 3 mm, pitch 1.5, spiral and standard reconstruction, in- plane resolution 0.9375mm (ACQSim, Philips, Eindhoven, The Netherlands) [343]. The femurs were scanned in a water basin, on top of a solid calibration phantom (Image Analysis, Columbia,

KY, USA).

6.2.3. Mechanical Testing

Following imaging, the specimens underwent mechanical testing in a hydraulic mechanical testing system (MTS) machine [343]. The setup was designed to simulate the single-limb stance- type loading conditions on the femur. A 30 mm diameter plastic cup with concave shape was used to apply the load to the femoral head. An axial load was applied on the head of the femur, with

10N/s from 0N until failure, while force and displacement of the plunger were recorded. The failure location of each femur was photographically documented. The details of sample preparation and mechanical setup have been explained in [343].

6.2.4. Finite Element Analysis

Three dimensional (3D) model of the femurs were constructed using MATLAB

(MathWorks Inc., Natick, MA, USA). The calcium-phosphate (Ca-P) density (ρCHA) of each pixel in the CT scan was calculated using the calibrating phantoms [345] (the pixels below 30 mg/ml were automatically removed) and a 3D model was constructed using mutually connected pixels

[346, 347]. Using the relationship 𝜌푎푠ℎ = 0.0633 + 0.887𝜌퐶퐻퐴 [348], the Ca-P densities were transformed to bone mineral density (𝜌푎푠ℎ). Then, the mineral densities are converted to tissue

131 elastic modulus using empirically derived constitutive equations for cancellous [210] and cortical

[349] bone. Elastic-perfectly plastic isotropic material behavior was assigned to model the mechanical behavior of bone material, and constrains and boundary conditions were applied to the model to mimic the mechanical testing experiments [343]. The developed models were imported to finite element (FE) software, ANSYS (Academic Research, Release 14.0, Cecil, PA, USA), which was used to perform the FE analysis. The failure load in FE analysis was defined as the maximum total reaction force achieved during loading under displacement control. Failure location in the computational models was defined by the location of plastic deformation at maximum total reaction force.

6.2.5. Structural Rigidity Analysis

CTRA determines bone rigidity and likelihood of failure based on the axial (EA), bending

(EI), and torsional (GJ) rigidities of the weakest cross-section in the bone (wehealen et. al.). These parameters are evaluated at each transaxial cross-section by summing the rigidity of all pixels

(figure 6.1).

By assuming the planes stay plane in deformation, Failure load for unsymmetrical axial loading can be defined as:

(6.1) 퐹푐 = 퐷푐휀푐 where 퐷푐 is the rigidity at the weakest bone cross-section and 휀푐 is the critical strain which identifies the fracture initiation. Using straight beam theory [350], 퐷푐 can be defined as:

1 퐷푐 = (6.2) 훾 − 훼푥푐 + 훽푦푐

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Figure 6-1. The curve beam model representation. (ABCD section from right figure is shown on the left before and after deformation with corresponding strains)

The 훾 parameter is associated with axial rigidity, and 훼 and 훽 are associated with bending rigidity of the-cross section. 푥푐푟 and 푦푐푟 are coordinates of critical location in the weakest cross- section and 훾, 훼 and 훽 are defined as:

1 훾 = (6.3) ∑ 퐸푖(𝜌) 퐴푖

퐷푥(퐸퐼)푥 + 퐷푦(퐸퐼)푥푦 (6.4) 훼 = 2 (퐸퐼)푥(퐸퐼)푦 − (퐸퐼)푥푦

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퐷푦(퐸퐼)푦 + 퐷푥(퐸퐼)푥푦 (6.5) 훽 = 2 (퐸퐼)푥(퐸퐼)푦 − (퐸퐼)푥푦 where 퐷푥 and 퐷푦 are the distances, from the geometric centroid, of the applied load in x and y directions. 퐸푖 and 𝜌 are the elastic modulus and density, respectively, at the ith location of the cross-section, and 퐴푖 is the incremental cross-sectional area (figure 6.1) and (퐸퐼)푥, (퐸퐼)푦 and

(퐸퐼)푥푦 correspond to the bending rigidities of the cross-section and can be defined as:

2 (6.6) (퐸퐼)푥 = ∑ 퐸푖(𝜌) 푦푖 퐴푖

2 (6.7) (퐸퐼)푦 = ∑ 퐸푖(𝜌) 푥푖 퐴푖

(6.8) (퐸퐼)푥푦 = ∑ 퐸푖(𝜌) 푥푖푦푖퐴푖

As the traditional CTRA is based on straight beam theory, its main disadvantage is that it does not account for the influence of intrinsic bone curvature. This influence is particularly important for structures where the ratio of the radius of curvature to the depth of a beam is less than 5 [351] and neglecting curvature effect can cause a meaningful underestimation of resulting stresses [350]. In the intertrochanteric region of the human femur has the highest curvature, this ratio is around 1 [352]. Therefore, in this scenario, it is essential to account for bone curvature in

CTRA analyses. Consider the sample human femur bone cross-section shown in figure 6.1.

Circumferential stress 𝜎휃휃 on the ABCD section can be found by balancing the resultant forces and moments (푁, 푀푥 and 푀푦) acting on the cross-section with circumferential stresses. Shear stress

𝜎푟휃 can be neglected compared to 𝜎휃휃 for thick cross sections [350]:

(6.9) 푁 = ∫ 𝜎휃휃 푑퐴

(6.10) 푀푥 = ∫ 𝜎휃휃 (푅 − 푟)푑퐴

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(6.11) 푀푦 = ∫ 𝜎휃휃 푥푑퐴

푟 is the distance of the center of curvature from an infinitesimal area (푑퐴), and 푅 is the distance of the center of curvature to the centroid of the whole cross-section. In figure 6.1, A’B’C’D’ is a deformed shape of a curved element ABCD. The relative movement of typical point in yz plane due to deformation (푑푒푦) is equal to (푅푛 − 푟)Δ(푑휃)푦, where 푅푛 neutral axis radius and Δ(푑휃)푦 is the angle between the deformed surface and the original surface in yz plane. The relative movement of the point in xz plane due to deformation (푑푒푥) is equal to 푥Δ(푑휃)푥, where Δ(푑휃)푥 is the angle between the deformed surface and the original surface in xz plane. Therefore, the total normal strain can be evaluated as:

푑푒푦 푑푒 (푅푛 − 푟)Δ(푑휃)푦 푥Δ(푑휃) (푅 − 푟) 푥 휖 = + 푥 = + 푥 = 푛 훼 + 훽 (6.12) 휃휃 푟푑휃 푟푑휃 푟푑휃 푟푑휃 푟 푟

The resultant force and moments can be written as:

푅 − 푟 푥 푁 = 퐹 cos 휓 = ∫ 퐸(푥, 푟) [ 푛 훼 + 훽] 푑푥푑푟 (6.13) 푟 푟 퐴 (6.14) 푅 − 푟 푥 푀 = 퐹퐷 = ∫ 퐸(푥, 푟) [ 푛 훼 + 훽] 푥 푑푥푑푟 푦 푥 푟 푟 퐴 (6.15) 푅푛 − 푟 푥 푀 = 퐹퐷 = ∫ 퐸(푥, 푟) [ 훼 + 훽] (푅 − 푟) 푑푥푑푟 푥 푦 푟 푟 퐴 where 퐹 is the external force applied to the bone, 퐷푥 and 퐷푦 are the distances of applied load to the centroid axes of the weakest cross-section, and 휓 is the angle between the external force and resultant normal force on the cross-section. Equations 6.13 to 6.15 should be solved

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simultaneously to find 훼, 훽 and 푅푛, which 푅푛 is the neutral axis radius. In the most general case

훼, 훽 and 푅푛 can be found as:

푅푛 = 푓(휓, 퐷푦, 퐷푥) 2 ((퐴퐴푝 − 퐴푛퐴푞)푅 + 퐴푞 − 퐴푝퐴푟) cos 휓 + (퐴푛퐴푞 − 퐴푝퐴)퐷푦 + (퐴푛퐴푟 − 퐴푞퐴)퐷푥 (6.16) = 2 2 (퐴푚퐴푝푅 − 퐴푛푅 − 퐴퐴푝 + 퐴푛퐴푞) cos 휓 + (퐴푛 − 퐴푚퐴푝)퐷푦 + (퐴퐴푛 − 퐴푚퐴푞)퐷푥

훽 = 푔1(훾, 퐷푦, 퐷푥)퐹 (6.17) 2 ((퐴푚퐴푞 − 퐴퐴푛)푅 + 퐴푛퐴푟 − 퐴퐴푞) cos 휓 + (퐴퐴푛 − 퐴푚퐴푞)퐷푦 + (퐴 − 퐴푚퐴푟) 퐷푥 = 2 2 2 퐹 퐴 퐴푝 − 2퐴퐴푛퐴푞 + 퐴푟퐴푛 + 퐴푚퐴푞 − 퐴푚퐴푝퐴푟

훼 = 푔2(훾, 퐷푦, 퐷푥)퐹 (6.18) 2 2 ((퐴푚퐴푝 − 퐴푛)푅 + 퐴푛퐴푞 − 퐴퐴푝) cos 휓 + (퐴푛 − 퐴푚퐴푝) 퐷푦 + (퐴퐴푛 − 퐴푚퐴푞)퐷푥 = 2 2 2 퐹 퐴 퐴푝 − 2퐴퐴푛퐴푞 + 퐴푟퐴푛 + 퐴푚퐴푞 − 퐴푚퐴푝퐴푟

1 푥푖 where 푅 is the radius of curvature, 퐴 = ∑ 퐸푖퐴푖, 퐴푚 = ∑ 퐸푖 퐴푖, 퐴푛 = ∑ 퐸푖 퐴푖, 퐴푞 = ∑ 퐸푖푥푖퐴푖, 푟푖 푟푖

2 푥푖 퐴푝 = ∑ 퐸푖 퐴푖 and 퐴푟 = ∑ 퐸푖푟푖퐴푖. If 푟 and 푥 are calculated from the modulus weighted surface 푟푖

centroid (푥̅푖 = ∑ 퐸푖푥푖퐴푖 / ∑ 퐸푖퐴푖 , 푦̅푖 = ∑ 퐸푖푦푖퐴푖 / ∑ 퐸푖퐴푖), 퐴푟 = ∑ 퐸푖푟퐴푖 = ∑ 퐸푖(푅 + 푦푖)퐴푖 =

푅퐴 and 퐴푞 = 0, then equations 6.16 to 6.18 are reduced to:

퐴(퐴푛푅퐷푥 − 퐴푝퐷푦) 푅 = 푓(휓, 퐷 , 퐷 ) = (6.19) 푛 푦 푥 2 2 (퐴푚퐴푝푅 − 퐴푛푅 − 퐴퐴푝) cos 휓 + (퐴푛 − 퐴푚퐴푝)퐷푦 + 퐴퐴푛퐷푥

퐴푛퐷푦 + (퐴 − 퐴푚푅)퐷푥 (6.20) 훽 = 푔1(휓, 퐷푦, 퐷푥)퐹 = 2 퐹 퐴퐴푝 + 퐴푛푅 − 퐴푚퐴푝푅

2 2 (6.21) ((퐴푚퐴푝 − 퐴푛)푅 − 퐴퐴푝) cos 휓 + (퐴푛 − 퐴푚퐴푝) 퐷푦 + 퐴퐴푛퐷푥 훼 = 푔 (휓, 퐷 , 퐷 )퐹 = 퐹 2 푦 푥 2 퐴(퐴퐴푝 + 퐴푛푅 − 퐴푚퐴푝푅)

Therefore, critical force can be found as:

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휀푐 퐹푐 = 푅푛푐 − 푟푐 푥푐 (6.22) ( 푔1(휓푐, 퐷푦푐, 퐷푥푐) + 푔2(휓푐, 퐷푦푐, 퐷푥푐)) 푟푐 푟푐 where subscript 푐 indicates the parameters values at the weakest location on the bone. The critical strain, where fracture is eminent, is set to 1.2% strain in compression, and 1% strain in tension

[203, 339, 353, 354].

6.3. An ideal case

The capability of the proposed model was examined by considering an ideal case of a human femur. The results of straight beam and curved beam models were compared with results from finite element analysis. To simulate the bone femur shape, a hollow shaft with inside diameter

(퐷1) of 16 mm and outside diameter (퐷2) of 32 mm was constructed. The finite element model was also constructed using ANSYS (Academic Research, Release 14.0, Cecil, PA, USA). The ratio of the radius of curvature to outside diameter (푅/퐷2) were changed from 3 to 1000 in the three models to determine the effect of curvature on the maximum principal strain at the critical location. A nominal pressure of 1 Pa was applied at the top surface, while the bottom surface was fully restrained (figure 6.2A). The results are shown in figure 6.2B, based on the difference of each model from FE analysis for maximum vertical strain. For low 푅/퐷2 ratios, the straight beam model is significantly biased, while the curved beam model exhibits a better correlation. As

푅/퐷2 increases, the two models converge. The strain contour plots for 푅/퐷2 = 3 are shown in figure 6.2C to 6.2E. Again, the curved beam model demonstrates a better correlation to FEA than the straight beam model.

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6.4. Results

The FE and curved beam CTRA models predicted the failure loads for intact femurs and femurs with lytic defects and exhibited a strong level of correlation with mechanical testing results (푅2 = 0.91 and 0.87 against the line of equality, respectively; figure 6.3). On the other hand, the straight beam CTRA model overestimated the failure load in almost all cases. The regression analysis gives the negative coefficient of determination of -0.9 against

138 the line of equality which shows the line of equality does not follow the trend between the failure load from mechanical testing (퐹푚푒푐ℎ) and straight CTRA (퐹푆푡푟푎푖𝑔ℎ푡).

Figure 6-3. Linear regression between failure loads predicted by A) straight beam model B) curved beam model and C) Finite element analysis versus failure load from mechanical testing.

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When using the best-fit linear regression, rigidity analysis obtained through curved beam CTRA demonstrated a strong correlation with the failure load obtained through mechanical testing (figure 6.4B, 6.4D and 6.4F). The coefficients of determination for the femurs were 0.89 for EA and 0.89 for EI and 0.73 for 퐺퐽, the torsional rigidity which is defined as the sum of 퐺퐼푥 and 퐺퐼푦 [340]. For straight beam CTRA, the rigidities are not well correlated with experimental failure load, even when using the best-fit regression, as the coefficients of determination were 0.51 for EA, 0.63 for EI and 0.59 for GJ (figure 6.4A, 6.4C and 6.4E). Note that figure 6.4C and 6.4D are plotted based on minimum bending rigidity

(퐸퐼푚푖푛) at each cross-section.

When considering all specimens, paired t-tests did not indicate differences between curved beam CTRA and mechanical testing with an average overestimation of 385 N for failure load (table 6.2, P = 0.067). FE analysis demonstrated a mean difference of −197 N compared to mechanical testing, which was not significant (P=0.24). Straight beam CTRA showed a mean difference of -3315 N when compared to mechanical testing (P<0.001).

The Bland–Altman technique was applied to assess the agreement of straight beam

CTRA, curved beam CTRA or FE-based failure load with the gold standard mechanical testing with limits of agreement determined as mean difference ± 1.96 standard deviations

(i.e., 95% confidence interval of the difference) [355]. Bland–Altman analysis revealed that the limits of agreement defined as 95% confidence intervals were reasonable for FE and curved beam CTRA models (figure 6.5A).

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Figure 6-4. Linear regression between failure load from mechanical testing versus straight beam model A) axial rigidity C) bending rigidity E) torsional rigidity and curved beam B) axial rigidity

D) bending rigidity F) torsional rigidity

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Table 6.2 Comparison of mechanical testing failure load vs. failure load found from FEA, Curve beam and straight beam model Intact Defect Sample No. Mech FEA Curve Straight Mech FEA Curve Straight 1 7852 7416 8533 9318 3002 4149 2856 6163 2 5007 4501 6474 6626 1853 3367 1619 2077 3 5031 5173 5046 10785 2181 2034 2465 7227 4 4728 5197 4192 5587 2806 3541 2201 6462 5 4141 3281 6347 9202 1237 1421 1898 5117 6 4660 4886 4765 8587 3960 4981 5332 10740 7 11034 11477 11047 13861 3980 4976 5419 7134 8 7970 7372 8462 9669 5985 6055 6988 11992 9 6821 7132 7974 13491 6547 7254 6733 11611 10 10470 9029 8852 9827 8815 8787 8594 8899 Failure Load 6771±2498 6547±2440 7169±2170 8719±2965 4037±2394 4656±2273 4410±2508 9695±2601 Average (N) Bland-Altman -1032 to -7541 to -2503 to 1707 -1732 to 493 -1776 to 1028 -8052 to 641 method, 95%Cl 1482 1693 Mean ± SD of difference vs. 225±641 -398±1074 -2924±2356 -619±568 -374±715 -3705±2218 mechanical testing P-value (paired t- 0.2965 0.2714 0.0035 0.0072 0.1327 0.0005 test) Kendall Tau ranking coefficients 0.82 0.73 0.51 0.91 0.87 0.56 (mechanical testing vs other methods) Total Failure Load 5404±2764 5601±2491 5790±2685 7742±3111 Average (N) Bland-Altman -1631 to -7749 to method (N), -2127 to 1355 1236 1120 95%Cl Mean ± SD of difference vs. -197±731 -385±888 -3315±2262 mechanical testing P-value (paired t- 0.2419 0.067 2.8e-6 test) Kendall Tau ranking coefficients 0.83 0.84 0.53 (mechanical testing vs other methods)

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For example, the mean difference of -385N for curved beam CTRA model failure load was associated with a precision between −2127 and 1355 N, implying that 95% of the time, curve beam model would provide an estimate of failure in this range compared to the gold standard. For failure load predicted by curved CTRA (퐹퐶푢푟푣푒), the bias was constant across the magnitude of failure load as judged by non-significant correlation between the average versus the difference (r=0.09, P=0.71). FE analysis showed more accurate estimates of failure load than each of the two CTRA models (all 푃 < 0.001, paired t-tests on the deltas versus mechanical testing). The limits of agreement in the Bland–Altman plot indicate that the FE estimated failure load on average is nearly the same as mechanical testing (mean difference of −197 N) and provides estimates that are within the range of −1631 to 1236 N (figure 6.5C, table 6.2). Moreover, the bias throughout the magnitude of possible failure loads is constant as indicated by a non-significant correlation between the average versus the difference

(r=0.38, P=0.10).

To further study differences in prediction of accuracy between the methods, the paired t-tests and Bland–Altman analysis were repeated for the intact and defect specimens separately (table 6.2). Paired t-tests then showed a significant difference between mechanical testing and straight beam CTRA for both intact femurs and femurs with lytic defects. The difference between mechanical testing and FE model for the femurs with lytic defects was also significant. In addition, for the curved beam CTRA the limits of agreement varied over the different analyses (total group and both subgroups), but the bias was constant. On the other hand, for FE model the limits of agreement were constant for all groups, while the biases were different.

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Figure 6-5. Bland–Altman plots for A) straight beam model (B) curved beam model and C) Finite element failure load versus mechanical testing failure load.

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For the defect femurs, straight beam CTRA showed the smallest bias (-374 N vs. −619

N for FE), whereas FE showed a higher agreement among predictions (SD 568 N vs. 715 N).

In addition, High Kendall rank correlations between the experiments and the predictions by either FE or curved beam CTRA (all significant at the P=0.05 level) were found (table 6.2).

This correlation was low for straight beam CTRA for all groups analyzed.

The fracture locations in the experiments were qualitatively compared to the fracture lines predicted by the FE model and failure load predicated by straight and curved beam

CTRA models (figure 6.6 provides a graphic presentation of representative specimen). The results indicated that the fracture locations were always directed through the lesion in the defect specimens. Overall, the fracture locations were reasonably well predicted by both FE and curve beam model as highlighted in figure 6.7. However, straight beam CTRA was inaccurate in four specimens with lytic defects and in all of the intact specimens.

6.5. Discussion

In recent years, different diagnostic tools have been developed to address the difficulties to predict fracture risk in patients with metastatic bone lesions. The ideal screening test should consider bone as a structure whose mechanical behavior depends on both material and geometric properties. This study evaluated the accuracy of straight and curved beam CTRA models and FE analysis to predict failure load, which was determined through mechanical testing in paired femurs with and without simulated lytic lesions. We were able to demonstrate that predicted failure loads from curved beam CTRA and FE analysis were highly correlated with the actual failure load obtained through mechanical testing. There were no significant differences in prediction accuracy between the two modeling techniques.

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A B C D

Figure 6-6. Fracture location as demonstrated by A) mechanical testing, B) Finite element analysis

C) Curved beam model and D) straight beam model for a representative sample with defect. For

straight and curved beam model the red areas shows the most critical locations which the fracture

is imminent.

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Figure 6-7. Fracture locations for all specimens as predicted by FEA, curve beam and straight beam models and failure load from mechanical testing.

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The correlation coefficients between the FE analysis predicted and the experimental failure loads (푅2 = 0.91) were similar to those obtained in other FE studies [343, 348, 356,

357]. Similarly, relatively high correlation coefficients between curved beam CTRA and mechanical testing data were evidenced (푅2 = 0.87). However, the correlation between straight beam CTRA and experimental failure load was very poor (푅2 = −0.9) and highly overestimated since the model was unable to capture the influence of bone curvature in critical cross-sections.

In addition, relatively high correlation coefficients between CTRA rigidities and mechanical testing data were evidenced (R2 = 0.82 and 0.86 for EA and EI respectively).

These results are comparable to those obtained by Hong et al. [339], who showed high coefficients of determination when comparing reductions in failure loads versus reductions in axial, bending and torsional rigidity (R2=0.84, 0.80 and 0.71, respectively) in samples from whale trabecular bone. Similarly, Whealan et al. [336] demonstrated the effectiveness of QCT derived measurements of rigidity for the prospective prediction of yield loads of vertebrae with simulated lytic lesions (푟푐=0.74). Finally, by assessing fracture prediction through benign skeletal lesions in children and young adults, Snyder et al. [337] indicated that bending and torsional rigidities were each highly significant predictors of fracture occurrence and combined, these measures could predict femoral fractures with 97% accuracy. As can be seen from the rigidity results (figure 6.4), the coefficient of determination for GJ is lower than those of EA and EI. The reason for this lower 푅2can be interpreted as bone structure tends bend around the axis with lower 퐸퐼 in unsymmetrical loading. Therefore, Since the GJ is in linear relationship with minimum and maximum bending rigidity (퐺퐽 = (퐸퐼푚푖푛 + 퐸퐼푚푎푥)/(2 + 2휈))

[340], the presence of 퐺퐼푚푎푥 in the summation reduces the correlation accuracy for GJ.

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In the specimens with a simulated defect, curved beam CTRA seemed to have a higher accuracy (as the bias was lowest), whereas FE analysis showed a higher precision (due to smaller limits of agreement). This could indicate that FE calculations need a correction for bias. In contrast, curved CTRA will provide more accurate estimates of failure load on the group level. However, further studies using larger numbers of specimens are warranted to confirm our findings.

Unlike previously proposed radiographic guidelines, both curved beam CTRA and FE models offer objective assessments of fracture risk by considering the material and geometric properties of bone. Curved beam CTRA is more accurate than straight beam CTRA when predicting both the magnitude of failure loads and the location of the failure. Both techniques are based on QCT imaging, but computational times differ considerably between the two methods. The estimated time for generating and running FE simulations in this study is of 8 hours per sample, and a sophisticated and relatively complex FE software is required to estimate fracture risk. In contrast, curved beam CTRA takes only approximately 10 minutes, to estimate fracture risk.

FE simulations are more appropriate for the implementation of complex loading conditions. The decrease in bone strength resulting from metastatic lesions is focal and displays a large degree of variability between patients. As a result, forces that insert on the femur close to the lesion site can be more dangerous than larger forces such as for example the hip contact force. The modelling of such potentially important anatomical characteristics might be more straightforward using FE analysis.

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Limitations of our study are shared with many previous works done in the field using ex vivo models for the assessment of failure load prediction using non-invasive imaging methods. Evident differences exist between the metastatic lytic lesions that were artificially simulated in this study and those seen in patients in the clinical practice. In our case, regularly shaped defects were limited to cortical lesions, while metastatic bone lesions generally show an irregular pattern and additionally involve trabecular tissue. However, QCT would be readily able to detect these irregularities and incorporate them into both algorithmic analytical processes, although accurately modeling the material properties of blastic metastatic tissue might be challenging. On a group level, both methods accurately predict the femoral load capacity, but on the individual level there can be rather large over- and under-estimations of the femoral strength. These subject-specific over- and underestimations should be improved before either of the methods can be implemented in clinical practice. Furthermore, prospective patient studies should resolve whether the two modeling techniques have equal prediction accuracy in clinical practice. That is, clinical experts have difficulties translating predicted biomechanical parameters, such as global strength or rigidity, to the clinical fracture risk of a specific patient.

In summary, the results of our study showed that non-invasive subject-specific fracture risk assessment techniques correlate well with actual failure loads measured in mechanical testing experiments. This suggests that curved beam CTRA could be further developed into a tool that can be used in clinical practice. When analyzing the defect femurs only, the results suggested that predictions by FEA are slightly more accurate on a subject-specific level, yet

CTRA analysis can be conducted expediently by non-expert operators. Validation in prospective clinical studies are needed to confirm these preliminary findings.

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Chapter 7 Concluding remarks

151

The effective elastic modulus, Poisson’s ratio and plastic collapse strength of anisotropic hierarchical honeycombs were obtained as a function of dimension ratios and wall angles of the structure. The analytical results for the elastic part were based on Castigliano’s second theorem, and numerical frame analysis was used to validate the analytical approach, and explore higher orders of hierarchy. The results show that a wide range of elastic modulus and strength ratio can be obtained for anisotropic hierarchical honeycombs by varying geometrical parameters. Table 1 summarizes the elastic modulus/strength values of anisotropic hierarchical honeycombs at different hierarchical orders. Increasing the hierarchy order improves the structural performance in terms of its elastic modulus and plastic collapse strength.

Also, a new class of fractal-appearing cellular meta-materials is introduced. The results show that the effective elastic modulus of the developed cellular material can be increased significantly by increasing the hierarchical order while preserving the structural density. The optimal hierarchical level has also shown to be increased by reducing the structural density. This particular case of hierarchical refinement can be served as a promising realization of performance enhancement due to hierarchy. Moreover, the current work provides insight into how incorporating hierarchy into the structural organization can play a substantial role in improving the properties and performance of the materials and structural systems and introduces new avenues for development of novel metamaterials with tailorable properties.

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Similar to hierarchical honeycombs, the hierarchical arrangement of structural properties in cortical and trabecular bone was explored and a mathematical model that accurately predicts the tissue’s mechanical properties as a function of these indices is determined. By using a variety of analytical techniques, we were able to characterize the structural and compositional properties of cortical and trabecular bones, as well as to determine the suitable mathematical model to predict the tissue’s mechanical properties using a continuum micromechanics approach. Our hierarchical analysis demonstrated that the differences between cortical and trabecular bone reside mainly at the micro- and ultrastructural levels. By gaining a better appreciation of the similarities and differences between the two bone types, we would be able to provide a better assessment and understanding of their individual roles, as well as their contribution to bone health overall.

In chapter 7, the results of our study showed that non-invasive subject-specific fracture risk assessment techniques correlate well with actual failure loads measured in mechanical testing experiments. This suggests that curved beam CTRA could be further developed into a tool that can be used in clinical practice. When analyzing the defect femurs only, the results suggested that predictions by FEA are slightly more accurate on a subject-specific level, yet CTRA analysis can be conducted expediently by non-expert operators. Validation in prospective clinical studies are needed to confirm these preliminary findings.

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Chapter 8 Appendix

A.1. P-tensor in a transversely isotropic matrix

For a detailed derivation of the P-tensor for anisotropic matrices, please refer to [70,98]. For a transversely isotropic material, the non-zero terms of the stiffness matrix C are C11 =

C22, C33, C12, C13 = C23, C44 = C55 and C66 = 1/2 (C11 − C22). For spherical inclusions, the non-zero terms of the P-tensor can be found as

1 1 푃 = ∫ ((2C2 − 6C2 − 5C 5C + 5C 5C + 5C C − 3C C + 8C C + 11 16 −1 13 44 11 33 12 33 11 44 12 44 33 44 6 2 2 8C33C44)푥 + (6C44 − 4C13 + 6C44 + 5C11C33 − 6C12C33 − 15C11C44 + 9C12C44 − 4 2 3 2 2 8C33C44)푥 + 5C11C33푥 + (2C13 − 6C44 + 3C12C33 + 15C11C44 − 9C12C44)푥 −

2 2 5C11C33푥 + 3C12C44 − 5C11C44) (푥 − 1)/퐷1푑푥 (A1)

1 1 푃 = ∫ ((2C − 2C2 − C 퐶 − 퐶 퐶 + 퐶 퐶 + 퐶 퐶 + 4C C )푥6 + (2C C − 12 16 −1 13 44 11 33 12 33 11 44 12 44 13 44 11 33 2 2 4 2 2 2C13 − 4C44 − 2C13 + 2C12C33 − 3C11C44 − 3C12C44 − 8C13C44)푥 + (2C13 + 2C44 − 2 C11C33 − C12C33 + 3 C11C44 + 3C12C44 + 4C13C44)푥 − C11C44 − C12C44)/퐷1푑푥 (A2)

1 1 푃 = ∫ ((C + C )푥4 + (−C − C )푥2)/퐷2푑푥 (A3) 13 4 −1 13 44 13 44

1 1 푃 = ∫ ((2C2 − C2 + C C + 2C C − 3C C + C C + 4C C + 8C C + 44 4 −1 13 11 11 12 11 13 11 33 12 33 11 44 13 44 6 2 2 4C33C44)푥 + (3C11 − 2C13 − 3C11C12 + 4C11C13 − 4C12C13 − C12C33 − 5C11C44 − 4 2 2 2 8C13C44)푥 + (3C11C12 − 3C11 − 2C11C13 + 2C12C13 + 4C11C44)푥 + C11 − C12C11)/ 퐷1푑푥 (A4) and

1 1 푃 = ∫ ((C − C )푥4 + C 푥2)/퐷2푑푥 (A5) 33 2 −1 44 11 11 where

154

2 2 2 2 2 2 2 퐷1 = (퐶12퐶13 + 퐶11퐶33 + 2C11C44 − C11C44 + 4C13C44 + 2C33C44 − C11C44 − C11C13 +

2C13C44 − C11C12C33 + C11C12C44 − 2C11C13C44 + 2C12C13C44 − 3C11C33C44 + 6 2 2 2 2 2 C12C33C44)푥 + (2C11C33 − 2C11C33 − 4C11C44 − 4C13C44 − 2C13C44 − 2C12C13 + 4 3C11C44 + 2C11C12C33 + 4C11C13C44 − 4C12C13C44 + 3C11C33C44 − C12C33C44)푥 + 2 2 2 2 2 (C12C13 − C11C13 + C11C33 + 2C11C44 − 3C11C44 − 2C11C44 − C11C12C33 + 3C11C12C44 + 2 2 2C12C13C44)푥 + C44C11 − 4C12C44C11 (A6) and

2 4 2 2 퐷2 = (C13 + 2C44C11 + C11C44 + C33C44)푥 + (−C13 − 2C44C13 + C11C33 − 2C11C44)푥 +

C11C44 − C11C33 (A7)

For cylindrical inclusions, the non-zero terms of the P-tensor are written as

1/8(5C11−3C12) 푃11 = 푃22 = 푃66 = (A8) C11(C11−C12)

−1/8(C11+C12) 푃12 = (A9) C11(C11−C12)

1 푃44 = 푃55 = (A10) 8C44

For platelet-shape-like inclusions the P-tensor becomes

1 푃33 = (A11) C33

A.2. P-tensor in an isotropic matrix

For isotropic material, the stiffness matrix can be written as 퐂 = 3푘퐈푣표푙 + 2휇퐈푑푒푣, where 푘 and

휇 are the bulk and shear modulus, respectively, and 퐈푣표푙and 퐈푑푒푣 are the volumetric and deviatoric

푣표푙 푑푒푣 푣표푙 part of the fourth-order unity tensor (퐈 = 1/3훿푖푗훿푘푙 and 퐈 = 퐈 − 퐈 . The 퐏-tensor in an isotropic matrix for cylindrical inclusions can be written as [277, 358]

155

퐸푠ℎ −1 푷푐푦푙 = 퐶푐푦푙 : 푪 (A12) where the Eshelby tensor has the following non-zero terms:

9/4(푘+ 휈) 푆퐸푠ℎ = 푆퐸푠ℎ = (A13) 11 22 (3푘+4휈)

1/4(3푘−5휈) 푆퐸푠ℎ = (A14) 12 (3푘+4휈)

1/2(3푘−2휈) 푆퐸푠ℎ = 푆퐸푠ℎ = (A15) 13 23 (3푘+4휈)

1 푆퐸푠ℎ = 푆퐸푠ℎ = (A16) 44 55 4

1/4(3푘+7 휈) 푆퐸푠ℎ = (A17) 66 (3푘+7 휈)

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