Mechanical Behavior of Cortical Bone

2 MECHANICAL BEHAVIOR OF CORTICAL BONE Theng P. Ng and Seyed S.R. Koloor CHAPTER OUTLINE 2.1 Introduction 19 2.2 Cortical Bone Structure and Composition 20 2.3 Linear Elasticity Behavior 21 2.3.1 Isotropic Elasticity 22 2.3.2 Orthotropic Elasticity 22 2.3.3 Transverse Elasticity 22 2.4 Mechanical Behavior of Bone 22 2.5 Anisotropic Behavior of the Cortical Bone 24 2.6 Compressive and Tensile Strength of Cortical Bone 24 2.7 Compressive and Tensile Strength in Longitudinal and Transverse Directions 25 2.8 Brittle Damaged Plasticity Model of Cortical Bone 25 2.9 Fractographic Analysis of Cortical Bone 27 2.10 Summary 28 2.11 Remind and Learn 28 References 29

2.1

Introduction

Studies on the mechanical behavior of cortical bone include investigations of elastic-plastic behavior to damage-evolution phenomena that leads to the interpretation of the failure process of bone structures. Such knowledge is beneficial for biomechanical evaluation of trauma plating fixation [1
4]. Optimal design of an implant is normally performed by studying on the behavior of plating fixation that includes both bone and implant. The complex geometry of the plating system normally causes stress concentration zones, which induce material nonlinear deformation; therefore, developing a constitutive damage model to predict the bone nonlinear behavior should be considered for better understanding of mechanical behavior in bone-implant fixation [2,5
7]. In previous researches, many Trauma Plating Systems. DOI: http://dx.doi.org/10.1016/B978-0-12-804634-0.00002-1 Copyright © 2017 Elsevier Inc. All rights reserved.

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studies have been focused on fracture mechanics of cortical bone [8
13], using linear elastic fracture mechanics (LEFM) to investigate on the characteristics of fracture phenomenon such as applied stress, crack growth rate, and fracture toughness with negligible plastic deformation. Other researchers have indicated that bone tissue behaves as anisotropic materials with a moderate amount of strain hardening prior to ultimate strength [8,10,14,15]. As a result of those investigations, the low accuracy of the LEFM model in failure analysis of cortical bone has become apparent; therefore, elastic-plastic fracture mechanics (EPFM) is recommended for damage analysis of cortical bone [8,12,13,16].

2.2

Cortical Bone Structure and Composition

The skeletal system is the assembly of numerous individual bones, joints and other connective tissues such as muscles, tendons, ligaments and cartilage that construct the framework of human body. The main functions of the skeletal system consist of (1) structural support, (2) mobility, (3) organs protection, and (4) mineral production and storage [17,18]. Dynamic and static overload conditions could gradually or rapidly damage the bone which leads to structural fracture and failure events. Such overload conditions are normally caused and amplified by excessive functioning, aging, bone deterioration, and abnormal loading such as road and sport accidents. Therefore, study on bone characteristics and behavior is one of the key points for preservation of bone quality and optimum design of implants [1,19
23]. The hierarchical organization of bone construction from nano- to macroscales is demonstrated in Fig. 2.1. The main constituents of bone at the nanoscale consist of collagen molecules and bone crystal. The assembly of the collagen molecules and crystal constituents produce collagen fibril and collagen fiber as shown in the nanostructure view (Fig. 2.1). In the microscopic point of view, collagen fibers are arranged in the concentric layers with a preferred orientation called lamellae. Each osteon consists of lamellae of bone tissues that are surrounded by haversian system including haversian canal, canaliculi, lacuna and osteocyte, which allows blood supply. In the macrostructural view, two main types of bone tissues are classified as trabecular (spongy form) and cortical (compact form) bones, which are different in volume fractions (porosity). Normally, the body skeleton is composed of 80% cortical and

Chapter 2 MECHANICAL BEHAVIOR OF CORTICAL BONE

Figure 2.1 Hierarchical organization of bone structure from macro-, micro- and nanoscales points of view [24].

20% trabecular bones. The high bone-volume fraction in the form of low porosity in cortical bone varies from 5% to 30%, while this factor ranges from 60% to 95% in trabecular bone. The cortical bone can generally be found on hard outer surfaces of bone structures such as femur and tibia bones that cover soft trabecular bone. Bone is composed of a connective tissue that is mainly constructed of three phases of materials: (1) ceramic-type materials as calcium phosphate or hydroxyapatite (Ca10(PO4)6(OH)2); (2) polymer-type composite (collagen); and (3) water. The hydroxyapatite is a brittle-like bone mineral or stiffening filler that contributes strength to the bone tissue, while collagen as a ductile or soft organic matrix that provides bone flexibility. The combination of brittle and ductile materials gives adequate mechanical characteristics to the bone for daily physiological activities.

2.3

Linear Elasticity Behavior

Linear deformation of cortical bone depends on defined material behavior. Complex geometry of bone structure and its hierarchical construction requires a constitutive model to represent the inherent behavior. In the following subsections, the outline of different linear elasticity behavior modes used to predict the linear deformation of bone structures are explained.

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2.3.1

Isotropic Elasticity

In the continuum mechanics assumption, a isotropic elasticity model describes the linear deformation of cortical bone using the simplified Hooke’s law. In this model, there are two constants as elastic modulus (EÞ and Poison’s ratio (vÞ that are used to predict the material deformation and internal stresses. The model is valid to compute stress within the yield limit. These assumption have been used for many materials including bone; however, recent investigations show a low accuracy of such a model for analysis of materials with specific microstructure composition and that act as inhomogeneous materials [21,25
27].

2.3.2

Orthotropic Elasticity

In the orthotropic elasticity definition, cortical bone behavior is defined based on three perpendicular planes symmetrical with the number of independent elastic constants. In this respect, the elastic modulus and Poison’s ratio parameters are characterized with different values in different principal directions. Composite laminate is an example of a material that exhibits orthotropic behavior [28]. Investigations have shown that cortical bone also behaves as orthotropic material and could be modeled using such a definition [25
27,29].

2.3.3

Transverse Elasticity

The transverse elasticity definition is used for the materials that behave with a form of subclass orthotropic behavior in which the material properties are the same in one plane. For instance, the elastic modulus parameter could be similar in axes 1 and 2 (E1 5 E2 ) as the plane of isotropic, and the other plane would be the transverse plane with a different property. In this aspect, the elastic constant parameters can be eliminated in comparison with the orthotropic definition, which could ease the analysis process and reduce the computational time for complex large geometries. Such behavior is also considered in many researches on stress analysis of bone structures [5,30].

2.4

Mechanical Behavior of Bone

The mechanical behavior of bone is classified as a quasibrittle material which is reflected by the properties of different type of constituents including (1) 69% of brittle-like bone

Chapter 2 MECHANICAL BEHAVIOR OF CORTICAL BONE

mineral crystal (calcium phosphate or hydroxyapatite, (Ca10(PO4)6(OH)2)); (2) 20% of ductile polymer-type (collagen) composite; and (3) 9% of water and other substances. Besides these, there are small amounts of impurities in the bone mineral crystal such as hydrogen phosphate, sodium, magnesium, citrate, carbonate, and other ions. The superb mechanical properties of bone are attributed to a combination of a brittle behavior of bone mineral crystal with high strength and a ductile behavior collagen having with high flexibility [31,32]. The good bio-functionality response of hydroxyapatite in bone is one of its most attractive feature for researcher and clinician when considering coating materials for metallic implants to improve the bioactivity and corrosion resistance. The mechanical behavior of cortical bone could be expressed using a stress-strain curve for different loading conditions, as illustrated in Fig. 2.2. Orthotropic behavior of bone is reflected by natural bone properties as composite materials with complex arrangement and orientation. Considering osteon to indicate the material orthotropic directions (Fig. 2.2A), the typical stress-strain curves for a human cortical bone under tension and compression loads are shown in Fig. 2.2B. The elastic modulus and strength of cortical bone along the longitudinal or osteon directions are higher in both compression and tension loading conditions compared to transverse direction [33
38]. It has shown that bone has higher elastic modulus and strength with lower strain in osteon direction in comparison to the radial or transverse direction. In addition, it was shown that the compressive strength of bone tissue is higher than tensile strength (Fig. 2.2B) [30,39].

Figure 2.2 Schematic view of (A) cortical bone structure with respect to osteon direction (longitudinal, L and transverse, T), (B) typical stress-strain curves of cortical bone [39].

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2.5

Anisotropic Behavior of the Cortical Bone

The complex hierarchical arrangement and orientation of the bone structure leads the bone to be heterogeneous and behave similar to an anisotropic material [14,24,34,35,40
44]. In mesoscale analysis, the functional unit of cortical bone material is the osteon constituent which is in the form of cylindrical structure. Anisotropic behavior of cortical bone is normally depend on the orientation of the osteons [37,41]. Numerous studies have been performed to investigate anisotropic behavior of bone [5,25
27,29,45,46]. Schneider et al. [29] has studied the mechanical behavior of cortical bone and with the assumption of orthotropic behavior as bone material property, achieved results from computational analysis that are close to experimental testing results compared to the isotropic material property. In a similar study by Baca et al. [45] it was shown that the material anisotropy description of cortical bone is substantially different from isotropic definition. In other work, the isotropic and orthotropic definitions of proximal femur were computationally analyzed and it was concluded that there are a significant differences in the prediction of von Mises stress distribution and nodal displacement at some part of proximal femur, but there were no notable changes in the mid-diaphysis of the femur bone [26]. Isaza et al. [25] proposed a finite element model to simulate real physiological loading conditions on proximal femur bone. They showed that isotropic material definition has significantly overestimated fracture load and implementation of isotropic definition in bone analysis is only suitable under specific loading and boundary conditions.

2.6

Compressive and Tensile Strength of Cortical Bone

In the continuum mechanics aspect, cortical bone behaves as elastic-plastic material under compressive loading. On the other hand, it displays behavior with negligible permanent or plastic deformation under tension loading. Such behavior has been considered in the quasi-brittle damage model that is used in many researches in the form of constitutive law to predict damage and failure phenomena of cortical bone [4,23,47
53]. Cortical bone normally shows higher elastic modulus and strength under compressive load than tensile load. The

Chapter 2 MECHANICAL BEHAVIOR OF CORTICAL BONE

difference of cortical bone strength under compression and tension is about 35% and 70% in the longitudinal and transverse directions of the osteon, respectively. This response is due to the porosity structure of the bone which creates stress raiser points in mesoscale. The existing voids in the porous bone get closed under compression, which leads the material to become more dense to sustain higher load. However, these voids provide stress concentration points under tension that result to the lower tensile strength.

2.7

Compressive and Tensile Strength in Longitudinal and Transverse Directions

Mechanical properties of cortical bone are not only dependent on the loading mode conditions as compression and tension, but also depend on orientation with respect to longitudinal and transverse osteon directions. Bird et al. [33] has shown that cortical bone has higher modulus and strength in longitudinal direction compared to the radial or transverse direction. This strength is achieved not only because of the porosity structure of the bone, but also due to the natural alignment of hierarchical cortical bone structure, different fracture toughening mechanisms, and energy consumed through the failure process. In the longitudinal direction of bone, more energy can be absorbed before fracture due to alignment of collagen fibrils. In addition, bone is stronger under compression load compared to tension load [30,34,39].

2.8

Brittle Damaged Plasticity Model of Cortical Bone

Brittle damage plasticity model is used to model behavior in many materials, including cortical bone [39,54,55]. The model for the cortical bone strain variation can be written using the total strain tensor εT, which consists of elastic and plastic strains and which is normally used to describe the variation of strain or deformation in the material. In the quasi-brittle material, the total strain tensor εT can be decomposed into elastic (εe ) and plastic parts (εpl ) as: εT 5 εe 1 εpl

ð2:1Þ

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The elastic stress-strain behavior of cortical bone can be written using Eq. (2.1) as: σ 5 Eεe 5 EðεT 2 εpl Þ

ð2:2Þ

where E is the elastic modulus, and σ is the stress tensor. The generation of voids in the cortical bone under deformation is considered as a microdamage event, which could be monitored using a scalar variable, d, as damage parameter corresponding to the effective stress, σ in the form of: σ 5 ð1 2 d Þσ;

0#d#1

ð2:3Þ

The damage variable d is defined between 0 (undamaged state) and 1 (full damage and failure). Then, the σ could be denoted using the initial elastic stiffness tensor, E0 as: σ 5 E0 ðεT 2 εpl Þ

ð2:4Þ

and the model dependent of damage parameter could be written by substituting Eq. (2.4) into Eq. (2.3) to create the following relationship: σ 5 ð1 2 dÞE0 ðεT 2 εpl Þ

ð2:5Þ

The typical stress-strain behavior of cortical bone under different loads based on the brittle damage plasticity model is displayed in Fig. 2.3. The parameters σc0 and σuc are initial yield and maximum/ultimate stresses under compressive load, and σut is the yield stress under tension load. The parameters dt and dc are the tensile and compressive damage variables, respectively. Once cortical bone experiences yielding in tension,

Figure 2.3 The stress-strain behavior of cortical bone: (left) in tension and (right) in compression loading conditions [39].

Chapter 2 MECHANICAL BEHAVIOR OF CORTICAL BONE

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or reaches to ultimate point in compression, the damage evolution and material softening process is initiated. The unloading behavior of cortical bone through the softening process is such that the elastic stiffness is reduced progressively as dictated by the damage variables (Eq. 2.5). Failure or crack initiation in cortical bone occur in the full damage state which is used in the form of the finite element code to assess the compressive behavior of cortical bone [39,56].

2.9

Fractographic Analysis of Cortical Bone

As can be seen in the microstructural point of view, osteon fibers strengthen cortical bone along the longitudinal direction (Fig. 2.2A). Fractographic images of failure in cortical bone have been captured in different directions to study crack deflection, osteon breakage, microcracking, and crack bridging [57,58]. Fig. 2.4 illustrates fracture of the cortical bovine bone along the transverse and longitudinal directions which are perpendicular and along the osteon direction, respectively. In the longitudinal fractured pattern (perpendicular to osteon direction), a crack is

Figure 2.4 Fractographic image of failure surface of bovine cortical bone in the surfaces perpendicular (A,B) and along (C,D) the osteon direction [56].

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propagated transverse to osteon’s direction which results cutting of the osteon fibers, which appears as fiber pull-out in Fig. 2.4A. On the other hand, in the transverse fractured pattern (surface along the osteon direction), the crack grows along osteon’s direction through the boundary of the fibers, as shown in Fig. 2.4D. The longitudinal fracture appears to be much rougher than the transverse fracture, which could indicate higher fracture energy is needed for cutting the osteon fibers in longitudinal direction compared to transverse direction [8,58,59].

2.10 Summary In this chapter, mechanical behavior of cortical bone was described with respect to its microscale construction. It was explained how the constructional features such as osteon constituent could affect the mechanical properties of cortical bone. Cortical bone with different constitutions in nano- micro- and macroscale bone hierarchical construction would provide different properties in each scale. Mechanical behavior of bone could be described in the form of a mathematical model for elastic, elastic-plastic, or elastic-to-damage behaviors, while experimental testing is crucial to match the mathematical model with actual results. Development of constitutive damage models could help for accurate prediction of bone mechanical behavior. Furthermore, a brittle damage model has been found to be beneficial in simulation of elastic-damage and failure behavior of cortical bone.

2.11 Remind and Learn P 2.1: Describe the composition of bone. P 2.2: Discuss the adequate linear elasticity model for simulation of cortical bone mechanical behavior in the elastic zone. P 2.3: Describe how cortical bone is quasi-brittle material. P 2.4: What is the behavior of cortical bone under compressive and tension loads? P 2.5: Express the mechanical behavior of cortical bone with respect to longitudinal and transverse direction of osteon. P 2.6: How could fractographic images express fracture of cortical bone with respect to longitudinal and transverse direction of osteon?

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