UNIVERSITY OF CALGARY Nonlinear Elasticity, Fluid Flow and Remodelling in Biological Tissues by Aleksandar Tomic A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING CALGARY, ALBERTA JUNE, 2012 Aleksandar Tomic 2012
Abstract Articular cartilage is a soft tissue with depth-dependent structure and composition that covers the ends of bones in diarthrodial joints. It transmits loads between bones and minimizes joint wear. The purpose of this work was to implement a large deformation non linear model for biological tissues with statistically oriented reinforcing fibres, using a biphasic formulation to model global articular cartilage behaviour. The implemented model takes into account the effect of the depth dependent variation in the collagen fibre orientation and the influence of the collagen fibres on the overall permeability of the tissue. This model was implemented using Finite Element software to determine the depth dependent deformation of articular cartilage. In addition, a separate investigation was conducted to determine the remodelling of the fibres in statistically oriented fibre reinforced materials as a response to an externally applied loading, using an arterial sample. ii
Acknowledgements First and foremost, I would like to thank my parents and my girlfriend for their everlasting support and care. I would not be where I am today without your sacrifice and your love. I would like to thank Dr. Salvatore Federico, my supervisor, for opening my eyes to the possibility and giving me an opportunity to pursue of graduate studies, and especially for his patience, guidance and friendship. I would like to thank Dr. Gabriel Wittum and Dr. Alfio Grillo for giving me with an opportunity to expand my horizons and knowledge by hosting me at Universität Frankfurt. I would especially like to thank Dr. Grillo for his supervision and care during my stay in Frankfurt. I would like to thank Jan Pajerski for sharing his knowledge on mathematical modelling of articular cartilage and programming, and for providing me help when needed. Your help was always appreciated. iii
iv Table of Contents Abstract....................................... ii Acknowledgements................................. iii Table of Contents.................................. iv List of Tables.................................... viii List of Figures.................................... ix 1 Introduction.................................. 1 1.1 Contribution.................................. 2 1.2 Outline of the Thesis............................. 3 2 Short Introduction to Continuum Mechanics................ 7 2.1 Bodies, Configurations, Fields, and Motion................. 8 2.1.1 Covariant and Contravariant Tensors................ 9 2.1.2 Covariant and Contravariant Transformations........... 11 2.1.3 Metric tensor............................. 12 2.1.4 Velocity and Acceleration...................... 13 2.2 Deformation Gradients and Strain Measures................ 14 2.2.1 Polar Decomposition......................... 16 2.2.2 Cauchy Green Deformation Tensors................. 17 2.2.3 Strain Measures............................ 19 2.2.4 Volumetric Distortional Decomposition............... 20 2.2.5 Velocity Gradient........................... 20 2.2.6 Rate of Deformation......................... 21 2.3 Stress and Conservation Laws........................ 22 2.3.1 Stress................................. 22 2.3.2 Conservation Laws.......................... 24 2.4 Constitutive Equations............................ 30 2.5 Hyperelasticity................................ 32 2.6 Growth and Remodeling........................... 33 2.6.1 Conservation Laws.......................... 34 2.6.2 Constitutive Equations........................ 36 2.7 Directional Derivative............................ 37 3 Introduction to Finite Element Analysis.................. 39 3.1 General Procedure for Solving a Problem by FEA............. 40 3.2 Weak Formulation for the Solid Materials.................. 43 3.2.1 Linearization............................. 43 3.2.2 Discretization............................. 45 3.3 Newton Raphson Method.......................... 47 3.3.1 Full Newton Method......................... 48 3.3.2 BFGS Method............................. 49 3.3.3 Line Search Method......................... 50 4 Composition and Structure of Articular Cartilage............. 52 4.1 Composition.................................. 54
4.1.1 Fluid.................................. 54 4.1.2 Collagen................................ 55 4.1.3 Proteoglycans............................. 56 4.1.4 Chondrocytes............................. 58 4.1.5 Other non collagenous proteins................... 59 4.2 Structure................................... 60 4.2.1 Superficial Zone............................ 62 4.2.2 Middle Zone.............................. 62 4.2.3 Deep Zone............................... 63 4.2.4 Calcified Zone............................. 64 5 Behaviour, Testing, and Material Models of Articular Cartilage...... 65 5.1 Mechanical Behaviour............................ 65 5.1.1 Viscoelasticity............................. 66 5.1.2 Compression.............................. 68 5.1.3 Tension................................ 70 5.1.4 Shear.................................. 71 5.1.5 Permeability.............................. 72 5.2 Mechanical Testing Methods......................... 74 5.2.1 Unconfined Compression....................... 75 5.2.2 Confined Compression........................ 76 5.2.3 Indentation.............................. 76 5.3 Material Models of Articular Cartilage................... 77 5.3.1 Monophasic Models.......................... 78 5.3.2 Biphasic Models............................ 82 5.3.3 Other Models of Articular Cartilage................. 92 6 A Theoretical Model of Articular Cartilage in Large Deformations.... 93 6.1 General Theory................................ 94 6.1.1 Stress................................. 94 6.1.2 Porosity................................ 95 6.1.3 Permeability.............................. 95 6.2 Transversely Isotropic Second Order Tensor Basis............. 96 6.3 Incompressible Solid Fluid Mixtures.................... 97 6.4 Hyperelastic Biphasic Mixtures....................... 99 6.5 Fibre Reinforced Biphasic Constitutive Model............... 102 6.5.1 Elasticity Formulation........................ 102 6.5.2 Permeability Formulation...................... 105 7 Numerical Implementation.......................... 109 7.1 Introduction to FEBio............................ 110 7.2 Implementation of the Elastic Formulation................. 111 7.2.1 Constitutive Functions for the Elastic Formulation......... 112 7.2.2 Derivation of the Spatial Stress and Elasticity Tensors...... 114 7.3 Implementation of the Permeability Formulation.............. 117 7.3.1 Constitutive Functions for the Permeability Formulation..... 119 7.3.2 Derivation of the Fourth Order Permeability Tangent Tensor... 120 v
7.4 Probability distribution and Depth Dependence.............. 123 7.4.1 Algorithm for the Elastic Formulation............... 128 7.4.2 Algorithm for the Permeability Formulation............ 129 7.5 Cartilage Modelling.............................. 132 7.5.1 Attachment to Subchondral Bone.................. 132 7.5.2 Articular Cartilage Composition................... 132 7.5.3 Articular Cartilage Properties.................... 133 7.5.4 Unconfined Compression....................... 136 8 Growth and Remodelling in Fibre Reinforced Biological Tissues..... 138 8.1 Mathematical Formulation.......................... 139 8.1.1 Geometry............................... 140 8.1.2 Boundary Conditions and Simplified Balance of Momentum... 141 8.1.3 Constitutive Model for the Fibre Reinforced Cylindrical Geometry 144 8.1.4 Remodelling Equation for Fibre Reinforced Materials....... 147 8.1.5 Determining the Hydrostatic Pressure p.............. 149 8.2 Numerical Implementation.......................... 150 9 Results and Discussions........................... 155 9.1 Articular Cartilage FE Model Validation.................. 155 9.1.1 Large Deformation Model With Statistically Oriented Reinforcing Fibres................................. 156 9.1.2 Holmes Mow Potential........................ 160 9.1.3 Permeability Formulation...................... 161 9.2 Analysis of Articular Cartilage........................ 166 9.2.1 Viscoelasticity............................. 167 9.2.2 Pore Pressure............................. 168 9.2.3 Stress Distribution.......................... 171 9.2.4 Fluid Flux............................... 173 9.2.5 Depth Dependent Strain....................... 175 9.2.6 Effect of Fibre Content on Zonal Strain............... 178 9.3 Fibre Remodelling in an Arterial Geometry................. 182 9.3.1 Evolution of the Mean Fibre Angle Over Time........... 182 9.3.2 Mean Fibre Angle Distribution in the Radial Direction...... 183 9.3.3 Stress Distribution in the Radial Direction............. 185 10 Conclusions.................................. 187 10.1 Implementation of the Articular Cartilage Model............. 188 10.2 General Conclusions............................. 190 10.2.1 Articular Cartilage Tissue Model.................. 190 10.2.2 Fibre Remodelling in an Arterial Geometry............ 192 10.3 Future Directions............................... 193 References...................................... 195 A Code for the Large Deformation Elasticity and Flow Model........ 214 A.1 Elasticity Model................................ 214 A.1.1 Header File.............................. 214 A.1.2 Source File.............................. 215 vi
A.2 Permeability Model.............................. 230 A.2.1 Header File.............................. 230 A.2.2 Source File.............................. 231 A.3 Remodelling Implementation......................... 247 A.3.1 Main program............................. 247 A.3.2 Constraint Equation Integral..................... 250 A.3.3 Calculation of the Hydrostatic Pressure p.............. 252 A.3.4 Calculation of the Radial Stress P rr................ 255 A.3.5 Calculation of the Tangential Stress P θθ.............. 256 A.3.6 Calculation of the Derivative of the Elastic Potential With Respect to the Remodelling Angle...................... 257 vii
List of Tables 3.1 Constructing spatial constitutive matrix from the components of fourth order spatial elasticity tensor using D IJ = Cijkl............... 47 7.1 Algorithm for anisotropic fibre contribution evaluation.......... 130 7.2 Algorithm for permeability evaluation.................... 131 8.1 Material parameters used in the remodelling implementation....... 153 8.2 Algorithm for the implementation of the remodelling of the fibres.... 154 9.1 Zonal deformation comparison for articular cartilage............ 177 viii
ix List of Figures 2.1 General motion of a deformable body.................... 10 2.2 Internal forces and the traction vector................... 23 2.3 Decomposition of a deformation gradient F................ 34 4.1 Illustration of a diarthrodial joint...................... 53 4.2 Hierarchical microstructure of a collagen fibre............... 56 4.3 Depth dependence of collagen and proteoglycan content.......... 57 4.4 Structure of aggrecan molecule........................ 58 4.5 Keratan and chondroitin sulfate in aggrecan molecule........... 59 4.6 Structure of proteoglycan macromolecule.................. 60 4.7 Extra cellular matrix structure....................... 61 4.8 Chondrocyte and collagen architecture relative to depth.......... 62 4.9 Articular cartilage zonal architecture.................... 63 5.1 Articular cartilage creep behaviour..................... 66 5.2 Articular cartilage stress relaxation behaviour............... 67 5.3 Fluid solid interaction during loading.................... 69 5.4 Tensile stress strain curve for articular cartilage.............. 70 5.5 Shear behaviour of articular cartilage.................... 72 5.6 Permeability flow test setup......................... 73 5.7 Permeability as a function of strain and applied pressure......... 74 5.8 Unconfined compression test......................... 75 5.9 Confined compression test.......................... 76 5.10 Indentation test................................ 77 5.11 Split lines in articular cartilage....................... 86 6.1 Function U(J(C)).............................. 101 6.2 Representative Element of Volume...................... 105 7.1 Probability distribution function and fibre orientation........... 125 7.2 Probability density functions......................... 127 7.3 Meshed Finite Element Model Geometry.................. 136 7.4 Normalized depth vs. concentration parameter in FE model....... 137 8.1 Boundary conditions for the geometry in the remodelling benchmark problem142 8.2 Benchmark problem geometry with fibre orientation............ 151 9.1 Normalized force as a function of stretch for the fibre model....... 157 9.2 Normalized force vs. the concentration parameter b............ 158 9.3 Validation of the implemented Holmes Mow potential........... 160 9.4 Axial fluid flux for a sample in unconfined compression.......... 162 9.5 Radial fluid flux for a sample in unconfined compression......... 164
9.6 Normalized fluid flux vs. the concentration parameter b.......... 165 9.7 Force vs. time for unconfined compression experiment........... 168 9.8 Force vs. time for unconfined compression experiment........... 169 9.9 Pore pressure in articular cartilage sample................. 170 9.10 Effective stress distribution in articular cartilage sample.......... 172 9.11 Axial and radial fluid flux in articular cartilage sample.......... 174 9.12 Zonal strain as a function of time for articular cartilage.......... 175 9.13 Zonal strain as a function of normalized fibre content for articular cartilage 179 9.14 Evolution of the mean fibre angle over time................ 183 9.15 Fibre angle distribution in the radial direction............... 184 9.16 Stress distribution in the radial direction.................. 185 x
1 Chapter 1 Introduction Articular cartilage is a soft tissue that covers the articulating ends of bones in diarthrodial joint. Its purpose is to distribute stress, therefore reducing friction and minimizing the mechanical joint wear (Athanasiou et al., 2009). The health of articular cartilage largely depends on the functioning of its cells, the chondrocytes, whose function is to synthesize the extracellular matrix (Poole et al., 1987). The biological response of the chondrocytes is directly influenced by their mechanical environment, and hence the understanding of the stress strain fields in articular cartilage, and how these fields affect the chondrocytes, is integral in understanding cartilage adaptation and degeneration. Modelling of articular cartilage is a challenging problem due to the complex tissue structure; articular cartilage is a biphasic, anisotropic, and inhomogeneous biological tissue. The volumetric fractions of its constituents vary with tissue depth, and consequently articular cartilage can be characterized a having depth dependent mechanical properties. The earliest theoretical studies of articular cartilage mechanics were based on monophasic, linear, isotropic, homogeneous material models (Hirsch, 1944; Sokoloff, 1966). This was followed by a number of biphasic, linear, isotropic models, the most well known being the model of Mow et al. (1980), which was based on the early work of Biot (1941). More recently, the Non Linear Transversely Isotropic, Transversely Homogeneous (NLTITH) model, introduced by Federico & Herzog (2008c) and Federico & Gasser (2010) and expanding on the work of Federico et al. (2005), has been used to model tissues with depth dependent collagen fibre distribution, such as articular cartilage. The main limitation of this model is that it an elasticity only model. In other words, this formulation does not take into account the effect of the collagen fibre orientation on the overall permeability
2 of the tissue. The most recent step towards a continuum model for articular cartilage is the large deformation non linear model for biological tissues with statistically oriented reinforcing fibres, developed by Federico & Grillo (2012). This model is a natural evolution of the model by Federico & Herzog (2008c), and it introduces the fibre dependent permeability formulation, which takes into account the orientation of the collagen fibres and how it affects the overall permeability of the tissue. 1.1 Contribution The main purpose of this Thesis was to numerically implement the large deformation non linear model for biological tissues with statistically oriented reinforcing fibres, developed by Federico & Grillo (2012). This includes the implementation of both the elasticity and permeability formulation, the latter assuming that the permeability depends on the fibre orientation and overall deformation of the tissue. Since commercial Finite Element packages do not allow explicit dependence of the permeability tensor on the deformation, the model has been implemented in an open source Finite Element package FEBio. Once the complete model has been successfully implemented, the main purpose of the work shifted to using the model framework to study the articular cartilage mechanics by comparing the results from the numerical simulations to experimental results from unconfined compression tests performed by Guilak et al. (1995). The secondary purpose of this Thesis was the development and implementation of an evolution equation for tissues with statistically oriented reinforcing fibres. The initial focus was the development and formulation of the appropriate constitutive and evolution equations. After this has been performed, the focus then shifted to the implementation of the developed model for the case of fibre remodelling with no growth. Finally, the
3 results obtained from the numerical simulations were compared to the results obtained by Olsson & Klarbring (2008). 1.2 Outline of the Thesis This thesis is divided into 10 chapters, including this introductory chapter. An effort was made to present different parts in a linear and coherent fashion. Mathematical background and introduction to Finite Element Analysis are presented first, as these topics are necessary to be able to understand to work in the subsequent chapters. Next, a description of cartilage composition and structure, along with the mechanical testing and existing cartilage models, are presented to give a background knowledge of the both the cartilage behaviour and the trends in cartilage modelling. Once the background has been presented, the focus of the Thesis shifts to the large deformation statistically oriented fibre reinforce articular cartilage model, and its numerical implementation. This is followed by a formulation for remodelling of fibre reinforced materials, and the numerical implementation of this formulation. The Thesis is concluded with reporting the results obtained from the models above, and the discussion of these results. Chapter 2 A brief introduction of Continuum Mechanics is presented in Chapter 2 in preparation for the subsequent chapters that deal with the mathematical framework of various models of articular cartilage, including the model implemented in this work. The chapter begins with an introduction to configuration map and motion of a continuum body, as well as the definition of covariant and contravariant objects. Next, the deformation gradient is introduced, followed by deformation and strain measures. Balance laws are introduced next, along with definition of stress, and this also leads to definition of constitutive equations and hyperelastic materials. The chapter is concluded with an introduction to
4 the treatment of growth and remodeling in continuum media, and finally by the definition of directional derivative, which is necessary operation in the non linear finite element method. Chapter 3 A brief introduction of the Finite Element (FE) method is presented in Chapter 3 in preparation for the chapters that deal with the numerical implementation of the constitutive model addressed in this Thesis. The chapter begins with an introduction to the general procedure for solving a problem through FE analysis, including the preparation of the problem, the considerations regarding the geometry of the problem, and the steps required to obtain a solution by FE method. This is followed by an introduction to the weak formulation for the solid material, and it includes both linearization and discretization of the domain. The chapter is concluded by considering iterative Newton Raphson methods used to obtain a solution of non linear FE problems. Chapter 4 Composition and structure of articular cartilage are outlined in chapter 4. This discussion covers the main constituents of articular cartilage, such as collagen fibres, proteoglycan macromolecules, chondrocytes, and interstitial fluid, as well as how these constituents change in content over the tissue depth. The structure of articular cartilage is also outlined, with a specific emphasis on the depth dependent microstructure and how this affects the behaviour of the tissue. Chapter 5 Mechanical behaviour, testing methods, and mathematical models used to describe the mechanical behaviour of articular cartilage are outlined in Chapter 5. The chapter begins with a discussion of the viscoelastic nature of articular cartilage, and is then followed by discussions on the behaviour of cartilage in compression, tension, and shear. Common
5 testing methods, such as unconfined and confined compression and the indentation tests, are outlined next, with a short discussion on how these tests can be used to determine the mechanical properties of articular cartilage. Finally, the mathematical models of articular cartilage are discussed. This starts with a discussion of early monophasic models and is followed by viscoelastic model discussion and a general background on the biphasic modelling. Chapter 6 The mathematical formulations used to model the articular cartilage in this Thesis are described and formulated in Chapter 6. The chapter starts with a general introduction and an introduction to the transversely isotropic basis, which is then followed by an introduction to the fibre reinforced biphasic constitutive model. The constitutive model formulation has two major sections; the first one provides the elasticity formulation with the appropriate constitutive equations, while the second provides an introduction to the permeability formulation, which depends on both the fibre orientation and the overall deformation. Chapter 7 The numerical implementation of the overall articular cartilage model is presented in Chapter 7. The chapter starts with a selection of the appropriate constitutive models for the proteoglycan matrix, collagen fibres, and the matrix permeability. This is then followed by the treatment of the depth dependent nature of the tissue, and the algorithm utilized in the implementation of the overall model. The chapter is concluded with an outline of articular cartilage modelling performed in this Thesis. Chapter 8 The growth and remodelling treatment for fibre reinforced materials with statistically oriented fibres is presented in Chapter 8. The chapter starts with a general introduction
6 to the evolution equations, and a description of the necessary simplifications. This is followed by the formulation of the geometry and the specific constitutive and remodelling equations. The chapter is concluded with an outline of the numerical procedure used to implement the remodelling in statistically oriented fibre reinforced tissue. Chapter 9 and 10 The results from both the implementation of the articular cartilage model and the remodelling of statistically oriented fibre reinforced materials are presented in Chapter 9. The chapter starts with a validation of the articular cartilage model by comparing it to the previously obtained numerical results (Holmes & Mow, 1990; Pajerski, 2010). Next, the unconfined compression tests performed by Guilak et al. (1995) were simulated and the numerical results were compared to the experimental results. The chapter is concluded by the presentation of the results obtained from the remodelling formulation, and the subsequent comparison to the results obtained by Olsson & Klarbring (2008). Chapter 10 outlines the general conclusions and the direction of future work.
7 Chapter 2 Short Introduction to Continuum Mechanics In order to allow detailed discussion on various mathematical models of articular cartilage it is essential to devote a chapter to an introduction of Continuum Mechanics. Continuum Mechanics is a branch of Mechanics that studies kinematics and material behaviour at a material scale at which the atomic and subatomic structure of the matter can be neglected, and the matter can be regarded as a continuum. Modeling the matter in this way is an approximation, albeit a highly accurate approximation at length scales greater than the inter atomic distances. The most obvious advantage of continuum approximation is that it is possible to apply mathematical structures of Differential Geometry and use tools such as Differential and Integral Calculus to describe the behaviour of the matter. Continuum Mechanics is the study of the laws of motion put forward by Isaac Newton (1643 1727) as applied to continuum media, and it can be further split into two major fields: Solid Mechanics, or the study of physics of continuous media with defined rest shape, and Fluid Mechanics, or the study of physics of continuous media which take shape of the surroundings. The theoretical framework of Continuum Mechanics was put forth by Leonhard Euler (1707 1783), and it was further developed by Augustin Louis Cauchy (1789 1857). Applications of Continuum Mechanics are numerous, and range anywhere from study of biological tissues such as arteries (Holzapfel et al., 2000) and articular cartilage (Mow et al., 1980) to structures such as skyscrapers and bridges. The brief outline of Continuum Mechanics presented in this chapter is based primarily on the textbooks Mathematica Foundations of Elasticity by Marsden & Hughes (1993), Mechanics of Continua by Eringen (1980), and Nonlinear Continuum Mechanics for
8 Finite Element Analysis by Bonet & Wood (2008), and it only includes the necessary background needed to understand the tissue models presented in this work. For a more detailed treatment of the topic, the Reader can refer to any of the textbooks listed above, or to one of many textbooks published in the field. 2.1 Bodies, Configurations, Fields, and Motion If a deformable continuous body occupies a certain region of physical space S = R 3 at a given time, this is called a configuration. By choosing a particular configuration B 0, usually called the reference configuration, and by ruling out the possibility of compenetrations or fractures, it is possible to put the reference configuration into a bijective correspondence with the current configuration at time t R + 0 and denoted by B(t) S by means of a diffeomorphism (a continuous and differentiable map, with continuous and differentiable inverse) called the configuration map χ: B 0 R + 0 S : (X, t) x = χ(x, t) (2.1) where X is the position of a point in the reference configuration and x is the position of the same point in the current configuration at time t. Points X and x are called material and spatial, respectively. Introducing the notion of tangent space, i.e., a replica of R 3 attached at each point in the body, allows for the definition of field quantities such as tensors of any order (e.g. 0 th order being scalars, 1 st order being vectors, etc.). A tangent space attached to a point X in the reference configuration is denoted by T X B 0, while a tangent space attached at point x in the current configuration is denoted by T x S, as illustrated in Figure 2.1. A physical quantity, or a field, f can be expressed in either the reference configuration as a function of X or in the spatial configuration as a function of x. In the former case, the Lagrangian formalism is adopted, where the observer tracks a certain particle defined
9 by the point X in the reference configuration. In the latter case, one adopts the Eulerian formalism, where the observer tracks a value of a physical quantity at a certain point x, regardless of which particle occupies that point in space. Hence, each field f can be defined in both Lagrangian and Eulerian formalisms, which are connected by the configuration map f(x, t) = e f(x, t) = e f(χ(x, t), t) (2.2) where f denotes Lagrangian form, while e f denotes the Eulerian form of the field (Federico, 2000). These two forms will be strictly differentiated in the remainder of this work in order to avoid confusion. 2.1.1 Covariant and Contravariant Tensors In tensor analysis, covariance and contravariance describe how the components of tensors transform with a change of basis between coordinate systems. If the coordinate systems are restricted to simple Euclidian coordinates and change of bases can be expressed as a set of rotations, covariant and contravariant objects are indistinguishable. However, once one considers more general coordinate systems, such as curvilinear coordinates and coordinates on differentiable manifolds, it becomes of critical importance to distinguish between the two types of objects. A linear form can be defined as α: V R α hom (V, R) = V (2.3) where hom (V, R) denotes a homeomorphism between the vector space V and the underlying scalar field R. Covector space V represents the dual space of V,which implies the principle of duality. Hence, α is often called a covector, linear form, or a 1 form. A vector can be defined as a linear form on covectors, i.e., v : V R v hom (V, R) = V (2.4)
10 Figure 2.1: General motion in the neighborhood of a particle within a continuous deformable body. Upper case variables indicate objects in the reference configuration, while lower case variables indicate objects in the current configuration. Configuration map χ is shown to map the points from the reference into the current configuration. Then tangent spaces T X B 0 and T x S as well as their attachment to a point in the body have also been illustrated. It should be noted that in this illustration it has been assumed that the coordinate bases of the two configurations are coincident, but this is generally not the case. The figure has been adapted from Bonet & Wood (2008). V denotes the vector space and V denotes the homeomorphism between the vector space V and the underlying scalar field R, and vice versa. Therefore, v is defined as a vector, as it is an object that resides in a vector space and operates on the homeomorphism of that space. In summary, covectors can be thought of as operators on vectors, while vectors can be thought of as operators on covectors. Similar to the vector space V, where the basis is defined as e a, the basis of the dual space V is defined as e a. Therefore, this implies basis conjugation as V, e a V, e a (2.5)
11 which leads to formal definition of the component decomposition of both objects in vector space, or vectors, and the objects in dual space, or covectors, as v = v a e a α = α a e a (2.6) Covariance and contravariance also applies to tensors of higher order, with one significant difference. Since tensors with order higher than 1 are mapping multiple spaces, or mapping from one space to another, and hence have more than one index, these tensors can have mixed covariant and contravariant indices. For example, the previously defined v is one time contravariant and zero time covariant, but tensors of higher order might be one time contravariant, one time covariant (a good example is the deformation gradient F, which will be defined later). 2.1.2 Covariant and Contravariant Transformations The most significant implication of the difference between contravariant and covariant objects arises when performing a change of basis. In order to demonstrate this difference and further reinforce the difference between the two types of objects, a simple basis change will be performed on both a vector and a covector. The basis change that will be performed is a rotation through a rotation tensor R. Therefore, defining e b = Ri b e a and expressing a vector v in components yields v = v a e a (2.7) = v b e b = v b R a be a Through simple association it can then be concluded that v a = R a bv b v b = (R 1 ) b av a (2.8) This is called a contravariant transformation, as the components transform with the inverse of the transformation tensor when compared to the basis of the vector space.
12 Now defining v = v a e a and expressing a covector α in components yields α(v) = α(v a e a ) = α(v b e b ) (2.9) = v a α a = v b α b Now applying the contravariant rotation to the vector components yields R a bv b α a = v b α b (2.10) Through simple association it can then be concluded that α a R a b = α b α b = α a R a b (2.11) This is called a covariant transformation, as the components transform in the same manner as the basis of the underlying vector space. 2.1.3 Metric tensor A metric tensor is a symmetric, positive definite bilinear form on a vector space, which takes as input a pair of tangent vectors x and y and produces a real number as g : V V R: ( x, y) g( x, y) (2.12) The metric tensor can also be thought of as a generalization of the familiar properties of the scalar product in Euclidian space. In a similar way to the scalar product in Euclidian space, the metric tensor is used to define the length of, and the angle between, vectors in the tangent space. In a given basis, the metric tensor is expressed as g = g ab e a e b (2.13) Another important application of the metric tensor is to relate the vector space to its dual space, i.e. g : V V, and to raise or lower indices, hence changing the nature of the object in question (e.g. fully covariant tensor fully contravariant tensor).
13 2.1.4 Velocity and Acceleration Since many nonlinear processes are time dependent, it is necessary to consider velocity and acceleration. However, even if the process is not time dependent, it is useful to establish these quantities as they are used to define the balance laws that govern the behaviour of the deformable continuous body. If the configuration map x = χ(x, t) is now considered, velocity can be defined as v(x, t) = χ t (X, t) va = χa (X, t) (2.14) t Velocity can also be defined in Eulerian formalism, where it is assumed that v a = e v a (χ, τ) where τ(x, t) = t is a function called the time map (Federico, 2000). This difference will become important later, when velocity is used to define other quantities. Hence, v = e v a (χ, τ) (2.15) In order to determine the acceleration, a derivative of velocity with respect to time has to be taken. Since velocity is a tensor of order 1, or a vector, it is necessary to perform a covariant derivative. Therefore, the acceleration is defined as a(x, t) = [ t v a (X, t)]e a (X, t) + v a (X, t)[ t e a (X, t)] (2.16) Rearranging equation (2.16) and introducing a an object called the Christhoffel symbol γbc a, the Lagrangian acceleration a : B R T x S can then be defined as a = a a e a = [ t v a + γcb a v b v c] e a, (2.17) which defines the covariant time derivative of the velocity as a a = t v a + γcb a v b v c. (2.18) Similarly, the Eulerian acceleration can be defined as e a = e a a e a = [ e t v a + e v a,b e v b + γcb a e v b e v c] e a, (2.19)
14 which defines the covariant time derivative of the spatial velocity as e a a = t e v a + e v a,b e v b + γ a cb e v b e v c (2.20) For more information on Christhoffel symbols, covariant derivatives, and the material time derivative, the Reader can refer to Marsden & Hughes (1993) or Bishop & Goldberg (1980). 2.2 Deformation Gradients and Strain Measures In Figure 2.1 on page 10, the position of the particle Q relative to X is given by the infinitesimal tangent vector X, which is an element in the tangent space T X B 0. After deformation, the position of the same particle relative to x is given by the infinitesimal tangent vector x, which is an element in the tangent space T x S. The position of point Q in the reference configuration can be expressed as, in components, χ a (X + X, t) = χ a (X, t) + χ a,a(x, t)[ X] A (2.21) This expression defines a tensor F (X, t) : T X B 0 T x S, with components F a A(X, t) = χ a,a(x, t) (2.22) which is called the deformation gradient and is a two point, once contravariant, once covariant tensor which maps vectors from the tangent space T X B 0 to the tangent space T x S. For example, the infinitesimal vector X in the reference configuration can be related to the infinitesimal vector x in the current configuration through the deformation gradient as x = F X (2.23) It should be noted that the configuration map χ(x, t) is used to map the point from the reference configuration B 0 to the current configuration B(t) S, while the
15 deformation gradient F (X, t) is used to map the vectors from the tangent space T X B 0 in the reference configuration to the tangent space T x S in the current configuration, and vice versa. Consequently, the transformation of a material vector into a spatial vector is known as a push forward, while the transformation of the spatial vector into a material vector is known as a pull back. Since, as previously mentioned, the configuration map χ(x, t) is a diffeomorphism, it can be stated that deformation gradient F is invertible and that its determinant J = det (F ) is strictly positive. Since the configuration map χ(x, t) is simply a change in coordinate reference between the reference and current configurations, J represents the change in volume. This is directly analogous to Radon Nikodym theorem on the change of measure in integrals. Hence, of a certain part of the body occupies a region in space R 0 in the reference configuration and the region R(t) in the current configuration, then the volume of region R(t) is given by e f(x, t) dv = J(X, t) e f (χ, τ) dv (2.24) R(t) R 0 where e f(x, t) is a field in Eulerian formalism and e f (χ, τ) = f. Similarly, a relation can be derived to relate the areas in the current configurations to the reference configuration. This relation is called Nanson s formula, and it is given by e e T n da = J(X, t) e T (χ, τ)f T N da (2.25) R 0 R(t) where e n is the normal covector to the surface R(t) in the current configuration, N is the normal covector in the reference configuration, e T is the Eulerian form of a spatial tensor of any order (the last index of which is contravariant), and e T (χ, τ) = T is its Lagrangian representation.
16 2.2.1 Polar Decomposition The deformation gradient F discussed in the previous sections transforms the vectors from the reference configuration tangent space T X B 0 into the vectors in the current configuration tangent space T x S, and vice versa. The role of F can be further enhanced by decomposing it into a product of an orthogonal times a symmetric tensor, or a symmetric times an orthogonal tensor through an application of Cauchy s polar decomposition theorem. Physically, this would correspond to decomposing deformation into a stretch followed by a rotation, or a rotation followed by a stretch. Mathematically, this can be expressed as F = R U = V R (2.26) or, in components, F a C = R a A G AB U BC = V ab g bc R c C (2.27) where the orthogonal two point tensor R is called the rotation tensor, the symmetric fully material tensor U is called the right stretch tensor, and the fully spatial tensor V is called the left stretch tensor. It should be noted that although a convention of using capital letter to denote material quantities and small letters to denote spatial quantities is used throughout this work, left stretch tensor V is an exception, as it is denoted by a capital letter. This is the only quantity that is not named according to the convention, in order to avoid confusion between the left stretch tensor V and the velocity v. Through manipulation of the expression in equation (2.26), it can be shown that the stretch tensors V and U are related by V = R U R T (2.28) or, in components, V ab = R a A G AC U CD G DE (R T ) b E (2.29)
17 Since it is clear from the expression in equation (2.28) that the stretch tensors U and V are just rotations of one another, it must be true that they have the same eigenvalues λ α and that their eigenvectors are related by a simple rotation, as n α = R N α. Hence, the two stretch tensor can be decomposed using the spectral theorem, which states that a symmetric tensor always has real eigenvalues and orthogonal eigenvectors. Therefore, the right stretch tensor U can be expressed as 3 U = λ α N α N α (2.30) α=1 while the left stretch tensor V can be expressed by 3 V = λ α n α n α (2.31) α=1 2.2.2 Cauchy Green Deformation Tensors In order to define a general measure of deformation of a certain body, consider the change in the scalar product of two material vectors X and Y, which are shown in Figure 2.1, as they deform to spatial vectors x and y, respectively. Recalling equation (2.23), it becomes possible to express the spatial scalar product x y in terms of the material vectors X, Y T X B 0 as x y = X C Y (2.32) where C is defined as the right Cauchy Green deformation tensor, and it is given in terms of the deformation gradient F as C = F T F (2.33) or, in components, C AB = (F T ) a A g ab F b B (2.34)
18 Alternatively, the material scalar product X Y can be expressed in terms of spatial vectors x, y T X S as X Y = x (b 1 ) y (2.35) where b is defined as the left Cauchy Green deformation tensor, and it is given in terms of the deformation gradient F as b = F F T (2.36) or, in components, b ab = (F ) a A G AB (F T ) B b (2.37) The right Cauchy Green deformation tensor C as well as the left Cauchy Green deformation tensor b are coaxial to stretch tensors U and V respectively, which implies that they can be expressed as C = U 2 and b = V 2, respectively. Due to the coaxial nature of these tensors, both left and right Cauchy Green deformation tensors also admit the spectral decompositions 3 3 C = λ 2 α N α N α, b = λ 2 α n α n α (2.38) α=1 α=1 which shows that they share the same eigenvalues λ α. This implies that the invariants, or quantities that remain unchanged under a certain set of transformations, of these two tensors are also the same, the importance of which will be discussed later in this work. The three invariants of the Cauchy Green deformation tensors are given as I 1 (C) = I 1 (b) = tr (C) = tr (b) (2.39a) I 2 (C) = I 2 (b) = 1 2 [ (tr (C)) 2 tr ( C 2)] = 1 2 [ (tr (b)) 2 tr ( b 2)] (2.39b) I 3 (C) = I 3 (b) = det (C) = det (b) (2.39c)
19 2.2.3 Strain Measures The change in a scalar product between material vectors X, Y T X B 0 and the spatial vectors x, y T X S can now be expressed as 1 2 [ x y X Y ] = 1 [ X C Y X G Y ] (2.40) 2 [ ] 1 = X (C G) Y 2 which defines a fully material measure of strain called the Lagrangian or Green strain tensor and defined as E = 1 [C G] (2.41) 2 Alternatively, the same change in the scalar product can be expressed with a reference to the spatial vectors x, y T X S as 1 2 [ x y X Y ] = 1 [ x g y x b 1 y ] (2.42) 2 [ 1 ( = x ) ] g b 1 y 2 which defines a fully spatial measure of strain called the Eulerian or Almansi strain tensor and defined as e = 1 2 [ g b 1 ] (2.43) It should be noted that the factor of 1/2 in both the definitions for E as well as e comes from the exponent 2 in definitions C = U 2 and b = V 2, respectively. It should also be noted that both the Green and Almansi strain admit the spectral decomposition as follows E = 3 α=1 1 2 (λ2 α 1) N α N α, e = 3 α=1 1(1 λ 2 2 α ) n α n α (2.44) The spectral decomposition of Green and Almansi strain motivates a generalization of the material and spatial strain measures of order n as (Hill, 1968) 3 3 E (n) 1 = n (λn α 1) N α N α, e (n) 1 = (1 λ n n α ) n α n α (2.45) α=1 α=1
20 In the particular case where n = 1, material and spatial nominal strains under large deformations are given. On the other hand, the limit n gives the logarithmic material and spatial strains that also admit large deformations. 2.2.4 Volumetric Distortional Decomposition When dealing with incompressible or nearly incompressible materials, it is often advantageous to separate the distortional, or isochoric, and volumetric components of deformation (Flory, 1961; Ogden, 1978). This separation must satisfy the condition that the distortional component, from here on denoted by F, does not imply any change in volume. Since the determinant of the deformation gradient F gives to volume strain ratio, the determinant of F must therefore satisfy det F = 1 in order to satisfy the separation criteria. Hence, the decomposition can be written as F = J 1/3 F (2.46) It is also possible to express the deformation gradient F in terms of the volumetric and distortional components J and F, respectively, as F = J 1/3 F (2.47) Using equation (2.47), it follows that C = J 2/3 C, b = J 2/3 b (2.48) along with the inverse relationships C = J 2/3 C, b = J 2/3 b (2.49) 2.2.5 Velocity Gradient By taking the gradient of the Eulerian form of velocity with respect to the spatial coordinates, one obtains the velocity gradient tensor e l e l = e v (2.50)
21 In components, equation (2.50) takes the form e l a b = e v a b (2.51) In order to be able to use the velocity gradient in the constitutive theory, it is necessary to make it fully covariant. This is done by lowering the first index and making it fully covariant, as follows e l = g e l, e l ab = g e ac l c b (2.52) Form this point forward, fully covariant velocity gradient will be denoted by just e l (i.e. e l = e l ) for simplicity. Since the velocity gradient is now a fully covariant second order tensor, it is possible to decompose it into two parts as e l = e d + e w, where e d is the deformation rate tensor and e w is the spin tensor. These two tensors are defined as e d = 1 2 ( e l + e l T ), e w = 1 2 ( e l e l T ) (2.53) It should be noted that the deformation rate tensor e d is the symmetric part of the velocity gradient, while the spin tensor e w is the skew symmetric part of the velocity gradient. 2.2.6 Rate of Deformation All measures of the rate of deformation are significantly simpler when expressed in terms of the velocity gradient l = v. It is simple to show that the rate of deformation gradient is given by F = lf (2.54) which also facilitates an alternate expression for the velocity gradient as l = F F 1 (2.55) Hence, using equations (2.33, 2.36, 2.41, 2.43), the rates of right and left Cauchy Green deformation tensors C and b as well as the Green and Almansi strain tensors E
22 and e, respectively, are given by Ė = 1 2Ċ = F T d F (2.56a) ė = 1 2ḃ 1 = 1 2 ( b 1 l + l T b 1) (2.56b) or, in components, Ė AB = 1 = (F T ) a 2ĊAB A d ab F b B (2.57a) [ ] ė ab = 1 2 (ḃ 1 ) ab = 1 2 (b 1 ) ac g cd l da + (l T ) ac g cd (b 1 ) db (2.57b) 2.3 Stress and Conservation Laws In the previous sections, the focus was on the Kinematics of Continua, i.e., the study of motion and deformation regardless or the causes of said motion or deformation. In this Section, the focus will shift to the Dynamics of Continua, or the relations between the external forces and the motion, or deformation, that is caused by these forces through the application of conservation laws. In order to apply the conservation laws, it is important to first introduce stress, or an object that describes the internal forces within a deformable body. 2.3.1 Stress Imagine a continuum body in the current configuration B(t) S, as shown in Figure 2.2, separated into two regions R 1 and R 2. In order to develop a concept of stress, it is necessary to study the action of internal forces exerted by one region R 1 onto the other region R 2, and vice versa, over the contact area. From Figure 2.2, it follows that on the point x on the cut surface between the two regions R 1 and R 2 there must be a force per unit area e t (n) (x, t) that restores equilibrium. The traction vector e t (n) (x, t) depends explicitly on the normal e n, and through the
23 Figure 2.2: An illustration of a general deformable body divided into two distinct regions, R 1 and R 2. The external force f applied on R 1 is shown, along with the internal forces exerted by region R 1 onto region R 2. The forces acting on the cut surface of region R 2 have been omitted for clarity. The figure has been adapted from Bonet & Wood (2008). principle of Cauchy tetrahedron it is possible to show that the relationship between the traction vector e t (n) (x, t) and the normal e n is linear as e t (n) = e σ e n (2.58) where e σ is the Cauchy stress tensor, and it can be expressed in components as 3 e σ = σ ab e e a e e b (2.59) a,b=1 where e e a and e e b represent the basis vectors and σ ab represents the components of the Cauchy stress tensor.
24 2.3.2 Conservation Laws Now that the concept of stress has been introduced, it is possible to formulate the conservation laws that will provide a connection between the applied external forces, internal forces, and the kinematics. There are four conservation laws that will be introduced, and these are Conservation of Mass Conservation of Linear Momentum Conservation of Angular Momentum Conservation of Energy An additional inequality, derived from the Second Principle of Thermodynamics and called the Clausius Duhem Inequality, will also be introduced, as an unilateral constraint. This inequality also has to be satisfied, as roughly it states that entropy cannot decrease, but only increase or remain constant. Conservation of Mass In the absence of growth, the mass of region R in side of a body B remains constant through motion. This property is referred to as the conservation of mass, and it is expressed as m = e ρ dv = J ρ dv = constant (2.60) R(t) R 0 where e ρ and ρ are the Eulerian and Lagrangian mass densities, respectively. The rate of mass change, in Eulerian form, can be written as ṁ = e t ρ dv (2.61) R(t) Applying Reynolds transport theorem to equation (2.61), the local Eulerian form of
25 the continuity equation can be expressed in two forms, and these are D t e ρ + e ρ div ( e v) = 0 t e ρ + div ( e ρ e v) = 0 (2.62a) (2.62b) where D t e ρ = t e ρ + e v grad( e ρ). These two equations show that the density can change with time in the current configuration. On the other hand, the conservation of mass can also be written in Lagrangian form. In this case, the equation takes the form ρ R (X) = J(X, t) ρ(x, t) (2.63) It should be noted that the referential mass density ρ R (X) does not depend on time in the absence of growth. Conservation of Linear Momentum Given a region R in a body B, as illustrated in Figure 2.2, with a density e ρ and external body force e f, and following the Cauchy s Separation axiom, the dynamical equilibrium is expressed by the conservation of linear momentum, which states that e ρ e a dv = e ρ e f dv + e t (n) da (2.64) R(t) R(t) R(t) Through the definition of stress, defined in equation (2.58), the linear momentum balance equation can be rewritten as e ρ e a dv = e ρ e f dv + e σ e n da (2.65) R(t) R(t) R(t) The application of the Gauss-Green divergence theorem gives the final integral form of the equation as e ρ e a dv = e ρ e f dv + div ( e σ) dv (2.66) R(t) R(t) R(t) Since the region R is arbitrary, equation (2.66) can be localized, taking the form e ρ e a = e ρ e f + div ( e σ) (2.67)
26 If the conservation of linear momentum is now expressed in the reference configuration by means of a Piola transformation, all quantities take the Lagrangian form and equation (2.67) becomes ρ R a = ρ R f + Div (P ) (2.68) where P is the first Piola Kirchhoff stress tensor, and it is given by P = JσF T, or, in components, P aa = J σ ab (F T ) A. As it can be seen from the definition of P, it is b a two point tensor, with the first leg in the spatial configuration and the second leg in the reference configuration. The first Piola Kirchhoff stress physically represents the nominal stress: current force over reference area. If the first leg of the first Piola Kirchhoff stress tensor P is also pulled back to the reference configuration, the second Piola Kirchhoff stress tensor S is obtained, and it is given by S = F 1 P (2.69) The second Piola Kirchhoff stress tensor S is the fully material counterpart of the Cauchy stress tensor σ, and it plays a crucial role in constitutive modeling. It can also be given in terms of the full Piola transformation as S = Jχ [σ] = JF 1 σf T (2.70) while the inverse full Piola transformation yields σ = J 1 χ [S] = J 1 F SF T (2.71) In components, equations (2.70) and (2.71) are expressed as S AB = J (F 1 ) A a σ ab (F T ) B b (2.72) and σ ab = J 1 F a A S AB F b B (2.73)
27 Conservation of Angular Momentum Given a region R in a body B illustrated in Figure 2.2, with a density e ρ and external body force e f, the rotational equilibrium about a point x, identified by the position vector x, is expressed by the conservation of angular momentum, which states that x e ρ e a dv = x e ρ e f dv + x e t (n) da (2.74) R(t) R(t) R(t) Using the definition of stress expressed in equation (2.58), the angular momentum balance can be rewritten as x e ρ e a dv = x e ρ e f dv + x e σ e n da (2.75) R(t) R(t) R(t) The application of Gauss Green divergence theorem, as well as slight mathematical manipulation, provides a rearrangement of equation (2.75) as x e ρ e a dv = x e ρ e f dv + x div ( e σ) dv + ε : e σ dv (2.76) R(t) R(t) R(t) R(t) where ε is the called the permutation tensor or the Levi Civita tensor, and it is defined as +1 if (i, j, k) is (1, 2, 3), (3, 1, 2) or (2, 3, 1), ε ijk = 1 if (i, j, k) is (1, 3, 2), (3, 2, 1) or (2, 1, 3), 0 if i = j or j = k or k = i After localization, equation (2.76) takes the form x e ρ e a = x e ρ e f + x div ( e σ) + ε : e σ (2.77) and after applying the conservation of linear momentum, the final form of the conservation of angular momentum is obtained to be ε : e σ = 0 (2.78) Equation (2.78) implies that e σ = e σ T, or that the Cauchy stress tensor must be symmetric. If this expression is transformed into reference configuration, it also implies that P F T = F P T and that S = S T.
28 Conservation of Energy The conservation of energy, which is more widely known as the First Law of Thermodynamics, is given by T + E = P ext + P therm (2.79) where T is the kinetic energy, E is the internal, or potential, energy, P ext is the power of external forces acting on the body, and P therm is the thermal power. These quantities are expressed as T = E = P ext = P therm = R(t) R(t) R(t) R(t) e ρ e a ev dv e ρ e Ė dv e ρ e f ev dv + [div ( e q) + e r] dv R(t) e t (n) ev da (2.80a) (2.80b) (2.80c) (2.80d) where e E is the internal energy per unit volume, e Ė = t e E, e q is the heat flux vector, and e r is the heat generation rate. The time derivatives for both the kinetic energy T and the internal energy E are obtained through Reynolds transport theorem and the continuity equation. It is now useful to attempt to eliminate the kinetic power T and the power due to the external forces P ext from equation (2.79) in favour of a term containing the power pertaining to stress. If equation (2.66) is multiplied by the velocity e v, and the terms are manipulated slightly, the following is obtained: e v eρ e a dv = e v eρ e f dv + e t (n) ev da + R(t) R(t) R(t) R(t) ε : e σ dv (2.81) Since it is known that T = P ext P int, through comparison between expressions for T and P ext and (2.81), it can be determined that P int = e σ : e l dv = R(t) R(t) e σ : e d dv (2.82)
29 It should be noted that P int can be written both in terms of an integral of e σ : e d and e σ : e l interchangeably as the product e σ : e w is equal to zero. It should also be noted that it can be said that the Cauchy stress e σ is power conjugated to the velocity gradient e l or the deformation rate e d. If the expression shown in equation (2.82) is pulled back to the reference configuration, an alternate expression for P int is obtained as P int = P : Ḟ dv = S : Ė dv (2.83) R 0 R 0 Equation (2.83) shows that the first Piola Kirchhoff stress P is power conjugated to the rate of deformation gradient Ḟ, while the second Piola Kirchhoff stress S is power conjugated to the rate of the Green Lagrange strain Ė. Hence, the final form for the conservation of energy becomes, using any of the three different power conjugations, E = P int + P therm (2.84) which can be localized into e ρ e Ė = e σ : e l + div ( e q) + e r ρ R Ė = P : Ḟ + Div (Q) + r R ρ R Ė = S : Ė + Div (Q) + r R (2.85a) (2.85b) (2.85c), respectively. Clausius Duhem Inequality Counting the number of conservation equations and the number of unknowns yields 8 equations and 17 unknowns. To make things even worse a thermo mechanical system is subject to an additional constraint, the Clausius Duhem inequality, which states that the entropy of the system must always stay constant or increase, and under no circumstances can it decrease. This inequality states that e ρ e γ e θ = e ρ ( e η e θ e Ė) + e σ : e l + 1 e e q grad( e θ) 0 (2.86) θ
30 where e γ is the entropy generation rate, e θ is the temperature, and e η is the entropy. The addition of the Clausius Duhem inequality increases the number of equations to 8, but the number of unknowns also increases to 19. In order to have a closed system, the number of unknowns must equal the number of equations. In order to get the additionally needed 11 equations, constitutive equations will be used. These equations are introduced in next Section. 2.4 Constitutive Equations As previously mentioned, constitutive equations are needed in order to complete the system. A general thermo mechanical system will be governed by 8 equations, which are the conservation laws and the Clausius Duhem inequality, that have 19 unknowns, which implied that additional 11 equations are needed to complete the system. These 11 equations are the constitutive equations, and they are of the form S(X, t) = Ŝ(χ(X, t ), θ(x, t ), X, t ) Q(X, t) = ˆQ(χ(X, t ), θ(x, t ), X, t ) E(X, t) = Ê(χ(X, t ), θ(x, t ), X, t ) η(x, t) = ˆη(χ(X, t ), θ(x, t ), X, t ) (2.87a) (2.87b) (2.87c) (2.87d) where Ŝ, ˆQ, Ê, and ˆη are the constitutive functions (there should be more constitutive equations listed, but they were omitted for clarity), and where X denotes all positions in the body X B 0 and t denotes all times before time t, t t. There are eight axioms that dictate the determination of constitutive equations (Eringen, 1980). These axioms are as follows: Axiom of Causality: It is considered that the motion and the temperature of the material point in the deformable body are the only observable quantities, and
31 hence all other quantities can be expressed in terms of these variables. Axiom of Determinism: The values of thermo mechanical equations at a material point X and at time t are determined by the history of motion and temperature of all material points in the body B. Axiom of Equipresence: All the constitutive equations should be expressed in terms of the same variables, until the contrary has been deduced. Axiom of Objectivity: Constitutive equations must be form invariant with respect to the rigid rotations of the frame of reference. Axiom of Material Invariance: Constitutive equations must be form invariant with respect to a group of orthogonal transformations and translations of material coordinates. Axiom of Neighborhood: The values of independent constitutive variables at distant material points from X do not affect the value of constitutive variables at material point X. Axiom of Memory: The value of constitutive variables at distant past do not affect the value of constitutive variables at present. Axiom of Admissability: All constitutive equations must comply with conservation laws. Considering now only purely mechanical elastic materials with no memory, the heat flux and the heat generation rate are assumed to be zero. Hence, the system of equations to be solved reduces to 0 = t e ρ + div ( e ρ e v) (2.88a) e ρ e a = e ρ e f + div ( e σ) (2.88b) 0 = ε : e σ (2.88c) e ρ e Ė = e σ : e l (2.88d)
32 This means that there are 8 equations containing 14 unknowns, implying that 6 constitutive equations are necessary to complete the system. 2.5 Hyperelasticity In hyperelastic materials, the work done by the stresses during deformation in a continuous deformable body is dependent only on the initial configuration of the body at time t 0 and on the current configuration of the body at time t, and not on the deformation history. Hence, it can be said that hyperelastic materials are path independent. The internal work performed by the stresses can be described by a function, called the elastic strain energy potential, and denoted by W. This is a function of the internal strain of the body, which implies that this elastic strain energy potential function can be expressed as a function of Green Lagrange strain E. On the other hand, the conjugated stress of the Green Lagrange strain E, the second Piola Kirchhoff stress S, can be expressed as the partial derivative of the strain potential W with respect to the Green Lagrange strain E, or S = W (E) (2.89) E Considering that E = 1 (C G), the elastic strain energy potential W can then also 2 be expressed in terms of right Cauchy Green deformation tensor C as S = 2 W (C) (2.90) C In order to obtain an expression for the fully spatial equivalent of the second Piola Kirchhoff stress S, or the Cauchy stress σ, an inverse full Piola transform σ = J 1 F SF T has to be performed. Hence, the expression for the Cauchy stress becomes σ = J 1 F [ ] [ ] W W E (E) F T = 2 J 1 F C (C) F T (2.91)
33 which in components becomes σ ab = J 1 F a A [ ] [ ] W W (E) (F T ) B b = 2 J 1 F a A (C) (F T ) B b (2.92) E AB C AB If the second derivative of the elastic strain energy potential is taken with respect to the deformation, as it is necessary when linearizing the stress through a Taylor approximation, a fourth order tensor is defined as C = 2 W E 2 (E) = W 4 2 2 (C) (2.93) C and is called the material elasticity tensor. A full inverse Piola transformation can be used to define the spatial elasticity tensor as [ ] [ ] C = J 1 χ [C] = J 1 2 W χ E 2 (E) = 4 J 1 2 W χ C 2 (C) (2.94) 2.6 Growth and Remodeling In Figure 2.3, the motion of each particle from the reference configuration B 0 to the current configuration B(t) S can be split up into two contributions: the contribution due to elastic deformation, and the contribution due to unconstrained growth. Following this logic, the deformation gradient F can be decomposed as F = F e G (2.95) where F e is the purely elastic contribution to the deformation gradient, and it performs the mapping F e : T X B 0 T x S, while G is the growth tensor, which takes into account the contribution to the deformation gradient due to growth, and it performs the mapping G: T X B 0 T X B 0. It should be noted that since mass is preserved along the path mapped by F e, this tensor is not directly related to growth (Ambrosi & Mollica, 2002). Both tensors F e and G behave in the same way as deformation gradient F, as all three are simply mapping a tangent space onto a different tangent space.
34 Figure 2.3: Illustration of the motion from the original unstressed configuration B 0 to the current configuration B(t) S. In addition, the decomposition of the deformation gradient F into the elastic and growth components F e and G, respectively, has been demonstrated. By using a definition of the velocity gradient in equation (2.55), it is possible to express that velocity gradient l in terms of the decomposition of the deformation gradient F as l = Ḟ F 1 (2.96) = l e + F e l g F e 1 where l e = Ḟ e F e 1 and l g = Ġ G 1. Through the application of the first order constitutive framework, it can easily be proven that the rate of growth of the medium is defined as R = tr(l g ). 2.6.1 Conservation Laws To define the equations of motion for a growing body, an approach taken by DiCarlo & Quiligotti (2002) and Olsson & Klarbring (2008) has been followed. In order to apply
35 the principle of virtual power principle, used to define the equations of motion, it is first necessary to define the time processes that are occurring in the deformable body. Hence, the primary processes defined are the standard velocity v = t χ(x, t) and the rate of change of the growth tensor V = Ġ. Furthermore, it is also possible to define additional materials parameters m, with a time rate of w = ṁ, a change is which would cause the tissue to change, or remodel. It should be noted that there may be a number of such material parameters that are collectively written as m = (m 1, m 2,...), with a rate of w = (w 1, w 2,...). Thus, the complete set of velocities for this problem is (v, V, w), and therefore the set of virtual velocities corresponding to these kinematically admissible quantities is (ˆv, ˆV, ŵ). Taking these virtual velocities into account and defining the volume force b and traction t acting on the body, the external virtual power is then written as ˆP e = B 0 ] [b ˆv + y : ˆV + µα ŵ α dv 0 + t ˆv ds (2.97) B 0 where y are the configurational forces associated with growth, and µ α are the configurational forces associated with remodeling. Defining two force like tensors P and Y and a scalar M α, the internal virtual power in given by ˆP i = B 0 [P : ˆd + Y : ˆV + Mα ŵ α ] dv 0 (2.98) where P is the first Piola Kirchhoff stress tensor, l is the velocity gradient, Y is the force like tensor associated with growth, and M α is the internal force associated with each remodeling parameter. Now assuming quasi static conditions, and knowing that in
36 this case ˆP e + ˆP i = 0, the following conservation laws are obtained 0 = Div (P ) + b (2.99a) t = P N P F T = F P T y = Y (2.99b) (2.99c) (2.99d) µ α = M α (2.99e) It should be noted that these conservation laws are similar to those derived previously for a system with no growth or remodeling, with addition of two equations. these two equations deal with growth and remodeling forces, respectively, and they state that the internal forces are equivalent to external forces. In order to further define both growth and remodeling and introduce the equations that define the evolution, it is necessary to introduce general theory of constitutive equations for growing and remodeling media. 2.6.2 Constitutive Equations Assuming hyperelastic formulation and that the strain energy elastic potential is a function of both the elastic part of the deformation gradient F e and the material remodeling parameter m, the total strain energy in the body becomes ψ(f e, G, m) = det G W (F e, m) dv 0 = B 0 W (Fe, m) dv 0 B 0 (2.100) where W (F e, m) is the elastic strain energy potential in the reference configuration. The time derivative of the elastic strain energy potential then becomes [ W (F e, m) = det G G 1 : Ġ W (F e, m) + W (F e, m): F e + W ] F e m (F e, m) ṁ α α (2.101) Now applying the second law of thermodynamics, where ψ(f e, G, m) + P i 0, and
37 localizing the result yields [ det (G) W (F e, m) P G T F e [ ] : F e + det (G)G 1 W (F e, m) Y F e T P [ ] ] det (G) W m (F e, m) M α α ṁ α + (2.102) Ġ 0 After applying incompressibility criterion det F e = 1 and some algebraic manipulation, as well as several assumptions, the following equations are obtained P = p F e T G T + det (G) W F e (F e, m)g T (2.103) and [ ] [ ] det (G) W m (F e, m) M α α ṁ α + det (G)G 1 W (F e, m) Y F T e P Ġ 0 (2.104) The inequality in equation (2.104) is known as the reduced dissipation inequality, and to ensure that it is satisfied for all evolutions of the system it is necessary to introduce a convex dissipation potential ϕ = ϕ(ġ, ṁ), which allows for the tissue to be analyzed as a generalized standard material Olsson & Klarbring (2008). If the convexity conditions of the dissipation potential are satisfied, then the evolution equations become ϕ Ġ(Ġ, ṁ) = F e T P (det G)G T W (F e, m) + Y ϕ ṁ (Ġ, ṁ) = M α α (det G) W m (F e, m) α (2.105a) (2.105b) These equations are the kinematically admissible equations that define the evolution of growth and the remodeling of the tissue. These can now be applied to any geometry and any type of material, as required by the problem. 2.7 Directional Derivative In order to introduce the basics of non linear finite element method in the next chapter, it is necessary to introduce the directional derivative and a differential, respectively, as f f(x + hv) f(x) (x) = lim v h 0 h Df(v)[u](x) = f (x) u (2.106) v
38 which are used in the linearization procedure necessary to solve the non-linear equations of the continuum mechanics problem at study. It should be noted that the directional derivative obeys the usual properties of derivatives, as it is nothing more than a derivative along a direction given be a certain vector. This concludes an introductory chapter on the basics of Continuum Mechanics. Net chapter will address the basics of the finite element method, where this method will be introduced and discussed.
39 Chapter 3 Introduction to Finite Element Analysis Finite Element Analysis (FEA), also known as Finite Element (FE) Method, is a numerical method used to solve field problems, which require the determination of the spatial distribution of one or more dependent variables. An example of such a problem is finding the stress field inside a cantilevered beam, or finding a spatial distribution of temperature in a piston of an internal combustion engine. Mathematically, a field problem can be governed by both a set of differential equations as well as a set of integral equations, both of which can be used to formulate finite elements. The Finite Element Method originated from the need for solving complex elasticity and structural analysis problems in civil and aeronautical engineering. The earliest development of FEM can be traced back to the work of Alexander Hrennikoff (1896 1984) and Richard Courant (1888 1972), who adapted the earlier methods of Rayleigh, Ritz, and Galerkin for solving Partial Differential Equation (PDE), and discretized to continuous domain into a set of discrete sub domains, usually called elements (Pelosi, 2007). This approach was further developed by Olgierd Zienkiewicz (1921 2009), who formulated the mathematical foundation of what we today call Finite Element Method (Stein, 2009). Further development of FEM occurred throughout the 1950s and in early 1960s through the work of John Argyris (1913 2004) and Ray Clough (1920 ) as applied to airframe and structural analysis, and civil engineering (Cook et al., 2001). In fact, it was Ray Clough that coined the term Finite Element in 1960. In 1963, FEM gained widespread credibility when it was recognized as a form of Rayleigh Ritz method, and not just a special trick used for stress analysis. Today, FEM is applied to problems in wide variety of engineering disciplines such as heat modeling, electromagnetism, stress
40 analysis, and fluid dynamics (Cook et al., 2001). Since 1970s, an increase in computational power and speed made FEM a viable design tool. Previously, due to the complex nature and a large number of equations generated by finite element formulation, FEM was mostly used to just verify a design or study a structure that has already failed (Cook et al., 2001). Today, it is not uncommon for FEM simulations to reach a million, or more, degrees of freedom. Such large systems require a tremendous amount of computational power, which is fortunately easily available today. The first section of this chapter will address the general procedure for solving a problem using Finite Element Method, including the problem formulation and the problem solution. This will be followed by a basic discussion of the Finite Element principles, including the weak formulation for solid materials and biphasic materials, and the concepts of linearization and discretization. The chapter will be concluded with a brief overview of the Newton-Raphson iterative procedure and its application to the non linear Finite Element Method. For more details on the formulation of Finite Element method used in this thesis, the Reader can refer to Maas et al. (2010). 3.1 General Procedure for Solving a Problem by FEA Solving a practical problem by Finite Element Method requires an in depth knowledge of the problem, preparation of the mathematical framework describing the problem, discretization, solution, and the post processing. Most often, more than one cycle through these steps is required. Generally, time spent by the computer performing the calculation is only a fraction of the time spent by the user implementing and configuring the model, as the user must have an in depth understanding of what the computer is doing. The process of obtaining a solution through Finite Element analysis can be broken down into several steps. These steps are very broad, and they are generally adapted, or
41 even omitted, as required by the problem at hand. They are presented below in order to give a general understanding of how a problem would be solved using the Finite Element Method. Problem Classification In order to solve the problem using Finite Elements, the user must understand the nature of the problem. Without this step, a proper model cannot be developed and the software cannot be told what to do. At present time, the software does not automatically decide the type of problem at hand, nor the solution method that should be used to solve the problem. Even though the general trend is for the software to be given greater decision making ability, the user should not abdicate control and let the computer decide. Ultimately, a computer is just a tool with limitations that can result in errors, and it has to be told what to do. Therefore, this is possibly the most important step when undertaking Finite Element Analysis, as it requires inherent understanding of the problem at hand. Mathematical Model Before performing discretization and a numerical solution, it is necessary to devise a mathematical model that will govern the physics of the problem at hand. Since Finite Element models can get very complex, it is important to decide which features of the problem are important and which can be omitted, and the mathematical model(s) that govern the behaviour. Hence, the geometry may be altered slightly to ignore the geometric irregularities, the loads and constraints may be altered slightly to simplify calculation while preserving physics, etc. It is important to note that the Finite Element Method will only be based on the mathematical model implemented, which implies that the mathematical model is usually selected based on the information sought by the user.
42 Preliminary Analysis Before obtaining the numerical solution by the means of Finite Element Method, it in important to compute a preliminary solution by whatever means necessary, be it a simple analytical solution, a handbook formulas, or experiments. These results can then be used to compare and verify the general accuracy and fine tune the FE model. It is important to perform these calculations before obtaining results through FEA, as the tendency to find the solution that supports the FE model increases significantly if the preliminary solution is found afterwards. Finite Element Analysis Use of a general, commercially available FEA software involves the following three steps: Pre processing: This step involves the input of data describing geometry, material properties, loads, and physical constraints. This step also involves meshing, which is a procedure for discretization of the domain to be solved. Numerical Analysis: This step involves the solution of the discretized domain through a solution of the matrices describing the system. Generally, these matrices are very large and must be solved by a computer, sometimes requiring significant computing power. Post processing: This final step involved the presentation of the solution in either tabular of graphical form. The user generally selects the data that should be displayed as well as the way that the data is displayed. In most cases, the above steps will have to be cycled through several times before a satisfactory solution is obtained. In fact, it is recommended to start with the simplest possible FE model, and then add details step by step. This reduces a possibility of an error, and it allows the use to find an existing error easier than implementing the full model in a single step.
43 3.2 Weak Formulation for the Solid Materials In general, the Finite Element formulation is established in terms of a weak form of the differential equation defining the system. In the context of solid mechanics this implies the use of the weak form of the virtual work equation as δw = e σ : δd dv e b δ e v dv e t δ e v da = 0 (3.1) B(t) B(t) B(t) where δd is the virtual rate of deformation tensor, δ e v is the virtual velocity, and e σ, e b, and e t are the Cauchy stress, body force per unit of volume, and the traction vector, respectively. Equation (3.1) is known as the spatial virtual work equation, since this formulation involves only quantities in spatial reference frame. The material reference frame equivalent of equation (3.1), called the material virtual work equation, can be expressed as δw = S : δ Ė dv B 0 b δ v dv B 0 t δ v da = 0 B 0 (3.2) where b is the body force per unit of undeformed volume, b = J e b, and t is the traction vector per unit of initial area. 3.2.1 Linearization Equation (3.1) can be thought of as the starting point for the non linear Finite Element method. It is a highly non linear equation and any method attempting to solve this equation, such as the Newton Raphson method, must necessarily be iterative. In order to linearize the Finite Element equations, the directional derivative of the virtual work in equation (3.1) must be calculated using the definition in equation (2.106). In an iterative procedure, a certain field quantity ϕ will be approximated by a trial solution ϕ k, and the calculations will be repeated until satisfactory accuracy is achieved.
44 The linearization of the virtual work equation around this trial solution gives δw(ϕ k, δ e v) + d δw (ϕ k, δ e v)[u] = 0 (3.3) The directional derivative of the virtual work will eventually lead to the definition of the stiffness matrix. In order to simplify the computation, it is convenient to separate the virtual work into an internal and an external component as d δw (ϕ, δ e v)[u] = d δw int (ϕ, δ e v)[u] d δw ext (ϕ, δ e v)[u] (3.4) where and δw ext (ϕ, δ e v) = δw int (ϕ, δ e v) = B(t) B(t) e b δ e v dv + e σ : δd dv (3.5) B(t) e t δ e v da (3.6) After lengthy calculations, as shown by Bonet & Wood (2008), the linearization of the internal virtual work is given by d δw int (ϕ, δ e v)[u] = B(t) δd: C: ε dv + B(t) e σ : [ ( u) T δ e v ] dv (3.7) where C is the spatial elasticity tensor and ε is the small deformation strain. It should be noted that this equation is symmetric with respect to both δ e v and u, which implies that after discretization a symmetric elasticity tensor will be obtained. The external virtual work has contributions from both the body forces and the surface tractions. Hence, the precise form of the linearized external virtual work depends on the form of these forces. If only pressure forces are considered to contribute to the surface tractions, the linearized external work takes the form given by d δw p ext (ϕ, δ e v)[u] = 1 2 1 2 A ξ p e x ξ A ξ p e x η [( ) ( )] u η δ e e v v + η u dξ dη (3.8) [( ) ( )] u ξ δ e e v v + ξ u dξ dη
45 where η and ξ represent the tangential vectors forming an orthogonal basis, and p represents the hydrostatic pressure. It should also be noted that discretization of this equation will also lead to a symmetric elasticity tensor. On the other hand, if only simple body forces are considered, such as gravity, the contribution to the linearized external work is zero, since the force is independent of the geometry. 3.2.2 Discretization The basic principle of the Finite Element Method is that the domain of the problem is divided into smaller sub units, called the finite elements. In the case of isoparametric elements, it is further assumed that each element has a local coordinate system, named the natural coordinates, and the coordinates and shape of the element are discretized using the same function. The discretization process is established by interpolating the geometry in terms of the coordinates X a of the nodes that define the geometry of the finite element, and the shape functions given by X = n N a (ξ 1, ξ 2, ξ 3 )X a (3.9) a=1 where n is the number of nodes and ξ i are the natural coordinates. Similarly, the motion of the body is described in terms of the current position x a (t) of the same particles as x(t) = n N a (ξ 1, ξ 2, ξ 3 )x a (t) (3.10) a=1 Other field quantities, such as displacement, velocity, and virtual velocity, can also be discretized in a similar manner as that shown in equation (3.10). When deriving the discretized equilibrium equations for solid materials, the integration performed over the entire domain can be expressed as a sum of the integrations constrained to the domain of the element. Due to this, the discretized equations are defined in terms of integrations over a particular element denoted by superscript (h).
46 The discretized equilibrium equations for this particular element h are then given by δw (h) (ϕ, N a δ e v) = δ e v a ( T (h) a ) F (h) a (3.11) where T (h) a = e σ N a dv (3.12) B (h) (t) F (h) a = N e a b dv + N e a t da (3.13) B (h) (t) B (h) (t) Equation (3.11) can be further simplified by splitting it into two components, the internal and external virtual work, and then simplifying the equations for specific loading scenarios. The internal virtual work can also be further split into two components, the constitutive and initial stress component, and it is then given by d δw int (ϕ, δ e v)[u] = δd: C: ε dv + e σ : [ ( u) T δ e v ] dv (3.14) B (h) (t) = d δw (h) c B (h) (t) (ϕ, δ e v)[u] + d δw σ (h) (ϕ, δ e v)[u] The constitutive component can be discretized as ( ) d δw c (h) (ϕ, δ e v)[u] = e v a B T a D B b dv u b (3.15) B (h) (t) where B is the linear strain displacement matrix and D is the spatial constitutive matrix constructed from the components of fourth order spatial elasticity tensor C as D IJ = Cijkl where the relationship between components is given in Table 3.1. The term in parentheses in equation (3.15) defines the constitutive component of the tangent matrix relating node a to node b in element e as K (h) c,ab = B T a D B b dv (3.16) B (h) (t) where the small strain tensor ε can be expressed as ε = n B a u a (3.17) a=1
47 Table 3.1: Constructing spatial constitutive matrix from the components of fourth order spatial elasticity tensor using D IJ = Cijkl I/J i/k j/l 1 1 1 2 2 2 3 3 3 4 1 2 5 2 3 6 1 3 The initial stress components, defined as the second part of the internal virtual work in equation (3.14), is defined as d δw σ (h) (ϕ, δ N e a v a )[N b u b ] = ( N a eσ N b ) g dv (3.18) B (h) (t) After discretization, the pressure component of the external virtual work defined in equation (3.6) is defined as d δw p (h) (ϕ, δ N e a v a )[N b u b ] = δ e v K (h) p,ab u b (3.19) where where ε is the Levi Civita tensor, and k (h) p,ab = 1 2 + 1 2 p e x A ξ ξ p e x A ξ η K (h) p,ab = ε k(h) p,ab (3.20) [ Na η N b N ] b η N a dξ dη (3.21) [ Na ξ N b N ] b ξ N a dξ dη 3.3 Newton Raphson Method The Newton Raphson method, also known as the Newton Method or the Full Newton Method, is the basis for solving non linear Finite Element equations. The following sections will describe the Full Newton method, the Broyden Fletcher Goldfarb Shanno
48 (BFGS) method, and a Line Search method, as well as the advantages and disadvantages of each method (Matthies & Strang, 1979). 3.3.1 Full Newton Method The Full Newton equation can be written in terms of the discretized equilibrium equations as δ e v T K u = δ e v T r (3.22) where K is the stiffness matrix and r is the residual vector. Since the virtual velocities δ e v are arbitrary, a discretized Full Newton scheme can be reformulated as K(x k ) u = r(x k ); x k+1 = x k + u (3.23) This is the basis of the Full Newton method. For each iteration k, both the stiffness matrix and the residual vector are re evaluated and a displacement increment u is calculated by pre multiplying both sides of equation (3.23) by an inverse of the stiffness matrix. This procedure is repeated until the convergence criteria is satisfied. The formulation of the stiffness matrix, as well as the calculation of its inverse, are computationally expensive and complex tasks. This is the main drawback of the Full Newton method, as for the complex systems the calculations require significant processing power. On the other hand, Quasi Newton methods do not require re evaluation of he stiffness matrix for every iteration. Instead, a quick update is calculated. One of the more successful such methods in the field of computational solid mechanics has been the BFGS method, which is described in detail in the next section.
49 3.3.2 BFGS Method The BFGS method updates the stiffness matrix, as well as it s inverse, to provide an approximation to the exact matrix. A displacement increment is thus defined as d k = x k x k 1 (3.24) and the increment in the residual vector is defined as g k = r k 1 r k (3.25) The updated stiffness matrix K k should then satisfy the Quasi Newton equation given by K k d k = g k (3.26) In order to perform the update using the BFGS method, the displacement increment is first calculated using u = K 1 k 1 r k 1 (3.27) This displacement vector defines a direction for the actual displacement increment. A Line Search method, which is discussed in the next section, can then be applied to determine the optimal displacement increment using a relationship given by x k = x k 1 + su (3.28) where s is determined from line search. With the updated position calculated, the residual vector r k can be evaluated. The stiffness update can then be expressed as K 1 k = A T k K 1 k 1 A k (3.29) where the matrix A is an n n matrix of the form A k = I + v k w T k (3.30)
50 where I is the identity matrix and the vectors v and w are given by [ d T ] 1/2 k g v k = k K k 1d k g k (3.31) K k 1 d k d T k w k = d k d T k g k (3.32) In order to avoid numerically unstable updates, a condition number c of the updating matrix A is calculated using the relationship [ d T ] 1/2 k g c = k (3.33) K k 1 d k d T k If this condition number exceeds a certain preset value, an update is not performed as it could result in computational errors. Therefore, the BFGS method offers several advantages when compared to the Full Newton method, the foremost being lower computational cost due to the fact that an update of the stiffness matrix is not necessary for every step. In order to make sure that this does not affect the accuracy or the speed of convergence, BFGS method is often used in conjunction with the Line Search Method, which is a powerful technique used to speed up convergence of Newton based methods. The Line Search method is described in more detail in the next section. 3.3.3 Line Search Method As Previously mentioned, Line Search method is a powerful technique used to speed up the convergence of Newton based methods. In this method, the direction of the displacement vector u is considered as optimal, but the magnitude of this vector is controlled by a parameter s, which implies that the increment is then given by x k+1 = x k + su (3.34) The value of parameter s is most often chosen so that the potential energy W(s) = W(x k + su) at the end of the iteration is minimized in the direction of the displacement
51 vector. This is equivalent to the requirement that the residual vector r(x k + su) at the end of the iteration is orthogonal to u, implying that R(s) = u T r(x k + su) = 0 (3.35) However, due to numerical and geometrical approximations, it is sufficient to obtain a value of parameter s such that R(s) < ρ R(0) (3.36) where ρ is typically given a value of 0.9. Under normal conditions, the value s = 1 automatically satisfies equation (3.36), which means that few extra operations are involved. However, when this is not the case, a more suitabe value for parameter s needs to be obtained. In this case it is convenient to approximate R(s) as a quadratic function of s given by R(s) (1 s)r(0) + R(1)s 2 = 0 (3.37) which yield a solution for parameter s as s = R(0) 2R(1) ± ( ) 2 R(0) R(0) 2R(1) R(1) (3.38) If r < 0, the quare root is positive and a first improved value for parameter s is obtained as ( s 1 = R(0) ) 2 R(0) 2R(1) ± R(0) 2R(1) R(1) (3.39) On the other hand, if r > 0, then the parameter s can be obtained by using a value that minimizes the quadratic function, i.e., s 1 = r/2. This procedure is now repeated with R(1) replaced by R(s 1 ) until equation (3.36) is satisfied.
52 Chapter 4 Composition and Structure of Articular Cartilage Cartilage is a flexible, avascular connective tissue found in the bodies of all vertebrate species, specifically is the joints, the rib cage, ear, nose, and intervertebral disks among others. It can be classified into three types, which differ in their biochemical composition, molecular microstructure, biomechanical properties, and biological and mechanical functions (Martini, 2005): Elastic Cartilage: This type of cartilage is present in outer ear, larynx, and epiglottis. The principal protein is elastin, hence this type of cartilage is more flexible than the others and able to sustain repeated bending without damage. Fibrocartilage: Fibrocartilage is found in the meniscus, the annulus fibrosus, and the temporo mandibular joint. Due to the high volumetric of fibres, it is very tough and is able to resist high loads. Hyaline Cartilage: This type of cartilage owes its name to its glass like appearance, and it constitutes the majority of cartilage in the human body. It can be found at the articulating ends of the bones, the ventral ends of the ribs, and in the larynx, trachea, and bronchi. The term articular cartilage refers to hyaline cartilage that lines the articulating ends of long bones in diarthrodial joints, as shown in Figure 4.1. The primary function of articular cartilage is to provide joints with an articulating surface having both low friction as well as high wear resistance, and that allows for relative bone to bone movement while supporting high compressive loads over the lifetime (Flik et al., 2007). At this moment in time, such unique and astonishing properties are unmatched by any synthetic material.
53 Figure 4.1: Schematic representation of a knee joint, which is a type of diarthrodial joint, with relevant components labeled. It can be seen that articular cartilage, shown in gray, lines the ends of articulating bones and prevents bone on bone articulation. The properties of articular cartilage are entirely due to its unique composition and structure. Hence, a basic understanding of said structure and composition is an essential step towards understanding its mechanical behavior. In the next section, the articular cartilage composition and molecular properties will be discussed in detail, while the following section will address the global depth dependent structure and the cell level architecture. It should be noted that the material properties for articular cartilage given in the subsequent sections are for humans; the material properties for articular cartilage in other animals may be substantially different. For a more detailed treatment of the composition and structure of articular cartilage, the Reader can refer to Athanasiou et al. (2009), Flik et al. (2007), or Mow & Hung (2001), as this topic is beyond the scope of this thesis.
54 4.1 Composition Articular cartilage is a white, dense tissue, with a firm, smooth, and slippery surface (Higginson et al., 1976), layering the articulating ends in diarthrodial joints, with a thickness that, in humans, can range from 1 mm to 5 mm. In biomechanical terms, articular cartilage is a complex, fibre reinforced material with non linear behaviour, that is made up of two phases: a solid organic matrix, consisting of roughly 20 25% of the total volume, and an interstitial fluid phase, consisting of roughly 75 80% of the total volume (Lipshitz et al., 1976; Maroudas, 1979; Stockwell, 1979; Mow et al., 1984; Wu & Herzog, 2002; Flik et al., 2007). The solid organic matrix is composed of approximately 65% collagen and 25% proteoglycans; the remaining 10% consists of glycoproteins, chondrocytes (cartilage cells), and lipids. The proteoglycans and collagen macromolecules are the primary structural and load bearing components (Flik et al., 2007), while the chondrocytes play an important role in overall cartilage health by synthesizing the extracellular matrix (Stockwell, 1979). 4.1.1 Fluid Water is the main fluid component in articular cartilage and in the synovial fluid that surrounds the articular cartilage and fills the synovial cavity. As previously mentioned, water accounts for roughly 75 80% of the total tissue volume, and this water volume is most concentrated near the articulating surface (approximately 80% volumetric fraction), and it decreases in a nearly linear fashion with increasing depth to a concentration of roughly 75% in the deep zone (Lipshitz et al., 1976; Maroudas, 1979; Mow & Hung, 2001). Articular cartilage is also a fully saturated tissue, where the interstitial fluid saturates and flows through the pores of the extra cellular matrix. It should also be noted that the interstitial fluid plays three important roles in the tissue, the first being the transport of nutrients and minerals through the tissue. In fact, the interstitial fluid
55 can be characterized as the primary carrier in the articular cartilage, due to the avascular nature of the tissue (Maroudas, 1979). The second role of the interstitial fluid in articular cartilage is to provide the tissue with high compressive strength. This effect results from the low permeability of the extracellular matrix, which is the result of the high frictional resistance to the flow within the matrix due to small pore sizes (Mow & Hung, 2001). The third important role of the interstitial fluid in articular cartilage is to provide lubrication and reduction of friction between surfaces during joint articulation. The frictional coefficient for cartilage on cartilage articulation is 0.005, which is lower than any synthetic material (Ateshian & Mow, 2005). 4.1.2 Collagen Collagen fibres are the primary components of connective tissues throughout the human body, and are the most abundant proteins in mammals. The primary function of collagen fibres in different tissues is to provide structural integrity, as it is a rather stiff and hard protein (Meyers et al., 2008). The structure of collagen is such that it is comprised of repeating amino acid sequences (glycine, proline, hydroxyproline, etc.) that exhibit a characteristic triple helix structure known as tropocollagen that self aggregates to firm collagen fibres. The hierarchical structure of collagen fibres is shown in Figure 4.2. In articular cartilage, collagen fibres are synthesized by the chondrocytes and are inhomogeneously distributed, giving the tissue a layered character (Ateshian & Mow, 2005), as illustrated in Figure 4.3. The layered character of the tissue will be addressed later on when the overall structure of the tissue is discussed, but at this point it is sufficient to say that the collagen fibres are the major components of articular cartilage dry weight, and that this dry weight varies with depth (Mow et al., 1992). Numerous types of collagen fibres are found in articular cartilage, including types II, V, VI, IX, X, and XI (Flik et al., 2007), however type II is predominant and it accounts
56 Figure 4.2: Hierarchical microstructure of the collagen fibre. Alpha chains, composed of repeating amino acid sequences, self aggregate into a triple helix, which in turn forms a collagen fibres. It should be noted that there are small gaps in the collagen fibres between the tropocollagen molecules, but these are accounted for through the quarter stagger array, as illustrated above. Adopted from Mow et al. (1992). for 90 95% of all collagen in this tissue (Flik et al., 2007; Athanasiou et al., 2009). Type II collagen is the collagen type that is responsible for the structural characteristics of the articular cartilage tissue, as it forms an interconnected fibre network that has a depth dependent orientation. While the definite roles of the other collagen types are not fully known, they are thought to play a role in intermolecular interactions and modulating the structure of type II collagen (Athanasiou et al., 2009). 4.1.3 Proteoglycans Proteoglycans are large macromolecules synthesized by the chondrocytes and composed of a protein core with attached polysaccharide side chains known as glycosaminoglycans (GAGs). Proteoglycans make up approximately 10 15% of the wet weight of articular cartilage (Flik et al., 2007), and Figure 4.3 illustrates the depth dependence of proteoglycan content. The primary proteoglycan in articular cartilage is called aggrecan, and it consists of a hyaluronan core with numerous negatively charged chondroitin and keratan sulfate GAG side chains (Flik et al., 2007; Athanasiou et al., 2009). The basic structure of an aggrecan molecule is shown in Figures 4.4, 4.5, and 4.6,
57 Figure 4.3: Collagen and proteoglycan content varies with depth from articular surface in articular cartilage. Image from Athanasiou et al. (2009). whereas Figures 4.5 and 4.6 further illustrates how the aggrecan molecules in cartilage bind to a single log chain of hyaluronan to form large proteoglycan aggregates. These proteoglycan aggregates can then be imagined a mesh type matrix that is interlaced through the collagen structure (Mow & Hung, 2001). The large size of this mesh type of matrix immobilizes and restrains it within the collagen network, as shown in Figure 4.7, while the carboxyl and sulfate groups give it a negative charge known as the fixed charge density (Athanasiou et al., 2009). It is due to this fixed charge density that the proteoglycan matrix absorbs interstitial fluid, resulting in the overall swelling of the tissue. This swelling stretches the collagen fibre network, until the balance between the swelling force and the tensile force in the collagen network is reached. The fixed charge density is also responsible for the overall cartilage stiffness in compression (Mow & Hung, 2001; Flik et al., 2007; Athanasiou et al., 2009).
58 Figure 4.4: The tube brush like structure of aggrecan is due to negatively charged keratan and chondroitin sulfate molecules, which due to negative charge repel each other. Image from Athanasiou et al. (2009). Other proteoglycans within articular cartilage include biglycan, decorin, and fibromodulin, which are also composed of core proteins with various GAGs attached as side chains (Roughley & Lee, 1994). As with the minor collagen types, the precise function of these proteoglycans are not fully known, but it is believed that they assist in the matrix assembly by associating with the collagen structure during development and repair. 4.1.4 Chondrocytes Articular cartilage has a low cell density when compared to other biological tissues. In fact, it contains only one type of cell, the chondrocyte, which literally means cartilage cell in Greek (Athanasiou et al., 2009). The diameter of chondrocytes ranges from 10µm to 13µm (Aydelotte & Kuettner, 1988), and they occupy generally less than 10% of the total volume in articular cartilage. It is also important to note that the chondrocyte volumetric fraction is depth dependent and decreases with depth from a maximum at the articulating surface (Clark et al., 2003).
59 Figure 4.5: Schematic depiction of aggrecan, which is composed of keratan sulfate and chondroitin sulfate chains bound covalently to a protein core molecule. The proteoglycan protein core has three globular regions as well as keratan sulfate rich and chondroitin sulfate rich regions. Adapted from Mow & Hung (2001). Chondrocytes are an integral part to the growth and maintenance of articular cartilage, as they are responsible for the synthesis of the various proteins that compose the extracellular matrix, such as collagen, proteoglycans, and various other non collagenous proteins (Stockwell, 1979; Athanasiou et al., 2009). The biosynthesis activity of the chondrocytes is often influenced by various environmental stimuli, such as growth factors, interleukins, and the mechanical environment. Altercation of these environmental stimuli, often occurring due to injury or disease, leads to articular cartilage adaptation, and in many cases degeneration (Flik et al., 2007). 4.1.5 Other non collagenous proteins In addition to the presence of collagen and proteoglycans, articular cartilage also contains a small fraction of non collagenous proteins. These include fibronectin, cartilage oligomeric protein, thrombospondin, tenascin, chondrocalcin, and superficial zone protein (Guilak et al., 2000; Athanasiou et al., 2009). The precise function of these proteins is not fully known at the present time and it is being investigated.
60 Figure 4.6: Schematic representation of a proteoglycan macromolecule. In the matrix, In the matrix, aggrecan non covalently bonds to hyaluronan to form a macromolecule, whereas this binding is stabilized through a link protein. Adapted from Mow & Hung (2001). 4.2 Structure Based on the previous section, it can be seen that articular cartilage composition is depth dependent. This logically leads to the assumption that the structure of articular cartilage is also depth dependent, which turn out to be the case. In fact, articular cartilage is most often described in terms of a zonal architecture that uses the depth from the articulating surface as a reference frame. Figure 4.8 shows this zonal arrangement in terms of chondrocyte shape and arrangement, as well as the collagen fibre orientation. These structural characteristics, in addition to the variations in articular cartilage composition, are the primary identifiers by which cartilage zones are identified. Articular cartilage is divided into three zones that extend from the articular surface to the tidemark at the start of the calcified zone, which marks the transition from cartilage to bone. These three zones have names that are based primarily on the collagen fibre orientation, and they are the superficial (or tangential) zone, middle (or transitional) zone, and the deep (or radial) zone. The calcified zone is the zone immediately below
61 Figure 4.7: Schematic representation of the molecular organization or articular cartilage. The structural components of cartilage, collagen and proteoglycans, interact to form porous composite fibre reinforced organic matrix that is swollen with interstitial fluid. Adapted from Mow & Hung (2001). the tidemark, and it represents the transitional area from articular cartilage to the subchondral bone (Mow et al., 1991). Figure 4.9 further illustrates this zonal arrangement of articular cartilage. It should also be noted that there exits large variations in this zonal architecture of articular cartilage that tie in directly to the overall function and loading, which is usually based on the location in the body and the joint in question (Athanasiou et al., 2009). The articulating surface of articular cartilage is covered by a very thin layer called the lamina splendens (Kumar et al., 2001). This acellular, non-fibrous region has an average thickness ranging from several hundred nanometers to a few micrometers, and its precise role is still not well known. Various hypotheses have suggested several possibilities for the function, or even existence, of this tissue, ranging from the facilitation of low friction and protection of the cartilage surface against wear to it being nothing more than a visual artifact that results from imaging and image processing (Athanasiou et al., 2009).
62 Figure 4.8: The architecture of articular cartilage is divided into zones identified by the chondrocyte cell shape and orientation, left, and collagen fibre orientation, middle. Adapted from Athanasiou et al. (2009). 4.2.1 Superficial Zone The superficial zone of articular cartilage extends from the articulating surface and ranges from 10 20% of the total tissue thickness. It is characterized by having small diameter, densely packed collagen fibres oriented parallel to the articulating surface (Meachim & Roy, 1969). The extra cellular matrix in this zone has a relatively low proteoglycan content as well as a low permeability (Muir et al., 1970; Maroudas, 1979). Chondrocyte cells are densely packed, exhibit a flattened shapes, and are oriented along the neighboring collagen fibres in the tangential direction (Eggli et al., 1988; Clark et al., 2003). Superficial zone chondrocyte cells also secrete specialized proteins that are thought to facilitate wear and frictional properties of the tissue in this zone (Flannery et al., 1999). 4.2.2 Middle Zone The middle zone occupies approximately 40 60% of the total tissue thickness. In this zone, the collagen fibres exhibit a random fibre orientation (Hunziker et al., 1997), while the proteoglycan content reaches a maximum in this zone (Venn & Maroudas, 1977), as illustrated in Figure 4.3. Chondrocyte cell density is significantly lower in this zone
63 Figure 4.9: The architecture of articular cartilage is divided into four different zones, each with different structural arrangement and hence different material properties. Image from Mow et al. (1991). than in the superficial zone, and the cell shape is close to spherical (Eggli et al., 1988; Athanasiou et al., 2009). 4.2.3 Deep Zone The deep zone is the last region of purely hyaline cartilage before reaching the tidemark, and it occupies the bottom 20 70% of the total tissue thickness (Maroudas, 1979). The collagen structure of the deep zone is characterized by large fibres that form bundles oriented perpendicular to the articular surface, and that subsequently anchor to the subchondral bone in the calcified zone (Muir et al., 1970). The proteoglycan content is significantly lower than in the middle zone (Venn & Maroudas, 1977), as shown in Figure 4.3, and the chondrocyte cell density is also the lowest compared to the three purely cartilaginous zones (Stockwell, 1971).These chondrocytes are slightly elongated in the direction of fibre orientation, and are generally arranged in vertical columns (Guilak,
64 1995; Buckwalter & Mankin, 1997). 4.2.4 Calcified Zone The region that extends from the tidemark to the subchondral bone is called the calcified zone. The purpose of this transitional region is to minimize the stiffness gradient between the elastic hyaline cartilage and rigid subchondral bone (Radin & Rose, 1986). Underlying this region of cartilage is the subchondral bone, which is the ultimate anchor point for articular cartilage as a whole. The overview of articular cartilage composition and structure presented in this chapter has attempted to provide a basic insight into complex composition and structure which characterize articular cartilage as a biological tissue. Since it has been noted that articular cartilage is highly anisotropic and inhomogeneous from both the compositional and the structural points of view, it is safe to assume that the mechanical behavior of articular cartilage is also highly anisotropic and inhomogeneous. The next chapter will discuss how the compositional and structural anisotropy affects the overall mechanical function and properties of cartilage, and different material models that attempt to take this anisotropic behavior into account.
65 Chapter 5 Behaviour, Testing, and Material Models of Articular Cartilage As previously mentioned, the primary function of articular cartilage is to transmit loads between the articulating ends of bones in diarthrodial joints while minimizing joint wear during loading that varies in both rate and magnitude. Since articular cartilage, along with numerous other tissues, has evolved over millions of years, it is safe to assume that the anisotropic composition and structure of this tissue is optimized for performance and longevity under the mechanical conditions that it experiences. In order to present the historical development of articular cartilage material models, as well as the model implemented in this work, it is first necessary to discuss the mechanical behaviour of cartilage under different loading conditions. This will then be followed by a discussion of various tests that can be conducted in order to determine the parameters describing the mechanical behaviour, and lastly a number of significant cartilage material models will be presented, starting with the earliest and concluding with the model implemented in this work. 5.1 Mechanical Behaviour This section will first discuss the viscoelastic nature of articular cartilage, including the behaviour under imposed loading and displacement as well as the evidence of its viscoelastic nature. Following the discussion of viscoelasticity, the behaviour of articular cartilage under three different loading scenarios (compression, tension, and shear) will be discussed.
66 5.1.1 Viscoelasticity As previously discussed, articular cartilage is a remarkable material with a host of unique properties. One of the most fundamental of these properties is its ability to store and dissipate mechanical energy during various activities, which is important to its role of efficiently transmitting loads while minimizing joint wear. This type of behaviour, called viscoelasticity or viscoelastic behaviour, is demonstrated in a number of different loading scenarios, namely tensile (Woo et al., 1980), compressive (Mow et al., 1980), and shear testing (Hayes & Mockros, 1971). There exist two fundamental responses of a viscoelastic material, and these are creep and stress relaxation. Creep occurs when a viscoelastic material is subjected to a constant loading. The typical response under such conditions is a rapid initial deformation followed by a slow, progressively increasing deformation known as creep, as shown in Figure 5.1. This time dependent, creeping deformation will occur until the equilibrium state has been reached. On the other hand, stress relaxation occurs when a viscoelastic material is subjected to a constant deformation. The typical response in this case would be a high initial stress followed by a slow, progressively decreasing stress required to maintain the deformation (Mow & Hung, 2001), as illustrated in Figure 5.2. Figure 5.1: Typical creep behaviour of articular cartilage under step loading. A step load, or force, is applied and then held constant. While the load is held constant, the displacement values change until equilibrium is reached. Illustration adapted from Lu & Mow (2008).
67 Figure 5.2: Typical stress relaxation behaviour of articular cartilage under applied displacement. The displacement is first applied as a ramp input and then, when desired displacement magnitude has been reached, held constant. The peak loading occurs at the end of the displacement ramp, which is followed by a period of stress dissipation until equilibrium has been reached. Illustration adapted from Lu & Mow (2008). The viscoelastic behaviour of articular cartilage arises from both a flow independent and a flow dependent mechanism (Mak et al., 1987). As the name implies, flow independent contribution measures the intrinsic viscoelasticity of the solid matrix (Setton et al., 1993). In articular cartilage, the flow independent contribution to viscoelastic behaviour is primarily a result of physical entanglements, intermolecular friction and chemical and electrostatic interaction between long molecular chains, such as collagen and proteoglycans (Setton et al., 1993; Zhu et al., 1996). The flow independent contribution is generally measured using shear testing, as this type of test does not apply a pressure gradient and therefore does not result in interstitial flow (Hayes et al., 1972). The flow dependent mechanism has shown to be the dominant mechanism responsible for the observed viscoelastic behaviour of articular cartilage (Mow & Lai, 1980), and it is the result of the flow of interstitial fluid through the porous matrix and the drag associated with this flow (Ateshian et al., 1997). This contribution is usually measured under conditions of confined compression, as this type of test minimizes the viscoelastic behaviour due to flow independent contributions (Ateshian et al., 1997).
68 5.1.2 Compression Compressive loading is one of the primary types of mechanical stresses experienced by articular cartilage. This type of loading is primarily governed by the movement of fluid through the interconnected pore structure of the extra cellular matrix (Athanasiou et al., 2009). Due to the structure and negative charge of the proteoglycan chains in the extracellular matrix, the brush like keratin and chondroitin sulfate molecules produce significant frictional resistance to fluid flow, resulting in low permeability. Consequently, high pressure is needed to overcome this high friction and produce fluid flow (Mow et al., 1984; Athanasiou et al., 2009). Since the extra cellular matrix permeability is low, the fluid pressure caused by the low permeability initially supports majority of the applied load (Maroudas & Bullough, 1968). As the majority of loading that articular cartilage experiences is dynamic loading of short duration (e.g., loading on cartilage during walking), more than 90% of load experienced during normal activities is supported by the interstitial fluid (Flik et al., 2007). This phenomenon is known as stress shielding of the extra cellular matrix. If the applied load is sustained and as the interstitial fluid starts to exude from the extra cellular matrix into the synovial cavity, the porous matrix begins to support more and more of the applied load, as illustrated if Figure 5.3. As a greater portion of the applied load is supported by the extracellular matrix, the proteoglycan chains are driven closer together, which results in further resistance to compression due to the repulsion of negatively charged keratin and chondroitin sulfate groups (Flik et al., 2007). The equilibrium stiffness of extracellular matrix in confined compression is primarily caused by this mechanism. A cartilaginous tissue with higher proteoglycan content will in general have higher compressive stiffness (Kempson et al., 1970). Hence, it is intuitive to conclude that the compressive stiffness also increases with depth from the articular surface (Schinagl
69 Figure 5.3: Initially the applied load is supported almost exclusively by the interstitial fluid L fluid, while the extracellular matrix, L matrix, supported only a small portion of the load. As the fluid exudes from the articular cartilage, the extra cellular matrix begins to support a greater portion of the load. As time approaches infinity, the load supported by the interstitial fluid essentially approaches zero. Illustration adapted from Athanasiou et al. (2009). et al., 1997; Chen et al., 2001), however this relationship is not linear and likely it depends on something more than just the proteoglycan content (Chen et al., 2001). Other factors, such as physical entanglement and mechanical friction between the protein chain molecules, likely contribute to this depth dependent compressive stiffness increase, especially in the deep zone where the solid fraction is at its maximum. As previously mentioned, compression and recovery of articular cartilage occurs at the micro scale level during normal joint loading since the interstitial fluid support majority of the applied load (Athanasiou et al., 2009). However, over the course of a day, the bulk cartilage tissue is compressed slightly compared to its initial state, and the resulting compressive strains are 15 20% (Armstong et al., 1979). On the other hand, a good period of inactivity (e.g., a night s sleep) will result in full recovery of the tissue.
70 5.1.3 Tension The mechanical behaviour of articular cartilage in tension is highly complex. In tension, the tissue is highly anisotropic, with different material properties parallel and perpendicular to the collagen fibres, and inhomogeneous, with greater tensile stiffness in the superficial zone than in the deep zone (Kempson, 1979; Roth & Mow, 1980). As the articular cartilage tissue is loaded in tension, the collagen fibres within the extra cellular matrix align and stretch along the axis of loading (Athanasiou et al., 2009). This change in the fibre orientation, in addition to the intrinsically viscoelastic nature of the interactions between collagen fibres and the surrounding proteoglycan matrix, lead to non linear tensile behaviour. A sample stress strain curve for articular cartilage tissue in a rate controlled tensile experiment is shown in Figure 5.4. In the toe region, the collagen fibres are realigning in the direction of the applied load, resulting in relatively low tensile modulus. As tensile strain increases, the fibres themselves begin to stretch, which results in a significant increase in the tensile stiffness (Roth & Mow, 1980). Figure 5.4: General stress strain curve for cartilage under tension. This type of tensile behaviour is common in many biological tissues that are composed of a large number of polymeric chains. The illustrations on the right show collagen fibres at various stages of loading. Illustration adapted from Mow & Hung (2001).
71 The anisotropic and inhomogeneous characteristics of articular cartilage are believed to be caused by the varying collagen and proteoglycan structural organization of the joint surface and the layering structural arrangement found within the tissue. Hence, as previously mentioned, the superficial zone exhibits the highest tensile stiffness, while the deep zone has the lowest tensile stiffness when the loading is applied in the direction perpendicular to the articulating surface (Setton et al., 1993; Mow & Hung, 2001). The reason for such a behaviour is that in the superficial region, a high volumetric fraction of collagen fibres are oriented in the direction of the applied load, therefore supporting that loading. This volumetric fraction decreases with depth in the tissue, resulting in lower tensile stiffness. Articular cartilage also exhibits viscoelastic behaviour in tension (Woo et al., 1987). The viscoelastic behaviour is attributable to both the internal friction associated with polymeric motion, as well as the flow of the interstitial fluid. During unconfined compression experiments, the collagen fibre network contributes significantly to the overall articular cartilage stiffness as the tissue experiences positive, tensile strains in the direction perpendicular to the direction of loading (Roth & Mow, 1980). 5.1.4 Shear Articular cartilage is usually subjected to the shearing deformation in the direction perpendicular to the articulating surface during rotational and translational movements in the joint (Athanasiou et al., 2009). Physically, pure sheer deformation causes no volume change in the tissue, and therefore the fluid flow is assumed to be zero (Hayes & Mockros, 1971). As a result of this, fluid dependent mechanisms do not play a role in the mechanical properties of articular cartilage during pure shear. The proteoglycan matrix and the collagen fibre network both contribute to shear properties of articular cartilage, however it is often assumed that the shear stiffness is
72 primarily influenced by the tensile properties of collagen fibre network (Zhu et al., 1993). This type of behaviour is further illustrated in Figure 5.5. Figure 5.5: A schematic representation of the structure of cartilage matrix in unloaded condition (left) and in pure shear (right). The tension of collagen fibres provides the shear stiffness. Illustration adapted from Zhu et al. (1993). 5.1.5 Permeability Permeability is a measure of the ease with which the fluid flows through a porous medium, and it is inversely proportional to the frictional drag force exerted by the porous medium onto the interstitial fluid. It is a property that is strongly dependent on the micro structure of the porous matrix, and it basically represents a measure of the resistive force required to cause a fluid flow of certain speed through a permeable medium, which in the case of articular cartilage is the extra cellular matrix (Mow & Hung, 2001). The permeability of articular cartilage is typically obtained by either curve fitting a biphasic model to compression test data, or by a direct measurement in a flow test, as illustrated in Figure 5.6. The low permeability of the articular cartilage has been associated with the proteoglycan matrix, or more specifically the brush like structures of the glycosaminoglycan macromolecules, which give rise to a significant frictional drag (Maroudas, 1968, 1976). It is also thought that the collagen fibre orientation plays a role in cartilage permeability,
73 Figure 5.6: Experimental setup used in measuring permeability of articular cartilage, involving the application of a pressure gradient across a tissue sample of thickness h. As the fluid pressure above the sample (P 1 ) is greater than the fluid pressure below the sample (P 2 ), fluid will flow through the tissue. By applying Darcy s law, the permeability of the tissue can easily be found. Illustration adapted from Mow & Hung (2001). specifically in the superficial zone, by directing the fluid flow (Maroudas & Bullough, 1968), and ultimately determining an anisotropic behaviour. It has been observed that the permeability of articular cartilage is highly dependent on the compressive strain and the pore pressure experienced by the tissue (Mansour & Mow, 1976), as shown in Figure 5.7. The study by Mansour & Mow (1976) was also the first instance where the permeability of articular cartilage has been found under compressive strain and at high physiological pressures, which closely resemble conditions found in diarthrodial joints during loading. The behaviour of the articular cartilage tissue sample considered in the experiment by Mow & Hung (2001), as shown in Figure 5.7, can be explained by the increased drag forces that arise when fluid velocity increases due to increased applied pressure and the increased drag forces that arise when fluid flows through a matrix with decreased pore size due to increased compressive strain, both of which result in decreased hydraulic permeability. In other words, when subjected to higher applied pressure or higher compressive strain, the tissue will appear stiffer and more resistant to fluid exudation. Several different analyses
74 Figure 5.7: Experimental curves for articular cartilage permeability demonstrate strong dependence on applied fluid pressure and compressive strain. Hydraulic permeability decreases with increasing fluid pressure and with increasing compressive strain. Illustration adapted from Mow & Hung (2001). of stress relaxation behaviour of articular cartilage have validated this concept and its importance in the capacity of interstitial fluid to support load (Ateshian et al., 1998; Soltz & Ateshian, 1998). 5.2 Mechanical Testing Methods The most common test methods used to determine the mechanical properties of articular cartilage are the unconfined compression, confined compression, and indentation. Combined, these test methods are used to determine most of the mechanical properties of cartilage tissue. In the next three subsections, these three techniques will be outlined in greater detail.
75 5.2.1 Unconfined Compression In the unconfined compression test, articular cartilage undergoes negative strain in the direction of the applied load and positive strain in the directions perpendicular to the direction of applied load. Due to the structure of articular cartilage, this type of loading results in both compressive and tensile stresses within the tissue. The proteoglycan matrix primarily resists negative strains in the loading direction, while the collagen fibre network resists the positive strains perpendicular to the loading direction. A schematic representation to this test is shown in Figure 5.8. Figure 5.8: A schematic representation of the experimental apparatus for an unconfined compression test. Generally the cartilage must be removed from the bone in order to perform the test. Illustration adapted from Mow & Guo (2002). Unconfined compression is a widely applied test, mostly to determine the intrinsic equilibrium compressive Young s modulus E through equilibrium stress strain response of the tissue, and the Poisson s ration ν through a direct measurement of the lateral expansion by using an optical method (Armstong et al., 1984). In addition to these two material coefficients, the hydraulic permeability k can be determined by fitting the stress relaxation or creep response curves of the loaded specimen to the biphasic constitutive theory (Mow et al., 1980).
76 5.2.2 Confined Compression The confined compression test generally results in an overall compressive state of stress in the articular cartilage tissue. Since the cartilage sample is laterally constrained and not allowed to expand, the effect of collagen fibres in tension is eliminated. This allows for determination of the equilibrium uniaxial strain elastic modulus, usually called the aggregate modulus H A in the cartilage research jargon. An illustration of a confined compression test apparatus is shown in Figure 5.9. Figure 5.9: A schematic representation of the experimental apparatus for an confined compression test. Tissue explants, where articular cartilage is still attached to the subchondral bone, are generally used in the test. Illustration adapted from Mow & Guo (2002). Confined compression can also be used to determine the hydraulic permeability k by fitting the transient stress relaxation or creep response curves of the loaded specimen to the biphasic constitutive theory (Mow et al., 1980). 5.2.3 Indentation Indentation is a very important test, as it allows for minimal disruption of articular cartilage microanatomy and testing in situ, or even possibly in vivo (Mow & Guo, 2002). However, an indentation test involves complex stress fields in articular cartilage under the indenter tip and within the tissue, and theoretical or numerical solutions of the
77 indentation problem using appropriate constitutive laws for cartilage must be used to determine the intrinsic mechanical properties of articular cartilage and to interpret the data (Elmore et al., 1963; Mow et al., 1989). An illustration of indentation test apparatus is shown in Figure 5.10. Figure 5.10: A schematic representation of the experimental apparatus for an indentation test. An important advantage of this test is that it can be conducted in situ, i.e., while the articular cartilage is still attached to the native bone. Illustration adapted from Mow & Guo (2002). Again, using the isotropic, homogeneous biphasic theory for articular cartilage, the intrinsic, equilibrium aggregate modulus H A, Poissons ratio ν, and hydraulic permeability k can be simultaneously determined (Mow & Guo, 2002). The cross-calibrations and agreement between different testing methodologies, such as unconfined compression, confined compression and indentation, give confidence not only in the experimental methods, but also in the validity of the biphasic theoretical approach to describe the stress-strain behavior of articular cartilage. In other words, it provides greater confidence in both the experimental results as well as the theoretical formulation. 5.3 Material Models of Articular Cartilage In order to study the behaviour of materials by applying the laws of Continuum Mechanics, it is necessary to provide constitutive equations characterizing the material behaviour
78 in order to close the system of differential equations. In other words, the constitutive equations are the mathematical equations that characterize the material model describing the behaviour of the material at study. It is very important to keep in mind that mathematical models are an abstraction and that they are only an approximation of reality; in other words, no mathematical model is a perfectly accurate representation of the real material. Generally, as mathematical models become more accurate in the representation of a physical behaviour, they tend to increase in complexity. Therefore, it is evident that generally a trade off has to be made between accuracy and complexity of the model, in order to be able to use the said model. A large number of mathematical models of articular cartilage have been developed over the last 70 years. These models have evolved from simple, monophasic, linear elastic models (Hirsch, 1944) to complex biphasic, triphasic, and even quadriphasic, models that take into account fluid, microstructural, and electrical effects (Mow & Lai, 1980; Lai et al., 1991; Huyghe & Janssen, 1997). The discussion of these models, presented in this section, focuses on the evolution of the mathematical models of articular cartilage, with particular focus on the models relevant to the work conducted in this thesis. 5.3.1 Monophasic Models The initial attempts to model the mechanical behaviour of articular cartilage generally involved the application of previously known constitutive framework for monophasic materials. Although significant simplifications were necessary, these early models allowed researchers to develop insight into the behaviour of articular cartilage during testing, which paved the way for more complex models. In the following two subsections, elastic and viscoelastic monophasic models, respectively, are outlined.
79 Elastic Models Initially, the studies of articular cartilage under mechanical loading centered around the determination of the elastic modulus of the tissue. Due to the anatomical configuration and the thinness of the tissue, the indentation test was the choice of many investigators (Hirsch, 1944; Sokoloff, 1966; Hayes et al., 1972), and the general procedure was to find the Young s modulus by fitting the data from these indentation experiments. Initial attempts to find the Young s modulus usually involved the application of Hertz s solution for the contact between two elastic bodies of infinite depth (Hirsch, 1944), however, the justification for this approach was vague (Mow et al., 1984). Another, and more sophisticated, approach was undertaken by Sokoloff (1966), who considered cartilage to be comparable to medium hard rubber in terms of its mechanical properties. Using this method, Sokoloff (1966) assumed that the Young s modulus E for the cartilage can be determined from the relation E = 3P 8w 0 a (5.1) where P is the constant applied load, w 0 is the depth of penetration of the indenter, and a is the radius of the plane ended cylindrical indenter. Equation (5.1) comes from the solution of the elastic punch problem for a linearly elastic medium of infinite depth at equilibrium. It should be noted that Sokoloff (1966) observed that the cartilage exhibited non linear behaviour with strains as low as 10%, which is contrary to the basic assumptions employed in this theory. An analysis for a plane ended, as well as spherical ended, indenter on a layer of linearly elastic material attached to a rigid foundation was also carried out by Hayes et al. (1972). Using an approach based on the reduction of the problem to the inversion of a Fredholm integral equation of the second kind, Hayes et al. (1972) were able to determine the displacement filed of the elastic layer at equilibrium. For the case of a
80 plane ended indenter, the Young s modulus was given as E = P (1 ν2 ) 2w o aκ(a/h, ν) (5.2) where h is the thickness of the elastic layer and the function κ(a/h, ν) comes from the solution of the Fredholm equation of the second kind. Assuming that cartilage is incompressible and using the theory of Hayes et al. (1972), it can be shown that for a cartilage layer on a rigid foundation the instantaneous Young s modulus was 30% less than that calculated by Sokoloff (1966). This shows that in the calculations of Sokoloff (1966), the stiffness of the underlying bone was lumped into the cartilage stiffness, giving an erroneously high Young s modulus (Mow et al., 1984). This occurred because bone is a stiffer material, hence increasing the stiffness of cartilage when included into the calculations. Viscoelastic Models The assumption that the articular cartilage is purely elastic applies, at best, only to equilibrium conditions, as there would be no dissipative effects due to the movement of interstitial fluid (Mow et al., 1984). The considerations of the effects of interstitial fluid brings into play the viscoelastic theory. In one of the first studies of viscoelasticity in articular cartilage, Elmore et al. (1963) showed that the creep response of the tissue in an indentation test is mostly due to the influx of interstitial fluid from the surrounding area. This explained the imperfect elasticity observed by Hirsch (1944). Upon examination of the results of indentation tests, Kempson et al. (1971) used an expression for what they referred to as two second creep modulus for cartilage, which was based on indentation studies of thin sheets of vulcanized natural rubber. For a plane ended indenter, it was assumed that E = P (1 ν2 ) φ(h/a) (5.3) 2w 0 a where φ(h/a) is an empirical viscoelastic function depending on the aspect ratio between the undeformed tissue thickness and the radius of the indenter.
81 The result of this equation is functionally equivalent to equation (5.2), i.e., this empirical approach tells nothing about the displacement field of the elastic layer and little about the boundary conditions. In any case, Kempson et al. (1971) measured the indentation two seconds after the application of the load, and then used equation (5.2) to determine E. This value of E is supposed to include the initial elastic response as well as a small portion of creep. However, this is inconsistent with the assumptions used to derive equation (5.2), and it provided inconsistent results. Further applications of the viscoelastic theory to articular cartilage in tension have been made by Woo et al. (1980) through comprehensive stretching experiments, from which they developed a quasi linear viscoelastic model, based on the assumption that the kernel of the stress strain history integral is a function of strain and time. This approach is very similar to that followed for other soft tissues (Fung, 1981). Through this model, Woo et al. (1980) found that they could accurately predict the relaxation and cyclic behavior, but at the same time they were unable to get accurate predictions for the elastic stress if the strain rates were high. At higher strain rates, the biphasic viscoelastic effects would begin to dominate, and the movement of interstitial fluid must be taken into account in order to make accurate predictions. In order to support this theory, Li et al. (1983) produced experimental evidence that correlated well with the biphasic viscoelastic treatment of cartilage in tension, and proved that the interstitial fluid flow is important if the strain rates are high enough. According to the theory by Li et al. (1983), the load per unit area for articular cartilage in tension under small strains is given by [ F (t) A = ε t E + h2 ε A 0 k 0 H A n α=1 ] A n exp( αn 2 H A k 0 t/h 2 ) αn 2 (5.4) where F (t) is load at time t, A is the cross sectional area, ε is the strain rate, t is the current time, h is the specimen thickness, k 0 is the permeability, α n and A 0...A n are the
82 constants depending on the Young s modulus of the solid phase and H A is the aggregate modulus of the tissue calculated by H A = λ s + 2µ s (5.5) where λ s and µ s are the Lamé parameters. The first parameter λ s has no immediate physical interpretation, but it serves to simplify the stiffness matrix in Hooke s law. On the other hand, the second parameter µ s serves as the shear modulus, and is also sometimes denoted as G. The two parameters together constitute a parametrization of the elastic moduli for homogeneous isotropic media. Monophasic models of articular cartilage are useful for providing a simple understanding of the behaviour of cartilage, but they are inherently limited in their description as they completely ignore the fluid phase in the tissue. Since interstitial fluid makes up approximately 65 80% of articular cartilage wet weight, and it contributes significantly to the mechanical behaviour of cartilage in different loading scenarios, a mathematical formulation involving both the solid and fluid phase of the tissue eventually emerged. 5.3.2 Biphasic Models The mathematical formulation that connected the interstitial fluid flow in articular cartilage to its mechanical behaviour was based on early models of soil consolidation poroelastic (Biot, 1941; Terzaghi, 1943) and biphasic continuum mixture theory (Craine et al., 1970; Bowen, 1976). The biphasic formulation was first introduced by Torzilli & Mow (1976), and then further extended by Mow & Lai (1979) and Mow et al. (1980). In these early models, articular cartilage was modeled as a mixture comprised of an incompressible fluid phase and an incompressible solid phase, which is both porous and permeable. Although poroelastic and biphasic mixture models has different conceptual bases, it has been shown that they are identical in the case of an inviscid fluid phase (Bowen, 1980;
83 Levenson et al., 1998). Since poroelastic formulations are available in most commercial Finite Element packages, unlike the biphasic formulations, they are often employed when implementing the models of articular cartilage tissue (Wu & Herzog, 2000; Li et al., 2002). This subsection will first introduce the general biphasic model, as introduced by Mow et al. (1980), upon which virtually all biphasic models of articular cartilage since the 1980 s have been based. Following this, anisotropic and hyperelastic models will be outlined and discussed. The subsection will be concluded with an outline of permeability modeling and fibre reinforced models, as upon these concepts the model implemented in this thesis is based. General Background Under the assumption that the interstitial fluid is macroscopically inviscid, i.e., it can only bear spherical (hydrostatic) states of stress but at the same time it generates frictional drag as it flows through the porous matrix, the Cauchy stress in the solid and the fluid, respectively, and the total Cauchy stress in the mixture are given by (Hassanizadeh, 1986) σ s = φ s p g 1 + σ c σ f = φ f p g 1 (5.6a) (5.6b) σ = p g 1 + σ c (5.6c) where φ s and φ f are the solid and fluid fractions in the spatial configuration, respectively, where the saturation condition (φ s +φ f = 1) must be met, g 1 is the spatial inverse metric tensor (the contravariant identity ), p is the pore pressure, a Lagrangian multiplier arising from the constraint of intrinsic incompressibility of the solid and fluid phases and the saturation condition, and σ c is the constitutive contribution to the solid stress, usually called the effective stress in soil mechanics (Terzaghi, 1943).
84 As an example, the constitutive stress in the linear biphasic theory formulated by Mow et al. (1980) is given by a linear elastic isotropic constitutive law σ c = λ tr (ε) g 1 + 2 µ ε (5.7) where ε is the infinitesimal strain tensor, λ is Lamé s first parameter, and µ is Lamé s second parameter, more commonly known as the shear modulus. The linear biphasic model was a significant step forward in modeling the mechanical behaviour of articular cartilage. For example, the viscoelastic behaviour of articular cartilage in compression could now be easily modeled and explained in terms of intrinsic physical characteristics. Due to the general nature of the formulation of the constitutive stress contribution, a different constitutive equation for constitutive stress can be used to account for different behaviour, such as anisotropy, transverse isotopy, and fibre reinforced materials (Mow et al., 1984; Almeida & Spilker, 1998; Federico et al., 2005; Federico & Grillo, 2012). Anisotropic Modeling The formulation of Mow et al. (1980) is often credited with the introduction of the biphasic mixture theory to the mathematical modeling of articular cartilage mechanical behaviour, while in fact this paper is a culmination of years of work by Torzilli & Mow (1976), Mow & Mansour (1977), and others. In fact, the general framework for anisotropic biphasic mathematical model of articular cartilage was first outlined by Mow & Lai (1979), while the work by Mow et al. (1980) simply presented the isotropic case of this formulation. Even though the anisotropic nature of articular cartilage has been observed for some time (Woo et al., 1980; Roth & Mow, 1980) and even though the anisotropic biphasic theoretical framework has existed for several decades, anisotropic models of articular cartilage have been implemented rather rarely due to high number of independent components of the elasticity tensor required to describe the material.
85 In fact, for a linearly elastic anisotropic material, the number of independent components of the elasticity tensor is 21, which is clearly too high of a number to handle easily and practically. This is the primary reason that most material models implement a higher level of symmetry, which results in a decrease in the required number of independent components of the elasticity tensor. For example, this number decreases to 9 for the case of orthotropy, which is the invariance under axis reflection, 5 for transverse isotropy, which is invariance under rotations about a given direction, and 2 for the case of isotropy, which is invariance under any arbitrary rotation. From the macroscopic point of view, it has been shown that articular cartilage is an orthotropic material (Wang et al., 2003; Chahine et al., 2004) due to the arrangement of the collagen fibres and the chondrocytes over the depth of the tissue. As previously mentioned, the collagen fibres are perpendicular to the articulating surface in the deep zone, approximately randomly oriented in the middle zone, and parallel to the articulating surface in the superficial zone (Hedlund et al., 1993). Similarly, the chondrocytes also have a varying arrangement as a function of tissue depth: they are flattened and parallel to the articular surface in the superficial zone, nearly spherical in the middle zone, and elongated and parallel to the depth direction in the deep zone (Guilak et al., 1995; Guilak, 1995). Due to this microstructural arrangement, the deep zone can be considered to be transversely isotropic in the direction of the tissue depth, the middle zone to be roughly isotropic, and the superficial zone to be orthotropic, with axes given by the direction of tissue depth, the split line direction, and an axis orthogonal to the former. The split line direction represents the primary fibre direction in the superficial zone and changes with the location on the articular cartilage surface, as illustrated in Figure 5.11. In order to simplify the mathematical modeling of articular cartilage, the tissue is often considered to be transversely isotropic (Bursać et al., 1999; Cohen et al., 1998) by assuming the fibres in the superficial zone to be randomly oriented in the plane parallel to the articulating
86 surface. This ensures that all three zones can be considered overall transversely isotropic in the direction of tissue depth. Figure 5.11: (A) Split lines (arrows) in human patellar articular cartilage are visible after pricking the articular surface with a conical awl, and oriented in (B) lateral-to-medial and (C) proximal-to-distal directions in the proximal and the lateral facets, respectively. Adapted from Bae et al. (2008). There are two possible strategies for building an anisotropic model, the first being the fitting of experimental data from a set of tests in various configurations in order to capture the anisotropic behaviour, and the other being the consideration of microstructure of the material and the material properties of all constituents being considered. For articular cartilage, the former path was followed by a large number of researchers, notably Barry & Aldis (1990), Cohen et al. (1998), Bursać et al. (1999), and others. The latter, microstructural, approach has been extensively studied by a number of researchers, notably Almeida & Spilker (1998), Federico et al. (2005), and Federico & Grillo (2012), and this is the approach that has been utilized in this thesis. Hence, the microstructural approach will be discussed in greater detail in the next chapter. Hyperelasticity As mentioned in Chapter 2, in a hyperelastic material the work done by the stresses during deformation depends only on the initial configuration of the body at time t 0 and on the current configuration of the body at time t, and not on the deformation history.
87 The stress in such a material can be described by the derivative of a certain function, called the elastic strain energy potential, with respect to a given measure of strain. A hyperelastic formulation is necessary when an elastic body undergoes large deformations. For articular cartilage, strains of up to 30% have been measured in situ during functional dynamic loading (Bingham et al., 2008), and even larger strains are possible during static loadings (Herberhold et al., 1999). Various hyperelastic formulations have been evaluated for their ability to predict the mechanical response of normal and osteoarthritic cartilage during indentation tests (Brown et al., 2009), and Yeoh and Mooney Rivlin models were deemed most appropriate, with the Mooney Rivlin model providing the best compromise between accuracy and required computational power. In addition to these standard hyperelastic formulations originally developed for finite strain of rubber like materials, a number of different models of soft biphasic tissues, including articular cartilage, have been developed. These include the isotropic model of Holmes & Mow (1990), the transversely isotropic model of Almeida & Spilker (1998) and Federico et al. (2005), and a orthotropic Conewise Linear Elasticity (CLE) model (Curnier et al., 1994) or Soltz & Ateshian (2000). Modeling of Permeability One of the first studies on cartilage permeability was conducted by McCutchen (1962), who found that permeability decreases as a function of the depth from the articulating surface. For the superficial zone, he found the average value of permeability to be 5.8 10 16 m 4 /N s. The depth dependence was also examined by Maroudas & Bullough (1968), who found permeability to increase from superficial region to the middle region by approximately 35%, and then to decrease from the middle region to the deep region by approximately 200%. This inhomogeneity has been attributed to the dense network of collagen fibres in the superficial region and to increased charge density of the proteoglycan
88 matrix in the deep zone (McCutchen, 1962). The dependence of permeability on the compressive strain was first demonstrated by Mansour & Mow (1976), which used an empirical application of Darcy s law to determine the apparent permeability of the tissue. This theory has its shortcomings, as the calculated permeability constant was only an average. The permeability, using Darcy s law for one dimensional flow, is given by k a = Qh P A (5.8) where k a is the apparent permeability, and is defined as the spatial average the permeability throughout the tissue, Q is the volume flux of permeated fluid through a permeating area A, and P A is the fluid pressure differential across the tissue. Realistically, under compression, the permeation through the tissue gives rise to a drag force of significant magnitude, which compacts the permeable solid matrix in a non uniform fashion. This compaction decreases the permeability within the tissue, and the decrease varies with the distance from the articulating surface, making the measured value k a in equation (5.8) an average, lumped parameter value. This is particularly important for a soft tissue such as articular cartilage, where significant compaction can occur and lead to significant difference between the measured average permeability and the local permeability at each point in the tissue. Mow & Lai (1980) took this line of research into the permeability of the tissue a step further, and determined an exponential relation between the compaction of the tissue and the tissue permeability. In order to incorporate this permeability model into the biphasic theory of articular cartilage, they introduced the concept of intrinsic permeability. This relation is given by k = k 0 ( ) φ s 0 φ f κ [ exp M J ] 2 1 (1 φ0 s ) φ s 2 (5.9)
89 where k 0 is the permeability of the undeformed tissue, φ s 0 is the solid fraction of the tissue in undeformed configuration, φ s and φ f are the solid and fluid fractions, respectively, in the deformed configuration, J is the determinant of the deformation gradient of the solid matrix (and J 2 = I 3 is the third invariant of the right Cauchy-Green deformation tensor), and κ and M are constants. κ is a positive parameter that determines how fast the permeability approaches zero when φ s 1 (which describes the configuration with pores closed, i.e., all fluid has escaped), and M is a parameter that comes from the observation that the permeability is an exponential function of the compressive strain. This relationship basically states that as the cartilage is compacted, the pore size decreases and as a result permeability decreases. This was later elaborated by Wu & Herzog (2000), who explicitly expressed the permeability in terms of the void ratio. Higginson et al. (1976) used permeability compression curves of a homogeneous polymeric porous filter to simulate compression induced anisotropy. When it is assumed that the permeability is isotropic in the undeformed configuration, these curves predicted that the permeability in the direction of the compression decreases faster that the permeability in the direction perpendicular to the direction of compression. This result was supported both experimentally (Reynauld & Quinn, 2006) and theoretically by a model of (Quinn et al., 2001), which is based on the reorientation of glycosaminoglycan chains under compression. These chains are randomly oriented in the undeformed configuration, but under compression they become transversely isotropic, with symmetry axis in the direction of compression. In the previously described models, the effect of collagen fibres on the anisotropy of permeability in articular cartilage was not addressed, even though the effect of collagen fibres on the elastic anisotropy has been extensively studied (Farquhar et al., 1990; Wu & Herzog, 2002; Wilson et al., 2004; Federico et al., 2005). A model by Federico & Herzog (2008a), based on the formulation developed in Federico & Herzog (2008b), takes into
90 account the effect of collagen fibres on the local permeability, where the permeability parallel to the fibres was found to be greater than the permeability orthogonal to the fibres. This model also evaluates the overall permeability of the tissue by through an averaging procedure when fibre orientation obeys a given statistical distribution (Federico et al., 2004a). The drawback of this model is that it is appropriate for only small deformations, and it is not appropriate for finite deformations. Recently, a finite deformation anisotropic permeability model was formulated by Ateshian & Weiss (2010). This model is not explicitly defined for fibre reinforced tissue with statistical fibre orientation, however it is possible to adapt the model reasonably easily. This model demonstrated that, in general, strain induces anisotropy in the permeability tensor under finite deformation, even if the permeability tensor was isotropic in the reference configuration, under various loading conditions. The model by Federico & Herzog (2008a) has recently been extended to the case of finite deformations by Federico & Grillo (2012), and this is the model of hydraulic permeability that has been implemented in this thesis. Fibre Reinforced Models As previously discussed, one of the main components of articular cartilage is the collagen fibre network, which is the primary structural component of cartilage in tension and shear. In the past several decades, a number of different fibre reinforced modes were formulated to model the mechanical behaviour of articular cartilage. Generally, these models can be split up into two categories: discrete spring models and continuum models. Spring models are based on Finite Elements (Soulhat et al., 1999; Li et al., 2002) and represent collagen fibres by means of springs connecting the nodes of the Finite Element mesh, which is an intrinsic geometric limitation. These types of models also suffer from an analytical limitation, as there is no direct correspondence between theory and Finite
91 Element implementation (Pajerski, 2010). In continuum models, there is no need for a mesh to be defined, as these models are built on the framework of Continuum Mechanics, independently of the subsequent Finite Element implementation. Early fibre reinforced continuum models (Farquhar et al., 1990; Wilson et al., 2004) were based on averaging the effect of finite number of fibre directions, each of which had to be described individually, and this resulted in great computation cost. This limitation was overcome with the introduction of the linear elastic formulation for a composite with statistically oriented inclusions (Federico et al., 2004a) and a non linear elastic model of biological tissues with statistically oriented fibres (Federico & Herzog, 2008c), as well as with the work of Ateshian et al. (2009). When these models are applied to articular cartilage, they result in a transversely isotropic material behaviour and transversely isotropic material structure. Hence, for articular cartilage, the above formulations become the linear transversely isotropic, transversely homogeneous (TITH) model (Federico et al., 2005) and the non linear transversely isotropic, transversely homogeneous (NLTITH) model (Federico & Gasser, 2010), which are based on averaging the elastic strain energy potential by means of an integral over the unit sphere weighted by a probability distribution of orientation (Federico et al., 2004a). The work of Ateshian et al. (2009) takes a similar approach, except that it utilized pre weighted ellipsoidal distributions to define the preferential fibre direction instead of an integral over a unit sphere weighted by a probability distribution. In a sense, the primary difference between the two approaches is in the implementation, as the formulation by Ateshian et al. (2009) jumps a step by using sets of pre weighted distributions. Most fibre reinforced mathematical models also take into account the tension compression non linearity of articular cartilage. Since the contribution of collagen fibre network is significantly higher in tension than it is in compression, many models (e.g., Wilson et al.
92 (2004)) eliminate fibre contribution in compression and instead give the fibres only tensile stiffness in the direction of the fibre axis. Since it is not realistic to assume that fibres have no contribution on compressive stiffness altogether, a number of other models give the fibres in compression a reduced, isotropic stiffness (Federico & Herzog, 2008c; Ateshian et al., 2009; Federico & Gasser, 2010). A model based on the former approach (Federico & Grillo, 2012) is also the subject of implementation is this thesis. 5.3.3 Other Models of Articular Cartilage Mathematical models of mechanical behaviour of articular cartilage that have more than two phases are often called mechano electrochemical models, and they take into account ion concentration and ion fluxes in addition to the cartilage solid and fluid phases. In other words, these models allow the swelling behaviour, which arises due to the repulsion of negatively charged proteoglycans and positively charged ions in the interstitial fluid (Maroudas et al., 1985), to be modelled. The most common osmotic swelling behaviour applied in these models is the Donnan osmotic pressure formulation (Mow & Guo, 2002). Lai et al. (1991) developed a small strain triphasic model, which was later generalized to finite deformations by Huyghe & Janssen (1997) in their quadriphasic model of articular cartilage.
93 Chapter 6 A Theoretical Model of Articular Cartilage in Large Deformations Complex mechanical phenomena, such as those observed in biological tissues, are often modelled using techniques that take advantage of modern computing capabilities, but still rely extensively on experimental determination of model parameters. These methods often give good approximations of the system s mechanical behaviour, but are restricted by how closely the structure of the test specimen resembles the material being modelled. Consequently, the feasibility of performing the necessary number of experiments to obtain the material parameters often becomes the limiting factor. On the other hand, theoretical modeling of materials, rather than seeking solutions and models that would fit specific and restrictive models, looks for a set of equations that translate the nature of the physical system into its analytical properties (Federico & Herzog, 2008c). Compared to purely numerical formulations, theoretical models usually retain more intuitive connections between the physical behaviour and structure of the material and its mathematical description. In this thesis, the behaviour of articular cartilage is described by means of the large deformation non linear model for biological tissues with statistically oriented reinforcing fibres developed by Federico & Grillo (2012). This model is a natural evolution of tissues models by Federico et al. (2005) and Federico & Herzog (2008c), with a formulation that allows for accounting for the effect of the content and orientation of collagen fibres on the elasticity and permeability of the tissue.
94 6.1 General Theory Articular cartilage is a biphasic porous material in which both the fluid and solid phases have a significant contribution to the overall mechanical behaviour of the tissue. The porous nature of the tissue is generally taken into account by two physical quantities: porosity and permeability. This section of the Thesis will address the non linear elastic modelling of the solid matrix along with the porous and permeable nature of the tissue, and the associated models. 6.1.1 Stress In this Thesis, the solid and fluid phases are taken into account by using a biphasic formulation described in greater detail in Chapter 5, in which the stress in a biphasic material is given by σ s = φ s p g 1 + σ c σ f = φ f p g 1 (6.1a) (6.1b) σ = p g 1 + σ c (6.1c) where the constitutive stress σ c (also known as the effective stress in soil mechanics (Terzaghi, 1943)) does not take into account the fluid pressure effects and represents the stresses in the solid phase resulting from overall tissue strain, and the inverse metric tensor, g 1 is used as the covariant identity. Since articular cartilage is comprised of multiple solid constituents, with the primary constituents being the proteoglycan matrix and the collagen fibres, σ c is obtained by superposition of the stress for each primary components as σ c = φ 0 σ 0 + φ 1 σ 1 (6.2) where φ indicates the volumetric fraction, σ indicates the Cauchy stress, and subscripts 0 and 1 represent the proteoglycan matrix and the collagen fibres, respectively. In a
95 similar manner, the spatial elasticity tensor Cc is given by Cc = φ 0 C0 + φ 1 C1 (6.3) where the subscripts follow the same convention as outlined above. In fact, this convention will be used for the rest of this document, unless otherwise noted. 6.1.2 Porosity Porosity is a non dimensional quantity, defined as the volumetric fraction of pores. When the saturation condition is respected (i.e., φ s + φ f = 1), the porosity coincides with the fluid fraction φ f. An alternative to porosity is the void ratio, defined as e = φ f φ s, e R = φ fr φ sr (6.4) where e and e R denote the void ratio in current and reference configurations, and φ f and φ s denote the fluid and solid volume fractions, respectively. These definitions of the void ratio are useful when using certain constitutive models, where it is useful to express J as 6.1.3 Permeability J = 1 + e 1 + e R (6.5) Permeability is the measure of the ease with which a fluid filtrates through a solid. When inertial effects and external body forces, such as gravity, are neglected, the balance of momentum for the fluid phase reduces to the statement that the divergence of the stress in the fluid is balanced by the drag force. Through some manipulation, this statement leads to Darcy s Law w = k grad p = k h (6.6) where tensor k is the spatial permeability of the tissue, and h = grad p is the hydraulic gradient. The permeability of a porous medium depends strongly on its microstructure
96 and is a macroscopic quantity that takes into account several microscopic phenomena (Podzniakov & Tsang, 2004). The magnitude of permeability depends on the size of the pores and the viscosity of the interstitial fluid, and its directional dependence is determined by the configuration of the flow paths, in this case defined by the fibre direction. The remaining part of this Chapter will present the mathematical formulation for the large deformation non linear model for biological tissues with statistically oriented reinforcing fibre, including the the mathematical framework of transversely isotropic materials, incompressible solid fluid mixtures, the actual fibre reinforced biphasic constitutive model, as well as the permeability model for materials with statistically oriented fibre inclusions. 6.2 Transversely Isotropic Second Order Tensor Basis Transverse isotropy is the symmetry under rotations about a given axis, called the axial direction, with the plane orthogonal to this axis being called the transverse plane. For the case of tensors in spatial reference frame, transverse isotropy is defined with respect to the spatial unit vector m in the spatial unit sphere S 2 x = {m T x S : m = 1} at the spatial point x, which represents all possible directions in the tangent space T x S. The tensor basis for transverse isotropy is obtained by decomposing the inverse metric g 1 into g 1 = a + t, with basis tensors a and t given by a = m m t = g 1 m m = g 1 a (6.7a) (6.7b) Since the unit vector m is in the axial direction, basis tensors a and t take the meaning of projection operators on the axial direction and the transverse plane, respectively. Henceforth, this allows decomposition of a vector w into the axial and transverse
97 components as w = a w w = t w = w a w (6.8a) (6.8b) Furthermore, any (contravariant) tensor y that is transversely isotropic with respect to m can be written as the linear combination y = y a + y t (6.9) where a and t are known as the axial and transverse projection operators, respectively. Tensor a is often called the structure tensor or fabric tensor, especially in the treatment of fibre reinforced materials, where m represents the fibre direction. The spatial structure tensor is proportional to the push forward of the material structure tensor. The latter is defined from the unit vector M in the material unit sphere S 2 X = {M T XB R : M = 1} at the material point X as A = M M (6.10) If it is considered that the normalized push forward of the material direction vector M must coincide with the spatial direction vector m, then the spatial structure tensor can be expressed in terms of the material structure tensor as a = [C : A] 1 F A F T (6.11) 6.3 Incompressible Solid Fluid Mixtures A saturated porous material consists of a solid and a fluid phase, and the two phases are assumed to coexist at each point in the system. Under the assumption of intrinsic incompressibility of both phases, the true mass densities ρ st and ρ ft remain constant
98 through the motion. Hence, the apparent mass densities ρ s and ρ f vary only as a response to the change in volumetric fractions as φ s = ρ s ρ st, φ f = ρ f ρ ft (6.12) where the volumetric fractions φ s and φ f always have to obey the saturation condition φ s + φ f = 1. It is important to note that such a system is globally compressible if the fluid exudation is permitted at the boundary, and locally compressible at a given material point if fluid is allowed to move away from that point. Hence the determinant J of the deformation gradient F is a measure of the variation of the volumetric fractions; when, at a given time t, J(X, t) > 1 the fluid has entered point X, while J(X, t) < 1 entails that fluid has moved away from point X. Rigorously, this is translated into the material form of the continuity equation for the solid phase, as ρ sr = J ρ s (6.13) where ρ sr is the value of the apparent mass density of the solid in the reference configuration. This definition can then be used to express the continuity equation for the solid in terms of the volumetric fractions as φ sr = J φ s (6.14) where φ sr is the solid volumetric fraction in the reference configuration. It is important to note that, when all the fluid has flown out of a given point X at a given time t, it implies that J(X, t) = φ sr, i.e., the point is now composed of only the incompressible solid and cannot undergo further compression. Hence, the system is subjected to the unilateral constraint of compaction expressed as J φ sr (6.15)
99 Using the definition of φ sr from equation (6.13) and the analogous fluid volumetric fraction in reference configuration, φ fr, the saturation condition φ s + φ f = 1 can be defined in the reference configuration as φ sr + φ fr = 1 (6.16) 6.4 Hyperelastic Biphasic Mixtures The basic mathematical framework for the model presented and implemented in this Thesis has been introduced as the Linear Transversely Isotropic, Transversely Homogeneous (TITH) model (Federico et al., 2005), and further expanded as the Non Linear Transversely Isotropic, Transversely Homogeneous (NLTITH) model developed by Federico & Herzog (2008c) and Federico & Gasser (2010), respectively, with the second work being implemented by Pajerski (2010). For monophasic (i.e., solid only) materials, incompressibility can be implemented by means of the decoupled elastic strain energy potential W (C) = U(J(C)) + W ( C) (6.17) where U(J(C)) is a function used to enforce compressibility, generally taken to be in a simple quadratic form U(J(C)) = 1 κ (J(C) 2 1)2, where the penalty number κ has the physical meaning of bulk modulus, and W ( C) is a function depending solely on the distortional measure of deformation C. Quasi incompressibility is attained for κ much larger than the shear moduli of the material (which feature as material parameters in W ( C). It is important to note that the potential in equation (6.17) is appropriate only in the case of incompressible or nearly incompressible materials, the case of the work by Federico & Gasser (2010), and would be problematic for the general case of compressible materials due to the absence of an interaction term, which should ba a function of both
100 J and C. Indeed, for anisotropic compressible materials, the lack of this interaction term would prevent existence of distortional deformations under hydrostatic states of stress and, viceversa, of volumetric dilatations under shear stress (Guo et al., 2008; Federico, 2010). Furthermore, it has also been shown (Federico, 2010) that the elasticity tensor would be incompatible with the linear elasticity tensor found through the experiments. Due to this, for a general mixture, it is necessary to replace the purely distortional part W ( C) in equation (6.17) by a non decoupled term V (J, C) depending on both the volumetric and distortional deformation. It is desired to have the referential solid volumetric fraction φ sr as a multiplicative factor of the potential, as this clarifies the limit passage to the case of monophasic incompressible materials (Federico & Grillo, 2012). With these considerations, the proposed elastic strain energy potential has the form W (J(C), C) = φ sr [ U(J(C)) + V (J(C), C) ] (6.18) where it is assumed that U(J(C)) is only active under volumetric contractions, i.e., for J(C) < 1, and has zero stiffness at zero strain. The choice of function U(J(C)) gives it the physical meaning of the correction term to the potential V (J(C), C) that accounts for the singularity condition at the compaction limit, while leaving the potential V (J(C), C) unaltered for positive dilatations and in the undeformed configuration (J(C) 1). This implies that, in the undeformed configuration, the elasticity tensor is nothing but the linear elasticity tensor and is independent of the choice of the function U(J(C)) (Federico & Grillo, 2012). A relatively simple form of function U(J(C)) that satisfies the requirements outlined by Federico & Grillo (2012) is expressed as U(J(C)) = H(J crit J(C))(J(C) J crit ) 2q (J(C) φ sr ) r (6.19) where H is the Heaviside step function, J crit ]φ sr, 1] is a certain critical value of the volumetric deformation under which the correction function U(J(C)) is active, q 2 is
101 a positive integer, and r ]0, 1] is another positive number. The two parameters J crit and φ sr are essentially geometrical, while the parameters q and r could be related to material properties (Federico & Grillo, 2012). Figure 6.1: Behaviour of function U(J(C)) along with its first and second derivatives with respect to J(C), for a value of the referential solid fraction φ sr = 0.2, a critical volumetric deformation J crit = 0.6, and exponents q = 2 and r = 1/2. Figure adapted from Federico & Grillo (2012). Figure 6.1 illustrates the behaviour of function U(J(C)) along with its first and second derivatives (which are respectively a pressure and a bulk modulus) with respect to J(C), which illustrates the correction nature of the function. In other words, function U(J(C)) engages as the value of J(C) decreases below the critical value J crit, and results in a significant increase in the elastic strain energy potential (and the bulk modulus), which accounts for the intrinsic incompressibility of the solid phase once the fluid has exuded.
102 6.5 Fibre Reinforced Biphasic Constitutive Model In fibre reinforced materials, the solid phase is composed of two sub phases, the proteoglycan matrix and the collagen matrix, denoted by indices 0 and 1, respectively. It is also assumed that these two sub phases share the same motion and the same deformation gradient F (Federico & Gasser, 2010). The volumetric fractions in the spatial and material pictures, respectively, must obey φ s = φ 0 + φ 1, φ sr = φ 0R + φ 1R (6.20) These equations then have to be coupled with the saturation condition in equation (6.16) to describe the volumetric partition of the system. Under the hypothesis of isotropic matrix, the anisotropy of the tissue arises due to the presence of the reinforcing fibres and their arrangement in space. The most general case, and the one applied in this Thesis, is that of the fibres oriented according to a normalized probability distribution ψ : S 2 X R + 0, S 2 X ψ(m) ds = 1 (6.21) which describes the probability to find, at a given material point X, a fibre oriented in direction M. The probability distribution in equation (6.21) is defined on the material unit sphere due to the fact that the orientation of the fibres is assumed to be known in the reference configuration, and must be derived after any motion has occured. In the following two sections, the specific strain energy potential and the permeability formulation used in this Thesis will be discussed in detail. 6.5.1 Elasticity Formulation By adopting the same approach that lead from the model of Walpole (1981) to that of Federico et al. (2004a), Federico & Herzog (2008c) generalized the superposition method
103 (Holzapfel et al., 2000) to the case of continuous infinity of fibre families. This method can then be modified to be admissible under the decomposition in equation (6.18), yielding the overall strain energy potential as W (C) = φ sr U(J(C)) + φ 0R V 0 (C) + φ 1R V e (C) (6.22) where V 0 (C) is the isotropic matrix contribution to the overall strain energy potential, generally taken as a simple Holmes Mow constitutive model (Holmes & Mow, 1990), while V e (C) is known as the ensemble fibre potential and is given by V e (C) = ψ(m)v 1 (C, A(M)) ds (6.23) S 2 X Collagen fibres, by their very nature, are significantly more stiff in tension than in compression, and that tension compression non linearity must be addressed by the tissue model if the model is to be an accurate representation of the physical scenario. Although this problem has been noted by Holzapfel et al. (2000), it has not proven relevant in their model as the arterial wall are always in tension. On the other hand, articular cartilage regularly experiences compression within its biological function, which means that the tension compression non symmetry becomes a primary issue. Rather than completely removing the contribution of the fibres in compression, as it is sometimes done (Wilson et al., 2004), a more realistic approach has been outlined by Federico & Herzog (2008c) and Federico & Gasser (2010), where the fibres have an isotropic and relatively small contribution in compression and a significantly larger anisotropic contribution in tension. This is achieved by separating the overall fibre potential V 1 (C, A(M)) as V 1 (C, A(M)) = V 1i (C) + V 1a (C, A(M)) (6.24) where V 1i (C) is the isotropic part, which is always active, and V 1a (C, A(M)) is the
104 anisotropic part. Substituting this fibre potential intro equation (6.23) yields V e (C) = V 1i (C) + ψ(m)v 1a (C, A(M)) ds (6.25) S 2 X where the integral of the isotropic part coincides with the isotropic part itself, as it does not depend on the direction. On the other hand, the anisotropic fibre potential, V 1a (C, A(M)), has to be switched off in the case of compression. This can be achieved by considering an invariant I 4 (C, A(M)), which is defined as I 4 (C, A(M)) = C : A(M) (6.26) This invariant can also be thought of as the square of the stretch in the fibre direction. In other words, when I 4 (C, A(M)) is greater than one, the fibre is in extension, and when I 4 (C, A(M)) is less than one, the fibre is in contraction. Following this idea, the anisotropic fibre potential can then be defined as V 1a (C, A(M)) = H(I 4 (C, A(M)) 1)V 1b (C, A(M)) (6.27) where V 1b (C, A(M)) is the basic anisotropic fibre potential, and H(I 4 (C, A(M)) 1) is the Heaviside step function where 1, H(I 4 (C, A(M)) 1) > 0 (fibres in extension), H(I 4 (C, A(M)) 1) = 0, H(I 4 (C, A(M)) 1) < 0 (fibres in contraction), (6.28) In this way, the ensemble fibre potential becomes V e (C) = V 1i (C) + ψ(m)h(i 4 (C, A(M)) 1)V 1b (C, A(M)) ds (6.29) S 2 X Equations (6.22) and (6.29) together form the overall strain energy potential for the fibre reinforced material, which in this case is articular cartilage. The specific constitutive functions for the reinforcing fibres as the proteoglycan matrix, which have been implemented in this Thesis, will be addressed in the next chapter, when the numerical implementation of the model is formulated and addressed.
105 6.5.2 Permeability Formulation If a fibre reinforced porous medium is considered (Federico & Herzog, 2008b), of which articular cartilage is a perfect example, it is convenient to define a Representative Element of Volume (REV) as a cylinder at a spatial point x, which consists of a rectilinear segment of a fibre oriented in the direction m S 2 x and the surrounding matrix, as shown in Figure 6.2. Figure 6.2: A porous fibre reinforced material, with a representative element of volume (REV) containing a small portion of an impermeable fibre (which can be considered as rectilinear), surrounded by the fluid saturated matrix. Figure adapted from Federico & Grillo (2012). Under the assumption of scale separation, at the fibre level, the matrix can be treated as a porous medium, i.e., the microscopic pores in the matrix are magnitudes smaller that the mesoscopic pores that are formed by the interstices between the fibres (Federico & Grillo, 2012). Therefore, at the REV level, the aim is to define an expression for the permeability of the REV, also known as the local permeability, and denoted by k REV. On the other hand, at the overall tissue level, the aim is to find the overall permeability
106 of the tissue, denoted by k, by means of a direction average Federico & Herzog (2008b). If an external hydraulic gradient h is applied on the REV, then the equilibrium of the flow obtained through the application of Darcy s Law and defined as w REV = k REV h, can be expressed by means of the rule of mixtures, where the contribution of each element in the mixture is weighted by its volumetric fraction as (Podzniakov & Tsang, 2004; Federico & Herzog, 2008b) k REV h = (1 φ 1 )k 0 h 0 h = (1 φ 1 )h 0 + φ 1 h 1 (6.30a) (6.30b) If an analogy is drawn between the Darcy s law, written as w = k h, and an equation of a linear dielectric material, written as d = ɛ e, where d is the dielectric displacement, ɛ is the dielectric permeability, and e is the electric field, it is possible to utilize the solution for the dielectric equation (Landau & Lifshitz, 1960) to obtain a solution for hydraulic gradient of the fibre inclusion in a matrix as (Federico & Herzog, 2008b) h 1 = [a + 2t] h (6.31) where a and t are the second order tensors that form orthogonal basis describing transverse isotropy in direction m, which represents the fibre direction. By combining equations (6.30a) (6.31), the REV permeability is obtained as k REV = k 0 [(1 φ 1 ) a + (1 2φ 1 ) t] (6.32) Since the solution (6.31) was found for an infinite cylinder (i.e., a fibre) in an infinite matrix (Landau & Lifshitz, 1960), it is valid only for small volumetric fractions of fibres. By means of differential methods for composite materials (McLaughlin, 1977; Norris, 1985), the REV permeability for arbitrary value of fibre volumetric fractions becomes (Federico & Herzog, 2008b) k REV = k 0 [(1 φ 1 ) a + (1 φ 1 ) 2 t ] (6.33)
107 By using a definition for t, as shown in equation (6.7b), it is possible to rewrite k REV in an alternative form as k REV = k 0 [(1 φ 1 ) φ 1 a + (1 φ 1 ) 2 g 1] (6.34) The spatial structure tensor a and the spatial fibre volumetric fraction φ 1 are not explicitly known, and it is therefore necessary to express these quantities in terms of their material counterparts. Hence, by using a definition for the spatial structure tensor in equation (6.11) and saying that φ 1 = J 1 φ 1R, the expression for k REV can be written purely in terms of material quantities as k REV (A(M)) = ˆk REV (C, A(M)) (6.35) = J 2 k 0 [(J φ 1R ) φ 1R [C : A(M)] 1 F A(M) F T + (J φ 1R ) 2 g 1] where the REV permeability k REV has been expressed as an explicit function of the material structure tensor A(M), and ˆk REV (C, A(M)) is the constitutive function, which depends explicitly on the material structure tensor and the right Cauchy Green deformation tensor C, associated with k REV. It is now possible to define the overall permeability of the material k as the directional average of the REV permeability as k = ˆk(C) = ψ(m) ˆk REV (C, A(M)) ds (6.36) S 2 X Substituting equation (6.35) into equation (6.36) yields an expression for the overall permeability k = ˆk(C) = J 2 k 0 [(J φ 1R ) φ 1R ẑ(c) + (J φ 1R ) 2 g 1] (6.37) where the linearity of the integral operation has been used to split the two terms of the sum in the expression of k REV in equation (6.35), and the tensor ẑ is given by [ ] ẑ(c) = F ψ(m) [C : A(M)] 1 A(M) ds F T (6.38) S 2 X
108 It should be noted that the integrand in equation (6.38) is a non affine function of the structure tensor A(M) and the right Cauchy-Green deformation tensor C, and therefore the deformation cannot be factorized out of the integral sign, but instead the integral has to be evaluated at each increment of the deformation, much like it has to be done for the case of elasticity. It is also important to note that the permeability has to be physically admissible, i.e., it has to approach zero as J φ sr. Therefore, it is essential to assign a suitable constitutive function to the permeability in undeformed configuration, k 0, in order for the overall permeability k to vanish at compaction limit. The simplest possible assignment of k 0 is to assume that it is isotropic under any deformation, i.e., that it has a form k 0 = ˆk 0 g 1, where ˆk 0 is a constitutive function that depends on the volumetric deformation J, such as the Holmes and Mow formulation (Holmes & Mow, 1990). In conclusion, the basis for the permeability model is formed by equations (6.37) and (6.38). As it was already mentioned, it is also necessary to select the appropriate constitutive function for the permeability in undeformed configuration, k 0. This will be addressed in the following chapter, when the numerical implementation of the model is formulated and addressed. This concludes the chapter on the theoretical formulation of the large deformation non linear model for biological tissues with statistically oriented reinforcing fibres, which includes both the elastic formulation as well as the formulation for the permeability tensor. In the following chapter, this model will be further expanded and prepared for implementation with the selection of the constitutive models for all constituents, the calculation of the elasticity tensor and the fourth order tangent of permeability, and a discussion and derivation of the implementation algorithm.
109 Chapter 7 Numerical Implementation The governing equations describing a physical system are often complex expressions for which analytical solutions only exist under simplifying assumptions, or do not exist at all. These assumptions are generally related to the system s boundary conditions, complexity, dimensionality, and/or symmetry. The solutions obtained using various assumptions are often helpful in gaining a general understanding of the behaviour of the system. However, even with simplifying assumptions, an analytical solution is often not available and numerical techniques must be used to find an approximate solution. The governing equations of the articular cartilage model presented in Chapter 6 of this Thesis cannot be solved analytically. Consequently, a numerical method known as the Finite Element Method (FEM), introduced in Chapter 3, is used to solve for the unknown field quantities in question, such as displacement, stress, strain, pore pressure, and fluid flux. This chapter begins with a brief introduction to FEBio, the open source FE package that was used to perform the Finite Element (FE) analysis in this Thesis. The focus of the chapter then shifts to the elastic formulation, where the specific constitutive models for the reinforcing fibres and the proteoglycan matrix are selected. This is followed by the derivation of the spatial stress and elasticity tensors, which is necessary to implement the model. The following section of the chapter addresses the permeability formulation, which includes the constitutive function selection, and the derivation of the fourth order tangent of permeability. This is followed by the treatment of the statistical fibre distribution and how this is related to the depthdependent nature of the tissue, when considering both the elasticity and the permeability. This section concludes with the description of the algorithms that has been used to implement the elasticity and permeability models. The
110 chapter concludes with an outline of the FE simulations conducted to test the articular cartilage model presented in this work and includes a summary of the FE model geometry, the assumptions made during the analysis, and the material parameters used. 7.1 Introduction to FEBio FEBio is a nonlinear FE solver that has been developed by the Musculoskeletal Research Laboratory (MRL) at the University of Utah (USA), and is specifically designed for biomedical applications. It offers modeling scenarios, constitutive models and boundary conditions that are relevant to many research areas in biomechanics. All features of FEBio can be used together seamlessly, giving the user a powerful tool for solving three dimensional problems in computational biomechanics. The software is open source, and pre compiled executables, as well as the open source code, are available for download at http://mrl.sci.utah.edu/software. FEBio supports two analysis types, namely quasi-static and dynamic. In a quasistatic analysis, the (quasi-) static response of the system is sought and the effects of inertia are ignored. In the presence of biphasic materials, a coupled solid-fluid problem is solved. In a dynamic analysis, the inertial effects are included in the governing equations to calculate the time dependent response of the system. Many nonlinear constitutive models are available, allowing the user to model the often complicated biological tissue behavior. Several isotropic constitutive models are supported, such as Neo Hookean, Mooney Rivlin, Ogden, Arruda Boyce and Veronda Westmann, to name a few. In addition to the isotropic models, there are several transversely isotropic constitutive models available. These models exhibit anisotropic behavior in a single preferred direction and are useful for representing biological tissues such as tendons, muscles and other tissues that contain fibres. FEBio also contains a rigid body
111 constitutive model. This model can be used to represent materials or structures whose deformation is negligible compared to that of other materials in the overall model. In addition to the built in models, FEBio allows for definition of new models, as was done in this Thesis, due to the open source nature of the program. This was the primary reason for the selection of this code, along with its biomechanics specific nature. Since FEBio is primarily a nonlinear FE solver, it does not have mesh generation capabilities. Therefore, the input files need to be generated by means of a pre processing software. The preferred preprocessor for FEBio is called PreView. PreView can convert several other formats to the FEBio input specification. For instance, NIKE3D and ABAQUS input files can be imported into PreView and can be exported from Pre- View as a FEBio input file. The post processing for FEBio is done in a program called PostView, which is the third and final part of the package that is used to perform FE analysis in FEBio. It offers similar capabilities to commercial FE post processors, but it is slightly less polished due to its open source nature. For more information of FEBio, as well as PreView and PostView, please refer to http://mrl.sci.utah.edu/software. 7.2 Implementation of the Elastic Formulation In order to implement the non linear, biphasic model for biological tissues with statistically oriented reinforcing fibres, developed by Federico & Grillo (2012) and introduced in Chapter 6, it is first necessary to prepare the model for implementation. The first task is to select appropriate constitutive functions for each constituent in the model. This is followed by the derivation of the spatial stress and elasticity tensors, which are necessary for the code to perform the iterations. On the other hand, a general implementation algorithm will be developed, and the statistical nature of the tissue will be addressed when the permeability model is introduced, since this treatment is common to both models.
112 7.2.1 Constitutive Functions for the Elastic Formulation In order to implement the model developed by Federico & Grillo (2012) and implemented in this Thesis, it is first necessary to select the appropriate constitutive functions for the anisotropic ensemble fibre contribution, isotropic fibre contribution, and the proteoglycan matrix. In order to simplify calculations, and following Pajerski (2010), it was chosen to model the proteoglycan matrix and the isotropic fibre contribution by using same constitutive functions while using different material parameters to account for different material properties. Anisotropic Ensemble Fibre Potential By defining the subset S 2 X(C) = {M S 2 X : H(I 4 (C, A(M)) 1) > 0} (7.1) of the material unit sphere on which the stretch is greater than one, i.e., selecting only those directions in which a fibre experiences extension only, the ensemble fibre potential defined in equation (6.29) becomes V e (C) = V 1i (C) + ψ(m)v 1b (C, A(M)) ds (7.2) S 2 X with the anisotropic ensemble fibre potential defined as V 1a (C) = ψ(m)v 1b (C, A(M)) ds (7.3) S 2 X This expression of the model was used by Federico & Gasser (2010) in order to avoid the discontinuity of the Heaviside function when computing the derivative of V 1a (C) when evaluating the stress and elasticity tensors. Consequently, the tension compression non symmetry is accounted for by using this conditional statement in the implementation. As for the base anisotropic fibre potential, following Federico & Gasser (2010), the chosen constitutive function has the form V 1b (C, A(M)) = 1 c 2 1b(I 4 (C, A(M)) 1) 2 (7.4)
113 where c 1b is a material parameter related to the Young s modulus of a collagen fibre. The value of this material parameter, along with the other material parameters, will be defined later in the chapter, when the material parameters are selected. Isotropic Fibre and Matrix Constitutive Function A formulation for modelling of the behaviour of soft hydrated tissues under finite deformations has been proposed by Holmes (1986) and further generalized by Holmes & Mow (1990). In fact, the form of the model in Holmes & Mow (1990) is specifically formulated for modelling articular cartilage in confined compression, where no fibres undergo tension. Hence, it can be safely assumed that the potential describes the behaviour of the proteoglycan matrix and the always active isotropic contribution of the fibres. Due to its isotropy, the original form of the Holmes Mow potential, as proposed by Holmes & Mow (1990), can be expressed as a function of the invariants of the right Cauchy Green deformation tensor C as V HM (C) = α exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] [I 3 (C)] β (7.5) where the constants α, a 1, and a 2 must be positive in order to ensure convexity, β = a 1 + 2 a 2 in order to ensure that stress is zero in the undeformed state, and β 1 for articular cartilage (Holmes & Mow, 1990). When it is imposed that the linear elasticity tensor derived from V HM (C) is congruent with the elasticity tensor of the linear theory, then the material parameters from the Holmes Mow model are related to the Lamé parameters by λ = 4 α a 2, µ = 2 α (a 1 + a 2 ) (7.6) As previously mentioned, it was assumed that the same constitutive model will represent both the proteoglycan matrix and the isotropic fibre contribution, but that some of the material parameters will take a different value. Therefore, both the proteoglycan matrix and the isotropic fibre contribution will use the same constants a 1, a 2, and β, but
114 will have a different value for α: α 0 and α 1i, respectively. The two potentials then take the form V 0 (C) = α 0 exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] [I 3 (C)] β (7.7a) V 1i (C) = α 1i exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] [I 3 (C)] β (7.7b) 7.2.2 Derivation of the Spatial Stress and Elasticity Tensors In order to calculate the Cauchy stress and the spatial elasticity tensor, the objects required by the FE solver to find the field solution, it is necessary to use the definitions for a hyperelastic material, described in Chapter 2, in equations (2.91) and (2.94) and defined as σ = 2 J 1 χ [S], C = 4 J 1 χ [C] (7.8) In order to compute the necessary derivatives and define the Cauchy stress and the spatial elasticity tensor, it is first necessary to combine the equations that form the elastic strain energy potential into a single equation. This in turn makes the process of taking a derivative more straight forward, as it is simpler to notice the dependent terms in the whole expression. Hence, the elastic strain energy potential for the overall material, defined in equations (6.19), (6.22) and (6.29), and the potentials for the isotropic fibre contribution and the proteoglycan contributions, defined in the previous section, can then be combined in a single potential to form the overall strain energy potential as W (C) = φ sr [ H(J crit J(C))(J(C) J crit ) 2q (J(C) φ sr ) r ] + φ 0R [ + φ 1R {[ + S 2 X ] exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] α 0 [I 3 (C)] β α 1i exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] [I 3 (C)] β ψ(m)h(i 4 (C, A(M)) 1)V 1b (C, A(M)) ds ] } (7.9)
115 After observing the complete strain energy potential, it is useful to separate the potential into several different terms and then take the derivatives separately, as this can be performed due to the additive nature of the potential. Therefore, if the definition of the Holmes Mow potential from equation (7.5) is used, then it is convenient to write this potential as V HM (C) = α exp [a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3) β ln (I 3 (C))] (7.10) and to introduce the function θ(c) = a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3) β ln (I 3 (C)) (7.11) in order for the Holmes Mow potential to read as V HM (C) = α exp [θ(c)] (7.12) Taking the derivatives of the Holmes Mow potential, the expression for the second Piola Kirchhoff stress and the material elasticity tensor take the form S HM = 2 α exp [θ(c)] θ (C) (7.13) C and where C HM = 4 α exp [θ(c)] [ ] θ θ (C) C C (C) + 2 θ C 2 (C) (7.14) θ C (C) = (a 1 + a 2 I 1 (C)) G 1 a 2 G 1 C G 1 β B (7.15) where I is the identity tensor and B = C 1, and 2 θ C 2 (C) = a 2 (G 1 G 1 ) a 2 I + β I (7.16) where I = G 1 G 1 and I = B B (Federico, 2012), both of which are defined using special tensor products defined by Curnier et al. (1994). Since it is necessary to provide
116 the spatial quantities for implementation, an inverse Piola transform can be performed to obtain the definitions for the Cauchy stress and the spatial elasticity tensor as σ HM = 2 α exp [θ(c)] [ (a 1 + a 2 I 1 (C)) b a 2 b 2 β g 1] (7.17) and CHM = 1 exp ( θ(c)) [σ α HM σ HM ] + 4 α exp (θ(c)) [ 3 a 2 K a 2 I + β ] I (7.18) where b is the left Cauchy Green deformation tensor, 3K = b b, I = b b is the push forward of I = G 1 G 1, and I = g 1 g 1 is the push forward of I = B B. By applying the expressions for the Cauchy stress and spatial elasticity tensor, the expression for these two objects for both the proteoglycan matrix and the isotropic fibre contribution can be formulated as σ 0 = 2 α 0 exp [θ(c)] [ (a 1 + a 2 I 1 (C)) b a 2 b 2 β g 1] σ 1i = 2 α 1i exp [θ(c)] [ (a 1 + a 2 I 1 (C)) b a 2 b 2 β g 1] (7.19a) (7.19b) and C0 = 1 α 0 exp ( θ(c)) [σ HM σ HM ] + 4 α 0 exp (θ(c)) Ca C1i = 1 α 1i exp ( θ(c)) [σ HM σ HM ] + 4 α 1i exp (θ(c)) Ca (7.20a) (7.20b) where Ca = 3 a 2 K a 2 I + β I. Having now found the expressions for the Cauchy stress and the spatial elasticity tensor for both the proteoglycan matrix and the isotropic fibre contribution, it is straight forward to find the expressions for these both the stress and elasticity tensors for the overall elastic strain energy potential. Therefore, the overall expressions for the Cauchy stress and the spatial elasticity tensor take the forms [ ] U σ = φ sr (J) g 1 + φ 0R σ 0 + φ 1R σ 1i (7.21) J ( ) + 2 J 1 φ 1R F H(I 4 (C, A) 1) c 1b ψ(m)[i 4 (C, A) 1] A(M) ds F T S 2 X
117 and ( ) C = φ sr [3 J 2 U U (J) + J 2 J (J) K 2 U ] (J) I + φ 0R C0 + φ 1R C1i (7.22) J ( ) + 4 J 1 φ 1R F: H(I 4 (C, A) 1) c 1b ψ(m) A(M) A(M) ds : F T S 2 X where 3K = B B, I = B B, F = F F and F T = F T F T using the notation of Federico (2012), and U/ J(J) and 2 U/ J 2 (J) represent the first and second derivatives of the penalty function U(J), defined in equation (6.19). These derivatives can be written, for every J < J crit, as U J (J) = H(J crit J) [ 2q(J J crit ) 2q 1 (J φ sr ) r (7.23) r(j J crit ) 2q (J φ sr ) (r+1)] and 2 U J 2 (J) = H(J crit J) [ 2q(2q 1)(J J crit ) 2q 2 (J φ sr ) r (7.24) 2(2q)(r)(J J crit ) 2q 1 (J φ sr ) (r+1) + r(r + 1)(J J crit ) 2q (J φ sr ) (r+2)] This concludes the section on the derivation of the Cauchy stress and the spatial elasticity tensor. In the following section, the permeability formulation will be introduced. This will be followed the depth dependent nature of the tissue, along with the method for defining the statistical orientation of the fibres, and the evaluation of the directional average integral, as this approach is common to both elasticity and permeability formulations. 7.3 Implementation of the Permeability Formulation The first part of this chapter consisted of the implementation of the elastic formulation formulated by Federico & Grillo (2012), which is a natural evolution of the works
118 by Federico et al. (2005), Federico & Herzog (2008c), and Federico & Gasser (2010). This section deals with the implementation of the permeability formulation developed by Federico & Herzog (2008b) and generalized to large deformations by Federico & Grillo (2012). In this model, the fibres are modelled as impermeable inclusions in a porous, permeable matrix, and they provide an effect of directing the flow. This results in a different permeability values in the axial direction and transverse direction, with respect to the fibres. A full derivation and explanation of this model is presented by Federico & Grillo (2012), while a detailed summary is provided in Chapter 6. To the best of the author s knowledge, this is the first Finite Element implementation of a model in which the fibres have a distinct effect on the permeability. Since commercial Finite Element packages, such as ABAQUS, do not allow full control over the permeability formulation and explicit dependence of the permeability tensor on the deformation gradient F, as required by the expression (6.37), it was necessary to use an open source FE package, namely FEBio. Even though this entailed large amount of programming, it was necessary as it allowed full control over the relevant variables and it provided the necessary freedom required to implement the permeability formulation. This section will start of with a selection of an appropriate constitutive function that is dependent on purely the volumetric deformation, in order to account for the porosity of the matrix. This will be followed by a derivation of the fourth order tensor, called the tangent of permeability, that defines how the permeability changes with deformation. The section will be concluded by a short discussion on the depth dependent nature of the tissue and the directional average formulation, and the algorithm used to implement this formulation.
119 7.3.1 Constitutive Functions for the Permeability Formulation As it can be observed in equation (6.37), it is necessary to select an appropriate constitutive function for the matrix permeability k 0 to implement the full permeability model. It is also important to note that the permeability has to be physically admissible, i.e., it has to approach zero as J φ sr. Therefore, it is essential to assign a suitable constitutive function to the matrix permeability k 0, in order for the overall permeability k to vanish at compaction limit. The simplest possible assignment of k 0 is to assume that it is isotropic under any deformation, i.e., that it has a form k 0 = ˆk 0 g 1, where ˆk 0 is a constitutive function that depends on the volumetric deformation J, such as the Holmes Mow formulation (Holmes & Mow, 1990). Holmes Mow permeability function (Holmes & Mow, 1990) can be defined as ˆk 0 (J) = k 0R ( J φsr 1 φ sr ) κ0 e M 0 (J2 1) 2 (7.25) where k 0R, κ 0, and M 0 are material constants determined from experiments. Indeed, as mentioned in the requirements, this function has the property that the permeability ˆk 0 (J) approaches zero as J φ sr. The physical meaning of this expression is that as the pores close and the true volume of the sample approaches the volume of the solid fraction, the permeability goes to zero. This is indeed physically admissible, as when the pores close, there is no permeation of fluid through the solid, which means that the permeability is zero. In the next section, it will be necessary to define a tangent of permeability, a measure of how permeability changes with deformation. When this quantity is defined, it will be necessary to evaluate the derivative of the Holmes Mow permeability function with respect to J. If a partial derivative of the ˆk 0 (J) is taken with respect to J, as needed in the expression for ˆD(C) in equation (7.36), the derivative is defined as ( ) ( ) J ˆk 0(J) 2 κ0 M 0 + κ 0 JM 0 φ sr φsr J = k 0R e M 0 (J2 1) 2 (7.26) J φ sr φ sr 1
120 The expressions in equations (7.25) and (7.26) for the permeability and its derivative with respect to J, respectively, can now be implemented into the equation for permeability, expressed in equation (6.37) and the equation for the tangent of permeability, which is derived in the next section. 7.3.2 Derivation of the Fourth Order Permeability Tangent Tensor In order to implement the permeability formulation discussed in this Thesis into FEBio, it is necessary to formulate the the problem in a way that can be easily programmed. One of the quantities that FEBio requires for implementation of permeability is an object that describes the change in permeability tensor in response to the deformation of the tissue, where the deformation can be expressed through the right Cauchy Green deformation tensor C. An object that fits this criteria is a partial derivative of the permeability with respect to the deformation, called the tangent of permeability, and defined as where K = D = ˆD(C) = J 1 F : [ C ( J F 1 ˆk(C) F T ) ] : F T (7.27) ( ) J F 1 ˆk(C) F T is the material permeability. Therefore, in order to define the tangent of permeability, it is first necessary to perform a full Piola transform of the spatial permeability, in order to have the measure of permeability in materials reference frame, then take the derivative with respect to the right Cauchy Green deformation tensor C, and finally perform an inverse Piola transform to get a spatial tangent of permeability using the fourth order deformation gradient, defined as (Federico, 2012) F = F F and F T = F T F T (7.28) Recalling equation (6.37), the spatial permeability is defined as k = ˆk(C) ] = J 2 k 0 [(J φ 1R ) φ 1R F Ẑ(C) F T + (J φ 1R ) 2 g 1 (7.29)
121 where Ẑ(C) = Performing a full Piola transform K = J F 1 k F T material permeability as K = ˆK(C) = J 1 ˆk0 (J) S 2 X ψ(m) [C : A(M)] 1 A(M) ds (7.30) yields an expression for the ) ( ) 2B (J φ 1R φ 1R Ẑ(C) + J 1 ˆk0 (J) J φ 1R (7.31) where B = C 1, k 0 = ˆk 0 (J) g 1, and ˆk 0 (J) is the Holmes Mow constitutive permeability model defined in the previous section. The next step in the derivation of the tangent of permeability, as outlined in equation (7.27), is to take the derivative of the material permeability with respect to right Cauchy Green deformation tensor C. This yields an expression for the material tangent of permeability, which can be written as ˆD(C) = 1 2 J 1 ˆk0 (J) (J φ 1R ) φ 1R Ẑ(C) B (7.32) + 1 2 ˆk 0(J) (J φ 1R ) φ 1R Ẑ(C) B + 1 2 ˆk 0 (J) φ 1R Ẑ(C) B + J 1 ˆk0 (J) (J φ 1R ) φ 1R Ẑ C (C) 1 2 J 2 ˆk0 (J) (J φ 1R ) 2 B B + 1 2 ˆk 0(J) (J φ 1R ) 2 B B + ˆk 0 (J) (J φ 1R ) B B + J 1 ˆk0 (J) (J φ 1R ) 2 B B Grouping the like terms in equation (7.32) together and stating that 3K = B B and that I = B B, which represent the pulled-back contravariant spherical and identity tensors, respectively, and which are defined by Federico (2012) in great detail, equation (7.32) simplifies to ( ( ˆD(C) = 1 φ 2 1R J 1 ˆk0 (J) + ˆk 0(J) )( ) ) J φ 1R + ˆk0 (J) Ẑ(C) B (7.33) ( + J 1 ˆk0 (J) ( ) ) Ẑ J φ 1R φ1r C (C) ( (ˆk 0(J) ˆk 0 (J) ) ( ) 2 J φ 1R + 2 ˆk0 (J) ( ) ) J φ 1R K + 3 2 + 2 J 1 ˆk0 (J) ( J φ 1R ) 2 I
122 where Ẑ(C) is defined in equation (7.30), ˆk 0 (J) is the Holmes Mow permeability function defined in equation (7.25), ˆk 0(J) is the derivative of the Holmes Mow permeability function, as defined in equation (7.26), and Ẑ C (C) = ψ(m)[c : A(M)] 2 A(M) A(M) ds (7.34) S 2 X Following the procedure in equation (7.27), the next step to obtaining the spatial tangent of permeability is to perform an inverse Piola transform ˆD(C) = J 1 F: ˆD(C): F T using the fourth order deformation gradient definition from equation (7.28). This provides an expression for the spatial tangent of permeability given by ( ( ˆD(C) = 1 J 1 φ 2 1R J 1 ˆk0 (J) + ˆk 0(J) )( ) ) J φ 1R + ˆk0 (J) ẑ(c) g 1 (7.35) + (J 2 ˆk0 (J) ( ) [ ] ) J φ 1R φ1r χ Ẑ C (C) + 3 2 J 1 ( (ˆk 0 (J) ˆk 0 (J) ) ( J φ 1R ) 2 + 2 ˆk0 (J) ( J φ 1R ) ) K + 2 J 2 ˆk0 (J) ( J φ 1R ) 2 I where K = 1 3 g 1 g 1 and I = g 1 g 1. If the identity tensor in equation (7.35) is further decomposed into spherical and deviatoric part as I = K + M, then the expression for the spatial tangent of permeability ˆD(C) simplifies into ( ( ˆD(C) = 1 J 1 φ 2 1R J 1 ˆk0 (J) + ˆk 0(J) )( ) ) J φ 1R + ˆk0 (J) ẑ(c) g 1 (7.36) + (J 2 ˆk0 (J) ( ) [ ] ) J φ 1R φ1r χ Ẑ C (C) [ ( 3 + J (ˆk 1 2 0 (J) ˆk 0 (J) ) ( ) 2 J φ 1R + 2 ˆk0 (J) ( ) ) J φ 1R + 2 J 2 ˆk0 (J) ( J φ 1R ) 2 ] K + 2 J 2 ˆk0 (J) ( J φ 1R ) 2 M where [ ] ẑ(c) = F ψ(m) [C : A(M)] 1 A(M) ds F T (7.37) S 2 X
123 and [ ] χ Ẑ C (C) = F : [ ] ψ(m) [C : A(M)] 2 A(M) A(M) ds : F T (7.38) S 2 X This concludes the section on the derivation of he spatial tangent of permeability, with equation (7.36) representing the final expression. In the following section, the depth-dependent nature of articular cartilage will be addressed, and it will be explained how this depth dependence will be incorporated into the permeability formulation. 7.4 Probability distribution and Depth Dependence As mentioned in Chapter 4, articular cartilage is a tissue with depth dependent structure and composition. Hence, any mathematical formulation that is used to model the tissue has to be able to recreate such a structure and composition. Therefore, when using a model that considers individual contributions of both fibres and the matrix, it is necessary for both of these constituents to have depth dependent nature. As previously mentioned, the Holmes Mow potential is used under the assumption of material isotropy and homogeneity. However, the compressive strength of articular cartilage has been shown to be depth dependent (Schinagl et al., 1997), and this depth dependence can be at least partially ascribed to the depth dependent volumetric fraction of proteoglycans and fibres, which make up the volumetric fraction of solid. Following the work of Pajerski (2010), the information regarding depth dependence of the tissue is included in the material parameters α 0 and α 1i, the material constant appearing in the potential for the proteoglycan matrix and the isotropic fibre contribution, respectively. By means of the material form of the continuity equation written in terms of the volumetric fractions, J takes the form J = φ 0R φ 0, J = φ 1R φ 1 (7.39)
124 Recalling that I 3 (C) = J 2, the expressions for the potentials of the proteoglycan matrix and the isotropic fibre contribution can be rewritten as V 0 (C) = α 0 φ V 1i (C) = φ 2β 0R α 1i 2β 1R φ 2β 0 exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] (7.40a) φ 2β 1 exp[a 1 (I 1 (C) 3) + a 2 (I 2 (C) 3)] (7.40b) This form of the Holmes Mow potential implies that the current volumetric fractions φ 2β 0 and φ 2β 1 carry the information of the volumetric deformation. On the other hand, the material constants A 0 = α 0 φ 2β 0R, A 1i = α 1i φ 2β 1R (7.41) are independent of the deformation, and can be thought to depend exclusively on the molecular structure of the proteoglycans and the fibres, respectively. The value for these constants can be retrieved from the homogeneous case as A 0 = φ α 0 hom 2β 0R hom, A 1i = α 1i hom φ 2β 1R hom (7.42) where α 0 hom and α 1i hom are the values found by fitting a small strain compression test based on the overall deformation of the sample, and φ 0R hom and φ 1R hom are the spatial averages of φ 0R and φ 1R, respectively, which, if the cartilage is inhomogeneous only in the direction of the tissue depth, can be found through an average theorem. Once A 0 and A 1i have been evaluated, the material parameters α 0 and α 1i in the Holmes Mow potentials for the proteoglycan matrix and the isotropic fibre contribution can be related to the respective volumetric fractions by means of a power law as α 0 = A 0 φ 2β 0R, α 1i = A 1i φ 2β 1R (7.43) where α 0 and α1i depend on the location in the tissue through the volumetric fractions φ 0R and φ 1R, respectively.
125 When considering the anisotropic fibre potential, it is necessary for the probability distribution function ψ(m) to be transversely isotropic in the direction W of the tissue depth, if the tissue is to be considered transversely isotropic and depth dependent. It is often convenient to select the global reference frame {E i } 3 i=1 such that the direction K coincides with E 1. Hence, any vector M can then be described in terms of the co latitude Θ and longitude Φ with respect to the polar direction W, as shown in Figure 7.1. A probability distribution is transversely isotropic with respect to direction W if, and only if, it depends uniquely on the co latitude angle Θ, but not on the longitude Φ. Figure 7.1: For all points in the material, the probability distribution function ψ(θ, Φ) represents the likelihood of finding the collagen fibre tangent (shown in blue) oriented in the direction of M. By integrating over the unit sphere S 2 X, all directions are taken into account and a probability density emerges. In order to visualize these probability densities, the Reader can refer to Figure 7.2 for a graphical representation. Figure adapted from Pajerski (2010). A function ψ(θ, Φ) (or ρ(θ), which is defined purely as a function of the co latitude angle Θ) obeying the condition outlined above is the Von Mises distribution, defined as ψ(θ, Φ) = ρ(θ) = 1 [ ] b exp [b(cos (2Θ) + 1)] π 2π erfi( (7.44) 2b) where erfi is the imaginary error function defined as erfi(x) = i erf(i x), and b is the concentration parameter quantifying how close the distribution is peaked around the
126 polar direction W. Figure 7.2 shows how the probability density distribution varies for values of b ranging from to +. As previously mentioned, the Holmes Mow permeability function is used under the assumption of material isotropy and homogeneity. In fact, this model does not take into account the presence of fibre and it is solely dependent on the void ratio, which is expressed through J. Since the Holmes Mow potential is used to define reference, undeformed, permeability, while equation (6.37) takes into account the presence of collagen fibres and how these fibres affect the fluid flux through the tissue, it is important to properly model the both the depth dependent matrix porosity, which is a function of both the applied strain and the depth of the tissue, and the depth dependent orientation of the fibres, which, as previously mentioned, are thought to direct the flux. In order to properly model the depth dependent characteristics of the matrix, which are defined by the Holmes Mow permeability model, shown in equation (7.25), and are explicitly dependent on J, the Holmes Mow potential is evaluated for every material point. Since the tissue has different solid and fluid fractions at different material points in the depth direction, it is implied that the referential, undeformed permeability k 0R featuring in the matrix permeability tensor k 0 will be different at different depths. This will be further discussed when addressing the selection of material parameters, later in this chapter. A concentration parameter tending towards will represent the fibres isotropically oriented in the transverse plane, while a concentration parameter of 0 will represent randomly oriented fibres. Lastly, a concentration parameter tending towards + will represent fibres oriented in the polar direction. These three cases can then be taken to represent the fibre orientation in the superficial, middle, and the deep zones of articular cartilage, respectively. In other words, to obtain the depth dependent nature of the contribution of the fibres, it is necessary to make the concentration parameter b a function
127 of the tissue depth that takes a large positive value in the deep zone, passes through zero in the middle zone, and takes a large negative value in the superficial zone. Therefore, following the previously defined analogy, different orientation of the fibres in different zones will result in different anisotropic fibre contribution in the elasticity formulation and different flux directions in the permeability models, since the fibres can direct the flow. Figure 7.2: Transversely isotropic probability density functions describing fibre orientation. Surface plots are defined by the vector ρ(θ)m, where M represents points on the unit sphere. Figure adapted from Federico & Gasser (2010). As shown in this section, both the potentials for the proteoglycan matrix and isotropic fibre contribution, along with the anisotropic fibre contribution and thee permeability formulation, are depth dependent in nature. This accounts for the depth dependent composition and structure of articular cartilage, and creates a more histologically accurate mathematical model. In the next section, the algorithms used to implement the
128 elastic model and the permeability formulation are described and outlined. 7.4.1 Algorithm for the Elastic Formulation In order to implement the elastic formulation, three different algorithms have to be utilized: one to implement the Holmes Mow potential for the proteoglycan matrix and isotropic fibre contribution, one to implement the anisotropic fibre potential, and the last algorithm to implement the penalty function U(J) along with its derivatives. The Holmes Mow potential has been extensively used to model cartilage behaviour, which means that there are a number of different algorithms available. Since the FE package that was used for this project, FEBio, is specifically written for biological applications, it has a number of different constitutive models, one being the Holmes Mow model. In order to make the implementation of the full elastic formulation as reliable as possible, and due to the amount of work required to implement the anisotropic fibre contribution, the built in Holmes Mow potential was used. For a detailed algorithm of this implementation, the Reader can refer to Maas et al. (2010). The algorithm used to implement the anisotropic fibre contribution was developed by Federico & Gasser (2010), and it is outlined in Table 7.1. Both the Cauchy stress and the spatial elasticity tensor involve the evaluation of an integral over an unit sphere. These integrals are evaluated through the method of spherical designs, a procedure that optimizes the distribution of a discrete set of points on a sphere so that the numerical integration over that sphere can proceed with equal weights (Hardin & Sloane, 1996). Once an integration order is defined, based on the value of concentration parameter b, a set of optimized points {M (α) } N α=1 S 2 X, known as a spherical design, is loaded. Each directional point M (α) corresponds to a material direction for which the collagen fibre stiffness contribution is evaluated. The spherical design is such that, for a function f on
129 S 2 X, it can be represented as S 2 x f(m) ds = 4π N N f(m (α) ) (7.45) The evaluation begins with the determination of invariant I 4 in order to determine whether the fibre corresponding to the material direction M is in tension or compression. If it is in tension, the evaluation of the fibre contribution to the spatial stress and elasticity tensor proceeds. If the fibre is not in tension, the fibre contribution for a given direction M is zero and the subroutine moves on to evaluate another direction of the unit sphere. Lastly, a simple algorithm was also used to implement the penalty function U(J). This algorithm consisted of a simple if statement that evaluates the function and its derivatives (shown in equations 6.19, 7.23, and 7.24) if J < J crit, and takes no action otherwise. This algorithm, along with the rest of the implementation, can be seen in the code reported in Appendix A.1. This concludes the section on the implementation of the elastic formulation. In the next section, the implementation of the permeability formulation is addressed, and an appropriate algorithm is developed. α=1 7.4.2 Algorithm for the Permeability Formulation The algorithm used to implement the permeability formulation is an adaptation of an algorithm used to implement the anisotropic ensemble fibre potential, developed by Federico & Gasser (2010) and introduced in the previous section, and it is outlined in Table 7.2. The evaluation begins with the determination of the fibre direction M, the structure tensor A, and the invariant I 4. These quantities are then used to evaluate the directional average integral, which defines the statistically dominant fibre direction. This fibre direction is then used to compute the overall permeability tensor. The code for the implementation of the permeability formulation into FEBio is presented is Appendix A.2.
130 This concludes the section on the implementation of the permeability formulation. The next section will contain the detailed description of the FE simulations that were performed, along with the appropriate geometries, boundary conditions, and assumptions. Table 7.1: The algorithm used for the finite element implementation of the anisotropic ensemble fibre potential in the implemented model. It has been implemented in FEBio (Maas et al., 2010) as a stand alone class, and it is called by the solver during each evaluation step. The algorithm was originally developed by Federico & Gasser (2010) GIVEN: deformation gradient F orientation distribution data K, b THEN: define integration order b load spherical design M (α), α {1, 2,..., N} Initialize stress and elasticity tensors σ 1ea 0, C1ea O DO: α = 1...N compute material structure tensor A (α) = M (α) M (α) compute fourth invariant I (α) 4 = C (α) : A (α) compute co latitude angle Θ (α) = arccos K M (α) IF (I (α) 4 > 1) THEN compute orientation density ρ (α) = 1 b π 2π erfi( 2b) IF (ρ (α) > ɛ) THEN compute V 1a (C, [ A(M)) contribution ] to stress and elasticity tensors σ 1ea = 2 J 1 F ρ (α) V 1b (I(α) 4 ) A (α) F T [ ] C1ea = 4 J 1 F : ρ (α) V 1b (I(α) 4 ) A (α) A (α) : F T update V 1a (C, A(M)) contribution to stress and elasticity tensors σ 1ea σ 1ea + 4π N σ(α) 1ea C1ea C1ea + 4π N C(α) 1ea END IF END IF END DO [ exp [b(cos (2Θ)+1)] ]
131 Table 7.2: The algorithm used for the finite element implementation of the permeability tensor in the implemented model. It has been implented in FEBio (Maas et al., 2010) as a stand alone class, and it is called by the solver during each evaluation step. The algorithm is an adaptation of an algorithm originally developed by Federico & Gasser (2010) for finding the anisotropic ensemble fibre potential GIVEN: deformation gradient F orientation distribution data K, b THEN: define integration order b load spherical design M (α), α {1, 2,..., N} Initialize permeability and tangent of permeability k 0, D O DO: α = 1...N compute material fibre direction M (α) compute material structure tensor A (α) = M (α) M (α) compute fourth invariant I (α) 4 = C (α) : A (α) compute co latitude angle Θ (α) = arccos [ K M (α) ] compute orientation density ρ (α) = 1 b exp [b(cos (2Θ)+1)] π 2π erfi( 2b) compute Z (α) = ρ (α) [C (α) : A (α) ] 1 A (α) Z (α) compute C = ρ (α) [C (α) : A (α) ] 2 A (α) A (α) update the integral Z Z + 4π N Z(α) Z update the integral Z + 4π Z C C N C END DO push forward the integrals z F Z F T, χ [ ] Z C Z C compute Holmes Mow potential ˆk0 (J), ˆk 0(J) evaluate k and D equations (6.37) and (7.36) (α)
132 7.5 Cartilage Modelling The theoretical model of articular cartilage presented in this Thesis has been implemented using the FE method to simulate the unconfined compression tests performed by Guilak et al. (1995), and to recreate the results implemented by Pajerski (2010) using FE method and a small deformation statistically oriented reinforcing fibre model. The numerical results from the simulations performed in this Thesis were then compared to experimental results and the previous numerical results. The assumptions made in the analysis regarding geometry, articular cartilage composition, boundary conditions, and material parameters are outlined in this section. 7.5.1 Attachment to Subchondral Bone In the unconfined compression experiments performed by Guilak et al. (1995), the articular cartilage test specimen were attached to the subchondral bone. In the simulations performed as a part of this Thesis, the calcified zone of the articular cartilage is ignored and it is assumed that the cartilage attaches directly to the bone, which is assumed to be rigid. This is a valid assumption as bone is significantly stiffer than cartilage. Since the FE geometry represent only the articular cartilage tissue, it is assumed that the bottom boundary of the geometry is fixed to simulate the attachment to the subchondral bone. 7.5.2 Articular Cartilage Composition The three components of the articular cartilage relevant to the FE model implemented in this Thesis are the fluid, collagen fibre, and proteoglycan matrix components. The values for the depth dependent volumetric fractions of each of the relevant components were based on literature values (Hedlund et al., 1993; Chen et al., 2001; Wilson et al., 2007; Athanasiou et al., 2009), and the fractions of each relevant component are given is terms of the volumetric fractions in reference configuration. These volumetric fractions
133 are given as a function of the normalized depth ξ, where it is assumed that ξ = 0 at the cartilage bone interface and ξ = 1 at the articulating surface. The expression for the volumetric fractions of the fluid, matrix, and fibre components are then respectively given by (Pajerski, 2010) φ fr (ξ) = 0.75 + 0.1ξ φ 0R (ξ) = 0.062ξ 2 + 0.038ξ + 0.046 φ 1R (ξ) = 0.062ξ 2 0.138ξ + 0.204 (7.46a) (7.46b) (7.46c) It should also be noted that the volumetric fraction of the solid, φ sr, can be found by adding the volumetric fractions of the matrix and the fibres as φ sr = φ 0R + φ 1R (7.47) 7.5.3 Articular Cartilage Properties The material properties of the proteoglycan matrix were obtained from Athanasiou et al. (1991). It should be noted that the obtained material properties are for the proteoglycan matrix/chondrocyte cell mixture, where the chondrocyte cell can be thought of as softer and more permeable inclusions in the proteoglycan matrix. Effectively, the inclusion of chondrocyte cells decreases the effective stiffness and increases the permeability of the tissue when compared to the pure proteoglycan matrix. Since the material properties used in this Thesis were obtained by fitting experimental results from indentation tests (Athanasiou et al., 1991) to a theoretical model of biphasic indentation developed by Mak et al. (1987), and since these indentation tests were carried out on intact cartilage samples containing both proteoglycan matrix and chondrocyte cells, it is assumed that the material parameters obtained take into account the entire chondrocyte cell/proteogylcan matrix mixture.
134 The material properties used to model the matrix are the aggregate modulus, H A0 exp = 0.5 MPa, and the Poisson s ratio ν 0 = 0.1. The subscript exp is added to the matrix stiffness to indicate that it is based on the experiments conducted on the global articular cartilage matrix. Consequently, the proteoglycan fraction is intrinsic to the value of H A0 exp, and, in order to maintain consistency with the formulation, the global stiffness H A0 exp was related to the actual matrix stiffness by φ 0R H A0 = H A0 exp. The material parameters used in the Holmes Mow potential were derived using H A0 and ν 0, and by enforcing that the Holmes-Mow potential reduces to the linear case of infinitesimal strain. Therefore, the material parameters for the Holmes Mow potential can be expressed as E 0 = H A0 (1 + ν 0 ) (1 2 ν 0 ) 1 ν 0 (7.48a) α 0 = 2 µ 0 + λ 0 4 β (7.48b) a 1 = β 2 µ 0 λ 0 2 µ 0 + λ 0 (7.48c) λ 0 a 2 = β 2 µ 0 + λ 0 (7.48d) β = a 1 + 2 a 2 = 1 (7.48e) where λ 0 and µ 0 are Lamé s first and second parameters for the proteoglycan matrix. The Young s modulus, E 1, and the Poisson s ratio, ν 1, of the collagen fibres were taken from published experimental data by Pins et al. (1997), and they have values of E 1 = 10 MPa and ν 1 = 0.3. The isotropic fibre contribution was given the same material parameters as the proteoglycan matrix, while the basic anisotropic fibre potential material parameter c 1b in anisotropic fibre potential V 1b in equation (7.4) was related to the collagen fibre material properties determined by Pins et al. (1997) by evaluating the spatial elasticity tensor C1, representing both isotropic and anisotropic contribution of hte fibres, at zero strain, and then selecting the appropriate axial components. The process
135 followed is outlined by Pajerski (2010), and it results in an expression for c 1b given as c 1b = 1 4 [ E 1 (1 ν 1 ) (1 + ν 1 ) (1 2 ν 1 ) E ] 0(1 3 ν 0 ) (1 + ν 0 ) (1 2 ν 0 ) (7.49) In order to implement the permeability formulation, it is necessary to determine the material parameters needed to implement the Holmes Mow permeability function, expressed in equation (7.25). The material parameters M and κ are determined experimentally and they were obtained from the literature (Holmes, 1986), where the values are M = 4.638 and κ = 0.0848. It is also necessary to define the initial, or referential, value of homogeneous permeability of cartilage, k 0R hom, where this value represents the homogeneous permeability of the tissue before any motion has occurred. From literature, this value was found to be approximately k 0R hom = 0.001 mm 4 /(N s) (Athanasiou et al., 1991). Since the initial void ratio, as well as J, vary with depth, it is assumed that the permeability also varies with depth. Therefore, the Holmes Mow permeability function, expressed in equation (7.25), can be slightly modified to define the the referential permeability as a function of the tissue depth. The modified expression then takes the form ( ) κ0 J(ξ) φsr k 0R (ξ) = k 0R hom e M 0 (J(ξ)2 1) 2 (7.50) 1 φ sr where J(ξ) = 1 + e R(ξ) 1 + e R hom (7.51) where e R (ξ) = φ fr (ξ)/φ sr (ξ), and e r hom, or the homogeneous void ratio for articular cartilage, is approximately 4.0 (Mow et al., 1984). Through equations (7.25) and (7.50), it is clear that the the initial permeability of articular cartilage ˆk 0 (J) will be both strain and depth dependent, respectively.
136 7.5.4 Unconfined Compression In their experiments, Guilak et al. (1995) conducted unconfined compression tests on canine articular cartilage samples from the patellofemoral joint. The scheme in Figure 7.3 shows the meshed FE model used to simulate these tests, both as a two dimensional schematic and a three dimensional model. In the FE model, the sample is 3 mm in diameter, 1 mm in height, and fixed at the base in all three principal directions e 1, e 2, and e 3 to simulate the attachment to the subchondral bone. The top and bottom boundaries are assumed to be impermeable, and the pore pressure at the side of the sample in Figure 7.3 is assumed to be zero, which implies that the side of the sample is permeable. An overall compressive strain of 15% in the e 3 direction was applied to the sample during a 30 second ramp, and the strain was then held constant up to a time of 1200 seconds. Figure 7.3: Two dimensional schematic and a three dimensional representation of the meshed FE model representing the unconfined compression tests performed by Guilak et al. (1995) The histological analysis performed by Guilak et al. (1995) indicated that the deep zone occupies approximately 50% of the total articular cartilage thickness, the middle zone occupies 40%, and the superficial zone occupies the remaining 10%. The depth
137 dependent collagen fibre orientation assumed in the FE analysis, used for both the elasticity and the permeability formulation, was based on these histological observations and the corresponding depth concentration parameter curve is shown in Figure7.4. Figure 7.4: Plot of normalized depth ξ versus the concentration parameter b, used to define the dominant collagen fibre orientation in bot hthe elastic and permeability formulation. The plot is based on the histological observations made by Guilak et al. (1995), and it has been adapted from Pajerski (2010) This concludes the chapter on the numerical implementation of the elasticity and permeability formulations introduced in Chapter 6. The following chapter will introduce the theory of growth and remodelling of biological tissues, and how it can be applied to fibre reinforced biological tissues.
138 Chapter 8 Growth and Remodelling in Fibre Reinforced Biological Tissues The mechanical behaviour of biological tissues is influenced by the presence of inclusions, and the multi phasic nature of the tissue. For example, in the case of articular cartilage, the mechanical properties and the orientation of the collagen fibres define the directional nature of the material properties, and make the tissue highly anisotropic. If it is assumed that these inclusions evolve over time, the tissue adapts to the external stimuli by rearranging the organization of the inclusions. This rearrangement of the internal structure, generally known as remodelling (Taber, 1995), implies that tissue symmetries, or the directional nature of the material properties, evolve over time. In a purely mechanical context, this phenomenon occurs as a response to an external loading: the remodelling of the tissue attempts to optimise the internal stresses. For this reason, an accurate characterization of the mechanical behaviour of the tissue should be able to predict how the inclusions evolve, and how their evolution is related to the quantities that determine the mechanical state of the tissue. Similarly, biological tissues may experience growth, or a change in mass, over time as a response to an external stimuli. It should be stressed here that this implies production or depletion of mass rather than a variation of mass due to influx or outflux of mass through the macroscopic boundaries of the body. In general, the change in the tissue that occurs during growth of an unloaded body may be due to, at the very least, two processes: material being added or removed, therefore changing the local stress free reference state of the tissue, and an elastic deformation that may be necessary to accommodate this change in tissue configuration and volume in order to make the total deformation compatible (i.e., so that material can undergo growth without inducing discontinuities in the
139 body). Since the growth process can be split into two parts, as previously mentioned, the total deformation can be decomposed into two parts: the change in the tissue stress free reference state, generally known as the growth, and the elastic deformation, as illustrated in equation (2.95). In order to define both growth and remodelling, it is necessary to formulate the evolution equations, which define how the tissue grows and remodels over time. For the general case, the evolution equations for growth and remodelling have been derived in Chapter 2, and they are specifically defined in equations (2.105a) and (2.105b), respectively. In the study of the benchmark problem considered in this Chapter, the effect of the growth is not considered, and the focus is shifted to the remodelling of the tissue. More specifically, the remodelling parameter considered is the mean fibre orientation angle ω, and the change in the mean fibre angle as a response to external stimuli. In the following sections, a benchmark problem will be introduced, and the mathematical formulation governing this model will be elucidated. The chapter will be concluded with a section regarding the numerical implementation of the mathematical formulation. 8.1 Mathematical Formulation In order to develop the mathematical formulation that will represent the behaviour of the physical tissue, it is first necessary to define the geometry. This Chapter deviates from the main biological theme of the Thesis, i.e., articular cartilage, and focuses on a problem of the stresses in an internally pressurised fibre reinforced cylindrical vessel. The reason for not using articular cartilage as the studied tissue in this benchmark problem are two fold; articular cartilage is a significantly more complex tissue, with depth dependent mechanical properties and a high degree of anisotropy. This, in turn, makes the formulation and implementation of the mathematical model significantly more complex and,
140 due to the inherent complexity of the remodelling and the inherent complexity of the statistical fibre orientation, computationally intensive. Therefore, it was decided that, as a demonstration of the effect of remodelling of statistically oriented fibre reinforced tissues, the geometry and loading scenarios are to be reasonably simple. The second reason for choosing a fibre reinforced hollow cylinder is for a direct comparison to the work of Olsson & Klarbring (2008), who simulated growth and remodelling of the arterial tissue using the Holzapfel Gasser Ogden constitutive model (Holzapfel et al., 2000) for fibre reinforced materials. The benchmark problem studied could then be further modified to more accurately represent the arterial geometry. In order to formulate the mathematical model, it is first necessary to define the geometry. This is followed by the application of the equilibrium equations, expressed in equation (2.99a), to the geometry, and application of the boundary conditions. Then the appropriate constitutive models are selected, and the remodelling equation is redefined from equation (2.105b). Finally, the appropriate initial conditions will be discussed and chosen. 8.1.1 Geometry The geometry is assumed to approximate a rotationally symmetric body. Hence, this problem is best suited to a cylindrical basis, where (E R, E Θ, E Z ) form the basis in B 0 and (e r, e θ, e z ) form the basis in B(t) S. The inner and outer radii in the hollow cylinder, represented in B 0, are denoted by R 0 and R 1, respectively, which implies that R [R 0, R 1 ] and Θ [0, 2π]. The deformation studied in this benchmark problem is pure inflation of a thick walled cylinder, which implied that the motion χ(x, t) is given by χ(, t) : (R, Θ, Z) (r(r, t), Θ, Z) (8.1)
141 and the deformation gradient F is represented as F = r R (R, t) e r E R r(r, t) + R e θ E Θ + e z E Z. (8.2) Due to the incompressible nature of the problem being studied, the radial deformation has to comply with the constraint det(f ) = 1, which results in a separable differential equation r (R, t) r(r, t) = R. (8.3) R By solving equation (8.3), an expression for the current radius and a physical constraint can be obtained as r(r, t) = R 2 + K(t) (8.4) where K(t) is an unknown time dependent integration constant. 8.1.2 Boundary Conditions and Simplified Balance of Momentum In order to simplify the balance of momentum, it is first necessary to define the boundary conditions, which can be defined through a force balance at the boundary of the body. Therefore, at the outer and inner walls, these force balances state that τ (r 1, t) g = λ 1 n 1, τ (r 0, t) g = λ 0 n 0 (8.5) where τ represents the distribution of the contact forces defined per unit surface area of the current configuration of the body, n 1 e r (r 1, t) and n 0 e r (r 0, t) are the unit vectors normal to the outer and inner walls, respectively, and λ 1 and λ 0 are scalar constants having the physical dimensions of pressure. These boundary conditions are illustrated in Figure 8.1. The surface traction t, featured in equation (2.99b), and defined per unit surface of the reference configuration of the body, is given by t = τ J N C 1 N (8.6)
142 Figure 8.1: A schematic representation of the boundary conditions in the inner and outer wall of the fibre reinforced hollow cylindrical geometry used in the study or a benchmark problem modelling the evolution of a fibre reinforced material under external stimuli in a form of an applied pressure λ. The subscripts 0 and 1 denote the inner and outer walls, respectively. Figure has been adapted from Olsson & Klarbring (2008) where J N C 1 N is the factor that accounts for the change of surface measure when passing from the current to the reference configuration (Bonet & Wood, 2008). Substituting equations (8.6) into equation (8.5), using Nanson s formula, and accounting for incompressibility constraint yields P N = λ 1 g 1 F T N on B 1, (8.7a) P N = λ 0 g 1 F T N on B 0 (8.7b) where G is the material metric tensor, and g 1 is the inverse spatial metric tensor. Assuming that the components of the stress tensor do not depend on the coordinates Θ and Z, the boundary conditions in equations (8.7a) and (8.7b), as well as the symmetry requirement of the Cauchy stress tensor, P F T = F P T, expressed in equation (2.99c), are sufficient to ensure that the only nonzero components of P are P rr and P θθ. Therefore,
143 the boundary conditions can be reformulated as R P rr (R 1, t) = P1 rr 2 (t) = λ 1 + K(t) 1 R 1 R P rr (R 0, t) = P0 rr 2 (t) = λ 0 + K(t) 0 R 0 (8.8a) (8.8b) It is also convenient to express the first Piola Kirchhoff stress tensor P as a decomposition of the volumetric part and the deviatoric part as P = p g 1 F T + P d, (8.9) where p is the hydrostatic pressure and P d is the deviatoric contribution of the first Piola Kirchhoff stress tensor. By expanding the momentum balance, expressed in equation (2.99a), the balance of momentum in radial direction is given by P rr R + 1 R ( P rr P θθ) = 0 (8.10) The radial momentum balance is the crucial balance equation in this problem, and the only one used, as the load is applied in the radial direction. Hence, it is the relation that will govern the system. In order to apply the boundary conditions to this equation, it is first necessary to integrate equation (8.10) with respect to R and then apply the boundary conditions. By assuming that no pressure acts on the outer surface, i.e., λ 1 = 0, using the constitutive relation (8.9), and by applying boundary conditions (8.8a) and (8.8b) to the momentum balance in equation (8.10), it is possible to obtain an expression for the inner pressure λ 0 in terms of the deviatoric stress components as λ 0 = R1 R 0 1 r(r, t) [ ] Pd θθ R2 r(r, t) P rr 2 d dr (8.11) In order to satisfy the boundary conditions, this equation has to be solved for the unknown stress fields P θθ d in equation (8.4). and Pd rr, as well as the unknown deformed radius r, expressed Since there is only one equation and three unknowns, the system
144 cannot be solved, and it is necessary to introduce additional equations in order to solve the constraint equation (8.11). These additional equations are the constitutive equation for the arterial material, defined in the next section, and the remodelling equation, defined after the following section. 8.1.3 Constitutive Model for the Fibre Reinforced Cylindrical Geometry Since, as previously mentioned, the geometry is being modelled as an incompressible solid, i.e., det F = 1, it is possible to use the a decoupled elastic strain energy potential in the form W(J, C) = U(J) + W ( C) (8.12) where U(J) = p(j 1), J 1 = 0 is an incompressibility constraint, and p is the hydrostatic pressure. The deviatoric contribution to the overall strain energy potential is given by W ( C) = φ 0R W0 ( C) + φ 1R S 2 ψ(m, ω) W 1 ( C, A(M)) ds (8.13) where ψ(m, ω) is a probability distribution function that defines the statistically dominant fibre direction, and it depends on both the fibre direction vector M and the mean fibre angle ω. If the fibre contribution W 1 ( C, A(M)) is further split into the isotropic and the anisotropic contribution, equation (8.13) can be rewritten as W ( C) = (φ 0R W0 ( C) + φ 1R W1i ( C)) + φ 1R S 2 ψ(m, ω) W 1a ( C, A(M)) ds (8.14) where the directional average of the isotropic contribution W 1i ( C)) coincides with the isotropic part itself, as it does not depend on the direction and the probability ψ is assumed to be normalised to one. On the other hand, the anisotropic fibre ensemble potential W1a ( C, A(M)) is the source of the anisotropy in the tissue, and it depends on the direction of the fibres. The expressions for these potential, along with isotropic fibre
145 and matrix potentials, are given by W 0 ( C) = 1 2 c 0 (Ī1( C 3)) W 1i ( C) = 1 2 c 1 (Ī1( C 3)) W 1a ( C, A(M)) = H(Ī4( C, A(M)) 1)c 2 (Ī4( C, A(M)) 1) 2 (8.15a) (8.15b) (8.15c) where both the matrix and the isotropic contribution of the fibres are modelled as incompressible Neo Hookean solids, the anisotropic fibre contribution is modelled by a quadratic term, and a Heaviside step function is used to account for stretched fibres only, as expressed in equation (6.28). The constants c 0, c 1, and c 2 act as the material parameters for the matrix, isotropic fibre contribution, and the anisotropic fibre contribution, respectively, and A(M) = M M is the material structure tensor. Since the second Piola Kirchhoff stress tensor is defined as S = 2 (W) C (J, C) for hyperelastic materials, the application of this derivative to the decoupled potential in equation (8.12) yields S = U J J C + W J J C + W C : C C (8.16) where p = U J simplified into is the hydrostatic pressure. Since W J S = 1 2 p J C 1 + W C : C C = 0, equation (8.16) can be (8.17) By evaluating the derivatives in equation (8.17) and using the constitutive model in equation (8.14), the expression for the second Piola Kirchhoff stress tensor becomes S = p J C 1 + J 2/3 (φ 0R c 0 + φ 1R c 1 ) [G 1 13 ] (G 1 : C) C 1 + 2 J 2/3 c 2 φ 1R ψ(m, ω) H [ Ī 4 ( C, A(M)) 1 ] S 2 [Ī4 ( C, A(M)) 1 ] [ A(M) 1 ] 3 (C : A(M)) C 1 ds (8.18)
146 Performing a push forward operation P = F S on equation (8.18), the expression for the first Piola Kirchhoff stress tensor P becomes P = p J g 1 F T + J 2/3 (φ 0R c 0 + φ 1R c 1 ) [F G 1 13 ] (G 1 : C) g 1 F T + 2 J 2/3 c 2 φ 1R ψ(m, ω) H [ Ī 4 ( C, AM) 1 ] S 2 [Ī4 ( C, A(M)) 1 ] [ F A(M) 1 ] 3 (C : A(M)) g 1 F T ds (8.19) Expressing the two components of P required by the constrain equation (8.11), P rr and P θθ, the expressions for these two components become [ P rr R = p r(r, t) + (φ 0R c 0 + φ 1R c 1 ) + 2 c 2 φ 1R ψ(m, ω) H [ Ī 4 ( C, A(M)) 1 ] S 2 R r(r, t) 1 3 (G 1 : C) ] r(r, t) R [Ī4 ( C, A(M)) 1 ] [ R r(r, t) ARR 1 r(r, t) (C : A(M)) 3 R ] ds (8.20) and P θθ = p [ R r(r, t) r(r, t) + (φ 0R c 0 + φ 1R c 1 ) R 1 3 (G 1 : C) ] R r(r, t) + 2 c 2 φ 1R ψ(m, ω) H [ Ī 4 ( C, A(M)) 1 ] S 2 [Ī4 ( C, A(M)) 1 ] [ r(r, t) R AΘΘ 1 3 (C : A(M)) R r(r, t) ] ds (8.21) In order to substitute the expressions for the radial and tangential first Piola Kirchhoff stress tensors P rr and P θθ, respectively, into the constraint equation (8.11), it is necessary to determine purely deviatoric component of both P rr and P θθ. This can be done using the constitutive relation (8.9), and it yields expressions for Pd rr [ Pd rr = (φ 0R c 0 + φ 1R c 1 ) + 2 c 2 φ 1R R r(r, t) 1 3 (G 1 : C) ] r(r, t) R and P θθ d ψ(m, ω) H [ Ī 4 ( C, A(M)) 1 ] S 2 [Ī4 ( C, A(M)) 1 ] [ R r(r, t) ARR 1 ] r(r, t) (C : A(M)) ds 3 R given by (8.22)
147 and [ r(r, t) Pd θθ = (φ 0R c 0 + φ 1R c 1 ) R 1 ] R 3 (G 1 : C) r(r, t) + 2 c 2 φ 1R ψ(m, ω) H [ Ī 4 ( C, A(M)) 1 ] S 2 [Ī4 ( C, A(M)) 1 ] [ r(r, t) R AΘΘ 1 ] 3 (C : A(M)) R r(r, t) ds (8.23) Substituting the expressions for P rr d and Pd θθ, given by equations (8.22) and (8.23), respectively, into the constraint equation (8.11), the simplified equation for the boundary constraint becomes λ 0 = R1 R 0 [ d f(r) + e ψ(m, ω) H [ Ī 4 1 ] [ Ī 4 1 ] ] g(r, M) ds dr (8.24) S 2 where d = (φ 0R c 0 + φ 1R c 1 ), e = 2 c 2 φ 1R, f(r) = 1 R R3, g(r, M) = 1 r(r,t) 4 R AΘΘ R 3 r(r,t) 4 A RR, and A ΘΘ and A RR are the second and first diagonal components, respectively, of the structure tensor A. After the constraint in equation (8.24) is satisfied for a given applied pressure at the inner boundary, the geometry of the deformed body at time t is known. This geometry can then be used to express the deviatoric stresses. In order to express the total stress at each material point, an additional term is necessary, as expressed in the constitutive relation (8.9). This is the hydrostatic pressure p, and the method used to find this term will be outlined after the next section, which will introduce the specific remodelling equation. 8.1.4 Remodelling Equation for Fibre Reinforced Materials The general form of the remodelling equation has been derived in Chapter 2, and the final form of is given in equation (2.105b), where m denotes the collection of the remodelling parameters. In order to apply this equation to the case of fibre reinforced materials, it is necessary to choose the appropriate remodeling parameter. We select the mean fibre angle ω as the remodelling parameter, as this angle is thought to change with deformation
148 in order to reduce internal stresses. In other words, the material will remodel through the change in fibre angle. After selecting the fibre angle as the remodelling parameter for fibre reinforced materials, the remodelling equation simplifies to ω = 1 Γ [ y ext W ] ω ( C) (8.25) where ω is the mean fibre angle, ω is the time derivative of the mean fibre angle, W( C) is the strain energy potential, given by equation (8.12), Γ is a remodeling constant and y ext is an external remodeling force. In order to take the derivative of the elastic strain energy potential, it is first necessary to define the appropriate probability distribution function ψ(m, ω). The most appropriate selection in the case of fibre reinforced materials is the Gaussian distribution, due to both simplicity and dependence on both the mean fibre angle ω and the true fibre angle α. The Gaussian distribution is given by ψ(m, ω) = exp exp S 2 [ ] (α ω) 2 2 σ 2 [ (α ω) 2 ] (8.26) 2 σ 2 where α is the true fibre angle, which defines the fibre direction vector M, and σ is the variance. Hence, if the Gaussian probability distribution function is substituted into the elastic strain energy potential in equation (8.12), the partial derivative of the strain energy potential with respect to the mean fibre angle takes the form where W ω ( α < α > C) = 2 c 2 φ 1 R ψ(m, ω) S σ 2 2 H(Ī4 1) (Ī4 1) 2 ds (8.27) < α >= α ψ(m, ω) ds S 2 (8.28) The remodeling equation (8.25) is then the relation that governs the remodeling of the fibres through the change in the mean fibre angle. This equation has to be evaluated once at each time interval, which is very computationally expensive due to the integral over all directions, defined by M and implicity defined by α.
149 In order to define the stress field, it is necessary to determine the hydrostatic pressure p, as required by the constitutive relation expressed in equation (8.9). In the next section, a method for determining p will be introduced, whereas this method relies on the solution of Volterra integral equations of the second kind. 8.1.5 Determining the Hydrostatic Pressure p The most convenient method for finding the hydrostatic pressure p is to take the momentum balance equation in radial direction, expressed in equation (8.10), and given as P rr R + 1 R ( P rr P θθ) = 0 (8.29) and integrate it with respect to R. This operation yields an integral expressed as p(r) r(r) R R R 0 K rr p(s) ds + P r(s) S2 d (R) P rr (R 0 ) + R R 0 P rr d (S) Pd θθ(s) ds = 0 S (8.30) where it is possible to isolate the terms that are still a function of R, but can be calculated in an explicit manner, into a function f(r) = P rr d (R) P rr (R 0 ) + R R 0 P rr d (S) Pd θθ(s) ds (8.31) S where K represents an integration constant from the physical constrain equation (8.4). If equation (8.30) is rewritten and the term p(r) is isolated, an expression for p(r) is given by p(r) = R r(r) f(r) R r(r) R R 0 K p(s) ds (8.32) r(s) S2 This type of equation is known as a Volterra equation of the second kind with degenerate kernel, and it has an analytical solution. An additional simplification to this equation can be done, and it is to let V (R) = R R 0 be rewritten as K r(s) S 2 p(s) ds. Henceforth, equation (8.32) can p(r) = R r(r) f(r) R V (R) (8.33) r(r)
150 Taking the derivative of V (R) yields an expression given by V (R) = K p(r) (8.34) r(r) R2 Equation (8.34) can now be isolated for p(r) and then substituted into equation (8.33) to to give an ordinary differential equation given by V (R) + K R (r(r)) 2 V (R) = K f(r) (8.35) R (r(r)) 2 with a boundary condition of V (R 0 ) = 0. If this ordinary differential equation is solved analytically, an expression for V (R) is obtained as R2 + K R [ ] K V (R) = f(s) R (S 2 + K) 3/2 R 0 ds (8.36) It is now possible to substitute for f(s) and,through equation (8.33), to obtain an analytical solution for the hydrostatic pressure p(r). This substitution yields p(r) = λ 0 + R [ ] R r(r) P d rr 1 (R) Pd θθ (S) S2 R 0 r(s) [r(s)] 2 P d rr (S) ds (8.37) where λ 0 is the hydrostatic pressure acting on the inner wall, P rr d is the deviatoric first Piola Kirchhoff stress in the radial direction, expressed in equation (8.22), and P θθ d is the deviatoric first Piola Kirchhoff stress in the tangential direction, as expressed in equation (8.23). Therefore, through a solution of equation (8.37), it is possible to solve for the hydrostatic pressure at each given radial point. The numerical implementation of this solution, along with the rest of the model, will be described in the following section, where the benchmark problem is implemented in MATLAB to validate the mathematical formulation. 8.2 Numerical Implementation The mathematical foundation of the benchmark problem studied in this chapter is given by the the geometrical constraint (8.4), the boundary constraint (8.11), the expressions
151 for the first Piola Kirchhoff stress (8.18), the remodelling equation (8.25), and the expression for the hydrostatic pressure (8.33). The solution will be based on the remodelling equation, where the mean fibre angle ω is treated as the primary unknown. The stress field, free energy, and the deformed radius will then be formulated as the functions of the mean fibre angle. The external remodelling force will be treated as a given datum, and it will be selected based on the the behaviour of the system. In other words, since the external remodelling force cannot be easily found, the value will be selected in a manner that results in a physically admissible remodelling of the system. Figure 8.2: A representative geometry used in the solution of the benchmark problem. A hollow cylinder is made a statistically oriented fibre reinforced material with a single family of fibres oriented in two direction and defined by the fibre angle α. Figure adapted from Olsson & Klarbring (2008). We recall that the benchmark problem in consideration is a fibre reinforced hollow cylinder, as shown in Figure 8.2, with an internally applied pressure, and two sets of reinforcing fibres. The reinforcing fibres are oriented in a direction given by the co latitude angle α, as shown, and they are oriented in a way that provides symmetry to the problem. In order to implement the statistically reinforced fibre potential, it is first necessary to define a way to express the fibre orientation, as additional parameters
152 are needed in addition to the fibre angle α due to the three dimensional nature of the problem. The components of the fibre direction vector M in cylindrical coordinates can then be expressed as M 1 M R = sin(α) cos(β) (8.38) M 2 M Θ R sin(α) sin(β) = R M 3 M Z = cos(α) where β is the longitude angle. It should be noted that M R, M Θ, M Z are the physical components of M, while M 1, M 2, M 3 are the representative components of M in cylindrical coordinates. The metric trace of right Cauchy Green deformation tensor, G 1 : C, as required by the expressions for P rr and P θθ in equations (8.19) and (8.20), is given by G 1 : C = R4 + r(r, t) 4 + r(r, t) 2 R 2 r(r, t) 2 R 2. (8.39) It is also necessary to define the fourth invariant Ī4, which is needed in calculation of the anisotropic fibre potential. This invariant is given by Ī 4 = R4 [sin(α) cos(β)] 2 + r(r, t) 4 [sin(α) sin(β)] 2 + r(r, t) 2 R 2 [cos(α)] 2 r(r, t) 2 R 2 (8.40) The expression for the fourth invariant is used in the Heaviside function H(Ī4 1), which is used to switch off the anisotropic fibre contribution in contraction. The presence of the Heaviside function is troublesome, as the argument of this function is unknown. This implies that it is not possible to know if the fibres are in extension or contraction before solving the problem, therefore rendering the purpose of the Heaviside function redundant. Fortunately, it is possible to define the domain where the Heaviside function is strictly positive. Therefore, the integration will only be performed over this domain. The Heaviside step function is strictly positive when α 0 and β [β 0, π/2[ ]π/2, π β 0 ] [π + β 0, 3π/2[ ]3π/2, 2π β 0 ] (8.41)
153 where ( ) R β 0 = tan 1 r(r, t) (8.42) It is now possible to select the material parameters to be used in the model, and to derive an algorithm for the implementation. The material parameters were selected to be similar to the material parameters used by Olsson & Klarbring (2008), in order to permit for a direct comparison between the results, and they are presented in Table 8.1. Table 8.1: Material parameters used in the implementation of the remodelling constitutive model for the remodelling of the fibres in the benchmark problem, with a statistically oriented fibre distribution. The parameters were selected to closely approximate the model of Olsson & Klarbring (2008) to allow for a direct comparison, and the evolution parameters Γ was selected in order make the evolution speed computable. Parameter Value R 0 (mm) 1.0 R 1 (mm) 2.0 β 0 2π α 0 π/2 φ 0R 0.8 φ 1R 0.2 c 0 (MPa) 0.03574 c 1 (MPa) 0.03574 c 2 (MPa) 0.3574 λ 0 (MPa) 0.020 σ 0.5 Γ 1 y ext 0 ω 0 (degrees) 45 The implementation of the mathematical formulation was performed in MATLAB. The algorithm for the implementation is presented in Table 8.2, and the complete code is presented in Appendix A.3. The code starts with the equilibrium constraint, expressed in equation (8.11), to find the integration constant K(t), and subsequently the deformed radius r(r, t). This is then followed by the calculation of the radial profile of the hy-
154 drostatic pressure, determined through equation (8.33), and the calculation of the radial profiles of both the radial and tangential stresses. These values are then used in calculation of the mean fibre angle at the next time step, and the whole process is repeated. In addition, all the integrals were calculated using the trapezoidal rule, and this was possible because the integral functions were separable (i.e. it was possible to separate a function of three variables into three single variable functions). For full details regarding the integration process, the Reader can refer to the code in Appendix A.3. Table 8.2: The algorithm used for the implementation of the remodelling constitutive model for the remodelling of the fibres in the benchmark problem, with a statistically oriented fibre distribution. The model was implemented in MATLAB due to simplicity of the implementation and the flexibility with array manipulations. The code for the implementation as provided in Appendix A.3 GIVEN: Discretized radial profile R 0 R i R 1 Inner boundary pressure λ 0 Initial mean fibre angle ω 0 DO: α = 1...t f Use the boundary constraint to find integration constant K(t) Equation (8.11) Find the deformed radius for all values of R Equation (8.4) Find the radial profile of the hydrostatic pressure p α Equation (8.33) Find the radial profile of radial first Piola Kirchhoff stress P rr α Find the radial profile of tangential first Piola Kirchhoff stress P θθ α Equation (8.19) Equation (8.20) Find the next time step value of the mean fibre angle ω α+1 Equation (8.25) END DO This concludes the chapter on the study of remodelling of the mean fibre angle in statistically oriented fibre reinforced biological tissues. This chapter covered both the mathematical formulation and numerical implementation of the remodelling problem. In the following chapter, the results from both the remodelling model, introduced in this chapter, and the FE model of articular cartilage, introduced in the Chapters 4 7, will be presented and discussed in detail.
155 Chapter 9 Results and Discussions This chapter is divided into three sections, each comprised of one set of results together with the relevant discussion. The first section outlines the validation of the numerical formulations implemented in this work to model articular cartilage. Next, the results from numerical simulations of the global unconfined compression tests conducted by Guilak et al. (1995) are presented, and the simulation results are compared to both the experimental results and the numerical simulations performed by Pajerski (2010), which have implemented the NLTITH model of Federico & Gasser (2010). Lastly, the numerical results from the evolution model, presented in Chapter 8, are summarized, and compared to the numerical work of Olsson & Klarbring (2008). 9.1 Articular Cartilage FE Model Validation The large deformation non linear model for biological tissues with statistically oriented reinforcing fibres, developed by Federico & Grillo (2012), with Holmes Mow potential (Holmes & Mow, 1990) describing the matrix and isotropic fibre contributions, and Holmes Mow permeability function (Holmes & Mow, 1990) describing the reference, undeformed, permeability, has been implemented using FEBio, an open source FE solver. The necessary code has been developed and implemented as a part of the work conducted in this Thesis, however it first had to be validated to ensure that the outputted results were accurate and consistent with the constitutive theories on which the code is based. Since the elastic formulation of the large deformation model (Federico & Grillo, 2012) is a natural evolution of the NLTITH model (Federico & Gasser, 2010), the code implemented
156 in this Thesis regarding the elastic formulation will be validated by direct comparisons to the implementation of the NLTITH model performed by Pajerski (2010). On the other hand, the Holmes Mow potential was developed and implemented by Holmes & Mow (1990), and these results will be used for the validation of the matrix and isotropic fibre contributions. Lastly, the permeability formulation will be validated by a comparison to the Holmes Mow permeability model (Holmes & Mow, 1990), and by observing the effects of varying the dominant fibre direction. 9.1.1 Large Deformation Model With Statistically Oriented Reinforcing Fibres The large deformation non linear model with statistically oriented reinforcing fibres is a hyperelastic material model developed to describe the effect of collagen fibre network on the mechanical behaviour of biological tissues. The collagen fibre network is primarily active in extension, as discussed in Chapter 6, and its tensile behaviour is demonstrated in the plot of the normalized force as a function stretch in Figure 9.1. The plot is based on a numerical simulation of a material sample subjected to an uniaxial tensile force, where the fibres in the sample are oriented in the direction of the force application. The results obtained in this numerical simulation have also been compared to the results obtained by Pajerski (2010) using the NLTITH model, and it is clear that they are in good agreement. It is also clear that the implemented model exhibits non linear behaviour, however this behaviour is very subtle and it could be approximated by a linear relation for tensile strains of up to 15%. Although the collagen fibre behaviour in tension, depicted in Figure 9.1, contributes to the overall non linearity of the implemented model, the major contributor to the non linear behaviour results from the variation of dominant direction of collagen fibre orientation within the tissue. The strength of the large deformation non linear model with statistically oriented reinforcing fibres lies in the fact that a continuous infinity
157 Figure 9.1: Plot of the normalized force a a function of the stretch of a generic fibre reinforced material modeled using the large deformation non linear model with statistically oriented reinforcing fibres. The concentration parameter, b, has been assumed to have a value of 5 throughout the whole material, therefore yielding fibres that are dominantly oriented in the direction of the force application. The force stretch relationship obtained using the model implemented in this Thesis, shown by the solid line, has also been compared to the relationship obtained by Pajerski (2010) using the NLTITH model, and shown by the dotted line. of fibre orientations can be described using this model, and consequently it is possible to describe any spatial fibre orientation by varying the concentration parameter b. This parameter acts to define the dispersion about the dominant direction of the fibre bundles, as demonstrated in Figure 7.2, and by varying the parameter the different zones of articular cartilage can be modelled using the same set of equations, as demonstrated in Figure 7.4. The change in the overall material stiffness in the unconfined compression and tension tests, which arises due to the variation in the fibre orientation, is depicted
158 in Figure 9.2. This variation in the concentration parameter could in turn be used to simulate different zones of articular cartilage, and create a non homogeneous global model of articular cartilage. Figure 9.2: Plot of the normalized force versus the concentration parameter b under the assumption that the load has been applied in the axial direction. The specimen being modelled was subjected to a 20% strain in either tension, depicted by the solid line, or compression, depicted by a dashed line, and the force required to attain the strain was recorded for concentration parameter values ranging from -8 to 8. The plot in Figure 9.2 is based on uniaxial tension/unconfined compression tests where the axis of transverse isotropy, K, is assumed to be oriented in the direction of the applied strain. A global strain of 20% is applied in both tension and compression for concentration parameter b ranging from -8 to 8. It is also important to note that the concentration parameter is assumed to be uniform throughout the whole tissue sample, therefore making the entire sample homogeneous.
159 When the fibres are oriented in the direction of the applied strain, the material stiffness in tension is significantly higher than material stiffness in compression due to the presence of the Heaviside step function H(I 4 1) in the constitutive model, as expressed in equations (6.28) and (6.29). The Heaviside step function is responsible for the non symmetric nature of the fibres in extension and contraction, as it acts to activate the anisotropic fibre ensemble potential in extension, and deactivate it in tension. The inclusion of the anisotropic fibre ensemble potential, in turn, significantly increases the overall stiffness in tension, as then the fibres are in extension. Basically, the fibres can be thought as highly resistant to tensile loading, but to buckle when in compression and hance have significantly lower stiffness. When the fibres are oriented perpendicular to the direction of the applied load (i.e., in the transverse direction), as is the case when the concentration parameter b is negative, the stiffness in compression is higher than material stiffness in tension due to the fact that global compression results in positive transverse strain, which activates the fibres in the transverse plane. As previously mentioned, this can easily be explained by stating that the fibres resist in extension and buckle in contraction. For the case of b = 0, the fibre orientation is equally weighted in all directions of the unit sphere and the material is initially isotropic, with anisotropy arising under deformation, due to the reorientation of the fibres. Also in this case, the material does not exhibit the same material stiffness in both tension and compression. This is due to the different fibre bundle activation in the case of tension and compression: the fibres closely aligned to the direction of the load application in the case of tension, and the fibres closely aligned to the transverse plane in the case of compression.
160 9.1.2 Holmes Mow Potential The Holmes Mow potential has been previously verified by Holmes & Mow (1990) using experimental results, and it has previously been implemented to model articular cartilage in literature (Wu & Herzog, 2000). In order to insure the validity of the code used to implement the Holmes Mow potential, the code was validated by a comparison to an analytical solution of the Holmes Mow potential for the case of confined compression (Holmes & Mow, 1990). This analytical solution is given by σ 22 = 1 2 H A ( ) λ 2 2 1 exp β (λ 2 2 1) (9.1) λ 2β+1 2 where H A is the aggregate modulus, β is a material parameter introduced in Chapter 7, and λ 2 is the uniaxial stretch in the direction of the loading e 2. Figure 9.3: Plot of the minimum principal stress as a function of the stretch, using the Holmes Mow potential (Holmes & Mow, 1990). A simulation of the uniaxial confined compression was conducted and the numerical results, depicted by the square points, were compared to the equation derived for the case of unaxial compression, depicted by the dashed line, and originally derived by Holmes & Mow (1990).
161 Figure 9.3 depicts a plot of the compressive stress as a function of the stretch for the analytical expression expressed in equation (9.1) and for the numerical simulations conducted with the implemented Holmes Mow potential. As demonstrated in the plot, the results from the simulation correlate directly with the analytical solution fr the case of uniaxial compression. Consequently, it is safe to assume that the FEBio implementation of the Holmes Mow potential is valid. 9.1.3 Permeability Formulation The permeability formulation developed by Federico & Grillo (2012), and described in Chapter 6, takes into account the presence of the impermeable inclusions, which, in the case of articular cartilage, are the collagen fibres, and the effect these inclusions have on the permeability of the tissue. In fact, the model takes a referential permeability function depending only on the porosity of tissue and then determines the overall permeability tensor, based on the fibre orientation and taking into account the fibre directed flow. In this thesis, the constitutive function selected for the undeformed, referential permeability is the Holmes Mow permeability function ˆk 0 (Holmes & Mow, 1990), expressed in equation (7.25), and it is assumed that the permeability remains isotropic under any deformation, i.e., that it has a form k 0 = ˆk 0 g 1, where k 0 is the referential permeability used in the permeability formulation expressed in equation (6.37). In order to validate the implementation of the permeability formulation, it is necessary to validate the effect of the presence of the fibres and the change in fibre orientation. The Holmes Mow permeability function will not be directly validated, as it is a part of the FEBio package and it has been extensively tested (Maas et al., 2010). In order to test the effect that the fibres have on the permeability of the tissue, the pure Holmes Mow permeability model will be compared to the permeability model implemented in this Thesis, with several different dominant fibre orientations. The comparison will be
162 performed by observing the fluid flux in the axial and the radial directions, and observing the effect of the fibre orientation on the fluid flux in those two directions. In order to test the effect of the fibre orientation on the permeability of the tissue, a normalized flux will be plotted as a function of the concentration parameter b for unconfined compression test. Figures 9.4 and 9.5 show the normalized axial and radial fluid fluxes, respectively, at an element located on the axis of symmetry at a depth of ξ = 0.5, respectively. The test is performed by ramp loading the sample in unconfined compression to a strain of 15% for the first 100 seconds, and then holding that strain for the subsequent 1100 seconds to observe the tissue relaxation. In both plots, the solid matrix has been modelled as a simple Neo Hookean material in order to focus on the permeability modelling. Figure 9.4: Plot of the axial fluid flux as a function of time in the case of unconfined compression for an element located on the axis of symmetry and ξ = 0.5. The plot shows a comparison between the axial fluid flux between the Holmes Mow permeability model and the implemented permeability formulation with three different values of the concentration parameter b: 0, 5, and -5.
163 Figure 9.4 plots a comparison between the axial fluid fluxes as a function of time for the Holmes Mow permeability model and the implemented permeability formulation with three different values of the concentration parameter b: 0, 5, and -5. As it can be observed in the plot, when the dominant fibre direction is in in the direction of the load application, the axial fluid flux is significantly higher than for the isotropic and transverse fibre direction. This can be explained by assuming that the fibres direct the permeation of the fluid in the matrix, and that the permeability parallel to the fibres is higher than the permeability perpendicular to the fibres. Similarly, when comparing the fluid flux for the case of transverse fibre orientation and the isotropic fibre orientation, the axial fluid flux in the case of transverse orientation is lower. This can also be explained by assuming that, since the fibres direct the flow, the flow in the axial direction is being restricted by the perpendicular fibre bundle orientation. As for the Holmes Mow permeability, it is interesting to note that equilibrium is reached much sooner than for the permeability formulation implemented in this Thesis. This can be explained by stating that the Holmes Mow permeability model does not take into account the presence of fibres, which are modelled as impermeable inclusions. Even though the referential permeability of the matrix, k 0R, has been adjusted to account for the presence of these inclusions, the inclusions (i.e. fibres) reorient after deformation and still have a significant effect on the overall permeability. Figure 9.5 plots a comparison between the axial fluid fluxes as a function of time for the Holmes Mow permeability model and the implemented permeability formulation with three different values of the concentration parameter b: 0, 5, and -5. Similarly to the axial fluid flux, the fibre orientation also has a subtle effect on the direction of the fluid flux. When the fibre bundle is oriented in the axial direction, the fluid flux is slightly less than in the case of isotropic fibre orientation. Conversely, the fluid flux for the transverse fibre bundle orientation is slightly higher that for the case of isotropic fibre
164 Figure 9.5: Plot of the radial fluid flux as a function of time in the case of unconfined compression for an element located on the axis of symmetry and ξ = 0.5. The plot shows a comparison between the radial fluid flux between the Holmes Mow permeability model and the implemented permeability formulation with three different values of the concentration parameter b: 0, 5, and -5. orientation. The effect of fibre orientation on the radial flux is less subtle than for the case of axial flux, but this is due to the boundary conditions; the fluid is only allowed to exude through the side walls, and the top and bottom are modelled as impermeable. Therefore, the fluid has to flow in the radial direction, and the fibres can only subtly shift the direction of the flux. For the case of the Holmes Mow permeability it is again evident that the system reaches equilibrium significantly faster. As previously mentioned, this is due to the fact that the Holmes Mow permeability model does not take into account the presence of fibres, modelled as impermeable inclusions, and the reorientation of fibres. The true strength of the permeability formulation implemented in this Thesis lies in the fact that a continuous infinity of fibre orientations can be described using this model,
165 and consequently it is possible to describe any spatial fibre orientation by varying the concentration parameter b, similar the the elastic formulation. This parameter acts to define the dominant direction of the fibre bundles, as demonstrated in Figure 7.2, and by varying the parameter the different zones of articular cartilage can be modelled using the same set of equations, as demonstrated in Figure 7.4. The anisotropy in permeability that arises due to the variation in the fibre orientation is depicted in Figure 9.6. Figure 9.6: Plot of the normalized fluid flux versus the concentration parameter b under the assumption that a displacement has been applied in the axial direction. The specimen being modelled was subjected to a 20% strain in unconfined compression and the axial and radial fluid fluxes, depicted by the solid and dashed lines, respectively, were recorded for concentration parameter values ranging from -8 to 8. The plot in Figure 9.6 is based on unconfined compression tests where the axis of transverse isotropy, K, is assumed to be oriented in the direction of the applied strain. A global strain of 20% is applied in both compression for concentration parameter b
166 ranging from -8 to 8. It is also important to note that the concentration parameter is assumed to be uniform throughout the whole tissue sample, therefore making the entire sample homogeneous. When the fibre bundle is oriented in the direction of the applied strain, the axial fluid flux is significantly higher than for the case of transverse fibre orientation. On the other hand, the radial fluid flux is lower, but the effect is much more subtle. The is due to the boundary conditions, as the fluid is only allowed to exude through the sides (top and bottom plates are impermeable), which implies that the radial fluid flux cannot change significantly. It is also important to note that the fluid flux in both directions is equal when the fibre distribution is isotropic; this makes sense, since the permeability depends on the fibre direction. Consequently, if the fibre orientation is isotropic, the permeability should be isotropic. On the other hand, it is important to note that the permeability tensor is dependent on the deformation, as expressed in equation (6.37), and that it will only be isotropic under no deformation. For the case of the plot in Figure 9.6, the deformation was small enough that this dependence on deformation is not be seen. 9.2 Analysis of Articular Cartilage The global analysis performed in this Thesis has been outlined in detail in Chapter 7. In summary, the non linear, biphasic model for biological tissues with statistically oriented reinforcing fibres was used to simulate the unconfined compression tests conducted by Guilak et al. (1995). The numerical results were then compared to the experimental results, along with the numerical results obtained by Pajerski (2010) for the same set of unconfined compression experiments. In order to ensure proper numerical analysis, the proper poroelastic elements were used. The mesh refinement was performed until convergence has been achieved, and
167 this convergence was established in terms of the zonal displacements. The solution was deemed to have converged when the zonal displacement results varied by less than 1% between successive mesh refinements. Due to the stress concentrations arising from a boundary conditions, which arises from the attachment of the cartilage sample to the subchondral bone, the stress field was not used as a convergence criterion. However, it should be noted that the stress field does converge to an unique value with successive mesh refinements. 9.2.1 Viscoelasticity In the outline of articular cartilage, presented in Chapter 4, it has been described that a large part of the viscoelastic behaviour of articular cartilage can be attributed to the filtration motion of the pore fluid through the tissue during compressive loading. Figure 9.7 shows a plot of the compressive force as a function of time for the unconfined compression test simulated in this Thesis, as well as a comparison to the results obtained by Pajerski (2010) for the same experiment. The plot clearly shows the stress relaxation behaviour typical of articular cartilage. Since the permeability of articular cartilage is low, the relaxation time is quite high and after 1200 s of constant strain, the sample is just reaching equilibrium. For the purpose of the discussion, the time of 1200 s will be treated as the steady state. It should also be noted that the results obtained in this Thesis have the same maximum compressive force, but lower equilibrium force than that obtained by Pajerski (2010). This can be attributed to higher rate of fluid exudation due to the fibre directed flow, and the fibres directing the flow through the unconfined sides of the sample. This is discussed in detail in the previous section, where the permeability formulation has been thoroughly validated.
168 Figure 9.7: Plot of the compressive force as a function of time for the unconfined compression simulations performed in this Thesis. The solid line represents the results obtained in this work, while the dashed line represents the results obtained by Pajerski (2010). 9.2.2 Pore Pressure The plot of the pore pressure with respect to the time for an element on the axis of symmetry at ξ = 0.5 is shown in Figure 9.8. Since the stress relaxation behaviour, depicted in Figure 9.7, is a result of the pore fluid exudation, it follows that the plot of the pore pressure as a function of time closely resembles the plot of force vs. time, as the primary driving force of pore fluid exudation os the pore pressure. The main difference between the two plots is that, in an ideal system, the pore pressure approaches zero at steady state, while the force approaches a non zero value that is equivalent to the force necessary to deform the solid fraction of the tissue to a specific strain. It should again be noted that when comparing the results obtained in this Thesis to the results obtained by Pajerski (2010), the two are in close agreement. The only
169 Figure 9.8: Plot of the pore pressure as a function of time of an element located on the axis of symmetry at ξ = 0.5 for the unconfined compression simulations performed in this Thesis. The solid line represents the results obtained in this work, while the dashed line represents the results obtained by Pajerski (2010). difference is that the results obtained in this Thesis relax in a more abrupt fashion, as there is a larger decrease in the pore pressure after the maximum. This can again be attributed to the fibre directed flow, and a higher rate of fluid exudation. A more detailed picture of pore pressure is presented by the contour plots in Figures 9.9(a) and 9.9(b), with 9.9(a) showing the pore pressure at the end of the ramp loading and 9.9(b) showing the pore pressure at 1200 s, generalized as the steady state in this Thesis. At 30 s, the pore pressure follows two general trends: it increases towards the axis of symmetry and it increases towards the base of the sample. These trends are mainly due to the system geometry and the fixed boundary at the base of the sample, respectively. At steady state, the pore pressure is zero.
170 (a) 30s (b) 1200s Figure 9.9: The pore pressure, in MPa, in the articular cartilage sample at the end of the loading ramp and at a time of 1200 s, characterized as equilibrium
171 The contour plots of the pore pressure also demonstrate the effect that the pore pressure has on the deformation of the sample. Since the displacements at the top and bottom boundaries are imposed, the high pore pressure at the end of the ramp load, shown in Figure 9.9(a), causes the sample to deform significantly in the transverse direction. These large transverse deformations have a direct effect on the stress distribution within the sample, which will be discussed in the next section. 9.2.3 Stress Distribution The effective stress distributions in the articular cartilage sample at the end of the 30 s ramp loading and at the steady state are shown in Figures 9.10(a) and 9.10(b), respectively. It is important to note that the values of the maximum stress at the end of the ramp loading are at least one order of magnitude larger than the values of maximum stress at the steady state. The high maximum stress at the end of the ramp loading is mostly due to the large tissue deformation in the transverse direction, resulting from the high pore pressure. As the pore pressure decreases with time, the deformation in the transverse direction also decreases, which results is significantly lower maximum stress at the steady state. It is important to note that the contour plots in Figures 9.10(a) and 9.10(b) indicate that the stress increases with distance from the bone cartilage interface, with the exception of the stress concentration due to fixed boundary condition. This relationship is especially apparent at the steady state, when it is clear that there is an increase in the effective stress with the distance from the bone cartilage interface, which can be attributed to the increase in the transverse strain. It is important to note that the stress distribution near the bone cartilage interface has a similar profile at both the end of the ramp and at the steady state (with the exception of magnitude), and this can be attributed to the boundary conditions at the bottom surface.
172 (a) 30s (b) 1200s Figure 9.10: The stress distribution contours, in MPa, in the articular cartilage sample in unconfined compression at the end of the loading ramp and at a time of 1200 s, characterized at equilibrium.
173 The general observations regarding the pore pressure and the effective stress distribution offer some insight into the complex relationship between the depth dependent composition and structure of articular cartilage, and the mechanical environment inside the tissue. In fact, they serve to provide a more intuitive way to understand how the constituents of articular cartilage interact to produce the complex mechanical behaviour observed in experiments. 9.2.4 Fluid Flux In order to characterize the overall permeability of the tissue, and the dependence of permeability on the fibre orientation, it is useful to plot the fluid flux in both axial and radial directions, as illustrated in Figures 9.11(a) and 9.11(b), respectively. The influence of fibre orientation on the direction of the fluid flux is illustrated in both figures, however it is more evident in Figure 9.11(b), which illustrates the radial fluid flux. In this contour, the radial fluid flux is the highest in the superficial zone. This makes sense, as in this zone the fibres are oriented parallel to the articulating surface, implying that the fluid encounters the least resistance in the radial direction. This is further reinforced when observing the radial fluid flux in the deep zone, where it is significantly smaller than in the superficial zone. In this zone, the dominant fibre direction is perpendicular to the articulating surface, and the fibres can be imagined to direct the flow in the axial direction. This is reinforced when observing the axial fluid flux in Figure 9.11(a). The general observations of the fluid flux in articular cartilage sample provide an insight into the complex relationship between the structure and composition of articular cartilage and its biphasic behaviour. It serves to help one develop a better understanding of how the tissue behaves under loading, and how the anisotropic permeability of the tissue affects the general mechanical behaviour of articular cartilage.
174 (a) Axial Fluid Flux (b) Radial Fluid Flux Figure 9.11: Axial and radial fluid fluxes, in mm/s, in the articular cartilage sample in unconfined compression at the end of the loading ramp, at 30 s. It should be noted that the positive directions in these contour plots are up and to the left.