The Elements of Continuum Biomechanics

Preface xi

Part One A one-dimensional context 1

1 Material bodies and kinematics 3

1.1 Introduction 3

1.2 Continuous vs. discrete 6

1.3 Configurations and deformations 9

1.4 The deformation gradient 14

1.5 Change of reference configuration 15

1.6 Strain 16

1.7 Displacement 18

1.8 Motion 19

1.9 The Lagrangian and Eulerian representations of fields 22

1.10 The material derivative 24

1.11 The rate of deformation 26

1.12 The cross section 27

2 Balance laws 29

2.1 Introduction 29

2.2 The generic Lagrangian balance equation 30

2.2.1 Extensive properties 30

2.2.2 The balance equation 31

2.3 The generic Eulerian balance equation 35

2.4 Case study: Blood flow as a traffic problem 37

2.5 Case study: Diffusion of a pollutant 39

2.5.1 Derivation of the diffusion equation 39

2.5.2 A discrete diffusion model 41

2.6 The thermo-mechanical balance laws 42

2.6.1 Conservation of mass 42

2.6.2 Balance of (linear) momentum 43

2.6.3 The concept of stress 44

2.7 Case study: Vibration of air in the ear canal 45

2.8 Kinetic energy 50

2.9 The thermodynamical balance laws 55

2.9.1 Introduction 55

2.9.2 Balance of energy 56

2.9.3 The entropy inequality 58

2.10 Summary of balance equations 59

2.11 Case study: Bioheat transfer and malignant hyperthermia 61

3 Constitutive equations 69

3.1 Introduction 69

3.2 The principle of determinism 70

3.3 The principle of equipresence 72

3.4 The principle of material frame-indifference 72

3.5 The principle of dissipation 75

3.6 Case study: Memory aspects of striated muscle 79

3.7 Case study: The thermo(visco)elastic effect in skeletal muscle 85

3.8 The theory of materials with fading memory 90

3.8.1 Groundwork 90

3.8.2 Fading memory 93

3.8.3 Stress relaxation 95

3.8.4 Finite linear viscoelasticity 96

4 Mixture theory 99

4.1 Introduction 99

4.2 The basic tenets of mixture theory 99

4.3 Mass balance 101

4.4 Balance of linear momentum 102

4.4.1 Constituent balances 102

4.4.2 Mixture balance 103

4.5 Case study: Confined compression of articular cartilage 106

4.5.1 Introduction 106

4.5.2 Empirical facts 107

4.5.3 Field equations 108

4.5.4 Nonlinear creep 112

4.5.5 Hysteresis 115

4.5.6 The linearized theory 115

4.6 Energy balance 121

4.6.1 Constituent balances 121

4.6.2 Mixture balance 123

4.7 The entropy inequality 124

4.8 Chemical aspects 125

4.8.1 Stoichiometry 125

4.8.2 Thermodynamics of homogeneous systems 129

4.8.3 Enthalpy and heats of reaction 131

4.8.4 The meaning of the Helmholtz free energy 134

4.8.5 Homogeneous mixtures 135

4.8.6 Equilibrium and stability 137

4.8.7 The Gibbs free energy as a Legendre transformation 138

4.9 Ideal mixtures 140

4.9.1 The ideal gas paradigm 140

4.9.2 Mixtures of ideal gases 141

4.9.3 Other ideal mixtures 145

4.10 Case study: Bone as a chemically reacting mixture 145

Part Two Toward three spatial dimensions 151

5 Geometry and kinematics 153

5.1 Introduction 153

5.2 Vectors and tensors 153

5.2.1 Why Linear Algebra? 153

5.2.2 Vector spaces 155

5.2.3 Linear independence and dimension 156

5.2.4 Linear operators, tensors, matrices 158

5.2.5 Inner-product spaces 161

5.2.6 The reciprocal basis 162

5.3 Geometry of classical space-time 164

5.3.1 A shortcut 164

5.3.2 R3 as a vector space 165

5.3.3 E3 as an affine space 166

5.3.4 Frames 166

5.3.5 Space-time and observers 169

5.3.6 Fields and the divergence theorem 170

5.4 Eigenvalues and eigenvectors 176

5.4.1 General concepts 176

5.4.2 More on principal invariants 178

5.4.3 The symmetric case 180

5.4.4 Functions of symmetric matrices 182

5.5 Kinematics 183

5.5.1 Material bodies 183

5.5.2 Configurations, deformations, motions 183

5.5.3 The deformation gradient 185

5.5.4 Local configurations 187

5.5.5 A word on notation 187

5.5.6 Decomposition of the deformation gradient 188

5.5.7 Measures of strain 193

5.5.8 The displacement field and its gradient 194

5.5.9 The geometrically linearized theory 196

5.5.10 Volume and area 198

5.5.11 The material derivative 201

5.5.12 Change of reference configuration 203

5.5.13 The velocity gradient 204

6 Balance laws and constitutive equations 207

6.1 Preliminary notions 207

6.1.1 Extensive properties 207

6.1.2 Transport theorem 208

6.2 Balance equations 210

6.2.1 The general balance equation 210

6.2.2 The balance equations of Continuum Mechanics 214

6.3 Constitutive theory 223

6.3.1 Introduction and scope 223

6.3.2 The principle of material frame-indifference and its applications 224

6.3.3 The principle of thermodynamic consistency and its applications 228

6.4 Material symmetries 231

6.4.1 Symmetries and groups 231

6.4.2 The material symmetry group 232

6.5 Case study: The elasticity of soft tissue 236

6.5.1 Introduction 236

6.5.2 Elasticity and hyperelasticity 236

6.5.3 Incompressibility 238

6.5.4 Isotropy 241

6.5.5 Examples 242

6.6 Remarks on initial and boundary-value problems 248

7 Remodelling, aging, growth 255

7.1 Introduction 255

7.2 Discrete and semi-discrete models 262

7.2.1 Challenges 262

7.2.2 Cellular automata in tumour growth 264

7.2.3 A direct model of bone remodelling 266

7.3 The continuum approach 268

7.3.1 Introduction 268

7.3.2 The balance equations of volumetric growth and remodelling 269

7.4 Case study: tumour growth 273

7.5 Case study: Adaptive elasticity of bone 277

7.5.1 The isothermal quasi-static case 281

7.6 Anelasticity 282

7.6.1 Introduction 282

7.6.2 The notion of material isomorphism 283

7.6.3 Non-uniqueness of material isomorphisms 286

7.6.4 Uniformity and homogeneity 287

7.6.5 Anelastic response 289

7.6.6 Anelastic evolution 290

7.6.7 The Eshelby stress 296

7.7 Case study: Exercise and growth 301

7.7.1 Introduction 301

7.7.2 Checking the proposed evolution law 301

7.7.3 A numerical example 303

7.8 Case study: Bone remodelling and Wolff's law 305

8 Principles of the Finite Element Method 309

8.1 Introductory remarks 309

8.2 Discretization procedures 310

8.2.1 Brief review of the method of finite differences 310

8.2.2 Non-traditional methods 313

8.3 The Calculus of Variations 313

8.3.1 Introduction 313

8.3.2 The simplest problem of the Calculus of Variations 315

8.3.3 The case of several unknown functions 321

8.3.4 Essential and natural boundary conditions 323

8.3.5 The case of higher derivatives 326

8.3.6 Variational problems with more than one independent variable 329

8.4 Rayleigh, Ritz, Galerkin 330

8.4.1 Introduction 330

8.4.2 The method of Rayleigh and Ritz 332

8.4.3 The methods of weighted residuals 334

8.4.4 Approximating differential equations by Galerkin's method 336

8.5 The finite element idea 341

8.5.1 Introduction 341

8.5.2 A piecewise linear basis 343

8.5.3 Automating the procedure 348

8.6 The FEM in Solid Mechanics 353

8.6.1 The Principle of Virtual Work 353

8.6.2 The principle of stationary potential energy 358

8.7 Finite element implementation 359

8.7.1 General considerations 359

8.7.2 An ideal element 360

8.7.3 Meshing, insertion maps and the isoparametric idea 362

8.7.4 The contractibility condition and its consequences 363

8.7.5 The element IVW routine 366

8.7.6 The element EVW routine 368

8.7.7 Assembly and solution 369