Multiscale Biomechanics Theory and Applications

Model biomechanical problems at multiple scales with this cutting-edge technology

Multiscale modelling is the set of techniques used to solve physical problems which exist at multiple scales either in space or time. It has been shown to have significant applications in biomechanics, the study of biological systems and their structures, which exist at scales from the macroscopic to the microscopic and beyond, and which produce a myriad of overlapping problems. The next generation of biomechanical researchers therefore has need of the latest multiscale modelling techniques.

Multiscale Biomechanics offers a comprehensive introduction to these techniques and their biomechanical applications. It includes both the theory of multiscale biomechanical modelling and its practice, incorporating some of the latest research and surveying a wide range of multiscale methods. The result is a thorough yet accessible resource for researchers looking to gain an edge in their biomechanical modelling.

Multiscale Biomechanics readers will find:

• Practical biomechanical applications for a variety of multiscale methods
• Detailed discussion of soft and hard tissues, and more
• An introduction to analysis of advanced topics ranging from stenting, drug delivery systems, and artificial intelligence in biomechanics

Multiscale Biomechanics is a useful reference for researchers and scientists in any of the life sciences with an interest in biomechanics, as well as for graduate students in mechanical, biomechanical, biomedical, civil, material, and aerospace engineering.

Soheil Mohammadi, PhD, is Professor of Computational Mechanics and Director of the High Performance Computing Lab in the School of Civil Engineering, University of Tehran, Iran. He has published extensively on contact mechanics, XFEM and meshless methods, multiscale physics, and related subjects.

Contents

Preface xiii

List of Abbreviations xvii

Part I Introduction 1

1 Introduction 3

1.1 Introduction to Biomechanics 3

1.2 Biology and Biomechanics 3

1.3 Types of Biological Systems 6

1.3.1 Biosolids 6

1.3.2 Biofluids 7

1.3.3 Biomolecules 8

1.3.4 Synthesized Biosystems 9

1.4 Biomechanical Hierarchy 10

1.4.1 Organ Level 10

1.4.2 Tissue Level 11

1.4.3 Cellular and Lower Levels 12

1.4.4 Complex Medical Procedures 13

1.5 Multiscale/Multiphysics Analysis 13

1.6 Scope of the Book 17

Part II Analytical and Numerical Bases 21

2 Theoretical Bases of Continuum Mechanics 23

2.1 Introduction 23

2.2 Solid Mechanics 23

2.2.1 Elasticity 24

2.2.2 Plasticity 28

2.2.3 Damage Mechanics 31

2.2.4 Fracture Mechanics 36

2.2.5 Viscoelasticity 53

2.2.6 Poroelasticity 59

2.2.7 Large Deformation 63

2.3 Flow, Convection and Diffusion 72

2.3.1 Thermodynamics 72

2.3.2 Fluid Mechanics 74

2.3.3 Gas Dynamics 78

2.3.4 Diffusion and Convection 81

2.4 Fluid–Structure Interaction 83

2.4.1 Lagrangian and Eulerian Descriptions 83

2.4.2 Fluid–Solid Interface Boundary Conditions 84

2.4.3 Governing Equations in the Eulerian Description 85

2.4.4 Coupled Lagrangian–Eulerian (CLE) 86

2.4.5 Coupled Lagrangian–Lagrangian (CLL) 87

2.4.6 Arbitrary Lagrangian–Eulerian (ALE) 88

3 Numerical Methods 93

3.1 Introduction 93

3.2 Finite Difference Method (FDM) 93

3.2.1 One-Dimensional FDM 94

3.2.2 Higher Order One-Dimensional FDM 95

3.2.3 FDM for Solving Partial Differential Equations 98

3.3 Finite Volume Method (FVM) 99

3.4 Finite Element Method (FEM) 102

3.4.1 Basics of FEM Interpolation 102

3.4.2 FEM Basis Functions/Shape Functions 103

3.4.3 Properties of the Finite Element Interpolation 105

3.4.4 Physical and Parametric Coordinate Systems 106

3.4.5 Main Types of Finite Elements 106

3.4.6 Governing Equations of the Boundary Value Problem 109

3.4.7 Numerical Integration 112

3.5 Extended Finite Element Method (XFEM) 113

3.5.1 A Review of XFEM Development 113

3.5.2 Partition of Unity 114

3.5.3 Enrichments 115

3.5.4 Signed Distance Function 115

3.5.5 XFEM Approximation for Cracked Elements 115

3.5.6 Boundary Value Problem for a Cracked Body 117

3.5.7 XFEM Discretisation of the Governing Equation 118

3.5.8 Numerical Integration 119

3.5.9 Selection of Enrichment Nodes for Crack Propagation 121

3.5.10 Incompatible Modes of XFEM Enrichments 122

3.5.11 The Level Set Method for Tracking Moving Boundaries 123

3.5.12 XFEM Tip Enrichments 124

3.5.13 XFEM Enrichment Formulation for Large Deformation Problems 132

3.6 Extended Isogeometric Analysis (XIGA) 133

3.6.1 Introduction 133

3.6.2 Isogeometric Analysis 133

3.6.3 Extended Isogeometric Analysis (XIGA) 136

3.6.4 XIGA Governing Equations 138

3.6.5 Numerical Integration 140

3.7 Meshless Methods 142

3.7.1 Why Going Meshless 142

3.7.2 Meshless Approximations 143

3.7.3 Meshless Solutions for the Boundary Value Problems 158

3.8 Variable Node Element (VNE) 166

4 Multiscale Methods 171

4.1 Introduction 171

4.2 Homogenization Methods 172

4.2.1 Introduction 172

4.2.2 Representative Volume Element (RVE) 173

4.2.3 Mathematical Homogenization 174

4.2.4 Computational Homogenization 181

4.3 Molecular Dynamics (MD) 195

4.3.1 Introduction 195

4.3.2 Statistical Mechanics 196

4.3.3 MD Equations of Motion 211

4.3.4 Models for Atomic Interactions – MD Potentials 215

4.3.5 Measures for Determining the State of MD Systems 222

4.3.6 Stress Computation in MD 223

4.3.7 Molecular Statics 226

4.3.8 Sample MD Simulation of a Polymer 227

4.4 Sequential Multiscale Method 229

4.4.1 Introduction 229

4.4.2 Multiscale Modelling of CNT Reinforced Concrete 230

4.4.3 Molecular Dynamics Simulation of CNTs 231

4.4.4 Simulation of CNT-Reinforced Calcium Silicate Hydrate 242

4.4.5 Micromechanical Simulation of CNT-Reinforced Cement 247

4.4.6 Mesoscale Simulation of CNT-Reinforced Concrete 250

4.4.7 Macroscale Simulation of CNT-Reinforced Concrete 256

4.5 Concurrent Multiscale Methods 258

4.5.1 Introduction 258

4.5.2 Quasi-Continuum Method (QC) 260

4.5.3 Bridging Domain Method (BDM) 267

4.5.4 Bridging Scale Method (BSM) 271

4.5.5 Disordered Concurrent Multiscale Method (DCMM) 272

4.5.6 Variable Node Multiscale Method (VNMM) 281

4.5.7 Enriched Multiscale Method (EMM) 288

Part III Biomechanical Simulations 297

5 Biomechanics of Soft Tissues 299

5.1 Introduction 299

5.2 Physiology of Soft Tissues 300

5.2.1 Soft Tissues, Skin 300

5.2.2 Artery 303

5.2.3 Heart Leaflet 303

5.2.4 Brain Tissue 304

5.3 Hyperelastic Models of Soft Tissues 305

5.3.1 Introduction 305

5.3.2 Description of Deformation and Definition of Invariants 307

5.3.3 Isotropic neo-Hookean Hyperelastic Model 309

5.3.4 Isotropic Mooney–Rivlin Hyperelastic Model 312

5.3.5 Hyperelastic Models for Multiscale Simulation of Tendon 313

5.3.6 Anisotropic Hyperelastic Models for Fibrous Tissues 316

5.3.7 Polyconvex Undamaged Functions for Fibrous Tissues 319

5.3.8 Damaged Soft Tissue 321

5.4 Multiscale Modelling of Undamaged Tendon 328

5.4.1 Fibril Scale 330

5.4.2 Fibre Scale 330

5.4.3 Tissue Scale 332

5.5 Multiscale Analysis of a Human Aortic Heart Valve 336

5.5.1 Introduction 336

5.5.2 Organ Scale Simulation 337

5.5.3 Simulation in the Tissue Scale 342

5.5.4 Cell Scale Analysis 347

5.6 Modelling of Ligament Damage 349

5.7 Modelling of the Peeling Test: Dissection of the Medial Tissue 355

5.8 Healing in Damaged Soft Tissue 359

5.8.1 Introduction 359

5.8.2 Physical Foundation of Tissue Healing 360

5.8.3 Solution Procedure 369

5.8.4 Numerical Analysis 372

5.9 Hierarchical Multiscale Modelling of a Degraded Arterial Wall 383

5.9.1 Definition of the Problem 383

5.9.2 Multiscale Model 387

5.9.3 Hyperelastic Material Models 389

5.9.4 Computational Framework of the Hierarchical Multiscale Homogenization 390

5.9.5 Numerical Results 394

5.10 Multiscale Modelling of the Brain 401

5.10.1 Introduction 401

5.10.2 Biomechanics of the Brain 402

5.10.3 Multiscale Modelling of the Brain (neo-Hookean Model) 403

5.10.4 Viscoelastic Modelling of the Brain 414

6 Biomechanics of Hard Tissues 423

6.1 Introduction 423

6.1.1 Hard Tissues 423

6.1.2 Chemical Composition of Bone 423

6.1.3 Multiscale Structure of Bone 423

6.1.4 Bone Remodelling 428

6.1.5 Contents of the Chapter 429

6.2 Concepts of Fracture Analysis of Hard Tissues 429

6.2.1 Numerical Studies of Bone Fracture 430

6.2.2 Constitutive Response of the Bone 433

6.2.3 Poroelastic Nature of Bone Tissues 433

6.2.4 Plasticity and Damage 433

6.2.5 Hyperelastic Response 435

6.3 Simulation of the Femur Bone at Multiple Scales 435

6.3.1 Microscale Simulation of the Trabecular Bone 436

6.3.2 Two-dimensional XFEM Mesoscale Fracture Simulation of the Cortical Bone 437

6.3.3 Macroscale Simulation of the Femur 443

6.4 Healing in Damaged Hard Tissue 446

6.4.1 Introduction 446

6.4.2 Physical Foundation of Bone Tissue Healing 448

6.4.3 Solution Procedure 455

6.4.4 Numerical Analysis 458

7 Supplementary Topics 467

7.1 Introduction 467

7.2 Shape Memory Alloy (SMA) Stenting of an Artery 468

7.2.1 Stenting Procedures 468

7.2.2 SMA Constitutive Equations 469

7.2.3 Contact Mechanics 471

7.2.4 Modelling of Stenting 471

7.2.5 Basics of Modelling 472

7.3 Multiscale Modelling of the Eye 474

7.4 Pulsatile Blood Flow in the Aorta 477

7.4.1 Description of the Problem 477

7.5 Shape Memory Polymer Drug Delivery System 479

7.6 Artificial Intelligence in Biomechanics 483

7.6.1 Artificial Intelligence and Machine Learning 483

7.6.2 Deep Learning 484

7.6.3 Physics-Informed Neural Networks (PINNs) 485

7.6.4 Biomechanical Applications of Artificial Intelligence 487

References 489

Index 519

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