Multiscale Biomechanics: Theory and Applications

Multiscale Biomechanics
Contents
Preface
List of Abbreviations
Part I Introduction
1 Introduction
1.1 Introduction to Biomechanics
1.2 Biology and Biomechanics
1.3 Types of Biological Systems
1.3.1 Biosolids
1.3.2 Biofluids
1.3.3 Biomolecules
1.3.4 Synthesized Biosystems
1.4 Biomechanical Hierarchy
1.4.1 Organ Level
1.4.2 Tissue Level
1.4.3 Cellular and Lower Levels
1.4.4 Complex Medical Procedures
1.5 Multiscale/Multiphysics Analysis
1.6 Scope of the Book
Part II Analytical and Numerical Bases
2 Theoretical Bases of Continuum Mechanics
2.1 Introduction
2.2 Solid Mechanics
2.2.1 Elasticity
2.2.2 Plasticity
2.2.3 Damage Mechanics
2.2.4 Fracture Mechanics
2.2.5 Viscoelasticity
2.2.6 Poroelasticity
2.2.7 Large Deformation
2.3 Flow, Convection and Diffusion
2.3.1 Thermodynamics
2.3.2 Fluid Mechanics
2.3.3 Gas Dynamics
2.3.4 Diffusion and Convection
2.4 Fluid–Structure Interaction
2.4.1 Lagrangian and Eulerian Descriptions
2.4.2 Fluid–Solid Interface Boundary Conditions
2.4.3 Governing Equations in the Eulerian Description
2.4.4 Coupled Lagrangian–Eulerian (CLE)
2.4.5 Coupled Lagrangian–Lagrangian (CLL)
2.4.6 Arbitrary Lagrangian–Eulerian (ALE)
3 Numerical Methods
3.1 Introduction
3.2 Finite Difference Method (FDM)
3.2.1 One-Dimensional FDM
3.2.2 Higher Order One-Dimensional FDM
3.2.3 FDM for Solving Partial Differential Equations
3.3 Finite Volume Method (FVM)
3.4 Finite Element Method (FEM)
3.4.1 Basics of FEM Interpolation
3.4.2 FEM Basis Functions/Shape Functions
3.4.3 Properties of the Finite Element Interpolation
3.4.4 Physical and Parametric Coordinate Systems
3.4.5 Main Types of Finite Elements
3.4.6 Governing Equations of the Boundary Value Problem
3.4.7 Numerical Integration
3.5 Extended Finite Element Method (XFEM)
3.5.1 A Review of XFEM Development
3.5.2 Partition of Unity
3.5.3 Enrichments
3.5.4 Signed Distance Function
3.5.5 XFEM Approximation for Cracked Elements
3.5.6 Boundary Value Problem for a Cracked Body
3.5.7 XFEM Discretisation of the Governing Equation
3.5.8 Numerical Integration
3.5.9 Selection of Enrichment Nodes for Crack Propagation
3.5.10 Incompatible Modes of XFEM Enrichments
3.5.11 The Level Set Method for Tracking Moving Boundaries
3.5.12 XFEM Tip Enrichments
3.5.13 XFEM Enrichment Formulation for Large Deformation Problems
3.6 Extended Isogeometric Analysis (XIGA)
3.6.1 Introduction
3.6.2 Isogeometric Analysis
3.6.3 Extended Isogeometric Analysis (XIGA)
3.6.4 XIGA Governing Equations
3.6.5 Numerical Integration
3.7 Meshless Methods
3.7.1 Why Going Meshless
3.7.2 Meshless Approximations
3.7.3 Meshless Solutions for the Boundary Value Problems
3.8 Variable Node Element (VNE)
4 Multiscale Methods
4.1 Introduction
4.2 Homogenization Methods
4.2.1 Introduction
4.2.2 Representative Volume Element (RVE)
4.2.3 Mathematical Homogenization
4.2.4 Computational Homogenization
4.3 Molecular Dynamics (MD)
4.3.1 Introduction
4.3.2 Statistical Mechanics
4.3.3 MD Equations of Motion
4.3.4 Models for Atomic Interactions – MD Potentials
4.3.5 Measures for Determining the State of MD Systems
4.3.6 Stress Computation in MD
4.3.7 Molecular Statics
4.3.8 Sample MD Simulation of a Polymer
4.4 Sequential Multiscale Method
4.4.1 Introduction
4.4.2 Multiscale Modelling of CNT Reinforced Concrete
4.4.3 Molecular Dynamics Simulation of CNTs
4.4.4 Simulation of CNT-Reinforced Calcium Silicate Hydrate
4.4.5 Micromechanical Simulation of CNT-Reinforced Cement
4.4.6 Mesoscale Simulation of CNT-Reinforced Concrete
4.4.7 Macroscale Simulation of CNT-Reinforced Concrete
4.5 Concurrent Multiscale Methods
4.5.1 Introduction
4.5.2 Quasi-Continuum Method (QC)
4.5.3 Bridging Domain Method (BDM)
4.5.4 Bridging Scale Method (BSM)
4.5.5 Disordered Concurrent Multiscale Method (DCMM)
4.5.6 Variable Node Multiscale Method (VNMM)
4.5.7 Enriched Multiscale Method (EMM)
Part III Biomechanical Simulations
5 Biomechanics of Soft Tissues
5.1 Introduction
5.2 Physiology of Soft Tissues
5.2.1 Soft Tissues, Skin
5.2.2 Artery
5.2.3 Heart Leaflet
5.2.4 Brain Tissue
5.3 Hyperelastic Models of Soft Tissues
5.3.1 Introduction
5.3.2 Description of Deformation and Definition of Invariants
5.3.3 Isotropic neo-Hookean Hyperelastic Model
5.3.4 Isotropic Mooney–Rivlin Hyperelastic Model
5.3.5 Hyperelastic Models for Multiscale Simulation of Tendon
5.3.6 Anisotropic Hyperelastic Models for Fibrous Tissues
5.3.7 Polyconvex Undamaged Functions for Fibrous Tissues
5.3.8 Damaged Soft Tissue
5.4 Multiscale Modelling of Undamaged Tendon
5.4.1 Fibril Scale
5.4.2 Fibre Scale
5.4.3 Tissue Scale
5.5 Multiscale Analysis of a Human Aortic Heart Valve
5.5.1 Introduction
5.5.2 Organ Scale Simulation
5.5.3 Simulation in the Tissue Scale
5.5.4 Cell Scale Analysis
5.6 Modelling of Ligament Damage
5.7 Modelling of the Peeling Test: Dissection of the Medial Tissue
5.8 Healing in Damaged Soft Tissue
5.8.1 Introduction
5.8.2 Physical Foundation of Tissue Healing
5.8.3 Solution Procedure
5.8.4 Numerical Analysis
5.9 Hierarchical Multiscale Modelling of a Degraded Arterial Wall
5.9.1 Definition of the Problem
5.9.2 Multiscale Model
5.9.3 Hyperelastic Material Models
5.9.4 Computational Framework of the Hierarchical Multiscale Homogenization
5.9.5 Numerical Results
5.10 Multiscale Modelling of the Brain
5.10.1 Introduction
5.10.2 Biomechanics of the Brain
5.10.3 Multiscale Modelling of the Brain (neo-Hookean Model)
5.10.4 Viscoelastic Modelling of the Brain
6 Biomechanics of Hard Tissues
6.1 Introduction
6.1.1 Hard Tissues
6.1.2 Chemical Composition of Bone
6.1.3 Multiscale Structure of Bone
6.1.4 Bone Remodelling
6.1.5 Contents of the Chapter
6.2 Concepts of Fracture Analysis of Hard Tissues
6.2.1 Numerical Studies of Bone Fracture
6.2.2 Constitutive Response of the Bone
6.2.3 Poroelastic Nature of Bone Tissues
6.2.4 Plasticity and Damage
6.2.5 Hyperelastic Response
6.3 Simulation of the Femur Bone at Multiple Scales
6.3.1 Microscale Simulation of the Trabecular Bone
6.3.2 Two-dimensional XFEM Mesoscale Fracture Simulation of the Cortical Bone
6.3.3 Macroscale Simulation of the Femur
6.4 Healing in Damaged Hard Tissue
6.4.1 Introduction
6.4.2 Physical Foundation of Bone Tissue Healing
6.4.3 Solution Procedure
6.4.4 Numerical Analysis
7 Supplementary Topics
7.1 Introduction
7.2 Shape Memory Alloy (SMA) Stenting of an Artery
7.2.1 Stenting Procedures
7.2.2 SMA Constitutive Equations
7.2.3 Contact Mechanics
7.2.4 Modelling of Stenting
7.2.5 Basics of Modelling
7.3 Multiscale Modelling of the Eye
7.4 Pulsatile Blood Flow in the Aorta
7.4.1 Description of the Problem
7.5 Shape Memory Polymer Drug Delivery System
7.6 Artificial Intelligence in Biomechanics
7.6.1 Artificial Intelligence and Machine Learning
7.6.2 Deep Learning
7.6.3 Physics-Informed Neural Networks (PINNs)
7.6.4 Biomechanical Applications of Artificial Intelligence
References
Index
EULA