Introduction to Continuum Biomechanics Synthesis Lectures on Biomedical Engineering

Kyriacos Athanasiou9781598296174, 1598296175

This book is concerned with the study of continuum mechanics applied to biological systems, i.e., continuum biomechanics. This vast and exciting subject allows description of when a bone may fracture due to excessive loading, how blood behaves as both a solid and fluid, down to how cells respond to mechanical forces that lead to changes in their behavior, a process known as mechanotransduction. We have written for senior undergraduate students and first year graduate students in mechanical or biomedical engineering, but individuals working at biotechnology companies that deal in biomaterials or biomechanics should also find the information presented relevant and easily accessible. Table of Contents: Tensor Calculus / Kinematics of a Continuum / Stress / Elasticity / Fluids / Blood and Circulation / Viscoelasticity / Poroelasticity and Thermoelasticity / Biphasic Theory

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Table of contents :
Introduction to Continuum Biomechanics……Page 2
Keywords……Page 5
Dedication……Page 6
Foreword……Page 8
Contents……Page 10
Introduction……Page 16
1.1.1 Summation (Dummy Indices)……Page 20
1.1.2 Multiple Equations (Free Indices)……Page 21
1.1.3 Kronecker Delta……Page 22
1.1.4 Manipulations……Page 23
1.2.2 Components of a Tensor……Page 24
1.2.3 Sum and Product of Tensors……Page 26
1.2.6 Orthogonal Tensor……Page 27
1.3.1 Symmetric vs. Antisymmetric Tensor……Page 30
1.3.2 Eigenvalues and Eigenvectors……Page 31
1.3.3 Principal Values and Principal Directions……Page 34
1.4.1 Scalar Invariants of a Tensor and the Cayley-Hamilton Theorem……Page 35
1.4.3 Tensor-Valued Functions of a Scalar……Page 36
1.4.4 Gradient and Divergence of Scalar, Vector, and Tensor Fields……Page 37
1.5 PROBLEMS……Page 38
2.1 DESCRIPTION OF THE MOTION OF A CONTINUUM……Page 44
2.2 MATERIAL VS. SPATIAL DESCRIPTION……Page 45
2.3 MATERIAL DERIVATIVE……Page 47
2.4 DEFORMATION-INDUCED STRAIN……Page 49
2.6 DILATATION……Page 54
2.7 RATE OF DEFORMATION……Page 55
2.8 CONTINUITY EQUATION (CONSERVATION OF MASS)……Page 56
2.9 PROBLEMS……Page 60
3.1 STRESS VECTOR (“TRACTION”)……Page 66
3.3 PRINCIPLE OF MOMENT OF MOMENTUM (PROOF OF STRESS TENSOR SYMMETRY)……Page 67
3.4 PRINCIPAL STRESSES……Page 68
3.5 MAXIMUM SHEAR STRESS……Page 69
3.6 EQUATIONS OF MOTION (CONSERVATION OF LINEAR MOMENTUM)……Page 72
*3.8 ALTERNATIVE STRESS DEFINITIONS……Page 73
3.9 DEMONSTRATIONS……Page 75
3.10 PROBLEMS……Page 78
4.1 SUMMARY UP TO NOW……Page 82
*4.2 GENERAL ELASTICITY……Page 83
4.2.1 Hyperelasticity……Page 84
4.2.2 Approximations Leading to Linear Elasticity……Page 87
4.3 EXPERIMENTAL OBSERVATIONS OF INFINITESIMAL LINEAR ELASTICITY……Page 88
4.4 LINEARLY ELASTIC SOLID……Page 90
4.5 ISOTROPIC LINEARLY ELASTIC SOLID……Page 91
4.6 MATERIAL PROPERTIES OF ELASTIC MATERIALS……Page 92
4.7 EQUATIONS OF THE INFINITESIMAL THEORY OF ELASTICITY……Page 94
4.8 COMPATIBILITY CONDITIONS FOR INFINITESIMAL STRAIN CONDITIONS……Page 95
4.9.1 Simple Infinitesimal Extension of a Linear Elastic Solid……Page 96
4.9.2 Pure Bending of a Beam……Page 99
4.9.3 Torsion of a Circular Cylinder……Page 102
4.10.1 Plane Strain……Page 107
*4.11 ANISOTROPIC LINEAR ELASTICITY……Page 109
4.12 PROBLEMS……Page 112
5.1 INTRODUCTION TO FLUIDS……Page 116
5.2 HYDROSTATICS……Page 117
5.3 NEWTONIAN VISCOUS FLUID……Page 119
5.4 MEANING OF l AND m……Page 120
5.6 NAVIER-STOKES EQUATIONS……Page 121
5.8 IMPORTANT DEFINITIONS……Page 122
5.9.1 Plane Couette Flow……Page 123
5.9.2 Plane Poiseuille Flow……Page 124
5.9.3 Extensions of Plane Poiseuille Flow……Page 127
*5.10 NON-NEWTONIAN FLUIDS……Page 128
*5.11 VORTICITY VECTOR……Page 130
*5.12 IRROTATIONAL FLOW……Page 131
5.12.1 Irrotational Flow of an Inviscid Incompressible Fluid……Page 132
5.13 PROBLEMS……Page 135
6.2 BASICS AND MATERIAL PROPERTIES OF BLOOD……Page 138
6.4 NON-NEWTONIAN BEHAVIOR OF BLOOD……Page 139
6.6 BLOOD RHEOLOGY……Page 141
6.7 SUMMARY……Page 143
6.8 LAMINAR FLOW OF BLOOD IN A TUBE……Page 144
6.9 PROBLEMS……Page 149
7.2 DEFINITION OF VISCOELASTICITY……Page 154
7.3 1-D LINEAR VISCOELASTICITY (DIFFERENTIAL FORM BASED ON MECHANICAL CIRCUIT MODELS)……Page 155
7.3.1 Maxwell Fluid……Page 156
7.3.2 Kelvin-Voigt Solid……Page 159
7.3.3 Standard Linear Solid……Page 160
7.3.4 Beyond the Canonical Models……Page 161
7.4 1-D LINEAR VISCOELASTICITY (INTEGRAL FORMULATION)……Page 163
7.5 3-D LINEAR VISCOELASTICITY……Page 165
*7.6 BOUNDARY VALUE PROBLEMS AND THE CORRESPONDENCE PRINCIPLE……Page 166
7.7 DYNAMIC BEHAVIOR OF VISCOELASTIC MATERIALS……Page 168
7.8 LIMITING CASES OF LINEAR VISCOELASTICITY ARE THE HOOKEAN SOLID AND NEWTONIAN VISCOUS FLUID……Page 171
7.9 PROBLEMS……Page 173
8.2.1 Terzaghi’s Principle of Effective Stress……Page 178
8.2.2 Darcy’s Law……Page 179
8.2.3 Constitutive Equations and Material Constants……Page 180
8.2.4 u-p Formulation of Poroelastic Governing Equations……Page 182
8.2.5 Consolidation of a Finite Layer (Terzaghi’s Problem)……Page 184
8.3.1 Introduction and Fourier’s Law……Page 187
8.3.2 Constitutive and Governing equations (“u-q” Formulation)……Page 188
8.3.3 Thermal Prestress/PreStrain……Page 190
8.4 PROBLEMS……Page 191
9.1 INTRODUCTION……Page 194
9.2 DEFINITIONS……Page 195
9.3 CONSERVATION OF MASS……Page 196
9.4 CONSERVATION OF MOMENTUM……Page 198
9.5 CONSTITUTIVE EQUATIONS……Page 200
9.6 SUMMARY AND EQUATIONS OF MOTION……Page 201
9.7 CONFINED COMPRESSION……Page 203
9.7.2 Stress Relaxation……Page 207
9.8 UNCONFINED COMPRESSION……Page 209
9.9 PROBLEMS……Page 211
Acknowledgments……Page 214
Afterword……Page 216
Bibliography……Page 218
Author Biography……Page 220