Biomechanics: Concepts and Computation

✍ Scribed by Cees Oomens et al.

• Publisher: Cambridge University Press
• Year: 2018
• Tongue: English
• Leaves: 422
• Edition: 2
• Category: Library

✦ Synopsis

Thoroughly revised and updated for the second edition, this comprehensive textbook
integrates basic and advanced concepts of mechanics with numerical methods and
biomedical applications. Coverage is expanded to include a complete introduction to
vector and tensor calculus, and new or fully updated chapters on biological materials and
continuum mechanics, motion, deformation and rotation, and the constitutive modelling
of solids and fluids. Topics such as kinematics, equilibrium, and stresses and strains are
also included, as well as the mechanical behaviour of fibres and the analysis of
one-dimensional continuous elastic media. Numerical solution procedures based on the
finite element method are presented, with accompanying MATLAB-based software and
dozens of new biomedical engineering examples and exercises allowing readers to
practise and improve their skills. Solutions for instructors are also available online. This is
the definitive guide for both undergraduate and graduate students taking courses in
biomechanics.

✦ Table of Contents

About the Cover page xiii
Preface to the First Edition xv
Preface to the Second Edition xvii
1 Vector and Tensor Calculus 1
1.1 Introduction 1
1.2 Definition of a Vector 1
1.3 Vector Operations 1
1.4 Decomposition of a Vector with Respect to a Basis 5
1.5 Some Mathematical Preliminaries on Second-Order Tensors 10
Exercises 13
2 The Concepts of Force and Moment 16
2.1 Introduction 16
2.2 Definition of a Force Vector 16
2.3 Newton’s Laws 18
2.4 Vector Operations on the Force Vector 19
2.5 Force Decomposition 20
2.6 Drawing Convention 24
2.7 The Concept of Moment 25
2.8 Definition of the Moment Vector 26
2.9 The Two-Dimensional Case 30
2.10 Drawing Convention for Moments in Three Dimensions 33
Exercises 34
3 Static Equilibrium 39
3.1 Introduction 39
3.2 Static Equilibrium Conditions 39
3.3 Free Body Diagram 42
Exercises 51
viii Contents
4 The Mechanical Behaviour of Fibres 56
4.1 Introduction 56
4.2 Elastic Fibres in One Dimension 56
4.3 A Simple One-Dimensional Model of a Skeletal Muscle 59
4.4 Elastic Fibres in Three Dimensions 62
4.5 Small Fibre Stretches 69
Exercises 73
5 Fibres: Time-Dependent Behaviour 79
5.1 Introduction 79
5.2 Viscous Behaviour 81
5.2.1 Small Stretches: Linearization 84
5.3 Linear Visco-Elastic Behaviour 85
5.3.1 Superposition and Proportionality 85
5.3.2 Generalization for an Arbitrary Load History 88
5.3.3 Visco-Elastic Models Based on Springs and Dashpots: Maxwell
Model 92
5.3.4 Visco-Elastic Models Based on Springs and Dashpots:
Kelvin–Voigt Model 96
5.4 Harmonic Excitation of Visco-Elastic Materials 97
5.4.1 The Storage and the Loss Modulus 97
5.4.2 The Complex Modulus 99
5.4.3 The Standard Linear Model 101
5.5 Appendix: Laplace and Fourier Transforms 106
Exercises 108
6 Analysis of a One-Dimensional Continuous Elastic
Medium 116
6.1 Introduction 116
6.2 Equilibrium in a Subsection of a Slender Structure 116
6.3 Stress and Strain 118
6.4 Elastic Stress–Strain Relation 121
6.5 Deformation of an Inhomogeneous Bar 122
Exercises 129
7 Biological Materials and Continuum Mechanics 133
7.1 Introduction 133
7.2 Orientation in Space 134
7.3 Mass within the Volume V 138
7.4 Scalar Fields 141
7.5 Vector Fields 144
ix Contents
7.6 Rigid Body Rotation 149
Exercises 151
8 Stress in Three-Dimensional Continuous Media 155
8.1 Stress Vector 155
8.2 From Stress to Force 156
8.3 Equilibrium 157
8.4 Stress Tensor 164
8.5 Principal Stresses and Principal Stress Directions 172
8.6 Mohr’s Circles for the Stress State 175
8.7 Hydrostatic Pressure and Deviatoric Stress 176
8.8 Equivalent Stress 177
Exercises 178
9 Motion: Time as an Extra Dimension 183
9.1 Introduction 183
9.2 Geometrical Description of the Material Configuration 183
9.3 Lagrangian and Eulerian Descriptions 185
9.4 The Relation between the Material and Spatial Time Derivatives 188
9.5 The Displacement Vector 190
9.6 The Gradient Operator 192
9.7 Extra Rigid Body Displacement 196
9.8 Fluid Flow 198
Exercises 199
10 Deformation and Rotation, Deformation Rate and Spin 204
10.1 Introduction 204
10.2 A Material Line Segment in the Reference and Current
Configurations 204
10.3 The Stretch Ratio and Rotation 210
10.4 Strain Measures and Strain Tensors and Matrices 214
10.5 The Volume Change Factor 219
10.6 Deformation Rate and Rotation Velocity 219
Exercises 222
11 Local Balance of Mass, Momentum and Energy 227
11.1 Introduction 227
11.2 The Local Balance of Mass 227
11.3 The Local Balance of Momentum 228
x Contents
11.4 The Local Balance of Mechanical Power 230
11.5 Lagrangian and Eulerian Descriptions of the Balance Equations 231
Exercises 233
12 Constitutive Modelling of Solids and Fluids 235
12.1 Introduction 235
12.2 Elastic Behaviour at Small Deformations and Rotations 236
12.3 The Stored Internal Energy 242
12.4 Elastic Behaviour at Large Deformations and/or
Large Rotations 244
12.4.1 Material Frame Indifference 244
12.4.2 Strain Energy Function 250
12.4.3 The Incompressible Neo-Hookean Model 252
12.4.4 The Incompressible Mooney–Rivlin Model 255
12.4.5 Compressible Neo-Hookean Elastic Solid 256
12.5 Constitutive Modelling of Viscous Fluids 261
12.6 Newtonian Fluids 262
12.7 Non-Newtonian Fluids 263
12.8 Diffusion and Filtration 264
Exercises 264
13 Solution Strategies for Solid and Fluid Mechanics Problems 270
13.1 Introduction 270
13.2 Solution Strategies for Deforming Solids 270
13.2.1 General Formulation for Solid Mechanics Problems 271
13.2.2 Geometrical Linearity 272
13.2.3 Linear Elasticity Theory, Dynamic 273
13.2.4 Linear Elasticity Theory, Static 273
13.2.5 Linear Plane Stress Theory, Static 274
13.2.6 Boundary Conditions 278
13.3 Solution Strategies for Viscous Fluids 280
13.3.1 General Equations for Viscous Flow 281
13.3.2 The Equations for a Newtonian Fluid 282
13.3.3 Stationary Flow of an Incompressible Newtonian Fluid 282
13.3.4 Boundary Conditions 283
13.3.5 Elementary Analytical Solutions 283
13.4 Diffusion and Filtration 285
Exercises 287
14 Solution of the One-Dimensional Diffusion Equation by
Means of the Finite Element Method 292
14.1 Introduction 292
xi Contents
14.2 The Diffusion Equation 293
14.3 Method of Weighted Residuals and Weak Form 295
14.4 Polynomial Interpolation 297
14.5 Galerkin Approximation 300
14.6 Solution of the Discrete Set of Equations 307
14.7 Isoparametric Elements and Numerical Integration 308
14.8 Basic Structure of a Finite Element Program 312
Exercises 319
15 Solution of the One-Dimensional Convection–Diffusion
Equation by Means of the Finite Element Method 327
15.1 Introduction 327
15.2 The Convection–Diffusion Equation 327
15.3 Temporal Discretization 330
15.4 Spatial Discretization 333
Exercises 338
16 Solution of the Three-Dimensional Convection–Diffusion
Equation by Means of the Finite Element Method 342
16.1 Introduction 342
16.2 Diffusion Equation 343
16.3 Divergence Theorem and Integration by Parts 344
16.4 Weak Form 345
16.5 Galerkin Discretization 345
16.6 Convection–Diffusion Equation 348
16.7 Isoparametric Elements and Numerical Integration 349
16.8 Example 353
Exercises 356
17 Shape Functions and Numerical Integration 363
17.1 Introduction 363
17.2 Isoparametric, Bi-Linear Quadrilateral Element 365
17.3 Linear Triangular Element 367
17.4 Lagrangian and Serendipity Elements 370
17.4.1 Lagrangian Elements 371
17.4.2 Serendipity Elements 373
17.5 Numerical Integration 373
Exercises 377
xii Contents
18 Infinitesimal Strain Elasticity Problems 382
18.1 Introduction 382
18.2 Linear Elasticity 382
18.3 Weak Formulation 384
18.4 Galerkin Discretization 385
18.5 Solution 391
Exercises 394
References 399
Index 401

✦ Subjects

Biomedical Engineering;Bioengineering;Engineering;Engineering & Transportation;Anatomy;Biological Sciences;Science & Math;Engineering;Aeronautical Engineering;Chemical Engineering;Civil Engineering;Electrical & Electronic Engineering;Environmental Engineering;Industrial Engineering;Mechanical Engineering;Nuclear Engineering;New, Used & Rental Textbooks;Specialty Boutique;Anatomy & Physiology;Biology & Life Sciences;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique