Biomechanics: Concepts and Computation

This quantitative approach integrates the classical concepts of mechanics and

computational modelling techniques, in a logical progression through a wide range of

fundamental biomechanics principles. Online MATLAB-based software, along with

examples and problems using biomedical applications, will motivate undergraduate

biomedical engineering students to practise and test their skills. The book covers topics

such as kinematics, equilibrium, stresses and strains, and also focuses on large

deformations and rotations and non-linear constitutive equations, including visco-elastic

behaviour and the behaviour of long slender ¬bre-like structures. This is the ¬rst textbook

that integrates both general and speci¬c topics, theoretical background and biomedical

engineering applications, as well as analytical and numerical approaches. This is the

de¬nitive textbook for students.

Cees Oomens is Associate Professor in Biomechanics and Continuum Mechanics at the

Eindhoven University of Technology, the Netherlands. He has lectured many different

courses ranging from basic courses in continuum mechanics at bachelor level, to courses

on mechanical properties of materials and advanced courses in computational modelling

at masters and postgraduate level. His current research focuses on damage and adaptation

of soft biological tissues, with emphasis on skeletal muscle tissue and skin.

Marcel Brekelmans is Associate Professor in Continuum Mechanics at the Eindhoven

University of Technology. Since 1998 he has also lectured in the Biomedical Engineering

Faculty at the University; here his teaching addresses continuum mechanics, basic level

and numerical analysis. He has published a considerable number of papers in well-known

journals, and his research interests in continuum mechanics include the modelling of

history-dependent material behaviour (plasticity, damage and fracture) in forming

processes.

Frank Baaijens is Full Professor in Soft Tissue Biomechanics and Tissue Engineering at

the Eindhoven University of Technology, where he has also been a part-time Professor in

the Polymer Group of the Division of Computational and Experimental Mechanics since

1990. He is currently Scienti¬c Director of the national research program on BioMedical

Materials (BMM), and his research focuses on soft tissue biomechanics and tissue

engineering.

CAMBRIDGE TEXTS IN BIOMEDICAL ENGINEERING

Series Editors

W. Mark Saltzman Yale University

Shu Chien University of California, San Diego

Series Advisors

William Hendee Medical College of Wisconsin

Roger Kamm Massachusetts Institute of Technology

Robert Malkin Duke University

Alison Noble Oxford University

Bernhard Palsson University of California, San Diego

Nicholas Peppas University of Texas at Austin

Michael Sefton University of Toronto

George Truskey Duke University

Cheng Zhu Georgia Institute of Technology

Cambridge Texts in Biomedical Engineering provides a forum for high-quality accessible

textbooks targeted at undergraduate and graduate courses in biomedical engineering. It will

cover a broad range of biomedical engineering topics from introductory texts to advanced

topics including, but not limited to, biomechanics, physiology, biomedical instrumentation,

imaging, signals and systems, cell engineering, and bioinformatics. The series blends theory

and practice, aimed primarily at biomedical engineering students, it also suits broader courses

in engineering, the life sciences and medicine.

Biomechanics

Concepts and Computation

Cees Oomens, Marcel Brekelmans, Frank Baaijens

Eindhoven University of Technology

Department of Biomedical Engineering

Tissue Biomechanics & Engineering

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

Information on this title: www.cambridge.org/9780521875585

© C. Oomens, M. Brekelmans and F. Baaijens 2009

This publication is in copyright. Subject to statutory exception and to the

provision of relevant collective licensing agreements, no reproduction of any part

may take place without the written permission of Cambridge University Press.

First published in print format 2009

ISBN-13 978-0-511-47927-4 eBook (EBL)

ISBN-13 978-0-521-87558-5 hardback

Cambridge University Press has no responsibility for the persistence or accuracy

of urls for external or third-party internet websites referred to in this publication,

and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate.

Contents

About the cover page xi

Preface xiii

1 Vector calculus 1

1.1 Introduction 1

1.2 De¬nition of a vector 1

1.3 Vector operations 1

1.4 Decomposition of a vector with respect to a basis 5

Exercises 8

2 The concepts of force and moment 10

2.1 Introduction 10

2.2 De¬nition of a force vector 10

2.3 Newton™s Laws 12

2.4 Vector operations on the force vector 13

2.5 Force decomposition 14

2.6 Representation of a vector with respect to a vector basis 17

2.7 Column notation 21

2.8 Drawing convention 24

2.9 The concept of moment 25

2.10 De¬nition of the moment vector 26

2.11 The two-dimensional case 29

2.12 Drawing convention of moments in three dimensions 32

Exercises 33

3 Static equilibrium 37

3.1 Introduction 37

3.2 Static equilibrium conditions 37

3.3 Free body diagram 40

Exercises 47

Contents

vi

4 The mechanical behaviour of ¬bres 50

4.1 Introduction 50

4.2 Elastic ¬bres in one dimension 50

4.3 A simple one-dimensional model of a skeletal muscle 53

4.4 Elastic ¬bres in three dimensions 55

4.5 Small ¬bre stretches 61

Exercises 66

5 Fibres: time-dependent behaviour 69

5.1 Introduction 69

5.2 Viscous behaviour 71

73

5.2.1 Small stretches: linearization

5.3 Linear visco-elastic behaviour 74

74

5.3.1 Continuous and discrete time models

5.3.2 Visco-elastic models based on springs and dashpots:

78

Maxwell model

5.3.3 Visco-elastic models based on springs and dashpots:

82

Kelvin“Voigt model

5.4 Harmonic excitation of visco-elastic materials 83

83

5.4.1 The Storage and the Loss Modulus

85

5.4.2 The Complex Modulus

87

5.4.3 The standard linear model

5.5 Appendix: Laplace and Fourier transforms 92

Exercises 94

6 Analysis of a one-dimensional continuous elastic medium 99

6.1 Introduction 99

6.2 Equilibrium in a subsection of a slender structure 99

6.3 Stress and strain 101

6.4 Elastic stress“strain relation 104

6.5 Deformation of an inhomogeneous bar 104

Exercises 111

7 Biological materials and continuum mechanics 114

7.1 Introduction 114

7.2 Orientation in space 115

7.3 Mass within the volume V 117

7.4 Scalar ¬elds 120

7.5 Vector ¬elds 122

7.6 Rigid body rotation 125

vii Contents

7.7 Some mathematical preliminaries on second-order tensors 127

Exercises 130

8 Stress in three-dimensional continuous media 132

8.1 Stress vector 132

8.2 From stress to force 133

8.3 Equilibrium 134

8.4 Stress tensor 142

8.5 Principal stresses and principal stress directions 146

8.6 Mohr™s circles for the stress state 149

8.7 Hydrostatic pressure and deviatoric stress 150

8.8 Equivalent stress 150

Exercises 152

9 Motion: the time as an extra dimension 156

9.1 Introduction 156

9.2 Geometrical description of the material con¬guration 156

9.3 Lagrangian and Eulerian description 158

9.4 The relation between the material and spatial time derivative 159

9.5 The displacement vector 161

9.6 The gradient operator 162

9.7 Extra displacement as a rigid body 164

9.8 Fluid ¬‚ow 166

Exercises 167

10 Deformation and rotation, deformation rate and spin 170

10.1 Introduction 170

10.2 A material line segment in the reference and current

con¬guration 170

10.3 The stretch ratio and rotation 173

10.4 Strain measures and strain tensors and matrices 176

10.5 The volume change factor 180

10.6 Deformation rate and rotation velocity 180

Exercises 183

11 Local balance of mass, momentum and energy 186

11.1 Introduction 186

11.2 The local balance of mass 186

11.3 The local balance of momentum 187

Contents

viii

11.4 The local balance of mechanical power 189

11.5 Lagrangian and Eulerian description of the balance equations 190

Exercises 192

12 Constitutive modelling of solids and ¬‚uids 194

12.1 Introduction 194

12.2 Elastic behaviour at small deformations and rotations 195

12.3 The stored internal energy 198

12.4 Elastic behaviour at large deformations and/or large rotations 200

12.5 Constitutive modelling of viscous ¬‚uids 203

12.6 Newtonian ¬‚uids 204

12.7 Non-Newtonian ¬‚uids 205

12.8 Diffusion and ¬ltration 205

Exercises 206

13 Solution strategies for solid and ¬‚uid mechanics

problems 210

13.1 Introduction 210

13.2 Solution strategies for deforming solids 210

211

13.2.1 General formulation for solid mechanics problems

212

13.2.2 Geometrical linearity

213

13.2.3 Linear elasticity theory, dynamic

213

13.2.4 Linear elasticity theory, static

214

13.2.5 Linear plane stress theory, static

218

13.2.6 Boundary conditions

13.3 Solution strategies for viscous ¬‚uids 220

221

13.3.1 General equations for viscous ¬‚ow

221

13.3.2 The equations for a Newtonian ¬‚uid

222

13.3.3 Stationary ¬‚ow of an incompressible Newtonian ¬‚uid

223

13.3.4 Boundary conditions

223

13.3.5 Elementary analytical solutions

13.4 Diffusion and ¬ltration 225