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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

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

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.
Concepts and Computation

Cees Oomens, Marcel Brekelmans, Frank Baaijens
Eindhoven University of Technology
Department of Biomedical Engineering
Tissue Biomechanics & Engineering
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

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.

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

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
5.2.1 Small stretches: linearization
5.3 Linear visco-elastic behaviour 74
5.3.1 Continuous and discrete time models
5.3.2 Visco-elastic models based on springs and dashpots:
Maxwell model
5.3.3 Visco-elastic models based on springs and dashpots:
Kelvin“Voigt model
5.4 Harmonic excitation of visco-elastic materials 83
5.4.1 The Storage and the Loss Modulus
5.4.2 The Complex Modulus
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

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
13.2.1 General formulation for solid mechanics problems
13.2.2 Geometrical linearity
13.2.3 Linear elasticity theory, dynamic
13.2.4 Linear elasticity theory, static
13.2.5 Linear plane stress theory, static
13.2.6 Boundary conditions
13.3 Solution strategies for viscous ¬‚uids 220
13.3.1 General equations for viscous ¬‚ow
13.3.2 The equations for a Newtonian ¬‚uid
13.3.3 Stationary ¬‚ow of an incompressible Newtonian ¬‚uid
13.3.4 Boundary conditions
13.3.5 Elementary analytical solutions
13.4 Diffusion and ¬ltration 225

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