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This page intentionally left blank **Biomechanics: Concepts and Computation** This quantitative approach integrates classical concepts of mechanics with computational modeling techniques in a logical progression through fundamental biomechanics principles. Online MATLAB-based software, along with examples and problems using biomedical applications, will motivate undergraduate biomedical engineering students to practice and test their skills. The book covers topics such as kinematics, equilibrium, stresses and strains, large deformations and rotations, and non-linear constitutive equations including visco-elastic behavior and the behavior of long slender fiber-like structures. This is the first textbook that integrates both general and specific topics, theoretical background, biomedical engineering applications, and analytical and numerical approaches. It is the definitive textbook for students. **Cees Oomens** is Associate Professor in Biomechanics and Continuum Mechanics at Eindhoven University of Technology, Netherlands. He has lectured many different courses ranging from basic courses in continuum mechanics at bachelor level to advanced courses in computational modeling at masters and postgraduate levels. 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 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 include modeling history-dependent material behavior (plasticity, damage and fracture) in forming processes. **Frank Baaijens** is Full Professor in Soft Tissue Biomechanics and Tissue Engineering at 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 Scientific 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 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) 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 Preface 1 Vector calculus 1.1 Introduction 1.2 Definition of a vector 1.3 Vector operations 1.4 Decomposition of a vector with respect to a basis Exercises 2 The concepts of force and moment 2.1 Introduction 2.2 Definition of a force vector 2.3 Newton’s Laws 2.4 Vector operations on the force vector 2.5 Force decomposition 2.6 Representation of a vector with respect to a vector basis 2.7 Column notation 2.8 Drawing convention 2.9 The concept of moment 2.10 Definition of the moment vector 2.11 The two-dimensional case 2.12 Drawing convention of moments in three dimensions Exercises 3 Static equilibrium 3.1 Introduction 3.2 Static equilibrium conditions 3.3 Free body diagram Exercises 4 The mechanical behaviour of fibres 5 Fibres: time-dependent behaviour 6 Analysis of a one-dimensional continuous elastic medium 7 Biological materials and continuum mechanics 8 Stress in three-dimensional continuous media 9 Motion: the time as an extra dimension 10 Deformation and rotation, deformation rate and spin **Contents** 4.1 Introduction 4.2 Elastic fibres in one dimension 4.3 A simple one-dimensional model of a skeletal muscle 4.4 Elastic fibres in three dimensions 4.5 Small fibre stretches Exercises 5.1 Introduction 5.2 Viscous behaviour 5.2.1 Small stretches: linearization 5.3 Linear visco-elastic behaviour 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 5.4.1 The Storage and the Loss Modulus 5.4.2 The Complex Modulus 5.4.3 The standard linear model Exercises 6 Analysis of a one-dimensional continuous elastic medium 99 7 Biological materials and continuum mechanics 114 8 Stress in three-dimensional continuous media 132 9 Motion: the time as an extra dimension 156 10 Deformation and rotation, deformation rate and spin 170 Exercises 18.2 Consider the mesh given in the figure below. The mesh consists of two linear triangular elements and a linear elasticity formulation applies to this element configuration. The solution u? is given by u?T _ _ u1 w1 u2 w2 u3 w3 u4 w4_ , where u and v denote the displacements in the x- and y-direction, respectively. (a) What is a possible top array of this element configuration assuming equal material and type identifiers for both elements? (b) What is the pos array for this element configuration? 4 3 (2) y x 1 (1) 2 (c) What is the dest array? (d) Based on the pos array, the non-zero entries of the stiffness matrix can be identified. Visualize the non-zero entries of the stiffness matrix. 326 Infinitesimal strain elasticity problems (e) Suppose that the boundary nodes in the array usercurves are stored as usercurves__ 13 34 24 1 2_ The solution u? is stored in the array sol. How are the displacements in the x-direction extracted from the sol array along the first usercurve? (f) Let the solution array sol and the global stiffness matrix q be given. Suppose that both displacements at nodes 1 and 2 are suppressed. Compute the reaction forces in these nodes. Compute the total reaction force in the y-direction along the boundary containing the nodes 1 and 2. 18.3 Consider the bi-linear element of the figure below. It covers exactly the domain -1 ? x ? 1 and -1 ? y ? 1. Assume that the plane strain condition applies. (a) Compute the strain displacement matrix B for this element in point (x, y). (b) Let the nodal displacements for this element be given by u?Te _ 0 0 0 0 1 0 1 0 . What are the strains in (x, y)? (c) If G = 1 and K = 2 what are the stress components ?xx, ?yy and ?xy at ( x, y) _ (-1, 1)? 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