Stephen C. Cowin Stephen B. Doty. Tissue Mechanics

Transcription

1 Tissue Mechanics

2 Stephen C. Cowin Stephen B. Doty Tissue Mechanics

3 Stephen C. Cowin Stephen B. Doty New York Center for Biomedical Hospital for Special Surgery Engineering 535 East 70th Street The City College of The City University New York, NY of New York Convent Avenue at 138th Street, New York, NY Library of Congress Control Number: ISBN-10: ISBN-13: Printed on acid-free paper Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights springer.com

4 The efforts of SCC in the preparation of this text are dedicated to his grandchildren. v

5 FOREWORD I was delighted when I learned in the fall of 2005 that Steve Cowin was working on a textbook in biomechanics. Steve and I were in the same department at Tulane University in the 1970s, and under his influence I learned the beauty and power of continuum mechanics as a means to better understand the musculoskeletal system. When I began teaching courses in biomechanics during that decade, it was natural to teach the material from a continuum mechanics perspective. Over the years I have used a variety of continuum mechanics texts, but, for the most part, I have had to find the biomedical examples I used directly from the research literature. I have now had a chance to review a draft of Tissue Mechanics by Cowin and Doty, and it exceeds my high expectations. The material includes a rigorous and comprehensive introduction to continuum mechanics oriented toward biomechanics. Indeed, all of the foundation topics for continuum models of biological materials are covered. This material is illustrated through applications to the hard and soft tissues of the human body. Steve Cowin is now one of the leading researchers in the mechanics of bone, so one would expect the chapters on bone tissue and bone tissue adaptation to be of a very high order. But the presentation on collagen and cartilage mechanics is also excellent. Their presentation of finite deformation mechanics and its application to tendons and ligaments is one of the most accessible in the literature. The text is enhanced by a dedicated website, which promises to continue to add to the text s richness. I presently teach a two-course sequence in continuum biomechanics to advanced undergraduates and graduate students at New Jersey Institute of Technology. I look forward to using Tissue Mechanics as the main text for the course. Cowin and Doty have performed a valuable service for the biomechanics community. For those of us who teach biomechanics, they have prepared the text that many of us have been looking for. But the principal beneficiaries of their efforts will be those who learn from this text with the aid of an instructor or on their own. As I did, thirty years ago, they will begin to appreciate the beauty and power of continuum mechanics as a way to better understand the biological world. I wholeheartedly recommend this text to the biomechanics community. William C. Van Buskirk, PhD Distinguished Professor of Biomedical & Mechanical Engineering Foundation Professor of Biomechanical Engineering New Jersey Institute of Technology Newark, New Jersey vii

6 PREFACE The objective of this book is to describe the methods of formulating continuum models for biological tissues. The text is structured in a stepwise hierarchical fashion such that only the mechanics necessary for the development of a topic is presented before the topic. The text begins with two chapters that are introductory. The first chapter describes the structure of tissues and reviews some of the facilitating unresolved problems of tissue formation by active and passive biological processes. The second chapter describes the spectrum of mechanical modeling of biological tissue structures, how one does it, and why it is done. Chapters 3 through 7 describe the development of linear anisotropic continuum mechanics models: a development of basic continuum kinematics is presented in Chapter 3, the continuum formulations of conservation laws is recounted in Chapter 4, the process of modeling material symmetry is explained in Chapter 5, the steps in the formulation of constitutive equations are enumerated in Chapter 6, and four linear continuum theories flow-through rigid porous media, elasticity, viscous fluid theory, and viscoelasticity are described in Chapter 7. These four continuum models are combined in different ways and applied at the microstructural level in Chapters 8 and 9, chapters in which the process of modeling material microstructure and poroelasticity, respectively, are explained. In Chapter 10 collagen is employed as a model protein to describe how proteins are manufactured and how collagen performs similar basic functions in many tissues akin to those of steel in a machine or a tall building. Chapters 11 and 12 describe bone tissue and the adaptation of bone tissue to mechanical stimulus, respectively. In Chapter 13 the theory for modeling of electrical effects in tissues is developed. The mechanics of various cartilage structures are recounted in Chapter 14. The material of Chapter 15 represents a return to mechanics; modeling of the kinematics and mechanics of large deformations is described in this chapter. In Chapter 16 these results are employed in the development of tendon and ligament mechanics. The presentations in the text differ from the customary presentations of these topics in many aspects, two of which are worth pointing out. First, all continuum models are developed for the anisotropic cases rather than the isotropic cases because most tissues are anisotropic in their material properties. Second, a slightly unconventional tensor-matrix notation is employed in this presentation. Its objective is to represent fourth-rank tensors as matrices that are composed of tensor components, something that the classical Voigt matrix notation for the anisotropic elasticity tensor does not achieve. In the notation employed here second- and fourth-rank tensors in three dimensions are represented as vectors and second-rank tensors, respectively, in six dimensions. Transformations in the six-dimensional space, corresponding to three-dimensional transformations, are six-by-six matrix multiplications that are easily entered and quickly computed with symbolic algebra software (Maple, Mathematica, MacSyma, and MatLab). In particular, the three-dimensional fourth-rank elasticity tensor is represented as a second-rank tensor in a space of six dimensions. This notation is described in the appendix on matrices and tensors. The material in this text is covered in two courses by the first author. The first course is Continuum Mechanics and the second Cell and Tissue Mechanics. The Continuum Mechanics course regularly draws students from Chemical, Civil, and Mechanical Engineering as well as ix

7 x PREFACE Biomedical Engineering. The material in the Continuum Mechanics course, in the order covered, is Appendix A, some of the middle of Chapter 2, then Chapters 3 through 9. The Continuum Mechanics course is a prerequisite for the Cell and Tissue Mechanics course. In Cell and Tissue Mechanics the material covered includes Chapter 1, all of Chapter 2, and Chapters 10 through 16. One of the significant requirements of the Cell and Tissue Mechanics course is that the student prepare a term paper and, during the semester, make a poster and then a PowerPoint presentation on the topic of the term paper. The term papers are required to have an emphasis on some aspect of tissue morphogenesis, molecular self-assembly, supramolecular assembly, and the growth and remodeling of a particular tissue. We would very much appreciate readers communicating with the authors on revisions to this book. In particular any corrections, comments, suggestions of material to be included (or excluded), and suggested problems with solutions for use as either examples or problems at the end of sections would be appreciated. Please these materials to scowin@earthlink.net. We will maintain a record of corrections, suggested additions, and suggested (HW) problems with solutions. These materials, as well as other supplemental text material, will be available from a website designed for that purpose, In particular, there are PowerPoint presentations of the material from each chapter on the website. We plan to develop additional chapters on various tissues and post them there. A problem solutions manual is available from Springer for instructors using this text in a course. Many people have been very helpful in the preparation of this text. Encouragement over the many years of preparation from Bill Van Buskirk, Michael Sacks, Jay Humphrey, Yi-Xian Qin, and Massimiliano Fraldi has been appreciated. The contributions from the students who took the courses in which the content of the book was contained in handouts presented have been most helpful; specifically, the contributions of Charles DaSalla, Jennifer Madeo, Mano Pahakis, Yuliya Vengrenyuk, and Liyun Wang stand out. Some of our colleagues have generously given their time to critique various chapters; in this regard we very much appreciate the efforts of Susannah Fritton, Weiyong Gu, Jacques Huyghe, Yi-Xian Qin, Agnès Rémond, Michael Sacks, Fred Silver, Yuliya Vengrenyuk, and Jeff Weiss. Monte Mehrabadi has made substantial indirect contributions to the text through his 35-year collaboration between one of us (SCC). Adam Curtis, Van Mow, Jean-Paul Revel, Jeff Weiss, and the late Art Winfree all contributed materials that were made into figures and we thank them. Finally, a special thanks to Tim Oliver, who configured the text submitted to Springer into the final product that you see, making the equations and, in particular, the figures more attractive in the process.

8 CONTENTS Foreword... Preface... v vii 1: The Structure of Tissues 1.1. Introduction The Adaptation of Tissues in the Species and in the Individual Tissue Types The Structure and Function of Cells The Hierarchical Structure of Tissues Plans for the Structure of a Tissue Structural Materials for the Tissue Assembly of the Tissue Structure References : Mechanical Modeling of Biological Structures 2.1. Introduction Models and the Real Physical World Guidelines for Modeling Biological Tissues and Solving Biomechanics Problems The Types of Models Used in Biomechanics The Particle Model The Rigid Object Model The Deformable Continuum Model Lumped Parameter Models Statistical Models Cellular Automata The Limits of Reductionism References A. Laplace Transform Refresher B. Direction Integration of First-Order Differential Equations C. Electrical Analogs of the Spring and Dashpot Models : Basic Continuum Kinematics 3.1. The Deformable Material Model, the Continuum Rates of Change and the Spatial Representation of Motion Infinitesimal Motions The Strain Conditions of Compatibility xi

9 xii CONTENTS 4: Continuum Formulations of Conservation Laws 4.1. The Conservation Principles The Conservation of Mass The State of Stress at a Point The Stress Equations of Motion The Conservation of Energy References : Modeling Material Symmetry 5.1. Introduction The Representative Volume Element (RVE) Crystalline Materials and Textured Materials Planes of Mirror Symmetry Characterization of Material Symmetries by Planes of Symmetry The Forms of the Symmetric Three-Dimensional Linear Transformation A The Forms of the Symmetric Six-Dimensional Linear Transformation Ĉ Curvilinear Anisotropy Symmetries that Permit Chirality Relevant Literature : Formulation of Constitutive Equations 6.1. Guidelines for the Formulation of Constitutive Equations Constitutive Ideas Localization Invariance under Rigid Object Motions Determinism Linearization Coordinate Invariance Homogeneous versus Inhomogeneous Constitutive Models Restrictions Due to Material Symmetry The Symmetry of the Material Coefficient Tensors Restrictions on the Coefficients Representing Material Properties Summary of Results Relevant Literature : Four Linear Continuum Theories 7.1. Formation of Continuum Theories The Theory of Fluid Flow through Rigid Porous Media The Theory of Elastic Solids The Theory of Viscous Fluids The Theory of Viscoelastic Materials Relevant Literature

10 CONTENTS xiii 8: Modeling Material Microstructure 8.1. Introduction The Representative Volume Element (RVE) Effective Material Parameters Effective Elastic Constants Effective Permeability Structural Gradients Tensorial Representations of Microstructure Relevant Literature : Poroelasticity 9.1. Poroelastic Materials The Stress Strain Pore Pressure Constitutive Relation The Fluid Content Stress Pore Pressure Constitutive Relation Darcy's Law Matrix Material and Pore Fluid Incompressibility Constraints The Undrained Elastic Coefficients Expressions of Mass and Momentum Conservation The Basic Equations of Poroelasticity The Basic Equations of Incompressible Poroelasticity Some Example Isotropic Poroelastic Problems An Example: The Unconfined Compression of an Anisotropic Disk Relevant Literature : Collagen Introduction The Extracellular Matrix (ECM) The Amino Acid Composition Sequence The Procollagen and Collagen Molecules Experimental and Theoretical Deduction of Collagen Structure The Axial Young's Modulus of a Collagen Molecule The Many Types and Classes of Collagens Collagen Structural Hierarchy Supramolecular Assembly Assembly of a Tendon References : Bone Tissue Introduction The Types of Bone Tissue Bone Interfaces, Porosities and Fluids or Gels Bone Cells The Elastic Symmetry of Cortical Bone

11 xiv CONTENTS The Poroelastic Model for Cortical Bone Electrokinetic Effects in Bone Cortical Bone Strength Cancellous Bone Architecture The Elastic Properties of Cancellous Bone The Literature of Bone Mechanics : Bone Tissue Adaptation Introduction Four Animal Experiments Representation of the Stimulus to Bone Remodeling A Phenomenological Theory of Surface Adaptation Estimating Remodeling Parameters Prediction of the Surface Adaptation Due to the Bending of Bone Summary of Surface Remodeling Results A One-Dimensional Phenomenological Internal Bone Strain Adaptation Model Three-Dimensional Phenomenological Bone Strain Adaptation Theories Possible Cellular Mechanisms for Bone Tissue Adaptation The Literature of Bone Tissue Adaptation : Modeling Poroelastic and Electrical Effects in Soft Tissues Introduction Kinematics of Mixtures The Conservation Laws for Mixtures A Statement of Irreversibility in Mixture Processes Donnan Equilibrium and Osmotic Pressure Continuum Model for a Charged Porous Medium: Constitutive Equations Linear Irreversible Thermodynamics and the Four-Constituent Mixture Modeling Swelling and Compression Experiments on the Intervertebral Disc Relevant Literature : Cartilage Introduction Hyaline Cartilage Elastic Cartilage Fibrocartilage The Perichondrium Common Properties of the Three Tissues: Intervertebral Disc, Articular Cartilage, and the Meniscus The Intervertebral Disc (IVD)

12 CONTENTS xv Articular Cartilage Lubrication in Synovial Joints The Mechanical Modeling of Articular Cartilage Behavior Menisci Relevant Literature : Kinematics and Mechanics of Large Deformations Large Deformations Large Homogeneous Deformations Polar Decomposition of the Deformation Gradients The Strain Measures for Large Deformations Measures of Volume and Surface Change in Large Deformations Stress Measures Finite Deformation Elasticity The Isotropic Finite Deformation Stress Strain Relation Finite Deformation Hyperelasticity Incompressible Elasticity Fung's Exponential Strain Energy Function Strain Energy Functions for Tissues Fung's Quasi-Linear Viscoelasticity (QLV) Relevant Literature : Tendon and Ligament Introduction The Constituents of Tendons and Ligaments The Cells and Cell Systems of the Tendon The Arrangement of Collagen Fibers in Tendons and Ligaments The Lubrication System in Tendons The Mechanical Properties of Tendons and Ligaments Constitutive Equations for Tendons and Ligaments Interstitial Fluid Flow The Insertions of Tendons and Ligaments Structural Adaptation References Appendix: Matrices and Tensors A.1. Introduction and Rationale A.2. Definition of Square, Column, and Row Matrices A.3. The Types and Algebra of Square Matrices A.4. The Algebra of N-Tuples A.5. Linear Transformations A.6. Vector Spaces A.7. Second-Order Tensors A.8. The Moment of Inertia Tensor

13 xvi CONTENTS A.9. The Alternator and Vector Cross-Products A.10. Connection to Mohr's Circles A.11. Special Vectors and Tensors in Six Dimensions A.12. The Gradient Operator and the Divergence Theorem A.13. Tensor Components in Cylindrical Coordinates Credit Lines Index