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1 CONTINUUM MECHANICS This is a modern textbook for courses in continuum mechanics. It provides both the theoretical framework and the numerical methods required to model the behavior of continuous materials. This self-contained textbook is tailored for advanced undergraduate or firstyear graduate students with numerous step-by-step derivations and worked-out examples. The author presents both the general continuum theory and the mathematics needed to apply it in practice. The derivation of constitutive models for ideal gases, fluids, solids, and biological materials and the numerical methods required to solve the resulting differential equations are also detailed. Specifically, the text presents the theory and numerical implementation for the finite difference and the finite element methods in the Matlab programming language. It includes thirteen detailed Matlab programs illustrating how constitutive models are used in practice. Dr. received his PhD in Mechanical Engineering from the Massachusetts Institute of Technology. He taught Mechanical Engineering at Drexel University from 2006 to He is currently an Associate Professor of Civil and Mechanical Engineering at Merrimack College. His research focuses on the modeling of biological and polymeric materials at various length scales.

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3 Continuum Mechanics CONSTITUTIVE MODELING OF STRUCTURAL AND BIOLOGICAL MATERIALS Merrimack College

4 CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore São Paulo, Delhi, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY , USA Information on this title: / This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Capaldi, Franco M., 1977 Continuum mechanics : constitutive modeling of structural and biological materials /. p. cm. Includes index. ISBN Continuum mechanics. I. Title. QA808.2.C dc ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

5 To Irene, Emma, and Nina with love.

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7 Contents Preface page xiii 1 Mathematics Vectors Second-Order Tensors Eigenvalues and Eigenvectors Spectral Decomposition of a Symmetric Tensor Coordinate Transformation Invariants Cayley-Hamilton Theorem Scalar, Vector, and Tensor Functions and Fields Integral Theorems 30 Exercises 32 Matlab Exercises 33 2 Kinematics Configurations Velocity and Acceleration Displacement Deformation Gradient Jacobian Nanson s Formula Homogenous Deformation, Isochoric Deformation, and Rigid Body Rotation Material and Spatial Derivatives Polar Decomposition of the Deformation Gradient Stretch Ratios Left and Right Cauchy Deformation Tensor Green Strain Tensor Almansi Strain Tensor Infinitesimal Strain Tensor Velocity Gradient, Rate of Deformation, Vorticity 65 vii

8 viii Contents 2.16 Reynolds Transport Theorem 68 Exercises 71 Matlab Exercises 73 3 The Stress Tensor Mass, Density, and Forces Traction Vector Cauchy Stress Tensor First Piola-Kirchhoff Stress Tensor Second Piola-Kirchhoff Stress Tensor Maximum Normal and Shear Stress Decomposition of the Stress Tensor 82 Exercises 83 4 Introduction to Material Modeling Forces and Fields Balance Laws Conservation of Mass Conservation of Linear Momentum Conservation of Angular Momentum Conservation of Energy The Second Law of Thermodynamics Summary of the Field Equations Stress Power Jump Conditions Constitutive Modeling Constitutive Modeling Principles Principle of Dissipation Principle of Material Frame Indifference Material Symmetry Isotropic Scalar-Valued Functions Isotropic Tensor-Valued Functions Internal Variables Thermodynamics of Materials Heat Transfer 120 Exercises Ideal Gas Historical Perspective Forces and Fields Balance Laws Constitutive Model Constraints Constitutive Relations Molecular Model of an Ideal Gas Governing Equations 131

9 Contents ix 5.6 Acoustic Waves Finite Difference Method Explicit Algorithm Implicit Algorithm Example Problem Matlab File Explicit Algorithm Matlab File Implicit Algorithm Fluids Historical Perspective Forces and Fields Balance Laws Constitutive Model Constraints Constitutive Relations for the Newtonian Fluid Stokes Condition Governing Equations Compressible Newtonian Fluid Incompressible Newtonian Fluid Irrotational Steady Flow of an Incompressible Newtonian Fluid Non-Newtonian Fluid Models Power Law Model Cross Model Bingham Model Couette Viscometer Newtonian Fluid Power Law Fluid Model General Non-Newtonian Fluid Elastic Material Models Historical Perspective Finite Thermoelastic Material Model Forces and Fields Balance Laws Constitutive Model Constraints Due to Material Frame Indifference Constraints Due to the Second Law of Thermodynamics Hyperelastic Material Model Balance Laws Constitutive Model Constraints Due to Material Frame Indifference Clausius-Duhem Inequality Material Symmetry Isotropic Materials Transversely Isotropic Materials 193

10 x Contents Incompressible Materials Common Hyperelastic Constitutive Models Freely Jointed Chain Linear Thermoelastic Material Model Balance Laws Constitutive Model Clausius-Duhem Inequality Linear Thermoelastic Constitutive Relation Material Symmetry Governing Equations for the Isotropic Linear Elastic Material Uniaxial Tension Test Kinematics Isotropic Linear Thermoelastic Material Incompressible Isotropic Neo-Hookean Model Continuum Mixture Theory Forces and Fields Balance Laws Conservation of Mass Conservation of Momentum Conservation of Angular Momentum Conservation of Energy Second Law of Thermodynamics Biphasic Model Isothermal Biphasic Model Application to Soft Tissue Confined Compression Experiment Unconfined Compression Growth Models Forces and Fields Balance Laws Conservation of Mass Reynolds Transport Theorem Conservation of Momentum Conservation of Angular Momentum Conservation of Energy Decomposition of the Deformation Gradient Summary of the Field Equations Constitutive Model Uniaxial Loading Kinematics Governing Equation 253

11 Contents xi Finite Difference Algorithm Example Problem Matlab File Parameter Estimation and Curve Fitting Propagation of Error Least Squares Fit Finite Element Method Introduction Element Types Natural Versus Global Coordinates for a Quadrilateral Element Field Variable Representation Within an Element Matrix Representation Integration of a Field Variable Gaussian Quadrature Differentiation of a Field Variable Formulation of the Governing Equations Plane Strain Deformation Statement of Virtual Work Discretization of Space Approximation of the Field Variables FEM Formulation Element Stiffness Tensor Body Force Vector Traction Force Vector Single Element Implementation Axisymmetric Deformation Statement of Virtual Work Discretization of Space Approximation of the Field Variables FEM Formulation Element Stiffness Tensor Body Force Vector Single Element Implementation Multiple Element Implementation Infinitesimal Plane Strain FEM with Material Nonlinearity Statement of Virtual Work Discretization of Space Approximation of the Field Variables FEM Formulation Plane Strain Finite Deformation Total Lagrangian Method Updated Lagrangian Method 323

12 xii Contents Updated Lagrangian Method Single Element Implementation Appendix Introduction to Matlab Reference Tables 334 Index 341

13 Preface This textbook is designed to give students an understanding and appreciation of continuum-level material modeling. The mathematics and continuum framework are presented as a tool for characterizing and then predicting the response of materials. The textbook attempts to make the connection between experimental observation and model development in order to put continuum-level modeling into a practical context. This comprehensive treatment of continuum mechanics gives students an appreciation for the manner in which the continuum theory is applied in practice and for the limitations and nuances of constitutive modeling. This book is intended as a text for both an introductory continuum mechanics course and a second course in constitutive modeling of materials. The objective of this text is to demonstrate the application of continuum mechanics to the modeling of material behavior. Specifically, the text focuses on developing, parameterizing, and numerically solving constitutive equations for various types of materials. The text is designed to aid students who lack exposure to tensor algebra, tensor calculus, and/or numerical methods. This text provides step-by-step derivations as well as solutions to example problems, allowing a student to follow the logic without being lost in the mathematics. The first half of the textbook covers notation, mathematics, the general principles of continuum mechanics, and constitutive modeling. The second half applies these theoretical concepts to different material classes. Specifically, each application covers experimental characterization, constitutive model development, derivation of governing equations, and numerical solution of the governing equations. For each material application, the text begins with the experimental observations, which outline the behavior of the material and must be captured by the material model. Next, we formulate the continuum model for the material and present general constitutive equations. These equations often contain parameters that must be determined experimentally. Therefore, the textbook has a chapter covering the theory and application of experimental error analysis and simple curve fitting. For each material class, the continuum model is then applied to a specific application and the resulting differential equations are solved numerically. Complete descriptions of the finite difference and finite element methods are included. Numerical solutions are implemented in Matlab and provided in the text along with flow charts illustrating the logic in the Matlab scripts. xiii

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