PHYSICAL FOUNDATIONS OF CONTINUUM MECHANICS s Physical Foundations of Continuum Mechanics will interest engineers, mathematicians, and physicists who study the macroscopic behaviour of solids and fluids or engage in molecular dynamical simulations. In contrast to standard works on the subject, Murdoch s book examines physical assumptions implicit in continuum modelling from a molecular perspective. In so doing, physical interpretations of concepts and fields are clarified by emphasising both their microscopic origin and sensitivity to scales of length and time. Murdoch expertly applies this approach to theories of mixtures, generalised continua, fluid flow through porous media, and systems whose molecular content changes with time. Elements of statistical mechanics are included, for comparison, and two extensive appendices address relevant mathematical concepts and results. This unique and thorough work is an authoritative reference for both students and experts in the field. is Professor Emeritus of Mathematics at the University of Strathclyde, Glasgow. His work on continuum mechanics has been widely published in such journals as the Archive for Rational Mechanics and Analysis, Proceedings of the Royal Society, Journal of Elasticity, International Journal of Engineering Science, Continuum Mechanics and Thermodynamics, and the Quarterly Journal of Mechanics & Applied Mathematics. He is the co-editor of two books: Modelling Macroscopic Phenomena at Liquid Boundaries and Modelling Coupled Phenomena in Saturated Porous Materials, and author of published lecture notes, Foundations of Continuum Modelling. Dr Murdoch has taught and lectured at many distinguished mathematics and engineering schools around the world.
Physical Foundations of Continuum Mechanics University of Strathclyde
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 10013-2473, USA Information on this title: /9780521765589 2012 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 Murdoch, A. I. Physical foundations of continuum mechanics /. pages cm Includes bibliographical references and index. ISBN 978-0-521-76558-9 1. Continuum mechanics. 2. Fluid mechanics. I. Title. QC155.7.M87 2013 531 dc23 2012015692 ISBN 978-0-521-76558-9 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.
For Frances, Duncan, and Margaret
Contents Preface page xiii 1 Introduction 1 1.1 Motivation 1 1.2 Contents 2 2 Some Elements of Continuum Mechanics 6 2.1 Preamble 6 2.2 Matter and Its Distribution 6 2.3 Motion of Matter: Kinematics and Material Points 7 2.4 The Formal (Axiomatic) Approach to Matter and Material Points 9 2.5 Mass Conservation 11 2.6 Dynamics I: Global Relations 14 2.6.1 Introduction 14 2.6.2 Linear Momentum Balance 14 2.6.3 Rotational Momentum Balance 16 2.6.4 Rigid Body Dynamics 17 2.7 Dynamics II: Local Relations 24 2.8 Thermomechanics 29 2.8.1 Global Balance of Energy 29 2.8.2 Aside on the Spin Vector Field w and Power Expended by Couples 29 2.8.3 Local Balance of Energy 30 3 Motivation for Seeking a Molecular Scale-Dependent Perspective on Continuum Modelling 33 3.1 Preamble 33 3.2 The Natural Continuum Prejudice 33 3.3 The Continuum Viewpoint on Mass Density ρ 34 3.4 Boundaries and the Scale Dependence of ρ 34 3.5 Continuity of ρ and the Discrete Nature of Matter 36 3.6 Velocity 37 3.7 The Pressure in a Gas 41 vii
viii Contents 3.8 Reproducibility 42 3.9 Summary of Conceptual Problems 42 3.10 Motivation for Space-Time Averaging of Molecular Quantities 43 4 Spatial Localisation, Mass Conservation, and Boundaries 44 4.1 Preamble 44 4.2 Weighted Averages and the Continuity Equation 44 4.3 The Simplest Choice w ɛ of Weighting Function 48 4.3.1 Definition of w ɛ 48 4.3.2 The Boundary Corresponding to w ɛ 48 4.3.3 Integration of ρ ɛ and pɛ over a Region 52 4.3.4 A Wrinkle to Be Resolved: Use of a Mollifier 53 4.3.5 Further Mollification Considerations 56 4.3.6 Regularity of Mollified Fields: Polynomial Mollifiers 58 4.3.7 Mollification as a Natural Consequence of Spatial Imprecision 60 4.4 Other Choices of Weighting Function 61 4.4.1 Cellular Averaging 61 4.4.2 Choices Associated with Repeated Averaging 62 4.4.3 Other Choices 67 4.5 Temporal Fluctuations 69 4.6 Summary 70 5 Motions, Material Points, and Linear Momentum Balance 71 5.1 Preamble 71 5.2 Motions and Material Points 72 5.3 Motions and Material Points for Non-Reacting Binary Mixtures 74 5.4 Linear Momentum Balance Preliminaries: Intermolecular Forces 76 5.5 Linear Momentum Balance 80 5.5.1 Derivation of the Balance Relation 80 5.5.2 The Thermal Nature of D w 82 5.5.3 Comparison of Contributions Tw and D w to T w 83 5.6 Determination of Candidate Interaction Stress Tensors 84 5.6.1 Preamble 84 5.6.2 Simple Form 84 5.6.3 Form for Pairwise-Balanced Interactions 85 5.6.4 Simple Choice of b ij for Pairwise-Balanced Interactions 85 5.6.5 Hardy-Type Choice of b ij for Pairwise-Balanced Interactions 86 5.6.6 Noll-Type Choice of b ij for Pairwise-Balanced Interactions 87 5.6.7 Conclusions 87 5.7 Calculation of Interaction Stresses for the Simplest Form of Weighting Function w ɛ 88 5.7.1 Determination of a i and Calculation of s Tw ɛ and sb Tw ɛ 88 5.7.2 Determination of ˆb H ij and Calculation of H T w ɛ 90 5.7.3 The Geometrical Complexity of b N ij 91 5.8 Comparison of Interaction Stress Tensors for the Simplest Form of Weighting Function w ɛ 91
Contents ix 5.8.1 Values for Two Simple Geometries 91 5.8.2 Integration over Planar Surfaces 92 5.9 Integrals of General Interaction Stress Tensors over the Boundaries of Regular Regions 95 5.9.1 Results for a General Choice of Weighting Function 95 5.9.2 Results for Choice w = wɛ 98 5.9.3 Further Remarks for Choice w = wɛ 100 6 Balance of Energy 102 6.1 Preamble 102 6.2 Derivation of Energy Balances 102 6.3 A Subatomic Perspective 110 7 Fine-Scale Considerations: Moments, Couple Stress, Inhomogeneity, and Energetics 115 7.1 Preamble 115 7.2 Generalised Moment of Momentum Balance 115 7.3 Inhomogeneity and Moment of Mass Conservation 121 7.4 Fine-Scale Energetics 123 7.5 Summary and Discussion 128 8 Time Averaging and Systems with Changing Material Content 130 8.1 Preamble 130 8.2 Motivation 130 8.3 Time Averaging 132 8.4 The Time-Averaged Continuity Equation 134 8.5 Time-Averaged Forms of Linear Momentum Balance 135 8.6 Time-Averaged Forms of Energy Balance 137 8.7 Systems with Changing Material Content I: General Global Considerations 139 8.8 Systems with Changing Material Content II: Specific Global Examples 146 8.8.1 Rocketry 146 8.8.2 Jet Propulsion 148 8.8.3 Falling Raindrop 150 8.9 Systems with Changing Material Content III: Local Evolution Equations at Specific Scales of Length and Time 150 8.9.1 Mass Balance 150 8.9.2 Linear Momentum Balance 153 8.9.3 Energy Balance 157 8.9.4 Concluding Remarks 164 8.10 Summary 165 9 Elements of Mixture Theory 167 9.1 Preamble 167 9.2 Mass Conservation and Material Points for a Non-Reacting Mixture Constituent 167
x Contents 9.3 Linear Momentum Balance for a Non-Reacting Mixture Constituent 169 9.4 On Relating Total Mixture Fields to Those of Constituents 174 9.5 A Paradox in Early Continuum Theories of Mixtures 177 9.6 Energy Balances 179 9.7 On Reacting Mixtures 184 9.7.1 General Considerations 184 9.7.2 A Simple Model of a Reacting Ternary Mixture 184 9.8 Concluding Remarks 187 10 Fluid Flow through Porous Media 188 10.1 Preamble 188 10.2 The General Forms of Mass Conservation and Linear Momentum Balance 189 10.3 Linear Momentum Balance at Scale ɛ = ɛ 1 with w = w ɛ1 192 10.4 Linear Momentum Balance at Scale ɛ = ɛ 2 with w = w ɛ2 193 10.5 Flow of an Incompressible Linearly Viscous Fluid through a Porous Body It Saturates 195 11 Linkage of Microscopic and Macroscopic Descriptions of Material Behaviour via Cellular Averaging 209 11.1 Preamble 209 11.2 Cellular Averaging 209 11.3 Concluding Remarks 223 12 Modelling the Behaviour of Specific Materials: Constitutive Relations and Objectivity 225 12.1 Preamble 225 12.2 Microscopic Considerations and the Key Role Played by Inertial Observers 226 12.3 Objectivity 237 12.3.1 Objectivity in General 237 12.3.2 Objectivity in Deterministic Continuum Mechanics 238 12.3.3 Elastic Behaviour 239 12.3.4 Simple Materials 242 12.3.5 Viscous Fluids 243 12.3.6 Other Materials and Considerations 247 12.4 Remarks on the mfi/isrbm Controversy 248 12.4.1 Introduction 248 12.4.2 Material Frame-Indifference 249 12.4.3 Invariance under Superposed Rigid Body Motions 249 12.4.4 Comparison of mfi, isr, isrbm, and Objectivity 249 12.4.5 A Personal History 252 12.4.6 A Final Remark 254 13 Comments on Non-Local Balance Relations 255 13.1 Preamble 255
Contents xi 13.2 Edelen s Non-Local Field Theories 255 13.3 Peridynamics 258 14 Elements of Classical Statistical Mechanics 263 14.1 Preamble 263 14.2 Basic Concepts in Classical Statistical Mechanics 263 14.2.1 Time Evolution in Phase Space of a System of Interacting Point Masses 263 14.2.2 Ensembles, Probability Density Functions, and Ensemble Averaging 264 14.3 Mass Conservation and Linear Momentum Balance 269 14.4 Generalisation of Irving and Kirkwood/Noll Results 276 14.5 Selection of a Probability Density Function: Projection Operator Methodology 281 15 Summary and Suggestions for Further Study 290 15.1 Preamble 290 15.2 Summary 290 15.3 Suggestions for Further Study 292 15.3.1 Interfacial Phenomena and Boundary Conditions 292 15.3.2 Generalised and Structured Continua 295 15.3.3 Configurational Forces 296 15.3.4 Reacting Mixtures 296 15.3.5 Electromagnetic Effects 297 15.3.6 Irreversibility 297 15.4 A Final Remark 302 Appendix A: Vectors, Vector Spaces, and Linear Algebra 303 Preamble 303 A.1 The Algebra of Displacements 304 A.2 Dimensionality 305 A.3 Angles, Magnitudes, and Euclidean Structure 306 A.4 Vectorial Entities and the Fundamental Space V 307 A.5 Products in V (Products of Physical Descriptors) 309 A.6 Unit Vectors, Orthonormal Bases, and Related Components 312 A.7 Linear Transformations on V and the General Definition of a Vector Space over R 314 A.8 The Transpose of a Linear Transformation on V and Tensor Products of Vectors 316 A.9 Orthonormal Bases and Matrix Representation of Vectors and Linear Transformations 318 A.10 Invertibility 321 A.11 Alternating Trilinear Forms on V 324 A.12 Principal Invariants of L LinV 326 A.12.1 The First Principal Invariant: I 1 (L) = tr L 326 A.12.2 The Second Principal Invariant: I 2 (L) 328 A.12.3 The Third Principal Invariant: I 3 (L) = detl 329
xii Contents A.13 Eigenvectors, Eigenvalues, and the Characteristic Equation for a Linear Transformation 331 A.14 A Natural Inner Product for LinV 332 A.15 Skew Linear Transformations and Axial Vectors 336 A.16 Orthogonal Transformations and Their Characterisation 338 A.17 Symmetric and Positive-Definite Linear Transformations 343 A.18 The Polar Decomposition Theorem 346 A.19 Third-Order Tensors and Elements of Tensor Algebra 347 A.20 Direct, Component, and Cartesian Tensor Notation 352 Appendix B: Calculus in Euclidean Point Space E 356 Preamble 356 B.1 Euclidean Point Space E 357 B.2 Cartesian Co-ordinate Systems for E 359 B.3 Deformations in E 359 B.3.1 Introduction 359 B.3.2 Isometries and Their Characterisation 360 B.3.3 Homogeneous Deformations 363 B.4 Generalisation of the Concept of a Derivative 366 B.4.1 Preamble 366 B.4.2 Differentiation of a Scalar Field 367 B.4.3 Differentiation of Point-Valued Fields 369 B.4.4 Differentiation of Vector Fields 371 B.4.5 Differentiation of Linear Transformation Fields 372 B.4.6 Remarks 373 B.4.7 Differentiation of Products and Compositions 373 B.4.8 Differentiation of the Determinant Function 376 B.5 Jacobians, Physically Admissible Deformations, and Kinematics 379 B.6 (Riemann) Integration over Spatial Regions 383 B.7 Divergences and Divergence Theorems 389 B.8 Calculations in Section 7.4 395 B.9 Proof of Results 10.5.1 396 B.10 Derivatives of Objective Fields 398 B.11 Calculus in Phase Space P When Identified with R 6N 400 B.11.1 Basic Concepts 400 B.11.2 Deformations and Differential Calculus in R 6N 402 B.11.3 Integration in R 6N 405 References 407 Index 413
Preface This work is intended to supplement and complement standard texts on continuum mechanics by drawing attention to physical assumptions implicit in continuum modelling. Particular attention is paid to linking continuum concepts, fields, and relations with underlying molecular behaviour via local averaging in both space and time. The aim is to clarify physical interpretations of concepts and fields and in so doing provide a sound basis for future studies. The contents should be of interest to engineers, mathematicians, and physicists who study macroscopic material behaviour. The contents are the result of a long-standing study of formal and axiomatic presentations of continuum mechanics. Some of the issues were first addressed in courses delivered under the auspices of CISM 1 (Udine, 1986, 1987), University of Cairo (1994, 1996), and AMAS 2 (Warsaw, 2002; Bydgoszcz, 2003), and other topics treated in published papers. Here the opportunity has been taken to elaborate upon and extend earlier works and to present a unified, more readily accessible treatment of the subject matter. Given the differing backgrounds of the intended readership, two extensive appendices have been included which develop relevant mathematical concepts and results. In particular, the use of direct (i.e., co-ordinate-free) notation is explained and related to that of Cartesian tensors. No work exists in isolation: the author is above all indebted to his teachers Mort Gurtin and Walter Noll who introduced him to the mathematical precision and clarity of exposition to be found in modern continuum mechanics. The use of weighting function methodology, central to much of the discussion, and the role of projection operators in statistical mechanics were explained at length to the author by Dick Bedeaux. Appreciation of porous media modelling was gained by interactions with Jozef Kubik and Majid Hassanizadeh. It is also a pleasure to acknowledge the support and encouragement over the years of Mort Gurtin, Peter Chadwick, Harley Cohen, Paolo Podio-Guidugli, Gianpietro del Piero, Angelo Morro, Gérard Maugin, Witold Kosiński, Antonio Romano, Ahmed Ghaleb, David Steigman, and 1 International Centre for Mechanical Sciences. 2 Centre of Excellence for Advanced Materials and Structures, Institute of Fundamental Technological Research, Polish Academy of Sciences. xiii
xiv Preface Eliot Fried. Extensive and comprehensive secretarial support for a TeX illiterate was provided in outstanding fashion by Mary McAuley. Finally, I am greatly indebted to my wife Margaret for her patience, support, and encouragement throughout the preparation of this work.