PHYSICAL FOUNDATIONS OF CONTINUUM MECHANICS

Transcription

1 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.

2

3 Physical Foundations of Continuum Mechanics University of Strathclyde

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 Murdoch, A. I. Physical foundations of continuum mechanics /. pages cm Includes bibliographical references and index. ISBN Continuum mechanics. 2. Fluid mechanics. I. Title. QC155.7.M 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 For Frances, Duncan, and Margaret

6

7 Contents Preface page xiii 1 Introduction Motivation Contents 2 2 Some Elements of Continuum Mechanics Preamble Matter and Its Distribution Motion of Matter: Kinematics and Material Points The Formal (Axiomatic) Approach to Matter and Material Points Mass Conservation Dynamics I: Global Relations Introduction Linear Momentum Balance Rotational Momentum Balance Rigid Body Dynamics Dynamics II: Local Relations Thermomechanics Global Balance of Energy Aside on the Spin Vector Field w and Power Expended by Couples Local Balance of Energy 30 3 Motivation for Seeking a Molecular Scale-Dependent Perspective on Continuum Modelling Preamble The Natural Continuum Prejudice The Continuum Viewpoint on Mass Density ρ Boundaries and the Scale Dependence of ρ Continuity of ρ and the Discrete Nature of Matter Velocity The Pressure in a Gas 41 vii

8 viii Contents 3.8 Reproducibility Summary of Conceptual Problems Motivation for Space-Time Averaging of Molecular Quantities 43 4 Spatial Localisation, Mass Conservation, and Boundaries Preamble Weighted Averages and the Continuity Equation The Simplest Choice w ɛ of Weighting Function Definition of w ɛ The Boundary Corresponding to w ɛ Integration of ρ ɛ and pɛ over a Region A Wrinkle to Be Resolved: Use of a Mollifier Further Mollification Considerations Regularity of Mollified Fields: Polynomial Mollifiers Mollification as a Natural Consequence of Spatial Imprecision Other Choices of Weighting Function Cellular Averaging Choices Associated with Repeated Averaging Other Choices Temporal Fluctuations Summary 70 5 Motions, Material Points, and Linear Momentum Balance Preamble Motions and Material Points Motions and Material Points for Non-Reacting Binary Mixtures Linear Momentum Balance Preliminaries: Intermolecular Forces Linear Momentum Balance Derivation of the Balance Relation The Thermal Nature of D w Comparison of Contributions Tw and D w to T w Determination of Candidate Interaction Stress Tensors Preamble Simple Form Form for Pairwise-Balanced Interactions Simple Choice of b ij for Pairwise-Balanced Interactions Hardy-Type Choice of b ij for Pairwise-Balanced Interactions Noll-Type Choice of b ij for Pairwise-Balanced Interactions Conclusions Calculation of Interaction Stresses for the Simplest Form of Weighting Function w ɛ Determination of a i and Calculation of s Tw ɛ and sb Tw ɛ Determination of ˆb H ij and Calculation of H T w ɛ The Geometrical Complexity of b N ij Comparison of Interaction Stress Tensors for the Simplest Form of Weighting Function w ɛ 91

9 Contents ix Values for Two Simple Geometries Integration over Planar Surfaces Integrals of General Interaction Stress Tensors over the Boundaries of Regular Regions Results for a General Choice of Weighting Function Results for Choice w = wɛ Further Remarks for Choice w = wɛ Balance of Energy Preamble Derivation of Energy Balances A Subatomic Perspective Fine-Scale Considerations: Moments, Couple Stress, Inhomogeneity, and Energetics Preamble Generalised Moment of Momentum Balance Inhomogeneity and Moment of Mass Conservation Fine-Scale Energetics Summary and Discussion Time Averaging and Systems with Changing Material Content Preamble Motivation Time Averaging The Time-Averaged Continuity Equation Time-Averaged Forms of Linear Momentum Balance Time-Averaged Forms of Energy Balance Systems with Changing Material Content I: General Global Considerations Systems with Changing Material Content II: Specific Global Examples Rocketry Jet Propulsion Falling Raindrop Systems with Changing Material Content III: Local Evolution Equations at Specific Scales of Length and Time Mass Balance Linear Momentum Balance Energy Balance Concluding Remarks Summary Elements of Mixture Theory Preamble Mass Conservation and Material Points for a Non-Reacting Mixture Constituent 167

10 x Contents 9.3 Linear Momentum Balance for a Non-Reacting Mixture Constituent On Relating Total Mixture Fields to Those of Constituents A Paradox in Early Continuum Theories of Mixtures Energy Balances On Reacting Mixtures General Considerations A Simple Model of a Reacting Ternary Mixture Concluding Remarks Fluid Flow through Porous Media Preamble The General Forms of Mass Conservation and Linear Momentum Balance Linear Momentum Balance at Scale ɛ = ɛ 1 with w = w ɛ Linear Momentum Balance at Scale ɛ = ɛ 2 with w = w ɛ Flow of an Incompressible Linearly Viscous Fluid through a Porous Body It Saturates Linkage of Microscopic and Macroscopic Descriptions of Material Behaviour via Cellular Averaging Preamble Cellular Averaging Concluding Remarks Modelling the Behaviour of Specific Materials: Constitutive Relations and Objectivity Preamble Microscopic Considerations and the Key Role Played by Inertial Observers Objectivity Objectivity in General Objectivity in Deterministic Continuum Mechanics Elastic Behaviour Simple Materials Viscous Fluids Other Materials and Considerations Remarks on the mfi/isrbm Controversy Introduction Material Frame-Indifference Invariance under Superposed Rigid Body Motions Comparison of mfi, isr, isrbm, and Objectivity A Personal History A Final Remark Comments on Non-Local Balance Relations Preamble 255

11 Contents xi 13.2 Edelen s Non-Local Field Theories Peridynamics Elements of Classical Statistical Mechanics Preamble Basic Concepts in Classical Statistical Mechanics Time Evolution in Phase Space of a System of Interacting Point Masses Ensembles, Probability Density Functions, and Ensemble Averaging Mass Conservation and Linear Momentum Balance Generalisation of Irving and Kirkwood/Noll Results Selection of a Probability Density Function: Projection Operator Methodology Summary and Suggestions for Further Study Preamble Summary Suggestions for Further Study Interfacial Phenomena and Boundary Conditions Generalised and Structured Continua Configurational Forces Reacting Mixtures Electromagnetic Effects Irreversibility 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

12 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 B.9 Proof of Results 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

13 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

14 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.