PRINCIPLES OF CONTINUUM MECHANICSA Study of Conservation Principles with Applications

by

J. N. ReddyDepartment of Mechanical Engineering

Texas A&M UniversityCollege Station, TX 77843-3123

Under contract to be published by

Cambridge University Press

July 2009

When even the brightest mind in our world has been trained up from childhood in a

superstition of any kind, it will never be possible for that mind, in its maturity,

to examine sincerely, dispassionately, and conscientiously any evidence or any

circumstance which shall seem to cast a doubt upon the validity of that superstition.

Mark Twain

The fact that an opinion has been widely held is no evidence whatever that it is

not utterly absurd; indeed in view of the silliness of the majority of mankind, a

widespread belief is more likely to be foolish than sensible. Bertrand Russell

Desire for approval and recognition is a healthy motive, but the desire to be

acknowledged as better, stronger, or more intelligent than a fellow being or fellow

scholar easily leads to an excessively egoistic psychological adjustment, which may

become injurious for the individual and for the community.

Albert Einstein

vi PREFACE

PrefaceYou cannot teach a man anything, you can only help him find it within himself.

Galileo Galilei

This book is a simplified version of the author’s book, An Introduction toContinuum Mechanics with Applications, published by Cambridge UniversityPress (New York, 2008), intended for use as an undergraduate text book.As most modern technologies are no longer discipline-specific but involvemultidisciplinary approaches, undergraduate engineering students should beeducated to think and work in such environments. Therefore, it is necessaryto introduce the subject of principles of mechanics (i.e., laws of physicsapplied to science and engineering systems) to undergraduate students sothat they have a strong background in the basic principles common to alldisciplines and be able to work at the interface of science and engineeringdisciplines. A first course on principles of mechanics provides an introductionto the basic concepts of stress and strain and conservation principles, andprepares engineer-scientists for advanced courses in traditional as well asemerging fields such as biotechnology, nanotechnology, energy systems, andcomputational mechanics. Undergraduate students with such backgroundmay seek advanced degrees in traditional (e.g., aerospace, civil, electrical,mechanical, physics, applied mathematics) as well as interdisciplinarydegrees programs (e.g., bioengineering, engineering physics, nanoscience andengineering, biomolecular engineering, and so on).There are not many books on principles of mechanics that are written

keeping the undergraduate engineering or science students in mind. Avast majority of books on the subject are written for graduate students ofengineering and tend to be more mathematical and too advanced to be ofuse for third year or senior undergraduate students. This book presentsthe subjects of mechanics of materials, fluid mechanics, and heat transfer inunified form using the conservation principles of mechanics. It is hoped thatthe book, which is simple and facilitates in presenting the main concepts ofthe previous three courses under a unified framework.With a brief discussion of the concept of a continuum in Chapter 1, a

review of vectors and tensors is presented in Chapter 2. Since the analyticallanguage of applied sciences and engineering is mathematics, it is necessaryfor all students of this course to familiarize themselves with the notation andoperations of vectors, matrices, and tensors that are used in the mathematical

PREFACE vii

description of physical phenomena. Readers who are familiar with the topicsof this chapter may refresh or skip and go to the next chapter. The subjectof kinematics, which deals with geometric changes without regard to theforces causing the deformation, is discussed in Chapter 3. Measures ofengineering normal and shear strains and definitions of mathematical strainsare introduced here. Both simple one-dimensional systems to as well as two-dimensional continua are used to illustrate the strain and strain-rate measuresintroduced. In Chapter 4, the concept of stress vector and stress tensor areintroduced. It is here, the readers are presented with entities that require twodirections - namely, the plane on which they are measured and the directionin which they act - to specify them. Transformation equations amongcomponents of stress tensor referred to two different orthogonal coordinatesystems are derived, and principal values and principal planes (i.e., eigenvalueproblems associated with the stress tensor) are also discussed.Chapter 5 is dedicated to the derivation of the governing equations of

mechanics using the conservation principles of continuum mechanics (orlaws of physics). The principles of conservation of mass, linear momentum,angular momentum, and energy are presented using one-dimensional systemsas well as general three-dimensional systems. The derivations are presentedin invariant (i.e., independent of a coordinate system) as well as in componentform. The equations resulting from these principles are those governingstress and deformation of solid bodies, stress and rate of deformation of fluidelements, and transfer of heat through solid media. Thus, this chapter formsthe heart of the course. Constitutive relations that connect the kinematicvariables (e.g., density, temperature, deformation) to the kinetic variables(e.g., internal energy, heat flux, and stresses) are discussed in Chapter 6 forelastic materials, viscous fluids, and heat transfer in solids.Chapter 7 is devoted to the application of the field equations derived in

Chapter 5 and constitutive models presented in Chapter 6 to problems ofheat conduction in solids, fluid mechanics (inviscid flows as well as viscousincompressible flows), diffusion, and solid mechanics (e.g., bars, beams, andplane elasticity). Simple boundary-value problems are formulated and theirsolutions are discussed. The material presented in this chapter illustrates howphysical problems are analytically formulated with the aid of the equationsresulting from the conservation principles.As stated previously, the present book is an undergraduate version of

the author’s book An Introduction to Continuum Mechanics (CambridgeUniversity Press, New York, 2008). The presentation herein is limited inscope when compared to the author’s graduate level textbook. The majorbenefit of a course based on this book is to present the governing equationsof diverse physical phenomena from a unified point of view, namely, fromthe conservation principles (or laws of physics) so that students of appliedscience and engineering see the physical principles as well as the mathematicalstructure common to diverse fields. Readers interested in advanced topicsmay consult the author’s continuum mechanics book cited above or othertitles listed in references therein.

viii PREFACE

The author is pleased to acknowledge the fact that the manuscript wastested with the undergraduate students in the College of Engineering atTexas A&M University as well as in the Engineering Science Programmeat the National University of Singapore. The students, in general, haveliked the contents and the simplicity with which the concepts are introducedand explained. They also expressed the feeling that the subject is morechallenging than most at the undergraduate level but a useful prerequisite tograduate courses in engineering.The book contains so many mathematical expressions that it is hardly

possible not to have typographical and other kinds of errors. The authorwishes to thank in advance those who are willing to draw the author’sattention to typos and errors, using the e-mail address jn_reddy@yahoo.com.

J. N. ReddyCollege Station

ix

Contents

Preface vi

1 INTRODUCTION 1

1.1 Continuum Mechanics 1

1.2 Objective of the Study 7

1.3 Summary 7

2 VECTORS AND TENSORS 11

2.1 Motivation 11

2.2 Definition of a Vector 11

2.3 Vector Algebra 122.3.1 Unit Vector 122.3.2 Zero Vector 122.3.3 Vector Addition 132.3.4 Multiplication of a Vector by a Scalar 142.3.5 Scalar Product of Vectors 152.3.6 Vector Product 162.3.7 Triple Products of Vectors 192.3.8 Plane Area as a Vector 212.3.9 Components of a Vector 22

2.4 Index Notation and Summation Convention 252.4.1 Summation Convention 252.4.2 Dummy Index 252.4.3 Free Index 262.4.4 Kronecker Delta and Permutation Symbols 262.4.5 Transformation Law for Different Bases 28

2.5 Theory of Matrices 312.5.1 Definition 312.5.2 Matrix Addition and Multiplication of a Matrix

by a Scalar 322.5.3 Matrix Transpose and Symmetric and Skew Symmetric

Matrices 332.5.4 Matrix Multiplication 332.5.5 Inverse and Determinant of a Matrix 35

x

2.6 Vector Calculus 382.6.1 The Del Operator 382.6.2 Divergence and Curl of a Vector 402.6.3 Cylindrical and Spherical Coordinate Systems 432.6.4 Gradient, Divergence and Curl Theorems 44

2.7 Tensors 452.7.1 Dyads 452.7.2 Nonion Form of a Dyad 462.7.3 Transformation of Components of a Dyad 472.7.4 Tensor Calculus 48

2.8 Summary 49Problems 49

3 KINEMATICS OF A CONTINUUM 55

3.1 Deformation and Configuration 55

3.2 Engineering Strains 563.2.1 Normal Strain 563.2.2 Shear Strain 57

3.3 General Kinematics of a Solid Continuum 603.3.1 Configurations of a Continuous Medium 603.3.2 Material and Spatial Descriptions 613.3.3 Displacement Field 64

3.4 Analysis of Deformation 653.4.1 Deformation Gradient Tensor 653.4.2 Various Types of Deformations 68

3.4.2.1 Pure dilatation 683.4.2.2 Simple extension 693.4.2.3 Simple shear 703.4.2.4 A nonhomogeneous deformation 70

3.4.3 Green Strain Tensor 713.4.4 Infinitesimal Strain Tensor 753.4.5 Principal Values and Principal Planes of Strains 77

3.5 Rate of Deformation and Vorticity Tensors 793.5.1 Velocity Gradient Tensor 793.5.2 Rate of Deformation Tensor 793.5.3 Vorticity Tensor and Vorticity Vector 80

3.6 Compatibility Equations 82

3.7 Summary 84

Problems 84

xi

4 STRESS VECTOR AND STRESS TENSOR 87

4.1 Introduction 91

4.2 Stress Vector, Stress Tensor, and Cauchy’s Formula 91

4.3 Transformation of Stress Components and Principal Stresses 1004.3.1 Transformation of Stress Components 1004.3.2 Principal Stresses and Principal Planes 102

4.4 Summary 104

Problems 105

5 CONSERVATION OF MASS, MOMENTA ANDENERGY 109

5.1 Introduction 109

5.2 Conservation of Mass 1105.2.1 Preliminary Discussion 1105.2.2 Conservation of Mass in Spatial Description 1105.2.3 Conservation of Mass in Material Description 1155.2.4 Reynolds Transport Theorem 116

5.3 Conservation of Momenta 1175.3.1 Principle of Conservation of Linear Momentum 1175.3.2 Principle of Conservation of Angular Momentum 132

5.4 Thermodynamic Principles 1335.4.1 Introduction 1335.4.2 Energy Equation for One-Dimensional Flows 1345.4.3 Energy Equation for a Three-Dimensional Continuum 137

5.5 Summary 139

Problems 140

6 CONSTITUTIVE EQUATIONS 147

6.1 Introduction 147

6.2 Elastic Solids 1486.2.1 Introduction 1486.2.2 Generalized Hooke’s Law for Orthotropic Materials 1486.2.3 Generalized Hooke’s Law for Isotropic Materials 151

6.3 Constitutive Equations for Fluids 1546.3.1 Introduction 1546.3.2 Ideal Fluids 1546.3.3 Viscous Incompressible Fluids 154

6.4 Heat Transfer 1556.4.1 General Introduction 1556.4.2 Fourier’s Heat Conduction Law 1556.4.3 Newton’s Law of Cooling 1566.4.4 Stefan—Boltzmann Law 156

6.5 Summary 157

Problems 157

xii

7 APPLICATIONS IN HEAT TRANSFER, FLUIDMECHANICS, AND SOLID MECHANICS 159

7.1 Introduction 159

7.2 Heat Transfer 1597.2.1 Governing Equation 1597.2.2 Analytical Solutions of One-Dimensional Heat Transfer 162

7.2.2.1 Steady-State Heat Transfer in a Cooling Fin 1627.2.2.2 Steady-State Heat Transfer in

a Surface Insulated Rod 1647.2.3 Axisymmetric Heat Conduction in a Circular Cylinder 1657.2.4 Two-Dimensional Heat Transfer 166

7.3 Fluid Mechanics 1697.3.1 Preliminary Comments 1697.3.2 Summary of Equations 1697.3.3 Inviscid Fluid Statics 1707.3.4 Parallel Flow (Navier-Stokes Equations) 172

7.3.4.1 Steady flow of viscous incompressible fluidbetween parallel plates 172

7.3.4.2 Steady flow of viscous incompressible fluidthrough a pipe 173

7.3.5 Diffusion Processes 175

7.4 Solid Mechanics 1787.4.1 Governing Equations 1787.4.2 Analysis of Bars 1807.4.3 Analysis of Beams 184

7.4.3.1 Principle of Superposition 1897.4.4 Analysis of Plane Elasticity Problems 192

7.4.5.1 Plane strain and plane stress problems 1927.4.5.2 Plane strain problems 1927.4.5.3 Plane stress problems 1937.4.5.4 Solution Methods 1947.4.5.5 Airy stress function 197

7.5 Summary 200

Problems 200

REFERENCES 211

ANSWERS TO SELECTED PROBLEMS 213

SUBJECT INDEX 225

xiii

About the Author

J. N. Reddy is a Distinguished Professor and the Holder of Oscar S. WyattEndowed Chair in the Department of Mechanical Engineering at Texas A&MUniversity (http://www.tamu.edu/acml).Professor Reddy is a renowned researcher and educator in the broad fields

of mechanics, applied mathematics, and computational engineering science.Professor Reddy’s research areas include theory and finite element analysisof problems in structural mechanics (composite plates and shells), fluiddynamics, and heat transfer; theoretical modelling of stress and deformationof biological cells and soft tissues; nanocomposites; and development ofrobust computational technology (including the K-version finite elementmodels based on the least-squares method in collaboration with ProfessorKaran Surana of the University of Kansas). He is the author of over 375journal papers and 16 books on these subjects. The books published by Dr.Reddy include Introduction to the Finite Element Method, 3rd ed., McGraw-Hill, 2006; Mechanics of Laminated Plates and Shells: Theory and Analysis,2nd ed., CRC Press, 2004; An Introduction to Nonlinear Finite ElementAnalysis, Oxford University Press, 2004; and An Introduction to ContinuumMechanics, Cambridge University Press, 2008.Dr. Reddy’s outstanding research credentials have earned him wide

international acclaim in the form of numerous professional awards, citations,fellowship in all major professional societies including AAM, AIAA,ASC, ASCE, ASME, IACM and USACM, membership on two dozenarchival journals, and numerous keynote and plenary lecture invitationsat international conferences. Dr. Reddy is the Editor-in-Chief of AppliedMechanics Reviews, Mechanics of Advanced Materials and Structures,International Journal of Computational Methods in Engineering Science andMechanics, and International Journal of Structural Stability and Dynamics.The extent of Dr. Reddy’s original and sustained contributions to

education, research, and professional service is substantial. As a result ofhis extensive publications of archival journal papers and books on a widerange of topics in applied sciences and engineering, Dr. Reddy is one of theselective few researchers in engineering around world who are recognized byISI Highly Cited Researchers with over 10,000 citations with H-index of over40. In February 2009 he was awarded a Honoris Causa (Honorary Doctorate)by the Technical University of Lisbon.