Fundamentals of continuum mechanics

CONTINUUM

MECHANICS

WITH APPLICATIONS TO MECHANICAL,

THERMOMECHANICAL, AND SMART MATERIALS

T h e r m o - M e c h a n ic a l E f f e c t s

MECHANICAL

STEPHEN E. BECHTEL

with

ROBERT L, LOWE

Fundamentals of

Continuum Mechanics

Fundamentals of

Continuum Mechanics

With Applications

to Mechanical,

"hermomechanical,

and Smart Materials

Stephen E. Bechtel

Robert L Lowe

ELSEVIER

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Contents

Prelace ......................................................................................................................................xiii

P A R T I THE B EG INN ING______________________________________

CHAPTER 1 What Is a Continuum? 3

CHAPTER 2 Our Mathematical Playground............................................... 5

2.1 Real numbers and Euclidean spacc............................................................5

2.1.1 Properties of real num bers...........................................................5

2 .1.2 Properties ol’ Euclidean space......................................................7

2.2 Tensor alg ebra...............................................................................................12

2.2.1 Second-order tensors, zero tensor, identity te n so r.............. 12

2.2.2 Product, transpose, .symmetry.................................................... 16

2.2..'^ Dyadic product .............................................................................22

2.2.4 Cartesian components, indicial notation, summation

convention .................................................................................... 24

2.2.5 Trace, scalar product, delerm inani...........................................35

2.2.6 Inverse, orthogonality, positive definiteness...........................39

2.2.7 Vector product, scalar triple product....................................... 42

2.3 Eigenvalues, eigenvectors, polar decom position, invariants ........ 44

2.4 Ten.sors of order three and fo u r................................................................47

2.5 Tensor c alcu lu s............................................................................................ 48

2.5.1 Partial derivatives.........................................................................48

2.5.2 Chain rule, gradient, divergence, curl.

divergence theorem .....................................................................52

2.5.3 Tensor calculus in Cartesian component fo rm .................... 56

2.6 Curvilinear coordinates.............................................................................59

2.6.1 Covariant and contravariant basis v ecto rs.............................60

2.6.2 Physical com ponents...................................................................63

2.6.3 Spatial derivatives: Covariant differentiation........................65

PART II K IN E M A TIC S , K IN ETICS, AND THE

FUNDAM ENTAL LAWS OF M E C H A N IC S AND

TH E R M O D Y N A M IC S __________________________________

CHAPTER 3 Kinematics: Motion and Deformation 75

3.1 Body, configuration, motion, displacem ent......................................... 75

3.2 Material derivative, velocity, acceleration ...........................................80

VII

viii Contents

3.3 Dcformalion and strain..........................................................................

3.3.1 Deformalion gradieni................................................................H5

3.3.2 Stretch, rotation. Green's deformalion tensor,

Cauchy deformalion te n so r.................................................... KS

3.3.3 Polar decomposition, stretch tensors, rotation tensor ....91

3.3.4 Principal stretches and principal directions........................ 94

3.3.5 Other measures of deformation and strain ..........................96

3.4 VekKity gradient, rate of deformation tensor, vorticity tensor .. I()4

3.5 Material point, material line, material surface.

material volum e.................................................................................. 109

3.6 Volume elements and surface elements in volume

and surface integrations..................................................................... 110

CHAPTER 4 The Fundamental Laws of Thermomechanics ii5

4.1 Mass ...................................................................................................... 1 15

4.2 Forces and moments, linear and angular momentum.................. 116

4.3 Equations of motion (mechanical conservation law s)................. 117

4 .4 The first law of thermodynamics (conservation of en erg y )....... 118

4.5 The transport and localization theorem s....................................... 120

4.5.1 The transport theorem ........................................................ 120

4.5.2 The k>eali/.ation theorem ................................................... 122

4.6 Cauchy stress tensor, heat Hux vector............................................ 124

4.7 The energy theorem and stress pow er............................................ 1.30

4.8 Local forms of the conservation la w s............................................ 131

4.9 Lagrangian forms of the integral conservation law s.................... 137

4.9.1 Mass, forces, moments, linear and angular

m om entum ............................................................................ 139

4.9.2 Conservation of mass, linear momentum, and

angular m om entum ............................................................. 140

4.9.3 First law of thermodynamics............................................ 141

4.9.4 Sum m ary............................................................................... 141

4 .1 0 Piola-Kirchhoff stress tensors, referential heal llux vector........ 142

4.10.1 Relations between spatial and referential quantities ... 142

4.11 The Lagrangian form of the energy theorem................................. 143

4.12 Lcxral conservation laws in Lagrangian fo rm ............................... 144

4.1 3 The second law of thermodynamics................................................ 148

PART ill CONSTITUTIVE MODELING_________________________

CHAPTER 5 Constitutive Modeling in Mechanics and

Thermomechanics..............................................................I57

Part I: M echanics.............................................................................................................. IS'?

5.1 Fundamcnial laws, conslilulive equations, a well-posed

inilial-value boundary-value prohleni............................................. 157

Contents

5.2 Rcslriclions on the constitutive cqualions....................................... 159

5.2.1 invariance under superposed rigid body m otion s......... 160

5.2.2 Material symmetry ............................................................... 171

Part 11: Therm omechanics................................................................................................ 174

5.3 Fundamental laws, conslilulive equations, thermomechanical

processes................................................................................................. 175

5.4 Restrictions on the conslitutive cqualions...................................... 178

5.4.1 Invariance under superposed rigid body m o tio n s.......... 178

CHAPTER 6 N o n lin e a r E l a s t i c i t y ................................................................... i8 i

6.1 Mechanical theory............................................................................... 181

6.2 Thermomechanical th eory ................................................................ 18.3

6 .2 .1 Restrictions imposed by ihe second law

of therm odynam ics.............................................................. 183

6.2.2 Restrictions imposed by invariance under

superposed rigid body motions and conservation

of angular m om entum ......................................................... 187

6.2.3 Resirictions imposed by malerial symmetry:

Isolropy .................................................................................. 191

6.3 Strain energy m odels.......................................................................... 194

CHAPTER 7 Fluid M e c h a n ic s ........................................................................... IM7

7.1 Mechanical theory............................................................................... 197

7.1.1 Viscous fluids......................................................................... 197

7.1.2 Invlscid fluids......................................................................... 205

7.2 Thermomechanical th eory ................................................................206

7.2.1 Viscous fluids......................................................................... 206

7.2.2 Invlscid fluids..........................................................................212

CHAPTER 8 In c o m p re s sib ility a n d T h e rm al E x p a n s io n ................... 215

8.1 Introduction........................................................................................... 215

8.1.1 Motion-temperature constraints.......................................... 216

8.1.2 Motion-entropy conslralnts.................................................217

8.2 Newtonian flu id s.................................................................................218

8.2.1 The compressible theory: A briel'review .........................218

8.2.2 Incompressiblllty...................................................................220

8.2.3 Incompresslbility as a constitutive limit: An

alternative perspective.........................................................227

8.2.4 Thermal expansion............................................................... 229

8.2.5 Thermal expansion as a constitutive limit: An

alternative perspective......................................................... 234

Contents

8.3 Nonlinear elaslic so lid s...................................................................... 236

8.3.1 The compressible theory: A brief review........................ 236

8.3.2 Incom pressibility.................................................................. 237

8.3.3 Incompressible strain energy m odels............................... 241

PART IV BEYOND M ECHANICS AND

THERIVIOMECHANICS

CHAPTER 9 Modeling of Thermo-Electro-Magneto￾Mechanical Behavior, with Application to Smart

M aterials............................................................................ 249

9.1 The fundamental laws of continuum electrodynamics;

Integral forms....................................................................................... 250

9.1.1 Notation and nomenclature ............................................... 250

9.1.2 Conservation of m ass...........................................................251

9.1.3 Balance of linear momentum ............................................ 252

9.1.4 Balance of angular m om entum .........................................256

9 .1.5 First law of thermodynamics............................................. 257

9.1.6 Second law of ihermodynamics......................................... 259

9.1.7 Conservation of electric charge........................................ 260

9.1.8 Faraday’s la w .........................T............................................262

9 .1.9 Gauss's law for magnetism................................................. 262

9.1.10 Gauss’s law for electricity................................................... 263

9.1.11 Ampere-Maxwell la w ......................................................... 264

9.1.12 Transformations between spatial and referential

TEMM quanlilies................................................................ 265

9.2 The fundamental laws of continuum electrodynamics:

Pointwise form s.................................................................................. 269

9.2.1 Eulerian fundamcnial law s....................................................269

9.2.2 Lagrangian fundamental laws...............................................275

9.3 Modeling of the effective electromagnetic fields............................277

9.3.1 Minkowski model ................................................................. 278

9.3.2 Lorenlz m txlel......................................................................... 278

9.3.3 Statistical m ixlel...................................................................... 278

9.3.4 Chu mcxiel ................................................................................279

9.3.5 A comparison of the four m odels........................................ 279

9.4 Modeling of the eleclromagneiically induced coupling

term s...................................................................................................... 280

9.4.1 An alternative approach......................................................... 281

9.5 Thermo-eleciro-magnelo-mechanical process.............................. 283

Contents

9.6 Conslitutive model developmenl lor

ihermo-eleetro-magneto-elastic materials:

Large-deformation theory.................................................................. 2X4

9.6.1 The reduced Clausius-Duhem inequality, work

conjugates............................................................................. 284

9.6.2 The all-extensive formulation........................................... 285

9.6.3 Otfier form ulations.............................................................. 288

9.6.4 Restrictions imposed by invariance under

superposed rigid body motions and conservation

of angular m om entum ........................................................293

9.7 Conslilulive model development for

ihemio-eleciro-magneto-elastic materials:

Small-def'ormation theory.................................................................. 294

9.7.1 Small-delomialion kinematics, kmetics.

electromagnelic fields, and fundamental laws.................294

9.7.2 Linear constitutive equations ............................................296

9.7.3 Material symmetry ..............................................................298

9.8 Linear, reversible, thermo-electro-magneio-mechanical

prtHJcsses................................................................................................299

9.9 Speciali/.ation of the small-deformation

ihermo-electro-magneio-elastic framework to piezoelectric

m aterials............................................................................................... 302

APPENDIX A Different Notions of Invariance .105

APPENDIX B The Physical Basis of Constitutive Assumptions 307

APPENDIX C Isotropic Tensors 309

APPENDIX D A Family of Thermomechanical Processes 3 1 1

APPENDIX E Energy Formulations and Stability Conditions for

Newtonian Fluids 3i3

E.1 Governing cqualions................................................................ 313

E.I.l Densily-entropy I'ormulalion......................................... 314

E. 1.2 Density-iemperature form ulation......................................315

E. 1.3 Pressure-eniropy formulation .............................................315

E. 1.4 Pressure-temperaiure formulation......................................316

E.2 Stability conditions .......................................................................... 317

APPENDIX F Additional Energy Formulations for

Thermo-Electro-Magneto-Mechanical M aterials 3i9

F.l Déformaiion-lemperature-electric displacement-magnetic

induction form ulation....................................................................... 319

Bibliography.......................................................................................................................321

In d e x .................................................................................................................................... 325

Preface

CONTINUUM MECHANICS: THE NEW PEDAGOGY

Since my days as a graduate student at iJerkeley In the early 1980s. the graduate

engineering mechanics curriculum has undergone Ihree major changes: l-irst. in

previous years, the curriculum left time lor courses in linear eiaslicily. linite elas￾ticity. plasticity, viscoelasticity. Inviscid Huid dynamics, and viscous Huid dynamics,

followed by a unifying course in continuum mechanics. Today, with the onslaught

of new materials and lechnological advances competing for space in the curriculum,

the reality Is that there Is no longer room for this many traditional mechanics courses.

Second, back In the day, much of the teaching and learning was accomplished through

homework exercises worked by the students on their own; graded and annotated by

the professor; and then returned to the students. Although the most effective, this

approach Is impractical given today's lime conslralnts on both the student and the

professor. As such, there is a risk that Important concepts may be overlooked If

they are not illuminated through worked examples. Third, mathematics and applied

mechanics have diverged, and this gap continues to widen. AsCourant and Hilbert j I )

mu.sed in their treatise Methods o f M athematical Physic.s. and is still the ca.se today:

"Since the seventeenth century, physical intuition has served us a vital source fo r

mathematical problems and methods. Recent trends and fashions have, however,

weakened the connecti<m between mathematics and physics: mathematicians,

turning away from the roots o f mathematics in intuition, have c<mcentrated on

refinement and emphasized the postulational side o f mathematics, and at times

have overlooked the unity o f their science with physics and other fields. In many

cases, physicists have ceased to appreciate the attitudes o f mathematicians. This

rift is unquestionably a serious threat to science as a whole: the broad stream of

scientific development may split into smaller and smaller rivulets and dry out"

This textbook adjusts to each of ihese realities: First, the material is covered in

the most time-efficient manner, that Is, by first giving the unified siluatli)n (continuum

mechanics), then applying It to special cases (iinite elasticity, viscous ffuid dynamics,

and so on). Because these special cases are presented in a single textbook, the

handoff between one subject and another is cleaner, and undue redundancy Is avoided.

Second, the majority of the problems In the textbook are presented as worked

examples with full, detailed solutions. Each of these problems Is designed to convey

an Important concept. Third, we place a strong emphasis on explicitly connecting

the mathematics to the continuum physics. Indicial notation Is jettisoned almost

entirely in favor of the more compact and elegant direct notation, allowing us to be

xiii