Notes in Continuum Mechanics

Noteson Continuum Mechanics

Eduardo W.V. Chaves

Lecture Noteson Numerical Methodsin Engineering and Sciences

Notes on Continuum Mechanics

Lecture Notes on Numerical Methods in Engineering and Sciences

Aims and Scope of the Series This series publishes text books on topics of general interest in the field of computational engineering sciences. The books will focus on subjects in which numerical methods play a fundamental role for solving problems in engineering and applied sciences. Advances in finite element, finite volume, finite differences, discrete and particle methods and their applications are examples of the topics covered by the series.

The main intended audience is the first year graduate student. Some books define the current state of a field to a highly specialised readership; others are accessible to final year undergraduates, but essentially the emphasis is on accessibility and clarity. The books will be also useful for practising engineers and scientists interested in state of the art information on the theory and application of numerical methods. Series Editor Eugenio Oate International Center for Numerical Methods in Engineering (CIMNE) School of Civil Engineering, Technical University of Catalonia (UPC), Barcelona, Spain Editorial Board Francisco Chinesta, Ecole Nationale Suprieure d'Arts et Mtiers, Paris, France Charbel Farhat, Stanford University, Stanford, USA Carlos Felippa, University of Colorado at Boulder, Colorado, USA Antonio Huerta, Technical University of Catalonia (UPC), Barcelona, Spain Thomas J.R. Hughes, The University of Texas at Austin, Austin, USA Sergio R. Idelsohn, CIMNE-ICREA, Barcelona, Spain Pierre Ladeveze, ENS de Cachan-LMT-Cachan, France Wing Kam Liu, Northwestern University, Evanston, USA Xavier Oliver, Technical University of Catalonia (UPC), Barcelona, Spain Manolis Papadrakakis, National Technical University of Athens, Greece Jacques Priaux, CIMNE-UPC Barcelona, Spain & Univ. of Jyvskyl, Finland Bernhard Schrefler, Universit degli Studi di Padova, Padova, Italy Genki Yagawa, Tokyo University, Tokyo, Japan Mingwu Yuan, Peking University, China Titles: 1. E. Oate, Structural Analysis with the Finite Element Method.

Linear Statics. Volume 1. Basis and Solids, 2009 2. K. Winiewski, Finite Rotation Shells. Basic Equations and

Finite Elements for Reissner Kinematics, 2010 3. E. Oate, Structural Analysis with the Finite Element Method. Linear Statics. Volume 2. Beams, Plates and Shells, 2013

4. E.W.V. Chaves. Notes on Continuum Mechanics. 2013

Notes on Continuum Mechanics

Eduardo W.V. Chaves School of Civil Engineering University of Castilla-La Mancha Ciudad Real, Spain

ISBN: 978-94-007-5985-5 (HB) ISBN: 978-94-007-5986-2 (e-book) Depsito legal: B-29347-2012 A C.I.P. Catalogue record for this book is available from the Library of Congress Lecture Notes Series Manager: M Jess Samper, CIMNE, Barcelona, Spain Cover page: Pall Disseny i Comunicaci, www.pallidisseny.com Printed by: Artes Grficas Torres S.A., Morales 17, 08029 Barcelona, Espaa www.agraficastorres.es Printed on elemental chlorine-free paper

Notes on Continuum Mechanics Eduardo W.V. Chaves First edition, 2013 International Center for Numerical Methods in Engineering (CIMNE), 2013 Gran Capitn s/n, 08034 Barcelona, Spain www.cimne.com No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

To my Parents

Contents

PREFACE .......................................................................................................................................................XIXABBREVIATIONS..........................................................................................................................................XXI OPERATORS AND SYMBOLS....................................................................................................................XXIII SI-UNITS ..................................................................................................................................................... XX INTRODUCTION ................................................................................................................. 1

1 MECHANICS...............................................................................................................................................1 2 WHAT IS CONTINUUM MECHANICS?....................................................................................................1

2.1 Hypothesis of Continuum Mechanics .........................................................................................1 2.2 The Continuum ...............................................................................................................................2

3 SCALES OF MATERIAL STUDIES.............................................................................................................3 3.1 Scale Study of Continuum Mechanics .........................................................................................3

4 THE INITIAL BOUNDARY VALUE PROBLEM (IBVP) .........................................................................6 4.1 Solving the IBVP.............................................................................................................................6 4.2 Simplifying the IBVP......................................................................................................................7

1 TENSORS.............................................................................................................................9

1.1 INTRODUCTION.....................................................................................................................................9 1.2 ALGEBRAIC OPERATIONS WITH VECTORS ....................................................................................10 1.3 COORDINATE SYSTEMS .....................................................................................................................16

1.3.1 Cartesian Coordinate System....................................................................................................16 1.3.2 Vector Representation in the Cartesian Coordinate System...............................................17 1.3.3 Einstein Summation Convention (Einstein Notation) ........................................................20

1.4 INDICIAL NOTATION .........................................................................................................................20 1.4.1 Some Operators..........................................................................................................................22

1.4.1.1 Kronecker Delta.............................................................................................................22 1.4.1.2 Permutation Symbol ......................................................................................................23

1.5 ALGEBRAIC OPERATIONS WITH TENSORS.....................................................................................28 1.5.1 Dyadic ..........................................................................................................................................28

1.5.1.1 Component Representation of a Second-Order Tensor in the Cartesian Basis...................................................................................................................................32

1.5.2 Properties of Tensors ................................................................................................................34 1.5.2.1 Tensor Transpose ..........................................................................................................34 1.5.2.2 Symmetry and Antisymmetry.......................................................................................36 1.5.2.3 Cofactor Tensor. Adjugate of a Tensor .....................................................................42 1.5.2.4 Tensor Trace...................................................................................................................42 1.5.2.5 Particular Tensors ..........................................................................................................44 1.5.2.6 Determinant of a Tensor ..............................................................................................45 1.5.2.7 Inverse of a Tensor........................................................................................................48 1.5.2.8 Orthogonal Tensors ......................................................................................................51

Contents

VII

V

NOTES ON CONTINUUM MECHANICS

V

1.5.2.9 Positive Definite Tensor, Negative Definite Tensor and Semi-Definite Tensors ............................................................................................................................. 52

1.5.2.10 Additive Decomposition of Tensors........................................................................ 53 1.5.3 Transformation Law of the Tensor Components ................................................................ 54

1.5.3.1 Component Transformation Law in Two Dimensions (2D) ................................. 61 1.5.4 Eigenvalue and Eigenvector Problem.................................................................................... 65

1.5.4.1 The Orthogonality of the Eigenvectors ..................................................................... 67 1.5.4.2 Solution of the Cubic Equation................................................................................... 69

1.5.5 Spectral Representation of Tensors ........................................................................................ 72 1.5.6 Cayley-Hamilton Theorem....................................................................................................... 76 1.5.7 Norms of Tensors ..................................................................................................................... 78 1.5.8 Isotropic and Anisotropic Tensor........................................................................................... 79 1.5.9 Coaxial Tensors.......................................................................................................................... 80 1.5.10 Polar Decomposition.............................................................................................................. 81 1.5.11 Partial Derivative with Tensors ............................................................................................. 83

1.5.11.1 Partial Derivative of Invariants ................................................................................. 85 1.5.11.2 Time Derivative of Tensors....................................................................................... 86

1.5.12 Spherical and Deviatoric Tensors ......................................................................................... 86 1.5.12.1 First Invariant of the Deviatoric Tensor.................................................................. 87 1.5.12.2 Second Invariant of the Deviatoric Tensor............................................................. 87 1.5.12.3 Third Invariant of Deviatoric Tensor ...................................................................... 89

1.6 THE TENSOR-VALUED TENSOR FUNCTION................................................................................. 91 1.6.1 The Tensor Series ...................................................................................................................... 91 1.6.2 The Tensor-Valued Isotropic Tensor Function ................................................................... 92 1.6.3 The Derivative of the Tensor-Valued Tensor Function ..................................................... 94

1.7 THE VOIGT NOTATION .................................................................................................................... 96 1.7.1 The Unit Tensors in Voigt Notation...................................................................................... 97 1.7.2 The Scalar Product in Voigt Notation.................................................................................... 98 1.7.3 The Component Transformation Law in Voigt Notation .................................................. 99 1.7.4 Spectral Representation in Voigt Notation.......................................................................... 100 1.7.5 Deviatoric Tensor Components in Voigt Notation........................................................... 101

1.8 TENSOR FIELDS ................................................................................................................................ 105 1.8.1 Scalar Fields .............................................................................................................................. 106 1.8.2 Gradient..................................................................................................................................... 106 1.8.3 Divergence ................................................................................................................................ 111 1.8.4 The Curl .................................................................................................................................... 113 1.8.5 The Conservative Field........................................................................................................... 115

1.9 THEOREMS INVOLVING INTEGRALS ............................................................................................ 117 1.9.1 Integration by Parts ................................................................................................................. 117 1.9.2 The Divergence Theorem ...................................................................................................... 117 1.9.3 Independence of Path ............................................................................................................. 120 1.9.4 The Kelvin-Stokes Theorem................................................................................................. 121 1.9.5 Greens Identities..................................................................................................................... 122

Appendix A: A GRAPHICAL REPRESENTATION OF A SECOND-ORDER TENSOR..... 125

A.1 PROJECTING A SECOND-ORDER TENSOR ONTO A PARTICULAR DIRECTION..................... 125 A.1.1 Normal and Tangential Components .................................................................................. 125 A.1.2 The Maximum and Minimum Normal Components........................................................ 127 A.1.3 The Maximum and Minimum Tangential Component .................................................... 128

A.2 GRAPHICAL REPRESENTATION OF AN ARBITRARY SECOND-ORDER TENSOR................... 130 A.2.1 Graphical Representation of a Symmetric Second-Order Tensor (Mohrs Circle)...... 134

A.3 THE TENSOR ELLIPSOID................................................................................................................ 138 A.4 GRAPHICAL REPRESENTATION OF THE SPHERICAL AND DEVIATORIC PARTS .................. 139

A.4.1 The Octahedral Vector .......................................................................................................... 139

III

CONTENTS

2 CONTINUUM KINEMATICS........................................................................................145 2.1 INTRODUCTION.................................................................................................................................145 2.2 THE CONTINUOUS MEDIUM ..........................................................................................................146

2.2.1 Kinds of Motion.......................................................................................................................147 2.2.1.1 Rigid Body Motion ......................................................................................................147

2.2.2 Types of Configurations .........................................................................................................149 2.2.2.1 Mass Density.................................................................................................................150

2.3 DESCRIPTION OF MOTION .............................................................................................................151 2.3.1 Material and Spatial Coordinates ...........................................................................................151 2.3.2 The Displacement Vector.......................................................................................................152 2.3.3 The Velocity Vector.................................................................................................................152 2.3.4 The Acceleration Vector .........................................................................................................152 2.3.5 Lagrangian and Eulerian Descriptions..................................................................................152

2.3.5.1 Lagrangian Description of Motion............................................................................152 2.3.5.2 Eulerian Description of Motion ................................................................................153 2.3.5.3 Lagrangian and Eulerian Variables............................................................................153

2.4 THE MATERIAL TIME DERIVATIVE ..............................................................................................156 2.4.1 Velocity and Acceleration in Eulerian Description ............................................................158 2.4.2 Stationary Fields .......................................................................................................................159 2.4.3 Streamlines ................................................................................................................................161

2.5 THE DEFORMATION GRADIENT ...................................................................................................163 2.5.1 Introduction ..............................................................................................................................163 2.5.2 Stretch and Unit Extension ....................................................................................................163 2.5.3 The Material and Spatial Deformation Gradient ................................................................165 2.5.4 Displacement Gradient Tensors (Material and Spatial) .....................................................168 2.5.5 Material Time Derivative of the Deformation Gradient. Material Time Derivative

of the Jacobian Determinant................................................................................................171 2.5.5.1 Material Time Derivative of F . The Spatial Velocity Gradient ..........................171 2.5.5.2 Rate-of-Deformation and Spin Tensors...................................................................172 2.5.5.3 The Material Time Derivative of 1F ......................................................................174 2.5.5.4 The Material Time Derivative of the Jacobian Determinant ................................174

2.6 FINITE STRAIN TENSORS.................................................................................................................176 2.6.1 The Material Finite Strain Tensor .........................................................................................177 2.6.2 The Spatial Finite Strain Tensor (The Almansi Strain Tensor) ........................................181 2.6.3 The Material Time Derivative of Strain Tensors ................................................................183

2.6.3.1 The Material Time Derivative of the Right Cauchy-Green Deformation Tensor .............................................................................................................................183

2.6.3.2 The Material Time Derivative of the Green-Lagrange Strain Tensor .................183 2.6.3.3 The Material Time Derivative of 1C ......................................................................184 2.6.3.4 Material Time Derivative of the Left Cauchy-Green Deformation Tensor.......184 2.6.3.5 The Material Time Derivative of the Almansi Strain Tensor ...............................185

2.6.4 Interpreting Deformation/Strain Tensors ...........................................................................186 2.6.4.1 The Relationship between the Strain and Stretch Tensors ...................................187 2.6.4.2 Change of Angle...........................................................................................................188 2.6.4.3 The Physical Interpretation of the Deformation/Strain Tensor

Components. The Right Stretch Tensor ...................................................................189 2.7 PARTICULAR CASES OF MOTION ...................................................................................................191

2.7.1 Homogeneous Deformation ..................................................................................................191 2.7.2 Rigid Body Motion...................................................................................................................192

2.8 POLAR DECOMPOSITION OF F .....................................................................................................195 2.8.1 Spectral Representation of Kinematic Tensors...................................................................197 2.8.2 Evolution of the Polar Decomposition................................................................................203

2.8.2.1 The Alternative Way to Express the Rate of Kinematic Tensors ........................208 2.9 AREA AND VOLUME ELEMENTS DEFORMATION ......................................................................215

2.9.1 Area Element Deformation....................................................................................................215

IX


2.9.1.1 The Material Time Derivative of the Area Element .............................................. 217 2.9.2 The Volume Element Deformation ..................................................................................... 218

2.9.2.1 The Material Time Derivative of the Volume Element ........................................ 219 2.9.2.2 Dilatation....................................................................................................................... 220 2.9.2.3 Isochoric Motion. Incompressibility ........................................................................ 220

2.10 MATERIAL AND CONTROL DOMAINS ........................................................................................ 220 2.10.1 The Material Domain............................................................................................................ 220 2.10.2 The Control Domain ............................................................................................................ 221

2.11 TRANSPORT EQUATIONS .............................................................................................................. 222 2.12 CIRCULATION AND VORTICITY................................................................................................... 224 2.13 MOTION DECOMPOSITION: VOLUMETRIC AND ISOCHORIC MOTIONS.............................. 225

2.13.1 The Principal Invariants ....................................................................................................... 227 2.14 THE SMALL DEFORMATION REGIME......................................................................................... 228

2.14.1 Introduction............................................................................................................................ 228 2.14.2 Infinitesimal Strain and Spin Tensors ................................................................................ 229 2.14.3 Stretch and Unit Extension.................................................................................................. 231 2.14.4 Change of Angle .................................................................................................................... 232 2.14.5 The Physical Interpretation of the Infinitesimal Strain Tensor ..................................... 232

2.14.5.1 Engineering Strain ..................................................................................................... 233 2.14.6 The Volume Ratio (Dilatation)............................................................................................ 235 2.14.7 The Plane Strain..................................................................................................................... 236

2.15 OTHER WAYS TO DEFINE STRAIN.............................................................................................. 239 2.15.1 Motivation............................................................................................................................... 239 2.15.2 The Logarithmic Strain Tensor ...........................................................................................241 2.15.3 The Biot Strain Tensor ......................................................................................................... 242 2.15.4 Unifying the Strain Tensors ................................................................................................. 242 2.15.5 One Dimensional Measurements of Strain (1D).............................................................. 243

2.15.5.1 Cauchys strain or Engineering strain or the Linear strain.................................. 243 2.15.5.2 The Logarithmic or True strain............................................................................... 243 2.15.5.3 The Green-Lagrange strain ...................................................................................... 243 2.15.5.4 The Almansi strain ....................................................................................................243 2.15.5.5 The Swaiger strain ..................................................................................................... 244 2.15.5.6 The Kuhn strain......................................................................................................... 244

3 STRESS .............................................................................................................................245

3.1 INTRODUCTION ................................................................................................................................ 245 3.2 FORCES ............................................................................................................................................... 245

3.2.1 Surface Forces (Traction) ....................................................................................................... 245 3.2.2 Gravitational Force (Body Force) .........................................................................................246

3.3 STRESS TENSORS...............................................................................................................................247 3.3.1 The Cauchy Stress Tensor...................................................................................................... 248

3.3.1.1 The Traction Vector....................................................................................................248 3.3.1.2 Cauchys Fundamental Postulate .............................................................................. 248

3.3.2 The Relationship between the Traction and the Cauchy Stress Tensor ......................... 252 3.3.3 Other Measures of Stress ....................................................................................................... 260

3.3.3.1 The First Piola-Kirchhoff Stress Tensor ................................................................. 260 3.3.3.2 The Kirchhoff Stress Tensor ..................................................................................... 262 3.3.3.3 The Second Piola-Kirchhoff Stress Tensor............................................................. 262 3.3.3.4 The Biot Stress Tensor ............................................................................................... 264 3.3.3.5 The Mandel Stress Tensor.......................................................................................... 264

3.3.4 Spectral Representation of the Stress Tensors.................................................................... 265 4 OBJECTIVITY OF TENSORS ........................................................................................269

4.1 INTRODUCTION ................................................................................................................................ 269

X

CONTENTS

4.2 THE OBJECTIVITY OF TENSORS.....................................................................................................270 4.2.1 The Deformation Gradient ....................................................................................................272 4.2.2 Kinematic Tensors...................................................................................................................273 4.2.3 Stress Tensors ...........................................................................................................................275

4.3 TENSOR RATES..................................................................................................................................277 4.3.1 Objective Rates.........................................................................................................................278

4.3.1.1 The Convective Rate ...................................................................................................279 4.3.1.2 The Oldroyd Rate ........................................................................................................279 4.3.1.3 The Cotter-Rivlin Rate ................................................................................................280 4.3.1.4 The Jaumann-Zaremba Rate ......................................................................................280 4.3.1.5 The Green-Naghdi Rate (Polar Rate) .......................................................................282

4.3.2 The Objective Rate of Stress Tensors ..................................................................................282 5 THE FUNDAMENTAL EQUATIONS OF CONTINUUM MECHANICS................. 285

5.1 INTRODUCTION.................................................................................................................................285 5.2 DENSITY .............................................................................................................................................285

5.2.1 Mass Density.............................................................................................................................286 5.3 FLUX....................................................................................................................................................286 5.4 THE REYNOLDS TRANSPORT THEOREM......................................................................................287

5.4.1 Reynolds Transport Theorem for Volumes with Discontinuities...................................288 5.5 CONSERVATION LAW.......................................................................................................................291 5.6 THE PRINCIPLE OF CONSERVATION OF MASS. THE MASS CONTINUITY EQUATION ........291

5.6.1 The Mass Continuity Equation in Lagrangian Description...............................................293 5.6.2 Incompressibility ......................................................................................................................295 5.6.3 The Mass Continuity Equation for Volume with Discontinuities ...................................295

5.7 THE PRINCIPLE OF CONSERVATION OF LINEAR MOMENTUM. THE EQUATIONS OF MOTION ...........................................................................................................................................297

5.7.1 Linear Momentum ...................................................................................................................297 5.7.2 The Principle of Conservation of Linear Momentum .......................................................297

5.7.2.1 The Equilibrium Equations........................................................................................298 5.7.3 The Equations of Motion with Discontinuities ..................................................................301

5.8 THE PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM. SYMMETRY OF THE CAUCHY STRESS TENSOR..............................................................................................................302

5.8.1 Angular Momentum ................................................................................................................302 5.8.2 The Principle of Conservation of Angular Momentum ....................................................303

5.9 THE PRINCIPLE OF CONSERVATION OF ENERGY. THE ENERGY EQUATION.....................307 5.9.1 Kinetic Energy..........................................................................................................................307 5.9.2 External and Internal Mechanical Power .............................................................................307 5.9.3 The Balance of Mechanical Energy.......................................................................................310 5.9.4 The Internal Energy.................................................................................................................312 5.9.5 Thermal Power .........................................................................................................................313 5.9.6 The First Law of Thermodynamics. The Energy Equation..............................................314

5.9.6.1 The Energy Equation in Lagrangian Description...................................................315 5.9.7 The Energy Equation with Discontinuity ............................................................................316

5.10 THE PRINCIPLE OF IRREVERSIBILITY. ENTROPY INEQUALITY.............................................318 5.10.1 The Second Law of Thermodynamics................................................................................318 5.10.2 The Clausius-Duhem Inequality..........................................................................................320 5.10.3 The Clausius-Planck Inequality............................................................................................321 5.10.4 The Alternative Form to Express the Clausius-Duhem Inequality ...............................321 5.10.5 The Alternative Form of the Clausius-Planck Inequality ................................................323 5.10.6 Reversible Process .................................................................................................................323 5.10.7 Entropy Inequality for a Domain with Discontinuity......................................................324

5.11 FUNDAMENTAL EQUATIONS OF CONTINUUM MECHANICS..................................................326 5.11.1 Particular Cases ......................................................................................................................327

5.11.1.1 Rigid Body Motion ....................................................................................................327

XI


5.11.1.2 Flux Problems ............................................................................................................ 327 5.12 FLUX PROBLEMS............................................................................................................................. 328

5.12.1 Heat Transfer ......................................................................................................................... 328 5.12.1.1 Thermal Conduction................................................................................................. 328 5.12.1.2 Thermal Convection Transfer ................................................................................. 330 5.12.1.3 Thermal Radiation.....................................................................................................330 5.12.1.4 The Heat Flux Equation........................................................................................... 330

5.13 FLUID FLOW IN POROUS MEDIA (FILTRATION) ...................................................................... 334 5.14 THE CONVECTION-DIFFUSION EQUATION ............................................................................. 335

5.14.1 The Generalization of the Flux Problem........................................................................... 338 5.15 INITIAL BOUNDARY VALUE PROBLEM (IBVP) AND COMPUTATIONAL MECHANICS ...... 338

6 INTRODUCTION TO CONSTITUTIVE EQUATIONS..............................................341

6.1 INTRODUCTION ................................................................................................................................ 341 6.2 THE CONSTITUTIVE PRINCIPLES................................................................................................... 343

6.2.1 The Principle of Determinism............................................................................................... 344 6.2.2 The Principle of Local Action ............................................................................................... 344 6.2.3 The Principle of Equipresence .............................................................................................. 344 6.2.4 The Principle of Objectivity................................................................................................... 344 6.2.5 The Principle of Dissipation .................................................................................................. 344

6.3 CHARACTERIZATION OF CONSTITUTIVE EQUATIONS FOR SIMPLE THERMOELASTIC MATERIALS...................................................................................................................................... 345

6.4 CHARACTERIZATION OF THE CONSTITUTIVE EQUATIONS FOR A THERMO-VISCOELASTIC MATERIAL............................................................................................................. 351

6.4.1 Constitutive Equations with Internal Variables.................................................................. 355 6.5 SOME EXPERIMENTAL EVIDENCE................................................................................................ 360

6.5.1 Behavior of Solids.................................................................................................................... 360 6.5.1.1 Temperature Effect .....................................................................................................362 6.5.1.2 Some Mechanical Properties of Solids ..................................................................... 362

6.5.2 Behavior of Fluids ................................................................................................................... 369 6.5.2.1 Viscosity ........................................................................................................................ 370

6.5.3 Behavior of Viscoelastic Materials ........................................................................................ 371 6.5.4 Rheological Models ................................................................................................................. 372

7 LINEAR ELASTICITY ....................................................................................................375

7.1 INTRODUCTION ................................................................................................................................ 375 7.2 INITIAL BOUNDARY VALUE PROBLEM OF LINEAR ELASTICITY............................................. 376

7.2.1 Governing Equations.............................................................................................................. 376 7.2.2 Initial and Boundary Conditions ........................................................................................... 377

7.3 GENERALIZED HOOKES LAW ...................................................................................................... 377 7.3.1 The Generalized Hookes Law in Voigt Notation ............................................................. 378 7.3.2 The Component Transformation Law for the Generalized Hookes Law .................... 379

7.3.2.1 The Matrix Transformation for Stress and Strain Components .......................... 380 7.3.2.2 The Transformation Matrix of the Elasticity Tensor Components .................... 381

7.4 THE ELASTICITY TENSOR............................................................................................................... 381 7.4.1 Anisotropy and Isotropy......................................................................................................... 381 7.4.2 Types of Elasticity Tensor Symmetry................................................................................... 382

7.4.2.1 Triclinic Materials ........................................................................................................ 382 7.4.2.2 Monoclinic Symmetry (One Plane of Symmetry)................................................... 383 7.4.2.3 Orthotropic Symmetry (Two Planes of Symmetry) ............................................... 384 7.4.2.4 Tetragonal Symmetry .................................................................................................. 384 7.4.2.5 Transversely Isotropic Symmetry (Hexagonal Symmetry).................................... 386 7.4.2.6 Cubic Symmetry........................................................................................................... 388 7.4.2.7 Symmetry in All Directions (Isotropy)..................................................................... 390

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CONTENTS

7.5 ISOTROPIC MATERIALS ....................................................................................................................392 7.5.1 Constitutive Equations............................................................................................................392 7.5.2 Experimental Determination of Elastic Constants.............................................................393

7.5.2.1 Youngs Modulus and Poissons Ratio .....................................................................393 7.5.2.2 The Shear and Bulk Moduli........................................................................................394

7.5.3 Restrictions on Elastic Mechanical Properties ....................................................................398 7.6 STRAIN ENERGY DENSITY..............................................................................................................399

7.6.1 Decoupling Strain Energy Density........................................................................................402 7.7 THE CONSTITUTIVE LAW FOR ORTHOTROPIC MATERIAL.......................................................404 7.8 TRANSVERSELY ISOTROPIC MATERIALS ......................................................................................405 7.9 THE SAINT-VENANTS AND SUPERPOSITION PRINCIPLES .......................................................406 7.10 INITIAL STRESS/STRAIN................................................................................................................408

7.10.1 Thermal Deformation ...........................................................................................................408 7.11 THE NAVIER-LAM EQUATIONS.................................................................................................410 7.12 TWO-DIMENSIONAL ELASTICITY................................................................................................410

7.12.1 The State of Plane Stress ......................................................................................................411 7.12.1.1 The Initial Strain.........................................................................................................412

7.12.2 The State of Plane Strain ......................................................................................................413 7.12.2.1 Thermal Strain ............................................................................................................415

7.12.3 Axisymmetric Solids ..............................................................................................................417 7.13 THE UNIDIMENSIONAL APPROACH............................................................................................418

7.13.1 Beam Structural Elements ....................................................................................................418 7.13.1.1 The Internal Normal Force and the Bending Moments .....................................420 7.13.1.2 The Shear Forces and the Torsional Moment ......................................................421 7.13.1.3 The Strain Energy ......................................................................................................422

8 HYPERELASTICITY...................................................................................................... 423

8.1 INTRODUCTION.................................................................................................................................423 8.2 CONSTITUTIVE EQUATIONS...........................................................................................................424

8.2.1 Elastic Tangent Stiffness Tensors .........................................................................................427 8.2.1.1 The Material Elastic Tangent Stiffness Tensor .......................................................427 8.2.1.2 The Spatial Elastic Tangent Stiffness Tensor..........................................................428 8.2.1.3 The Instantaneous Elastic Tangent Stiffness Tensor.............................................430 8.2.1.4 The Elastic Tangent Stiffness Pseudo-Tensor ........................................................431

8.3 ISOTROPIC HYPERELASTIC MATERIALS .......................................................................................432 8.3.1 The Constitutive Equation in terms of Invariants..............................................................434

8.3.1.1 The Constitutive Equation in terms of and ........................................................434 8.3.1.2 The Constitutive Equation in terms of ..................................................................436

8.3.2 Series Expansion of the Energy Function ...........................................................................436 8.3.3 Constitutive Equations in terms of the Principal Stretches ..............................................437

8.4 COMPRESSIBLE MATERIALS ............................................................................................................440 8.4.1 The Stress Tensors...................................................................................................................442 8.4.2 Compressible Isotropic Materials..........................................................................................445

8.4.2.1 Compressible Isotropic Material in terms of the Invariants .................................446 8.5 INCOMPRESSIBLE MATERIALS ........................................................................................................447

8.5.1 Geometrical Interpretation.....................................................................................................449 8.5.2 Isotropic Incompressible Hyperelastic Materials................................................................450

8.5.2.1 Series Expansion of the Energy Function for an Isotropic Incompressible Hyperelastic Materials ..................................................................................................451

8.6 EXAMPLES OF HYPERELASTIC MODELS ......................................................................................451 8.6.1 The Neo-Hookean Material Model.......................................................................................452 8.6.2 The Ogden Material Model ....................................................................................................452

8.6.2.1 The Incompressible Ogden Material Model............................................................452 8.6.2.2 The Hadamard Material Model..................................................................................453

8.6.3 The Mooney-Rivlin Material Model......................................................................................453

XIII


XIV

8.6.3.1 Strain Energy Density ................................................................................................. 453 8.6.3.2 The Stress Tensor ........................................................................................................ 454

8.6.4 The Yeoh Material Model ...................................................................................................... 454 8.6.4.1 Strain Energy Density ................................................................................................. 454 8.6.4.2 The Stress Tensor ........................................................................................................ 454

8.6.5 The Arruda-Boyce Material Model ....................................................................................... 454 8.6.6 The Blatz-Ko Hyperelastic Model ........................................................................................ 455 8.6.7 The Saint Venant-Kirchhoff Model ..................................................................................... 455

8.6.7.1 Strain Energy Density ................................................................................................. 455 8.6.7.2 The Stress Tensor ........................................................................................................ 456 8.6.7.3 The Elastic Tangent Stiffness Tensor ...................................................................... 456

8.6.8 The Compressible Neo-Hookean Material Model ............................................................. 457 8.6.8.1 Strain Energy Density ................................................................................................. 457 8.6.8.2 The Stress Tensor ........................................................................................................ 457 8.6.8.3 The Elastic Tangent Stiffness Tensor ...................................................................... 458

8.6.9 The Gent Model ...................................................................................................................... 460 8.6.10 The Statistical Model............................................................................................................. 460 8.6.11 The Eight-Parameter Model ................................................................................................ 461

8.7 ANISOTROPIC HYPERELASTICITY ................................................................................................. 462 8.7.1 Transversely Isotropic Material ............................................................................................. 463

9 PLASTICITY.....................................................................................................................465

9.1 INTRODUCTION ................................................................................................................................ 465 9.2 THE YIELD CRITERION................................................................................................................... 467

9.2.1 The Yield Surface for Anisotropic Materials....................................................................... 468 9.2.1.1 The Yield Surface Gradient........................................................................................ 468

9.2.2 The Yield Surface for Isotropic Materials............................................................................ 468 9.2.3 The Yield Surface for Materials Independent of Pressure................................................ 471

9.2.3.1 The von Mises Yield Criterion .................................................................................. 471 9.2.3.2 The Tresca Yield Criterion......................................................................................... 475

9.2.4 The Yield Criteria for Pressure-Dependent Materials ....................................................... 477 9.2.4.1 The Mohr-Coulomb Criterion................................................................................... 478 9.2.4.2 The Drucker-Prager Yield Criterion......................................................................... 482 9.2.4.3 The Rankine Yield Criterion...................................................................................... 486

9.2.5 Evolution of the Yield Surface .............................................................................................. 489 9.3 PLASTICITY MODELS IN SMALL DEFORMATION REGIME (UNIAXIAL CASES) ..................... 491

9.3.1 Rate-Independent Plasticity Models (Uniaxial Case) ......................................................... 491 9.3.1.1 Perfect Elastoplastic Behavior................................................................................... 491 9.3.1.2 Isotropic Hardening Elastoplastic Behavior ........................................................... 495 9.3.1.3 Kinematic Hardening Elastoplastic Behavior ......................................................... 500 9.3.1.4 Isotropic-Kinematic Elastoplastic Behavior............................................................ 502

9.4 PLASTICITY IN SMALL DEFORMATION REGIME (THE CLASSICAL PLASTICITY THEORY).......................................................................................................................................... 503

9.4.1 The Infinitesimal Strain Tensor and Constitutive Equation............................................. 504 9.4.2 Helmholtz Free Energy .......................................................................................................... 505 9.4.3 Internal Energy Dissipation and the Evolution of the Internal Variables ..................... 505 9.4.4 The Elastoplastic Tangent Stiffness Tensor........................................................................ 507 9.4.5 The Classical Flow Theory................................................................................................... 512

9.4.5.1 Perfect Plasticity........................................................................................................... 512 9.4.5.2 Isotropic-Kinematic Hardening Plasticity ............................................................... 513

9.5 PLASTIC POTENTIAL THEORY ....................................................................................................... 515 9.6 PLASTICITY IN LARGE DEFORMATION REGIME........................................................................ 518 9.7 LARGE-DEFORMATION PLASTICITY BASED ON THE MULTIPLICATIVE

DECOMPOSITION OF THE DEFORMATION GRADIENT.......................................................... 518 9.7.1 Kinematic Tensors................................................................................................................... 518

CONTENTS

9.7.1.1 Deformation and Strain Tensors...............................................................................520 9.7.1.2 Area and Volume Elements Deformation...............................................................524 9.7.1.3 The Spatial Velocity Gradient ....................................................................................525 9.7.1.4 The Oldroyd Rate ........................................................................................................528 9.7.1.5 The Cotter-Rivlin Rate ................................................................................................529

9.7.2 The Stress Tensors...................................................................................................................531 9.7.2.1 Stress Tensor Rates......................................................................................................532

9.7.3 The Helmholtz Free Energy...................................................................................................533 9.7.3.1 Decoupling the Helmholtz Free Energy ..................................................................533 9.7.3.2 The Objectivity Principle for the Helmholtz Free Energy....................................533 9.7.3.3 The Isotropic Helmholtz Free Energy .....................................................................534 9.7.3.4 The Rate of Change of the Isotropic Helmholtz Free Energy.............................534

9.7.4 The Plastic Potential and the Yield Criterion ......................................................................536 9.7.5 The Dissipation and the Constitutive Equation..................................................................537 9.7.6 Evolution of the Internal Variables.......................................................................................538 9.7.7 The Elastoplastic Tangent Stiffness Tensors.......................................................................539

9.7.7.1 The Elastoplastic Tangent Stiffness Tensor ............................................................540 9.7.8 The Hyperelastoplastic Model with von Mises Yield Criterion........................................542

9.7.8.1 The Helmholtz Free Energy ......................................................................................542 9.7.8.2 The Stress Tensor ........................................................................................................543 9.7.8.3 Formulation Considering the Transformation as an Isochoric

Transformation..............................................................................................................544 9.7.8.4 The Rate of Change of the Helmholtz Free Energy ..............................................545 9.7.8.5 Yield Criterion and Evolution of the Internal Variables .......................................546

10 THERMOELASTICITY ................................................................................................ 547

10.1 THERMODYNAMIC POTENTIALS .................................................................................................547 10.1.1 The Specific Internal Energy ...............................................................................................548 10.1.2 The Specific Helmholtz Free Energy .................................................................................548 10.1.3 The Specific Gibbs Free Energy .........................................................................................549 10.1.4 The Specific Enthalpy ...........................................................................................................550

10.2 THERMOMECHANICAL PARAMETERS .........................................................................................552 10.2.1 Isothermal and Isentropic Processes ..................................................................................552 10.2.2 Specific Heats and Latent Heat Tensors ............................................................................553

10.3 LINEAR THERMOELASTICITY .......................................................................................................556 10.3.1 Linearization of the Constitutive Equations......................................................................556

10.3.1.1 The Linearized Piola-Kirchhoff Stress Tensor .....................................................557 10.3.1.2 The Linearized Heat Flux Vector............................................................................558 10.3.1.3 Linearized Entropy ....................................................................................................560 10.3.1.4 The Helmholtz Free Energy Approach .................................................................560 10.3.1.5 Linearization of the Constitutive Equations..........................................................561 10.3.1.6 Linear Thermoelasticity in a Small Deformation Regime ...................................561 10.3.1.7 Linear Thermoelasticity in a Small Deformation Regime ...................................562

10.4 THE DECOUPLED THERMO-MECHANICAL PROBLEM IN A SMALL DEFORMATION REGIME ............................................................................................................................................565

10.4.1 The Purely Thermal Problem...............................................................................................567 10.4.2 The Purely Mechanical Problem..........................................................................................568

10.5 THE CLASSICAL THEORY OF THERMOELASTICITY IN FINITE STRAIN (LARGE DEFORMATION REGIME) .............................................................................................................569

10.5.1 The Coupled Heat Flux Equation.......................................................................................570 10.5.2 The Specific Helmholtz Free Energy .................................................................................572

10.6 THERMOELASTICITY BASED ON THE MULTIPLICATIVE DECOMPOSITION OF THE DEFORMATION GRADIENT..........................................................................................................573

10.6.1 Kinematic Tensors.................................................................................................................574 10.6.2 The Stress Tensor ..................................................................................................................576

XV


10.6.3 Area and Volume Elements................................................................................................. 576 10.6.4 Isotropic Materials................................................................................................................. 578 10.6.5 The Constitutive Equations ................................................................................................. 579

10.6.5.1 The Constitutive Equation for Energy .................................................................. 579 10.6.5.2 The Constitutive Equations for Stress ................................................................... 580 10.6.5.3 The Constitutive Equation for Entropy ................................................................ 582

10.7 THERMOPLASTICITY IN A SMALL DEFORMATION REGIME................................................... 583 10.7.1 The Specific Helmholtz Free Energy ................................................................................. 583 10.7.2 Internal Energy Dissipation................................................................................................. 584

11 DAMAGE MECHANICS ................................................................................................587

11.1 INTRODUCTION.............................................................................................................................. 587 11.2 THE ISOTROPIC DAMAGE MODEL IN A SMALL DEFORMATION REGIME .......................... 588

11.2.1 Description of the Isotropic Damage Model in Uniaxial Cases .................................... 588 11.2.1.1 The Constitutive Equation....................................................................................... 589

11.2.2 The Three-Dimensional Isotropic Damage Model.......................................................... 590 11.2.2.1 Helmholtz Free Energy ............................................................................................ 590 11.2.2.2 Internal Energy Dissipation and the Constitutive Equations ............................ 591 11.2.2.3 Ingredients of the Damage Model...................................................................... 593 11.2.2.4 The Hardening/Softening Law ............................................................................... 599

11.2.3 The Elastic-Damage Tangent Stiffness Tensor ................................................................ 601 11.2.4 The Energy Norms................................................................................................................ 602

11.2.4.1 The Symmetrical Damage Model (Tension-Compression) Model I.............. 602 11.2.4.2 The Tension-Only Damage Model Model II .................................................... 603 11.2.4.3 The Non-Symmetrical Damage Model Model III ............................................ 604

11.3 THE GENERALIZED ISOTROPIC DAMAGE MODEL................................................................. 605 11.3.1 The Strain Energy Function................................................................................................. 606 11.3.2 Spherical and Deviatoric Effective Stress.......................................................................... 607 11.3.3 Thermodynamic Considerations ......................................................................................... 607 11.3.4 The Elastic-Damage Tangent Stiffness Tensor ................................................................ 608

11.4 THE ELASTOPLASTIC-DAMAGE MODEL IN A SMALL DEFORMATION REGIME ................ 609 11.4.1 The Elasto-Plastic Damage Model by Sim&Ju (1987) in a Small Deformation

Regime..................................................................................................................................... 610 11.4.1.1 Helmholtz Free Energy ............................................................................................ 610 11.4.1.2 Internal Energy Dissipation. Constitutive Equations. Thermodynamic

Considerations............................................................................................................... 611 11.4.1.3 Damage Characterization ......................................................................................... 612 11.4.1.4 The Elastic-Damage Tangent Stiffness Tensor .................................................... 612 11.4.1.5 Characterization of the Plastic Response. The Elastoplastic-Damage

Tangent Stiffness Tensor....................................................................................... 613 11.5 THE TENSILE-COMPRESSIVE PLASTIC-DAMAGE MODEL...................................................... 615

11.5.1 Helmholtz Free Energy ........................................................................................................ 616 11.5.2 Damage Characterization ..................................................................................................... 617 11.5.3 Evolution of the Damage Parameters................................................................................ 618 11.5.4 Evolution of the Plastic Strain Tensor............................................................................... 619 11.5.5 Internal Energy Dissipation................................................................................................. 619

11.6 DAMAGE IN A LARGE DEFORMATION REGIME ...................................................................... 621 11.6.1 Gurtin & Francis One-Dimensional Model ..................................................................... 622 11.6.2 The Rate Independent 3D Elastic-Damage Model.......................................................... 622 11.6.3 The Damage Variable. Damage Evolution........................................................................ 623 11.6.4 The Plastic-Damage Model by Sim & Ju (1989) ............................................................ 624

11.6.4.1 Specific Helmholtz Free Energy ............................................................................. 624 11.6.4.2 Internal Energy Dissipation. Constitutive Equations. Thermodynamic

Considerations............................................................................................................... 624 11.6.4.3 Damage Characterization ......................................................................................... 626

XVI

CONTENTS

XVII

11.6.4.4 The Hyperelastic-Damage Tangent Stiffness Tensor ..........................................626 11.6.4.5 Characterization of the Plastic Response. The Effective Elastoplastic-

Damage Tangent Stiffness Tensor .............................................................................627 11.6.4.6 The Elastoplastic-Damage Tangent Stiffness Tensor..........................................628

11.6.5 The Plastic-Damage Model by Ju(1989).............................................................................628 11.6.5.1 Helmholtz Free Energy.............................................................................................629 11.6.5.2 Internal Energy Dissipation. Constitutive Equation. Thermodynamic

Considerations...............................................................................................................629 11.6.5.3 Characterization of Damage. The Tangent Damage Hyperelasticity Tensor ..630 11.6.5.4 The Elastic-Damage Tangent Stiffness Tensor ....................................................630 11.6.5.5 Characterization of Plastic Response. The elastoplastic Tangent Stiffness

Tensor. ............................................................................................................................631 11.6.5.6 The Elastoplastic-Damage Tangent Stiffness Tensor..........................................632

12 INTRODUCTION TO FLUIDS.................................................................................... 635

12.1 INTRODUCTION ..............................................................................................................................635 12.2 FLUIDS AT REST AND IN MOTION...............................................................................................636

12.2.1 Fluids at Rest ..........................................................................................................................636 12.2.2 Fluids in Motion.....................................................................................................................637

12.3 VISCOUS AND NON-VISCOUS FLUIDS.........................................................................................637 12.3.1 Non-Viscous Fluids (Perfect Fluids) ..................................................................................638 12.3.2 Viscous Fluids.........................................................................................................................638

12.4 LAMINAR TURBULENT FLOW.......................................................................................................639 12.5 PARTICULAR CASES ........................................................................................................................640

12.5.1 Incompressible Fluids ...........................................................................................................640 12.5.2 Irrotational Flow ....................................................................................................................641 12.5.3 Steady Flow.............................................................................................................................641

12.6 NEWTONIAN FLUIDS .....................................................................................................................642 12.6.1 The Stokes Condition...........................................................................................................645

12.7 STRESS, DISSIPATED AND RECOVERABLE POWERS.................................................................645 12.8 THE FUNDAMENTAL EQUATIONS FOR NEWTONIAN FLUIDS...............................................647

12.8.1 The Navier-Stokes-Duhem Equations of Motion............................................................648 12.8.1.1 Alternative Form of the Fundamental Equations for Newtonian Fluids .........648 12.8.1.2 The Fundamental Equations for Incompressible Newtonian Fluid..................649

12.8.2 The Navier-Stokes Equations of Motion...........................................................................650 12.8.3 The Euler Equations of Motion..........................................................................................650

12.8.3.1 Non-Viscous and Incompressible Fluids...............................................................651 12.8.3.2 Bernoullis Equation..................................................................................................652

12.8.4 The Equation of Vorticity ....................................................................................................653 BIBLIOGRAPHY .................................................................................................................... 659 INDEX .................................................................................................................................. 667

Preface

The Continuum Mechanics is a key subject to several degrees based on physical science, such as: Civil Engineering, Industrial Engineering, Meteorology, Magnetism, Oceanography, Aerodynamics, Hydrodynamics, Marine Engineering, etc.

This book grew out of notes for the course Introduction to Continuum Mechanics of the career of Civil Engineering of the University of Castilla-La Mancha (Spain), and is intended for students who are initiating a university degree based on physical science, and is also intended for PhD students as well researchers.

In order to provide greater clarity for students, this book presents a thorough detail at the time of the demonstration of the equations. At the time of writing the book, the author has had a big concern for trying to unify the existing nomenclature, and to this end has consulted numerous articles and books on the subject. With respect to the notation, the developments of the equations are indiscriminately presented in tensorial, inditial and Voigt notations. Another aspect is that the book is self-contained, so that the concepts used are defined in the text.

Finally, I would like to express my gratitude to: Houzeaux (Guillaume), Vzquez (Mariano), Gallego (Inmaculada), Pulido (Loli), Bentez (Jos Mara), Casati (Mara Jesus), Vlez (Eduardo), Solares (Cristina), Olivares (Miguel ngel), Escobedo (Fernando), Simarro (Gonzalo), Sanz (Ana), for aid to the revision of the first edition in Spanish. I would also like to thank Toby Wakely for reviewing the English.

I would also to thank two Professors who marked my teaching and research career: Prof. Xavier Oliver and Prof. Wilson Venturini (in memoriam).

Eduardo W. V. Chaves

Ciudad Real-Spain, October 2012

Preface

XIX

Abbreviations

IBVP Initial Boundary Value Problem BVP Boundary Value Problem FEM Finite Element Method BEM Boundary Element Method FDM Finite Difference Method Latin i.e. id est that is et al. et alii and the others e.g. exempli gratia for example etc. et cetera and so on v., vs. versus versus viz. vidilicet namely

Abbreviations

XXI

Operators and Symbols

2xx x Macaulay bracket

x Euclidian norm of x )(xTr trace of )(x

T)(x transpose of )(x 1)( x inverse of )(x Tx)( inverse of the transpose of )(x sym)(x symmetric part of )(x skew)(x antisymmetric (skew-symmetric) part of )(x sph)(x spherical part of )(x dev)(x deviatoric part of )(x

x module of x > @> @x jump of x scalar product x{xdet determinant of x

x{x DtD material time derivative of x

)(xcof cofactor of x ; xAdj adjugate of x xTr trace of x

: double scalar product (or double contraction or double dot product) 2 Scalar differential operator

tensorial product )(x{x grad gradient of x )(x{x div divergence of x

vector product (or cross product)

Operators and Symbols

XXIII

SI-Units

length m - metro energy, work, heat NmJ - Joules mass kg - kilogram power W

sJ { watt

time s - second temperature K - Kelvin permeability 2m

velocity sm dynamic viscosity sPa u

acceleration 2sm mass flux

smkg

2

energy NmJ - Joules energy flux smJ2

force N - Newton thermal conductivity mKW

pressure, stress 2mNPa { - Pascal mass density 3m

kg

Prefix Symbol n10 Prefix Symbol n10 pico p 1210 kilo k 310 nano K 910 Mega M 610 micro P 610 Giga G 910 mili m 310 Tera T 1210 centi c 210 deci d 10

SI-Units

XXV

Continuum Mechanics

Introduction

1 Mechanics

Broadly speaking, Mechanics is the branch of physics that studies the behavior of a body when it is subjected to forces, (e.g. deformation) and how it evolves over time. In general, Mechanics can be classified into: Theoretical Mechanics; Applied Mechanics; Computational Mechanics.

Theoretical Mechanics establishes the laws that govern a particular physical problem based on fundamental principles. Applied Mechanics transfers theoretical knowledge to use it in scientific and engineering problems. Computational Mechanics solves problems by simulation with numerical tools implemented in the computer. In this book we focus our attention to the Theoretical and Applied Mechanics.

2 What is Continuum Mechanics?

Broadly speaking, Continuum Mechanics is the branch of Mechanics that studies motion (deformation) of a medium that consists of matter subjected to forces. For example, how would a wooden and a concrete beam deform when the same force is applied to them? Another example we can look at is fluids, e.g. for a given pressure, how does water (or oil) flow in a pipeline?

2.1 Hypothesis of Continuum Mechanics

As we know, a physical body consists of small molecules (an agglomeration of two or more atoms). Then, by means of sophisticated experiments, we can observe that these constituents are not distributed homogeneously, that is, there are gaps (voids) between them. However, within the scope of Continuum Mechanics these phenomenological

Introduction

1, Notes on Continuum Mechanics, Lecture Notes on Numerical Methods in Engineering and Sciences 4, DOI 10.1007/978-94-007-5986-2_1,

International Center for Numerical Methods in Engineering (CIMNE), 2013

E.W.V. Chaves


2

characteristics are ignored. For example, if we are dealing with a fluid in Continuum Mechanics, the properties: mass density, pressure and velocity are assumed to be continuous function. Treating a system of molecules as a continuous medium is valid if we compare the mean free path of molecules (/ ) (average distance particles travel before colliding with each other) with the characteristic physical length scale ( C" ). For example, for solids and liquids we have cm710|/ and for gases cm610|/ , Chung (1996). Then the ratio

C"/ is known as the Knudsen number ( Kn ). If this number is much smaller than

unity, the domain can be treated as a continuum; otherwise we must use statistical mechanics to obtain the governing equations of the problem whereby we can establish that:

approach cmicroscopi1

approach cmacroscopi1

!/

/

"

"Kn

Kn

The fundamental hypothesis in Continuum Mechanics is that the matter of which the medium is made up is continuously distributed and that the variables involved in the problem (e.g. velocity, acceleration, pressure, mass density, etc.) are continuous functions. Then, by means of approximations or additional equations to that initially proposed for the problem, we can characterize a continuum with discontinuous variables associated with the problem, e.g. fracture problem and shock waves among others.

2.2 The Continuum

In general, when we apply force to solids they are able to recover their original states when said force is removed. However, this is not the case with fluids, i.e. solids and fluids apparently act very differently. Therefore, traditionally, continuum mechanics has been divided into two groups: solids and fluids (liquids and gases). As we will see throughout this book, the fundamental equations of Continuum Mechanics are the same for both of these.

For many decades, solid and fluid mechanics have been treated independently from each other. However, nowadays, it is not advisable to work like this. Firstly, it is necessary to simulate more complex materials, e.g. materials that have characteristics of solids and fluids simultaneously. These materials, besides presenting elastic properties, (obeying the constitutive law for solids), also exhibit characteristics of fluids due to their viscosity, for example: viscoelastic materials. Secondly, the need to simulate the problem of fluid-solid interaction has improved the relationship between fluids and solids.

Recently, a third branch of continuum mechanics has emerged, which is related to multiphysics problems, characterized by phase change, e.g. from solid to liquid phase or vice versa, and which includes mechanical systems that transcend classical mechanical boundary of solids and fluids. Then, traditionally, continuum mechanics can be divided into:

csMultiphysiGasesLiquids

Fluids

Solids

Continuum The

INTRODUCTION

3

3 Scales of Material Studies

According to Willam(2000), materials science can be studied on different scales, (see Figure 1), namely: Metric level

At this level, we include most problems posed in Civil, Mechanical, Aerospace Engineering.

Millimeter level At this level, it may enroll the specimen used to measure the material mechanical properties in the laboratory.

Micrometer level Micro-structural characteristic, such as micro-defects and cement hydration products, are observed at this scale.

Nanometer level At this level, we contemplate atomic and molecular processes.

Figure 1: Multiscale in Material Mechanics, Willam(2000).

3.1 Scale Study of Continuum Mechanics

The continuum mechanics is raised at a macroscopic level. That is, the variables of the problem at a macroscopic level are considered as being the average of these variables at a

Nano Mechanics

Meso Mechanics

Macro Mechanics

Structural Mechanics

Micro Mechanics

m0101u

m3101 u

m6101 u

m9101 u


4

mesoscale level. Let us take, for example, blood, which can be treated in different ways, depending on the scale under consideration. At a m610 scale, we consider blood flows around a blood cell where the deformation of the cell walls is taken into consideration. Then, at a m410 scale, we can consider the fluid flow through a set of blood cells, which thus allows us to observe the fluid effects on cells. Next, at a m310 scale (macroscopic level), we can consider the fluid flow through arteries or veins (ignoring the individual cells) as being a fluid with certain macroscopic properties (e.g. velocity, pressure, etc.), (see Figure 2).

Figure 2: Scale levels in blood.

Another example we can use is a material made up of a mixture of materials such as concrete, which is fundamentally formed by mixing cement, aggregates, and water. At the

m910 scale, we can distinguish the atomic structure of the cement and aggregates. Then, at a m610 scale it is possible to identify individual cement grains before hydration and grains of calcium silicate and calcium hydroxide can be appreciated, upon hydration. Finally, at the m310 millimeter scale, we can distinguish individually each of the aggregates and pores (gaps). Note, at this level, the interaction between parts of cement and aggregates is important. On the m10 metric scale and on the m1 laboratory scale, the concrete internal structure can be examined to ensure that its properties are identical in all directions and at all its points, which is what characterizes a homogenous and isotropic material. Another example for understanding in which scale continuum mechanics is raised is by measuring mass density (S ), which is a macroscopic variable for continuum mechanics.

Blood flow in an artery

(macro scale 10-3m)

Meso scale - 10-4 m

Micro scale - 10-5 m

INTRODUCTION

5

We can determine the mass density of a cube (with sides a ) by dividing its total mass by its volume. So, let us consider a new cube (with sides ac ) whose volume is less than the first one. In Figure 3(b) we can observe that, depending on the position of the new cube, we can obtain different values for mass density, as different position contain different amounts of matter and voids. That is, if we can vary the a -dimension from a very small size, we will notice that the mass density value will oscillate, (see Figure 4). However, there will be a a -dimension region in which the mass density value maintains constant. The continuum mechanics is raised into this interval.

Figure 3: Mass density measurement.

It is possible to extend the continuum mechanics to other scales by adding certain hypothesis, such as the so-called scale effect, but this is not a subject covered in this book.

Figure 4: Mass density. )(alog

S

Scale of Continuum Mechanics

a

a

ac

ac cube with less matter

(more empty)

cube with more matter (less empty)

(a) (b)

. . . ...

...

...

...

...


6

4 The Initial Boundary Value Problem (IBVP)

Continuum mechanics, based on certain principles, attempts to formulate the equations that govern given physical problems by means of partial differential equations. To these we must add the boundary and initial conditions in order to guarantee the uniqueness of the problem. This set of partial differential equations and the boundary initial conditions make up the Initial Boundary Value Problem (IBVP), (see Figure 5). With a static or quasi-static problem the IBVP becomes a Boundary Value Problem (BVP) where the initial conditions are redundant.

Figure 5: Statement and solution of the problem.

4.1 Solving the IBVP

Once the physical problem is stated, it can be solved and the IBVP solution can be analytical (exact solution), or numerical (approximated solution), (see Figure 5). In practice, obtaining the analytical solution of the IBVP is very difficult or even impossible because of the problem complexity (e.g. due to its geometry, forces, or boundaries),

Notes in Continuum Mechanics

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