Notes

on Continuum Mechanics

Eduardo W.V. Chaves

Lecture Notes

Lecture Noteson Numerical Methodsin Engineering and Sciences

 

Notes on Continuum Mechanics

 

 

Lecture Notes on Numerical Methods inEngineering and Sciences

Aims and Scope of the Series

This series publishes text books on topics of general interest in the field of computationalengineering sciences.

The books will focus on subjects in which numerical methods play a fundamental role forsolving problems in engineering and applied sciences. Advances in finite element, finitevolume, finite differences, discrete and particle methods and their applications are examplesof the topics covered by the series.

The main intended audience is the first year graduate student. Some books define thecurrent state of a field to a highly specialised readership; others are accessible to final yearundergraduates, but essentially the emphasis is on accessibility and clarity.

The books will be also useful for practising engineers and scientists interested in state ofthe art information on the theory and application of numerical methods.

Series EditorEugenio OñateInternational Center for Numerical Methods in Engineering (CIMNE)School of Civil Engineering, Technical University of Catalonia (UPC), Barcelona, Spain

Editorial BoardFrancisco Chinesta, Ecole Nationale Supérieure d'Arts et Métiers, Paris, FranceCharbel Farhat, Stanford University, Stanford, USACarlos Felippa, University of Colorado at Boulder, Colorado, USAAntonio Huerta, Technical University of Catalonia (UPC), Barcelona, SpainThomas J.R. Hughes, The University of Texas at Austin, Austin, USASergio R. Idelsohn, CIMNE-ICREA, Barcelona, SpainPierre Ladeveze, ENS de Cachan-LMT-Cachan, FranceWing Kam Liu, Northwestern University, Evanston, USAXavier Oliver, Technical University of Catalonia (UPC), Barcelona, SpainManolis Papadrakakis, National Technical University of Athens, GreeceJacques Périaux, CIMNE-UPC Barcelona, Spain & Univ. of Jyväskylä, FinlandBernhard Schrefler , Università degli Studi di Padova, Padova, ItalyGenki Yagawa, Tokyo University, Tokyo, JapanMingwu Yuan, Peking University, China

Titles:

1. E. Oñate, Structural Analysis with the Finite Element Method.Linear Statics. Volume 1. Basis and Solids, 2009

2. K. Wiśniewski, Finite Rotation Shells. Basic Equations andFinite Elements for Reissner Kinematics, 2010

3. E. Oñate, 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. ChavesSchool of Civil EngineeringUniversity of Castilla-La ManchaCiudad Real, Spain

 

 

ISBN: 978-94-007-5985-5 (HB)ISBN: 978-94-007-5986-2 (e-book)

Depósito legal: B-29347-2012

 A C.I.P. Catalogue record for this book is available from the Library of Congress

Lecture Notes Series Manager: Mª Jesús Samper, CIMNE, Barcelona, Spain 

Cover page: Pallí Disseny i Comunicació, www.pallidisseny.com 

Printed by: Artes Gráficas Torres S.A.,Morales 17, 08029 Barcelona, Españawww.agraficastorres.es

Printed on elemental chlorine-free paper

Notes on Continuum MechanicsEduardo W.V. Chaves

First edition, 2013

 International Center for Numerical Methods in Engineering (CIMNE), 2013Gran Capitán s/n, 08034 Barcelona, Spainwww.cimne.com

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise,without written permission from the Publisher, with the exception of any material suppliedspecifically for the purpose of being entered and executed on a computer system, for exclusiveuse by the purchaser of the work.

 

 

 To my Parents

 

 

 

Contents

PREFACE .......................................................................................................................................................XIX  ABBREVIATIONS..........................................................................................................................................XXIOPERATORS AND S YMBOLS....................................................................................................................XXIIISI-UNITS ..................................................................................................................................................... XX

INTRODUCTION ................................................................................................................. 11 MECHANICS...............................................................................................................................................12  W HAT IS CONTINUUM MECHANICS?....................................................................................................1

2.1 Hypothesis of Continuum Mechanics ............... ................ ................ ............... ................ ........... 12.2 The Continuum ................. ................ ................ ................. ................ ................ ........................... ..2

3 SCALES OF M ATERIAL S TUDIES.............................................................................................................33.1 Scale Study of Continuum Mechanics ................ ................ ................. ................ ................ ........3

4  THE INITIAL BOUNDARY V  ALUE PROBLEM (IBVP).........................................................................64.1 Solving the IBVP.............................................................................................................................64.2 Simplifying the IBVP......................................................................................................................7

1  TENSORS........................................................................................................................ .....91.1 INTRODUCTION.....................................................................................................................................91.2  ALGEBRAIC OPERATIONS WITH V ECTORS ....................................................................................101.3 COORDINATE S YSTEMS .....................................................................................................................16

1.3.1 Cartesian Coordinate System....................................................................................................161.3.2 Vector Representation in the Cartesian Coordinate System...............................................171.3.3 Einstein Summation Convention (Einstein Notation) .............. ............... ............... ............ 20

1.4 INDICIAL NOTATION .........................................................................................................................201.4.1 Some Operators..........................................................................................................................22

1.4.1.1 Kronecker Delta.............................................................................................................221.4.1.2 Permutation Symbol......................................................................................................23

1.5  ALGEBRAIC OPERATIONS WITH TENSORS.....................................................................................281.5.1 Dyadic ................ ................. ................ ................ ................. ................ ................ ........................ 28

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

1.5.2 Properties of Tensors................................................................................................................341.5.2.1 Tensor Transpose ............... ................ ................. ................ ................ ................. .........341.5.2.2 Symmetry and Antisymmetry.......................................................................................361.5.2.3 Cofactor Tensor. Adjugate of a Tensor .............. ................ ................ ............... ........421.5.2.4 Tensor Trace...................................................................................................................421.5.2.5 Particular Tensors .............. ................. ................ ................ ................. ................ ..........441.5.2.6 Determinant of a Tensor .............. ................ ............... ................ ............... ................ ..451.5.2.7 Inverse of a Tensor........................................................................................................481.5.2.8 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........................................................................531.5.3 Transformation Law of the Tensor Components................................................................54

1.5.3.1 Component Transformation Law in Two Dimensions (2D).................................611.5.4 Eigenvalue and Eigenvector Problem....................................................................................65

1.5.4.1 The Orthogonality of the Eigenvectors.....................................................................671.5.4.2 Solution of the Cubic Equation...................................................................................69

1.5.5 Spectral Representation of Tensors........................................................................................721.5.6 Cayley-Hamilton Theorem.......................................................................................................761.5.7 Norms of Tensors ................ ................. ................ ................ ................. ................ ................... 781.5.8 Isotropic and Anisotropic Tensor...........................................................................................791.5.9 Coaxial Tensors..........................................................................................................................801.5.10 Polar Decomposition..............................................................................................................811.5.11 Partial Derivative with Tensors.............................................................................................83

1.5.11.1 Partial Derivative of Invariants ................ ................ ................ ................ ................ . 851.5.11.2 Time Derivative of Tensors.......................................................................................86

1.5.12 Spherical and Deviatoric Tensors.........................................................................................861.5.12.1 First Invariant of the Deviatoric Tensor..................................................................871.5.12.2 Second Invariant of the Deviatoric Tensor.............................................................871.5.12.3 Third Invariant of Deviatoric Tensor .............. ................ ............... ................ ......... 89

1.6  THE TENSOR -V  ALUED TENSOR FUNCTION................................................................................. 911.6.1 The Tensor Series......................................................................................................................911.6.2 The Tensor-Valued Isotropic Tensor Function ................. ................ ................ ................. . 921.6.3 The Derivative of the Tensor-Valued Tensor Function .............. ............... ............... ......... 94

1.7  THE V OIGT NOTATION ....................................................................................................................961.7.1 The Unit Tensors in Voigt Notation......................................................................................971.7.2 The Scalar Product in Voigt Notation....................................................................................981.7.3 The Component Transformation Law in Voigt Notation..................................................991.7.4 Spectral Representation in Voigt Notation..........................................................................1001.7.5 Deviatoric Tensor Components in Voigt Notation...........................................................101

1.8  TENSOR FIELDS ................................................................................................................................1051.8.1 Scalar Fields ................ ................ ................. ................ ................ ................. ........................... . 1061.8.2 Gradient.....................................................................................................................................1061.8.3 Divergence................................................................................................................................1111.8.4 The Curl ................ ................ ................ ................. ................ ................ ............................ ....... 1131.8.5 The Conservative Field...........................................................................................................115

1.9  THEOREMS INVOLVING INTEGRALS ............................................................................................1171.9.1 Integration by Parts.................................................................................................................1171.9.2 The Divergence Theorem ............... ................ ............... ................ ............... ................ .........1171.9.3 Independence of Path.............................................................................................................1201.9.4 The Kelvin-Stokes’ Theorem.................................................................................................1211.9.5 Green’s Identities.....................................................................................................................122

 Appendix A : A GRAPHICAL REPRESENTATION OF A SECOND-ORDER TENSOR.....125 A.1 PROJECTING A SECOND-ORDER TENSOR ONTO A P ARTICULAR 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 R EPRESENTATION OF AN ARBITRARY SECOND-ORDER TENSOR ...................130 A.2.1 Graphical Representation of a Symmetric Second-Order Tensor (Mohr’s Circle)...... 134

 A.3  THE TENSOR ELLIPSOID................................................................................................................138 A.4 GRAPHICAL R EPRESENTATION OF THE SPHERICAL AND DEVIATORIC P ARTS..................139

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

III