Introduction to Continuum Mechanics - kashanu.ac.ir

549 Pages·2009·4.19 MB·English

Introduction to
Continuum Mechanics
Fourth Edition
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Introduction to
Continuum Mechanics
Fourth Edition
W. Michael Lai
Professor Emeritus of Mechanical Engineering and Orthopedic Bioengineering
Columbia University, New York, NY, USA
David Rubin
Senior Scientist
Weidlinger Associates, Inc., New York, NY, USA
Erhard Krempl
Professor Emeritus of Mechanical Engineering
Rensselaer Polytechnic Institute, Troy, NY, USA
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Table of Contents
Preface to the Fourth Edition...........................................................................................................................xiii
Chapter 1 Introduction...........................................................................................1
1.1 Introduction...............................................................................................................................1
1.2 What Is Continuum Mechanics?..............................................................................................1
Chapter 2 Tensors.................................................................................................3
Part A: Indicial Notation..................................................................................................................3
2.1 Summation Convention, Dummy Indices................................................................................3
2.2 Free Indices...............................................................................................................................4
2.3 The Kronecker Delta.................................................................................................................5
2.4 The Permutation Symbol..........................................................................................................6
2.5 Indicial Notation Manipulations...............................................................................................7
Problems for Part A............................................................................................................................8
Part B: Tensors..................................................................................................................................9
2.6 Tensor: A Linear Transformation.............................................................................................9
2.7 Components of a Tensor.........................................................................................................11
2.8 Components of a Transformed Vector...................................................................................14
2.9 Sum of Tensors.......................................................................................................................16
2.10 Product of Two Tensors.........................................................................................................16
2.11 Transpose of a Tensor............................................................................................................18
2.12 Dyadic Product of Vectors.....................................................................................................19
2.13 Trace of a Tensor....................................................................................................................20
2.14 Identity Tensor and Tensor Inverse........................................................................................20
2.15 Orthogonal Tensors.................................................................................................................22
2.16 Transformation Matrix Between Two Rectangular Cartesian Coordinate
Systems....................................................................................................................................24
2.17 Transformation Law for Cartesian Components of a Vector................................................26
2.18 Transformation Law for Cartesian Components of a Tensor...............................................27
2.19 Defining Tensor by Transformation Laws.............................................................................29
2.20 Symmetric and Antisymmetric Tensors.................................................................................31
2.21 The Dual Vector of an Antisymmetric Tensor......................................................................32
2.22 Eigenvalues and Eigenvectors of a Tensor............................................................................34
2.23 Principal Values and Principal Directions of Real Symmetric Tensors...............................38
2.24 Matrix of a Tensor with Respect to Principal Directions.....................................................39
2.25 Principal Scalar Invariants of a Tensor..................................................................................40
Problems for Part B...........................................................................................................................41
vi Table of Contents
Part C: Tensor Calculus.................................................................................................................45
2.26 Tensor-Valued Functions of a Scalar.....................................................................................45
2.27 Scalar Field and Gradient of a Scalar Function....................................................................47
2.28 Vector Field and Gradient of a Vector Function...................................................................50
2.29 Divergence of a Vector Field and Divergence of a Tensor Field........................................50
2.30 Curl of a Vector Field............................................................................................................52
2.31 Laplacian of a Scalar Field.....................................................................................................52
2.32 Laplacian of a Vector Field....................................................................................................52
Problems for Part C...........................................................................................................................53
Part D: Curvilinear Coordinates...................................................................................................54
2.33 Polar Coordinates....................................................................................................................54
2.34 Cylindrical Coordinates..........................................................................................................59
2.35 Spherical Coordinates.............................................................................................................61
Problems for Part D..........................................................................................................................67
Chapter 3 Kinematics of a Continuum...................................................................69
3.1 Description of Motions of a Continuum................................................................................69
3.2 Material Description and Spatial Description........................................................................72
3.3 Material Derivative.................................................................................................................74
3.4 Acceleration of a Particle.......................................................................................................76
3.5 Displacement Field.................................................................................................................81
3.6 Kinematic Equation for Rigid Body Motion.........................................................................82
3.7 Infinitesimal Deformation.......................................................................................................84
3.8 Geometrical Meaning of the Components of the Infinitesimal Strain
Tensor......................................................................................................................................88
3.9 Principal Strain........................................................................................................................93
3.10 Dilatation.................................................................................................................................93
3.11 The Infinitesimal Rotation Tensor.........................................................................................94
3.12 Time Rate of Change of a Material Element........................................................................95
3.13 The Rate of Deformation Tensor...........................................................................................95
3.14 The Spin Tensor and the Angular Velocity Vector...............................................................98
3.15 Equation of Conservation of Mass.........................................................................................99
3.16 Compatibility Conditions for Infinitesimal Strain Components..........................................101
3.17 Compatibility Condition for Rate of Deformation Components.........................................104
3.18 Deformation Gradient...........................................................................................................105
3.19 Local Rigid Body Motion.....................................................................................................106
3.20 Finite Deformation................................................................................................................106
3.21 Polar Decomposition Theorem.............................................................................................110
3.22 Calculation of Stretch and Rotation Tensors from the Deformation
Gradient.................................................................................................................................111
3.23 Right Cauchy-Green Deformation Tensor...........................................................................114
3.24 Lagrangian Strain Tensor.....................................................................................................118
Table of Contents vii
3.25 Left Cauchy-Green Deformation Tensor.............................................................................121
3.26 Eulerian Strain Tensor..........................................................................................................125
3.27 Change of Area Due to Deformation...................................................................................128
3.28 Change of Volume Due to Deformation..............................................................................129
3.29 Components of Deformation Tensors in Other Coordinates...............................................131
3.30 Current Configuration as the Reference Configuration.......................................................139
Appendix 3.1: Necessary and Sufficient Conditions for Strain Compatibility............................140
Appendix 3.2: Positive Definite Symmetric Tensors....................................................................143
Appendix 3.3: The Positive Definite Root of U2 ¼ D.................................................................143
Problems for Chapter 3...................................................................................................................145
Chapter 4 Stress and Integral Formulations of General
Principles..........................................................................................155
4.1 Stress Vector.........................................................................................................................155
4.2 Stress Tensor.........................................................................................................................156
4.3 Components of Stress Tensor...............................................................................................158
4.4 Symmetry of Stress Tensor: Principle of Moment of Momentum.....................................159
4.5 Principal Stresses..................................................................................................................162
4.6 Maximum Shearing Stresses.................................................................................................162
4.7 Equations of Motion: Principle of Linear Momentum........................................................168
4.8 Equations of Motion in Cylindrical and Spherical Coordinates.........................................170
4.9 Boundary Condition for the Stress Tensor..........................................................................171
4.10 Piola Kirchhoff Stress Tensors.............................................................................................174
4.11 Equations of Motion Written with Respect to the Reference Configuration.....................179
4.12 Stress Power..........................................................................................................................180
4.13 Stress Power in Terms of the Piola-Kirchhoff Stress Tensors............................................181
4.14 Rate of Heat Flow Into a Differential Element by Conduction..........................................183
4.15 Energy Equation....................................................................................................................184
4.16 Entropy Inequality.................................................................................................................185
4.17 Entropy Inequality in Terms of the Helmholtz Energy Function.......................................186
4.18 Integral Formulations of the General Principles of Mechanics..........................................187
Appendix 4.1: Determination of Maximum Shearing Stress and the Planes on
Which It Acts.........................................................................................................191
Problems for Chapter 4...................................................................................................................194
Chapter 5 The Elastic Solid...............................................................................201
5.1 Mechanical Properties...........................................................................................................201
5.2 Linearly Elastic Solid...........................................................................................................204
Part A: Isotropic Linearly Elastic Solid.....................................................................................207
5.3 Isotropic Linearly Elastic Solid............................................................................................207
5.4 Young’s Modulus, Poisson’s Ratio, Shear Modulus, and Bulk Modulus...........................209
5.5 Equations of the Infinitesimal Theory of Elasticity............................................................213
viii Table of Contents
5.6 Navier Equations of Motion for Elastic Medium................................................................215
5.7 Navier Equations in Cylindrical and Spherical Coordinates...............................................216
5.8 Principle of Superposition....................................................................................................218
A.1 Plane Elastic Waves............................................................................................................218
5.9 Plane Irrotational Waves.......................................................................................................218
5.10 Plane Equivoluminal Waves.................................................................................................221
5.11 Reflection of Plane Elastic Waves.......................................................................................226
5.12 Vibration of an Infinite Plate...............................................................................................229
A.2 Simple Extension, Torsion and Pure Bending.................................................................231
5.13 Simple Extension..................................................................................................................231
5.14 Torsion of a Circular Cylinder.............................................................................................234
5.15 Torsion of a Noncircular Cylinder: St. Venant’s Problem..................................................239
5.16 Torsion of Elliptical Bar.......................................................................................................240
5.17 Prandtl’s Formulation of the Torsion Problem....................................................................242
5.18 Torsion of a Rectangular Bar...............................................................................................246
5.19 Pure Bending of a Beam......................................................................................................247
A.3 Plane Stress and Plane Strain Solutions..........................................................................250
5.20 Plane Strain Solutions...........................................................................................................250
5.21 Rectangular Beam Bent by End Couples.............................................................................253
5.22 Plane Stress Problem............................................................................................................254
5.23 Cantilever Beam with End Load..........................................................................................255
5.24 Simply Supported Beam Under Uniform Load...................................................................258
5.25 Slender Bar Under Concentrated Forces and St. Venant’s Principle.................................260
5.26 Conversion for Strains Between Plane Strain and Plane Stress Solutions.........................262
5.27 Two-Dimensional Problems in Polar Coordinates...............................................................264
5.28 Stress Distribution Symmetrical About an Axis..................................................................265
5.29 Displacements for Symmetrical Stress Distribution in Plane Stress Solution....................265
5.30 Thick-Walled Circular Cylinder Under Internal and External Pressure.............................267
5.31 Pure Bending of a Curved Beam.........................................................................................268
5.32 Initial Stress in a Welded Ring............................................................................................270
5.33 Airy Stress Function ’ ¼ f(r) cos ny and ’ ¼ f(r) sin ny.................................................270
5.34 Stress Concentration Due to a Small Circular Hole in a Plate under Tension..................274
5.35 Stress Concentration Due to a Small Circular Hole in a Plate under Pure Shear.............276
5.36 Simple Radial Distribution of Stresses in a Wedge Loaded at the Apex..........................277
5.37 Concentrated Line Load on a 2-D Half-Space: The Flamont Problem..............................278
A.4 Elastostatic Problems Solved with Potential Functions.................................................279
5.38 Fundamental Potential Functions for Elastostatic Problems...............................................279
5.39 Kelvin Problem: Concentrated Force at the Interior of an Infinite Elastic Space.............290
5.40 Boussinesq Problem: Normal Concentrated Load on an Elastic Half-Space.....................293
5.41 Distributive Normal Load on the Surface of an Elastic Half-Space..................................296
5.42 Hollow Sphere Subjected to Uniform Internal and External Pressure...............................297
Table of Contents ix
5.43 Spherical Hole in a Tensile Field.........................................................................................298
5.44 Indentation by a Rigid Flat-Ended Smooth Indenter on an Elastic Half-Space................300
5.45 Indentation by a Smooth Rigid Sphere on an Elastic Half-Space......................................302
Appendix 5A.1: Solution of the Integral Equation in Section 5.45..............................................306
Problems for Chapter 5, Part A......................................................................................................309
Part B: Anisotropic Linearly Elastic Solid................................................................................319
5.46 Constitutive Equations for an Anisotropic Linearly Elastic Solid......................................319
5.47 Plane of Material Symmetry.................................................................................................321
5.48 Constitutive Equation for a Monoclinic Linearly Elastic Solid..........................................323
5.49 Constitutive Equation for an Orthotropic Linearly Elastic Solid........................................324
5.50 Constitutive Equation for a Transversely Isotropic Linearly Elastic Material...................325
5.51 Constitutive Equation for an Isotropic Linearly Elastic Solid............................................327
5.52 Engineering Constants for an Isotropic Linearly Elastic Solid...........................................328
5.53 Engineering Constants for a Transversely Isotropic Linearly Elastic Solid.......................329
5.54 Engineering Constants for an Orthotropic Linearly Elastic Solid......................................330
5.55 Engineering Constants for a Monoclinic Linearly Elastic Solid........................................332
Problems for Part B.........................................................................................................................333
Part C: Isotropic Elastic Solid Under Large Deformation......................................................334
5.56 Change of Frame...................................................................................................................334
5.57 Constitutive Equation for an Elastic Medium Under Large Deformation..........................338
5.58 Constitutive Equation for an Isotropic Elastic Medium......................................................340
5.59 Simple Extension of an Incompressible Isotropic Elastic Solid.........................................342
5.60 Simple Shear of an Incompressible Isotropic Elastic Rectangular Block..........................343
5.61 Bending of an Incompressible Isotropic Rectangular Bar...................................................344
5.62 Torsion and Tension of an Incompressible Isotropic Solid Cylinder.................................347
Appendix 5C.1: Representation of Isotropic Tensor-Valued Functions.......................................349
Problems for Part C.........................................................................................................................351
Chapter 6 Newtonian Viscous Fluid....................................................................353
6.1 Fluids.....................................................................................................................................353
6.2 Compressible and Incompressible Fluids.............................................................................354
6.3 Equations of Hydrostatics.....................................................................................................355
6.4 Newtonian Fluids..................................................................................................................357
6.5 Interpretation of l and m.......................................................................................................358
6.6 Incompressible Newtonian Fluid..........................................................................................359
6.7 Navier-Stokes Equations for Incompressible Fluids............................................................360
6.8 Navier-Stokes Equations for Incompressible Fluids in Cylindrical and Spherical
Coordinates............................................................................................................................364
6.9 Boundary Conditions............................................................................................................365
6.10 Streamline, Pathline, Steady, Unsteady, Laminar, and Turbulent Flow.............................365
6.11 Plane Couette Flow...............................................................................................................368