Introduction to Continuum Mechanics1

Front Cover

Introduction to Continuum Mechanics

Copyright Page

Table of Contents

Preface to the Fourth Edition

Chapter 1: Introduction

1.1 Introduction

1.2 What is Continuum Mechanics?

Chapter 2: Tensors

Part A: Indicial Notation

2.1 Summation Convention, Dummy Indices

2.2 Free Indices

2.3 The Kronecker Delta

2.4 The Permutation Symbol

2.5 Indicial Notation Manipulations

Problems For Part A

Part B: Tensors

2.6 Tensor: A Linear Transformation

2.7 Components of a Tensor

2.8 Components of a Transformed Vector

2.9 Sum of Tensors

2.10 Product of Two Tensors

2.11 Transpose of A Tensor

2.12 Dyadic Product of Vectors

2.13 Trace of A Tensor

2.14 Identity Tensor and Tensor Inverse

2.15 Orthogonal Tensors

2.16 Transformation Matrix Between Two Rectangular Cartesian Coordinate Systems

2.17 Transformation Law for Cartesian Components of A Vector

2.18 Transformation Law for Cartesian Components of a Tensor

2.19 Defining Tensor by Transformation Laws

2.20 Symmetric and Antisymmetric Tensors

2.21 The Dual Vector of an Antisymmetric Tensor

2.22 Eigenvalues and Eigenvectors of a Tensor

2.23 Principal Values and Principal Directions of Real Symmetric Tensors

2.24 Matrix of a Tensor with Respect to Principal Directions

2.25 Principal Scalar Invariants of a Tensor

Problems for Part B

Part C: Tensor Calculus

2.26 Tensor-Valued Functions of a Scalar

2.27 Scalar Field and Gradient of a Scalar Function

2.28 Vector Field and Gradient of a Vector Function

2.29 Divergence of a Vector Field and Divergence of a Tensor Field

2.30 Curl of a Vector Field

2.31 Laplacian of a Scalar Field

2.32 Laplacian of a Vector Field

Problems for Part C

Part D: Curvilinear Coordinates

2.33 Polar Coordinates

2.34 Cylindrical Coordinates

2.35 Spherical Coordinates

Problems for Part D

Chapter 3: Kinematics of a Continuum

3.1 Description of Motions of a Continuum

3.2 Material Description and Spatial Description

3.3 Material Derivative

3.4 Acceleration of a Particle

3.5 Displacement Field

3.6 Kinematic Equation for Rigid Body Motion

3.7 Infinitesimal Deformation

3.8 Geometrical Meaning of the Components of the Infinitesimal Strain Tensor

3.9 Principal Strain

3.10 Dilatation

3.11 The Infinitesimal Rotation Tensor

3.12 Time Rate of Change of a Material Element

3.13 The Rate of Deformation Tensor

3.14 The Spin Tensor and the Angular Velocity Vector

3.15 Equation of Conservation of Mass

3.16 Compatibility Conditions for Infinitesimal Strain Components

3.17 Compatibility Condition for Rate of Deformation Components

3.18 Deformation Gradient

3.19 Local Rigid Body Motion

3.20 Finite Deformation

3.21 Polar Decomposition Theorem

3.22 Calculation of Stretch and Rotation Tensors from the Deformation Gradient

3.23 Right Cauchy-Green Deformation Tensor

3.24 Lagrangian Strain Tensor

3.25 Left Cauchy-Green Deformation Tensor

3.26 Eulerian Strain Tensor

3.27 Change of Area Due to Deformation

3.28 Change of Volume Due to Deformation

3.29 Components of Deformation Tensors in Other Coordinates

3.30 Current Configuration as the Reference Configuration

Appendix 3.1: Necessary and Sufficient Conditions for Strain Compatibility

Appendix 3.2: Positive Definite Symmetric Tensors

Appendix 3.3: The Positive Definite Root of U2 = D

Problems for Chapter 3

Chapter 4: Stress and Integral Formulations of General Principles

4.1 Stress Vector

4.2 Stress Tensor

4.3 Components of Stress Tensor

4.4 Symmetry of Stress Tensor: Principle of Moment of Momentum

4.5 Principal Stresses

4.6 Maximum Shearing Stresses

4.7 Equations of Motion: Principle of Linear Momentum

4.8 Equations of Motion in Cylindrical and Spherical Coordinates

4.9 Boundary Condition for the Stress Tensor

4.10 Piola Kirchhoff Stress Tensors

4.11 Equations of Motion Written with Respect to the Reference Configuration

4.12 Stress Power

4.13 Stress Power in Terms of the Piola-Kirchhoff Stress Tensors

4.14 Rate of Heat Flow into a Differential Element by Conduction

4.15 Energy Equation

4.16 Entropy Inequality

4.17 Entropy Inequality in Terms of the Helmholtz Energy Function

4.18 Integral Formulations of the General Principles of Mechanics

Appendix 4.1: Determination of Maximum Shearing Stress and the Planes on Which It Acts

Problems for Chapter 4

Chapter 5: The Elastic Solid

5.1 Mechanical Properties

5.2 Linearly Elastic Solid

Part A: Isotropic Linearly Elastic Solid

5.3 Isotropic Linearly Elastic Solid

5.4 Young's Modulus, Poisson's Ratio, Shear Modulus, and Bulk Modulus

5.5 Equations of the Infinitesimal Theory of Elasticity

5.6 Navier Equations of Motion for Elastic Medium

5.7 Navier Equations in Cylindrical and Spherical Coordinates

5.8 Principle of Superposition

A.1 Plane Elastic Waves

5.9 Plane Irrotational Waves

5.10 Plane Equivoluminal Waves

5.11 Reflection of Plane Elastic Waves

5.12 Vibration of an Infinite Plate

A.2 Simple Extension, Torsion, and Pure Bending

5.13 Simple Extension

5.14 Torsion of a Circular Cylinder

5.15 Torsion of a Noncircular Cylinder: St. Venant's Problem

5.16 Torsion of Elliptical Bar

5.17 Prandtl's Formulation of the Torsion Problem

5.18 Torsion of a Rectangular Bar

5.19 Pure Bending of a Beam

A.3 Plane Stress and Plane Strain Solutions

5.20 Plane Strain Solutions

5.21 Rectangular Beam Bent by End Couples

5.22 Plane Stress Problem

5.23 Cantilever Beam with End Load

5.24 Simply Supported Beam Under Uniform Load

5.25 Slender Bar Under Concentrated Forces and St. Venant's Principle

5.26 Conversion for Strains Between Plane Strain and Plane Stress Solutions

5.27 Two-Dimensional Problems in Polar Coordinates

5.28 Stress Distribution Symmetrical About an Axis

5.29 Displacements for Symmetrical Stress Distribution in Plane Stress Solution

5.30 Thick-Walled Circular Cylinder Under Internal and External Pressure

5.31 Pure Bending of a Curved Beam

5.32 Initial Stress in a Welded Ring

5.33 Airy Stress Function phivf(r)cosntheta and phivf(r)sinntheta

5.34 Stress Concentration Due to a Small Circular Hole in a Plate Under Tension

5.35 Stress Concentration Due to a Small Circular Hole in a Plate Under Pure Shear

5.36 Simple Radial Distribution of Stresses in a Wedge Loaded at the Apex

5.37 Concentrated Line Load on a 2-D Half-Space: the Flamont Problem

A.4 Elastostatic Problems Solved with Potential Functions

5.38 Fundamental Potential Functions for Elastostatic Problems

5.39 Kelvin Problem: Concentrated Force at the Interior of an Infinite Elastic Space

5.40 Boussinesq Problem: Normal Concentrated Load on an Elastic Half-Space

5.41 Distributive Normal Load On The Surface Of An Elastic Half-Space

5.42 Hollow Sphere Subjected to Uniform Internal and External Pressure

5.43 Spherical Hole in a Tensile Field

5.44 Indentation by a Rigid Flat-Ended Smooth Indenter on an Elastic Half-Space

5.45 Indentation by a Smooth Rigid Sphere on an Elastic Half-Space

Appendix 5A.1: Solution of the Integral Equation in Section 5.45

Problems for Chapter 5, Part A, Sections 5.1-5.8

Problems for Chapter 5, Part A, Sections 5.9-5.12 (A.1)

Problems for Chapter 5, Part A, Sections 5.13-5.19 (A.2)

Problems for Chapter 5, Part A, Sections 5.20-5.37 (A.3)

Problems for Chapter 5, Part A, Sections 5.38-5.46 (A.4)

Part B: Anisotropic Linearly Elastic Solid

5.46 Constitutive Equations for an Anisotropic Linearly Elastic Solid

5.47 Plane of Material Symmetry

5.48 Constitutive Equation for a Monoclinic Linearly Elastic Solid

5.49 Constitutive Equation for an Orthotropic Linearly Elastic Solid

5.50 Constitutive Equation for a Transversely Isotropic Linearly Elastic Material

5.51 Constitutive Equation for an Isotropic Linearly Elastic Solid

5.52 Engineering Constants for an Isotropic Linearly Elastic Solid

5.53 Engineering Constants for a Transversely Isotropic Linearly Elastic Solid

5.54 Engineering Constants for an Orthotropic Linearly Elastic Solid

5.55 Engineering Constants for a Monoclinic Linearly Elastic Solid

Problems for Part B

Part C: Isotropic Elastic Solid Under Large Deformation

5.56 Change of Frame

5.57 Constitutive Equation for an Elastic Medium Under Large Deformation

5.58 Constitutive Equation for an Isotropic Elastic Medium

5.59 Simple Extension of an Incompressible Isotropic Elastic Solid

5.60 Simple Shear of an Incompressible Isotropic Elastic Rectangular Block

5.61 Bending of an Incompressible Isotropic Rectangular Bar

5.62 Torsion and Tension of an Incompressible Isotropic Solid Cylinder

Appendix 5C.1: Representation of Isotropic Tensor-Valued Functions

Problems for Part C

Chapter 6: Newtonian Viscous Fluid

6.1 Fluids

6.2 Compressible and Incompressible Fluids

6.3 Equations of Hydrostatics

6.4 Newtonian Fluids

6.5 Interpretation of lambda and mu

6.6 Incompressible Newtonian Fluid

6.7 Navier-Stokes Equations for Incompressible Fluids

6.8 Navier-Stokes Equations for Incompressible Fluids in Cylindrical and Spherical Coordinates

6.9 Boundary Conditions

6.10 Streamline, Pathline, Steady, Unsteady, Laminar, and Turbulent Flow

6.11 Plane Couette Flow

6.12 Plane Poiseuille Flow

6.13 Hagen-Poiseuille Flow

6.14 Plane Couette Flow of Two Layers of Incompressible Viscous Fluids

6.15 Couette Flow

6.16 Flow Near an Oscillating Plane

6.17 Dissipation Functions for Newtonian Fluids

6.18 Energy Equation for a Newtonian Fluid

6.19 Vorticity Vector

6.20 Irrotational Flow

6.21 Irrotational Flow of an Inviscid Incompressible Fluid of Homogeneous Density

6.22 Irrotational Flows as Solutions of Navier-Stokes Equation

6.23 Vorticity Transport Equation for Incompressible Viscous Fluid with a Constant Density

6.24 Concept of a Boundary Layer

6.25 Compressible Newtonian Fluid

6.26 Energy Equation in Terms of Enthalpy

6.27 Acoustic Wave

6.28 Irrotational, Barotropic Flows of an Inviscid Compressible Fluid

6.29 One-Dimensional Flow of a Compressible Fluid

6.30 Steady Flow of a Compressible Fluid Exiting a Large Tank Through a Nozzle

6.31 Steady Laminar Flow of a Newtonian Fluid in a Thin Elastic Tube: An Application to Pressure-Flow Relation in a Pulmonar...

Problems for Chapter 6

Chapter 7: The Reynolds Transport Theorem and Applications

7.1 Green's Theorem

7.2 Divergence Theorem

7.3 Integrals Over a Control Volume and Integrals Over a Material Volume

7.4 The Reynolds Transport Theorem

7.5 The Principle of Conservation of Mass

7.6 The Principle of Linear Momentum

7.7 Moving Frames

7.8 A Control Volume Fixed with Respect to a Moving Frame

7.9 The Principle of Moment of Momentum

7.10 The Principle of Conservation of Energy

7.11 The Entropy Inequality: The Second Law of Thermodynamics

Problems for Chapter 7

Chapter 8: Non-Newtonian Fluids

Part A: Linear Viscoelastic Fluid

8.1 Linear Maxwell Fluid

8.2 A Generalized Linear Maxwell Fluid with Discrete Relaxation Spectra

8.3 Integral Form of the Linear Maxwell Fluid and of the Generalized Linear Maxwell Fluid with Discrete Relaxation Spectra

8.4 A Generalized Linear Maxwell Fluid with a Continuous Relaxation Spectrum

8.5 Computation of Relaxation Spectrum and Relaxation Function

Part B: Nonlinear Viscoelastic Fluid

8.6 Current Configuration as Reference Configuration

8.7 Relative Deformation Gradient

8.8 Relative Deformation Tensors

8.9 Calculations of the Relative Deformation Tensor

8.10 History of the Relative Deformation Tensor and Rivlin-Ericksen Tensors

8.11 Rivlin-Ericksen Tensors in Terms of Velocity Gradient: The Recursive Formula

8.12 Relation between Velocity Gradient and Deformation Gradient

8.13 Transformation Law for the Relative Deformation Tensors Under a Change of Frame

8.14 Transformation Law for Rivlin-Ericksen Tensors Under a Change of Frame

8.15 Incompressible Simple Fluid

8.16 Special Single Integral-Type Nonlinear Constitutive Equations

8.17 General Single Integral-Type Nonlinear Constitutive Equations

8.18 Differential-Type Constitutive Equations for Incompressible Fluids

8.19 Objective Rate of Stress

8.20 Rate-Type Constitutive Equations

Part C: Viscometric Flow of an Incompressible simple Fluid

8.21 Viscometric Flow

8.22 Stresses in Viscometric Flow of an Incompressible Simple Fluid

8.23 Channel Flow

8.24 Couette Flow

Appendix 8.1: Gradient of Second-Order Tensor for Orthogonal Coordinates

Problems for Chapter 8

References

Answers to Problems

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Index