Continuum Mechanics for Engineers, Third Edition

Description

This new edition provides a complete, concise, and accessible introduction to advanced engineering mechanics. It explores the basic concepts behind continuum mechanics, linear and nonlinear elasticity, and viscoelasticity, and demonstrates their application in engineering practice. Foreword Contents 1 Description of Motion 1.1 Definition of the Continuous Medium 1.2 Equations of Motion 1.3 Descriptions of Motion 1.3.1 Material Description 1.3.2 Spatial Description 1.4 Time Derivatives: Local, Material and Convective 1.5 Velocity and Acceleration 1.6 Stationarity 1.7 Trajectory 1.7.1 Differential Equation of the Trajectories 1.8 Streamline 1.8.1 Differential Equation of the Streamlines 1.9 Streamtubes 1.9.1 Equation of the Streamtube 1.10 Streaklines 1.10.1 Equation of the Streakline 1.11 Material Surface 1.12 Control Surface 1.13 Material Volume 1.14 Control Volume Problems and Exercises 2 Strain 2.1 Introduction 2.2 Deformation Gradient Tensor 2.2.1 Inverse Deformation Gradient Tensor 2.3 Displacements 2.3.1 Material and Spatial Displacement Gradient Tensors 2.4 Strain Tensors 2.4.1 Material Strain Tensor (Green-Lagrange Strain Tensor) 2.4.2 Spatial Strain Tensor (Almansi Strain Tensor) 2.4.3 Strain Tensors in terms of the Displacement (Gradients) 2.5 Variation of Distances: Stretch and Unit Elongation 2.5.1 Stretches, Unit Elongations and Strain Tensors 2.6 Variation of Angles 2.7 Physical Interpretation of the Strain Tensors 2.7.1 Material Strain Tensor 2.7.2 Spatial Strain Tensor 2.8 Polar Decomposition 2.9 Volume Variation 2.10 Area Variation 2.11 Infinitesimal Strain 2.11.1 Strain Tensors. Infinitesimal Strain Tensor 2.11.2 Stretch. Unit Elongation 2.11.3 Physical Interpretation of the Infinitesimal Strains 2.11.4 Engineering Strains. Vector of Engineering Strains 2.11.5 Variation of the Angle between Two Differential Segments in Infinitesimal Strain 2.11.6 Polar Decomposition 2.12 Volumetric Strain 2.13 Strain Rate 2.13.1 Velocity Gradient Tensor 2.13.2 Strain Rate and Spin Tensors 2.13.3 Physical Interpretation of the Strain Rate Tensor 2.13.4 Physical Interpretation of the Rotation Rate Tensor 2.14 Material Time Derivatives of Strain and Other Magnitude Tensors 2.14.1 Deformation Gradient Tensor and its Inverse Tensor 2.14.2 Material and Spatial Strain Tensors 2.14.3 Volume and Area Differentials 2.15 Motion and Strains in Cylindrical and Spherical Coordinates 2.15.1 Cylindrical Coordinates 2.15.2 Spherical Coordinates Problems and Exercises 3 Compatibility Equations 3.1 Introduction 3.2 Preliminary Example: Compatibility Equations of a Potential Vector Field 3.3 Compatibility Conditions for Infinitesimal Strains 3.4 Integration of the Infinitesimal Strain Field 3.4.1 Preliminary Equations 3.4.2 Integration of the Strain Field 3.5 Compatibility Equations and Integration of the Strain Rate Field Problems and Exercises 4 Stress 4.1 Forces Acting on a Continuum Body 4.1.1 Body Forces 4.1.2 Surface Forces 4.2 Cauchy's Postulates 4.3 Stress Tensor 4.3.1 Application of Newton's 2nd Law to a Continuous Medium 4.3.2 Stress Tensor 4.3.3 Graphical Representation of the Stress State in a Point 4.4 Properties of the Stress Tensor 4.4.1 Cauchy Equation. Internal Equilibrium Equation 4.4.2 Equilibrium Equation at the Boundary 4.4.3 Symmetry of the Cauchy Stress Tensor 4.4.4 Diagonalization. Principal Stresses and Directions 4.4.5 Mean Stress and Mean Pressure 4.4.6 Decomposition of the Stress Tensor into its Spherical and Deviatoric Parts 4.4.7 Tensor Invariants 4.5 Stress Tensor in Curvilinear Orthogonal Coordinates 4.5.1 Cylindrical Coordinates 4.5.2 Spherical Coordinates 4.6 Mohr's Circle in 3 Dimensions 4.6.1 Graphical Interpretation of the Stress States 4.6.2 Determination of the Mohr's Circles 4.7 Mohr's Circle in 2 Dimensions 4.7.1 Stress State on a Given Plane 4.7.2 Direct Problem: Diagonalization of the Stress Tensor 4.7.3 Inverse Problem 4.7.4 Mohr's Circle for Plane States (in 2 Dimensions) 4.7.5 Properties of the Mohr's Circle 4.7.6 The Pole of Mohr's Circle 4.7.7 Mohr's Circle with the Soil Mechanics Sign Criterion 4.8 Mohr's Circle for Particular Cases 4.8.1 Hydrostatic Stress State 4.8.2 Mohr's Circles for a Tensor and its Deviator 4.8.3 Mohr's Circles for a Plane Pure Shear Stress State Problems and Exercises 5 Balance Principles 5.1 Introduction 5.2 Mass Transport or Convective Flux 5.3 Local and Material Derivatives of a Volume Integral 5.3.1 Local Derivative 5.3.2 Material Derivative 5.4 Conservation of Mass. Mass continuity Equation 5.4.1 Spatial Form of the Principle of Conservation of Mass. Mass Continuity Equation 5.4.2 Material Form of the Principle of Conservation of Mass 5.5 Balance Equation. Reynolds Transport Theorem 5.5.1 Reynolds' Lemma 5.5.2 Reynolds' Theorem 5.6 General Expression of the Balance Equations 5.7 Balance of Linear Momentum 5.7.1 Global Form of the Balance of Linear Momentum 5.7.2 Local Form of the Balance of Linear Momentum 5.8 Balance of Angular Momentum 5.8.1 Global Form of the Balance of Angular Momentum 5.8.2 Local Spatial Form of the Balance of Angular Momentum 5.9 Power 5.9.1 Mechanical Power. Balance of Mechanical Energy 5.9.2 Thermal Power 5.10 Energy Balance 5.10.1 Thermodynamic Concepts 5.10.2 First Law of Thermodynamics 5.11 Reversible and Irreversible Processes 5.12 Second Law of Thermodynamics. Entropy 5.12.1 Second Law of Thermodynamics. Global form 5.12.2 Physical Interpretation of the Second Law of Thermodynamics 5.12.3 Reformulation of the Second Law of Thermodynamics 5.12.4 Local Form of the Second Law of Thermodynamics. Clausius-Planck Equation 5.12.5 Alternative Forms of the Second Law of Thermodynamics 5.13 Continuum Mechanics Equations. Constitutive Equations 5.13.1 Uncoupled Thermo-Mechanical Problem Problems and Exercises 6 Linear Elasticity 6.1 Hypothesis of the Linear Theory of Elasticity 6.2 Linear Elastic Constitutive Equation. Generalized Hooke's Law 6.2.1 Elastic Potential 6.3 Isotropy. Lamé's Constants. Hooke's Law for Isotropic Linear Elasticity 6.3.1 Inversion of Hooke's Law. Young's Modulus. Poisson's Ratio 6.4 Hooke's Law in Spherical and Deviatoric Components 6.5 Limits in the Values of the Elastic Properties 6.6 The Linear Elastic Problem 6.6.1 Governing Equations 6.6.2 Boundary Conditions 6.6.3 Quasi-Static Problem 6.7 Solution to the Linear Elastic Problem 6.7.1 Displacement Formulation: Navier's Equation 6.7.2 Stress Formulation: Beltrami-Michell Equation 6.8 Unicity of the Solution to the Linear Elastic Problem 6.9 Saint-Venant's Principle 6.10 Linear Thermoelasticity. Thermal Stresses and Strains 6.10.1 Linear Thermoelastic Constitutive Equation 6.10.2 Inverse Constitutive Equation 6.10.3 Thermal Stresses and Strains 6.11 Thermal Analogies 6.11.1 First Thermal Analogy (Duhamel-Newman Analogy) 6.11.2 Second Thermal Analogy 6.12 Superposition Principle in Linear Thermoelasticity 6.13 Hooke's Law in terms of the Stress and Strain ``Vectors" Problems and Exercises 7 Plane Linear Elasticity 7.1 Introduction 7.2 Plane Stress State 7.2.1 Strain Field. Constitutive Equation 7.2.2 Displacement Field 7.3 Plane Strain 7.3.1 Strain and Stress Fields 7.4 The Plane Linear Elastic Problem 7.5 Problems Typically Assimilated to Plane Elasticity 7.5.1 Plane Stress 7.5.2 Plane Strain 7.6 Representative Curves of Plane Elasticity 7.6.1 Isostatics or stress trajectories 7.6.2 Isoclines 7.6.3 Isobars 7.6.4 Maximum Shear Stress or Slip Lines Problems and Exercises 8 Plasticity 8.1 Introduction 8.2 Previous Notions 8.2.1 Stress Invariants 8.2.2 Spherical and Deviatoric Components of the Stress Tensor 8.3 Principal Stress Space 8.3.1 Normal and Shear Octahedral Stresses 8.4 Rheological Models 8.4.1 Elastic Element (Spring Element) 8.4.2 Frictional Element 8.4.3 Elastic-Frictional Model 8.4.4 Frictional Model with Hardening 8.4.5 Elastic-Frictional Model with Hardening 8.5 Elastoplastic Phenomenological Behavior 8.5.1 Bauschinger Effect 8.6 Incremental Theory of Plasticity in 1 Dimension 8.6.1 Additive Decomposition of Strain. Hardening Variable 8.6.2 Elastic Domain. Yield Function. Yield Surface 8.6.3 Constitutive Equation 8.6.4 Hardening Law. Hardening Parameter 8.6.5 Elastoplastic Tangent Modulus 8.6.6 Uniaxial Stress-Strain Curve 8.7 Plasticity in 3 Dimensions 8.7.1 Constitutive Equation 8.8 Yield Surfaces. Failure Criteria 8.8.1 Von Mises Criterion 8.8.2 Tresca Criterion or Maximum Shear Stress Criterion 8.8.3 Mohr-Coulomb Criterion 8.8.4 Drucker-Prager Criterion Problems and Exercises 9 Constitutive Equations in Fluids 9.1 Concept of Pressure 9.1.1 Hydrostatic Pressure 9.1.2 Mean Pressure 9.1.3 Thermodynamic Pressure. Kinetic Equation of State 9.2 Constitutive Equations in Fluid Mechanics 9.3 Constitutive Equation in Viscous Fluids 9.4 Constitutive Equation in Newtonian Fluids 9.4.1 Relation between the Thermodynamic and Mean Pressures 9.4.2 Constitutive Equation in Spherical and Deviatoric Components 9.4.3 Stress Power, Recoverable Power and Dissipative Power 9.4.4 Thermodynamic Considerations 9.4.5 Limitations in the Viscosity Values 10 Fluid Mechanics 10.1 Governing Equations 10.2 Hydrostatics. Fluids at Rest 10.2.1 Hydrostatic Equations 10.2.2 Gravitational Force. Triangular Pressure Distribution 10.2.3 Archimedes' Principle 10.3 Fluid Dynamics: Barotropic Perfect Fluids 10.3.1 Equations of the Problems 10.3.2 Resolution of the Mechanical Problem under Potential Body Forces. Bernoulli's Trinomial 10.3.3 Solution in a Steady-State Regime 10.3.4 Solution in Transient Regime 10.4 Fluid Dynamics: (Newtonian) Viscous Fluids 10.4.1 Navier-Stokes Equation 10.4.2 Energy Equation 10.4.3 Governing Equations of the Fluid Mechanics Problem 10.4.4 Physical Interpretation of the Navier-Stokes and Energy Equations 10.4.5 Reduction of the General Problem to Particular Cases 10.5 Boundary Conditions in Fluid Mechanics 10.5.1 Velocity Boundary Conditions 10.5.2 Pressure Boundary Conditions 10.5.3 Mixed Boundary Conditions 10.5.4 Boundary Conditions on Free Surfaces 10.6 Laminar and Turbulent Flows 10.6.1 Laminar Flow 10.6.2 Turbulent Flow 10.7 Fluid Mechanics Formulas 10.7.1 Stress tensor for Newtonian fluids 10.7.2 Continuity Equation 10.7.3 Navier-Stokes Equation Problems and Exercises 11 Variational Principles 11.1 Governing Equations 11.1.1 Functionals. Functional Derivatives 11.1.2 Extrema of the Functionals. Variational Principles. Euler-Lagrange Equations 11.2 Virtual Work Principle (Theorem) 11.2.1 Interpretation of the Virtual Work Principle 11.2.2 Virtual Work Principle in terms of the Stress and Strain Vectors 11.3 Potential Energy. Minimum Potential Energy Principle Problems and Exercises Bibliography Blank Page