Esben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer

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1 Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer

2 Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics Like I Need Another Hole in My Head" The Main Emphasis of this Book xxvii xxv xxv xxvi How to Read this Book xxviii Expected Prerequisites What this Book Is About and what it Is Not Part I, Continuum Mechanics What Are these Other Parts About? xxviii xxviii xxviii xxviii Part II, Specialized Continua xxix Part III, Beams with Cross Sections xxix Part IV, Buckling xxix Part V, Introduction to the Finite Element Method xxix Part VI, Mathematical Preliminaries Some Comments on Notation xxx xxx Some Comments on Length xxx I Continuum Mechanics 1 1 The Purpose of Continuum Mechanics 3 2 Large Displacements and Large Strains 5 21 Introduction 5 22 Kinematics and Deformation Kinematics and Strain Kinematic Field Equations Lagrange Strain 9

3 Table of Contents 2221 "Fiber" Elongation Change of Angle Infinitesimal Strains and Infinitesimal Rotations Compatibility Equations Kinematic Boundary Conditions Equilibrium Equations Static Field Equations Properties of the Stress Vector Static Boundary Conditions Principle of Virtual Displacements The Budiansky-Hutchinson Dot Notation Generalized Strains and Stresses Principle of Virtual Forces Constitutive Relations Hyperelastic Materials Plastic Materials Potential Energy Linear Elasticity Complementary Energy Static Equations by the Principle of Virtual Displacements 30 3 Kinematically Moderately Nonlinear Theory 33 4 Infinitesimal Theory Introduction Kinematics and Deformation Kinematics and Strain Strain Compatibility Equations Kinematic Boundary Conditions Interpretation of Strain Components Both Indices Equal Different Indices Transformation of Strain Transformation of Coordinates Transformation of Components of Displace ment and Components of Strain Principal Strains Equilibrium Equations 47 Esben Byskov Continuum Mechanics for Everyone August 14, 2012

4 Table of Contents xi 431 Interpretation of Stress Components Internal Equilibrium Static Boundary Conditions Transformation of Stress Principal Stresses Potential Energy Linear Elasticity Principle of Virtual Forces Complementary Strain Energy Function Complementary Energy 58 5 Constitutive Relations Rearrangement of Strain and Stress Components Linear Elasticity Isotropic Linear Elasticity The Value of Poisson's Ratio 66 Ex 5-1 Expression for the Bulk Modulus 68 Ex 5-2 Is Our Expression for the Strain Energy Valid? 69 Ex 5-3 Special Two-Dimensional Strain and Stress States in Elastic Bodies Nonlinear Constitutive Models Plasticity One-Dimensional Case Rigid, Perfect Plasticity Steel Concrete Wood Strain Hardening Unloading Reloading Multi-Axial Plastic States von Mises' "Law" Tresca's "Law" Unloading Reloading Kinematic Hardening Isotropic Hardening 85 August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

5 xii Table of Contents II Specialized Continua 87 6 The Idea of Specialized Continua 89 7 Plane, Straight Beams Beam Deformation Modes Axial Deformation Shear Deformation Bending Deformation The Three Fundamental Beam Strains Choice of Deformation Modes Fully Nonlinear Beam Theory Kinematics Kinematically Moderately Nonlinear Straight Bernoulli-Euler Beams Kinematics 97 Ex 7-1 Rigid Rotation of a Beam Equilibrium Equations Interpretation of the Static Quantities Kinematically Moderately Nonlinear Straight Timoshenko Beams Kinematics Axial Strain Shear Strain Curvature Strain Equilibrium Equations Kinematically Linear Straight Bernoulli-Euler Beams Kinematically Linear Straight Timoshenko Beams Kinematics Ill 7611 Axial Strain Ill 7612 Shear Strain Ill 7613 Curvature Strain Ill 7614 Interpretation of Shear Strain Ill 7615 Interpretation of Operators Generalized Stresses Equilibrium Equations Plane, Straight Elastic Bernoulli-Euler Beams 115 Ex 7-2 When Is the Linear Theory Valid? 116 Byskov Continuum Mechanics for Everyone August 14, 2012

6 Table of Contents xiii Ex 7-3 The Euler Column Plane, Straight Elastic Timoshenko Beams 122 Ex 7-4 A Cantilever Timoshenko Beam Plane, Curved Bernoulli-Euler Beams Kinematically Fully Nonlinear Curved Bernoulli-Euler Beams Geometry and Kinematics Displacements and Displacement Derivatives Length of the Line Element 129 Ex 8-1 Comparison With Straight Beam Rotation of the Beam Curvature of the Beam Generalized Strains Axial Strain Curvature Strain Equilibrium Equations Constitutive Relations 133 Ex 8-2 The Elastica Kinematically Moderately Nonlinear Curved Bernoulli-Euler Beams Kinematics Generalized Strains Axial Strain Curvature Strain Comparison Between Straight and Curved Beams Budiansky-Hutchinson Notation Equilibrium Equations Interpretation of Static Quantities Kinematically Linear Curved Bernoulli-Euler Beams Kinematics Generalized Strains Axial Strain Curvature Strain Generalized Stresses Equilibrium Equations 147 Ex 8-3 Bending Instability of Circular Tubes 148 August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

7 xiv Table of Contents 9 Plane Plates Kincmatically Moderately Nonlinear Plates Kinematic Description Budiansky-Hutchinson dot Notation Internal Virtual Work External Virtual Work Principle of Virtual Displacements Equilibrium Equations Interpretation of Static Quantities Plane Elastic Plates Generalized Quantities Constitutive Relations for Isotropic Plates Differential Equations Boundary Conditions Kinematic Boundary Conditions Static Boundary Conditions The Airy Stress Function Other Stress Functions Kinematically Linear Plates Kinematic Description Equilibrium Equations Interpretation of Static Quantities 185 Ex 9-1 Linear Plate Example Kinematically Linear vs Nonlinear Plate Theory 193 III Beams with Cross-Sections and Plates with Thick ness Introduction to "Beams with Cross-Sections" Bending and Axial Deformation of Linear Elastic Beam Cross-Sections Linear Elastic Material Purpose Beam Cross-Section and Beam Fibers Pure Axial Strain 201 Esben Byskov Continuum Mechanics for Everyone August 14, 2012

8 Table of Contents xv 1114 Both Axial and Curvature Strain in Bernoulli-Euler Beams Axial Force, Zeroth-and First-Order Moments Bending Moment and Second-Order Moments Summary of Linear Elastic Stress-Strain Relations Cross-Sectional Axes Beam Axis and Center of Grav ity The Beam Axis at the Neutral Axis Independence of Results of Choice of Beam Axis Distribution of Axial Strain and Axial Stress Examples of Moments of Inertia 210 Ex 11-1 Rectangular Cross-Section 210 Ex 11-2 Circular Cross-Section 211 Ex 11-3 T-Shaped Cross-Section '212 Ex 11-4 Thin-Walled I-Shaped Cross-Section 214 Ex 11-5 Circular Tube Ring-Shaped Cross-Section Shear Deformation of Linear Elastic Beam Cross-Sections Without and With a Cross-Section Formulas for Shear Stresses in Beams A Little Continuum Mechanics Axial and Transverse Equilibrium Moment Equilibrium 222 Ex 12-1 Where to Load a Beam Examples of Shear Stress Computations 225 Ex 12-2 Rectangular Cross-Section 225 Ex 12-3 Circular Cross-Section 226 Ex 12-4 Thin-Walled I-Shaped Cross-Section 228 Ex 12-5 Circular Tube Ring-Shaped Cross-Section Shear Stiffness Rectangular Cross-Section Case (a) Solution by Timoshenko & Goodier Case (b) Solution of Rotated "Beam" Timoshenko Beam Theory Values of the Effective Area A Simple Lower Bound 237 Ex 12-6 Rectangular Cross-Section 239 Ex 12-7 Circular Cross-Section 239 August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

9 xvi Table of Contents 1237 Concluding Remarks Unconstrained Torsion Introduction 241 Ex 13-1 One-Dimensional Torsion Structural Problem Geometry Kinematics Strains Statics Stress Function Equilibrium Compatibility Torsional Moment Torsional Moment -Simply Connected Re gion Torsional Moment Multiply Connected Re gion Linear Elasticity Compatibility Warping Function Examples of Elastic Torsion 255 Ex 13-2 Circular Cross-Section 255 Ex 13-3 Elliptic Cross-Section 257 Ex 13-4 Circular Tube Ring-Shaped Cross-Section 260 Ex 13-5 Equilateral Triangle Cross-Section 263 Ex 13-6 Rectangular Cross-Section Concluding Remarks Introduction to "Plates with Thickness" Bending and In-Plane Deformation of Linear Elastic Plates Linear Elastic Plates Outline of Procedure Kinematic Relations Three-Dimensional Constitutive Relations Constitutive Relations for Two-Dimensional Plate 275 Esben Byskov Continuum Mechanics for Everyone August 14, 2012

10 Table of Contents xvii IV Buckling Stability Buckling Stability Concepts Static Stability and Instability Phenomena Limit Load Buckling Snap-Through Bifurcation Buckling Classical Critical Load Further Comments Criteria and Methods for Determination of Stability and In stability Static Neighbor Equilibrium Stability Criterion Energy-Based Static Stability Criterion Dynamic Stability Criterion Introductory Example 285 Ex 17-1 Model Column Elastic Buckling Problems with Linear Prebuckling Nonlinear Prebuckling Some Prerequisites Linear Prebuckling Principle of Virtual Displacements Bifurcation Buckling Classical Critical Load Higher Bifurcation Loads 301 Ex 18-1 The Euler Column 302 Ex 18-2 A Pinned-Pinned Column Analyzed by Timoshenko Theory 306 Ex 18-3 Buckling of an Elastic Plate Expansion Theorem Numerical and Approximate Solutions, the Rayleigh Quotient Stationarity of the Rayleigh Quotient Minimum Property of the Rayleigh Quotient? The Rayleigh-Ritz Procedure Another Finite Element Notation 326 Ex 18-4 Interpretation of J2k([jk}{v}T[G)TlG}{v}"j^ A Word of Caution Examples of Application of the Rayleigh Quotient and the Rayleigh-Ritz Procedure 329 August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

11 xviii Table of Contents Ex 18-5 Roorda's Frame Application of the Rayleigh-Ritz Procedure 329 Ex 18-6 Plate Buckling Rayleigh-Ritz Procedure Concluding Comments on the Examples Above Initial Postbuckling with a Unique Buckling Mode Selected Formulas from Chapter General Formulas Fundamental Path Prebuckling Buckling Bifurcation Initial Postbuckling First-Order Problem Buckling Problem Second-Order Problem Third-Order Problem Solubility Conditions on the Second- and Third-Order Problems 351 Ex 19-1 Ex 19-2 Postbuckling of Roorda's Frame and the First-Order Postbuckling Coefficient 351 Postbuckling of Symmetric Two-Bar Frame Second-Order Postbuckling Coefficient Imperfection Sensitivity Imperfection Sensitivity and a Single Buckling Mode Non-Vanishing First Order Postbuckling Constant 2012 Vanishing First Order Postbuckling Constant, Non- Vanishing Second Order Postbuckling Constant 2013 Is the Imperfection Detrimental? 360 Ex 20-1 Geometrically Imperfect Euler Column 202 Mode Interaction and Geometric Imperfections 366 Ex 20-2 Mode Interaction in a Truss Column Elastic-Plastic Buckling The Shanley Column Introduction Columns of Elastic-Plastic Materials Constitutive Model Engesser's First Proposal (1889) Engesser's Second Proposal Which Load Is the Critical One? Shanley's Experiments and Observations (1947) 380 Esben Byakov Continuum Mechanics for Everyone August 14, 2012

12 Table of Contents xix 213 The Shanley Model Column Kinematic Relations Static Relations Constitutive Relations Elastic Model Column Tangent Modulus Load Reduced Modulus Load Shanley's Analysis and Proposal Prebuckling Bifurcation Kinematic Relations Static Relations Constitutive Relations Reduced Modulus Load Possible Bifurcations Both Springs Load Further Both Springs Unload Spring 1 Unloads, Spring 2 Loads Further 388 V Introduction to the Finite Element Method About the Finite Element Method An Introductory Example in Several Parts Generalized Quantities and Potential Energy The Exact Solution The Simplest Approximation More Terms? The case a 3 = The case a 1 = Focus on the System, Simple Elements Interpretation of the System Stiffness Matrix Focus on the Simple Elements Focus on the "Real" Elements Beam Elements 1 and Interpretation of the Element Stiffness Matrix Spring Elements 3 and August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

13 XX Table of Contents Assembling of the Element Matrices Right-Hand Side Vector Right-Hand Side Vector for Distributed Loads Potential Energy, System of Finite Element Equations 2311 Element Displacement Vectors Generalized Strains and Stresses Summary of the Procedure of the Introductory Example Plate Finite Elements for In-Plane States Introduction A Rectangular Plate Finite Element for In-Plane States Displacement Field Strain Distribution Constitutive Assumption Stiffness Matrix for Isotropy A Deficiency of the Melosh Element Internal Nodes and Their Elimination Introduction Structural Problem Nondimensional Quantities Displacement and Displacement Interpolation Strain Distribution Matrix Stiffness Matrix Elimination of Internal Nodes 437 No Free Lunches Program written in maxima Circular Beam Finite Elements, Problems and Solutions Introduction Strains Stresses Linear Elasticity Potential Energy Matrix Formulation Discretization Stiffness Matrix Internal Mismatch Locking 457 Esben Byskov Continuum Mechanics for Everyone August 14, 2012

14 Table of Contents xxi 267 Rigid-Body Displacements Self-Straiiiing Modified Potential Energy Justification of the Modified Potential Other Ways to Handle "Locking" Matrix Formulation Discretization Elimination of Lagrange Multipliers and Strain Pa rameters Rigid-Body Displacements Self-Strain Analytic Results Matrices Fundamental Matrices for 6 Displacement Degrees of Freedom 468 Ex 26-1 Numerical Example Concluding Comments Modified Complementary Energy and Stress Hybrid Finite Elements Modified Complementary Energy Establishing of a Modified Complementary Energy Stress Hybrid Finite Elements Discretization Elimination of Stress Field Parameters A Rectangular Stress Hybrid Finite Element 489 Ex 27-1 Comparison Between a Stress Hybrid El ement and the Melosh Element Why Does the Hybrid Element Perform so Well in Bending? Isoparametric Version Linear Elastic Finite Element Analysis of Torsion A Functional for Torsion Discretization 500 Ex 28-1 Ex 28-2 A Simple Rectangular Finite Element for Torsion 501 An Eight-Nodes Rectangular Finite Ele ment for Torsion 503 Ex 28-3 Finite Element Results 503 August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

15 xxii Table of Contents VI Mathematical Preliminaries Introduction Notation Overbar Tilde Indices Vectors and Matrices Fields Operators Index Notation, the Summation Convention, and a Little About Tensor Analysis Index Notation Comma Notation Summation Convention Lowercase Greek Indices Symmetric and Antisymmetric Quantities Product of a Symmetric and an Antisym metric Matrix Product of a Symmetric and a General Matrix Summation Convention Results in Brevity Generalized Coordinates Vectors as Generalized Coordinates Functions as Generalized Coordinates Introduction to Variational Principles Introduction Functional 522 Ex 32-1 A Broken Pocket Calculator Variations Systems with a Finite Number of Degrees of Freedom 532 Ex 32-2 A "Structure" with One Degree of Freedom 533 Ex 32-3 A "Structure" with Two Degrees of Freedom Systems with Infinitely Many Degrees of Freedom 537 Esben Byskov Continuum Mechanics for Everyone August 14, 2012

16 Table of Contents xxiii Ex 32-4 A Structure with Infinitely Many Degrees of Freedom Lagrange Multipliers 545 Ex 32-5 A Structure with Auxiliary Conditions The Euler Column General Treatment Budiansky-Hutchinson Notation Linear, Quadratic and Bilinear Operators Principle of Virtual Displacements Variation of a Potential Potential Energy for Linear Elasticity Stationarity of lip for Linearity min(iip) for Linearity min(5iip) for Linearity Too Stiff Behavior Single Point Force Complementary Energy for Linear Elasticity Minimum Complementary Energy min(5iic) for Linearity => Too Flexible Behavior Single Point Force 564 Ex 33-1 A Clamped-Clamped Beam Auxiliary Conditions Lagrange Multipliers Principle of Virtual Displacements Budiansky-Hutchinson Notation for Selected Examples Interpretations Related to Example Ex Interpretations Related to Example Ex Interpretations Related to Example Ex Interpretations Related to Sections 73 and Interpretations Related to Section Bibliography 571 Index 579 Continuum Mechanics for Everyone Esben Byskov

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