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Foreword xiii Series Preface xv List of Symbols and Abbreviations xvii Introduction xxiii About the Companion Website xxxv 1 Elastic Anisotropic Behavior of Composite Materials 1 1.1 Anisotropic Elasticity of Composite Materials 1 1.1.1 Fourth Rank Tensor Notation of Hooke’s Law 1 1.1.2 Voigt’s Matrix Notation of Hooke’s Law 2 1.1.3 Kelvin’s Matrix Notation of Hooke’s Law 5 1.2 Unidirectional Fiber Bundle 7 1.2.1 Components of a Unidirectional Fiber Bundle 7 1.2.2 Elastic Properties of a Unidirectional Fiber Bundle 7 1.2.3 Effective Elastic Constants of Unidirectional Composites 8 1.3 Rotational Transformations of Material Laws, Stress and Strain 10 1.3.1 Rotation of Fourth Rank Elasticity Tensors 11 1.3.2 Rotation of Elasticity Matrices in Voigt’s Notation 11 1.3.3 Rotation of Elasticity Matrices in Kelvin’s Notation 13 1.4 Elasticity Matrices for Laminated Plates 14 1.4.1 Voigt’s Matrix Notation for Anisotropic Plates 14 1.4.2 Rotation of Matrices in Voigt’s Notation 15 1.4.3 Kelvin’s Matrix Notation for Anisotropic Plates 15 1.4.4 Rotation of Matrices in Kelvin’s Notation 16 1.5 Coupling Effects of Anisotropic Laminates 17 1.5.1 Orthotropic Laminate Without Coupling 17 1.5.2 Anisotropic Laminate Without Coupling 17 1.5.3 Anisotropic Laminate With Coupling 17 1.5.4 Coupling Effects in Laminated Thin-Walled Sections 18 1.6 Conclusions 18 References 19 2 Phenomenological Failure Criteria of Composites 21 2.1 Phenomenological Failure Criteria 21 2.1.1 Criteria for Static Failure Behavior 21 2.1.2 Stress Failure Criteria for Isotropic Homogenous Materials 21 2.1.3 Phenomenological Failure Criteria for Composites 22 2.1.4 Phenomenological Criteria Without Stress Coupling 23 2.1.4.1 Criterion of Maximum Averaged Stresses 23 2.1.4.2 Criterion of Maximum Averaged Strains 24 2.1.5 Phenomenological Criteria with Stress Coupling 24 2.1.5.1 Mises-Hill Anisotropic Failure Criterion 24 2.1.5.2 Pressure-Sensitive Mises-Hill Anisotropic Failure Criterion 26 2.1.5.3 Tensor-Polynomial Failure Criterion 27 2.1.5.4 Tsai-Wu Criterion 30 2.1.5.5 Assessment of Coefficients in Tensor-Polynomial Criteria 30 2.2 Differentiating Criteria 33 2.2.1 Fiber and Intermediate Break Criteria 33 2.2.2 Hashin Strength Criterion 33 2.2.3 Delamination Criteria 35 2.3 Physically Based Failure Criteria 35 2.3.1 Puck Criterion 35 2.3.2 Cuntze Criterion 36 2.4 Rotational Transformation of Anisotropic Failure Criteria 37 2.5 Conclusions 40 References 40 3 Micromechanical Failure Criteria of Composites 45 3.1 Pullout of Fibers from the Elastic-Plastic Matrix 45 3.1.1 Axial Tension of Fiber and Matrix 45 3.1.2 Shear Stresses in Matrix Cylinders 51 3.1.3 Coupled Elongation of Fibers and Matrix 53 3.1.4 Failures in Matrix and Fibers 54 3.1.4.1 Equations for Mean Axial Displacements of Fibers and Matrix 54 3.1.4.2 Solutions of Equations for Mean Axial Displacements of Fibers and Matrix 56 3.1.5 Rupture of Matrix and Pullout of Fibers from Crack Edges in a Matrix 57 3.1.5.1 Elastic Elongation (Case I) 57 3.1.5.2 Plastic Sliding on the Fiber Surface (Case II) 58 3.1.5.3 Fiber Breakage (Case III) 58 3.1.6 Rupture of Fibers, Matrix Joints and Crack Edges 59 3.2 Crack Bridging in Elastic-Plastic Unidirectional Composites 60 3.2.1 Crack Bridging in Unidirectional Fiber-Reinforced Composites 60 3.2.2 Matrix Crack Growth 61 3.2.3 Fiber Crack Growth 62 3.2.4 Penny-Shaped Crack 65 3.2.4.1 Crack in a Transversal-Isotropic Medium 65 3.2.4.2 Mechanisms of the Fracture Process 66 3.2.4.3 Crack Bridging in an Orthotropic Body With Disk Crack 66 3.2.4.4 Solution to an Axially Symmetric Crack Problem 68 3.2.5 Plane Crack Problem 72 3.2.5.1 Equations of the Plane Crack Problem 72 3.2.5.2 Solution to the Plane Crack Problem 74 3.3 Debonding of Fibers in Unidirectional Composites 75 3.3.1 Axial Deformation of Unidirectional Fiber Composites 75 3.3.2 Stresses in Unidirectional Composite in Cases of Ideal Debonding or Adhesion 79 3.3.2.1 Equations of an Axially Loaded Unidirectional Compound Medium (A) 79 3.3.2.2 Total Debonding (B) 82 3.3.2.3 Ideal Adhesion (C) 83 3.3.3 Stresses in a Unidirectional Composite in a Case of Partial Debonding 84 3.3.3.1 Partial Radial Load on the Fiber Surface 84 3.3.3.2 Partial Radial Load on the Matrix Cavity Surface 84 3.3.3.3 Partial Debonding With Central Adhesion Region (D) 85 3.3.3.4 Partial Debonding With Central Debonding Region (E) 88 3.3.3.5 Semi-Infinite Debonding With Central Debonding Region (F) 89 3.3.4 Contact Problem for a Finite Adhesion Region 89 3.3.5 Debonding of a Semi-Infinite Adhesion Region 93 3.3.6 Debonding of Fibers from a Matrix Under Cyclic Deformation 95 3.4 Conclusions 98 References 98 4 Optimization Principles for Structural Elements Made of Composites 105 4.1 Stiffness Optimization of Anisotropic Structural Elements 105 4.1.1 Optimization Problem 105 4.1.2 Optimality Conditions 106 4.1.3 Optimal Solutions in Anti-Plane Elasticity 109 4.1.4 Optimal Solutions in Plane Elasticity 109 4.2 Optimization of Strength and Loading Capacity of Anisotropic Elements 110 4.2.1 Optimization Problem 110 4.2.2 Optimality Conditions 113 4.2.3 Optimal Solutions in Anti-Plane Elasticity 114 4.2.4 Optimal Solutions in Plane Elasticity 114 4.3 Optimization of Accumulated Elastic Energy in Flexible Anisotropic Elements 116 4.3.1 Optimization Problem 116 4.3.2 Optimality Conditions 117 4.3.3 Optimal Solutions in Anti-Plane Elasticity 118 4.3.4 Optimal Solutions in Plane Elasticity 119 4.4 Optimal Anisotropy in a Twisted Rod 119 4.5 Optimal Anisotropy of Bending Console 122 4.6 Optimization of Plates in Bending 123 4.7 Conclusions 125 References 125 5 Optimization of Composite Driveshaft 129 5.1 Torsion of Anisotropic Shafts With Solid Cross-Sections 129 5.2 Thin-Walled Anisotropic Driveshaft with Closed Profile 132 5.2.1 Geometry of Cross-Section 132 5.2.2 Main Kinematic Hypothesis 133 5.3 Deformation of a Composite Thin-Walled Rod 135 5.3.1 Equations of Deformation of a AnisotropicThin-Walled Rod 135 5.3.2 Boundary Conditions 138 5.3.2.1 Ideal Fixing 138 5.3.2.2 Ideally Free End 138 5.3.2.3 Boundary Conditions of the Intermediate Type 140 5.3.3 Governing Equations in Special Cases of Symmetry 140 5.3.3.1 Orthotropic Material 140 5.3.3.2 Constant Elastic Properties Along the Arc of a Cross-Section 140 5.3.4 Symmetry of Section 140 5.4 Buckling of Composite Driveshafts Under a Twist Moment 141 5.4.1 Greenhill’s Buckling of Driveshafts 141 5.4.2 Optimal Shape of the Solid Cross-Section for Driveshaft 143 5.4.3 Hollow Circular and Triangular Cross-Sections 144 5.5 Patents for Composite Driveshafts 146 5.6 Conclusions 150 References 150 6 Dynamics of a Vehicle with Rigid Structural Elements of Chassis 155 6.1 Classification of Wheel Suspensions 155 6.1.1 Common Designs of Suspensions 155 6.1.2 Types of Twist-Beam Axles 156 6.1.3 Kinematics of Wheel Suspensions 157 6.2 Fundamental Models in Vehicle Dynamics 159 6.2.1 Basic Variables of Vehicle Dynamics 159 6.2.2 Coordinate Systems of Vehicle and Local Coordinate Systems 161 6.2.2.1 Earth-Fixed Coordinate System 161 6.2.2.2 Vehicle-Fixed Coordinate System 162 6.2.2.3 Horizontal Coordinate System 162 6.2.2.4 Wheel Coordinate System 162 6.2.3 Angle Definitions 162 6.2.4 Components of Force and Moments in Car Dynamics 163 6.2.5 Degrees of Freedom of a Vehicle 163 6.3 Forces Between Tires and Road 167 6.3.1 Tire Slip 167 6.3.2 Side Slip Curve and Lateral Force Properties 168 6.4 Dynamic Equations of a Single-Track Model 170 6.4.1 Hypotheses of a Single-Track Model 170 6.4.2 Moments and Forces in a Single-Track Model 171 6.4.3 Balance of Forces and Moments in a Single-Track Model 173 6.4.4 Steady Cornering 174 6.4.4.1 Necessary Steer Angle for Steady Cornering 174 6.4.4.2 Yaw Gain Factor and Steer Angle Gradient 175 6.4.4.3 Classification of Self-Steering Behavior 176 6.4.5 Non-Steady Cornering 179 6.4.5.1 Equations of Non-Stationary Cornering 179 6.4.5.2 Oscillatory Behavior of Vehicle During Non-Steady Cornering 180 6.4.6 Anti-Roll Bars Made of Composite Materials 181 6.5 Conclusions 182 References 182 7 Dynamics of a Vehicle With Flexible, Anisotropic Structural Elements of Chassis 183 7.1 Effects of Body and Chassis Elasticity on Vehicle Dynamics 183 7.1.1 Influence of Body Stiffness on Vehicle Dynamics 183 7.1.2 Lateral Dynamics of Vehicles With Stiff Rear Axles 184 7.1.3 Induced Effects on Wheel Orientation and Positioning of Vehicles with Flexible Rear Axle 185 7.2 Self-Steering Behavior of a Vehicle With Coupling of Bending and Torsion 188 7.2.1 Countersteering for Vehicles with Twist-Beam Axles 188 7.2.1.1 Countersteering Mechanisms 188 7.2.1.2 Countersteering by Anisotropic Coupling of Bending and Torsion 190 7.2.2 Bending-Twist Coupling of a Countersteering Twist-Beam Axle 192 7.2.3 Roll Angle of Vehicle 193 7.2.3.1 Relationship Between Roll Angle and Centrifugal Force 193 7.2.3.2 Lateral Reaction Forces on Wheels 193 7.2.3.3 Steer Angles on Front Wheels 194 7.2.3.4 Steer Angles on Rear Wheels 194 7.3 Steady Cornering of a Flexible Vehicle 196 7.3.1 Stationary Cornering of a Car With a Flexible Chassis 196 7.3.2 Necessary Steer Angles for Coupling and Flexibility of Chassis 196 7.3.2.1 Limit Case: Lateral Acceleration Vanishes 196 7.3.2.2 Absolutely Rigid Front and Rear Wheel Suspensions 197 7.3.2.3 Bending and Torsion of a Twist Member Completely Decoupled 197 7.3.2.4 General Case of Coupling Between Bending and Torsion of a Twist Member 198 7.3.2.5 Neutral Steering Caused by Coupling Between Bending and Torsion of a Twist Member 198 7.4 Estimation of Coupling Constant for a Twist Member 199 7.4.1 Coupling Between Vehicle Roll Angle and Twist of Cross-Member 199 7.4.2 Stiffness Parameters of a Twist-Beam Axle 200 7.4.2.1 Roll Spring Rate 200 7.4.2.2 Lateral Stiffness 201 7.4.2.3 Camber Stiffness 203 7.5 Design of the Countersteering Twist-Beam Axle 203 7.5.1 Requirements for a Countersteering Twist-Beam Axle 203 7.5.2 Selection and Calculation of the Cross-Section for the Cross-Member 205 7.5.3 Elements of a Countersteering Twist-Beam Axle 208 7.6 Patents on Twist-Beam Axles 211 7.7 Conclusions 214 References 214 8 Design and Optimization of Composite Springs 217 8.1 Design and Optimization of Anisotropic Helical Springs 217 8.1.1 Forces and Moments in Helical Composite Springs 217 8.1.2 Symmetrically Designed Solid Bar With Circular Cross-Section 220 8.1.3 Stiffness and Stored Energy of Helical Composite Springs 223 8.1.4 Spring Rates of Helical Composite Springs 225 8.1.5 Helical Composite Springs of Minimal Mass 228 8.1.5.1 Optimization Problem 228 8.1.5.2 Optimal Composite Spring for the Anisotropic Mises-Hill Strength Criterion 228 8.1.6 Axial and Twist Vibrations of Helical Springs 231 8.2 Conical Springs Made of Composite Material 233 8.2.1 Geometry of an Anisotropic Conical Spring in an Undeformed State 233 8.2.2 Curvature and Strain Deviations 235 8.2.3 Thin-Walled Conical Shells Made of Anisotropic Materials 236 8.2.4 Variation Principle 237 8.2.5 Structural Optimization of a Conical Spring Due to Ply Orientation 239 8.2.6 Conical Spring Made of Orthotropic Material 241 8.2.7 Bounds for Stiffness of a Spring Made of Orthotropic Material 243 8.3 Alternative Concepts for Chassis Springs Made of Composites 244 8.4 Conclusions 248 References 249 9 Equivalent Beams of Helical Anisotropic Springs 255 9.1 Helical Compression Springs Made of Composite Materials 255 9.1.1 Statics of the Equivalent Beam for an Anisotropic Spring 255 9.1.2 Dynamics of an Equivalent Beam for an Anisotropic Spring 258 9.2 Transverse Vibrations of a Composite Spring 260 9.2.1 Separation of Variables 260 9.2.2 Fundamental Frequencies of Transversal Vibrations 262 9.2.3 Transverse Vibrations of a Symmetrically Stacked Helical Spring 264 9.3 Side Buckling of a Helical Composite Spring 265 9.3.1 Buckling Under Axial Force 265 9.3.2 Simplified Formulas for Buckling of a Symmetrically Stacked Helical Spring 266 9.4 Conclusions 267 References 267 10 Composite Leaf Springs 269 10.1 Longitudinally Mounted Leaf Springs for Solid Axles 269 10.1.1 Predominantly Bending-Loaded Leaf Springs 269 10.1.2 Moments and Forces of Leaf Springs in a Pure Bending State 270 10.1.3 Optimization of Leaf Springs for an Anisotropic Mises-Hill Criterion 272 10.2 Leaf-Tension Springs 275 10.2.1 Combined Bending and Tension of a Spring 275 10.2.2 Forces and Rates of Leaf-Tension Springs 277 10.3 Transversally Mounted Leaf Springs 278 10.3.1 Axle Concepts of Transverse Leaf Springs 278 10.3.2 Analysis of a Transverse Leaf Spring 280 10.3.3 Examples and Patents for Transversely Mounted Leaf Springs 283 10.4 Conclusions 286 References 287 11 Meander-Shaped Springs 289 11.1 Meander-Shaped Compression Springs for Automotive Suspensions 289 11.1.1 Bending Stress State of Corrugated Springs 289 11.1.2 \"Equivalent Beam\" of a Meander Spring 292 11.1.3 Axial and Lateral Stiffness of Corrugated Springs 292 11.1.4 Effective Spring Constants of Meander and Coil Springs for Bending and Compression 293 11.2 Multiarc-Profiled Spring Under Axial Compressive Load 294 11.2.1 Multiarc Meander Spring With Constant Cross-Section 294 11.2.2 Multiarc Meander Spring With Optimal Cross-Section 297 11.2.3 Comparison of Masses for Fixed Spring Rate and Stress 298 11.3 Sinusoidal Spring Under Compressive Axial Load 299 11.3.1 Sinusoidal Meander Spring With Constant Cross-Section 299 11.3.2 Sinusoidal Meander Spring With Optimal Cross-Section 301 11.3.3 Comparison of Masses for Fixed Spring Rate and Stress 302 11.4 Bending Stiffness of Meander Spring With a Constant Cross-Section 303 11.4.1 Bending Stiffness of a Multiarc Meander Spring With a Constant Cross-Section 303 11.4.2 Bending Stiffness of a Sinusoidal Meander Spring with a Constant Cross-Section 303 11.5 Stability of Corrugated Springs 304 11.5.1 Euler’s Buckling of an Axially Compressed Rod 304 11.5.2 Side Buckling of Meander Springs 306 11.6 Patents for Chassis Springs Made of Composites in Meandering Form 307 11.7 Conclusions 314 References 315 12 Hereditary Mechanics of Composite Springs and Driveshafts 317 12.1 Elements of Hereditary Mechanics of Composite Materials 317 12.1.1 Mechanisms of Time-Dependent Deformation of Composites 317 12.1.2 Linear Viscoelasticity of Composites 318 12.1.3 Nonlinear Creep Mechanics of Anisotropic Materials 319 12.1.4 Anisotropic Norton-Bailey Law 321 12.2 Creep and Relaxation of Twisted Composite Shafts 322 12.2.1 Constitutive Equations for Relaxation in Torsion of Anisotropic Shafts 322 12.2.2 Torque Relaxation for an Anisotropic Norton-Bailey Law 322 12.3 Creep and Relaxation of Composite Helical Coiled Springs 323 12.3.1 Compression and Tension Composite Springs 323 12.3.2 Relaxation of Helical Composite Springs 324 12.3.3 Creep of Helical Composite Compression Springs 324 12.4 Creep and Relaxation of Composite Springs in a State of Pure Bending 325 12.4.1 Constitutive Equations for Bending Relaxation 325 12.4.2 Relaxation of the Bending Moment for the Anisotropic Norton-Bailey Law 326 12.4.3 Creep in a State of Bending 326 12.5 Conclusions 327 References 327 Appendix A Mechanical Properties of Composites 331 A.1 Fibers 331 A.1.1 Glass Fibers 331 A.1.2 Carbon Fibers 331 A.1.3 Aramid Fibers 331 A.2 Physical Properties of Resin 332 A.3 Laminates 334 A.3.1 Unidirectional Fiber-Reinforced Composite Material 334 A.3.2 Fabric 334 A.3.3 Non-Woven Fabric 334 References 335 Appendix B Anisotropic Elasticity 337 B.1 Elastic Orthotropic Body 337 B.2 Distortion Energy and Supplementary Energy 338 B.3 Plane Elasticity Problems 339 B.3.1 Plane Strain State 339 B.3.2 Plane Stress State 339 B.4 Generalized Airy Stress Function 340 B.4.1 Plane Stress State 340 B.4.2 Plane Strain State 340 B.4.3 Rotationally Symmetric Elasticity Problems 340 Appendix C Integral Transforms in Elasticity 343 C.1 One-Dimensional Integral Transform 343 C.2 Two-Dimensional Fourier Transform 344 C.3 Potential Functions for Plane Elasticity Problems 344 C.4 Rotationally Symmetric, Spatial Elasticity Problems 346 C.5 Application of the Fourier Transformation to Plane Elasticity Problems 348 C.6 Application of the Hankel Transformation to Spatial, Rotation-Symmetric Elasticity Problems 349 Index 351

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