Design and Analysis of Composite Structures for Automotive Applications

VLADIMIR KOBELEV, PHD, is a professor of mechanical engineering at the University of Siegen in Germany. He is a member of the International Society for Structural and Multidisciplinary Optimization and EUROMECH. He has authored three other books, including Durability of Springs, and has contributed over 60 articles to international scientific journals.

• Foreword xiiiSeries Preface xvList of Symbols and Abbreviations xviiIntroduction xxiiiAbout the Companion Website xxxv1 Elastic Anisotropic Behavior of Composite Materials 11.1 Anisotropic Elasticity of Composite Materials 11.1.1 Fourth Rank Tensor Notation of Hooke’s Law 11.1.2 Voigt’s Matrix Notation of Hooke’s Law 21.1.3 Kelvin’s Matrix Notation of Hooke’s Law 51.2 Unidirectional Fiber Bundle 71.2.1 Components of a Unidirectional Fiber Bundle 71.2.2 Elastic Properties of a Unidirectional Fiber Bundle 71.2.3 Effective Elastic Constants of Unidirectional Composites 81.3 Rotational Transformations of Material Laws, Stress and Strain 101.3.1 Rotation of Fourth Rank Elasticity Tensors 111.3.2 Rotation of Elasticity Matrices in Voigt’s Notation 111.3.3 Rotation of Elasticity Matrices in Kelvin’s Notation 131.4 Elasticity Matrices for Laminated Plates 141.4.1 Voigt’s Matrix Notation for Anisotropic Plates 141.4.2 Rotation of Matrices in Voigt’s Notation 151.4.3 Kelvin’s Matrix Notation for Anisotropic Plates 151.4.4 Rotation of Matrices in Kelvin’s Notation 161.5 Coupling Effects of Anisotropic Laminates 171.5.1 Orthotropic Laminate Without Coupling 171.5.2 Anisotropic Laminate Without Coupling 171.5.3 Anisotropic Laminate With Coupling 171.5.4 Coupling Effects in Laminated Thin-Walled Sections 181.6 Conclusions 18References 192 Phenomenological Failure Criteria of Composites 212.1 Phenomenological Failure Criteria 212.1.1 Criteria for Static Failure Behavior 212.1.2 Stress Failure Criteria for Isotropic Homogenous Materials 212.1.3 Phenomenological Failure Criteria for Composites 222.1.4 Phenomenological Criteria Without Stress Coupling 232.1.4.1 Criterion of Maximum Averaged Stresses 232.1.4.2 Criterion of Maximum Averaged Strains 242.1.5 Phenomenological Criteria with Stress Coupling 242.1.5.1 Mises–Hill Anisotropic Failure Criterion 242.1.5.2 Pressure-Sensitive Mises–Hill Anisotropic Failure Criterion 262.1.5.3 Tensor-Polynomial Failure Criterion 272.1.5.4 Tsai–Wu Criterion 302.1.5.5 Assessment of Coefficients in Tensor-Polynomial Criteria 302.2 Differentiating Criteria 332.2.1 Fiber and Intermediate Break Criteria 332.2.2 Hashin Strength Criterion 332.2.3 Delamination Criteria 352.3 Physically Based Failure Criteria 352.3.1 Puck Criterion 352.3.2 Cuntze Criterion 362.4 Rotational Transformation of Anisotropic Failure Criteria 372.5 Conclusions 40References 403 Micromechanical Failure Criteria of Composites 453.1 Pullout of Fibers from the Elastic-Plastic Matrix 453.1.1 Axial Tension of Fiber and Matrix 453.1.2 Shear Stresses in Matrix Cylinders 513.1.3 Coupled Elongation of Fibers and Matrix 533.1.4 Failures in Matrix and Fibers 543.1.4.1 Equations for Mean Axial Displacements of Fibers and Matrix 543.1.4.2 Solutions of Equations for Mean Axial Displacements of Fibers and Matrix 563.1.5 Rupture of Matrix and Pullout of Fibers from Crack Edges in a Matrix 573.1.5.1 Elastic Elongation (Case I) 573.1.5.2 Plastic Sliding on the Fiber Surface (Case II) 583.1.5.3 Fiber Breakage (Case III) 583.1.6 Rupture of Fibers, Matrix Joints and Crack Edges 593.2 Crack Bridging in Elastic-Plastic Unidirectional Composites 603.2.1 Crack Bridging in Unidirectional Fiber-Reinforced Composites 603.2.2 Matrix Crack Growth 613.2.3 Fiber Crack Growth 623.2.4 Penny-Shaped Crack 653.2.4.1 Crack in a Transversal-Isotropic Medium 653.2.4.2 Mechanisms of the Fracture Process 663.2.4.3 Crack Bridging in an Orthotropic Body With Disk Crack 663.2.4.4 Solution to an Axially Symmetric Crack Problem 683.2.5 Plane Crack Problem 723.2.5.1 Equations of the Plane Crack Problem 723.2.5.2 Solution to the Plane Crack Problem 743.3 Debonding of Fibers in Unidirectional Composites 753.3.1 Axial Deformation of Unidirectional Fiber Composites 753.3.2 Stresses in Unidirectional Composite in Cases of Ideal Debonding or Adhesion 793.3.2.1 Equations of an Axially Loaded Unidirectional Compound Medium (A) 793.3.2.2 Total Debonding (B) 823.3.2.3 Ideal Adhesion (C) 833.3.3 Stresses in a Unidirectional Composite in a Case of Partial Debonding 843.3.3.1 Partial Radial Load on the Fiber Surface 843.3.3.2 Partial Radial Load on the Matrix Cavity Surface 843.3.3.3 Partial Debonding With Central Adhesion Region (D) 853.3.3.4 Partial Debonding With Central Debonding Region (E) 883.3.3.5 Semi-Infinite Debonding With Central Debonding Region (F) 893.3.4 Contact Problem for a Finite Adhesion Region 893.3.5 Debonding of a Semi-Infinite Adhesion Region 933.3.6 Debonding of Fibers from a Matrix Under Cyclic Deformation 953.4 Conclusions 98References 984 Optimization Principles for Structural Elements Made of Composites 1054.1 Stiffness Optimization of Anisotropic Structural Elements 1054.1.1 Optimization Problem 1054.1.2 Optimality Conditions 1064.1.3 Optimal Solutions in Anti-Plane Elasticity 1094.1.4 Optimal Solutions in Plane Elasticity 1094.2 Optimization of Strength and Loading Capacity of Anisotropic Elements 1104.2.1 Optimization Problem 1104.2.2 Optimality Conditions 1134.2.3 Optimal Solutions in Anti-Plane Elasticity 1144.2.4 Optimal Solutions in Plane Elasticity 1144.3 Optimization of Accumulated Elastic Energy in Flexible Anisotropic Elements 1164.3.1 Optimization Problem 1164.3.2 Optimality Conditions 1174.3.3 Optimal Solutions in Anti-Plane Elasticity 1184.3.4 Optimal Solutions in Plane Elasticity 1194.4 Optimal Anisotropy in a Twisted Rod 1194.5 Optimal Anisotropy of Bending Console 1224.6 Optimization of Plates in Bending 1234.7 Conclusions 125References 1255 Optimization of Composite Driveshaft 1295.1 Torsion of Anisotropic Shafts With Solid Cross-Sections 1295.2 Thin-Walled Anisotropic Driveshaft with Closed Profile 1325.2.1 Geometry of Cross-Section 1325.2.2 Main Kinematic Hypothesis 1335.3 Deformation of a Composite Thin-Walled Rod 1355.3.1 Equations of Deformation of a AnisotropicThin-Walled Rod 1355.3.2 Boundary Conditions 1385.3.2.1 Ideal Fixing 1385.3.2.2 Ideally Free End 1385.3.2.3 Boundary Conditions of the Intermediate Type 1405.3.3 Governing Equations in Special Cases of Symmetry 1405.3.3.1 Orthotropic Material 1405.3.3.2 Constant Elastic Properties Along the Arc of a Cross-Section 1405.3.4 Symmetry of Section 1405.4 Buckling of Composite Driveshafts Under a Twist Moment 1415.4.1 Greenhill’s Buckling of Driveshafts 1415.4.2 Optimal Shape of the Solid Cross-Section for Driveshaft 1435.4.3 Hollow Circular and Triangular Cross-Sections 1445.5 Patents for Composite Driveshafts 1465.6 Conclusions 150References 1506 Dynamics of a Vehicle with Rigid Structural Elements of Chassis 1556.1 Classification of Wheel Suspensions 1556.1.1 Common Designs of Suspensions 1556.1.2 Types of Twist-Beam Axles 1566.1.3 Kinematics of Wheel Suspensions 1576.2 Fundamental Models in Vehicle Dynamics 1596.2.1 Basic Variables of Vehicle Dynamics 1596.2.2 Coordinate Systems of Vehicle and Local Coordinate Systems 1616.2.2.1 Earth-Fixed Coordinate System 1616.2.2.2 Vehicle-Fixed Coordinate System 1626.2.2.3 Horizontal Coordinate System 1626.2.2.4 Wheel Coordinate System 1626.2.3 Angle Definitions 1626.2.4 Components of Force and Moments in Car Dynamics 1636.2.5 Degrees of Freedom of a Vehicle 1636.3 Forces Between Tires and Road 1676.3.1 Tire Slip 1676.3.2 Side Slip Curve and Lateral Force Properties 1686.4 Dynamic Equations of a Single-Track Model 1706.4.1 Hypotheses of a Single-Track Model 1706.4.2 Moments and Forces in a Single-Track Model 1716.4.3 Balance of Forces and Moments in a Single-Track Model 1736.4.4 Steady Cornering 1746.4.4.1 Necessary Steer Angle for Steady Cornering 1746.4.4.2 Yaw Gain Factor and Steer Angle Gradient 1756.4.4.3 Classification of Self-Steering Behavior 1766.4.5 Non-Steady Cornering 1796.4.5.1 Equations of Non-Stationary Cornering 1796.4.5.2 Oscillatory Behavior of Vehicle During Non-Steady Cornering 1806.4.6 Anti-Roll Bars Made of Composite Materials 1816.5 Conclusions 182References 1827 Dynamics of a Vehicle With Flexible, Anisotropic Structural Elements of Chassis 1837.1 Effects of Body and Chassis Elasticity on Vehicle Dynamics 1837.1.1 Influence of Body Stiffness on Vehicle Dynamics 1837.1.2 Lateral Dynamics of Vehicles With Stiff Rear Axles 1847.1.3 Induced Effects on Wheel Orientation and Positioning of Vehicles with Flexible Rear Axle 1857.2 Self-Steering Behavior of a Vehicle With Coupling of Bending and Torsion 1887.2.1 Countersteering for Vehicles with Twist-Beam Axles 1887.2.1.1 Countersteering Mechanisms 1887.2.1.2 Countersteering by Anisotropic Coupling of Bending and Torsion 1907.2.2 Bending-Twist Coupling of a Countersteering Twist-Beam Axle 1927.2.3 Roll Angle of Vehicle 1937.2.3.1 Relationship Between Roll Angle and Centrifugal Force 1937.2.3.2 Lateral Reaction Forces on Wheels 1937.2.3.3 Steer Angles on Front Wheels 1947.2.3.4 Steer Angles on Rear Wheels 1947.3 Steady Cornering of a Flexible Vehicle 1967.3.1 Stationary Cornering of a Car With a Flexible Chassis 1967.3.2 Necessary Steer Angles for Coupling and Flexibility of Chassis 1967.3.2.1 Limit Case: Lateral Acceleration Vanishes 1967.3.2.2 Absolutely Rigid Front and Rear Wheel Suspensions 1977.3.2.3 Bending and Torsion of a Twist Member Completely Decoupled 1977.3.2.4 General Case of Coupling Between Bending and Torsion of a Twist Member 1987.3.2.5 Neutral Steering Caused by Coupling Between Bending and Torsion of a Twist Member 1987.4 Estimation of Coupling Constant for a Twist Member 1997.4.1 Coupling Between Vehicle Roll Angle and Twist of Cross-Member 1997.4.2 Stiffness Parameters of a Twist-Beam Axle 2007.4.2.1 Roll Spring Rate 2007.4.2.2 Lateral Stiffness 2017.4.2.3 Camber Stiffness 2037.5 Design of the Countersteering Twist-Beam Axle 2037.5.1 Requirements for a Countersteering Twist-Beam Axle 2037.5.2 Selection and Calculation of the Cross-Section for the Cross-Member 2057.5.3 Elements of a Countersteering Twist-Beam Axle 2087.6 Patents on Twist-Beam Axles 2117.7 Conclusions 214References 2148 Design and Optimization of Composite Springs 2178.1 Design and Optimization of Anisotropic Helical Springs 2178.1.1 Forces and Moments in Helical Composite Springs 2178.1.2 Symmetrically Designed Solid Bar With Circular Cross-Section 2208.1.3 Stiffness and Stored Energy of Helical Composite Springs 2238.1.4 Spring Rates of Helical Composite Springs 2258.1.5 Helical Composite Springs of Minimal Mass 2288.1.5.1 Optimization Problem 2288.1.5.2 Optimal Composite Spring for the Anisotropic Mises–Hill Strength Criterion 2288.1.6 Axial and Twist Vibrations of Helical Springs 2318.2 Conical Springs Made of Composite Material 2338.2.1 Geometry of an Anisotropic Conical Spring in an Undeformed State 2338.2.2 Curvature and Strain Deviations 2358.2.3 Thin-Walled Conical Shells Made of Anisotropic Materials 2368.2.4 Variation Principle 2378.2.5 Structural Optimization of a Conical Spring Due to Ply Orientation 2398.2.6 Conical Spring Made of Orthotropic Material 2418.2.7 Bounds for Stiffness of a Spring Made of Orthotropic Material 2438.3 Alternative Concepts for Chassis Springs Made of Composites 2448.4 Conclusions 248References 2499 Equivalent Beams of Helical Anisotropic Springs 2559.1 Helical Compression Springs Made of Composite Materials 2559.1.1 Statics of the Equivalent Beam for an Anisotropic Spring 2559.1.2 Dynamics of an Equivalent Beam for an Anisotropic Spring 2589.2 Transverse Vibrations of a Composite Spring 2609.2.1 Separation of Variables 2609.2.2 Fundamental Frequencies of Transversal Vibrations 2629.2.3 Transverse Vibrations of a Symmetrically Stacked Helical Spring 2649.3 Side Buckling of a Helical Composite Spring 2659.3.1 Buckling Under Axial Force 2659.3.2 Simplified Formulas for Buckling of a Symmetrically Stacked Helical Spring 2669.4 Conclusions 267References 26710 Composite Leaf Springs 26910.1 Longitudinally Mounted Leaf Springs for Solid Axles 26910.1.1 Predominantly Bending-Loaded Leaf Springs 26910.1.2 Moments and Forces of Leaf Springs in a Pure Bending State 27010.1.3 Optimization of Leaf Springs for an Anisotropic Mises–Hill Criterion 27210.2 Leaf-Tension Springs 27510.2.1 Combined Bending and Tension of a Spring 27510.2.2 Forces and Rates of Leaf-Tension Springs 27710.3 Transversally Mounted Leaf Springs 27810.3.1 Axle Concepts of Transverse Leaf Springs 27810.3.2 Analysis of a Transverse Leaf Spring 28010.3.3 Examples and Patents for Transversely Mounted Leaf Springs 28310.4 Conclusions 286References 28711 Meander-Shaped Springs 28911.1 Meander-Shaped Compression Springs for Automotive Suspensions 28911.1.1 Bending Stress State of Corrugated Springs 28911.1.2 “Equivalent Beam” of a Meander Spring 29211.1.3 Axial and Lateral Stiffness of Corrugated Springs 29211.1.4 Effective Spring Constants of Meander and Coil Springs for Bending and Compression 29311.2 Multiarc-Profiled Spring Under Axial Compressive Load 29411.2.1 Multiarc Meander Spring With Constant Cross-Section 29411.2.2 Multiarc Meander Spring With Optimal Cross-Section 29711.2.3 Comparison of Masses for Fixed Spring Rate and Stress 29811.3 Sinusoidal Spring Under Compressive Axial Load 29911.3.1 Sinusoidal Meander Spring With Constant Cross-Section 29911.3.2 Sinusoidal Meander Spring With Optimal Cross-Section 30111.3.3 Comparison of Masses for Fixed Spring Rate and Stress 30211.4 Bending Stiffness of Meander Spring With a Constant Cross-Section 30311.4.1 Bending Stiffness of a Multiarc Meander Spring With a Constant Cross-Section 30311.4.2 Bending Stiffness of a Sinusoidal Meander Spring with a Constant Cross-Section 30311.5 Stability of Corrugated Springs 30411.5.1 Euler’s Buckling of an Axially Compressed Rod 30411.5.2 Side Buckling of Meander Springs 30611.6 Patents for Chassis Springs Made of Composites in Meandering Form 30711.7 Conclusions 314References 31512 Hereditary Mechanics of Composite Springs and Driveshafts 31712.1 Elements of Hereditary Mechanics of Composite Materials 31712.1.1 Mechanisms of Time-Dependent Deformation of Composites 31712.1.2 Linear Viscoelasticity of Composites 31812.1.3 Nonlinear Creep Mechanics of Anisotropic Materials 31912.1.4 Anisotropic Norton–Bailey Law 32112.2 Creep and Relaxation of Twisted Composite Shafts 32212.2.1 Constitutive Equations for Relaxation in Torsion of Anisotropic Shafts 32212.2.2 Torque Relaxation for an Anisotropic Norton–Bailey Law 32212.3 Creep and Relaxation of Composite Helical Coiled Springs 32312.3.1 Compression and Tension Composite Springs 32312.3.2 Relaxation of Helical Composite Springs 32412.3.3 Creep of Helical Composite Compression Springs 32412.4 Creep and Relaxation of Composite Springs in a State of Pure Bending 32512.4.1 Constitutive Equations for Bending Relaxation 32512.4.2 Relaxation of the Bending Moment for the Anisotropic Norton–Bailey Law 32612.4.3 Creep in a State of Bending 32612.5 Conclusions 327References 327Appendix A Mechanical Properties of Composites 331A.1 Fibers 331A.1.1 Glass Fibers 331A.1.2 Carbon Fibers 331A.1.3 Aramid Fibers 331A.2 Physical Properties of Resin 332A.3 Laminates 334A.3.1 Unidirectional Fiber-Reinforced Composite Material 334A.3.2 Fabric 334A.3.3 Non-Woven Fabric 334References 335Appendix B Anisotropic Elasticity 337B.1 Elastic Orthotropic Body 337B.2 Distortion Energy and Supplementary Energy 338B.3 Plane Elasticity Problems 339B.3.1 Plane Strain State 339B.3.2 Plane Stress State 339B.4 Generalized Airy Stress Function 340B.4.1 Plane Stress State 340B.4.2 Plane Strain State 340B.4.3 Rotationally Symmetric Elasticity Problems 340Appendix C Integral Transforms in Elasticity 343C.1 One-Dimensional Integral Transform 343C.2 Two-Dimensional Fourier Transform 344C.3 Potential Functions for Plane Elasticity Problems 344C.4 Rotationally Symmetric, Spatial Elasticity Problems 346C.5 Application of the Fourier Transformation to Plane Elasticity Problems 348C.6 Application of the Hankel Transformation to Spatial, Rotation-Symmetric Elasticity Problems 349Index 351