Title Details

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 been published in journals more than 60 times.

List of symbols 1

Introduction 1

Composites in automotive chassis and drivetrain 1

Physical properties of composite materials 4

Structure of the book 9

Target audience of the book 13

References 13

Chapter 1 Elastic anisotropic behavior of composite materials 21

1.1 Anisotropic elasticity of composite materials 21

1.1.1 4th rank tensor notation of Hooke’s law 21

1.1.2 Voigt’s matrix notation of Hooke’s law 22

1.1.3 Kelvin’s matrix notation of Hooke’s law 25

1.2 Unidirectional fiber bundle 26

1.2.1 Components of unidirectional fiber bundle 26

1.2.2 Elastic properties of unidirectional fiber bundle 27

1.2.3 Effective elastic constants of unidirectional composites 30

1.3 Rotational transformations of material laws, stress and strain 32

1.3.1 Rotation of 4th rank elasticity tensors 33

1.3.2 Rotation of elasticity matrices in Voigt’s notation 34

1.3.3 Rotation of elasticity matrices in Kelvin’s notation 36

1.4 Elasticity matrices for laminated plates 37

1.4.1 Voigt’s matrix notation for anisotropic plates 37

1.4.2 Rotation of matrices in Voigt’s notation 39

1.4.3 Kelvin’s matrix notation for anisotropic plates 40

1.4.4 Rotation of matrices in Kelvin’s notation 42

1.5 Coupling effects of anisotropic laminates 43

1.5.1 Orthotropic laminate without coupling 43

1.5.2 Anisotropic laminate without coupling 43

1.5.3 Anisotropic laminate with coupling 44

1.5.4 Coupling effects in laminated thin-walled sections 44

Conclusions 45

References 45

Chapter 2 Phenomenological failure criteria of composites 45

2.1 Phenomenological failure criteria 45

2.1.1 Criteria for static failure behavior 45

2.1.2 Stress failure criteria for isotropic homogenous materials 45

2.1.3 Phenomenological failure criteria for composites 46

2.1.4 Phenomenological criteria without stress coupling 47

2.1.5 Phenomenological criteria with stress coupling 49

2.2 Differentiating criteria 61

2.2.1 Fiber and intermediate break criteria 61

2.2.2 Hashin strength criterion 62

2.2.3 Delamination criteria 64

2.3 Physically based failure criteria 65

2.3.1 Puck criterion 65

2.3.2 Cuntze criterion 66

2.4 Rotational transformation of anisotropic failure criteria 67

Conclusions 72

References 72

Chapter 3 Micromechanical failure criteria of composites 81

3.1. Pull-Out of fibers from elastic-plastic matrix 81

3.1.1 Axial tension of fiber and matrix 81

3.1.2 Shear stresses in matrix cylinders 91

3.1.3 Coupled elongation of fibers and matrix 94

3.1.4 Failures in matrix and fibers 95

3.1.6 Rupture of fibers, matrix joints crack edges 104

3.2 Crack bridging in elastic-plastic unidirectional composites 106

3.2.1 Crack bridging in unidirectional fiber reinforced composites 106

3.2.2 Matrix crack growth 107

3.2.3 Fiber crack growth 108

3.2.4 Penny-shaped crack 114

3.2.5 Plane crack problem 125

3.3. Debonding of fibers in unidirectional composites 131

3.3.1 Axial deformation of unidirectional fiber composites 131

3.3.2 Stresses in unidirectional composite in cases of ideal debonding or adhesion 138

3.3.3 Stresses in unidirectional composite in case of partial debonding 143

3.3.4 Contact problem for finite adhesion region 152

3.3.5 Debonding of semi-infinite adhesion region 160

3.3.6 Debonding of fibers from matrix under cyclic deformation 164

Conclusions 168

References 168

Chapter 4 Optimization principles for structural elements made of composites 151

4.1 Stiffness optimization of anisotropic structural elements 151

4.1.1 Optimization problem 151

4.1.2 Optimality conditions 153

4.1.3 Optimal solutions in anti-plane elasticity 157

4.1.4 Optimal solutions in plane elasticity 157

4.2 Optimization of strength and loading capacity of anisotropic elements 160

4.2.1 Optimization problem 160

4.2.2 Optimality conditions 162

4.2.3 Optimal solutions in anti-plane elasticity 163

4.2.4 Optimal solutions in plane elasticity 164

4.3 Optimization of accumulated elastic energy in flexible anisotropic elements 167

4.3.1 Optimization problem 167

4.3.2 Optimality conditions 168

4.3.3 Optimal solutions in anti-plane elasticity 169

4.3.4 Optimal solutions in plane elasticity 170

4.4 Optimal anisotropy in twisted rod 172

4.5 Optimal anisotropy of bending console 175

4.6 Optimization of plates in bending 177

Conclusions 179

References 180

Chapter 5 Optimization of composite driveshaft 181

5.1 Torsion of anisotropic shafts with solid cross-section 181

5.2 Thin-walled anisotropic driveshaft with closed profile 186

5.2.1 Geometry of cross-section 186

5.2.2 Main kinematic hypothesis 188

5.3 Deformation of composite thin-walled rod 191

5.3.1 Equations of deformation of anisotropic thin-walled rod 191

5.3.2. Boundary conditions 196

5.3.4 Symmetry of section 200

5.4 Buckling of composite driveshafts under twist moment 201

5.4.1 Greenhill’s buckling of driveshafts 201

5.4.2 Optimal shape of the solid cross-section for driveshaft 204

5.4.3 Hollow circular and triangle cross-sections 206

5.5 Patents for composite driveshafts 208

Conclusions 214

References 214

Chapter 6 Dynamics of vehicle with rigid structural elements of chassis 213

6.1. Classification of wheel suspensions 213

6.1.2 Common design of suspensions 213

6.1.3 Types of twist-beam axles 215

6.1.4 Kinematics of wheel suspension 216

6.2 Fundamental models in vehicle dynamics 218

6.2.1 Basic variables of vehicle dynamics 218

6.2.2 Coordinate systems of vehicle and local coordinate systems 221

6.2.3 Angle definitions 223

6.2.4 Components of force and moments in car dynamics 228

6.2.5 Degrees of freedom of vehicle 230

6.3 Forces between tires and road 231

6.3.1 Tire slip 231

6.3.2 Side slip curve and lateral force properties 233

6.4 Dynamic equations of single-track model 235

6.4.1 Hypotheses of single-track model 235

6.4.2 Moments and forces in single-track model 237

6.4.3 Balance of forces and moments in single-track model 241

6.4.4 Steady cornering 242

6.4.5 Non-steady cornering 248

6.4.6 Anti-roll bars made of composite materials 251

Conclusions 253

References 253

Chapter 7 Dynamics of vehicle with flexible, anisotropic structural elements of chassis 255

7.1 Effects of body and chassis elasticity on vehicle dynamics 255

7.1.1 Influence of body stiffness on vehicle dynamics 255

7.1.2 Lateral dynamics of vehicles with stiff rear axle 256

7.1.3 Induced effects on wheel orientation and positioning of vehicles with flexible rear axle 258

7.2 Self-steering behavior of the vehicle with coupling of bending and torsion 264

7.2.1 Countersteering for vehicles with twist-beam axes 264

7.2.2 Bending-twist coupling of countersteering twist-beam axle 267

7.2.3 Roll angle of vehicle 269

7.3 Steady cornering of flexible vehicle 273

7.3.1 Stationary cornering of car with flexible chassis 273

7.3.2 Necessary steer angles for coupling and flexibility of chassis 274

7.4 Estimation of coupling constant for twist member 279

7.4.1 Coupling between vehicle roll angle and twist of cross-member 279

7.4.2 Stiffness parameters of twist-beam axle 280

7.5 Design of the countersteering twist-beam axle 287

7.5.1 Requirements for countersteering twist-beam axle 287

7.5.2 Selection and calculation of cross-section for the cross-member 292

7.5.3 Elements of countersteering twist-beam axle 293

7.6 Patents on twist-beam axes 301

Conclusions 305

References 306

Chapter 8 Design and optimization of composite springs 309

8.1 Design and optimization of anisotropic helical springs 309

8.1.1 Forces and moments in helical composite springs 309

8.1.2 Symmetrically designed solid bar with circular cross-section 317

8.1.3 Stiffness and stored energy of helical composite springs 319

8.1.4 Spring rates of helical composite springs 323

8.1.5 Helical composite springs of minimal mass 326

8.1.6 Axial and twist vibrations of helical springs 330

8.2 Conical springs made of composite material 333

8.2.1 Geometry of anisotropic conical spring in undeformed state 333

8.2.2 Curvature and strain deviations 336

8.2.3 Thin-walled conical shells made of anisotropic materials 337

8.2.4. Variation principle 339

8.2.5. Structural optimization of conical spring due to ply orientation 342

8.2.6 Conical spring made of orthotropic material 344

8.2.7. Bounds for stiffness of spring made of orthotropic material 346

8.3 Alternative concepts for chassis springs made of composites 355

Conclusions 363

References 367

Chapter 9 “Equivalent Beams” of helical anisotropic springs 365

9.1 Helical compression springs made of composite materials 365

9.1.1 Statics of “equivalent beam” for anisotropic spring 365

9.1.2 Dynamics of “equivalent beam” for anisotropic spring 368

9.2 Transverse vibrations of composite spring 371

9.2.1 Separation of variables 371

9.2.2 Fundamental frequencies of transversal vibrations 374

9.2.3 Transverse vibrations of symmetrically stacked helical spring 376

9.3 Side buckling of helical composite spring 377

9.3.1 Buckling under axial force 377

9.3.2 Simplified formulas for buckling of symmetric stacked helical spring.. 379

Conclusions 379

References 380

Chapter 10 Composite leaf springs 393

10.1 Longitudinally mounted leaf springs for solid axes 393

10.1.1 Predominantly bending-loaded leaf springs 393

10.1.2 Moments and forces of leaf springs in pure bending state 395

10.1.3 Optimization of leaf springs for anisotropic Mises-Hill criterion 396

10.2. Leaf-tension springs 401

10.2.1 Combined bending and tension of spring 401

10.2.2 Forces and rates of leaf-tension springs 403

10.3 Transversally mounted leaf springs 407

10.3.1 Axle concepts of transverse leaf spring 407

10.3.2 Analysis of transverse leaf spring 409

10.3.3 Examples and patents for transversely mounted leaf springs 412

Conclusions 415

References 416

Chapter 11 Meander-shaped springs 421

11.1 Meander-shaped compression springs for automotive suspensions 421

11.1.1 Bending stress state of corrugated springs 421

11.1.2 “Equivalent beam” of meander spring 426

11.1.3 Axial and lateral stiffness of corrugated springs 426

11.1.4 Effective spring constants of meander and coil springs for bending and compression 429

11.2 Multiarc-profiled spring under axial compressive load 429

11.2.1 Multiarc meander spring with constant cross-section 429

11.2.2 Multiarc meander spring with optimal cross-section 433

11.2.3 Comparison of masses for fixed spring rate and stress 435

11.3 Sinusoidal spring under compressive axial load 438

11.3.1 Sinusoidal meander spring with constant cross-section 438

11.3.2 Sinusoidal meander spring with optimal cross-section 440

11.3.3 Comparison of masses for fixed spring rate and stress 442

11.4 Bending stiffness of meander spring with constant cross-section 443

11.4.1 Bending stiffness of multiarc meander spring with constant cross-section 443

11.4.2 Bending stiffness of sinusoidal meander spring with constant cross-section 444

11.5 Stability of corrugated springs 448

11.5.1 Euler’s buckling of axially compressed rod 448

11.5.2 Side buckling of meander springs 448

11.6 Patents for chassis springs made of composites in meandering form 450

Conclusions 460

References 460

Chapter 12 Hereditary mechanics of composite springs and driveshafts 460

12.1 Elements of hereditary mechanics of composite materials 460

12.1.1 Mechanisms of time-dependent deformation of composites 460

12.1.2 Linear viscoelasticity of composites 462

12.1.3 Nonlinear creep mechanics of anisotropic materials 463

12.1.4 Anisotropic Norton-Bailey law 466

12.2 Creep and relaxation of twisted composite shafts 467

12.2.1 Constitutive equations for relaxation in torsion of anisotropic shafts 467

12.2.2 Torque relaxation for anisotropic Norton-Bailey law 468

12.3 Creep and relaxation of composite helical coiled springs 470

12.3.1 Compression and tension composite spring 470

12.3.2 Relaxation of helical composite springs 471

12.3.3 Creep of helical composite compression springs 472

12.4 Creep and relaxation of composite springs in state of pure bending 473

12.4.1 Constitutive equations for bending relaxation 473

12.4.1 Relaxation of bending moment for anisotropic Norton-Bailey law 474

12.4.3 Creep in state of bending 475

Conclusions 476

References 476

Appendices i

Appendix A Mechanical properties of composites i

A.1 Fibers i

A.1.1 Glass Fibers i

A.1.2 Carbon fibers i

A.1.3 Aramid Fibers i

A.2 Physical properties of resin iii

A.3 Laminates iv

A.3.2 Unidirectional fiber-reinforced composite material iv

A.3.1 Fabric iv

A.3.3 Non-Woven Fabric iv

Appendix B Anisotropic elasticity v

B.1 Elastic Orthotropic Body v

B.2 Distortion energy and supplementary energy viii

B.3 Plane elasticity problems ix

B.3.2 Plane strain state ix

B.3.3 Plane stress state ix

B.4 Generalized Airy stress function x

B.4.1 Plane stress state x

B.4.2 Plane strain state x

B.4.3 Rotationally symmetric elasticity problems x

Appendix C Integral transforms in elasticity xi

C.1 One-dimensional integral transform xi

C.2 Two-dimensional Fourier-Transform xiii

C.3 Potential functions for plane elasticity problems xiii

C.4 Rotationally symmetric, spatial elasticity problems xvii

C.5 Application of the Fourier transformation to plane elasticity problems xix

C.6 Application of the Hankel transformation to spatial, rotation-symmetrical elasticity problems xxi

Index xxiv