Preface xix

Acknowledgements xxix

**Part A Constitutive Equations of Materials**** 1**

**Chapter 1 Elements of Anisotropic Elasticity and Complements on Previsional Calculations ****3**

*Yvon CHEVALIER*

1.1 Constitutive equations in a linear elastic regime 4

1.2 Technical elastic moduli 7

1.3 Real materials with special symmetries 10

1.4 Relationship between compliance Sij and stiffness Cij for orthotropic materials 23

1.5 Useful inequalities between elastic moduli 24

1.6 Transformation of reference axes is necessary in many circumstances 27

1.7 Invariants and their applications in the evaluation of elastic constants 28

1.8 Plane elasticity 35

1.9 Elastic previsional calculations for anisotropic composite materials 38

1.10 Bibliography 51

1.11 Appendix 52

Appendix 1.A Overview on methods used in previsional calculation of fiber-reinforced composite materials 52

**Chapter 2 Elements of Linear Viscoelasticity ****57**

*Yvon CHEVALIER*

2.1 Time delay between sinusoidal stress and strain 59

2.2 Creep and relaxation tests 60

2.3 Mathematical formulation of linear viscoelasticity 63

2.4 Generalization of creep and relaxation functions to tridimensional constitutive equations 71

2.5 Principle of correspondence and Carson-Laplace transform for transient viscoelastic problems 74

2.6 Correspondence principle and the solution of the harmonic viscoelastic system 82

2.7 Inter-relationship between harmonic and transient regimes 83

2.8 Modeling of creep and relaxation functions: example 87

2.9 Conclusion 100

2.10 Bibliography 100

**Chapter 3 Two Useful Topics in Applied Viscoelasticity: Constitutive Equations for Viscoelastic Materials ****103**

*Yvon CHEVALIER and Jean Tuong VINH*

3.1 Williams-Landel-Ferry’s method 104

3.2 Viscoelastic time function obtained directly from a closed-form expression of complex modulus or complex compliance 112

3.3 Concluding remarks 136

3.4 Bibliography 137

3.5 Appendices 139

Appendix 3.A Inversion of Laplace transform 139

Appendix 3.B Sutton’s method for long time response 143

**Chapter 4 Formulation of Equations of Motion and Overview of their Solutions by Various Methods ****145**

*Jean Tuong VINH*

4.1 D’Alembert’s principle 146

4.2 Lagrange’s equation 149

4.3 Hamilton’s principle 157

4.4 Practical considerations concerning the choice of equations of motion and related solutions 159

4.5 Three-, two- or one-dimensional equations of motion? 162

4.6 Closed-form solutions to equations of motion 163

4.7 Bibliography 164

4.8 Appendices 165

Appendix 4.A Equations of motion in elastic medium deduced from Love’s variational principle 165

Appendix 4.B Lagrange’s equations of motion deduced from Hamilton’s principle 167

**Part B Rod Vibrations ****173**

**Chapter 5 Torsional Vibration of Rods ****175**

*Yvon CHEVALIER, Michel NUGUES and James ONOBIONO*

5.1 Introduction 175

5.1.1 Short bibliography of the torsion problem 176

5.1.2 Survey of solving methods for torsion problems 176

5.1.3 Extension of equations of motion to a larger frequency range 179

5.2 Static torsion of an anisotropic beam with rectangular section without bending – Saint Venant, Lekhnitskii’s formulation 180

5.3 Torsional vibration of a rod with finite length 199

5.4 Simplified boundary conditions associated with higher approximation equations of motion [5.49] 204

5.5 Higher approximation equations of motion 205

5.6 Extension of Engström’s theory to the anisotropic theory of dynamic torsion of a rod with rectangular cross-section 207

5.7 Equations of motion 212

5.8 Torsion wave dispersion 215

5.9 Presentation of dispersion curves 219

5.10 Torsion vibrations of an off-axis anisotropic rod 225

5.11 Dispersion of deviated torsional waves in off-axis anisotropic rods with rectangular cross-section 235

5.12 Dispersion curve of torsional phase velocities of an off-axis anisotropic rod 240

5.13 Concluding remarks 241

5.14 Bibliography 242

5.15 Table of symbols 244

5.16 Appendices 246

Appendix 5.A Approximate formulae for torsion stiffness 246

Appendix 5.B Equations of torsional motion obtained from Hamilton’s variational principle 250

Appendix 5.C Extension of Barr’s correcting coefficient in equations of motion 257

Appendix 5.D Details on coefficient calculations for θ (z, t) and ζ (z, t) 258

Appendix 5.E A simpler solution to the problem analyzed in Appendix 5.D 263

Appendix 5.F Onobiono’s and Zienkievics’ solutions using finite element method for warping function *φ *265

Appendix 5.G Formulation of equations of motion for an off-axis anisotropic rod submitted to coupled torsion and bending vibrations 273

Appendix 5.H Relative group velocity versus relative wave number 279

**Chapter 6 Bending Vibration of a Rod**** 291**

*Dominique LE NIZHERY*

6.1 Introduction 291

6.1.1 Short bibliography of dynamic bending of a beam 292

6.2 Bending vibration of straight beam by elementary theory 293

6.3 Higher approximation theory of bending vibration 299

6.4 Bending vibration of an off-axis anisotropic rod 313

6.5 Concluding remarks 324

6.6 Bibliography 326

6.7 Table of symbols 327

6.8 Appendices 328

Appendix 6.A Timoshenko’s correcting coefficients for anisotropic and isotropic materials 328

Appendix 6.B Correcting coefficient using Mindlin’s method 333

Appendix 6.C Dispersion curves for various equations of motion 334

Appendix 6.D Change of reference axes and elastic coefficients for an anisotropic rod 337

**Chapter 7 Longitudinal Vibration of a Rod ****339**

*Yvon CHEVALIER and Maurice TOURATIER*

7.1 Presentation 339

7.2 Bishop’s equations of motion 343

7.3 Improved Bishop’s equation of motion 345

7.4 Bishop’s equation for orthotropic materials 346

7.5 Eigenfrequency equations for a free-free rod 346

7.6 Touratier’s equations of motion of longitudinal waves 350

7.7 Wave dispersion relationships 367

7.8 Short rod and boundary conditions 393

7.9 Concluding remarks about Touratier’s theory 395

7.10 Bibliography 396

7.11 List of symbols 397

7.12 Appendices 399

Appendix 7.A an outline of some studies on longitudinal vibration of rods with rectangular cross-section 399

Appendix 7.B Formulation of Bishop’s equation by Hamilton’s principle by Rao and Rao 401

Appendix 7.C Dimensionless Bishop’s equations of motion and dimensionless boundary conditions 405

Appendix 7.D Touratier’s equations of motion by variational calculus 408

Appendix 7.E Calculation of correcting factor q (Cijkl) 409

Appendix 7.F Stationarity of functional J and boundary equations 419

Appendix 7.G On the possible solutions of eigenvalue equations 419

**Chapter 8 Very Low Frequency Vibration of a Rod by Le Rolland-Sorin’s Double Pendulum ****425**

*Mostefa ARCHI and Jean-Baptiste CASIMIR*

8.1 Introduction 425

8.2 Short bibliography 427

8.3 Flexural vibrations of a rod using coupled pendulums 427

8.4 Torsional vibration of a beam by double pendulum 434

8.5 Complex compliance coefficient of viscoelastic materials 436

8.6 Elastic stiffness of an off-axis rod 443

8.7 Bibliography 449

8.8 List of symbols 450

8.9 Appendices 452

Appendix 8.A Closed-form expression of θ_{1} or θ_{2} oscillation angles of the pendulums and practical considerations 452

Appendix 8.B Influence of the highest eigenfrequency *ω*3 on the pendulum oscillations in the expression of *θ*_{1} (t) 457

Appendix 8.C Coefficients a of compliance matrix after a change of axes for transverse isotropic material 458

Appendix 8.D Mathematical formulation of the simultaneous bending and torsion of an off-axis rectangular rod 460

Appendix 8.E Details on calculations of s_{35} and ϑ_{13} of transverse isotropic materials 486

**Chapter 9 Vibrations of a Ring and Hollow Cylinder ****493**

*Jean Tuong VINH*

9.1 Introduction 493

9.2 Equations of motion of a circular ring with rectangular cross-section 494

9.3 Bibliography 502

9.4 Appendices 503

Appendix 9.A Expression u (θ) in the three subintervals delimited by the roots of equation [9.33] 503

**Chapter 10 Characterization of Isotropic and Anisotropic Materials by Progressive Ultrasonic Waves**** 513**

*Patrick GARCEAU*

10.1 Presentation of the method 513

10.2 Propagation of elastic waves in an infinite medium 515

10.3 Progressive plane waves 516

10.4 Polarization of three kinds of waves 518

10.5 Propagation in privileged directions and phase velocity calculations 519

10.6 Slowness surface and wave propagation through a separation surface 528

10.7 Propagation of an elastic wave through an anisotropic blade with two parallel faces 535

10.8 Concluding remarks 542

10.9 Bibliography 543

10.10 List of Symbols 544

10.11 Appendices 546

Appendix 10.A Energy velocity, group velocity, Poynting vector 546

Appendix 10.B Slowness surface and energy velocity 553

**Chapter 11 Viscoelastic Moduli of Materials Deduced from Harmonic Responses of Beams ****555**

*Tibi BEDA, Christine ESTEOULE, Mohamed SOULA and Jean Tuong VINH*

11.1 Introduction 555

11.2 Guidelines for practicians 557

11.3 Solution of a viscoelastic problem using the principle of correspondence 558

11.4 Viscoelastic solution of equation of motions 564

11.5 Viscoelastic moduli using equations of higher approximation degree 579

11.6 Bibliography 588

11.7 Appendices 589

Appendix 11.A Transmissibility function of a rod submitted to longitudinal vibration (elementary equation of motion) 589

Appendix 11.B Newton-Raphson’s method applied to a couple of functions of two real variables 1 and 2 components of 590

Appendix 11.C Transmissibility function of a clamped-free Bernoulli’s rod submitted to bending vibration 591

Appendix 11.D Complex transmissibility function of a clamped-free Bernoulli’s rod and its decomposition into two functions of real variables 593

Appendix 11.E Eigenvalue equation of clamped-free Timoshenko’s rod 594

Appendix 11.F Transmissibility function of clamped-free Timoshenko’s rod 595

**Chapter 12 Continuous Element Method Utilized as a Solution to Inverse Problems in Elasticity and Viscoelasticity ****599**

*Jean-Baptiste CASIMIR*

12.1 Introduction 599

12.2 Overview of the continuous element method 601

12.3 Boundary conditions and their implications in the transfer matrix 608

12.4 Extensional vibration of straight beams (elementary theory) 609

12.5 The direct problem of beams submitted to bending vibration 612

12.6 Successive calculation steps to obtain a transfer matrix and simple displacement transfer function 620

12.7 Continuous element method adapted for solving an inverse problem in elasticity and viscoelasticity 622

12.8 Bibliography 624

12.9 Appendices 624

Appendix 12.A Wavenumbers deduced from Timoshenko’s equation 624

List of Authors 629

Index 631