Over the last 50 years, the various available methods of investigating dynamic properties of materials have resulted in significant advances in this area of materials science. Dynamic tests have also recently proven to be as efficient as static tests, and have the advantage that they are often easier to use at lower frequency. This book explores dynamic testing, the methods used, and the experiments performed, placing a particular emphasis on the context of bounded medium elastodynamics.
The book initially focuses on the complements of continuum mechanics before moving on to the various types of rod vibrations: extensional, bending and torsional. In addition, chapters contain practical examples alongside theoretical discussion to facilitate the reader's understanding. The results presented are the culmination of over 30 years of research by the authors and will be of great interest to anyone involved in this field.
Table of Contents:
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 θ1 or θ2 oscillation angles of the pendulums and practical considerations 452
Appendix 8.B Influence of the highest eigenfrequency ω3 on the pendulum oscillations in the expression of θ1 (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 s35 and ϑ13 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
About the Author :
Yvon Chevalier is Emeritus Professor at the Institut Superieur de Mécanique de Paris (SUPMECA), France. Since 2000 he has been co-editor in chief Mecanique et Industries journal, supported by the French Association of Mechanics. He is a well-known expert in the dynamics of composite materials and propagation of waves in heterogeneous materials. He also has great experience in the areas of hyper-elasticity and non-linear viscoelasticity of rubber materials. Jean Tuong Vinh is Emeritus University Professor of Mechanical Engineering at the University of Paris VI in France. He carries out research into theoretical viscoelasticity, non-linear functional Volterra series, computer algorithms in signal processing, frequency Hilbert transform, special impact testing, wave dispersion on rods and continuous elements and solution of related inverse problems.
