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Mechanical Characterization of Materials and Wave Dispersion


Mechanical Characterization of Materials and Wave Dispersion


1. Aufl.

von: Yvon Chevalier, Jean Vinh Tuong

266,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 04.03.2013
ISBN/EAN: 9781118623152
Sprache: englisch
Anzahl Seiten: 639

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Beschreibungen

<p>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. <p>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.
<p>Preface xix</p> <p>Acknowledgements xxix</p> <p><b>Part A Constitutive Equations of Materials</b><b> 1</b></p> <p><b>Chapter 1 Elements of Anisotropic Elasticity and Complements on Previsional Calculations </b><b>3<br /></b><i>Yvon CHEVALIER</i></p> <p>1.1 Constitutive equations in a linear elastic regime 4</p> <p>1.2 Technical elastic moduli 7</p> <p>1.3 Real materials with special symmetries 10</p> <p>1.4 Relationship between compliance Sij and stiffness Cij for orthotropic materials 23</p> <p>1.5 Useful inequalities between elastic moduli 24</p> <p>1.6 Transformation of reference axes is necessary in many circumstances 27</p> <p>1.7 Invariants and their applications in the evaluation of elastic constants 28</p> <p>1.8 Plane elasticity 35</p> <p>1.9 Elastic previsional calculations for anisotropic composite materials 38</p> <p>1.10 Bibliography 51</p> <p>1.11 Appendix 52</p> <p>Appendix 1.A Overview on methods used in previsional calculation of fiber-reinforced composite materials 52</p> <p><b>Chapter 2 Elements of Linear Viscoelasticity </b><b>57<br /></b><i>Yvon CHEVALIER</i></p> <p>2.1 Time delay between sinusoidal stress and strain 59</p> <p>2.2 Creep and relaxation tests 60</p> <p>2.3 Mathematical formulation of linear viscoelasticity 63</p> <p>2.4 Generalization of creep and relaxation functions to tridimensional constitutive equations 71</p> <p>2.5 Principle of correspondence and Carson-Laplace transform for transient viscoelastic problems 74</p> <p>2.6 Correspondence principle and the solution of the harmonic viscoelastic system 82</p> <p>2.7 Inter-relationship between harmonic and transient regimes 83</p> <p>2.8 Modeling of creep and relaxation functions: example 87</p> <p>2.9 Conclusion 100</p> <p>2.10 Bibliography 100</p> <p><b>Chapter 3 Two Useful Topics in Applied Viscoelasticity: Constitutive Equations for Viscoelastic Materials </b><b>103<br /></b><i>Yvon CHEVALIER and Jean Tuong VINH</i></p> <p>3.1 Williams-Landel-Ferry’s method 104</p> <p>3.2 Viscoelastic time function obtained directly from a closed-form expression of complex modulus or complex compliance 112</p> <p>3.3 Concluding remarks 136</p> <p>3.4 Bibliography 137</p> <p>3.5 Appendices 139</p> <p>Appendix 3.A Inversion of Laplace transform 139</p> <p>Appendix 3.B Sutton’s method for long time response 143</p> <p><b>Chapter 4 Formulation of Equations of Motion and Overview of their Solutions by Various Methods </b><b>145<br /></b><i>Jean Tuong VINH</i></p> <p>4.1 D’Alembert’s principle 146</p> <p>4.2 Lagrange’s equation 149</p> <p>4.3 Hamilton’s principle 157</p> <p>4.4 Practical considerations concerning the choice of equations of motion and related solutions 159</p> <p>4.5 Three-, two- or one-dimensional equations of motion? 162</p> <p>4.6 Closed-form solutions to equations of motion 163</p> <p>4.7 Bibliography 164</p> <p>4.8 Appendices 165</p> <p>Appendix 4.A Equations of motion in elastic medium deduced from Love’s variational principle 165</p> <p>Appendix 4.B Lagrange’s equations of motion deduced from Hamilton’s principle 167</p> <p><b>Part B Rod Vibrations </b><b>173</b></p> <p><b>Chapter 5 Torsional Vibration of Rods </b><b>175<br /></b><i>Yvon CHEVALIER, Michel NUGUES and James ONOBIONO</i></p> <p>5.1 Introduction 175</p> <p>5.1.1 Short bibliography of the torsion problem 176</p> <p>5.1.2 Survey of solving methods for torsion problems 176</p> <p>5.1.3 Extension of equations of motion to a larger frequency range 179</p> <p>5.2 Static torsion of an anisotropic beam with rectangular section without bending – Saint Venant, Lekhnitskii’s formulation 180</p> <p>5.3 Torsional vibration of a rod with finite length 199</p> <p>5.4 Simplified boundary conditions associated with higher approximation equations of motion [5.49] 204</p> <p>5.5 Higher approximation equations of motion 205</p> <p>5.6 Extension of Engström’s theory to the anisotropic theory of dynamic torsion of a rod with rectangular cross-section 207</p> <p>5.7 Equations of motion 212</p> <p>5.8 Torsion wave dispersion 215</p> <p>5.9 Presentation of dispersion curves 219</p> <p>5.10 Torsion vibrations of an off-axis anisotropic rod 225</p> <p>5.11 Dispersion of deviated torsional waves in off-axis anisotropic rods with rectangular cross-section 235</p> <p>5.12 Dispersion curve of torsional phase velocities of an off-axis anisotropic rod 240</p> <p>5.13 Concluding remarks 241</p> <p>5.14 Bibliography 242</p> <p>5.15 Table of symbols 244</p> <p>5.16 Appendices 246</p> <p>Appendix 5.A Approximate formulae for torsion stiffness 246</p> <p>Appendix 5.B Equations of torsional motion obtained from Hamilton’s variational principle 250</p> <p>Appendix 5.C Extension of Barr’s correcting coefficient in equations of motion 257</p> <p>Appendix 5.D Details on coefficient calculations for θ (z, t) and ζ (z, t) 258</p> <p>Appendix 5.E A simpler solution to the problem analyzed in Appendix 5.D 263</p> <p>Appendix 5.F Onobiono’s and Zienkievics’ solutions using finite element method for warping function <i>φ </i>265</p> <p>Appendix 5.G Formulation of equations of motion for an off-axis anisotropic rod submitted to coupled torsion and bending vibrations 273</p> <p>Appendix 5.H Relative group velocity versus relative wave number 279</p> <p><b>Chapter 6 Bending Vibration of a Rod</b><b> 291<br /></b><i>Dominique LE NIZHERY</i></p> <p>6.1 Introduction 291</p> <p>6.1.1 Short bibliography of dynamic bending of a beam 292</p> <p>6.2 Bending vibration of straight beam by elementary theory 293</p> <p>6.3 Higher approximation theory of bending vibration 299</p> <p>6.4 Bending vibration of an off-axis anisotropic rod 313</p> <p>6.5 Concluding remarks 324</p> <p>6.6 Bibliography 326</p> <p>6.7 Table of symbols 327</p> <p>6.8 Appendices 328</p> <p>Appendix 6.A Timoshenko’s correcting coefficients for anisotropic and isotropic materials 328</p> <p>Appendix 6.B Correcting coefficient using Mindlin’s method 333</p> <p>Appendix 6.C Dispersion curves for various equations of motion 334</p> <p>Appendix 6.D Change of reference axes and elastic coefficients for an anisotropic rod 337</p> <p><b>Chapter 7 Longitudinal Vibration of a Rod </b><b>339<br /></b><i>Yvon CHEVALIER and Maurice TOURATIER</i></p> <p>7.1 Presentation 339</p> <p>7.2 Bishop’s equations of motion 343</p> <p>7.3 Improved Bishop’s equation of motion 345</p> <p>7.4 Bishop’s equation for orthotropic materials 346</p> <p>7.5 Eigenfrequency equations for a free-free rod 346</p> <p>7.6 Touratier’s equations of motion of longitudinal waves 350</p> <p>7.7 Wave dispersion relationships 367</p> <p>7.8 Short rod and boundary conditions 393</p> <p>7.9 Concluding remarks about Touratier’s theory 395</p> <p>7.10 Bibliography 396</p> <p>7.11 List of symbols 397</p> <p>7.12 Appendices 399</p> <p>Appendix 7.A an outline of some studies on longitudinal vibration of rods with rectangular cross-section 399</p> <p>Appendix 7.B Formulation of Bishop’s equation by Hamilton’s principle by Rao and Rao 401</p> <p>Appendix 7.C Dimensionless Bishop’s equations of motion and dimensionless boundary conditions 405</p> <p>Appendix 7.D Touratier’s equations of motion by variational calculus 408</p> <p>Appendix 7.E Calculation of correcting factor q (Cijkl) 409</p> <p>Appendix 7.F Stationarity of functional J and boundary equations 419</p> <p>Appendix 7.G On the possible solutions of eigenvalue equations 419</p> <p><b>Chapter 8 Very Low Frequency Vibration of a Rod by Le Rolland-Sorin’s Double Pendulum </b><b>425<br /></b><i>Mostefa ARCHI and Jean-Baptiste CASIMIR</i></p> <p>8.1 Introduction 425</p> <p>8.2 Short bibliography 427</p> <p>8.3 Flexural vibrations of a rod using coupled pendulums 427</p> <p>8.4 Torsional vibration of a beam by double pendulum 434</p> <p>8.5 Complex compliance coefficient of viscoelastic materials 436</p> <p>8.6 Elastic stiffness of an off-axis rod 443</p> <p>8.7 Bibliography 449</p> <p>8.8 List of symbols 450</p> <p>8.9 Appendices 452</p> <p>Appendix 8.A Closed-form expression of θ<sub>1</sub> or θ<sub>2</sub> oscillation angles of the pendulums and practical considerations 452</p> <p>Appendix 8.B Influence of the highest eigenfrequency <i>ω</i>3 on the pendulum oscillations in the expression of <i>θ</i><sub>1</sub> (t) 457</p> <p>Appendix 8.C Coefficients a of compliance matrix after a change of axes for transverse isotropic material 458</p> <p>Appendix 8.D Mathematical formulation of the simultaneous bending and torsion of an off-axis rectangular rod 460</p> <p>Appendix 8.E Details on calculations of s<sub>35</sub> and ϑ<sub>13</sub> of transverse isotropic materials 486</p> <p><b>Chapter 9 Vibrations of a Ring and Hollow Cylinder </b><b>493<br /></b><i>Jean Tuong VINH</i></p> <p>9.1 Introduction 493</p> <p>9.2 Equations of motion of a circular ring with rectangular cross-section 494</p> <p>9.3 Bibliography 502</p> <p>9.4 Appendices 503</p> <p>Appendix 9.A Expression u (θ) in the three subintervals delimited by the roots of equation [9.33] 503</p> <p><b>Chapter 10 Characterization of Isotropic and Anisotropic Materials by Progressive Ultrasonic Waves</b><b> 513<br /></b><i>Patrick GARCEAU</i></p> <p>10.1 Presentation of the method 513</p> <p>10.2 Propagation of elastic waves in an infinite medium 515</p> <p>10.3 Progressive plane waves 516</p> <p>10.4 Polarization of three kinds of waves 518</p> <p>10.5 Propagation in privileged directions and phase velocity calculations 519</p> <p>10.6 Slowness surface and wave propagation through a separation surface 528</p> <p>10.7 Propagation of an elastic wave through an anisotropic blade with two parallel faces 535</p> <p>10.8 Concluding remarks 542</p> <p>10.9 Bibliography 543</p> <p>10.10 List of Symbols 544</p> <p>10.11 Appendices 546</p> <p>Appendix 10.A Energy velocity, group velocity, Poynting vector 546</p> <p>Appendix 10.B Slowness surface and energy velocity 553</p> <p><b>Chapter 11 Viscoelastic Moduli of Materials Deduced from Harmonic Responses of Beams </b><b>555<br /></b><i>Tibi BEDA, Christine ESTEOULE, Mohamed SOULA and Jean Tuong VINH</i></p> <p>11.1 Introduction 555</p> <p>11.2 Guidelines for practicians 557</p> <p>11.3 Solution of a viscoelastic problem using the principle of correspondence 558</p> <p>11.4 Viscoelastic solution of equation of motions 564</p> <p>11.5 Viscoelastic moduli using equations of higher approximation degree 579</p> <p>11.6 Bibliography 588</p> <p>11.7 Appendices 589</p> <p>Appendix 11.A Transmissibility function of a rod submitted to longitudinal vibration (elementary equation of motion) 589</p> <p>Appendix 11.B Newton-Raphson’s method applied to a couple of functions of two real variables 1 and 2 components of 590</p> <p>Appendix 11.C Transmissibility function of a clamped-free Bernoulli’s rod submitted to bending vibration 591</p> <p>Appendix 11.D Complex transmissibility function of a clamped-free Bernoulli’s rod and its decomposition into two functions of real variables 593</p> <p>Appendix 11.E Eigenvalue equation of clamped-free Timoshenko’s rod 594</p> <p>Appendix 11.F Transmissibility function of clamped-free Timoshenko’s rod 595</p> <p><b>Chapter 12 Continuous Element Method Utilized as a Solution to Inverse Problems in Elasticity and Viscoelasticity </b><b>599<br /></b><i>Jean-Baptiste CASIMIR</i></p> <p>12.1 Introduction 599</p> <p>12.2 Overview of the continuous element method 601</p> <p>12.3 Boundary conditions and their implications in the transfer matrix 608</p> <p>12.4 Extensional vibration of straight beams (elementary theory) 609</p> <p>12.5 The direct problem of beams submitted to bending vibration 612</p> <p>12.6 Successive calculation steps to obtain a transfer matrix and simple displacement transfer function 620</p> <p>12.7 Continuous element method adapted for solving an inverse problem in elasticity and viscoelasticity 622</p> <p>12.8 Bibliography 624</p> <p>12.9 Appendices 624</p> <p>Appendix 12.A Wavenumbers deduced from Timoshenko’s equation 624</p> <p>List of Authors 629</p> <p>Index 631</p>
<b>Yvon Chevalier</b> is Emeritus Professor at the Institut Superieur de Mécanique de Paris (SUPMECA), France. Since 2000 he has been co-editor in chief <i>Mecanique et Industries</i> 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.  <p><b>Jean Tuong Vinh</b> 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.</p>

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