Mechanical Characterization of Materials and Wave Dispersion

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.

• Preface xixAcknowledgements xxixPart A Constitutive Equations of Materials 1Chapter 1 Elements of Anisotropic Elasticity and Complements on Previsional Calculations 3Yvon CHEVALIER1.1 Constitutive equations in a linear elastic regime 41.2 Technical elastic moduli 71.3 Real materials with special symmetries 101.4 Relationship between compliance Sij and stiffness Cij for orthotropic materials 231.5 Useful inequalities between elastic moduli 241.6 Transformation of reference axes is necessary in many circumstances 271.7 Invariants and their applications in the evaluation of elastic constants 281.8 Plane elasticity 351.9 Elastic previsional calculations for anisotropic composite materials 381.10 Bibliography 511.11 Appendix 52Appendix 1.A Overview on methods used in previsional calculation of fiber-reinforced composite materials 52Chapter 2 Elements of Linear Viscoelasticity 57Yvon CHEVALIER2.1 Time delay between sinusoidal stress and strain 592.2 Creep and relaxation tests 602.3 Mathematical formulation of linear viscoelasticity 632.4 Generalization of creep and relaxation functions to tridimensional constitutive equations 712.5 Principle of correspondence and Carson-Laplace transform for transient viscoelastic problems 742.6 Correspondence principle and the solution of the harmonic viscoelastic system 822.7 Inter-relationship between harmonic and transient regimes 832.8 Modeling of creep and relaxation functions: example 872.9 Conclusion 1002.10 Bibliography 100Chapter 3 Two Useful Topics in Applied Viscoelasticity: Constitutive Equations for Viscoelastic Materials 103Yvon CHEVALIER and Jean Tuong VINH3.1 Williams-Landel-Ferry’s method 1043.2 Viscoelastic time function obtained directly from a closed-form expression of complex modulus or complex compliance 1123.3 Concluding remarks 1363.4 Bibliography 1373.5 Appendices 139Appendix 3.A Inversion of Laplace transform 139Appendix 3.B Sutton’s method for long time response 143Chapter 4 Formulation of Equations of Motion and Overview of their Solutions by Various Methods 145Jean Tuong VINH4.1 D’Alembert’s principle 1464.2 Lagrange’s equation 1494.3 Hamilton’s principle 1574.4 Practical considerations concerning the choice of equations of motion and related solutions 1594.5 Three-, two- or one-dimensional equations of motion? 1624.6 Closed-form solutions to equations of motion 1634.7 Bibliography 1644.8 Appendices 165Appendix 4.A Equations of motion in elastic medium deduced from Love’s variational principle 165Appendix 4.B Lagrange’s equations of motion deduced from Hamilton’s principle 167Part B Rod Vibrations 173Chapter 5 Torsional Vibration of Rods 175Yvon CHEVALIER, Michel NUGUES and James ONOBIONO5.1 Introduction 1755.1.1 Short bibliography of the torsion problem 1765.1.2 Survey of solving methods for torsion problems 1765.1.3 Extension of equations of motion to a larger frequency range 1795.2 Static torsion of an anisotropic beam with rectangular section without bending – Saint Venant, Lekhnitskii’s formulation 1805.3 Torsional vibration of a rod with finite length 1995.4 Simplified boundary conditions associated with higher approximation equations of motion [5.49] 2045.5 Higher approximation equations of motion 2055.6 Extension of Engström’s theory to the anisotropic theory of dynamic torsion of a rod with rectangular cross-section 2075.7 Equations of motion 2125.8 Torsion wave dispersion 2155.9 Presentation of dispersion curves 2195.10 Torsion vibrations of an off-axis anisotropic rod 2255.11 Dispersion of deviated torsional waves in off-axis anisotropic rods with rectangular cross-section 2355.12 Dispersion curve of torsional phase velocities of an off-axis anisotropic rod 2405.13 Concluding remarks 2415.14 Bibliography 2425.15 Table of symbols 2445.16 Appendices 246Appendix 5.A Approximate formulae for torsion stiffness 246Appendix 5.B Equations of torsional motion obtained from Hamilton’s variational principle 250Appendix 5.C Extension of Barr’s correcting coefficient in equations of motion 257Appendix 5.D Details on coefficient calculations for θ (z, t) and ζ (z, t) 258Appendix 5.E A simpler solution to the problem analyzed in Appendix 5.D 263Appendix 5.F Onobiono’s and Zienkievics’ solutions using finite element method for warping function φ 265Appendix 5.G Formulation of equations of motion for an off-axis anisotropic rod submitted to coupled torsion and bending vibrations 273Appendix 5.H Relative group velocity versus relative wave number 279Chapter 6 Bending Vibration of a Rod 291Dominique LE NIZHERY6.1 Introduction 2916.1.1 Short bibliography of dynamic bending of a beam 2926.2 Bending vibration of straight beam by elementary theory 2936.3 Higher approximation theory of bending vibration 2996.4 Bending vibration of an off-axis anisotropic rod 3136.5 Concluding remarks 3246.6 Bibliography 3266.7 Table of symbols 3276.8 Appendices 328Appendix 6.A Timoshenko’s correcting coefficients for anisotropic and isotropic materials 328Appendix 6.B Correcting coefficient using Mindlin’s method 333Appendix 6.C Dispersion curves for various equations of motion 334Appendix 6.D Change of reference axes and elastic coefficients for an anisotropic rod 337Chapter 7 Longitudinal Vibration of a Rod 339Yvon CHEVALIER and Maurice TOURATIER7.1 Presentation 3397.2 Bishop’s equations of motion 3437.3 Improved Bishop’s equation of motion 3457.4 Bishop’s equation for orthotropic materials 3467.5 Eigenfrequency equations for a free-free rod 3467.6 Touratier’s equations of motion of longitudinal waves 3507.7 Wave dispersion relationships 3677.8 Short rod and boundary conditions 3937.9 Concluding remarks about Touratier’s theory 3957.10 Bibliography 3967.11 List of symbols 3977.12 Appendices 399Appendix 7.A an outline of some studies on longitudinal vibration of rods with rectangular cross-section 399Appendix 7.B Formulation of Bishop’s equation by Hamilton’s principle by Rao and Rao 401Appendix 7.C Dimensionless Bishop’s equations of motion and dimensionless boundary conditions 405Appendix 7.D Touratier’s equations of motion by variational calculus 408Appendix 7.E Calculation of correcting factor q (Cijkl) 409Appendix 7.F Stationarity of functional J and boundary equations 419Appendix 7.G On the possible solutions of eigenvalue equations 419Chapter 8 Very Low Frequency Vibration of a Rod by Le Rolland-Sorin’s Double Pendulum 425Mostefa ARCHI and Jean-Baptiste CASIMIR8.1 Introduction 4258.2 Short bibliography 4278.3 Flexural vibrations of a rod using coupled pendulums 4278.4 Torsional vibration of a beam by double pendulum 4348.5 Complex compliance coefficient of viscoelastic materials 4368.6 Elastic stiffness of an off-axis rod 4438.7 Bibliography 4498.8 List of symbols 4508.9 Appendices 452Appendix 8.A Closed-form expression of θ1 or θ2 oscillation angles of the pendulums and practical considerations 452Appendix 8.B Influence of the highest eigenfrequency ω3 on the pendulum oscillations in the expression of θ1 (t) 457Appendix 8.C Coefficients a of compliance matrix after a change of axes for transverse isotropic material 458Appendix 8.D Mathematical formulation of the simultaneous bending and torsion of an off-axis rectangular rod 460Appendix 8.E Details on calculations of s35 and ϑ13 of transverse isotropic materials 486Chapter 9 Vibrations of a Ring and Hollow Cylinder 493Jean Tuong VINH9.1 Introduction 4939.2 Equations of motion of a circular ring with rectangular cross-section 4949.3 Bibliography 5029.4 Appendices 503Appendix 9.A Expression u (θ) in the three subintervals delimited by the roots of equation [9.33] 503Chapter 10 Characterization of Isotropic and Anisotropic Materials by Progressive Ultrasonic Waves 513Patrick GARCEAU10.1 Presentation of the method 51310.2 Propagation of elastic waves in an infinite medium 51510.3 Progressive plane waves 51610.4 Polarization of three kinds of waves 51810.5 Propagation in privileged directions and phase velocity calculations 51910.6 Slowness surface and wave propagation through a separation surface 52810.7 Propagation of an elastic wave through an anisotropic blade with two parallel faces 53510.8 Concluding remarks 54210.9 Bibliography 54310.10 List of Symbols 54410.11 Appendices 546Appendix 10.A Energy velocity, group velocity, Poynting vector 546Appendix 10.B Slowness surface and energy velocity 553Chapter 11 Viscoelastic Moduli of Materials Deduced from Harmonic Responses of Beams 555Tibi BEDA, Christine ESTEOULE, Mohamed SOULA and Jean Tuong VINH11.1 Introduction 55511.2 Guidelines for practicians 55711.3 Solution of a viscoelastic problem using the principle of correspondence 55811.4 Viscoelastic solution of equation of motions 56411.5 Viscoelastic moduli using equations of higher approximation degree 57911.6 Bibliography 58811.7 Appendices 589Appendix 11.A Transmissibility function of a rod submitted to longitudinal vibration (elementary equation of motion) 589Appendix 11.B Newton-Raphson’s method applied to a couple of functions of two real variables 1 and 2 components of 590Appendix 11.C Transmissibility function of a clamped-free Bernoulli’s rod submitted to bending vibration 591Appendix 11.D Complex transmissibility function of a clamped-free Bernoulli’s rod and its decomposition into two functions of real variables 593Appendix 11.E Eigenvalue equation of clamped-free Timoshenko’s rod 594Appendix 11.F Transmissibility function of clamped-free Timoshenko’s rod 595Chapter 12 Continuous Element Method Utilized as a Solution to Inverse Problems in Elasticity and Viscoelasticity 599Jean-Baptiste CASIMIR12.1 Introduction 59912.2 Overview of the continuous element method 60112.3 Boundary conditions and their implications in the transfer matrix 60812.4 Extensional vibration of straight beams (elementary theory) 60912.5 The direct problem of beams submitted to bending vibration 61212.6 Successive calculation steps to obtain a transfer matrix and simple displacement transfer function 62012.7 Continuous element method adapted for solving an inverse problem in elasticity and viscoelasticity 62212.8 Bibliography 62412.9 Appendices 624Appendix 12.A Wavenumbers deduced from Timoshenko’s equation 624List of Authors 629Index 631