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 coeditor-in-chief of 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 extensive 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 in rods and continuous elements and solution of related inverse problems.

• Preface xxiAcknowledgements xxxiPART I - MECHANICAL AND ELECTRONIC INSTRUMENTATION 1Chapter 1. Guidelines for Choosing the Experimental Set-up 3Jean Tuong VINH1.1. Choice of matrix coefficient to be evaluated and type of wave to be adopted 41.2. Influence of frequency range 81.3. Dimensions and shape of the samples 91.4. Tests at high and low temperature 101.5. Sample holder at high temperature 101.6. Visual observation inside the ambient room 111.7. Complex moduli of viscoelastic materials and damping capacity measurements 111.8. Previsional calculation of composite materials 111.9. Bibliography 11Chapter 2. Review of Industrial Analyzers for Material Characterization 13Jean Tuong VINH2.1. Rheovibron and its successive versions 142.2. Dynamic mechanical analyzer DMA 01dB–Metravib and VHF 104 Metravib analyzer 172.3. Bruel and Kjaer complex modulus apparatus (Oberst Apparatus) 182.4. Dynamic mechanical analyzer DMA – Dupont de Nemours 980 202.5. Elasticimeter using progressive wave PPM 5 222.6. Bibliography 24Chapter 3. Mechanical Part of the Vibration Test Bench 25Jean Tuong VINH3.1. Clamping end 253.2. Length correction 293.3. Supported end 333.4. Additional weight or additional torsion lever used as a boundary condition 343.5. Free end 343.6. Pseudo-clamping sample attachment 353.7. Sample suspended by taut threads 383.8. Sample on foam rubber plate serving as a mattress 413.9. Climatic chamber 413.10. Vacuum system 413.11. Bibliography 42Chapter 4. Exciters and Excitation Signals 43Jean Tuong VINH4.1. Frequency ranges 434.2. Power 434.3. Nature and performance of various exciters 444.4. Room required for exciter installation 474.5. Details for electrodynamic shakers 484.6. Low cost electromagnetic exciters with permanent magnet 544.7. Piezoelectric and ferroelectric exciters 554.8. Design of special ferroelectric transducers 674.9. Power piezoelectric exciters 694.10. Technical details concerning ultrasonic emitters for the measurement of material stiffness coefficients on ultrasonic test benches 704.11. Bibliography 744.12. Appendix 4A. Example of ferroelectric plates and disks 74Chapter 5. Transducers 77Jean Tuong VINH and Michel NUGUES5.1. Introduction 775.2. Transducers and their principal performance 785.3. The main classes of fixed reference transducers 795.4. Condenser-type transducer 825.5. Inductance transducers 895.6. Mutual inductance transducer 925.7. Differential transformer transducer 935.8. Contactless inductance transducer with a permanent magnet 935.9. Eddy current transducer 945.10. Seismic transducers 975.11. Piezoresistive accelerometer 1095.12. Other transducers 1105.13. Force transducers 1115.14. Bibliography 1135.15. Appendix 5A. Condenser with polarization 1135.16. Appendix 5B. Eigenfrequencies of some force transducers: Rayleigh and Rayleigh-Ritz upper bound methods 1155B.1. Rayleigh’s method 1165B.2. Rayleigh-Ritz’s method 1175B.3. Preliminary experimental test on the force transducer 117Chapter 6. Electronic Instrumentation, Connecting Cautions and Signal Processing 119Jean Tuong VINH6.1. Preamplifiers and signal conditioners following the transducers 1206.2. Cables and wiring considerations 1216.3. Transducer selection and mountings 1236.4. Transducer calibration 1296.5. Digital signal processing systems: an overview 1336.6. Other signal processing programs 1416.7. Reasoned choice of excitation signals 1426.8. Bibliography 1466.9. Appendix 6A. The Shannon theorem and aliasing phenomenon 1476.10. Appendix 6B. Time window (or weighting function)1506B.1. Kaiser-Bessel window 1516B.2. Hamming window 152Chapter 7. The Frequency Hilbert Transform and Detection of Hidden Non-linearities in Frequency Responses 155Jean Tuong VINH7.1. Introduction 1557.2. Mathematical expression of the Hilbert transform 1577.3. Kramer-Kronig’s relationships 1627.4. Causal signal and Fourier transform 1637.5. Hilbert transform of a truncated transfer function 1647.6. Impulse response of a system. Non-causality due to measurement defects 1727.7. Summary of principal result in sections 7.5 and 7.6 1747.8. Causalized Hilbert transform 1757.9. Some practical aspects of Hilbert transform computation 1767.10. Conclusion 1817.11. Bibliography 1817.12. Appendix 7A. Line integral of complex function and Cauchy’s integral 1827A.1. Analyticity of a function f(z) of complex variable z 1827A.2. Expression of Cauchy’s integral of the function f(z)/(z- 1837.13. Appendix 7B. Hilbert transform obtained directly by Guillemin’s method 184Chapter 8. Measurement of Structural Damping 187Jean Tuong VINH8.1. Introduction 1878.2. Overview of various methods used to evaluate damping ratios in structural dynamics 1908.3. Measurement of structural damping coefficient by multimodal analysis 1978.4. The Hilbert envelope time domain method 2018.5. Detection of hidden non-linearities 2038.6. How to relate material damping to structural damping? 2038.7. Concluding remarks 2078.8. Bibliography 208PART II - REALIZATION OF EXPERIMENTAL SET-UPS AND INTERPRETATION OF MEASUREMENTS 209Chapter 9. Torsion Test Benches: Instrumentation and Experimental Results 211Michel NUGUES9.1. Introduction 2119.2. Industrial torsion test bench 2119.3. Parasitic bending vibration of rod 2159.4. Shear moduli of transverse isotropic materials 2159.5. Elastic moduli obtained for various materials 2209.6. Experimental set-up to obtain dispersion curves in a large frequency range 2229.7. Experimental results obtained on short samples 2249.8. Experimental wave dispersion curves obtained by torsional vibrations of a rod with rectangular cross-section 2279.9. Frequency spectrum for isotropic metallic materials (aluminum and steel alloy) 2309.10. Impact test on viscoelastic high damping material 2329.11. Concluding remarks 2389.12. Bibliography 2399.13. Appendix 9A. Choice of equations of motion 2409A.1. Circular cross-section 2409A.2. Square cross-section 2419A.3. Rectangular cross-section 2419A.4. Ratio of Young’s modulus to shear modulus 2419A.5. Special experimental studies of wave dispersion phenomenon 2429.14. Appendix 9B. Complementary information concerning formulae used to interpret torsion tests 2429B.1. Quick overview of Saint Venant’s theory applied to the problem of dynamic Torsion 2429.15. Appendix 9C. Details concerning the βΤ(c) function in the calculation of rod stiffnessCT 2459.16. Appendix 9D. Compliments concerning the solution of equations of motion with first order theory 2469D.1. Displacement field 2469D.2. Relations between two sets of coefficients 2469D.3. Equations giving the two sets of coefficients Aa, Ba, Ca, Da deduced from the four boundary conditions 2489D.4. Evaluation of coefficients in [9D.6] 2489D.5. Equations in Aa, Ba, Ca, Da deduced from the four boundary conditions 249Chapter 10. Bending Vibration of Rod Instrumentation and Measurements 255Dominique LE NIZHERY10.1. Introduction 25510.2. Realization of an elasticimeter 25510.3. How to conduct bending tests 26210.4. Concluding remarks 26710.5 Bibliography 26810.6. Appendix 10A. Useful formulae to evaluate the Young’s modulus by bending vibration of rods 26810A.1. Bernoulli-Euler’s equation 26810A.2. Timoshenko-Mindlin’s equation 26910A.3. Boundary conditions and wave number equation 26910A.4. Important parameters in rod bending vibration 26910A.5. Expression of the wave number 27010A.6. Young’s modulus (Bernoulli’s theory) 27010A.7. Young’s modulus (Timoshenko-Mindlin’s equation) 270Chapter 11. Longitudinal Vibrations of Rods: Material Characterization and Experimental Dispersion Curves 271Yvon CHEVALIER and Jean Tuong VINH11.1. Introduction 27111.2. Mechanical set-up 27211.3. Electronic set-up 27211.4. Estimation of phase velocity 27411.5. Short samples and eigenvalue calculations for various materials 28011.6. Experimental results interpreted by the two theories 28311.7. Influence of slenderness (δL = 2L/h) on eigenfrequency 29111.8. Experimental results obtained with short rod 29211.9. Concluding remarks 29211.10. Bibliography 29511.11. Appendix 11A. Eigenvalue equation for rod of finite length 29611.12. Appendix 11B. Additional information concerning solutions of Touratier’s  equations 30011B.1. Eigenequation with elementary theory of motion 301Chapter 12. Realization of Le Rolland-Sorin’s Double Pendulum and Some Experimental Results 305Mostefa ARCHI and Jean-Baptiste CASIMIR12.1. Introduction 30512.2. Principal mechanical parts of the double pendulum system 30512.3. Instrumentation 31212.4. Experimental precautions 31512.5. Details and characteristics of the elasticimeter 31712.6. Some experimental results 31812.7. Damping ratio estimation by logarithmic decrement method 32212.8. Concluding remarks 32412.9. Bibliography 32512.10. Appendix 12A. Equations of motion for the set (pendulums, platform and sample) and Young’s modulus calculation deduced from bending tests 32612A.1. Equations of motion 32612A.2. Solutions for pendulum oscillations 32812A.3. Relationship between beating period τ and sample stiffness k 32912A.4. Young’s modulus calculation 33012.11. Appendix 12B. Evaluation of shear modulus by torsion tests 33112B.1. Energy expression 331Chapter 13. Stationary and Progressive Waves in Rings and Hollow Cylinders 335Yvon CHEVALIER and Jean Tuong VINH13.1. Introduction 33513.2. Choosing the samples based on material symmetry 33613.3. Practical realization of a special elasticimeter for curved beams and rings: in plane bending vibrations 33713.4. Ultrasonic benches 34213.5. Experimental results and interpretation 34313.6. List of symbols 35813.7. Bibliography 35913.8. Appendix 13A. Evaluation of Young’s modulus by using in plane bending motion of the ring 35913.9. Appendix 13B. Determination of inertia moment of a solid by means of a three-string pendulum 36013B.1. Principle of the method 36013B.2. Calculations 36113.10. Appendix 13C. Necessary formulae to evaluate Young’smodulus of a straight beam 364Chapter 14. Ultrasonic Benches: Characterization of Materials by Wave Propagation Techniques 367Patrick GARCEAU14.1. Introduction 36714.2. Ultrasonic transducers 36714.3. Pulse generator 36914.4. Mechanical realization of ultrasonic benches 37114.5. Experimental interpretation of phase velocity and group velocity 37514.6. Some experimental results on composite materials 38014.7. Viscoelastic characterization of materials by ultrasonic waves 38314.8. Bibliography 38814.9. Appendix 14A. Oblique incidence and energy propagation direction 38914.10. Appendix 14B. Water immersion bench, measurement of coefficients of stiffness matrix 39214B.1. Expression of phase velocity in the sample 39314B.2. Phase velocity measurement by propagation time (?·?nt ) evaluation 39414B.3. Phase velocity evaluation without time measurements 394Chapter 15. Wave Dispersion in Rods with a Rectangular Cross-section: Higher Order Theory and Experimentation 397Maurice TOURATIER15.1. Introduction 39715.2. Summary table of some wave dispersion research 39815.3. Longitudinal wave dispersion: influence of the material and geometry of the bounded medium 39915.4. Bending wave dispersion 40315.5. First order for torsional motion in a transverse isotropic rod 40815.6. Interest in theories with higher degrees of approximation 41415.7. Experimental set-ups to visualize stationary waves in rods 41615.8. Electronic set-up and observed signals on a multi-channel oscilloscope 42115.9. Presentation of experimental results 42415.10. Concluding remarks 42715.11. Bibliography 42815.12. Appendix 15A. Touratier’s theory using Hellinger–Reissner’s mixed fields 42915A.1. Outline of Touratier’s mixed field theory 42915A.2. General equations deduced from the two fields principle 43215A.3. Formulation of the boundary condition problem 43215A.4. Symmetry considerations concerning the three kinds of motion 43315A.5. Truncating process for one dimensional theories: extensional waves 43715A.6. Equations of motion for extensional movement 43815A.7. Effective front velocity and wave front velocity 43915A.8. Bending equations of motion 44115A.9. Equations of motion: torsional vibration 44415.13. Appendix 15B. Third order Touratier’s theory 44515B.1. Extensional waves with nine evaluated modes 44615B.2. Geometrical characteristics of displacement components uj mn and physical interpretation 44715B.3. Bending mode in the direction x – geometrical interpretation 44815B.4. Shear motion around longitudinal rod axis 450List of Authors 453Index 455