PART A. CONSTITUTIVE EQUATIONS OF MATERIALS

Transcription

1 Preface... xix Acknowledgements... xxix PART A. CONSTITUTIVE EQUATIONS OF MATERIALS Chapter 1. Elements of Anisotropic Elasticity and Complements on Previsional Calculations... 3 Yvon CHEVALIER 1.1. Constitutive equations in a linear elastic regime Symmetry applied to tensors s ijkl and c ijkl Constitutive equations under matrix form Technical elastic moduli Tension tests with one normal stress component σ Shear test Real materials with special symmetries Change of reference axes Orthotropic materials possess two orthogonal planes of symmetry Quasi-isotropic transverse (tetragonal) material Transverse isotropic materials (hexagonal system) Quasi-isotropic material (cubic system) Isotropic materials Relationship between compliance S ij and stiffness C ij for orthotropic materials Useful inequalities between elastic moduli Orthotropic materials Quasi transverse isotropic materials Transverse isotropic, quasi-isotropic, and isotropic materials... 26

2 vi Mechanics of Viscoelastic Materials and Wave Dispersion 1.6. Transformation of reference axes is necessary in many circumstances Practical examples Components of stiffness and compliance after transformation Remarks on shear elastic moduli G ii (ij = 23, 31, 12) and stiffness constants C ii (with i = 4, 5, 6) The practical consequence of a transformation of reference axes Invariants and their applications in the evaluation of elastic constants Elastic constants versus invariants Practical utilization of invariants in the evaluation of elastic constants Plane elasticity Expression of plane stress stiffness versus compliance matrix Plane stress stiffness components versus three-dimensional stiffness components Elastic previsional calculations for anisotropic composite materials Long fibers regularly distributed in the matrix Stratified composite materials Reinforced fabric composite materials Bibliography Appendix Appendix 1.A. Overview on methods used in previsional calculation of fiber-reinforced composite materials Chapter 2. Elements of Linear Viscoelasticity Yvon CHEVALIER 2.1. Time delay between sinusoidal stress and strain Creep and relaxation tests Creep test Relaxation test Ageing and non-ageing viscoelastic materials Viscoelastic materials with fading memory Mathematical formulation of linear viscoelasticity Linear system Superposition (or Boltzmann s) principle Creep function in a functional constitutive equation Relaxation function in functional constitutive equations Properties of relaxation and creep functions Generalization of creep and relaxation functions to tridimensional constitutive equations Relaxation function as components in a stiffness matrix

3 vii Creep function as components in a compliance matrix Some remarks on the mathematical formulation of linear viscoelatic behavior of materials Principle of correspondence and Carson-Laplace transform for transient viscoelastic problems Carson-Laplace s transform Complex moduli Properties of the complex relaxation and creep components matrix versus circular frequency Correspondence principle and the solution of the harmonic viscoelastic system Inter-relationship between harmonic and transient regimes Modeling of creep and relaxation functions: example Basic rheological cells General parametric modeling Modeling with three rheological parameters: examples Modeling with four rheological parameters: fractional derivative Conclusion Bibliography Chapter 3. Two Useful Topics in Applied Viscoelasticity: Constitutive Equations for Viscoelastic Materials Yvon CHEVALIER and Jean Tuong VINH 3.1. Williams-Landel-Ferry s method The effect of temperature change Williams-Landel-Ferry s formulation Experimental procedure to obtain the master curve Examples of master curves Applicability of the method of reduced variables Utilization of another representation of complex moduli (or compliances) Extension of Williams-Landel-Ferry s methods to composite materials Viscoelastic time function obtained directly from a closed-form expression of complex modulus or complex compliance Overview of state-of-the-art in viscoelasticity and structural dynamics Polynomial functions of frequency used to express complex moduli Fractional derivatives

4 viii Mechanics of Viscoelastic Materials and Wave Dispersion Use of inverse Carson-Laplace transform to obtain a closed-form expression of transient response Applications Concluding remarks Williams-Landel-Ferry s method Polynomial quotient and fractional derivatives are used to mathematically express the dynamic response of viscoelastic materials Bibliography Appendices Appendix 3.A. Inversion of Laplace transform Appendix 3.B. Sutton s method for long time response Chapter 4. Formulation of Equations of Motion and Overview of their Solutions by Various Methods Jean Tuong VINH 4.1. D Alembert s principle Generalized coordinates Principle of virtual work Equation of motion by D Alembert s principle Lagrange s equation System subjected to r holonomic geometric constraints Generalized forces Introduction to kinetic energy Lagrange s equation Potential function Lagrangian function with Rayleigh dissipation function Hamilton s principle Practical considerations concerning the choice of equations of motion and related solutions Boundary conditions and approximate equations of motion Choice of equations of motion The degree of approximation Three-, two- or one-dimensional equations of motion? Three-dimensional equations of motion Two- or one-dimensional equations of motion Wave dispersion Closed-form solutions to equations of motion Degree of differential equations of six or less Degree of differential equations exceeding six Bibliography

5 ix 4.8. Appendices Appendix 4.A. Equations of motion in elastic medium deduced from Love s variational principle Appendix 4.B. Lagrange s equations of motion deduced from Hamilton s principle PART B. ROD VIBRATIONS Chapter 5. Torsional Vibration of Rods Yvon CHEVALIER, Michel NUGUES and James ONOBIONO 5.1. Introduction Short bibliography of the torsion problem Survey of solving methods for torsion problems Extension of equations of motion to a larger frequency range Static torsion of an anisotropic beam with rectangular section without bending Saint Venant, Lekhnitskii s formulation Airy s function and field of displacement ψ Solution of equation [5.10] with Airy s function Expressions of torsion moment M t versus torsion angle α and stiffness C T Approximate formulae for M T and C T Dynamic torsion of a rod with rectangular section using the elementary equation of motion Pure dynamic torsion of a rod with rectangular cross-section using first-degree approximation Torsional vibration of a rod with finite length Closed-form solution of θ (z, t) Evaluation of spatial solution Θ ( z ) Evaluation of coefficients θ (z, t) and ζ (z, t) Simplified boundary conditions associated with higher approximation equations of motion [5.49] Higher approximation equations of motion Slenderness Ratio E/G ij Flatness ratio s = width/thickness = b/h Saint Venant s theory versus higher approximation theory Extension of Engström s theory to the anisotropic theory of dynamic torsion of a rod with rectangular cross-section Equations of motion Matricial form Decoupled equation of motion and correcting coefficient

6 x Mechanics of Viscoelastic Materials and Wave Dispersion 5.8. Torsion wave dispersion Expressions of angular displacement α (z, t) and axial displacement ψ (z, t) Phase velocity for the two first elastodynamic modes Presentation of dispersion curves Isotropic rod with circular section Isotropic steel rod with rectangular section Anisotropic composite rods Some remarks Practical remark Comparison with other theories (Barr and Engström) Torsion vibrations of an off-axis anisotropic rod Displacement field Equations of motion Eigenfrequency equations of motion Solutions of decoupled equations For a very long rod Dispersion of deviated torsional waves in off-axis anisotropic rods with rectangular cross-section Dispersion curve of torsional phase velocities of an off-axis anisotropic rod Concluding remarks Characterization of shear moduli and non-diagonal coefficients of a compliance matrix Dispersion of torsional phase velocity Saint Venant s warping theory A more elaborate warping theory Dispersion of torsional phase velocity Nugue s theory Dispersion of phase and group velocities Bibliography Table of symbols Appendices Appendix 5.A. Approximate formulae for torsion stiffness Appendix 5.B. Equations of torsional motion obtained from Hamilton s variational principle Appendix 5.C. Extension of Barr s correcting coefficient in equations of motion Appendix 5.D. Details on coefficient calculations for θ (z, t) and ζ (z, t) Appendix 5.E. A simpler solution to the problem analyzed in Appendix 5.D

7 xi Appendix 5.F. Onobiono s and Zienkievics solutions using finite element method for warping function φ Appendix 5.G. Formulation of equations of motion for an off-axis anisotropic rod submitted to coupled torsion and bending vibrations Appendix 5.H. Relative group velocity versus relative wave number Chapter 6. Bending Vibration of a Rod Dominique LE NIZHERY 6.1. Introduction Short bibliography of dynamic bending of a beam Bending vibration of straight beam by elementary theory Bernoulli-Euler s equation of motion Solutions of Bernoulli-Euler s equations Higher approximation theory of bending vibration Formulation of the equations of motion Coupled equations of motion Decoupled equation of motion Solution to the practical problem Stationary bending vibration Non-dimensional equation of motion Equation of motion with reduced variables Expression of flexural displacement and shear ϕ Coefficient calculations Evaluation of Young s modulus by solution of Timoshenko s equation Dispersion curves related to various theories of bending waves in a rod Influence of rotational inertia and transverse shear Bending vibration of an off-axis anisotropic rod Preliminary considerations Flexural vibration of an off-axis rod Equations of motion Reduced variables and characteristic frequency equations for stationary waves Computer program using trial and error method to evaluate Poisson s number Concluding remarks Choice of equations of motion Test sample slenderness and influence of shear effect Strong influence of shear effect for composite materials: accuracy of Young s modulus and non-diagonal technical modulus evaluation

8 xii Mechanics of Viscoelastic Materials and Wave Dispersion Comments on computational effort to evaluate Young s modulus for off-axis samples Bibliography Table of symbols Appendices Appendix 6.A. Timoshenko s correcting coefficients for anisotropic and isotropic materials Appendix 6.B. Correcting coefficient using Mindlin s method Appendix 6.C. Dispersion curves for various equations of motion Appendix 6.D. Change of reference axes and elastic coefficients for an anisotropic rod Chapter 7. Longitudinal Vibration of a Rod Yvon CHEVALIER and Maurice TOURATIER 7.1. Presentation Elementary equation of motion Boundary conditions Bishop s equations of motion Isotropic material Transversely isotropic material Longitudinal wave dispersion Improved Bishop s equation of motion Bishop s equation for orthotropic materials Eigenfrequency equations for a free-free rod Harmonic solution for dimensionless Bishop s equation Boundary conditions and eigenvalue equations Touratier s equations of motion of longitudinal waves General considerations Dispersion curves according to various theories Displacement field and boundary conditions Strain and stress components Energy functionals and Hamilton s principle Equations of motion Boundary equations Wave dispersion relationships Phase velocity in composite materials Extensional waves in an anisotropic composite rod with rectangular cross-section dispersion curves Natural boundary conditions Various types of solutions for the equations of motion

9 xiii 7.8. Short rod and boundary conditions Elementary theory Boundary conditions Concluding remarks about Touratier s theory Bibliography List of symbols Appendices Appendix 7.A. an outline of some studies on longitudinal vibration of rods with rectangular cross-section Appendix 7.B. Formulation of Bishop s equation by Hamilton s principle by Rao and Rao Appendix 7.C. Dimensionless Bishop s equations of motion and dimensionless boundary conditions Appendix 7.D. Touratier s equations of motion by variational calculus. 408 Appendix 7.E. Calculation of correcting factor q (C ijkl ) Appendix 7.F. Stationarity of functional J and boundary equations Appendix 7.G. On the possible solutions of eigenvalue equations Chapter 8. Very Low Frequency Vibration of a Rod by Le Rolland-Sorin s Double Pendulum Mostefa ARCHI and Jean-Baptiste CASIMIR 8.1. Introduction Frequency range Simplicity and ease Short bibliography Flexural vibrations of a rod using coupled pendulums Lagrange equations of motion Eigenvalue equation Solutions for pendulum oscillations Relationship between beating period and sample stiffness k Torsional vibration of a beam by double pendulum Equations of torsional motions Complex compliance coefficient of viscoelastic materials General consideration Expression of (ω 1 -ω 2 ) Utilization of correspondence principle to obtain complex sample rigidity Elastic stiffness of an off-axis rod Elementary equations of motion Higher approximation equations of motion

10 xiv Mechanics of Viscoelastic Materials and Wave Dispersion 8.7. Bibliography List of symbols Appendices Appendix 8.A. Closed-form expression of θ 1 or θ 2 oscillation angles of the pendulums and practical considerations Appendix 8.B. Influence of the highest eigenfrequency ω 3 on the pendulum oscillations in the expression of θ 1 (t) Appendix 8.C. Coefficients a of compliance matrix after a change of axes for transverse isotropic material Appendix 8.D. Mathematical formulation of the simultaneous bending and torsion of an off-axis rectangular rod Appendix 8.E. Details on calculations of s 35 and υ 13 of transverse isotropic materials Chapter 9. Vibrations of a Ring and Hollow Cylinder Jean Tuong VINH 9.1. Introduction Equations of motion of a circular ring with rectangular cross-section Generalized displacement and force Bending equations Strain components Force components Equations of motion Eigenvalue equations Solution of characteristics equation Equations of motion of the ring submitted to in plane forced vibrations Expression of Young s modulus versus Q Bibliography Appendices Appendix 9.A. Expression u (θ) in the three subintervals delimited by the roots λ of equation [9.33] Chapter 10. Characterization of Isotropic and Anisotropic Materials by Progressive Ultrasonic Waves Patrick GARCEAU Presentation of the method Use of ultrasonic waves Practical considerations of ultrasonic waves Propagation of elastic waves in an infinite medium

11 xv Progressive plane waves Christoffel tensor Christoffel s equation Eigenvalues of Γ ik are real Eigenvectors of Γ ik are mutually orthogonal Polarization of three kinds of waves Longitudinal wave Transverse or shear wave Quasi transverse (QT) wave Propagation in privileged directions and phase velocity calculations Wave propagation along material symmetry axes Elastic constants obtained with p colinear with one of the three material directions of symmetry Wave propagation along a non-privileged direction Slowness surface and wave propagation through a separation surface Slowness surface representation Slowness surfaces for transverse isotropic composite material Propagation of an elastic wave through an anisotropic blade with two parallel faces Direct transducer couplings with sample Water immersion bench Concluding remarks Bibliography List of Symbols Appendices Appendix 10.A. Energy velocity, group velocity, Poynting vector Appendix 10.B. Slowness surface and energy velocity Chapter 11. Viscoelastic Moduli of Materials Deduced from Harmonic Responses of Beams Tibi BEDA, Christine ESTEOULE, Mohamed SOULA and Jean Tuong VINH Introduction Trial and error method Are wave dispersion phenomena to be taken into account or not? Williams-Landel-Ferry s method Validity of elementary equations of motion Choice of equations of motion without recourse to Williams-Landel-Ferry s method Guidelines for practicians Viscoelastic complex moduli and frequency range

12 xvi Mechanics of Viscoelastic Materials and Wave Dispersion Solution of a viscoelastic problem using the principle of correspondence First step: elastic solution Second step: estimation of dispersion curves Searching for a solution to viscoelastic moduli using characteristic functions Viscoelastic solution of equation of motions Experimental dynamic responses must be available Elementary equations of motion and closed-form expression for the transmissibility function for an extensional wave Torsional vibration of a viscoelastic rod Bending vibration of a viscoelastic rod Viscoelastic moduli using equations of higher approximation degree Inertia and shear effects Torsional vibration Bending vibration of a rod (higher order approximation) Bibliography Appendices Appendix 11.A. Transmissibility function of a rod submitted to longitudinal vibration (elementary equation of motion) Appendix 11.B. Newton-Raphson s method applied to a couple of functions of two real variables β 1 and β 2 components of β Appendix 11.C. Transmissibility function of a clamped-free Bernoulli s rod submitted to bending vibration Appendix 11.D. Complex transmissibility function of a clamped-free Bernoulli s rod and its decomposition into two functions of real variables β 1, β Appendix 11.E. Eigenvalue equation of clamped-free Timoshenko s rod Appendix 11.F. Transmissibility function of clamped-free Timoshenko s rod Chapter 12. Continuous Element Method Utilized as a Solution to Inverse Problems in Elasticity and Viscoelasticity Jean-Baptiste CASIMIR Introduction Overview of the continuous element method Equilibrium equations and the force-displacement relationship Explicit continuous element State vector and dynamic transfer matrix

13 xvii Transfer matrix and dynamic stiffness presented in suitable form for matrix calculation Eigenvalues and eigenvectors of dynamic stiffness Boundary conditions and their implications in the transfer matrix Extensional vibration of straight beams (elementary theory) The direct problem of beams submitted to bending vibration Euler-Bernoulli s transfer matrix and dynamic stiffness matrix Numerical simulation for a free-free Bernoulli s rod Bending vibration of a free-free Timoshenko s beam Dynamic transfer matrix for Timoshenko s beam Successive calculation steps to obtain a transfer matrix and simple displacement transfer function Eigenvalue calculations Eigenvector calculation of [Dω] Continuous element method adapted for solving an inverse problem in elasticity and viscoelasticity Bibliography Appendices Appendix 12.A. Wavenumbers β 1, β 2 deduced from Timoshenko s equation List of Authors Index