Mechanics of Solids: Volume IV: Waves in Elastic and Viscoelastic Solids (Theory and Experiment)

Mechanics of Solids: Volume IV: Waves in Elastic and Viscoelastic Solids (Theory and Experiment)

Paperback(Softcover reprint of the original 1st ed. 1974)

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Reissue of Encyclopedia of Physics / Handbuch der Physik, Volume VIa The mechanical response of solids was first reduced to an organized science of fairly general scope in the nineteenth century. The theory of small elastic deformations is in the main the creation of CAUCHY, who, correcting and simplifying the work of N AVIER and POISSON, through an astounding application of conjoined scholarship, originality, and labor greatly extended in breadth the shallowest aspects of the treatments of par­ ticular kinds of bodies by GALILEO, LEIBNIZ, JAMES BERNOULLI, PARENT, DANIEL BER­ NOULLI, EULER, and COULOMB. Linear elasticity became a branch of mathematics, culti­ vated wherever there were mathematicians. The magisterial treatise of LOVE in its second edition, 1906 - clear, compact, exhaustive, and learned - stands as the summary of the classical theory. It is one of the great "gaslight works" that in BOCHNER'S words! "either do not have any adequate successor[s] '" or, at least, refuse to be super­ seded . . . ; and so they have to be reprinted, in ever increasing numbers, for active research and reference", as long as State and Society shall permit men to learn mathe­ matics by, for, and of men's minds. Abundant experimentation on solids was done during the same century. Usually the materials arising in nature, with which experiment most justly concerns itself, do not stoop easily to the limitations classical elasticity posits.

Product Details

ISBN-13: 9783540131632
Publisher: Springer Berlin Heidelberg
Publication date: 07/02/1984
Edition description: Softcover reprint of the original 1st ed. 1974
Pages: 334
Product dimensions: 6.69(w) x 9.61(h) x 0.03(d)

Table of Contents

Wave Propagation in Nonlinear Viscoelastic Solids..- I. Introduction.- 1. Background. Scope of this article.- 2. Plan of this article. Notation.- II. Experimental methods in one-dimensional wave propagation.- 3. Introductory remarks.- 4. Planar impact loading configuration.- 5. Measurement of the transmitted wave profiles.- ?) The displacement interferometer.- ?) The velocity interferometer.- ?) Diffuse surface interferometry.- ?) In-material interferometry.- 6. Determination of wave front velocities.- III. One-dimensional motions in materials with memory.- 7. Kinematics and balance laws.- 8. Classification of waves.- 9. Simple materials with fading memory.- IV. Propagation of acoustic waves.- 10. Infinitesimal sinusoidal progressive waves.- 11. Determination of the stress relaxation function from acoustic wave experiments.- V. Propagation of one-dimensional steady waves.- 12. The existence of steady waves.- 13. Experimental observation of steady shock waves.- 14. Finite linear viscoelasticity and the evaluation of material response functions.- 15. Steady shock wave solutions.- VI. Growth and decay of one-dimensional shock waves.- 16. The shock amplitude equation.- 17. The critical acceleration.- 18. Shock pulse attenuation.- 19. Some remarks on the evolution of steady shock waves.- VII. Growth and decay of one-dimensional acceleration waves.- 20. The amplitude of waves in homogeneously deformed regions.- 21. Expansive waves in precompressed regions.- ?) Acceleration wave.- ?) Expansive wave decay.- 22. Compressive wave growth in undeformed regions at rest.- 23. Toward a three-dimensional characterization of nonlinear viscoelastic solids.- ?) Dilatational response of isotropic finite linear viscoelastic solids.- ?) The velocity of acceleration waves in isotropic solids subject to hydrostatic pressure.- ?) Experimental determination of the instantaneous and equilibrium pressure-density curves.- VIII. Thermodynamic influences on viscoelastic wave propagation.- 24. Thermodynamics of materials with memory.- ?) Constitutive assumption with temperature as the independent thermodynamic variable.- ?) Constitutive assumption with entropy as the independent thermodynamic; variable.- 25. Propagation of steady shock waves.- ?) Steady waves in thermoviscoelastic solids.- ?) A specific thermoviscoelastic constitutive assumption and the evaluation of material response functions.- ?) Steady shock wave solutions.- 26. The growth and decay of shock waves.- ?) General properties of shock transition.- ?) The shock amplitude equation.- ?) Shock wave behavior in a particular thermoviscoelastic solid.- 27. The growth and decay of acceleration waves.- 28. Stress-energy response.- References.- Waves in Solids..- A. Introduction.- 1. Scope.- 2. Notation.- B. Foundations.- 3. Motion and deformation.- 3.1. Geometry of deformation. Strain.- 3.2. Derivatives with respect to time.- 3.3. Useful identities involving J.- 3.4. Transport theorem.- 3.5. More identities.- 4. Stress.- 5. Energy flux and distributed sources.- 6. Equations of balance.- 6.1. Mass balance.- 6.2. A useful formula.- 6.3. Equation of motion.- 6.4. Energy balance.- C. Equilibrium states.- 7. Motivation. Mutative and nonmutative processes.- 8. Thermostatic assumptions. Basis of classical thermostatics.- 9. Tensor and abbreviated notation.- 10. Thermostatic relations and coefficients.- 10.1. Thermodynamic potentials.- 10.2. Specific heats.- Table 10.1. Thermodynamic potentials and their derivatives.- 10.3. Maxwell coefficients.- 10.4. The differences.- 10.5. The difference Cr — Cv and the ratio.- 10.6. Strain dependence of the specific heats and expansion coefficients.- 10.7. Internal energy as a function of other variables.- 10.8. Internal energy as an independent variable.- Table 10.2. Internal energy as a function of (V?, T), (??, T), (??, S).- Table 10.3. S, T, ?? as functions of (V?,U).- Table 10.4. S, T, V? as functions of (??, U).- 10.9. Grüneisen numbers.- 11. Thermostatics under hydrostatic pressure.- 11.1. Thermodynamic potentials.- Table 11.1. Thermodynamic potentials and their derivatives under spherical stress.- 11.2. Compressiblity, bulk modulus, volumetric expansion coefficient.- 11.3. Specific heats at constant volume and constant pressure.- 11.4. Arbitrariness of reference pressure.- 11.5. Maxwell coefficients.- 11.6. BS/B=Cp/CJ.- 11.7. Grüneisen number ?.- 11.8. Pressure derivatives of the specific heats and expansion coefficient.- 11.9. Internal energy as a function of other variables.- 11.10. Internal energy as an independent variable.- Table 11.2. Internal energy as a function of (J, T), (p, T), (p, S).- Table 11.3. S, T, p as functions of (J, U).- Table 11.4. S, T, J as functions of (p, U.- 11.11. Other derivatives with respect to pressure.- Table 11.5. Expressions for pressure derivatives evaluated at zero pressure.- 11.12. Derivatives with respect to temperature.- 11.13. Thermal expansion at constant pressure.- D. Electromechanical interactions.- 12. Basic equations of electromagnetic theory in a material representation.- 12.1. Maxwell’s field equations.- 12.2. Units.- 12.3. Effect of particle velocity.- 12.4. Material representations of fields and their properties.- 12.5. Material representation of the field equations.- 12.6. Integral forms of the equations.- 12.7. Electrodynamic potentials.- 12.8. Poynting’s theorem in spatial and material representations.- 13. Results based on the electrodynamical theory of Tiersten and Tsai.- 13.1. Introduction.- 13.2. The theory of Tiersten and Tsai.- 13.3. The total stored energy.- 13.4. Thermostatics.- 13.5. Equation of motion.- 14. Extension of thermostatics to include electromechanical interactions.- 14.1. Material coefficients.- Table 14.1. Derivatives of U(D, B, S, Vij).- Table 14.2. Thermodynamic potentials including electrical variables (after Mason, 1966).- Table 14.3. Connections among the various coefficients.- Table 14.4. Replacements for obtaining pyroelectric relations from thermoelastic relations.- 14.2. Linear piezoelectric equations.- E. Material symmetry.- 15. Isotropy groups, Laue groups, and crystal point groups.- Table 15-1- Laue groups, generators of associated rotation groups, and point groups included in each Laue group.- Table 15.2. The thirty-two crystal point groups.- Table 15.3. Order of the point groups and number of symmetry operations of each kind.- Table 15.4. Essential symmetry of the crystal systems.- 16. Effect of symmetry on material coefficients.- Table 16.1. Forms of a first-rank polar tensor referred to the conventional cartesian system.- Table 16.2. Second-rank polar tensors.- Table 16.3. Third-rank polar tensors.- Table 16.4. Fourth-rank polar tensors.- Table 16.5. Fifth-rank polar tensors.- Table 16.6. Third-order elastic constants for the eleven Laue groups and for isotropic media.- F. Exponentially damped plane waves.- 17. Complex representation of waves.- 17.1. Waves sinusoidal in time, attenuated in space.- 17.2. Waves sinusoidal in space decaying in time.- 17.3. Inhomogeneous plane waves.- 18. Stress and deformation in exponentially damped plane sinusoidal waves.- G. Linear viscoelastic interactions.- 19. The linear viscoelastic model.- 20. One-dimensional linear viscoelastic models.- 20.1. Introduction.- 20.2. Sinusoidal time variations.- 20.3. Decaying time variations.- Table 20.1. Functions associated with five special models of viscoelasticity.- 20.4. Special cases of linear viscoelasticity.- 21. A difficulty: Thermal effects.- H. Thermoviscoelastic media.- 22. General relations.- 23. A special model for thermoviscoelasticity.- 24. Linearized steady-state response.- I. Small-amplitude waves that are sinusoidal in time.- 25. Thermoviscoelastic medium.- 26. Elastic medium with Newtonian viscosity and heat conduction according to Fourier’s law.- 27. Idealized thermoelastic medium. Elastic medium.- 28. Initially stressed elastic medium.- 28.1. Introduction.- Table 28.1. Description of reference and present states.- 28.2. Linearization of the equation of motion.- 28.3. Solutions for small-amplitude waves.- 28.4. Propagation direction and velocity.- J. Ultrasonic measurements as a function of static initial stress.- 29. Determination of third-order elastic coefficients.- 29.1. Introduction.- 29.2. Initial derivative of QN W2 in terms of material properties.- 30. Effective elastic coefficients.- 30.1. Introduction.- 30.2. Lack of uniqueness of coefficients in equation of motion.- 30.3. Symmetry.- 30.4. Essential difference between wave propagation in unstressed and anisotropically stressed media.- 30.5. Effective elastic coefficients under hydrostatic pressure.- 30.6. Interpretation of the effective elastic coefficients ? ijkm as coefficients in a linearized stress-deformation relation.- 30.7. Relations of effective elastic coefficients to the bulk modulus and compressibility.- 30.8. Measurement of effective elastic coefficients.- 31. Pressure derivatives of elastic coefficients.- 31.1. Introduction.- 31.2. Pressure derivatives of thermodynamic coefficients.- 31.3. Pressure derivatives of effective coefficients.- 31.4. Relation between effective and thermodynamic elastic coefficients.- 31.5. Relation between pressure derivatives of effective and thermodynamic coefficients.- Table 31.1. ??v under hydrostatic pressure (monoclinic and triclinic classes excluded).- Table 31.2. Difference of pressure derivatives of effective and thermodynamic elastic coefficients, (monoclinic and triclinic classes excluded).- Table 31.3. Pressure derivatives of effective and thermodynamic elastic coefficients.- K. Analysis of ultrasonic measurements as a function of temperature.- 32. Elastic coefficients as a function of temperature.- 33. Derivatives with respect to temperature.- L. Examples.- 34. Elastic waves in crystals.- 34.1. Introduction.- 34.2. Special forms of the coefficients for cubic crystals.- 34.3. Conditions for a positive definite strain energy in cubic crystals.- 34.4. Acoustical tensor for arbitrary propagation directions in a cubic crystal.- 34.5. Directions for purely longitudinal and transverse waves in cubic crystals.- Table 34.1. Pure modes in cubic crystals.- 34.6. Determination of elastic constants.- 35. Thermoviscoelastic waves in cubic crystals.- 36. Piezoelectrically excited vibrations.- 36.1. Thickness-shear vibrations of an infinite plate.- 36.2. Electromechanical coupling coefficient.- 36.3. Electrical impedance of a vibrating piezoelectric plate.- 36.4. Equivalent circuit of a piezoelectric transducer.- 36.5. Thickness-longitudinal vibrations of an infinite piezoelectric plate.- 37. Radial motion of thin circular piezoelectric ceramic disks.- 38. Waves of finite amplitude in elastic media.- 38.1. Equation of motion.- 38.2. Characteristics of the equation of one-dimensional longitudinal motion.- 38.3. Simple-wave solution.- 38.4. The discontinuity distance.- 38.5. Particle velocity in the oscillating simple wave.- 38.6. Solution for the displacement.- 38.7. Relation of harmonic growth to higher-order elastic coefficients.- 38.8. Hypothetical linear medium for one-dimensional longitudinal motion.- 38.9. One-dimensional longitudinal stress-extension relation.- 39. Longitudinal shock waves in elastic solids.- 39.1. Relations that apply across a shock propagating into a medium at rest in its reference configuration.- 39.2. The connection of the stress-extension relation to the curve of shock velocity versus particle velocity.- 39.3. Thermodynamic considerations.- 40. Reflection of longitudinal waves at normal incidence.- 40.1. Reflection of a continuous disturbance.- 40.2. Reflection of a shock wave.- 41. Longitudinal motion of a piezoelectric material.- 41.1. Series expansions of the stress and electric field.- 41.2. Entropy jump across a shock in a piezoelectric material.- 41.3. Approximate solution for the passage of a shock wave through a shortcircuited piezoelectric slab.- “Uncoupled” approximation.- “Coupled approximation”.- 41.4. Determination of coefficients.- References.- Namenverzeichnis. — Author Index.- Sachverzeichnis (Deutsch-Englisch).- Subject Index (English-German).

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