Theory of Elastic Waves in Crystals

ISBN-13: 9781475712773 / Angielski / Miękka / 2012 / 375 str.

Fedor I. Fedorov

Opis

The translation into English of Academician Fedorov's ex- cellent treatise on elastic wave propagation in solids has come at an opportune time. His systematic exposition of all aspects of this field is most lucid and straightforward. The author has gone to considerable pains to develop in his mathematical background a consistent tensor framework which acts as a unifying motif through- out the various aspects of the subject. In many respects his approach will appear quite novel as his treatment introduces several concepts and parameters previously unfamiliar to the literature of the West. Extensive tables in the final chapters illustrate the application of these ideas to the exist- ing body of experimental data. The book is both extensive and comprehensive in al1 phases of the subject. Workers in the fields of ultrasonic propagation and elastic properties will find this treatise of great interest and direct concern. H. B. Huntington Rensselaer Polytechnic Institute Troy, New York November 1967 v Preface to the American Edition In preparing this edition I have corrected various misprints and errors appearing in the Russian edition, but I have also incorpo- rated some substantial changes and additions, the latter representing some results I and my colleagues have recently obtained and pub_ lished in Russian journals. For example, in section 32 I have added a general derivation of the equation for the seetion of the wave surface by a symmetry plane for cubic, hexagonal, tetragonal, and orthorhombic crystals.

Spis treści

1. General Equation s of the Theory of Elasticity.- 1. Deformation tensor.- 2. Stress tensor.- 3. Equilibrium conditions and the equation of an elastic medium.- 4. Hooke’s law.- 5. Energy of a deformed elastic body.- 6. Tensor for the elastic moduli.- 7. Crystal symmetry.- 8. Elastic moduli of crystals of the lower systems.- 9. Elastic moduli of crystals of the higher systems.- 2. Elements of Linear Algebra and Direct Tensor Calculus.- 10. Vectors and matrices in n-dimensional space..- 11. Three-dimensional tensors and dyads.- 12. The Levi-Civita tensor and its applications.- 13. Eigenvalues and eigenvectors of a second-rank tensor.- 14. Tensor relations in a plane.- 3. General Laws of Propagation of Elastic Waves in Crystals.- 15. Plane waves and Christoffel’s equation.- 16. General properties of the A tensor and forms of plane elastic waves in crystals.- 17. Special directions for elastic waves in crystals.- 18. Longitudinal normals and acoustic axes.- 19. Form of the ? tensor for various crystal systems.- 20. Convoluted tensor for the elastic moduli.- 4. Energy Flux and Wave Surfaces.- 21. The energy-flux vector and the ray velocity.- 22. Energy vector with acoustic axes.- 23. Elliptical polarization in elastic waves and the instantaneous energy-flux vector.- 24. Wave surfaces.- 25. Sections of the wave surfaces by symmetry planes.- 5. General Theory of Elastic Waves in Crystals Based on Comparison with an Isotropic Medium.- 26. Mean elastic anisotropy of a crystal.- 27. Comparison with an isotropic medium.- 28. Special directions.- 29. Approximate theory of quasilongitudinal waves.- 30. Another form of the approximate theory.- 6. Elastic Waves in Transversely Isotropic Media.- 31. Covariant form of the ? tensor.- 32. Phase velocities and displacements.- 33. Comparison of a hexagonal crystal with an isotropic medium.- 34. Mean transverse anisotropy.- 35. Comparison with a transversely isotropic medium.- 7. Elastic Waves in Crystals of the Higher Systems.- 36. Cubic crystals.- 37. Approximate theory for cubic crystals.- 38. Tetragonal crystals.- 39. Comparison with a hexagonal crystal.- 40. Trigonal crystals.- 8. Reflection and Refraction of Elastic Waves.- 41. Boundary conditions for plane elastic waves.- 42. Reflection of elastic waves at the free boundary of an isotropic medium.- 43. Reflection at the free boundary of a crystal.- 44. The complex refraction vector and inhomogeneous plane waves.- 45. Invariant characteristics of the polarization of plane waves.- 46. Inhomogeneous waves at a free boundary.- 9. Elastic Waves and the Thermal Capacity of a Crystal.- 47. Statistical theory of the thermal capacity of a solid.- 48. Computation of the Debye temperature.- 49. Averaging of the products of components of unit vector.- 50. Debye temperatures of cubic crystals.- 51. Debye temperatures of hexagonal crystals.- Literature Cited.