Theory of viscoelasticity, plasticity, elastic waves, and elastic stability

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Theory of viscoelasticity, plasticity, elastic waves, and elastic stability

editor, C. Truesdell ; contributions by P.J. Chen ... [et al.]

（Mechanics of solids / editor, C. Truesdell, v. 3）

Springer-Verlag, 1984, c1973

  • : pbk. us
  • : pbk. gw

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注記

"This book originally appeared in hardcover as volume VIa/3 of Encyclopedia of physics"--T.p. verso

Includes bibliographies and indexes

内容説明・目次

内容説明

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 NAVIER 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 l 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.

目次

  • The Linear Theory of Viscoelasticity.- A. Introduction.- 1. Plan and scope of this article.- 2. Notation. Vectors, tensors, and linear transformations.- 3. Processes and histories.- 4. Convolutions.- 5. The Boltzmann operator.- B. Foundations of the linear theory.- 6. Linear hereditary laws.- 7. Boltzmann laws. Definitions.- 8. Characterization of Boltzmann laws.- 9. Constitutive relations. Linearly viscoelastic materials.- 10. Constitutive equations. Boltzmann laws. Stress relaxation.- 11. Relaxation and creep laws.- 12. Isotropic materials.- 13. Additional properties of Boltzmann laws. Mechanical forcing.- 14. Differential operator laws.- 15. Relaxation times and differential operator laws.- 16. Special differential operators.- 17. Field equations.- C. Quasi-static linear viscoelasticity.- 18. The quasi-static assumption.- 19. Quasi-static viscoelastic processes.- 20. Displacement equations of equilibrium.- 21. Equations of compatibility.- 22. Boundary data.- 23. The quasi-static boundary value problem.- 24. Synchronous and separable boundary data.- 25. Initial response. Elastic states.- 26. The past-history problem.- 27. Integral theorems.- 28. Uniqueness of quasi-static viscoelastic processes.- 29. Existence of quasi-static viscoelastic processes.- 30. Quasi-static variational principles.- 31. Elastic-viscoelastic correspondence.- 32. Stress functions for quasi-static viscoelastic processes.- 33. Singular solutions.- 34. Green's processes and integral solutions.- 35. Saint-Venant's principle.- D. Dynamic linear viscoelasticity.- 36. Dynamic viscoelastic processes.- 37. Field equations.- 38. Complete dynamic displacement generating functions.- 39. Power and energy.- 40. Uniqueness. Boundary value problem.- 41. Waves. Singular surfaces.- 42. Initial value problem. Uniqueness and existence of solutions.- 43. Oscillatory displacement processes. Free vibrations.- 44. Dynamic variational principles.- References.- Theory of Elastic Stability.- A. Introduction.- B. Abstract dynamical systems.- 1. Introduction.- 2. General features of a dynamical system.- 3. Definition of a dynamical system.- 4. The set of initial data and a related mapping.- C. Definitions of stability.- 5. Introduction.- 6. Definition of Liapounov stability.- 7. Further definitions.- 8. Continuous dependence.- 9. Instability in the sense of Liapounov.- 10. Boundedness and Liapounov stability.- 11. Instability in the sense of Lagrange.- 12. Stability and uniqueness.- D. Stability theorems for abstract dynamical systems.- 13. Introduction.- 14. Maximum principles.- 15. Liapounov's theorems on stability. (The second method).- 16. Discussion of the theorems.- 17. Examples.- 18. Relation of stability to the calculus of variations.- 19. Theorems on instability.- 20. Boundedness and asymptotic stability.- E. Eigenfunction analyses.- F. Stability of elastic bodies.- 21. Introduction.- 22. Derivation of basic equations.- 23. The equations of perturbed motion. Isothermal linear elasticity.- 24. Incremental equations for thermoelasticity.- 25. Some causes of perturbations.- G. Liapounov functions for finite thermoelasticity.- 26. Introduction.- 27. Liapounov functions from the energy balance equation.- 28. Liapounov functions from the entropy production inequality.- H. Liapounov stability in the class of non-linear perturbations.- 29. Introduction.- 30. Sufficiency theorems.- I. The energy criterion for stability.- 31. Statement and history of the criterion.- 32. Necessary conditions.- 33. Sufficient conditions.- 34. Criticism of the energy criterion.- J. Stability for a fixed surface and under dead loads in the class of small incremental displacements.- 35. Stability of equilibrium.- 36. Stability of a body with time-dependent elasticities.- 37. Discussion of choice of measure.- 38. Stability with multipolar elasticity.- 39. Incompressible media.- K. Stability under dead surface loads in the class of small incremental displacements.- 40. Introduction.- 41. Stability analysis I.- 42. Stability analysis II.- 43. Note on Korn's inequality.- L. Instability under dead surface loads from the equations of linear incremental displacement.- 44. Instability from negative-definite total energy.- 45. Non-uniqueness and instability.- 46. The method of adjacent equilibrium.- 47. History and application of the test.- M. Logarithmic convexity.- 48. Introduction.- 49. Convexity of the function F(t
  • ?,t0).- 50. Applications.- N. Extension of stability analysis for traction boundary conditions.- 51. Stability without an axis of equilibrium.- 52. Stability with an axis of equilibrium.- 53. Instability analysis.- 54. Incompressible media.- O. Stability in special traction boundary value problems.- 55. Introduction.- 56. Isotropic compressible material under hydrostatic stress.- 57. Incompressible elastic material.- P. Stability in the class of linear thermoelastic displacements under dead loads.- 58. Introduction.- 59. Liapounov stability.- 60. Instability.- 61. Hoelder stability.- 62. Asymptotic stability.- Q. Classification of stability problems with non-dead loading.- 63. Introduction.- 64. Group (a): Persistent stability.- 65. Group (b): Motion-dependent data.- R. Stability under weakly conservative loads.- 66. Definitions.- 67. Characterisations of weakly conservative forces.- 68. Stability analyses.- S. Stability with time-dependent and position-dependent data.- 69. Introduction.- 70. Prescribed surface traction with zero initial data.- 71. Prescribed surface traction with non-zero initial data.- 72. Prescribed surface displacement.- 73. Prescribed body force.- 74. Variation in the elasticities.- 75. Change in initial data under dead loading.- 76. Convexity arguments.- 77. Further arguments.- T. Stability under follower forces.- 78. Introduction.- 79. Examples using the Liapounov theory.- 80. Adjacent equilibrium method. Instability by divergence.- 81. Eigenfunction expansions. Analyses depending upon separation of variables.- U. Dissipative forces.- 82. Introduction.- 83. Snap-through.- References.- Growth and Decay of Waves in Solids.- I. Introduction.- 1. Nature of this article.- 2. General scheme of notation.- II. Preliminaries.- 3. Basic kinematical concepts.- 4. Theory of singular surfaces.- 5. Definition of shock waves and acceleration waves.- III. Acceleration waves in elastic bodies.- 6. Longitudinal waves in anisotropic elastic bodies.- 7. Transverse waves in anisotropic elastic bodies.- 8. Thermodynamic influences on waves in anisotropic elastic bodies.- 9. Waves in isotropic elastic bodies.- 10. Waves of arbitrary shape in isotropic elastic bodies.- 11. Thermodynamic influences on waves in isotropic elastic bodies.- IV. One dimensional waves in bodies of material with memory.- 12. Acceleration waves in bodies of material with memory.- 13. The local and global behavior of the amplitudes of acceleration waves.- 14. Acceleration waves entering homogeneously deformed bodies of material with memory.- 15. Thermodynamic influences on acceleration waves in bodies of material with memory.- 16. Shock waves entering unstrained bodies of material with memory.- 17. Consequences of the existence of steady shock waves.- V. One dimensional waves in elastic bodies.- 18. Acceleration waves in elastic bodies.- 19. Shock waves in elastic bodies.- VI. One dimensional waves in elastic non-conductors of heat.- 20. Acceleration waves in elastic non-conductors of heat.- 21. Acceleration waves entering deformed elastic non-conductors.- 22. Shock waves in elastic non-conductors of heat.- 23. Shock waves entering deformed elastic non-conductors.- VII. One dimensional waves in inhomogeneous elastic bodies.- 24. Acceleration waves in inhomogeneous elastic bodies.- 25. Acceleration waves in inhomogeneous elastic bodies at rest.- 26. Shock waves in inhomogeneous elastic bodies.- 27. Shock waves in inhomogeneous elastic bodies at rest.- 1. Existence of the one dimensional kinematical condition of compatibility.- 2. Proofs of Theorems 13.2, 13.3, 13.4 and 13.5.- 3. Derivation of (16.12).- References.- List of works cited.- Additional references.- Ideal Plasticity.- A. The basic equations.- I. The three-dimensional problem.- 1. Quadratic yield condition.- 2. Some basic formulas. Mohr circles.- 3. Plastic potential.- 4. Tresca's yield criterion. "Singular" yield conditions.- 5. "Compatibility" relations.- 6. The flow equations of Prandtl and Reuss.- 7. Further stress strain laws.- 8. Remarks on some three-dimensional problems.- 8 bis. Remarks on uniqueness for rigid plastic solids.- II. Discontinuous solutions.- a) Characteristics. Application to the three-dimensional problem of the perfectly plastic body.- 9. Introduction.- 10. Examples.- 11. Systems of differential equations.- 12. Characteristics of the v. Mises plasticity equations.- 13. Further results and comments.- b) Continuation.- 14. Characteristic surfaces. Characteristic condition.- 15. Compatibility conditions.- 16. Discontinuous solutions.- 17. Preliminary comments on discontinuous solutions in plasticity.- c) Hadamard's theory.- 18. Moving surfaces.- 19. Geometrical and kinematical discontinuity conditions.- 19 bis. Continuation.- 20. Application to a system of equations.- 21. Compatibility conditions.- d) Shock conditions. Stress discontinuities.- 22. " Shock conditions ".- 23. On the classification of discontinuities.- 24. Stress discontinuities.- B. Plane problems.- I. Plane strain, plane stress, and generalizations.- 25. Plane strain with v. Mises' or with Tresca's yield condition derived from three-dimensional problem.- 26. Plane strain under general yield condition.- 27. Plane stress with quadratic yield condition.- 27 bis. Plane stress. Continued.- 28. Generalized plane stress.- II. The theory of plane strain.- a) Differential relations.- 29. Basic equations.- 30. Continuation. Slip line field.- b) Integration. Particular solutions.- 31. Integration.- 32. Examples of exact particular solutions.- 33. Discontinuities.- C. The general plane problem.- I. Basic theory.- a) The equations.- 34. Linearization.- 35. Various yield conditions.- b) Characteristics of the complete plane problem.- 36. Characteristic directions and compatibility relations.- 37. Continuation. Relation to O. Mohr's theory. Differential equations in characteristic coordinates.- 38. Examples for Sects. 36 and 37.- c) Remarks on integration. Examples.- 39. On integration.- II. Singular solutions and various remarks.- a) Limit line singularities and branch line singularities.- 40. Limit line singularities.- 41. Limit line singularities. Continuation.- 42. Branch line singularities.- b) Simple waves.- 43. Definition.- 44. Simple waves. Continuation.- 45. Simple waves for particular yield conditions.- c) Various remarks.- 46. Remarks on the approximate solution of initial-value problems.- 47. Summary remarks on some further problems.- D. Boundary-value problems.- I. Some elastic-plastic problems.- a) The torsion problem.- 48. Fully elastic and fully plastic torsion.- 49. Elastic-plastic torsion.- 50. Examples, further problems and concluding remarks.- b) The thick walled tube.- 51. Expansion of a cylindrical tube.- 52. Partly plastic tube.- 53. Further solutions. Comments.- c) Flat ring and flat sheet in plane stress. Further elastic-plastic problems.- 54. Flat ring radially stressed as a problem of plastic-elastic equilibrium.- 55. Continuation: Plastic-elastic equilibrium.- 56. Expansion of a circular hole in an infinite sheet.- 57. A few further elastic-plastic problems.- II. Some plastic-rigid problems.- a) Various remarks.- 58. The plastic-rigid body.- 58 bis. Axial symmetry. A few remarks.- b) Wedge with pressure on one face.- 59. General discussion and velocity distribution.- c) Plastic mass between rough rigid plates.- 60. Infinite slab.- 61. Slab of material with overhanging ends between rough plates.- Some reference books.- Topics in the Mathematical Theory of Plasticity.- A. Introduction.- B. Foundation of the theory.- I. Thermodynamics for elastic-plastic materials.- 1. Motion and deformation history.- 2. Constitutive functionals.- 3. Elastic-plastic resolution of deformations.- 4. Consequences of the Clausius-Duhem inequality.- 5. Rate independent materials.- 6. Isotropy and principle of frame indifference.- II. Prandtl-Reuss theory.- 7. Flow rule.- 8. Prandtl-Reuss equations.- III. St. Venant-Levy-v. Mises theory.- IV. Theory of Hencky.- C. General theorems.- I. Intrinsic formulation of the variational principle.- II. Examples.- 9. Elastic-plastic torsion in the sense of Hencky.- 10. Stationary creep deformation of a plate.- 11. Plane stress and plane deformation problems.- III. Existence of certain steady plastic flows.- 12. Formulation of a weak problem.- 13. Existence of solution of certain weak problems.- D. Torsion problems.- I. Completely plastic torsion.- 14. Variational formulation of the problem.- 15. Solution of the problem.- 16. St. Venant's conjecture and its extension.- II. Elastic-plastic torsion.- 17. Formal statement of the problem.- 18. Variational formulation of the problem.- 19. Existence and uniqueness of the extremal.- 20. Holder continuity of the extremal.- 21. The existence of an elastic core.- 22. Continuity of stress.- 23. Natural partition of the cross section.- 24. Some properties of E and P.- 25. Elastic-plastic boundary.- 26. Dependence upon angle of twist.- 27. Dependence upon the yield strength.- 28. Dependence upon the angle of twist, continued.- References.- Namenverzeichnis - Author Index.- Sachverzeichnis (Deutsch-Englisch).- Subject Index (English-German).

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