Linear theories of elasticity and thermoelasticity ; Linear and nonlinear theories of rods, plates, and shells

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Linear theories of elasticity and thermoelasticity ; Linear and nonlinear theories of rods, plates, and shells

editor, C. Truesdell ; contributions by S.S. Antman ... [et al.]

（Mechanics of solids / editor, C. Truesdell, v. 2）

Springer-Verlag, 1984, c1973

  • pbk. : U.S.
  • pbk. : Germany

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"This book originally appeared in hardcover as volume VIa/2 of Encyclopedia of physics"--T.p. verso

Includes bibliographies and index

Description and Table of Contents

Description

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 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.

Table of Contents

The Linear Theory of Elasticity.- A. Introduction.- 1. Background. Nature of this treatise.- 2. Terminology and general scheme of notation.- B. Mathematical preliminaries.- I. Tensor analysis.- 3. Points. Vectors. Second-order tensors.- 4. Scalar fields. Vector fields. Tensor fields.- II. Elements of potential theory.- 5. The body B. The subsurfaces L1 and L2 of ?B.- 6. The divergence theorem. Stokes' theorem.- 7. The fundamental lemma. Rellich's lemma.- 8. Harmonic and biharmonic fields.- III. Functions of position and time.- 9. Class CM,N.- 10. Convolutions.- 11. Space-time.- C. Formulation of the linear theory of elasticity.- I. Kinematics.- 12. Finite deformations. Infinitesimal deformations.- 13. Properties of displacement fields. Strain.- 14. Compatibility.- II. Balance of momentum. The equations of motion and equilibrium.- 15. Balance of momentum. Stress.- 16. Balance of momentum for finite motions.- 17. General solutions of the equations of equilibrium.- 18. Consequences of the equation of equilibrium.- 19. Consequences of the equation of motion.- III. The constitutive relation for linearly elastic materials.- 20. The elasticity tensor.- 21. Material symmetry.- 22. Isotropic materials.- 23. The constitutive assumption for finite elasticity.- 24. Work theorems. Stored energy.- 25. Strong ellipticity.- 26. Anisotropic materials.- D. Elastostatics.- I. The fundamental field equations. Elastic states. Work and energy.- 27. The fundamental system of field equations.- 28. Elastic states. Work and energy.- II. The reciprocal theorem. Mean strain theorems.- 29. Mean strain and mean stress theorems. Volume change.- 30. The reciprocal theorem.- III. Boundary-value problems. Uniqueness.- 31. The boundary-value problems of elastostatics.- 32. Uniqueness.- 33. Nonexistence.- IV. The variational principles of elastostatics.- 34. Minimum principles.- 35. Some extensions of the fundamental lemma.- 36. Converses to the minimum principles.- 37. Maximum principles.- 38. Variational principles.- 39. Convergence of approximate solutions.- V. The general boundary-value problem. The contact problem.- 40. Statement of the problem. Uniqueness.- 41. Extension of the minimum principles.- VI. Homogeneous and isotropic bodies.- 42. Properties of elastic displacement fields.- 43. The mean value theorem.- 44. Complete solutions of the displacement equation of equilibrium.- VII. The plane problem.- 45. The associated plane strain and generalized plane stress solutions.- 46. Plane elastic states.- 47. Airy's solution.- VIII. Exterior domains.- 48. Representation of elastic displacement fields in a neighborhood of infinity.- 49. Behavior of elastic states at infinity.- 50. Extension of the basic theorems in elastostatics to exterior domains.- IX. Basic singular solutions. Concentrated loads. Green's functions.- 51. Basic singular solutions.- 52. Concentrated loads. The reciprocal theorem.- 53. Integral representation of solutions to concentrated-load problems.- X. Saint-Venant's principle.- 54. The v. Mises-Sternberg version of Saint-Venant's principle.- 55. Toupin's version of Saint-Venant's principle.- 56. Knowles' version of Saint-Venant's principle.- 56a. The Zanaboni-Robinson version of Saint-Venant's principle.- XI. Miscellaneous results.- 57. Some further results for homogeneous and isotropic bodies.- 58. Incompressible materials.- E. Elastodynamics.- I. The fundamental field equations. Elastic processes. Power and energy. Reciprocity.- 59. The fundamental system of field equations.- 60. Elastic processes. Power and energy.- 61. Graffi's reciprocal theorem.- II. Boundary-initial-value problems. Uniqueness.- 62. The boundary-initial-value problem of elastodynamics.- 63. Uniqueness.- III. Variational principles.- 64. Some further extensions of the fundamental lemma.- 65. Variational principles.- 66. Minimum principles.- IV. Homogeneous and isotropic bodies.- 67. Complete solutions of the field equations.- 68. Basic singular solutions.- 69. Love's integral identity.- V. Wave propagation.- 70. The acoustic tensor.- 71. Progressive waves.- 72. Propagating surfaces. Surfaces of discontinuity.- 73. Shock waves. Acceleration waves. Mild discontinuities.- 74. Domain of influence. Uniqueness for infinite regions.- VI. The free vibration problem.- 75. Basic equations.- 76. Characteristic solutions. Minimum principles.- 77. The minimax principle and its consequences.- 78. Completeness of the characteristic solutions.- References.- Linear Thermoelasticity.- A. Introduction.- 1. The nature of this article.- 2. Notation.- B. The foundations of the linear theory of thermoelasticity.- 3. The basic laws of mechanics and thermodynamics.- 4. Elastic materials. Consequences of the second law.- 5. The principle of material frame-indifference.- 6. Consequences of the heat conduction inequality.- 7. Derivation of the linear theory.- 8. Isotropy.- C. Equilibrium theory.- 9. Basic equations. Thermoelastic states.- 10. Mean strain and mean stress. Volume change.- 11. The body force analogy.- 12. Special results for homogeneous and isotropic bodies.- 13. The theorem of work and energy. The reciprocal theorem.- 14. The boundary-value problems of the equilibrium theory. Uniqueness.- 15. Temperature fields that induce displacement free and stress free states.- 16. Minimum principles.- 17. The uncoupled-quasi-static theory.- D. Dynamic theory.- 18. Basic equations. Thermoelastic processes.- 19. Special results for homogeneous and isotropic bodies.- 20. Complete solutions of the field equations.- 21. The theorem of power and energy. The reciprocal theorem.- 22. The boundary-initial-value problems of the dynamic theory.- 23. Uniqueness.- 24. Variational principles.- 25. Progressive waves.- List of works cited.- Existence Theorems in Elasticity.- 1. Prerequisites and notations.- 2. The function spaces $$\mathop H\limits^ \circ$$m and Hm.- 3. Elliptic linear systems. Interior regularity.- 4. Results preparatory to the regularization at the boundary.- 5. Strongly elliptic systems.- 6. General existence theorems.- 7. Propagation problems.- 8. Diffusion problems.- 9. Integro-differential equations.- 10. Classical boundary value problems for a scalar 2nd order elliptic operator.- 11. Equilibrium of a thin plate.- 12. Boundary value problems of equilibrium in linear elasticity.- 13. Equilibrium problems for heterogeneous media.- Boundary Value Problems of Elasticity with Unilateral Constraints.- 1. Abstract unilateral problems: the symmetric case.- 2. Abstract unilateral problems: the nonsymmetric case.- 3. Unilateral problems for elliptic operators.- 4. General definition for the convex set V.- 5. Unilateral problems for an elastic body.- 6. Other examples of unilateral problems.- 7. Existence theorem for the generalized Signorini problem.- 8. Regularization theorem: interior regularity.- 9. Regularization theorem: regularity near the boundary.- 10. Analysis of the Signorini problem.- 11. Historical and bibliographical remarks concerning Existence Theorems in Elasticity.- The Theory of Shells and Plates.- A. Introduction.- 1. Preliminary remarks.- 2. Scope and contents.- 3. Notation and a list of symbols used.- B. Kinematics of shells and plates.- 4. Coordinate systems. Definitions. Preliminary remarks.- 5. Kinematics of shells: I. Direct approach.- ?) General kinematical results.- ?) Superposed rigid body motions.- ?) Additional kinematics.- 6. Kinematics of shells continued (linear theory): I. Direct approach.- ?) Linearized kinematics.- ?) A catalogue of linear kinematic measures.- ?) Additional linear kinematic formulae.- ?) Compatibility equations.- 7. Kinematics of shells: II. Developments from the three-dimensional theory.- ?) General kinematical results.- ?) Some results valid in a reference configuration.- ?) Linearized kinematics.- ?) Approximate linearized kinematic measures.- ?) Other kinematic approximations in the linear theory.- C. Basic principles for shells and plates.- 8. Basic principles for shells: I. Direct approach.- ?) Conservation laws.- ?) Entropy production.- ?) Invariance conditions.- ?) An alternative statement of the conservation laws.- ?) Conservation laws in terms of field quantities in a reference state.- 9. Derivation of the basic field equations for shells: I. Direct approach.- ?) General field equations in vector forms.- ?) Alternative forms of the field equations.- ?) Linearized field equations.- ?) The basic field equations in terms of a reference state.- 10. Derivation of the basic field equations of a restricted theory: I. Direct approach.- 11. Basic field equations for shells: II. Derivation from the three-dimensional theory.- ?) Some preliminary results.- ?) Stress-resultants, stress-couples and other resultants for shells.- ?) Developments from the energy equation. Entropy inequalities.- 12. Basic field equations for shells continued: II. Derivation from the three-dimensional theory.- ?) General field equations.- ?) An approximate system of equations of motion.- ?) Linearized field equations.- ?) Relationship with results in the classical linear theory of thin shells and plates.- 12 A. Appendix on the history of derivations of the equations of equilibrium for shells.- D. Elastic shells.- 13. Constitutive equations for elastic shells (nonlinear theory): I. Direct approach.- ?) General considerations. Thermodynamical results.- ?) Reduction of the constitutive equations under superposed rigid body motions.- ?) Material symmetry restrictions.- ?) Alternative forms of the constitutive equations.- 14. The complete theory. Special results: I. Direct approach.- ?) The boundary-value problem in the general theory.- ?) Constitutive equations in a mechanical theory.- ?) Some special results.- ?) Special theories.- 15. The complete restricted theory: I. Direct approach.- 16. Linear constitutive equations: I. Direct approach.- ?) General considerations.- ?) Explicit results for linear constitutive equations.- ?) A restricted form of the constitutive equations for an isotropic material.- ?) Constitutive equations of the restricted linear theory.- 17. The complete theory for thermoelastic shells: II. Derivation from the three-dimensional theory.- ?) Constitutive equations in terms of two-dimensional variables. Thermodynamical results.- ?) Summary of the basic equations in a complete theory.- 18. Approximation for thin shells: II. Developments from the three-dimensional theory.- ?) An approximation procedure.- ?) Approximation in the linear theory.- 19. An alternative approximation procedure in the linear theory: II. Developments from the three-dimensional theory.- 20. Explicit constitutive equations for approximate linear theories of plates and shells: II. Developments from the three-dimensional theory.- ?) Approximate constitutive equations for plates.- ?) The classical plate theory. Additional remarks.- ?) Approximate constitutive relations for thin shells.- ?) Classical shell theory. Additional remarks.- 21. Further remarks on the approximate linear and nonlinear theories developed from the three-dimensional equations.- 21 A. Appendix on the history of the derivation of linear constitutive equations for thin elastic shells.- 22. Relationship of results from the three-dimensional theory and the theory of Cosserat surface.- E. Linear theory of elastic plates and shells.- 23. The boundary-value problem in the linear theory.- ?) Elastic plates.- ?) Elastic shells.- 24. Determination of the constitutive coefficients.- ?) The constitutive coefficients for plates.- ?) The constitutive coefficients for shells.- 25. The boundary-value problem of the restricted linear theory.- 26. A uniqueness theorem. Remarks on the general theorems.- F. Appendix: Geometry of a surface and related results.- A.1. Geometry of Euclidean space.- A.2. Some results from the differential geometry of a surface.- ?) Definition of a surface. Preliminaries.- ?) First and second fundamental forms.- ?) Covariant derivatives. The curvature tensor.- ?) Formulae of Weingarten and Gauss. Integrability conditions.- ?) Principal curvatures. Lines of curvature.- A.3. Geometry of a surface in a Euclidean space covered by normal coordinates.- A.4. Physical components of surface tensors in lines of curvature coordinates.- References.- The Theory of Rods.- A. Introduction.- 1. Definition and purpose of rod theories. Nature of this article.- 2. Notation.- 3. Background.- B. Formation of rod theories.- I. Approximation of three-dimensional equations.- 4. Nature of the approximation process.- 5. Representation of position and logarithmic temperature.- 6. Moments of the fundamental equations.- 7. Approximation of the fundamental equations.- 8. Constitutive relations.- 9. Thermo-elastic rods.- 10. Statement of the boundary value problems.- 11. Validity of the projection methods.- 12. History of the use of projection methods for the construction of rod theories.- 13. Asymptotic methods.- II. Director theories of rods.- 14. Definition of a Cosserat rod.- 15. Field equations.- 16. Constitutive equations.- III. Planar problems.- 17. The governing equations.- 18. Boundary conditions.- C. Problems for nonlinearly elastic rods.- 19. Existence.- 20. Variational formulation of the equilibrium problems.- 21. Statement of theorems.- 22. Proofs of the theorems.- 23. Straight and circular rods.- 24. Uniqueness theorems.- 25. Buckled states.- 26. Integrals of the equilibrium equations. Qualitative behavior of solutions.- 27. Problems of design.- 28. Dynamical problems.- References.- Namenverzeichnis. - Author Index.- Sachverzeichnis (Deutsch-Englisch).- Subject Index (English-German).

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