Antiplane Elastic Systems

ISBN-13: 9783540028055 / Angielski / Miękka / 1962 / 266 str.

Louis M. Milne-Thomson

Opis

The term antiplane was introduced by L. N. G. FlLON to describe such problems as tension, push, bending by couples, torsion, and flexure by a transverse load. Looked at physically these problems differ from those of plane elasticity already treated * in that certain shearing stresses no longer vanish. This book is concerned with antiplane elastic systems in equilibrium or in steady motion within the framework of the linear theory, and is based upon lectures given at the Royal Naval College, Greenwich, to officers of the Royal Corps of Naval Constructors, and on technical reports recently published at the Mathematics Research Center, United States Army. My aim has been to tackle each problem, as far as possible, by direct rather than inverse or guessing methods. Here the complex variable again assumes an important role by simplifying equations and by introducing order into much of the treatment of anisotropic material. The work begins with an introduction to tensors by an intrinsic method which starts from a new and simple definition. This enables elastic properties to be stated with conciseness and physical clarity. This course in no way commits the reader to the exclusive use of tensor calculus, for the structure so built up merges into a more familiar form. Nevertheless it is believed that the tensor methods outlined here will prove useful also in other branches of applied mathematics.

Spis treści

I. The Law of Elasticity.- 1.1. Continued dyadic products.- 1.12. Definition of a tensor.- 1.14. Properties of tensors.- 1.2. The stress tensor.- 1.22. Directions of principal stress.- 1.3. The deformation tensor.- 1.32. Deformation tensor in terms of displacement.- 1.34. Virtual displacements.- 1.36. Variation of the Lagrangian deformation tensor.- 1.4. The equation of motion.- 1.5. Internal energy.- 1.54. Energy of deformation.- 1.6. Elastic deformation.- 1.62. The law of elasticity.- 1.64. Form of the strain-energy density for elastic deformation.- 1.66. The tensor H(4).- 1.7. Hooke’s law.- 1.72. Matrix expression of Hooke’s law.- 1.8. Anisotropy.- 1.82. The strain-energy function.- 1.9. Elastic symmetry.- 1.92. Plane of elastic symmetry.- 1.94. Axis of elastic symmetry.- 1.95. Axis of alternating symmetry.- 1.98. Isotropic material.- Examples I.- II. Stress functions and complex stresses.- 2.0. Introductory notions.- 2.01. Definition of an antiplane system.- 2.1. Stress functions and fundamental stress combinations.- 2.12. Case where the body-vector derives from a potential.- 2.2. The strain coefficients for antiplane stress.- 2.24. The stress combination ?.- 2.3. The displacement.- 2.32. The elimination of zz.- 2.4. The strain-energy function.- 2.5. The elimination of the displacements.- 2.52. The equations satisfied by the stress functions.- 2.56. Notation for indefinite integration.- 2.6. The complex stresses.- 2.62. The characteristic equation has no real roots.- 2.7. Expression of the fundamental stress combinations in terms of the complex stresses.- 2.72. The displacement in terms of the complex stresses.- 2.73. Induced circuits.- 2.74. The cyclic properties of the complex stresses.- 2.75. Boundary conditions for the complex stresses.- 2.76. The form of the complex stresses at infinity.- 2.77. Determinateness of the complex stresses.- 2.8. Effective stress functions.- 2.9. The shear function.- Examples II.- III. Isotropic beams.- 3.1. The boundary conditions for a prismatic beam.- 3.2. The isotropic beam.- 3.22. The moment equations.- 3.24. Determination of the longitudinal stress component zz.- 3.25. The area theorem.- 3.26. The moment M3.- 3.27. The position of the load point.- 3.3. Classification of certain antiplane problems.- 3.4. The equations which give the displacement in pure antiplane stress.- 3.41. Expressions for the displacement in pure antiplane stress.- 3.5. The boundary condition for the pure antiplane problem for isotropic beams.- 3.51. Solution of the pure antiplane problem for isotropic beams.- 3.52. Expression of shears in curvilinear coordinates.- 3.6. Simple extension.- 3.62. Suspended cylinder.- 3.7. Bending by terminal couples.- 3.8. Circular cylinder pushed into a hole.- 3.81. Displacement for the pushed cylinder.- 3.82. Lines of principal stress for the pushed cylinder.- 3.83. Cylindrical roller gripped between parallel planes.- Examples III.- IV. The torsion of isotropic beams.- 4.1. The torsion problem.- 4.2. Lines of shearing stress.- 4.22. The displacement.- 4.24. Hydrodynamic analogies.- 4.26. Maximum shearing stress.- 4.28. Change of axis of twist.- 4.3. The twisting moment.- 4.32. The twisting moment in terms of ?.- 4.34. Torsional rigidity.- 4.36. Maximum torsional rigidity.- 4.37. A minimum property of the torsional rigidity.- 4.38. A maximum property of the torsional rigidity.- 4.39. Potential energy.- 4.4. Solution by conformal mapping.- 4.42. Cross-section a circle.- 4.44. Cross-section a cardioid.- 4.46, Cross-section one loop of Bernoulli’s lemniscate.- 4.48. Mapping on a semicircle.- 4.5. The $$ z\bar z $$method.- 4.52. Cross-section an ellipse.- 4.53. Cross-section an equilateral triangle.- 4.54. Cross-section a lune.- 4.55. Cross-section an epitrochoid.- 4.56. Cross-section Booth’s lemniscate.- 4.6. Boundary conditions.- 4.61. Rectangular cross-section.- 4.63. Confocal elliptic ring.- 4.7. A uniqueness theorem.- 4.71. The principle of virtual displacements.- 4.72. Application of the principle of virtual displacements to torsion.- 4.73. Elliptic cross-section.- 4.74. Rectangular cross-section.- 4.75. Parabolic cross-section.- 4.77. The membrane analogy.- 4.8. The principle of virtual stresses.- 4.81. Application of the principle of virtual stresses to torsion.- 4.83. Composite profiles.- 4.85. Cross-section a parabolic crescent.- 4.87. Segmental cross-section.- 4.88. Triangular cross-section.- 4.9. Torsion of a compound bar of isotropic materials.- 4.92. Circular shaft with an inserted circular shaft.- Examples IV.- V. The flexure of isotropic beams.- 5.1. The flexure problem.- 5.11. The stress component zz in the isotropic case.- 5.12. The geometrical parameters of the cross-section.- 5.14. The shears xzyz.- 5.16. The displacement.- 5.2. The centre of flexure.- 5.22. Cross-section a circle.- 5.23. Cross-section a circular annulus.- 5.24. Cross-section a limaçon.- 5.26. Cross-section one loop of Bernoulli’s lemniscate.- 5.3. Half-sections.- 5.4. Shear stress functions.- 5.41. Timoshenko’s stress function.- 5.42. Cross-section an ellipse.- 5.44. Cross-section an equilateral triangle.- 5.46. Gauss’s theorem and integration by parts.- 5.47. The principle of virtual stresses applied to Timoshenko’s stress function.- 5.48. Cross-section an ellipse (virtual stresses).- 5.5. de St. Venant’s flexure function.- 5.51. Cross-section a rectangle.- 5.53. Application of the principle of virtual work.- 5.54. Cross-section an ellipse.- 5.56. Cross-section a rectangle (virtual work).- Examples V.- VI. Antiplane of elastic symmetry.- 6.1. Bending by couples.- 6.2. Boundary conditions.- 6.3. A device for transforming integrals.- 6.32. Relations between the applied forces.- 6.33. Properties of the couples M1, M2, M3.- 6.4. Simplifying assumptions.- 6.41. Twisting by an axial couple.- 6.42. Twisting with simultaneous bending couples.- 6.44. Twisting of an elliptic cylinder by an axial couple.- 6.46. Pure torsion of an elliptic cylinder.- 6.5. Antiplane of elastic symmetry.- 6.52. An affine change of variables.- 6.53. Reduction to the isotropic case.- 6.56. Solution of the equation satisfied by ?.- 6.6. The striess component zz.- 6.61. Determrnation of the constants A1, B1, C1, A2, B2, C2.- 6.62. Bounday conditions.- 6.63. Flexure of an elliptic cylinder with an antiplane of elastic symmetry.- 6.64. The twisting moment.- 6.65. The displacement.- 6.7. Orthotropic material.- 6.72. Torsion of an orthotropic elliptic cylinder.- 6.74. Torsion of an orthotropic rectangular beam.- 6.76. Flexure of an orthotropic elliptic cylinder.- 6.8. Methods of approximation.- Examples VI.- VII. General linear and cylindrical anisotropy.- 7.1. Generalized plane deformation.- 7.12. The complex stresses for generalized plane deformation.- 7.2. Line force applied to an elastic half-plane.- 7.3. Induced mappings for the region exterior to an ellipse.- 7.32. Elliptic cylindrical hole in an infinite elastic space.- 7.33. Determination of the complex stresses.- 7.35. Elliptic hole under hydrostatic pressure.- 7.37. Unloaded elliptic cylindrical hole in a space under stress.- 7.4. Bending of a cantilever by a transverse force at the free end.- 7.41. Centre of flexure.- 7.42. Timoshenko’s stress function.- 7.44. Flexure of a cylinder with elliptic or circular cross-section.- 7.5. Cylindrical anisotropy.- 7.52. The displacement in cylindrical anisotropy.- 7.54. Lateral and end conditions.- 7.6. Equations satisfied by the stress functions.- 7.7. Circular tube under pressure.- 7.71. Determination of the stresses.- 7.72. Lamé’s problem of the tube under pressure.- Examples VII.- References.