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. I. The Law of Elasticity.- 1.1. Continued dyadic products.- 1.2. The stress tensor.- 1.3. The deformation tensor.- 1.4. The equation of motion.- 1.5. Internal energy.- 1.6. Elastic deformation.- 1.7. Hooke's law.- 1.8. Anisotropy.- 1.9. Elastic symmetry.- Examples I.- II. Stress functions and complex stresses.- 2.0. Introductory notions.- 2.1. Stress functions and fundamental stress combinations.- 2.3. The displacement.- 2.4. The strain-energy function.- 2.5. The elimination of the displacements.- 2.6. The complex stresses.- 2.7. Expression of the fundamental stress combinations in terms 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.- 3.3. Classification of certain antiplane problems.- 3.4. The equations which give the displacement in pure antiplane stress.- 3.5. The boundary condition for the pure antiplane problem for isotropic beams.- 3.6. Simple extension.- 3.7. Bending by terminal couples.- 3.8. Circular cylinder pushed into a hole.- Examples III.- IV. The torsion of isotropic beams.- 4.1. The torsion problem.- 4.2. Lines of shearing stress.- 4.3. The twisting moment.- 4.4. Solution by conformal mapping.- 4.5. The $$ z\\bar z $$method.- 4.6. Boundary conditions.- 4.7. A uniqueness theorem.- 4.8. The principle of virtual stresses.- 4.9. Torsion of a compound bar of isotropic materials.- Examples IV.- V. The flexure of isotropic beams.- 5.1. The flexure problem.- 5.2. The centre of flexure.- 5.3. Half-sections.- 5.4. Shear stress functions.- 5.5. de St. Venant's flexure function.- Examples V.- VI. Antiplane of elastic symmetry.- 6.1. Bending by couples.- 6.2. Boundary conditions.- 6.3. A device for transformingintegrals.- 6.4. Simplifying assumptions.- 6.5. Antiplane of elastic symmetry.- 6.6. The striess component zz.- 6.7. Orthotropic material.- 6.8. Methods of approximation.- Examples VI.- VII. General linear and cylindrical anisotropy.- 7.1. Generalized plane deformation.- 7.2. Line force applied to an elastic half-plane.- 7.3. Induced mappings for the region exterior to an ellipse.- 7.4. Bending of a cantilever by a transverse force at the free end.- 7.5. Cylindrical anisotropy.- 7.6. Equations satisfied by the stress functions.- 7.7. Circular tube under pressure.- Examples VII.- References.
