By Louis M. Milne-Thomson
The time period antiplane used to be brought via L. N. G. FlLON to explain such difficulties as stress, push, bending via undefined, torsion, and flexure through a transverse load. checked out bodily those difficulties vary from these of airplane elasticity already handled * in that sure shearing stresses now not vanish. This publication is anxious with antiplane elastic platforms in equilibrium or in regular movement in the framework of the linear conception, and relies upon lectures given on the Royal Naval university, Greenwich, to officials of the Royal Corps of Naval Constructors, and on technical experiences lately released on the arithmetic study middle, usa military. My goal has been to take on each one challenge, so far as attainable, by way of direct instead of inverse or guessing tools. right here the advanced variable back assumes an enormous function through simplifying equations and via introducing order into a lot of the remedy of anisotropic fabric. The paintings starts with an advent to tensors by means of an intrinsic strategy which begins from a brand new and straightforward definition. this allows elastic houses to be acknowledged with conciseness and actual readability. This path by no means commits the reader to the unique use of tensor calculus, for the constitution so equipped up merges right into a extra established shape. however it is thought that the tensor equipment defined right here will turn out worthy additionally in different branches of utilized arithmetic.
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Extra info for Antiplane Elastic Systems (Ergebnisse der angewandten Mathematik)
Introductory notions When the external force is parallel to the generators we have the problem of extension (or compression), and when the external force is perpendicular to the generators we have the problem of flexure. If the external forces reduce to twisting couples about a longitudinal axis we have the torsion problem, while the problem of bending by couples arises when the external forces reduce to couples about an axis perpendicular to the generators. Naturally all the above sets of forces may be applied simultaneously to give a mixed problem of extension, flexure, torsion and bending.
A third class besides the identical operation allows the operation of reflexion in a plane only. There are a further 24 classes for which a principal axis exists, that is to say every axis of symmetry other than the principal axis is at right angles to the principal axis, and every plane of symmetry either passes through the principal axis or is at right angles to that axis. The 5 remaining classes are characterized by the presence of 4 axes of symmetry for which the rotation is 2n/3 and which are equally inclined to one another like the diagonals of a cube.
There are a further 24 classes for which a principal axis exists, that is to say every axis of symmetry other than the principal axis is at right angles to the principal axis, and every plane of symmetry either passes through the principal axis or is at right angles to that axis. The 5 remaining classes are characterized by the presence of 4 axes of symmetry for which the rotation is 2n/3 and which are equally inclined to one another like the diagonals of a cube. Not all these classes of geometrical symmetry correspond with distinct classes of elastic symmetry, indeed from the point of view of elastic symmetry a division into 9 classes suffices *.