Ergebnisse der angewandten Mathematik – serie

It is not my intention to present a treatise of elasticity in the follow­ ing pages. The size of the volume would not permit it, and, on the other hand, there are already excellent treatises. Instead, my aim is to develop some subjects not considered in the best known treatises of elasticity but nevertheless basic, either from the physical or the analytical point of view, if one is to establish a complete theory of elasticity. The material presented here is taken from original papers, generally very recent, and concerning, often, open questions still being studied by mathematicians. Most of the problems are from the theory of finite deformations [non-linear theory], but a part of this book concerns the theory of small deformations [linear theory], partly for its interest in many practical questions and partly because the analytical study of the theory of finite strain may be based on the infinitesimal one.

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.

In an epoch-making paper entitled "On an approximate solution for the bending of a beam of rectangular cross-section under any system of load with special reference to points of concentrated or discontinuous loading", received by the Royal Society on June 12, 1902, L. N. G. FlLON introduced the notion of what was subsequently called by LovE "general­ ized plane stress". In the same paper FlLO~ also gave the fundamental equations which express the displacement (u, v) in terms of the complex variable. The three basic equations of the theory of KoLOsov (1909) which was subsequently developed and improved by MUSKHELISHVILI (1915 and onwards) can be derived directly from Filon's equations. The derivation is indicated by FlLO)!E~KO-BoRODICH. Although FILO)! proceeded at once to the real variable, historically he is the founder of the modern theory of the application of the complex variable to plane elastic problems. The method was developed independently by A. C. STEVEXSOX in a paper received by the Royal Society in 1940 but which was not published, for security reasons, until 1945.

Das vorliegende Buch behandelt die unter dem Namen der konflu­ enten hypergeometrischen Funktion bekannte höhere transzendente Funktion, der in den physikalischen und technischen Anwendungen der Mathematik eine besonders in den letzten beiden Jahrzehnten ständig steigende Bedeutung zukommt. Es steht außer Zweifel, daß sich diese Tendenz in der Zukunft noch wesentlich verstärken wird, und so wie zunächst die Zylinderfunktionen nur von einigen Wenigen zuverlässig gehandhabt werden konnten, bis sie heute selbst schon dem rechnenden Ingenieur vertraut geworden sind, so wird auch die Theorie der all­ gemeineren konfluenten hypergeometrischen Funktion sehr bald einem immer größeren Kreis von Physikern geläufig sein. In diese Entwick­ lung soll das vorliegende Buch fördernd eingreifen. Die große praktische Bedeutung der hier behandelten Funktion bedarf schon deswegen kaum einer eingehenden Begründung, weil sie einmal eine große Zahl einfacherer spezieller Funktionen, die schon seit langem zum täglichen Werkzeug des Physikers gehören, als Sonderfälle umfaßt. Es genügt, an dieser Stelle zu erwähnen, daß dazu u. a. der Integrallogarithmus, der Integralsinus und -cosinus, das Fehlerintegral, die Fresnelschen Integrale, die Zylinderfunktionen und endlich die Funktionen des parabolischen Zylinders gehören. Es hat also derjenige, der sich die Mühe macht, die konfluente hypergeometrische Funktion eingehender zu studieren, den nicht hoch genug einzuschätzenden Vorteil, daß ihm die Theorie dieser Funktion die Eigenschaften der aus ihr ableit­ baren Funktionen sozusagen von einer höheren Warte aus zu über­ blicken gestattet.