Systematic compilation of integral equations of the Rizzo type and of Kupradze's functional equations for boundary value problems of plane elastostatics

Abstract

With the aid of the singularity method (indirect method) 16 integral equations of the Rizzo type and the corresponding functional equations of the Kupradze type for the geometrical, the statical and the mixed boundary value problem of plane elastostatics are established. Up to now only a few of them and not even in all cases the best ones have been used for the numerical treatment of technical problems. The equations have the same kernels as those compiled in [16], however, they differ fundamentally from those by the fact that the boundary values are subjected to an integral transformation before being prescribed as the "right hand sides". Usually such equations are established with the aid of Green's theorem (direct method). By virtue of the systematic deduction of the equations using layers of edge dislocations, edge dislocation dipoles, forces and force dipoles and prescribing the boundary displacements and their derivatives and the boundary tractions and their integrals all equations can be inserted in a uniform and closed scheme of classification. The integral equations of the Rizzo type are represented as special cases of functional equations of the Kupradze type. Because of the different structures of the kernels not all equations are suited for numerical calculations in the same way. To be able to choose the most suitable formulation for every individual problem the conditions of the equations are investigated and compared with those of integral equations of other important types. Since all functions occurring in the equations can be interpreted mechanically in an elementary manner it is possible to understand and apply these equations without special mathematical knowledge. Chapter 1.2 contains an extensive survey of the paper in hand. ZUSAMMENFASSUNGMit Hilfe der Singularit~itenmethode (indirekten Methode) werden 16 Integralgleichungen vom RizzoTyp und 16 Kupradzesche Funktionalgleichungen fiir das geometrische, das statische und das gemischte Randwertproblem der ebenen Elastostatik aufgestellt. Von den Gleichungen wurden bisher nur wenige und nicht einmal in allen F~illen die besten zur numerischen Behandlung technischer Aufgaben benutzt. Die Gleichungen besitzen zwar dieselben Kerne wie die in [16] angegebenen, unterscheiden sich aber yon jenen grundlegend dadurch, dab die Randwerte einer Integraltransformation unterzogen werden, bevor man sie auf der "rechten Seite" vorschreibt, l)blicherweise werden solche Gleichungen mit Hilfe des Greenschen Satzes aufgestellt (direkte Methode). Dank der systematischen Herleitung der Gleichungen unter Verwendung yon Belegungen yon Versetzungen, Versetzungsdipolen, Kriiften und Kriiftedipolen bei Vorgabe der Randverschiebungen und ihrer Ableitungen sowie der Randspannungen und ihrer Integrale lassen sich alle Gleichungen in ein einheitliches und abgeschlossenes Klassifikationsschema einordnen. Die Integralgleichungen vom Rizzo-Typ werden darin als Spezialf~ille yon Kupradzeschen Funktionalgleichungen dargestellt. Wegen ihrer unterschiedlichen Struktur eignen sich ni...

Solution of integral equations for plane elastostatical problems with discontinuously prescribed boundary values

“…The equations are of the first and of the second kind and have regular kernels or kernels with poles of the first order. Integral equations with logarithmically singular kernels are not investigated at all and equations representing direct boundary element method (see [13][14][15][16][17][18]) statements are only investigated concisely by the example of Rizzo's equation for the problem with given boundary tractions.…”

“…However, effects similar to those described below can also be observed with integral equations of the direct method. As an example we consider here Rizzo's equation for the statical problem [13] (see also [15] Eqs. (6.2), (11.1)).…”