generalized point models in structural mechanics

[...]... Rewriting these waves in spherical coordinates and analysing the phase functions one nds out that saddle point exists only for the integral describing interaction of re ected wave with the spherical wave The phase function in that integral is = ;1 + cos # cos #0 sin ' + sin # sin #0 and the saddle point is at # = #0, ' = =2 Applying the formula (1.23) to the integrals 1 and letting R ! +1 yields (i)... uid loaded plate is the coincidence frequency, that is such frequency fc for which cf = ca Lines 100 and 200 present the wavenumbers k in water and in air correspondingly Their crossing points with the horizontal line mark coincidence frequencies j January 24, 2002 12 0:23 WorldScienti c/ws-b8-5x6-0 BOOK-IVA Vibrations of Thin Elastic Plates and Classical Point Models Above coincidence frequency the... 4.1 General properties of models 4.1.1 Generalized models in two dimensions 4.1.2 Structure of generalized models in three dimensions 4.1.3 Generalized models in the plate with in nite crack 4.2 Extending the model of narrow crack to oblique incidence 4.2.1 Reformulation of the model 4.2.2 Edge waves propagating along a narrow crack 4.3... and f(s) into Taylor series near the saddle point and computing integral of the principal order terms yields the saddle January 24, 2002 0:23 16 WorldScienti c/ws-b8-5x6-0 BOOK-IVA Vibrations of Thin Elastic Plates and Classical Point Models point asymptotics{ Z eip (s)f(s)ds s 2 f exp ip + i sign ( 00) : 4 pj 00j (1.22) Here the phase function , its second order derivative 00 and function f in the right-hand... w(s) 42 ; k0 g ; g 42 ; k0 w(s) dx dy Integrating by parts in the last integral in a similar way as in the Section 1.1.2 allows this integral to be reduced to a contour integralk Due to the second radiation condition in (1.16) and asymptotics of the Green function g(x y r0) the integral over the circle of large radius R tends to zero with R ! +1 Finally the required integral representation takes the form... the right-hand side are calculated in the saddle point Analogously one can derive the saddle point asymptotics in the case of double integral ZZ 2 i eip (s1 s2 ) f(s1 s2 )ds1 ds2 p 00 f exp ip + 4 sgn( 00) : (1.23) p j j Again functions and f and the matrix 00 = 00 of second order derivatives @ 2 =@s2 @ 2 =@s1 @s2 1 =@s1 @s2 @ 2 =@s2 2 @2 are computed in the saddle point which is de ned by the system... Here n is the internal normal to the surface of the scatterer, 0 is the domain where the obstacle is joint to the plate (in the case of holes, 0 is union of all the holes), @ 0 is the contour of 0 on the plate and is the internal normal to this contour, overline stands for complex conjugation Summation in (1.17) is carried on all corner points of @ 0 and in the case of nonsmooth @ the integral is assumed... of interaction can be calculated by the saddle point method which gives exact result for R ! 1 Consider the integral 1 (U (g) U (s)) According to (1.34) the scattered eld at large distances splits into the sum of spherical wave Usph and cylindrical surface wave Usurf Let the incident eld be a plane wave (1.32) The geometrical part in this case is the sum of incident and re ected plane waves Rewriting... afterwards regularization allows stress intensity coe cients introduced as coe cients of singular terms, to characterize the probability of crack growth See details in section 3.3.1 January 24, 2002 0:23 8 WorldScienti c/ws-b8-5x6-0 Vibrations of Thin Elastic Plates and Classical Point Models w becomes complexy Introducing the wave number of exural waves k0 in an isolated plate by the formula 2 %h... isolated plate 3.4.3 Generalized point model 3.4.4 Other models of circular holes 3.5 Model of narrow joint of two semi -in nite plates 3.5.1 Problem formulation 3.5.2 Isolated plate 3.5.3 Generalized model 3.5.4 Scattering by the generalized model of narrow joint Chapter 4 Discussions and recommendations . A. L. Smirinov Generalized Point Models in Structural Mechanics Author: I. V. Andronov Mathematical Problems of Control Theory: An Introduction Author: G. A. Leonov Vibrational Mechanics: . Cataloguing -in- Publication Data A catalogue record for this book is available from the British Library GENERALIZED POINT MODELS IN STRUCTURAL MECHANICS Copyright 0 2002 by World Scientific Publishing. for research specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics. Selected Volumes in Series B Proceedings of the First International Congress on