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Mathematics and Its Applications, Managing Editor, M HAZEWINKEL. Centre for Mathematics and Computer Science Amsterdam The Netherlands. Volume 476, The Theory of Anisotropic, Elastic Plates. Tamaz S Vashakmadze, Vekua Institute of Applied Mathematics VIAM. Faculty of Applied Mathematics and Computer Science. of lavakhishvili Tbilisi State University, SPRINGER SCIENCE BUSINESS MEDIA B V. A C I P Catalogue record for this book is available from the Library of Congress. ISBN 978 90 481 5215 5 ISBN 978 94 017 3479 0 eBook. DOI 10 1007 978 94 017 3479 0, Printed on acidjree paper.

AII Rights Reserved, 1999 Springer Science Business Media Dordrecht. Originally published by Kluwer Academic Publishers in 1999. Softcover reprint of the hardcover 1st edition 1999. No part of the material protected by this copyright notice may be reproduced or. utilized in any form or by any means electronic or mechanical. including photocopying recording or by any information storage and. retrieval system without written permis sion from the copyright owner. This monograph is dedicated to the memory of, Solomon Michlin Ilya Vekua and my parents. Anna and Sergy, Foreword Xl, Notations XlV, Introduction 1. 1 The Basic Equations and Boundary Value Problems, in the Theory of Elasticity of Anisotropic Bodies 1. 1 1 The Basic Equations 1, 1 2 Classification of transition methods literature review 2.

Chapter I Refined Theories 5, 2 The Method of the Construction of Refined. Theories without Simplifying Hypotheses 5, 2 1 Isotropic case 6. 2 2 Anisotropic case 19, 2 3 Basic results 25, 2 4 Variational formulation for refined theories generalized. Hellinger Reissner variational principle 29, 3 On the Construction of Refined Theories. for Nonhomogeneous Plates and on the, Problems of Boundary Conditions 32.

3 1 Nonhomogeneous case 32, 3 2 Dynamic case 36, 3 3 Problems connected with lateral boundary conditions. paradoxes of classical refined theories 36, 4 On Construction of Refined Theories of. Elastic Plates with Variable Thickness 39, 4 1 Models corresponding to bending problems 40. 4 2 Models corresponding to plane stress strain problems 45. 5 On Unimprovable Estimates on the Class of Functions. for Transition Errors for Refined Theories 48, Chapter II Theories with Regular Processes 55. 6 The Construction and Investigation, of Vekua s Two Dimensional Models 55.

6 1 Review of related works and basic equations 55. 6 2 Convergence of incomplete Fourier Legendre series. for domains with angles and edges Pollard s, result on limited density 56. 6 3 Construction and justification of Vekua type systems in the case. when plates surfaces are free Korn type inequality 60. 6 4 Construction and justification of Vekua type systems in. the case when the boundary conditions on surfaces are. Newtonian type Korn type inequalities 77, 6 5 Three point operator equations and the method of. differential factorization of such systems plates, of variable thickness 82. 6 6 The case of mixed boundary conditions 89, 6 7 Vekua system for an anisotropic nonhomogeneous. elastic plate 90, 7 On One New Model of Elastic Plates 92.

7 1 Models corresponding to bending problems 93, 7 2 Models corresponding to plane stress strained problems 100. 7 3 Some generalizations 102, 8 The Application of Vekua s Method. Extensions and Examples 103, 8 1 Vekua Kantorovich projective method for self infinitive. and infinitive intervals 103, 8 2 Generalization of the Vekua Kantorovich method for. elastic quasi cylindrical bodies 107, 8 3 Dynamic case 109.

8 4 Investigation of the boundary value problem for. thermodynamical stress strained state of, isotropic bodies Example 1 110. 8 5 The problems of definition of the stress strained state. of an orthotropical elastic plates Example 2 112, 8 6 Isotropic circular cylinder of finite length. Example 3 113, 8 7 Some generalizations 115, 9 Refined Theories for Piezoelectric and Electrically. Conductive Elastic Plates 117, 10 Some New Mathematical Problems of the Theory. of Nonlinear Elasticity 121, 10 1 On the problems of thermoelasticity 121.

10 2 On homogeneity of Vekua theory of plates and shallow. shells 124, 10 3 Research design and unsolved problems 127. 11 A Brief Mathematical Review, Some Justifications of the Vekua Theory for Cusped. Non Shallow and Nonhomogeneous Shells 128, 11 1 Elastic bodies with non smooth boundaries cusped. plates and shells 128, 11 2 On nonlinear non shallow isotropic shells 130. 11 3 To investigation of Vekua theory for isotropic. thickwalled shells of nonhomogeneous structure 136. 11 4 Shell theory using two basic surfaces 142, Chapter III Some Approximate Methods and.

Numerical Realizations 143, 12 Methods of Solving Two Dimensional. Boundary Value Problems 144, 12 1 Variant of a variation discrete method 144. 12 2 On solving two dimensional problems in an unbounded domain 152. 12 3 Disctere difference schemes of approximate solving. two dimensional boundary value problems 158, 12 4 Continuous analogue of alternating direction method 160. 13 To a Numerical Solution of One Dimensional, Boundary Value Problems 164. 13 1 P and Q formulae 165, 13 2 A solution of the boundary value problem 13 3 13 2 169.

13 3 A solution of the boundary value problem 13 1 13 2 176. 14 Generalized Factorization Method 180, 15 Nonlinear Case with Newton s Boundary Conditions 184. 16 To an Analysis of Numerical Methods for Solving Boundary. Value Problems for Second Order Linear Differential. Equations with a Small Parameter 189, 16 1 Estimations of derivatives of a solution by using. asymptotic expansion 189, 16 2 On estimating the remaider term of a multipoint method 195. 17 Some Numerical Realizations 200, 17 1 Numerical results for solving boundary value problems with. boundary layers for second order ordinary differential equations 200. 17 2 Numerical design of shearing forces for elastic plates 205. 17 3 Standard programs for design boundary layer, effects one dimentional case 211.

Bibliography 221, The main purpose of this work is construction of the mathematical theory of. elastic plates and shells by means of which the investigation of basic boundary. value problems of the spatial theory of elasticity in the case of cylindrical do. mains reduces to the study of two dimensional boundary value problems BVP. of comparatively simple structure, In this respect in sections 2 5 after the introductory material methods of re. duction known in the literature as usually being based on simplifying hypotheses. are studied Here in contradiction to classical methods the problems connected. with construction of refined theories of anisotropic nonhomogeneous plates with. variable thickness without the assumption of any physical and geometrical re. strictions are investigated The comparative analysis of such reduction methods. was carried out and in particular in section 5 the following fact was established. the error transition occuring with substitution of a two dimensional model for the. initial problem on the class of assumed solutions is restricted from below. Further in section 6 Vekua s method of reduction containing regular pro. cess of study of three dimensional problem is investigated In this direction the. problems connected with solvability convergence of processes and construction. of effective algorithms of approximate solutions are studied. The investigation of these different methods of reduction and application of. some necessary representations obtained in sections 2 3 made it possible to con. struct in section 7 the theory of elastic plates with the following values. a simplicity and physical evidence based on hypotheses and the possibility. of investigating the problems of error estimation and convergence of the process. characterizing methods containing regular processes are combined ibid. b the differential operator corresponding to this theory is factorized in such. a way that the search for an approximate solution of the initial three dimensional. problem is carried out by means of a parallel procedure i e by inversion of simple. uniform operators being the subject of investigation in two dimensional theory of. elasticity, In section 8 the application of some results from sections 3 4 6 7 is represented. in some more general problems of the theory of elasticity 1 three dimensional. problems for unbounded cylindrical bodies from the point of view of the appli. cation of projective methods from section 6 are studied with substantiation of. Saint Venant principle 2 problems for domains of quasi cylindrical form corre. sponding to shell bodies are considered ibid and the methods of calculation. based on Schwarz alternative technique are proposed 3 algorithms for problems. in the case of thick orthotropic plates are constructed and realized when a hori. zontal section D appears to be single or bi connected boundary domain 4 the. methods set forth are extended to dynamical problems of thermoelasticity 5 two. dimensional models for multi layer plates are suggested 6 the possibility of the. application of methods known in literature as the theory of cracks and problems. of magnetoelasticity for plates constructed in sections 2 4 is discussed. In section 9 we consider the application of some results of Chapter I when. piezoelectric and electric conductive thermodynamic elastic plates are anisotropic. and nonhomogeneous, In section 10 research designs and a list of some unsolved mathematical prob. lems of non linear solid mechanics are given Here also refined theories are con. structed for some linear BVPs of thermoelasticity and methods are developed for. solving three point operator equations arised and in particular for Vekua system. when Oh are nonhomogeneuos with variable thickness of plates or shallow shells. In the section 11 there are briefly described results for developing some parts. of section 6 in the case of non shallow nonhomogeneous isotropic elastic shells. In chapter III methods are considered for the approximate solution of two. dimentional boundary value problems for the system of differential equations. which also arise as a necessary step for solving the BVPs for chapter I and II. In section 12 methods are considered for solving two dimensional BVPs for. strongly elliptical differential systems, In sections 13 16 a method is considered for solving a one dimensional BVP for.

second order nonlinear ordinary differential equations with Newtonian boundary. conditions, The section 17 has applicable character and here are given numerical results. and correspondently programs of design some concrete one and two dimentional. BVP by methods developting in sections 12 16, The material of this investigation is based in particular on the monographs. Package of applied programs of designing spatial constructions 1982j Some prob. lems of mathematical theory of anisotropic elastic plates 1986j as well as on some. of the author s articles published between 1964 to 1996. The material of this book was reported systematically in the Seminar of VIAM. from 1968 till 1997 on VI VII VIII X XI Symposiums of ISIMM and at the. conferences Boundary integral equations problems programms numerical realiza. tions Puschino Moscow 1984 1989 Some part of this monograph contains this. special course of lectures for students of Tbilisi State University and in particular. in September 1997 the author read the cycle of lectures on Tbilisi International. Centre of Mathematics and Informatics TICMI, The author wishes to express his thanks to all participants of these meet. ings especially to Sergei Ambartsumyan Grigori Barenblatt Oleg Belotsercov. ski Aleksander Chekin David Gordeziani George Jaiani Allan Jeffrey Alek. sander Khvoles Karl Kirchgassner Ekkehart Kroner Michail Lazarev Thomas. Levinski Leo Magnaradze George Manjavidze Hamlet Meladze Sitiro Mina. gava Lev Nikitin Petr Perlin Vladimir Petviashvili Jose Rodrigues Wolfgang. Schneider Hans Troger Natali Vadchva Ram Vadchva Iosif Vorovich Wolfgang. Wendland Leonidas Xantis Vakhtang Zgenti Franz Ziegler Henrick Zorski. The author takes this opportunity to express his thanks to a group of math. ematicians from the Vekua Institute of Applied Mathematics especially to Mary. Bitsadze Tsitsinb Gabeskiria lurii Morozov Aliki Muradova Eka Shavlakadze. Tamar Siboshvili Diana and Tamriko Vashakmadze for their careful revision of the. manuscript and valuable contributions to the improvement of this work. The author wishes to thank the ISF Foundation of George Soros Grant KZB. 200 and the Rector of Tbilisi State University Roin Metreveli for their cardinal. Lastly it is a pleasure to express my deep gratitude to Ivo Babushka Philippe. Ciarlet Michiel Hazewinkel Anatolii Fomenko Tengiz Meunargia. Tamaz Vashakmadze, March Octouber 1997, Below we introduce the list of basic notations which will be used afterwards. The repetition of an the index denotes summation small Latin and Greek in. dices assume the values of 1 2 3 and 1 2 accordingly unless otherwise stipulated. In a reference to a subsection of a section the first number denotes the number. of the section the second one denotes the subsection. x Xl X2 X3 or X y z are rectangular Cartesian coordinates. X r cp z are cylindrical coordinates, Oh X D XI X2 x h h h h XI X2 i.

Some Justifications of the Vekua Theory for Cusped Non Shallow and Nonhomogeneous Shells 128 11 1 Elastic bodies with non smooth boundaries cusped plates and shells 128 11 2 On nonlinear non shallow isotropic shells 130 11 3 To investigation of Vekua theory for isotropic thickwalled shells of nonhomogeneous structure 136 11 4 Shell theory