Foundations of Engineering Mechanics. A. N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies

Transcription

1 Foundations of Engineering Mechanics A. N. Guz, Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies

2 Springer-Verlag Berlin Heidelberg GmbH

3 A.N.Guz Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies Translated by M. Kashtalian With 26 Figures ' Springer

4 Series Editors: Prof. V. 1. Babitsky, DSc Loughborough University Department of Mechanical Engineering LEII 3TU Loughborough, Leicestershire United Kingdom Prof. Dr. J. Wittenburg Universităt Karlsruhe (TH) Institut fur Mechanik KaiserstraBe 12 D Karlsruhe I Germany Author: Prof. Dr. A.N. Guz National Academy of Sciences of Ukraine Institute of Mechanics Nesterov Str Kiev I Ukraine Translator: Dr. M. Kashtalian Imperial College of Science, Technology and Medicine Department of Aeronautics Prince Consort Road SW7 2BY London I United Kingdom Library of Congress Cataloging-in-Publication Data Guz',Aleksandr Nikolaevich, [Trekhmernaia teoriia ustoichivosti deformiruemykh tel. English] Fundamentals of the three-dimensional theory of stability of deformable bodies I A.N. Guz; translated by Maria Kashtalian. p. cm. -- (Foundations of engineering mechanies) Includes bibliographical references and index. ISBN (alk. paper) I.Elastie solids--mathematieal models. 2. Deformations (Mechanies)--Mathematical models. 3. Thin-walled structuresa. Structural stability.1. Title. II. Series. TA418.G ' ddc CIP ISBN DOI / ISBN (ebook) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is coneerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadeasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publieation or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999 Softcover reprint of the hardcover 1 st edition 1999 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready copyfrom translator Cover-Design: de'blik, Berlin SPIN Printed on acid-free paper

5 To my wife Galina for her patience and understanding

6 Preface to the Russian edition At the present time stability theory of deformable systems has been developed into a manifold field within solid mechanics with methods, techniques and approaches of its own. We can hardly name a branch of industry or civil engineering where the results of the stability theory have not found their application. This extensive development together with engineering applications are reflected in a flurry of papers appearing in periodicals as well as in a plenty of monographs, textbooks and reference books. In so doing, overwhelming majority of researchers, concerned with the problems of practical interest, have dealt with the loss of stability in the thin-walled structural elements. Trying to simplify solution of the problems, they have used two- and one-dimensional theories based on various auxiliary hypotheses. This activity contributed a lot to the preferential development of the stability theory of thin-walled structures and organisation of this theory into a branch of solid mechanics with its own up-to-date methods and trends, but left three-dimensional linearised theory of deformable bodies stability (TL TDBS), methods of solving and solutions of the three-dimensional stability problems themselves almost without attention. It must be emphasised that by threedimensional theories and problems in this book are meant those theories and problems which do not draw two-dimensional plate and shell and one-dimensional rod theories. Recent advances in technologies and material science along with intrinsic trends within solid mechanics gave a powerful incentive to the development of TL TDBS as a separate field of solid mechanics. Stability of thick-walled metal structures and thin-walled composite structures with low shear strength, fracture of composite and prestressed materials (materials with initial stresses), stability of rubber-like materials and structural elements, wave propagation in prestressed solids and non-destructive detection of residual stresses, related problems of rock mechanics (e.g., stability of rock workings), geophysics (e.g., folding in the earth's crust) and biomechanics are only few examples of the problems which are worth considering or which need to be considered within the scope of TL TDBS. Three-dimensional theory of deformable bodies stability (TL TDBS), as it is treated by the author in this book, embraces setting up the problems and developing of general issues, methods of solving and solutions of selected groups of problems in the context of the three-dimensionallinearised equations (i.e. without reduction of them to two- and one-dimensional ones with the help of certain hypotheses) using stability criteria, which correspond to the approved stability criteria in the stability theory of thin-walled structures. In so doing, investigation of some groups of problems concerning particular three-dimensional or twodimensional (in plane and axisymmetrical problems) states, which cannot be solved for one or another reason within the scope of the stability theory of thinwalled structures, is one of the important tasks of TLTDBS. The other is the estimation of accuracy of the stability theory of thin-walled structures in the prob-

7 VIII Preface to the Russian edition!ems, where the rigour of its application is not quite obvious. Thus, employment of methods of the theory of stability of motion is regarded as a separate issue of TL TBDS and is not considered in this book. Deformable bodies, including natural and man-made constructive materials, may be divided into two categories: the first for the materials which can undergo large deformations, for example, elastomers and some materials under extreme conditions; the second for materials which can undergo only small deformations (the most of constructive materials including comparatively rigid composite materials under usual working conditions). This division, although rather artificial yet commonly accepted, is also reflected within the scope of TL TDBS, where two theories have been developed: one describes finite (large) precritical deformations and is suitable for materials of the first category, and the other, existing in several variants, describes small precritical deformations and is suitable for materials of the second category. Of course, the finite precritical deformations theory is more general and yields the small precritical deformations theory. However, from the standpoint of applications, the small precritical deformations theory has gained much more acceptance, since it suits to the sufficiently wide range of materials and because the behaviour of materials under large deformations is not comprehensively studied yet. Consideration of materials under small deformations within the scope of the finite precritical deformations theory seems unreasonable due to the fact that in this case the theories differ by the very small quantities. The difference of such magnitude is usually ignored while processing experimental data. These circumstances contributed to the more detailed exposition in this book of TL TDBS for small precritical deformations as applied to various types of solids. TLTDBS for finite deformations, consistently expounded in this book in the general enough form for elastic solids, is undoubtedly necessary, in particular, for materials of the first category, and worth developing further. Development of TLTDBS began in the first half of this century with the investigations of elastic stability, which were later carried out also for non-elastic solids. Since three-dimensional linearised equations of motion and boundary conditions are common for elastic and non-elastic solids, we shall not distinguish TLTDBS for elastic and non-elastic solids while discussing historical aspects. Equations of TLTDBS for small precritical deformations and the uniform precritical state were originally obtained in 1913 by Southwell [105] on the physical grounds. Later Biezeno and Hencky [85], again on the physical grounds, developed linearised equations for an arbitrary (non-uniform) precritical state. The equations, derived in [105] and [85] in Cartesian and cylindrical co-ordinates, are also given in [7]. It was Biot who originally derived in [86, 87] the equations of TLTDBS under small precritical deformations in Lagrangian co-ordinates, which coincide with Cartesian ones in undeformed state, by linearisation of equations of the nonlinear elasticity theory. In so doing, displacement derivatives were assumed to be small along with the strains. The main results of the mechanics of incremental deformations, a counterpart of TL TDBS, obtained by Biot in his numerous papers, were summarised in his book [88], the first ever monograph on TLTDBS. Equations of TLTDBS in curvilinear co-ordinates were constructed with the help of tensor analysis by Neuber [99], who also obtained equations in terms of displacements for the particular form of dependence between stresses and strains and considered some methods of their integration in the case of the uniform precritical state. Results of [99] in Cartesian co-ordinates correspond in a certain sense to [86-88].

8 Preface to the Russian edition IX Trefftz was the first who derived in [107, 108] equations of TLTDBS for small precritical deformations using variational methods. In these papers he also obtained sufficient conditions of stability. Trefftz's approach was extended by Kappus, who formulated in [97] equations of motion of the elasticity theory under large deformations in Lagrangian co-ordinates, which before deformation coincide with Cartesian ones. Besides that, he was the first who obtained in 1939 the equations of TL TDBS under finite (large) precritical deformations, consistently simplified them for a particular form of elastic stress-strain relations and made a transition to the small precritical deformations theory [97]. Equations of TL TDBS under small precritical deformations were also obtained by Novozhilov [68], who simplified them for small rotation angles. Using tensor representation, Green, Rivlin and Shield in 1952 provided the most general form of equations of TL TDBS under finite precritical deformations for an isotropic elastic solid with an arbitrary form of elastic potential [90]. They also considered a number of particular cases. In their paper linearised elasticity theory was referred to as the theory of small elastic deformations superposed on finite elastic deformations, the term subsequently used by other authors. It must be emphasised that although the above papers [105, 86, 87, 99, 97, 68, 90] dealt with three-dimensional linearised equations of elastic stability, linearised equations of motion in terms of stresses and boundary conditions are common for all types of solids. They can be specified by substituting into them linearised constitutive equations for a particular type of a solid. More detailed account of historic development oftltdbs is given in the author's paper [21]. Thus, in the first half of this century linearised equations of motion in terms of stresses and boundary conditions were obtained in a rather rigorous way for finite and small precritical deformations. In the second half of the century development of TL TDBS was focussed on general issues such as variational principles, general problem statement, construction of solutions, their existence and uniqueness, and on solving particular groups of problems. The obtained results were published in monographs [2, 20, 22, 27, 29, 36, 37-41, 49, 63, 80, 88, 109] and outlined in surveys [1, 3-5, 21, 34, 35, 39, 44, 45, 84], partially they are also included in wellknown books on continuum mechanics [12, 68, 81]. An intensive development of TL TDBS with its promising applications as well as the existence of sufficient amount of publications summarising achievements in separate branches of TL TDBS emphasise the necessity in a book, in which the fundamentals of TL TDBS would be expounded in a systematic and consistent way and which could be used by advanced students. To the best of author's knowledge, at present the book of this kind is not available, and he makes an attempt to fill, a! least partly, the gap. The present book consists of four chapters. The list of references contains only papers devoted to the general issues of TL TDBS, well-known books on general issues of solid mechanics and selected papers on related topics. Besides that, it includes all monographs and surveys on TL TDBS known to the author. While writing the book, the author tried to follow the style, notations and apparatus accepted in the home scientific publications. In the case of existence of several versions of exposition of the subject, the author gave preference to his own works and those of his disciples in order to achieve integrity of treatise. Kiev, 1985 A.N. Guz

9 Preface to the English translation The present book is devoted to the fundamentals of three-dimensional linearised theory of deformable bodies stability (TL TDBS) and includes formulation of basic equations, stability criteria and variational principles for various types of solids, representation of general solutions and analysis of the simplest groups of problems with detection of mechanical effects. TL TDBS is one of constituents of three-dimensionallinearised solid mechanics (TLSM), along with statics and dynamics of prestressed solids (solids with initial stresses). Statics of prestressed solids deals, for example, with brittle and quasibrittle fracture, contact interactions and some other problems for prestressed solids, whereas dynamics is concerned, for instance, with wave propagation in prestressed solids (including physical foundations of non-destructive ultra-sonic methods of detecting stresses in solids), dynamic problems of brittle fracture of prestressed solids etc. The common feature of all these groups of problems is utilisation of the basic equations of TLSM, expounded in Chap. 2 of the present book. This fact considerably expands the domain of applicability of the results of the present book to the field of solid mechanics. The first results, obtained by the author on TLSM, were published as early as in By 1986, i.e. the time of publication of the present book in Russian, anumber of books on TLMS had been already written by the author and his disciples [20, 22, 27, 29, 36-38, 40, 41]. In the subsequent years the following books on the subject have appeared: Guz AN (1986) Elastic waves in solids with initial stresses, vols. 1-2 (in Russian). Naukova Dumka, Kiev. Vol. I. General theory, 376pp. Vol. 2. Propagation regularities, 536pp Guz AN (1990) Mechanics of fracture of composite materials in compression (in Russian). Naukova Dumka, Kiev, 632pp Guz AN, Babich SY, Rudnitskii VB (1995) Contact interaction of elastic solids with initial stresses (in Ukrainian). Vyshcha Shkola, Kiev, 304pp Besides that, the following books on TLSM are worth mentioning: Guz AN, Babich IY (1985) Three-dimensional theory of stability of deformable bodies (in Russian). Naukova Dumka, Kiev, 280pp. Vol. 4 in: Guz AN (ed) ( ) Three-dimensional problems of elasticity and plasticity, vols. 1-6 (in Russian). Naukova Dumka, Kiev Guz AN (1991) Brittle fracture of materials with initial stresses (in Russian). Naukova Dumka, Kiev, 288pp. Vol. 2 in: Guz AN (ed) ( ) Nonclassical problems of fracture mechanics, vols. 1-4 (in Russian). Naukova Dumka, Kiev Guz AN, Dyshel MS, Nazarenko VM (1992) Fracture and stability of materials with cracks (in Russian). Naukova Dumka, Kiev, 456pp. Vol. 4, book 1 in: Guz AN (ed) ( ) Non-classical problems of fracture mechanics, vols. 1-4 (in Russian). Naukova Dumka, Kiev

10 Preface to the English translation XI The recent monograph Guz AN (1998) Dynamics of compressible viscous fluid (in Russian). A.S.K., Kiev, 310pp is devoted to the related problems of hydrodynamics. The author believes that the above bibliographic data will help the Englishspeaking readers to be informed about the results obtained by Ukrainian scientists in Kiev, at the SP Timoshenko Institute of Mechanics, within the last three decades. Some results on one of the above problems, published in English, may be found in a special issue of "Applied Mechanics Review": Guz AN (ed) (1992) Micromechanics of composite materials: Focus on Ukrainian Research. Appl Mech Review 45(2): Finally, the Supplement has been added to the English translation. It is worth noting that the methods of solving problems in TL TDBS differ substantially in the case of uniform and non-uniform precritical states. If the precritical state is nonuniform, numerical methods and methods based on variational principles are mainly used [27], whereas in the case of uniform precritical or initial state general solutions and based on them methods of complex potentials are applied. With the help of the latter a number of exact solutions of linearised problems of solid mechanics has been found in the recent years. It is these results that are outlined in the Supplement. For the sake of convenience, the Supplement is provided with a separate list of references. The author would like to express his deep gratitude to Igor A. Guz, D.Sc., for his much appreciated help in preparing the English edition of the book and correcting misprints in the Russian original, and Dr. Maria Kashtalyan for her most careful translation of the book which contributed to improvement of the exposition, and her tremendous work on preparation of the English manuscript. Kiev, June 1998 A.N. Guz

11 Translator's Preface I hope that this translation will enable many others to share the pleasure the original has given to me. The present monograph is just one of the tremendous collection of more than 330 books in Ukrainian, Russian and English, published over eight decades by the scientists of the SP Timoshenko Institute of Mechanics since its foundation by Professor Timoshenko in Unfortunately, only a part of them is known to the English-speaking researchers. I believe that this translation and the recent paper Guz AN, Rushchitsky JJ (1998) Main Ukrainian historical and modern sources on mechanics: Focus on the SP Timoshenko Institute of Mechanics. Appl Mech Review 51(3): which provides the unique bibliographical information, will help to fill, to some extent, the existing gap. While translating the list of citations, I tried to adapt it as much as possible for the English-speaking readers. Since the list contains plenty of papers, published originally in Russian in the journals (e.g., Prikladnaya Mekhanika, Fiziko Khimicheskaya Mekhanika Materialov and others) which have being translated into English, I have made references to them in accordance with the English translation by the publishers. Besides that, the references to English editions of Russian monographs are added if available, and the references to Russian translations of English monographs are changed to those to the originals. In so doing, the original order of citation is retained, so that by Russian-speaking readers familiar with the original could easily use the English edition. I am the most grateful to Igor A. Guz, D.Sc., for his assistance in the arduous task of proof-reading the manuscript. Cambridge-Kiev-London, June 1998 Maria Kashtalyan

12 Table of contents Preface to the Russian edition VII Preface to the English translation X Translator's preface... XII Chapter 1. Fundamentals of nonlinear solid mechanics Essentials of tensor analysis Notations and co-ordinate systems Base vectors, metric tensor Geometrical objects. Invariants of the second order tensor Covariant differentiation. Physical components Particular cases Description of state of strain Strain tensor and its invariants Changes in geometrical objects Definition of,tracking" loads Simplifications for small deformations Simplifications for small rotation angles Description of state of stress Stress tensors Equations of motion. Boundary conditions Virtual work of external forces Simplification for small deformations Feasible generalisation of the small deformations theory Elastic solids General elastic solid Hyperelastic solid Linear elastic anisotropic solid On the development of the simplest nonlinear elasticity theory Plastic solids Deformation theory (theory of small elastoplastic deformations) Flow theory (general equations) Flow theory of strain-hardening solids (inversion of relationships, particular cases) Theory of the perfectly plastic solid

13 XIV Table of contents 6 Solids with rheological properties The simplest relationships of linear theory Linear theory of viscoelasticity (integral form) Nonlinear theory of viscoelasticity (nonlinear creep) Complex media. Constitutive equations for viscoelastoplastic solids Chapter 2. Fundamentals of linearised solid mechanics States of stress and strain Principles of construction of linearised theory Geometrical relationships Equations of motion. Boundary and initial conditions Simplifications for small deformations. Classification of linearised problem statements Definition of,tracking" load Uniform states Feasible generalisations of the small initial deformations theory Elastic solids Compressible solids Incompressible solids On the analogy with the linear elasticity theory Theorem on the uniqueness of solution in the linearised elasticity Sufficient conditions of stability for compressible bodies Sufficient conditions of stability for incompressible bodies Non-elastic solids Deformation theory (theory of small elastoplastic deformations) Flow theory of strain-hardening plastic solids Theory of the perfectly plastic solids Viscoelastic solids Viscous solids. Inversion of constitutive equations Viscoelastoplastic solids Chapter 3. General issues of three-dimensional linearised theory of deformable bodies stability (TL TDBS) Stability criteria for deformable bodies On surface and volume forces. Classification of problems Stability criteria and statement of the problem for elastic bodies Criterion of stability of the state of equilibrium and problem statement for plastic solids. Generalised concept of continuing loading Criterion of stability of deformation. Comparative analysis of two criteria of stability Stability criterion and problem statement for bodies with rheological properties General statement of stability problem for deformable bodies

14 Table of contents XV 11.1 Representation of linearised constitutive equations for solids with rheological properties. Non-uniform initial state Representation of linearised constitutive equations for solids with rheological properties. Uniform precritical state General statement of stability problem Application of Galerkin's method Sufficient conditions of applicability of the static method Compressible bodies. General case Incompressible bodies. General case ,Tracking, loads applied to the whole surface of a body ,Tracking, loads applied to the part of body surface Variational principles Compressible bodies under,dead, loads. Particular case Compressible bodies under,dead, loads. General case Incompressible bodies under,dead, loads. Particular case Incompressible bodies under,dead, loads. General case ,Tracking, loads applied to the whole surface of a body ,Tracking, loads applied to the part of body surface General solutions for uniform precritical states Compressible bodies. General relationships Compressible bodies. Plane and anti-plane problems Compressible bodies. Three-dimensional problems Incompressible bodies. General relationships Incompressible bodies. Plane and anti-plane problems Incompressible bodies. Three-dimensional problems On the representations of solutions for one feasible generalisation of the small initial deformations theory Determination of coefficients for particular types of solids Approximate approach in three-dimensional theory of stability Chapter 4. Analysis of the simplest problems All-round compression of isotropic simply connected body. Application of the integral stability criteria Application of sufficient conditions of stability in the integral form to compressible bodies. Analysis of results for various types of bodies Implementation of the problem statement in the differential form to compressible bodies. Analysis of results for various types of bodies Application of sufficient conditions of stability in the integral form to incompressible bodies. Analysis of results for various types of bodies Implementation of the problem statement in the differential form to incompressible bodies. Analysis of results for various types of bodies Internal (structural) instability. Properties of the basic system of simultaneous equations Compressible bodies Incompressible bodies

15 XVI Table of contents 17.3 Bodies with rheological properties Implementation of the approximate approach Near-the-surface instability. Problems for semi-restricted regions Half-plane. Analysis of results for various compressible and incopressible bodies Half-space. Analysis of results for various compressible and incompressible bodies Compression of a strip (plane strain problem) Incompressible strips Compressible strips Asymptotic analysis of thin-walled strip Approximate approach. Analysis of results for isotropic and composite strips Analysis of results for a thick-walled strip Compression of high-elastic non-circular cylindrical body. Implementation of variational principles Basic equations. Implementation of variational principles Bar mode of stability loss. Numerical examples Supplement. Exact solutions of mixed plane problems of llnearised solid mechanics References References supplement Biography