RUSSIAN ACADEMY OF SCIENCES INSTITUTE FOR PROBLEMS IN MECHANICAL ENGINEERING P. A. Zhilin ADVANCED PROBLEMS IN MECHANICS Selection of articles presented at the Annual International Summer School – Conference “Advanced Problems in Mechanics” Volume 2 Saint Petersburg 2006 P.A. Zhilin. Advanced Problems in Mechanics. Selection of articles presented at the Annual Summer School – Conference “Advanced Problems in Mechanics”. Volume 2. St. Petersburg: Edition of the Institute for Problems in Mechanical Engineering of Russian Academy of Sciences. 2006. 306 p. Editorial board: D.A. Indeitsev, E.A. Ivanova, A.M. Krivtsov. In this selection you may find presentations by P.A. Zhilin given from 1994 to 2005 at the International Summer School – Conference “Advanced Problems in Mechanics”. This book consists of two volumes: the first one containes articles in Russian, and the second one in English. The scope of the questions under discussion is wide. It includes fundamental laws of mechanics, direct tensor calculus, dynamics of rigid bodies, nonlinear theory of rods, general theory of inelastic media including the theory of plasticity, consolidating granular media, phase transitions, and also piezoelectricity, ferromagnetism, electrodynamics, and quantum mechanics. This selection of papers by P.A. Zhilin, in fact, presents method of construction of continual theories with rotational degrees of freedom, mathematical technique necessary for this purpose, and also examples of application of the theories mentioned above to the description of various physical phenomena. For researchers, PhD students, undergraduate students of last years specializing in mechanics and theoretical physics. ISBN c O.P. Zhilina c Institute for Problems in Mechanical Engineering of Russian Academy of Sciences Introduction 3 Introduction In this selection you can find the papers, presented either by P.A. Zhilin or by his co-authors at the APM Summer-School Conference in the period from 1994 to 2005. The book is issued in two volumes: the first one contains articles in Russian, the second one contains articles in English. In both volumes the articles are listed in chronological order. The range of questions discussed is wide. It includes fundamental laws of mechan- ics, direct tensor calculus, rigid body dynamics, nonlinear rod theory, general theory of non-linear media, including plasticity, consolidating granular media, phase transitions, as well as piezoelectricity, ferromagnetism, electrodynamics and quantum mechanics. At first sight it seems that the papers are not related one to another. But this is not so. Let us show a few examples. Rigid body oscillator, introduced in the article related to the absolute rigid body dynamics, is used further as fundamental model when constructing inelastic media theory, piezoelectricity theory, and theory of magnetoelastic materials. Methods of description of the spinor motion, based on use of the direct tensor calculus, are used and developed both for solving rigid body dynamics problems and for solving nonlinear rod theory problems. The same methods are used when constructing various continuum models, which take into account rotational degrees of freedom. The symmetry theory and tensor invariant theory, which are presented in the book, dedicated to this topic, are being actively used and developed when constructing rod theory, as well as for other continuum theories. Two papers are dedicated to the formulation of fundamental laws of the Eulerian mechanics — mechanics of a general body, consisting of particles with rotational degrees of freedom. All continuum theories, presented in the digest, including electrodynamics, are built adhering to the same positions based on the fundamental laws of mechanics. When building continuum models both for elastic and inelastic media, the theory of strains is used, which is based on the idea of using the reduced energy balance equation for defining measures of deformation. By the elementary examples of discrete systems mechanics the notions of internal energy, chemical potential, temper- ature and entropy are introduced. Definition of these quantities is given by means of pure mechanical arguments, which are based on using special mathematical formulation of energy balance equation. The same method of introducing the basic thermodynamics notions indicated above is used when building different continuum theories. In fact the selected papers of P.A. Zhilin represent the method for constructing continuum theories with rotational degrees of freedom together with the necessary mathematical apparatus, as well as examples of using the mentioned theories when describing different physical phenomena. Among others, the first volume of the digest includes two papers, dedicated 4 P. A. Zhilin. Advanced Problems in Mechanics to the fundamental laws of mechanics, which were written with big time interval, and two articles on the rod theory, which were also written in the different periods of time. The Reader can take advantage of following the development of scientific ideas. The first paper dedicated to the fundamental laws of mechanics, is a quite perfect, logically rigorous theory. Nevertheless, after many years, author returns to this topic. The aim was not to change something in the original variant, but to complete it by including in it thermodynamical ideas. The mentioned above can equally be pertained to the two papers on the rod theory. Not every physical theory permits including of new notions in it. Often, when needed to describe a new phenomenon, one is forced to reject an old theory and build a new one instead. The theories presented in this selection have an ability to be developed. This is their great advantage, and that is one of the important reasons why they attract attention of researchers. The editorial board is grateful to N.A. Zhilina for help with preparing this book for publication; to S.N. Gavrilov, E.F. Grekova, I.I. Neygebauer, and E.V. Shishkina for translation to English of the introductional parts and the list of publications. D.A. Indeitsev, E.A. Ivanova, A.M. Krivtsov. 5 Contents P.A. Zhilin — searching for Truth 6 Short biography and scientific results of P.A. Zhilin 9 Classical and Modified Electrodynamics (1996) 32 A General Model of Rigid Body Oscillator (1998) 43 Ferromagnets and Kelvin’s Medium: Basic Equations and Magnetoacoustic Resonance (1998) 66 On the Painleve Paradoxes (2000) 89 The Main Direction of the Development of Mechanics for XXI Century (2000) 112 A.I. Lurie — Works on Mechanics (2001) 126 Phase Transitions and General Theory of Elasto-Plastic Bodies (2002) 140 Generalized Continuum and Linear Theory of Piezoelectric Materials (2002) 153 A Micro-Polar Theory for Binary Media with Application to Flow of Fiber Suspensions (2003) 165 Symmetries and Orthogonal Invariants in Oriented Space (2005) 204 Nonlinear Theory of Thin Rods (2005) 227 A Micro-Polar Theory for Piezoelectric Materials (2005) 250 List of basic publications by P.A. Zhilin 262 6 P. A. Zhilin. Advanced Problems in Mechanics P. A. Zhilin — searching for Truth “There is no action without reason in nature; comprehend the reason and you won’t need the experience.” Leonardo da Vinci The most significant features of scientific society in the end of XX — beginning of XXI century are pragmatism and particular specialization. To the least degree this can be applied to Pavel Andreevich Zhilin. Sincere interest, willing to perceive the Truth and to bring his knowledge to people were the solely motives for his work. Breadth of scientific interests of P.A. Zhilin is impressing — having fundamental character, his works cover practically all areas of mechanics and are extended to electrodynamics and quantum physics. Hardly anyone can express the views of P.A. Zhilin on science better than himself: “Aim of each science is in perception of the Reality. At the same time the science investigates not the Reality itself, but the simplified models of the Reality. Approaching to the true Reality can be achieved by broadening the model. But to construct a model we need to know at least, what exactly we are going to model. In other words we have to have an a priori idea of the Reality. So we have a vicious circle — to perceive the Reality we need a science, and to construct a science we need to know the Reality. Fortunately the solution of this one would think unsolvable problem is integrated in the human mind, which has two qualitatively different categories: a) intuition and b) intellect. Intuition is the ability of a human being to sense the world around us directly, which can not be reduced to the five basic senses. This is what poets, musicians, painters and other artists are conscious of. Intuition may be trained as well as every other ability of human being, but it requires permanent and purposeful efforts. Intellect is an ability of human being to think logically, basing on an a priori knowl- edge, “built in” the intellect “memory”. A powerful modern computer is a practically perfect analogue of intellect.” From the paper “Reality and Mechanics” P. A. Zhilin — searching for Truth 7 Doctor of Science, professor, author of more than 200 scientific papers, many of which were published in the key scientific journals, a Teacher who educated more than one generation of disciples — both PhD and Dr. Sci, P.A. Zhilin was a mind of a wide scope and of great erudition. Being by his position an adherent of the rigorous Science, he was also deeply interested in Eastern philosophies. Fundamental scientific ideas of Pavel Andreevich, concerning the importance of spinor motions when describing events at the micro-level and modelling the electromagnetic field, are in correspondence with different metaphysical concepts of the origin of the World. These ideas in various forms were proposed by the great classics of science, whose works Pavel Andreevich studied in a deep and detailed way. The achievement of Pavel Andreevich is the translation of these ideas from a vague general form of words and intuitive assumptions into a rigourous form of mathematical models.