The statistical problem of turbulence
This is the classical example of an application of statistical and probability theory in continuum mechanics. The stochastic nature of the turbulent flow follows from the fact that the particle velocities fluctuate over distances which are small compared
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L E C T U R ES - No.
92
EKKEHART KRÖNER UNIVERSITY OF
STUTTGART
STATISTICAL CONTINUUM MECHANICS
COURSE HELD AT THE DEPARTMENT OF GENERAL MECHANICS OCTOBER 1971
UDINE 1971
SPRINGER-VERLAG WIEN GMBH
This wodt ilmiject to copyright All rights are reserved, whether the whole or part of the material il concemed specifically those of translation, reprinting, re-uae of illustrations, broadcasting, reproduction by photocopying machine or 8imiiar meiDll, and storllpl in data banks.
©
1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81129-0 DOI 10.1007/978-3-7091-2862-6
ISBN 978-3-7091-2862-6 (eBook)
PREFACE
The scientific branch of statistical continuum mechanics, as the name says, serves the purpose of treating statistical problems in continuum mechanics. The first important successes were achieved in the field of turbulence in the thierties of this century. In recent time the statistical continuum mechanics has experienced a Zarge extension by including the complex field of materials of a heterogeneaus constitution such as polycristalline aggregates, multiphase mixtures, composites etc .. It is the main aim of the statistical theory in this field to calculate the effective properties from the given properties of the constituents and from some information about their structural arrangement. Important progress has been made in the case of elastic materials. Here the names of Beran, Molyneux, Lomakin, Volkov, Klinskikh, McCoy, and Mazilu deserve particular mention. The great fluctuations in stress and strain which accompany plastic deformation Iead to the expectation that the final theory of plasticity, too, will contain statistical elements. Investigations in this field are still very preliminary, however. So it is not clear at all whether the statistical theory should be developed along the lines of approach to turbulence or whether it should be a statistics of dislocations which, as is well-known, are the fundamental sources of the stress and strain fluctuations. In this course I give, after the introduction in chapter I, a somewhat restricted but, I hope, still sufficiently detailed outline of the mathematical theory of probability and statistics ( chapter II). Today, all the concepts and notations presented here are standard, hence I have not tried to be particularly original in this part of the course. The systematics of chapter II and of chapter III in which the mathematical theory is combined with physical laws into the formulation of statistical continuum mechanics follow largely the excellent monograph of M. Beran "Statistical Continuum Theories", Interscience Publishers 1968. Throughout this courseI have tried, as far as possible, to employ the notation of Beran.
4
Prefaae
This should facilitate for the reader of the present text the study of the more comprehensive presentation of Beran. In chapter IV the field of statistical turbulence which may already be considered classical is outlined very briejly, just to accustom the reader to the
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