The Second Method of Solving the System of Smoluchowski Equations (3.12) [1,4]
The idea of this method is similar to the one used by Kolmogorow in his work on stochastic processes. Compared with the latter, i is a generalization with real physical sense.
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A ND
J.
L E C T U R ES
No.
93
LITWINISZYN
MJNJNG OOLLEGE, CRAOOW
STOCHASTIC METHODS IN MECHANICS OF GRANULAR BODIES
COURSE HELD AT THE DEPARTMENT OF GENERAL MECHANICS OCTOBER 1972
UDINE 1974
SPRINGER-VERLAG WIEN GMBH
This work is subject to copyrighl All rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
©
1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1972
ISBN 978-3-211-81310-2 DOI 10.1007/978-3-7091-2836-7
ISBN 978-3-7091-2836-7 (eBook)
P R E F A C E
When disaussing the meahanias of soil, roaks and loose media the models of the so aalled meahanias of aontinuous media are in general use. This model as sumes the invariant of the aontaat relations between the elements of the media. In aase of the above media being in motion the invariant relation of the aontaats is not maintained. Contaats between these elements ahange, the ordered relation is not maintained, and the elements intermingle. The motion of the medium is aharaaterized by the mass aharaater of random ahanges in aontaat relations and aonsequently by random displaaement of the medium elements. The movement of such a aolleation of elements depends on their meahanical properties only in a small degree, being mainly dependent on their spatial struature. Sinae the interaation of the elements has a mass and random charaater, the summary effeat of displaaements of elements is defined by random laws in agreement with the aentral limiting theorems. These heuristia aonsiderations suggest the idea of desaribing the displaaements of a loose medium on the basis of a model different from the model of a model different from the model of a aontinuous medium.
4
Preface
That model may be regarded as a system of integral equations whiah are generalizations of the Smoluahowski equation desaribing the stoahastia proaesses of the Markov type. In partiaular~ from this system a parabolia system of differential equations~ defining the mean values of displaaement aomponents of a loose medium~ aan be obtained. Solutions for a number of aases of boundary aonditions of this system have been given. The results have been aompared with the displaaement measurements obtained in experiments aarried out in a loose medium in whiah the aorresponding boundary aonditions have been realized.
J. Litwiniszyn Udine~
Oatober 1971
STOCHASTIC METHODS IN MECHANICS OF GRANULAR BODIES The mechanical phenomena in so called continuous media explained by means of a model based on the concept of tinuous include phenomena for the explanation of which a
co~
cont~
uous model is inadequate. In some cases we may feel that the mathematical model by means of which we describe the phenomenon is continuous, whereas the actual phenomenon described by the model is not continuous. The concept of noncontinuity seems to be inherent in the world of events and unavo
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