Table Of ContentLecture Notes in Applied
and Computational Mechanics
Volume 1S
Series Editors
Prof. Dr.-Ing. Friedrich Pfeiffer
Prof. Dr.-Ing. Peter Wriggers
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Micropolar Theory
of Elasticity
Janusz Dyszlewicz
, Springer
Professor JANUSZ DYSZLEWICZ
Institute of Mathematics
Faculty of Fundamental Problems of Technology
Wroclaw University of Technology
Wybrzeze Wyspianskiego 27
50-370 Wroclaw
Poland
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ISBN 978-3-642-07528-5 ISBN 978-3-540-45286-7 (eBook)
DOI 10.1007/978-3-540-45286-7
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This paper is dedicated to the
memory of PROFESSOR WITOLD NO W ACKI
Preface
This monograph contains the results of my research in the area of asymmet
ric theory of elasticity, conducted from 1969 to 1986 under the direction of
PROFESSOR WITOLD NOWACKI.
I am indebted to PROFESSOR NOWACKI, thanks to whose invaluable and
very kind research assistance I obtained the results which were the foundation
of this monograph. Therefore, I would like to express my deepest gratitude to
him and honour his memory. He will remain in my thoughts with due respect.
During my research assistantship at the Institute of Mechanics at the Uni
versity of Warsaw in 1970-1973 I had the opportunity to participate in sem
inars and conferences, study critical reviews and carryon numerous discus
sions and conversations. All this resulted in many valuable remarks included
in this monograph. In this connection, I would like to thank Professor J6zef
Ignaczak and Professor Marek Sokolowski from the Institute of Fundamental
Problems of Technology at the Polish Academy of Sciences, as well as Pro
fessor Zbigniew Olesiak and Professor Adam Piskorek from the Institute of
Mechanics at the University of Warsaw.
In 1974-1986 I participated in the seminar "Mathematical Models of De
formable Media" at the Institute of Mathematics of the Technical University
of Wrodaw (TUW). My thanks go to the organizers of the seminar, Professor
Bertold Lysik from the Institute of Mathematics, Professor Otton Difbrowski
from the Institute of Civil Engineering, and Professor Igor Kisiel from the In
stitute of Geotechnics, for valuable discussions, in particular those concerning
practical aspects of the results obtained.
I would also like to thank Professor Zbigniew Romanowicz from the Insti
tute of Mathematics as well as Professor Wadaw Kasprzak and the Senate
of the Technical University of Wrodaw, for granting me support for my re
search in 1984-1985. Thanks are also due to Professor Jaroslaw Stefaniak
from the Institute of Applied Mechanics at the Technical University of Poz
nan for a number of valuable consultations in 1990 concerning my research
pu blications.
Continuing the extensive research activity of My ADVISOR, PROFESSOR
NOWACKI, I included in this monograph the results obtained by me and my
students after 1986. Thus, this work is an extended version of my earlier
work entitled Boundary and Initial-Boundary Value Problems of Equations
VIII Preface
of Micropolar Elastostatics and Elastodynamics, published in Polish by the
Publishing House of the Technical University ofWroclaw in 1990 in the mono
graph series. It appears under [.58] in the references. I would like to stress that
the final form of this monograph was shaped during my stay as a research
fellow at the Institute of Fundamental Problems of Technology of the Polish
Academy of Sciences in the academic year 1986/1987, under the direction
and with immense help of Professor J6zef Ignaczak.
The manuscript of [58], listed as [49] in the references under the tenta
tive title, Boundary Value Problems of Micropolar Theory of Elasticity, was
highly evaluated by My ADVISOR in a review from 1984. The main stream
of this manuscript was the subject of the monograph lectures which I gave
in 1980-1990 to the exclusive classes of students of mechanics at the Faculty
of Fundamental Problems of Technology and the Faculty of Civil Engineer
ing at TUW. In those classes I taught students, with whom I later wrote
a number of valuable papers (quoted in the references), namely Czeslaw
Kolodziej, Bozena Slotwinska, Monika Czub, Jacek Wytrifzek, Pawel Zal,
Jacek Bienkowski and Marcin Sikora. Later, I did joint research work on the
Cosserats media with my Ph.D student, Mountajab AI-Hasan, who, after
graduating from Al Baath University from the Syrian Arab Republic, ob
tained a scholarship to the Wroclaw University of Technology. I would like
to add that the seminars of Professor Jan Langer from the Institute of Civil
Engineering at TUW and Professor Bertold Lysik had a great influence on
the high level of the research papers written in cooperation with my students.
This monograph concerns analytical methods of solving boundary and
initial-boundary value problems for linear differential equations with partial
derivatives describing the models of continuous media with microstructure of
Cosserats type, such as the model of isotropic hemitropic medium of Aero
Kuvshinski (A-K), the model of micropolar body of Eringen-Nowacki (E-N),
the model of couple-stress medium of Koiter-Mindlin (K-M), the model of hy
pothetical medium (HM) and the model of Hooke's medium (H). Within the
frameworks of elastostatics, theromoelastostatics and elastodynamics, with
the fields of body loadings and distortions taken into account, we present
many methods of solving the fundamental partial differential equations of
the models mentioned above, giving basic formulations of the initial-boundary
problems of the dynamics and boundary value problems of the statics. These
are the following methods: the method of stress equations of Beltrami-Michell
type, the method of stress equations of motion of Ignaczak type (SEMP),
the method of direct integration of the equations in displacements and ro
tations, the method of stress function, the methods of generalized repre
sentations of solutions, namely, of Galerkin, Iacovache (Cauchy-Kovalevski
Somigliana), Papkovich-Neuber, Love, then, the method of generalized poten
tials of Nowacki (Green-Lame), and finally, the method of superposition by
means of the generalized Schaefer vector. The methods mentioned above are
used for solving boundary- and initial-boundary value problems, for plane,
Preface IX
axially-symmetric and three-dimensional states of strain, as well as for the
determination of the Green functions. In particular, we solve and discuss in
detail a certain important class of boundary-value problems, containing the
generalized problems of Kirsch, Goodier, Lamb, the boundary value prob
lems for half-spaces of Dirichlet and Neumann; among others, we have here
the solved problems of Boussinesq, Cerruti and Boussinesq-Mindlin. With
great attention we treat the problems of stress concentration and singular
ities of physical fields. Of primary concern to us is micropolar theory (E-N
model), which is treated not only as a limit of he mitro pic theory (A-K model),
but also as a theory, from which one can obtain the results of couple-stress
theory (K-M model), the results of the theory of pure rotations (HM) as well
as those of the classical theory of elasticity, by taking the limits of the micro
polar parameters. Therefore, great emphasis is put on the original schemes of
performing limits from one theory to another, within the framework of fun
damental equations, general representations of solutions, singular solutions
and particular boundary value problems.
The monograph consists of the Introduction and four chapters. In the In
troduction we discuss the models studied in this work, namely (A-K), (E-N),
(K-M), (HM) and (H) models, as well as the fundamental equations, the
boundary conditions, the initial-boundary conditions, and finally, the mate
rial constants for all five models mentioned above. Chapter 1 refers to three
dimensional problems. In Chapter 2, we discuss axially-symmetric problems.
In Chapter 3, we consider two-dimensional problems in the plane state of
strain. In Chapter 4, we analyse hemitropic media and vector equations.
From the mathematical point of view, we use the Kupradse theorems on
the existence of solutions, the Fourier and Hankel integral transformations,
and the general theory of linear ordinary differential equations.
The referees viewed this work as a monograph. It contains both the results
of theoretical and practial character and extends our knowledge of elastic
materials with microstructure, acquired from scientific literature. This work
is the first monograph on the subject that has appeared after the monograph
of Professor Witold Nowacki, Theory of Asymmetric Elasticity, published
in 1986 by Polish Scientific Publishers & Pergamon Press. In particular, it
constitutes an exhaustive supplement and an extension of Professor Nowacki's
work, thus it can be of interest to specialists in such areas as solid mechanics
or partial differential equations, theoretical engineers and students taking
courses in advanced mechanics at universities and polytechnics.
This work has had many sponsors, which has been a necessary condition for
it to come out in print. The main support came from the Scientific Research
Committee (KBN) in Warsaw. Also, the financial support from the Dean of
the Faculty of Civil Engineering at TUW, Professor Otton D~browski, and
the Dean of the Faculty of Fundamental Problems of Technology at TUW.
Professor Jerzy Czerwonko, is gratefully acknowledged.
X Preface
Finally, I would like to warmly express my thanks to Professor Czeslaw
Ryll-Nardzewski, Professor Zbigniew Olszak, Professor Aleksander Weron,
and the Scientific and Educational Board of the Institute of Mathematics,
as well as my colleagues from the Institute of Mathematics, the Institute of
Civil Engineering and the Institute of Geotechnics at TUW, for their kind
interest in my research and the present monograph that I experienced over
the years. In particular, I am grateful to Professor Romuald Lenczewski and
Dr Artur Rozwadowski for translating the manuscript from Polish, and Pro
fessor Marek Sokolowski - for verifying the text. Moreover, I would like to
thank Dr Przemyslaw Scherwentke and Monika Kaczmarz for type-setting
the text. Thanks are also due to Dr Stanislaw Kroczak, who prepared the
final versions of all figures and to Professor Wojciech Glabisz for consulta
tions. Last but not least, I would like to thank Professor Wojciech Kordecki
for his highly proffesional supervision of the typesetting of the manuscript.
Wroclaw, September .5, 2003 Janusz Dyszlewicz
