Table Of ContentFoundations of Engineering Mechanics
L.I. Manevitch ·LV. Andrianov · V.G. Oshrnyan, Mechanics of
Periodically Heterogeneous Structures
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
http:// www.springer.de/engine/
L.I. Manevitch · I.V .. Andrianov
V.G. Oshmyan
Mechanics of Periodically
Heterogeneous Structures
With 76 Figures
' Springer
Series Editors:
V.I. Babitsky J. Wittenburg
Department of Mechanical Engineering Institut fiir Technische Mechanik
Loughborough University Universitiit Karlsruhe (TH)
LE 11 3TU Lough borough, Leicestershire Kaiserstr.12
Great Britain 76128 Karlsruhe I Germany
Authors:
Leonid I. Manevitch Igor V. Andrianov
Institute of Chemical Physics Institute of Chemical Physics
4 Kosygin Street 4 Kosygin Street
117977 Moscow I Russia 117977 Moscow I Russia
Victor G. Oshmyan
Institute of Chemical Physics
4 Kosygin Street
117977 Russia
Library of Congress Cataloging-in-Publication Data
Manevitch, Leonid 1.:
Mechanics of Periodically Heterogeneous Structures I L.l. Manevitch; I. V. Andrianov;
V.G. Oshmyan.-Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris;
Tokyo: Springer, 2002
(Foundations of Engineering Mechanics)
ISBN 978-3-642-07489-9 ISBN 978-3-540-44571-5 (eBook)
DOI 10.1007/978-3-540-44571-5
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Preface
Relieve the problem of inessential detail
and reduce it to the simplest elements.
Rene Descartes
Heterogeneity of a material or construction can be caused by two main reasons.
Non-uniformity of certain physical characteristics (density, elastic modulus, con
ductivity, etc.) is the first. Two- or multi-phase composites are typical examples of
this type of material. The second origin of heterogeneity is a geometrical one. Re
inforcement of the shells and plates by stringers, discrete supports and other con
structive elements is widely used in numerous applications. Both reasons cause
heterogeneity of the stress-strain state and the descriptions of the mechanical re
sponses meet very similar mathematical difficulties. Therefore, it is natural to ana
lyze and solve the corresponding boundary-value problems applying a similar and
in some cases identical technique.
Many problems in modern composites, heterogeneous plates and shells theory
are governed by partial differential equations with rapidly changing and mostly
discontinuous coefficients. Obviously, there are two opposite limits in which the
direct application of conventional technique is efficient.
The first limit is a small number of heterogeneities. It means that the scale, l, of
inhomogeneity (inclusion diameter, distance between stringers, etc.) is of the same
order as the typical outer size, L, of the structure, L '7 l. Direct numerical methods
(finite elements, finite differences, etc.) should be appl,ied in this case. The high
level of modern digital computing power provides precise results in numerous
complicated problems of composites, plates and shell mechanics.
However, even modern computers [65,71,158] capnot efficiently assist in solv
ing the problems corresponding to the mechanics of heterogeneous media in the
opposite limit L >> l. This is a reason for an application of certain kind of homog
enization technique in this limit: effective media theory (EMT, the term used in
composite mechanics) or structurally orthotropic theory (SOT, the term used in
plate and shell mechanics) in particular. The replacement of heterogeneous media
by the homogeneous continuum, which is characterized by certain effective consti
tutive equations, is the basic instrument for both EMT and SOT. Four important
questions should be resolved in the context of homogenization of the media.
(1) Clearly, the limit Ul ~ oo is a necessary condition for the possibility of cor
rect approximation of heterogeneous media by the homogeneous one. However, it
is not clear at all whether this condition is sufficient. Numerous mathematical stud
ies (mathematical homogenization theory, MHT) are devoted to this subject. The
VI Preface
rigorous proof of the existence of the homogenization limit is the main aim of cor
responding publications and very important results have been obtained in this
field.
(2) Let us suppose the possibility of the homogenization. The next problem then
arises: how to determine the effective constitutive equations (composite elastic
moduli, particularly). Direct averaging is obviously wrong, but is often applied for
homogenization. Effective values of constitutive constants can be bounded with
the help of various variational theorems. Unfortunately, the bounds obtained are
very wide and cannot be improved in the practically important case of a sharp dif
ference in component properties. Self-consistent approaches are popular and effi
cient techniques for the estimation of effective elastic constants. The rigorous
MHT reduces the calculation of the effective properties to the solution of bound
ary-value problems for the periodicity cell in the case of a periodic structure,
which allows application of conventional numerical techniques.
(3) Effective medium theories are able to describe the main terms of displace
ment fields only, but not the local stress-strain state. Corresponding information
has to be determined if one is going to estimate fracture parameters.
(4) Numerous important applications (especially of plate and shell theory) deal
with the case of a finite L/l ratio. In this intermediate region, the effective uniform
description is obviously insufficient and high-order asymptotic expansions should
be determined.
Rigorous presentation of MHT is the subject of numerous papers, reviews and
manuscripts, and we do not pursue the goal of extending and improving this ac
count. However, we see a gap in the analytical and/or numerical performance of
the corresponding asymptotic analysis of heterogeneous system static and dynamic
behaviors. The main purpose of this book is to fill this gap. Numerous applications
to composite media, heterogeneous plates and shells are considered. We include a
lot of details, numerical results for cell problem solutions, calculations of high
order terms of asymptotic expansions, boundary layer analysis, etc. The represen
tation is mainly based on the original results of the authors in collaboration with
A. Diskovsky, A. Givental, L. Givental, M. Guy, E. Kholod, N. Knunyantz,
S. Koblik, N. Kozhina, G. Krizhevsk , V. Lesnichaya, V. Loboda, A. Pavlenko,
lA.
Pisankq, A. Sharnrovskii, . Shevchenk , G. Starushenko and S. Timan.
The authors are grateful to V. Babitsky for the suggestion to write this book,
and to V. Danishevs'kyy, E. Gusarova and 0. Manevitch for the technical assis
tance.
Contents
Preface........................................................................... V
0 Introduction ..................................................... .
0.1 Numerical and asymptotic procedures in the theory of ........... .
heterogeneous materials................................................ 1
0.2 Mathematical standpoint................................................ 3
0.3 Physical statements of the homogenization problem................ 6
1 Definitions, assumptions and theorems
in homogenization problems.................................. 7
1.1 Definitions for homogenization problems in solid of periodic ... .
microctructure.............................. .................. ... ... ...... 7
1.2 Cell problems and cell solutions for an elastic solid of periodic
microstructure............................................................ 10
1.3 Asymptotic series in homogenization problems of ................ ..
periodic microstructure .. . .. . .. . .. . .. . .. . .. . .. . .. .. .. . .. . .. . .. . .. . .. . .. . 14
2 Application of cell functions for the calculation
of binary composite elastic moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 Laminated composite .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Particulate-filled composite............................................ 26
2.2.1 Structural model................................................... 26
2.2.2 Boundary-value problems and a numerical technique for ...
their solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 Elastic properties of a binary composite of periodic ........ .
structure with perfectly bonded components. . . . . . . . . . . . . . . . . . . 3 3
2.2.4 Effect of adhesion on the effective elastic moduli of ........ .
a binary composite of periodic structure........................ 44
2.2.5 Analysis ofmicromechanical field distributions.............. 48
VIII Contents
3 Asymptotic study of linear vibrations of a stretched ... .
beam with concentrated masses and discrete elastic ... .
supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
3.1 Statement of the problem............................................... 58
3.2 Asymptotic analysis...................................................... 61
3.2.1 Empty frequency domains....................................... 61
3.2.2 Low-frequency region, a=O. Long-wave modes . . . . . . . . . . . . . 64
3.2.3 Medium-frequency region, a=2. Tooth-like wave modes... 67
3.2.4 High-frequency region, a=2.5. Vibrations of the beam ..... .
between immobile heavy masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.5 Conclusions .......... 00 . ..................... 0. ..... 0000.0.0 .. 0.... 73
4 Reinforced plates .. . . . . . ..... .. . . . . . . . . . . .. . .. . . . . . . . .. . .. . . .. . . . 76
4.1 Flexural vibrations of a rectangular reinforced plate on ........... .
an elastic foundation .... 00 00............................................. 76
4.2 Static problem ............ 00. 0. .. 0. ....... 00 . .... 0. .......... 0. .......... 00 92
4.3 Flexural vibrations and equilibrium state of circular plates ....... ..
reinforced by radial ribs . 0. .............. 0. .. 0. . 0. 0. 0• ......... 0. . . . . . . . . 97
4.4 Geometrically nonlinear fiexural vibrations of rectangular ...
0 •• 0 ••
reinforced plates ..... 0. ......... 0. ........ 00 ....... 00 . ................... 0 106
4.5 Account of ribs torsion rigidity .. 0. .... 0. ................ 0. 0. 0. . . . . . . . .. 112
4.6 Account ofribs eccentricity ........... 0000............................ ... 116
4.7 Homogenization for plates with wide ribs .. .. .... .. .. . .. .. .. .. .. .. ... 122
5 Problems of elasticity theory for reinforced ..
0 .............. .
orthotropic plates .... 0. ............................. 0 0 128
o o •••••••• o ••••••••
5.1 Reinforced orthotropic strip ........ 00...... ........................ .... 128
5.2 Force transfer to a stringer orthotropic strip via an elastic ........ .
element ....................... 0. . 00 ...................................... 00 146
5.3 Plane vibrations of circular cylindrically orthotropic plates with
radial ribs .............. 0.. 00 0 0. ... 00 ..... 00 0 0 ... 0. .. 0. ............. 00. 151
o ....
6 Reinforced shells ......... 00............................ .... .... ........... 156
6.1 Stringer cylindrical shells ........ 00000 ...................... 0000.. ...... 156
6.2 Shells of revolution with meridional ribs .. .. .. .. .. . .. .. . .. . .. .. .. .. 166
6.3 Cross-reinforced shells................................................................ 174
6.4 Waffle- and ring-reinforced shells.............................................. 176
6.5 Cylindrical shells rarely reinforced using stringers ..................... 178
Contents IX
7 Corrugated plates . . . . . . . . . .. . .. . .. . . . . . . . . . . . . . .. . . .. . . . . . . . . . . .. .. . . . . .. 188
7.1 Model problem................................................................ 190
7.2 Transformation of basic equations .. . . . . . . . . . . . . . . . . . . .. . .. . .. . .. .. . .. . .. . 190
7.3 Axisymmetrical deformation of a circular corrugated plate......... 194
7.4 Rectangular corrugated plate........................................................ 203
7.5 Axisymmetrical vibrations of a circular corrugated plate............ 209
8 Other periodic structures............................................. 212
8.1 Vibrations of a cylindrical shell with a large number of apparent .
masses.................................................................... 212
8.2 Plates on an elastic foundation with strip-shaped and support-free
par ........................................................................... 216
8.3 Laminated structures....................................................... 218
8.4 Multisupported plates.................................................................. 221
8.5 Plates and shells with a periodic system of hinges.................... 225
8.6 Simplified nonlinear equations for smooth plates and shells....... 228
9 Perforated plates and shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.1 Bending of rectangular plates with periodic square perforations.. 233
9.2 Eigenvalue problem for a perforated plate............................ 241
9.3 Analytical approach for a large hole.................................... 242
9.4 Matching of asymptotic solutions by means of two-point Pade ...
approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.5 The plane theory of elasticity in a perforated domain............... 248
9.6 Perforated shallow shells................................................ 251
Concluding remarks. Perspectives and open problems .. 254
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
