Mathematical Modeling in Mechanics of Granular Materials

396 Pages·2012·10.273 MB·English

by  Oxana Sadovskaya, Vladimir Sadovskii (auth.)

Advanced Structured Materials
Volume 21
Series Editors
Andreas Öchsner
Lucas F. M. da Silva
Holm Altenbach
For furthervolumes:
http://www.springer.com/series/8611
Oxana Sadovskaya Vladimir Sadovskii
•
Organized by Holm Altenbach
Mathematical Modeling
in Mechanics of Granular
Materials
123
OxanaSadovskaya Vladimir Sadovskii
ICM SBRAS ICM SBRAS
Akademgorodok 50/44 Akademgorodok 50/44
Krasnoyarsk Krasnoyarsk
Russia660036 Russia660036
HolmAltenbach
Magdeburg
Germany
ISSN 1869-8433 ISSN 1869-8441 (electronic)
ISBN 978-3-642-29052-7 ISBN 978-3-642-29053-4 (eBook)
DOI 10.1007/978-3-642-29053-4
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012938145
(cid:2)Springer-VerlagBerlinHeidelberg2012
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Foreword
The new monograph ‘‘Mathematical Modeling in Mechanics of Granular Mate-
rials’’ written by Oxana & Vladimir Sadovskii is based on a previous Russian
version published in 2008. The Russian version was significantly revised and
extended. The References were updated with respect to the readers not being
familiar with the Russian language. Instead of eight chapters of the Russian ori-
ginalversiontherearenowtenchapters—anewchapterdevotedtocontinuawith
independent rotational degrees offreedom is added.
Looking on the basics of this book it is obvious that the starting point is the
method of rheological models. In Continuum Mechanics one can split the
approaches in material modeling into three different directions:
• the deductive approach (top-down modeling), which starts with some general
mathematical structures restricted by the constitutive axioms and after that
special cases will be deduced,
• the inductive approach (bottom-up modeling), which starts with special cases
that are generalized step by step to derive more complex models, and
• last but not least the method of rheological modeling lying in-between the first
and the second approaches.
The last approach is related to a pure phenomenological modeling without
takingintoaccountthemicrostructuralbehavior.Ontheotherhand,thisapproach
isanengineeringmethodinmaterialmodelingsincetheparameteridentificationis
very simple and can be computer-assisted performed.
Since the new monograph is based on the method of rheological models the
questionariseswhyweneedanewbookonrheologicalmodels.Inthisfieldthere
exist a lot of outstanding monographs, among them being:
• Deformation,StrainandFlow:anElementaryIntroductiontoRheology,written
byMarkusReinerandpublishedbyH.K.Lewis(London,1960)andwhichwas
translated later into German and Russian,
v
vi Foreword
• Vibrations of Elasto-plastic Bodies, written by Vladimir A. Pal’mov and pub-
lishedbySpringer(Berlin,1998),whichisbasedontheoriginalRussianedition
from 1976,
• Materialtheorie—Mathematische Beschreibung des phänomenologischen ther-
momechanischen Verhaltens (Theory of Materials—Mathematical Description
of the Phenomenological Thermo-mechanical Behavior), written by Arnold
Krawietz and published by Springer (Berlin et al., 1986),
• Phänomenologische Rheologie—eine Einführung (Phenomenological Rheol-
ogy—an Introduction), written by Hanswalter Giesekus and published by
Springer (Berlin et al., 1994),
• Continuum Mechanics and Theory of Materials, written by Peter Haupt and
published by Springer (Berlin et al., 2002, 2nd edition).
The newmonograph is an excellent additiontothe existingliterature since the
following items are new and have not been discussed in the previous books:
• a new rheological model (the rigid contact model) is introduced,
• the application fields of rheological models are extended to granular materials,
• aconsequentandnewmathematicaldescription,necessaryforthenewelement,
is given and used also for the plastic rheological model, and
• several new examples are introduced, solved, and discussed.
Itisdesirablethatthismonographwillbeacceptedbythescientificcommunity
as well as the other monographs in this field.
Magdeburg, Germany, January 2012 Holm Altenbach
Preface
This monograph contains original results in the field of mathematical and
numerical modeling of mechanical behavior of granular materials and materials
with different strengths. Zones of the strains localization are defined by means of
proposedmodels.Theprocessesofpropagationofelasticandelastic-plasticwaves
in loosened materials are analyzed. Mixed type models, describing the flow of
granular materials in the presence of quasi-static deformation zones, are con-
structed. Numerical realizations of mechanics models of granular materials on
multiprocessor computer systems are considered.
The book is intended for scientific researchers, university lecturers, post-
graduates, and senior students, who specialize in the field of the mechanics of
deformable bodies, mathematical modeling, and adjacent fields of applied math-
ematics and scientific computing.
This monograph is a revised and supplemented edition of the book ‘‘Mathe-
maticalModelingintheProblemsofMechanicsofGranularMaterials’’,published
by‘‘Fizmatlit’’(Moscow)in2008inRussian.ComparedwiththeRussianedition,
its content is expanded by a new Chap. 10, devoted to mathematical modeling of
dynamic deformations of structurally inhomogeneous media, taking into account
therotationaldegreesoffreedomoftheparticles.Besides,inChap.7theSect.7.4,
containing new results on the analysis of wave motions in layered media with
viscoelasticinterlayers,isadded,andChap.9,Sect.9.8isaddedwiththeresultsof
solving the problem of radial expansion of spherical and cylindrical layers of a
granular material under finite strains.
Theresultspresentedinthemonographwereusedwhenreadingspecialcourses
in the Siberian Federal University. The work was performed at the Institute of
ComputationalModelingoftheSiberianBranchofRussianAcademyofSciences.
It was partially supported by the Russian Foundation for Basic Research (grants
no. 04–01–00267, 07–01–07008, 08–01–00148, 11–01–00053), the Krasnoyarsk
Regional Science Foundation (grant no. 14F45), the Complex Fundamental
Research Program no. 17 ‘‘Parallel Computations on Multiprocessor Computer
Systems’’ofthePresidiumofRAS,theProgramno.14‘‘FundamentalProblemsof
Informavtics and Informational Technologies’’ of the Presidium of RAS, the
vii
viii Preface
Program no. 2 ‘‘Intelligent Information Technologies, Mathematical Modeling,
System Analysis andAutomation’’ofthe Presidium ofRAS,the Interdisciplinary
Integration Project no. 40 of the Siberian Branch of RAS, the grant no. MK–
982.2004.1 of the President of Russian Federation, and the grant of the Russian
Science Support Foundation.
TheauthorswishtoacknowledgeB.D.Annin,A.A.Burenin,S.K.Godunov,
M. A. Guzev, A. M. Khludnev, A. S. Kravchuk, A. G. Kulikovskii, V. N. Ku-
kujanov, N. F. Morozov, V. P. Myasnikov, A. I. Oleinikov, B. E. Pobedrya, A.
F. Revuzhenko, and E. I. Shemyakin for discussions of the results forming the
basis of this book.
It should be noted that significant improvements in the presentation of the
material in comparison with the Russian edition was achieved through the atten-
tiveparticipationofthescientificeditorofthemonograph—Prof.HolmAltenbach,
who has made many invaluable comments on the content.
Lastbutnotleasttheauthorswishtoexpressspecialthanks,forsupportingthis
project, to Dr. Christoph Baumann as a responsible person from Springer Pub-
lishers Group.
Krasnoyarsk, Russia, January 2012 Oxana Sadovskaya
Vladimir Sadovskii
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Rheological Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Granular Material With Rigid Particles. . . . . . . . . . . . . . . . . 7
2.2 Elastic-Visco-Plastic Materials. . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Cohesive Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Computer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Fiber Composite Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Porous Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7 Rheologically Complex Materials. . . . . . . . . . . . . . . . . . . . . 41
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Mathematical Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Convex Sets and Convex Functions . . . . . . . . . . . . . . . . . . . 49
3.2 Discrete Variational Inequalities. . . . . . . . . . . . . . . . . . . . . . 61
3.3 Subdifferential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Kuhn–Tucker’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Duality Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Spatial Constitutive Relationships. . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 Granular Material With Elastic Properties . . . . . . . . . . . . . . . 101
4.2 Coulomb–Mohr Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Von Mises–Schleicher Cone . . . . . . . . . . . . . . . . . . . . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
ix
x Contents
5 Limiting Equilibrium of a Material With Load
Dependent Strength Properties. . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1 Model of a Material With Load Dependent
Strength Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 Static and Kinematic Theorems . . . . . . . . . . . . . . . . . . . . . . 133
5.3 Examples of Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.5 Plane Strain State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6 Elastic–Plastic Waves in a Loosened Material. . . . . . . . . . . . . . . 171
6.1 Model of an Elastic–Plastic Granular Material. . . . . . . . . . . . 171
6.2 A Priori Estimates of Solutions . . . . . . . . . . . . . . . . . . . . . . 177
6.3 Shock-Capturing Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.4 Plane Signotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.5 Cumulative Interaction of Signotons. . . . . . . . . . . . . . . . . . . 208
6.6 Periodic Disturbing Loads. . . . . . . . . . . . . . . . . . . . . . . . . . 212
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7 Contact Interaction of Layers. . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.1 Formulation of Contact Conditions. . . . . . . . . . . . . . . . . . . . 223
7.2 Algorithm of Correction of Velocities. . . . . . . . . . . . . . . . . . 234
7.3 Results of Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.4 Interaction of Blocks Through Viscoelastic Layers. . . . . . . . . 247
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8 Results of High-Performance Computing. . . . . . . . . . . . . . . . . . . 259
8.1 Generalization of the Method. . . . . . . . . . . . . . . . . . . . . . . . 259
8.2 Distinctive Features of Parallel Realization . . . . . . . . . . . . . . 265
8.3 Results of Two-Dimensional Computations . . . . . . . . . . . . . . 272
8.4 Numerical Solution of Three-Dimensional Problems. . . . . . . . 275
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
9 Finite Strains of a Granular Material. . . . . . . . . . . . . . . . . . . . . 289
9.1 Dilatancy Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9.2 Basic Properties of the Hencky Tensor . . . . . . . . . . . . . . . . . 297
9.3 Model of a Viscous Material with Rigid Particles. . . . . . . . . . 304
9.4 Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
9.5 Couette Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
9.6 Motion Over an Inclined Plane. . . . . . . . . . . . . . . . . . . . . . . 314
9.7 Plane-Parallel Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
9.8 Radial Expansion of Spherical and Cylindrical Layers . . . . . . 321
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329