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Advanced Structured Materials
Volume 21
Series Editors
Andreas Öchsner
Lucas F. M. da Silva
Holm Altenbach
For further volumes:
http://www.springer.com/series/8611
Oxana Sadovskaya
•
Vladimir Sadovskii
Organized by Holm Altenbach
Mathematical Modeling
in Mechanicsof Granular
Materials
123
Oxana Sadovskaya
ICM SB RAS
Akademgorodok 50/44
Krasnoyarsk
Russia 660036
Holm Altenbach
Magdeburg
Germany
Vladimir Sadovskii
ICM SB RAS
Akademgorodok 50/44
Krasnoyarsk
Russia 660036
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
Springer Heidelberg New York Dordrecht London
Library of Congress Control Number: 2012938145
Ó Springer-Verlag Berlin Heidelberg 2012
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
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Foreword
The new monograph ‘‘Mathematical ModelinginMechanicsofGranular 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-
ginal version there are now ten chapters—a new chapter devoted to continua with
independent rotational degrees of freedom 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
taking into account the microstructural behavior. On the other hand, this approach
is an engineering method in material modeling since the parameter identification is
very simple and can be computer-assisted performed.
Since the new monograph is based on the method of rheological models the
question arises why we need a new book on rheological models. In this field there
exist a lot of outstanding monographs, among them being:
• Deformation, Strain and Flow: an Elementary Introduction to Rheology, written
by Markus Reiner and published by H. K. Lewis (London, 1960) and which was
translated later into German and Russian,
v
• Vibrations of Elasto-plastic Bodies, written by Vladimir A. Pal’mov and pub-
lished by Springer (Berlin, 1998), which is based on the original Russian edition
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 new monograph is an excellent addition to the existing literature 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,
• a consequent and new mathematical description, necessary for the new element,
is given and used also for the plastic rheological model, and
• several new examples are introduced, solved, and discussed.
It is desirable that this monograph will be accepted by the scientific community
as well as the other monographs in this field.
Magdeburg, Germany, January 2012 Holm Altenbach
vi Foreword
Preface
This monograph contains original results in the field ofmathematical and
numerical modelingof mechanical behavior ofgranularmaterials and materials
with different strengths. Zones of the strains localization are defined by means of
proposed models. The processes of propagation of elastic and elastic-plastic waves
in loosened materials are analyzed. Mixed type models, describing the flow of
granular materialsin the presence of quasi-static deformation zones, are con-
structed. Numerical realizations ofmechanics models ofgranularmaterials 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-
matical Modelingin the Problems ofMechanicsofGranular Materials’’, published
by ‘‘Fizmatlit’’ (Moscow) in 2008 in Russian. Compared with the Russian edition,
its content is expanded by a new Chap. 10, devoted to mathematicalmodeling of
dynamic deformations of structurally inhomogeneous media, taking into account
the rotational degrees of freedom of the particles. Besides, in Chap. 7 the Sect. 7.4,
containing new results on the analysis of wave motions in layered media with
viscoelastic interlayers, is added, and Chap. 9, Sect. 9.8 is added with the results of
solving the problem of radial expansion of spherical and cylindrical layers of a
granular material under finite strains.
The results presented in the monograph were used when reading special courses
in the Siberian Federal University. The work was performed at the Institute of
Computational Modelingof the Siberian Branch of Russian Academy of Sciences.
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’’ of the Presidium of RAS, the Program no. 14 ‘‘Fundamental Problems of
Informavtics and Informational Technologies’’ of the Presidium of RAS, the
vii
Program no. 2 ‘‘Intelligent Information Technologies, Mathematical Modeling,
System Analysis and Automation’’ of the Presidium of RAS, 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.
The authors wish to acknowledge B. 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-
tive participation of the scientific editor of the monograph—Prof. Holm Altenbach,
who has made many invaluable comments on the content.
Last but not least the authors wish to express special thanks, for supporting this
project, to Dr. Christoph Baumann as a responsible person from Springer Pub-
lishers Group.
Krasnoyarsk, Russia, January 2012 Oxana Sadovskaya
Vladimir Sadovskii
viii Preface
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 GranularMaterials . . . . . . . . . . . . . . . . . . . . . . . . 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
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
x Contents
10 Rotational Degrees of Freedom of Particles 333
10.1 A Model of the Cosserat Continuum. . . . . . . . . . . . . . . . . . . 333
10.2 Computational Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
10.3 Generalization of the Model . . . . . . . . . . . . . . . . . . . . . . . . 366
10.4 Finite Strains of a Medium With Rotating Particles . . . . . . . . 377
10.5 Finite Strains of the Cosserat Medium . . . . . . . . . . . . . . . . . 382
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Contents xi
[...]... transportation ofgranular materials of minerals industry and agriculture production, problems of design of storage bunkers and grain tanks, problems of design of chemical machines with a boiling granular layer, problems of modelingof avalanching, etc In spite of the fact that the foundations of the theory have been laid even at the beginning of the development of continuum mechanicsin the classical...Chapter 1 Introduction The theory ofgranularmaterials is among the most interesting and intensively developing fields ofmechanics because the area of its application is very wide It involves problems ofmechanicsof geomaterials (soils and rocks) related to the estimation of strength and stability of mine openings, bases and slopes when performing designed construction engineering work, problems of transportation... constructed with the help of rheological schemes including a special element called rigid contact, is worked out By the combination of this element with traditional ones (elastic spring, viscous damper, and plastic hinge), special mathematical models ofmechanicsofgranularmaterials taking into account features of the deformation process are obtained The static and kinematic theorems of the limit equilibrium... multiple solution of the system (2.16) may be eliminated To this end, all components of the vector U except stresses of plastic hinges are determined from Eqs (2.16) and the equations for Vik+1 involved in (2.17) Stresses of plastic hinges are assumed to be arbitrary Strains of rigid contacts remain undetermined as well More exactly, a basis of the space of solutions of the system of linear algebraic... application to soil mechanics have a similar disadvantage [6, 12, 30, 34] because tension and compression states in them differ from one another in sign of instantaneous strain rate rather than in sign of total strain The equations of uniaxial dynamic deformation of a granular material with elastic particles, correct from the mechanical point of view, being a limiting case of the equations of heteromodular... problems of geophysics (seismicity) are worked out A model of mixed type taking into account stagnation regions of quasi-static deformation in a moving flow of a loosened granular material is constructed In the context of this model, an exact solution describing the Couette stationary rotational flow between coaxial cylinders is obtained Nonstationary avalanche-like motion of a granular material along an inclined... its original position More complex rheological properties of particles and the binder are taken into account in the scheme involving four elements of different types shown in Fig 2.14 This is probably the only version of the configuration of four elements which results in a model correct in the mechanical sense Judging by this scheme, in the tension state, where εc = εv − ε p > 0, a plastic hinge has... determined in terms of stress by integrating the differential equation (2.9) with respect to ε and stress is determined in terms of strain with the help of the same equation with respect to σ In this case the general solution is given by the integral 1 t − t0 + σ = s(t) ≡ σ0 exp − aη a t exp − t0 t − t1 dε(t1 ), aη (2.10) 2.3 Cohesive GranularMaterials 19 where the integration constant σ0 is determined... property Because of this, further this question is related to correctness of a computational algorithm being applied An example of a rheological scheme involving four base elements of different types which is correct in this sense is given in Fig 2.14 of the previous section In the general case a rheological scheme involving n elements is subdivided into m levels depending on the position of connective... materials, as a rule, depend on a number of side factors such as inhomogeneity in size of particles and in composition, anisotropy, fissuring, moisture, etc This results in low accuracy of experimental measurements of phenomenological parameters of models At the present time, two classes ofmathematical models corresponding to two different conditions of deformation of a granular material (quasistatic conditions . problems of design of
storage bunkers and grain tanks, problems of design of chemical machines with a
boiling granular layer, problems of modeling of avalanching,. monograph contains original results in the field of mathematical and
numerical modeling of mechanical behavior of granular materials and materials
with