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 Mechanics of 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. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) 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- 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 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 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 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- matical Modeling in the Problems of Mechanics of Granular 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 mathematical modeling 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 Modeling of 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 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 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 of granular 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 modeling of 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 mechanics in the classical...Chapter 1 Introduction The theory of granular materials is among the most interesting and intensively developing fields of mechanics because the area of its application is very wide It involves problems of mechanics of 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 of mechanics of granular materials 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 Granular Materials 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 of mathematical 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