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Über dieses Buch

This monograph contains original results in the field of mathematical and numerical modeling of mechanical behavior of granular materials and materials with different strengths. It proposes new models helping to define zones of the strain localization. The book shows how to analyze processes of the propagation of elastic and elastic-plastic waves in loosened materials, and constructs models of mixed type, describing the flow of granular materials in the presence of quasi-static deformation zones. In a last part, the book studies a numerical realization of the models on multiprocessor computer systems.

The book is intended for scientific researchers, lecturers of universities, post-graduates and senior students, who specialize in the field of the deformable materials mechanics, mathematical modeling and adjacent fields of applied and calculus mathematics.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
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 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.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 2. Rheological Schemes

Abstract
The traditional rheological method is supplemented by a new element—rigid contact, which serves to take into account different resistance of a material to tension and compression. A rigid contact describes mechanical properties of an ideal granular material involving rigid particles for an uniaxial stress state. Combining it with elastic, plastic, and viscous elements, one can construct rheological models of different complexity.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 3. Mathematical Apparatus

Abstract
Basic notions of convex analysis required for the generalization of constitutive relationships of uniaxial deformation of granular materials to the spatial case are considered. Proofs of some theorems from subdifferential calculus and duality theory which are used below in the study of models of the spatial stress-strain state are presented.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 4. Spatial Constitutive Relationships

Abstract
The constitutive relationships for a granular material involving absolutely rigid and elastic particles as well as the constitutive equations for a heteromodular elastic material are generalized to the case of a spatial stress-strain state under small strains. Versions of constraints on admissible stress tensors for an isotropic material, which are defined with the help of the Coulomb–Mohr and von Mises–Schleicher cones, are considered. Dual cones of admissible strain tensors are constructed. The projection operators, which are used further in algorithms for numerical realization of spatial models, are presented.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 5. Limiting Equilibrium of a Material With Load Dependent Strength Properties

Abstract
Solvability of static boundary-value problems within the framework of a model describing small strains in a material with load dependent strength properties, for example at tension and compression, is studied. A generalization of the static and kinematic theorems of the theory of limiting equilibrium is given. As an example of the application of the kinematic theorem, an upper estimate of the limit load and of the angle of departure of linear zone of the strain localization for the problem on discontinuity of a notched sample under the action of pressure on the edges of a notch is found. It is shown that logarithmic spirals serve as localization lines. With the help of two-sided estimates, an expression for the angle of natural slope of an ideal granular material is obtained. For the numerical solution of boundary-value problems, an iterative algorithm based on the finite-element approximation of a model is worked out. Results of computations which confirm the obtained estimating solutions are presented.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 6. Elastic–Plastic Waves in a Loosened Material

Abstract
A priori estimates for solutions in characteristic cones, which provide assurance that a boundary-value problem with initial data and dissipative boundary conditions is well-posed in the framework of a model describing dynamic deformation of an elastic–plastic granular material, are obtained. Shock adiabatic curves for plane longitudinal compression waves, propagating in an unbounded body, are constructed for various combinations of mechanical parameters of a material. Computational algorithm for the analysis of propagation of shock waves of small amplitude in a granular material, based on the method of splitting with respect to physical processes and with respect to spatial variables, is proposed. The results of two-dimensional computations of interaction of signotons in an inhomogeneous loosened material accompanied by a transverse cumulative ejection as well as the results of modeling of the “dry boiling” process (formation of continuity jumps in a material under the action of periodic load and their collapse) are presented.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 7. Contact Interaction of Layers

Abstract
Algorithms for numerical implementation of conditions of dynamic contact interaction of deformable materials with a beforehand unknown zone of contact which varies in the process of motion are constructed. These algorithms take into account the influence of friction forces in a contact zone. On the basis of these algorithms a method for the numerical modeling of deformation of a body of a granular material in the presence of sliding surfaces is worked out. Results of testing an algorithm and results of the numerical solution of a problem for two layers of a medium consisting of an elastic-plastic material are presented. Computational algorithms are developed that simulate the dynamic interaction of elastic blocks through thin viscoelastic layers in structurally inhomogeneous media such as rocks.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 8. Results of High-Performance Computing

Abstract
Algorithms for numerical implementation of the shock-capturing method for solving the problems of dynamics of a granular material are constructed. In these algorithms computations are parallelized at the stage of splitting a problem with respect to spatial variables. Different ways of distribution of a computational domain among parallel computational nodes are considered. It is shown that the minimal number of exchanges between nodes is achieved when a domain is decomposed into regular cubes. Numerical results for propagation of elastic–plastic waves in two-dimensional and three-dimensional formulations obtained with the help of multiprocessor computer systems of the MVS series are presented.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 9. Finite Strains of a Granular Material

Abstract
A mathematical model of developed flow of a granular material is considered. On the phenomenological level, elastic properties characteristic for a compacted material and viscous properties appearing in loosening are taken into account. Exact solutions of problems on rotational and plane-parallel motion of a material with stagnant zones are constructed. Using them, influence of viscosity on a flow pattern is analyzed.
Oxana Sadovskaya, Vladimir Sadovskii

Chapter 10. Rotational Degrees of Freedom of Particles

Abstract
On the basis of a mathematical model of the Cosserat continuum and a generalized model, that describes the different resistance of a material with respect to tension and compression, the influence of rotational motion of particles onto the stress-strain state of a granular material is studied. It is shown that a couple-stress elastic medium has the resonance frequency, coinciding with the frequency of natural oscillations of rotational motion of the particles. The solution of the problem of uniform shear of a granular material, having rotational degrees of freedom, is analyzed in the framework of linear and nonlinear models.
Oxana Sadovskaya, Vladimir Sadovskii
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