Aleksandr F. Revuzhenko

Mechanics of Granular Media

e Springer

Contents

INTRODUCTION ....................•.•........................•...........•...........•..•........•• 1

Chapter 1. THEORETICAL SUBSTANTIATION OF EXPERIMENTAL METHODS TO STUDY GRANULAR MATERIALS ........................................................................................... 19

§ 1.1. PROBLEM STATEMENT„„„„„„.„„„„„„„„„„„„„„„„„„„„„„19 § 1.2. HOMOGENEOUS SHEAR FLOWS „„„„„„„„„„„„„„„„„„„„. 22

1.2.1. Superposition of simple shears „„„ „ „„ „. „ „ „„. „ „„„ „ „ „„„ „ „. 22 1.2.2. Elliptical and hyperbolic flows ....... „.„ .„„„„.„ .. „ ............. „„„ 25 1.2.3. Spatial flows .... „„ .... „.„„.„„„„„„„„„„„„„„„„„„„.„„„„„„„„ 32

§ 1.3. COMPLEX LOADING AS A SUPERPOSITION OF HOMOGENEOUS TENSIONS AND COMPRESSIONS „„„„ 38

1.3.1. Plane strain „„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„.„38 1.3.2. Spatial deformation .„„.„„„„„„„.„„„„.„„ ..... „ ... „„„ ... „„ .. „„. 50

Cbapter 2. DILATANCY AND PLASTICITY OF GRANULAR MATERIALS UNDER SIMPLE SHEAR ............................................• 57

§ 2.1. UNIFORM SHEAR DEVICE ... „ ...................... „ .......... „ ......... „57 § 2.2. DILATANCY ...... „.„.„.„ ..... „.„.„„ .... „„ ...... „„ ....... „ ......... „ .... „. 61 § 2.3. PLASTICITY ............. „„„ .... „„ ........... „ ..... „„ ............. „ .......... „. 69 § 2.4. KINKED TRAJECTORY LOADING ..... „„.„„ ..... „.„„ ..... „„„„ 76

Chapter 3. COMPLEX LOADING WITH THE CONTINUO US ROTATION OF THE STRAIN TENSOR AXES ..............•.•.........•.•.••. 85

§ 3.1. COMPLEX LOADING DEVICE. PROCEDURE AND RESULTS „ .......... „„ ........ „ ... „ ...... „.„ ....... „ ......... „ .. „„ .. „„ .. „.„.85

§ 3.2. EFFECT OF DIRECTIONAL MASS MOTION UNDER THE COMPLEX LOADING .„„„„„„„„„„.„ .... „.„„„„„ .... „„„„„ ... „.100 § 3.3. SPATIAL COMPLEX LOADING ..... „.„.„„.„„„ ....... „ ........ „„107

Chapter 4. LOCALIZATION OF STRAINS AND PROCESSES OF SELF-ORGANIZATION IN GRANULAR MATERIALS ...•..... 111

§ 4.1. ALGORITHM FOR EXPERIMENT AL REGULAR STRUCTURES ... „„„„ ..... „„„.„„„„„„„.„„„.„„.„„„ ..... „„„„„„ .. „„„„ 111

VIII Contents

4.1.1. Problem statement ............................... ................................. 112 4.1.2. Algorithm ............................................................................. 113

§ 4.2. BLOCK STRUCTURE UNDER THE SIMPLE SHEAR ........ 116 § 4.3. STRUCTURES IN GRANULAR MATERIALS WITH COHESION UNDER THE COMPLEX LOAD ING ......................... 124

4.3.1. Polygonal structures ............................................................. 124 4.3.2. Eddy structures ..................................................................... 131

§ 4.4. STRUCTURES IN COHESIONLESS GRANULAR MATERIALS UNDER THE COMPLEX LOADING ....................... 133

4.4.1. Spatial-time structure associated with the localization in the systems of planes ............................................................ „ .... 135 4.4.2. Spiral slip line systems ......................................................... 138 4.4.3. Turbulent plastic flow ........................................................... 144 4.4.4. Structures associated with slip lines of the type of the Cassinian ovals and Bernoulli lemniscate ...................................... 145 4.4.5. Structures associated with the fatigue localization ............... 149

§ 4.5. STRUCTURES ARISING IN THE SYMMETRIC PRIMARY F.LOWS ........................................................................... 151 § 4.6. SOME STRUCTURES ARISING UNDER MASS FORCES. 158 § 4.7. DISASTROUS, LONG-TERM WEAK INF.LUENCE ON A FREE-F.LOWING MEDIUM .................................................. 160 § 4.8. SELF-ORGANIZATION OF GRANULAR MATERIALS IN CYLINDRICAL ROTARY RESERVOIR ......................................... 164

Chapter S. MATHEMATICAL MODELS OF DEFORMATION OF GRANULAR MATERIALS .................•... „ .•. „ .•.•..•......•.•.............•.•..... 167

§ 5.1. DILATANCY ............................................................... „ .......... 167 § 5.2. INTERN AL FRICTION ................................... „ ...................... 175 § 5.3. MATHEMATICAL MODELS ............. „ ........ „ ....................... 177 § 5.4. CLOSED SYSTEM OF EQUATIONS ............ „ ...................... 181 § 5.5. ENERGY DISSIPATION ........................................................ 186

Chapter 6. BOUNDARY VALUE PROBLEMS .•....•.. „ .. „ .••...•..•. „ ...•. 189 § 6.1. SOME TEST PROBLEMS ................. „ . .... .. ......... . .................. 189 § 6.2. EXACT SOLUTIONS TO THE EQUATIONS OF LIMIT EQUILIBRIUM OF GRANULAR MATERIAL IN ELEMENTARY FUNCTIONS ................................................ „ .................................... 190

6.2.1. Solutions for perfectly connected medium „ .. „ ............... „ .... 191 6.2.2 Solutions for material with intemal friction „ .......... „„ .......... 203

§ 6.3. TAKING INTO ACCOUNT THE DILATANCY IN THE BASIC REFERENCE FORMULAE OF THE GRANULAR MATERIAL MECHANICS ............................................................... 205

Contents IX

§ 6.4. LATERAL THRUST COEFFICIENT OF GRANULAR MA IBRIAL .............. „ ....... „ ............. „ ..................... „ ..... „ .......... „ ...... 209 § 6.5. JANSSEN'S PROBLEM ................................ „ ......... „ ........ „ ... 215 § 6.6. PRESSURE EXERTED BY GRANULAR MATERIALS ON THE YIELDING BOUNDING STRUCTURES .......... „.„ ...... „ .. 230 § 6.7. FLOWS WITH THE LOCALIZATION OF SHEARS IN THE CONVERGING CHANNELS ....................... „ .......... „ ......... 237 § 6.8. MECHANICS OF GRANULAR MATERIALS IN MINING.240

Chapter 7. SOME PROBLEMS OF GRANULAR (POWDER) MATERIAL TREATMENT „ .... „„„„„„„ •.. „„„„.„„„„„„„„„„„„„„„„. 243

§ 7.1. MIXING OF GRANULAR (POWDER) MATERIALS .......... 243 7 .1.1. Problem statement .. „ ... „ ... „„„ .„„ ... „ .„ ....... „„ .... „ .„„ ..... „ .... 244 7 .1.2. Petfect lllixing .... „ ......... „ . ... „ ..... „ ..... „ ...... „ ...... „ ... „„ ..... „ ... 245 7 .1.3. Principles for realizing the petfect lllixing ............. „ ............ 250

§ 7 .2. PSEUDO-VISCOSITY OF GRANULAR MATERIALS AND MINERAL BENEFICIATION ........................... „ .................... 251 § 7.3. PRECISION BATCHING OFTHE POWDER MATERIALS ....................... „.„ ...... „ .................................. „ ........ „ ... 255 § 7 .4. DILAT ANCY COMPACTION OF THE POWDER MATERIALS ............................................................................. „ ...... 256

Chapter 8. PRACTICALITY OF THE RESULTS IN THE MECHANICS-RELA TED FIELDS„„„„„„„„„„„„„„„„„„ .• 259

§ 8.1. GENERAL THEORY OF VISCOMETRIC FLOWS. PROCEDURES FOR ANAL YZING THE CONSITUTIVE EQUATIONS OF COMPOUND MEDIA .„ ...................... „ .............. 259 § 8.2. PROBLEM ON BALL FORMATION FROM A FREE VOLUME OF NON-NEWTONIAN LIQUID ...................... „ ........... 263 § 8.3. ROCK MECHANICS .„ ........................... „ ............ „„ ...... „ .. „ ... 264 § 8.4. MECHANICS OF THE ELASTO-PLASTIC MEDIA ............ 267 § 8.5. APPLICATIONS OF THE DIRECTIONAL MOTION OF MASSES .............. „ ...................................... „„.„ .. „ ...... „ .............. 269

8.5.1. Processing of plastic and powder materials.„„„ ... „„.„.„„„„269 8.5.2. Tidal waves and directional motion of the Earth mass ..... „ .. 272

References ............................................................................................... 27 5