Micromechanics of granular materials:
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ISTE [u.a.]
2009
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| Physical Description: | XV, 346 S. Ill., graph. Darst. |
| ISBN: | 9781848210752 |
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| 020 | |a 9781848210752 |c (hbk.) : GBP 78.95 |9 978-1-8482-1075-2 | ||
| 035 | |a (OCoLC)316513972 | ||
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| 130 | 0 | |a Micromécanique des matériaux granulaires | |
| 245 | 1 | 0 | |a Micromechanics of granular materials |c ed. by Bernard Cambou ; Mechel Jean ; Farhang Radjaï |
| 264 | 1 | |a London [u.a.] |b ISTE [u.a.] |c 2009 | |
| 300 | |a XV, 346 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Micromechanics | |
| 650 | 4 | |a Granular materials | |
| 650 | 0 | 7 | |a Mikromechanik |0 (DE-588)4205811-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Granulärer Stoff |0 (DE-588)4256351-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Granulärer Stoff |0 (DE-588)4256351-3 |D s |
| 689 | 0 | 1 | |a Mikromechanik |0 (DE-588)4205811-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Cambou, Bernard |4 edt | |
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| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017097036&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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| adam_text | Contents
Introduction
..................................... xiii
Chapter
1.
Experimental and Numerical Analysis of Local Variables in
Granular Materials
................................. 1
Farhang Radjai and Jack
Lanier
1.1.
Introduction
.................................. 1
1.2.
Description of granular texture
....................... 3
1.2.1.
Particle connectivity
.......................... 3
1.2.2.
Contact network anisotropy: fabric tensors
............. 5
1.2.2.1.
General case
........................... 5
1.2.2.2.
Case of 2D data
......................... 7
1.2.2.3.
Case of
3D
data
......................... 11
1.2.3.
Branch vectors
............................. 13
1.2.4.
Evolution of granular texture
..................... 14
1.2.5.
Space partition: tessellation
...................... 17
1.2.5.1.
Voronoi cells
........................... 17
1.2.5.2.
Dirichlet cells
........................... 17
1.2.5.3.
General case
........................... 18
1.2.5.4.
Neighborhoods and local void ratios
.............. 18
1.3.
Granular kinematics
............................. 20
1.3.1.
Particle displacements and rotations
................. 20
1.3.2.
Rolling versus sliding
......................... 21
1.3.3.
Fluctuating displacement fields
.................... 23
1.3.3.1.
Uniform strain and fluctuations
................. 23
1.3.3.2.
Probability densities
....................... 25
1.3.3.3.
Spatial correlations
........................ 26
1.3.3.4.
Granulence
............................ 27
1.3.4.
Local and global strains
........................ 27
1.3.4.1.
Particle-scale strain
....................... 28
vi Micromechanics
of
Granular Materials
1.3.4.2.
Strain localization........................
29
1.4.
Force transmission
.............................. 31
1.4.1.
Probability density functions
..................... 32
1.4.2.
Bimodal character of stress transmission
.............. 39
1.4.3.
Force anisotropy
............................ 42
1.5.
Conclusion
.................................. 44
1.6.
Bibliography
................................. 45
Chapter
2.
The Stress Tensor in Granular Media and in other Mechanical
Collections
...................................... 51
Jean-Jacques MOREAU
2.1.
Introduction
.................................. 51
2.1.1.
Motivation
............................... 52
2.1.2.
The theoretical background
...................... 53
2.1.3.
Dynamics
................................ 55
2.1.4.
Pertinence
................................ 56
2.2.
Efforts and virtual power
.......................... 59
2.2.1.
Resultant and moment of an effort
.................. 59
2.2.2.
Internal efforts
............................. 60
2.2.3.
Forces
.................................. 62
2.2.4.
Efforts of order greater than zero
................... 62
2.2.5.
Contact actions
............................. 64
2.3.
Equilibrium
.................................. 65
2.3.1.
Main equalities
............................. 65
2.3.1.1.
Case of a continuous body
................... 65
2.3.1.2.
Case of a granular material
................... 66
2.3.2.
Classical continuous body
....................... 67
2.3.3.
Piece of string
............................. 68
2.3.4.
Finite collection of points
....................... 71
2.3.5.
Interaction bridges
........................... 73
2.3.6.
Saturated soil
.............................. 75
2.4.
Comparison with the pair-by-pair approach
................ 76
2.4.1.
The classical definition
........................ 76
2.4.1.1.
The pair-by-pair calculation
................... 77
2.4.2.
Numerical discussion of a tri-axial test
................ 79
2.5.
Directions of cut
............................... 83
2.5.1.
Force transmitted across a cut
..................... 83
2.5.2.
Proof of the cutting direction law
................... 84
2.5.3.
2D bank
................................. 87
2.5.3.1.
Free surface law
......................... 87
2.5.4.
Conical pile
............................... 88
2.6.
Coarse graining the equation of Statics
................... 90
2.6.1.
The divergence operator
........................ 90
Contents
vii
2.7.
One step into Dynamics
........................... 91
2.7.1.
Introducing the acceleration field
................... 91
2.7.2.
Rigid bodies
.............................. 92
2.7.2.1.
Introducing the mass center
................... 92
2.7.2.2.
Introducing principal axes
.................... 93
2.7.2.3.
Invoking rigid body dynamics
................. 94
2.7.2.4.
Spherical inertia
......................... 95
2.7.2.5.
2D models
............................ 95
2.7.3.
Percussions
............................... 96
2.8.
Bibliography
................................. 97
Chapter
3.
Multiscale Techniques for Granular Materials
.......... 101
Bernard
Cambou, Alexandre
Danescu
3.1.
Introduction
.................................. 101
3.2.
Scale change and fabric tensors
....................... 102
3.2.1.
Solid particles description
....................... 102
3.2.1.1.
Size of particles
......................... 102
3.2.1.2.
Shape of particles
........................ 102
3.2.2.
Fabric description for a granular sample
............... 106
3.2.2.1.
Coordination number and compactness
............ 106
3.2.2.2.
Definition of the overall anisotropy of a sample
....... 108
3.2.3.
Voids description
............................
Ill
3.3.
Change of scale for static variables
..................... 112
3.4.
Change of scale for kinematic variables in granular materials
...... 115
3.4.1.
Definition of local kinematic variables
................ 115
3.4.2.
Method based on an energetic approach
............... 117
3.4.3.
Definition of strain from a discrete equivalent continuum
..... 118
3.4.3.1.
Strain proposed by Kruyt and Rothenburg in 2D
....... 118
3.4.3.2.
Strain proposed by
Cambou
et al.
in 2D
............ 121
3.4.3.3.
Strain proposed by
Bagi
..................... 123
3.4.4.
Strain defined from best-fit methods
................. 123
3.4.4.1.
Strain proposed by Cundal
................... 123
3.4.4.2.
Strain proposed by Liao
et al...................
124
3.4.4.3.
Strain proposed by
Cambou
et al................
126
3.4.5.
Analysis of the different
microstructural
definitions of strain
and comparison with the macro strain defined at the considered
sample scale
............................... 26
3.5.
Statistical homogenization in granular materials
............. 131
3.5.1.
Elastic behavior of a granular sample
................ 132
3.5.1.1.
Model based on kinematic localization
............ 132
3.5.1.2.
Model based on static localization
............... 133
3.5.2.
Elastic behavior of a granular sample
................ 135
3.5.2.1.
Voigt-type hypothesis for kinematic localization
....... 135
viii Micromechanics of
Granular
Materials
3.5.2.2.
Static
localization hypothesis..................
136
3.5.3.
Extension to nonlinear elasticity
................... 139
3.5.4.
Definition of a yield surface from
alocai
criterion
......... 140
3.5.4.1.
Remarks on the validity of the yield criteria
.......... 143
3.5.5.
Difficulties and limitations for statistical homogenization in
granular materials
........................... 143
3.6.
Bibliography
................................. 145
Chapter
4.
Numerical Simulation of Granular Materials
........... 149
Michel Jean
4.1.
Introduction
.................................. 149
4.2.
The actors of a contact problem
....................... 152
4.2.1.
Bodies, contactors and candidates to contact
............ 153
4.2.2.
Some bodies and contactors used in numerical simulation
..... 156
4.2.3.
Sorting
.................................. 164
4.3.
Kinematic relations
.............................. 167
4.3.1.
Usual rigid body kinematics
...................... 167
4.3.2.
Local variables
............................. 168
4.3.3.
The distance function
......................... 169
4.3.4.
Relations between generalized and local variables
......... 170
4.3.4.1.
The mappings
H
and H*
.................... 172
4.3.4.2.
Non-uniqueness
......................... 172
4.3.5.
Boundary conditions, driven or locked degrees of freedom
.... 173
4.4.
The dynamical equation
........................... 174
4.4.1.
2D or
3D
bodies
............................ 175
4.4.2.
Deformable bodies
........................... 176
4.4.3.
Shocks, momentum, impulses and percussions
........... 176
4.4.4.
Energy formulae
............................ 178
4.5.
Frictional contact laws
............................ 179
4.5.1.
Unilaterality
............................... 180
4.5.1.1.
Signorini
conditions
....................... 180
4.5.1.2.
Complementary relation and convex analysis
......... 181
4.5.1.3.
Flexibility models
........................ 182
4.5.1.4.
Shock laws
............................ 184
4.5.2.
Friction laws
.............................. 186
4.5.2.1.
Coulomb s law
.......................... 186
4.5.2.2.
Coulomb s law and convex analysis
.............. 188
4.5.2.3.
Coulomb-type friction laws, strongly viscous at slow sliding
speeds
............................... 189
4.5.2.4.
Dynamical friction, static friction coefficients
........ 189
4.5.3.
Choosing a frictional contact law
................... 191
4.5.4.
Cohesive behavior
........................... 192
4.5.4.1.
Mohr-Coulomb law
....................... 195
Contents ix
4.5.4.2.
Rolling
grains,
welded
grains
.................. 195
4.6.
The equations governing
a collection
of contacting bodies
........ 196
4.7.
Preparing numerical samples
........................ 198
4.7.1.
Boundary conditions
.......................... 199
4.7.2.
Initial state
............................... 203
4.7.3.
Size of samples
............................. 205
4.8.
Smooth DEM numerical methods
...................... 206
4.8.1.
Molecular dynamics methods
..................... 206
4.8.2.
Smooth DEM methods
......................... 207
4.8.2.1.
Discretizing the dynamical equation
.............. 207
4.8.2.2.
Discretizing the function reac (q,u)
.............. 208
4.8.3.
PCDEM methods
............................ 208
4.8.3.1.
Discrete form of the frictional law Reac
............ 211
4.8.3.2.
Proof of equation
(4.40)..................... 212
4.8.3.3.
Numerical scheme
........................ 212
4.8.3.4.
Remarks
.............................. 213
4.8.3.5.
Damping
С = аМ........................
214
4.8.4.
Choosing the time-step
........................ 214
4.9.
Non-smooth DEM numerical methods
................... 216
4.9.1.
Event-driven methods
......................... 216
4.9.1.1.
Computing a collision
...................... 216
4.9.1.2.
Remark
.............................. 220
4.9.2.
Non-smooth contact dynamics method
................ 220
4.9.2.1.
Discretization of the dynamical equation
........... 220
4.9.2.2.
Discrete form of kinematic relations
.............. 221
4.9.2.3.
Discrete forms of frictional contact relations
......... 222
4.9.2.4.
Restriction of the dynamical equation to candidates to contact
223
4.9.2.5.
Signorini
μ
-Coulomb
standard problem
............ 224
4.9.2.6.
Remarks
.............................. 224
4.9.2.7.
Solving the
Signorini
¿¿-Coulomb standard problem
..... 224
4.9.2.8.
Solution of the 2D
Signorini ¿i-Coulomb
standard problem
. 226
4.9.2.9.
Solving the frictional contact problem, Gauss
Seidel
nesting
226
4.10.
Some illustrating examples
......................... 227
4.10.1.
The bouncing ball problem
..................... 227
4.10.2.
Frictional contact examples by explicit or implicit methods
. . . 230
4.10.2.1.
Example
1............................ 231
4.10.2.2.
Example
2............................ 231
4.10.2.3.
Example
3............................ 233
4.10.2.4.
Example
4............................ 233
4.11.
Quasi-static evolutions, equilibrium dedicated methods
........ 234
4.11.1.
A strongly viscous contact law
................... 235
4.11.2.
Flexibility models
...........................
236
4.11.3.
Rigid bodies and
Signorini, /¿-Coulomb
law
............ 237
χ
Micromechanics
of Granular Materials
4.11.4.
Quasi-static evolutions versus dynamics
.............. 238
4.12.
Accuracy criteria
.............................. 241
4.12.1.
Implicit methods
........................... 242
4.12.2.
Explicit methods
........................... 243
4.12.3.
Some accuracy estimators
...................... 243
4.12.3.1.
Mean and quadratic violations
................ 243
4.12.3.2.
Bipotential violation
...................... 245
4.13.
Indétermination
in granular materials
................... 246
4.13.1.
A wedged disk example
....................... 248
4.13.1.1.
A rigid wedged disk example
................. 248
4.13.1.2.
Analyzing the kinematic
indétermination
.......... 249
4.13.1.3.
A classical example of deformable model
.......... 250
4.13.1.4.
Investigating
indétermination,
numerical experiments
... 253
4.13.1.5.
The single wedged disk
.................... 253
4.13.1.6.
Loading experiment, domains of attraction
......... 255
4.13.1.7.
Another view of domains of attraction: rigid model
..... 257
4.13.1.8.
Loading-unloading experiments
................ 260
4.13.1.9.
Indétermination
in the deformable model
.......... 260
4.13.2.
Three wedged disks,
200
packed disks examples
......... 261
4.13.2.1.
Three wedged disks
...................... 261
4.13.2.2. 200
disks sample
........................ 263
4.13.3.
Conclusions
.............................. 265
4.14.
Stability
................................... 265
4.14.1.
Perturbations
............................. 266
4.14.2.
Coulomb stable sample
........................ 267
4.14.3.
Left reactions perturbations of the single wedged disk
...... 268
4.14.4.
Left reactions perturbations of a
2,400
polygon sample
..... 269
4.14.5.
Further comments
........................... 274
4.15.
Numerical integration schemes
...................... 275
4.15.1.
θ
method
................................ 276
4.15.2.
Consistency of the discrete approximations
............ 278
4.15.3.
Newmark method
........................... 280
4.15.4.
Deformable grains
.......................... 281
4.15.5.
Further comments
........................... 283
4.16.
More non-smooth DEM methods
..................... 284
4.16.1.
The NSCD method, Gauss-Seidel nesting
............. 284
4.16.2.
The NSCD method, Jacobi nesting
................. 285
4.16.3.
The bi-potential method
....................... 286
4.16.4.
The generalized Newton method
.................. 287
4.16.5.
Gradient-type methods
........................ 289
4.16.6.
Mathematical programming methods
................ 290
4.16.7.
Multigrid computation
........................ 290
4.16.8.
Parallel computation
......................... 291
Contents xi
4.17.
Signorini
^-Coulomb
derived laws
.................... 292
4.17.1.
Status
.................................. 293
4.17.1.1.
Status
SIGNORINIJľONTACT
................. 293
4.17.1.2.
Status COULOMB.SLIDING and COULOMB.STICKING
. . 294
4.17.2.
Change of variables
.......................... 294
4.17.2.1.
Remarks
............................. 296
4.17.3.
Algorithm NSCD and derived laws
................. 296
4.17.4.
Gap
Signorini
condition and Coulomb s law
............ 297
4.17.5.
Inelastic quasi-choc law and Coulomb s law
............ 297
4.17.6.
Velocity
Signorini
condition and Coulomb s law
......... 298
4.17.7.
Restitution shock law together with Coulomb s law
........ 298
4.17.8.
Velocity
Signorini
condition and Coulomb s law with static or
dynamic friction coefficient
...................... 298
4.17.9.
Flexible unilateral conditions and Coulomb s law
......... 299
4.17.10.
Mohr Coulomb cohesive law
.................... 299
4.17.11.
A simple cohesive example
..................... 300
4.17.12.
Fiber-like materials
......................... 301
4.18.
Conclusion
.................................. 301
4.19.
Appendix: basic convex analysis
..................... 303
4.19.1.
Convex sets and cones
........................ 303
4.19.2.
Convex functions, conjugates, subdifferential
........... 303
4.19.3.
Standard
Signorini
relation
..................... 305
4.19.4.
Standard Coulomb s law
....................... 305
4.19.5.
Bipotentials
.............................. 306
4.20.
Bibliography
................................. 307
Chapter
5.
Frictionless Unilateral Multibody Dynamics
........... 317
Patrick Ballard
5.1.
Introduction
.................................. 317
5.2.
The dynamics of rigid body systems
.................... 318
5.2.1.
The geometric description
....................... 318
5.2.2.
Formulation of the dynamics
..................... 319
5.2.3.
Well-posedness of the dynamics
................... 321
5.3.
The dynamics of rigid body systems with perfect bilateral constraints
. 322
5.3.1.
The geometric description
....................... 322
5.3.2.
Formulation of the dynamics
..................... 323
5.3.3.
Well-posedness of the dynamics
................... 325
5.4.
The dynamics of rigid body systems with perfect unilateral constraints
326
5.4.1.
The geometric description
....................... 326
5.4.2.
Formulation of the dynamics
..................... 327
5.4.2.1.
Equation of motion
....................... 327
5.4.2.2.
The impact constitutive equation
................ 329
5.4.2.3.
The evolution problem
...................... 333
xii Micromechanics
of Granular Materials
5.4.3.
Well-posedness of the dynamics
................... 334
5.4.3.1.
Comments
............................ 339
5.5.
Bibliography
................................. 340
List of Authors
.................................... 343
Index
.......................................... 345
îThe
main focus of
Micromechanics
of
Granular Materials
is on static
¡and dynamic loading applied to dense materials; rapid flows and
widely dispersed media are also covered in less detail.
^Three essential areas are covered:
χ
Local variable analysis: contact forces, displacements and
rotations, orientation of contacting particles and fabric tensors are
all examples of local variables. Their statistical distributions, such
as spatial distribution and possible localized effects, are analyzed,
taking into account experimental results or numerical simulations.
¿Change of scales procedures: also known as homogenization
techniques , these procedures make it possible to construct
Icontinuum laws to be used in a continuum mechanics approach or
Ito
perform smaller scale analyses.
^Numerical modeling: several methods designed to calculate
■^approximate solutions of dynamical equations together with
¡unilateral contact and frictional laws are presented, including
fmolecular dynamics, the distinct element method and non-smooth
contact dynamics. Numerical examples are provided and the
equality of numerical approximations is discussed.
¡¡Bernard
Cambou
is a Professor of Civil and Mechanical
«¡Engineering at the ECL,
Lyon,
France.
{¡Michel Jean
is Emeritus Director of Research at
Laboratoire de
¡¡Mécanique et d Acoustique LMA-CNRS, Marseille, France.
{
JFarhang Radjaï
is
CNRS Research
Director at the University of;
ÍMontpellier2,
France.
|
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| discipline | Physik |
| format | Book |
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| id | DE-604.BV035292006 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:30:35Z |
| institution | BVB |
| isbn | 9781848210752 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017097036 |
| oclc_num | 316513972 |
| open_access_boolean | |
| owner | DE-703 DE-29T |
| owner_facet | DE-703 DE-29T |
| physical | XV, 346 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | ISTE [u.a.] |
| record_format | marc |
| spelling | Micromécanique des matériaux granulaires Micromechanics of granular materials ed. by Bernard Cambou ; Mechel Jean ; Farhang Radjaï London [u.a.] ISTE [u.a.] 2009 XV, 346 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Micromechanics Granular materials Mikromechanik (DE-588)4205811-9 gnd rswk-swf Granulärer Stoff (DE-588)4256351-3 gnd rswk-swf Granulärer Stoff (DE-588)4256351-3 s Mikromechanik (DE-588)4205811-9 s DE-604 Cambou, Bernard edt Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017097036&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017097036&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Micromechanics of granular materials Micromechanics Granular materials Mikromechanik (DE-588)4205811-9 gnd Granulärer Stoff (DE-588)4256351-3 gnd |
| subject_GND | (DE-588)4205811-9 (DE-588)4256351-3 |
| title | Micromechanics of granular materials |
| title_alt | Micromécanique des matériaux granulaires |
| title_auth | Micromechanics of granular materials |
| title_exact_search | Micromechanics of granular materials |
| title_full | Micromechanics of granular materials ed. by Bernard Cambou ; Mechel Jean ; Farhang Radjaï |
| title_fullStr | Micromechanics of granular materials ed. by Bernard Cambou ; Mechel Jean ; Farhang Radjaï |
| title_full_unstemmed | Micromechanics of granular materials ed. by Bernard Cambou ; Mechel Jean ; Farhang Radjaï |
| title_short | Micromechanics of granular materials |
| title_sort | micromechanics of granular materials |
| topic | Micromechanics Granular materials Mikromechanik (DE-588)4205811-9 gnd Granulärer Stoff (DE-588)4256351-3 gnd |
| topic_facet | Micromechanics Granular materials Mikromechanik Granulärer Stoff |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017097036&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017097036&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | UT micromecaniquedesmateriauxgranulaires AT camboubernard micromechanicsofgranularmaterials |