Failure of granular assemblies [Elektronische Ressource] / vorgelegt von Philipp Welker

Sujets

• Naturwissenschaften

Informations

• Publié par: universitat_stuttgart
• Publié le: 01 janvier 2011
• Nombre de lectures: 16
• Langue: Deutsch
• Poids de l'ouvrage: 10 Mo

Extrait

PhilippWelker
Failureofgranularassemblies
Fakultat¨ fur¨ MathematikundPhysik
Universitat¨ Stuttgart
2011Failure of granular assemblies
Von der Fakultat¨ Mathematik und Physik der Universitat¨ Stuttgart
¨zur Erlangung der Wurde eines Doktors
der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Philipp Welker
aus Nurnber¨ g
Hauptberichter : Apl. Prof. Dr. Dr. Rudolf Hilfer
Zweitberichter : Prof. Dr. Gunther¨ Wunner
Drittberichter : Habil. Dr. Sean McNamara
Prufungsv¨ orsitzender : Prof. Dr. Jor¨ g Wrachtrup
Tag der mundlichen¨ Prufung¨ : 11. Januar 2011
Institut fur¨ Computerphysik der Universitat¨ Stuttgart
2011List of publications
Articles
P. R. Welker and S. C. McNamara, “What triggers failure in frictional granular assemblies?,”
Physical Review E, vol. 79, no. 6, pp. 061305, 2009.
P. R. Welker and S. C. McNamara, “Precursors of failure and weakening in a biaxial test,”
Granular Matter, vol. 13, no. 1, pp. 93, 2010.
P. Welker and S. McNamara, “An increasing length scale associated with the failure of gran-
ular solids.” submitted to Europhyics Letters (EPL), 2011.
Proceedings
P. Welker and S. McNamara, “Trigger of failure in granular assemblies,” in Tagungsband zur
DPG Frhjahrstagung der Sektion Kondensierte Materie (SKM), 2009.
P. Welker and S. McNamara, “Failure in small granular assemblies,” in Powders and Grains
(M. Nakagawa and S. Luding, eds.), pp. 289–292, Springer, 2009.
P. Welker and S. McNamara, “Precursors of failure in granular systems,” in Tagungsband
zur DPG Frhjahrstagung der Sektion Kondensierte Materie (SKM), 2010.
iiiContents
Zusammenfassung vii
1 Introduction 1
2 Model, analysis, and simulation procedure 7
2.1 Particle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Membrane like boundary conditions . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 Which grains are part of the boundary? . . . . . . . . . . . . . . . 12
2.4.2 Fixing the membrane . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Units and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Rattlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Global hyperstatic number . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8.2 Derivation from the equations of motion . . . . . . . . . . . . . . . 20
2.8.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.4 Closer inspection of the stability limit . . . . . . . . . . . . . . . . 23
2.8.5 Inversion and particle velocities . . . . . . . . . . . . . . . . . . . 25
2.8.6 Inspection of energy terms in kinetic and potential energy . . . . . 26
2.9 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.10 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.11 Loading the sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.12 Investigation of the quasi-rigid limit with simulations . . . . . . . . . . . . 31
2.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Trigger of failure in small two-dimensional systems 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Leading up to failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Sliding contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iiiiv CONTENTS
3.2.2 Macroscopic and microscopic stiffness . . . . . . . . . . . . . . . 38
3.3 Example of a stiffness transition . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 The three triggers of failure . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Mechanical instability (vkv< 0) . . . . . . . . . . . . . . . . . . 41
3.4.2 Null-mode trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.3 Excursion: Fit of the kinetic energy at null modes . . . . . . . . . . 48
3.4.4 Ambushed transitions . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Compound Status changes . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Relative occurrence of the trigger mechanisms . . . . . . . . . . . . . . . . 55
3.7 Triggers in larger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8 Testing the accuracy of the stiffness matrix description . . . . . . . . . . . 58
3.9 What drives failure? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.9.1 The exponential rise of the kinetic energy . . . . . . . . . . . . . . 62
3.9.2 Contributions to the kinetic energy . . . . . . . . . . . . . . . . . . 62
3.10 Scaling of the rise in kinetic energy at failure . . . . . . . . . . . . . . . . 63
3.11 The failure staircase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Properties of failure for different system sizes 67
4.1 Is failure an isostatic transition? . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Small systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 Larger . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 External force at failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Dependence on the particle stiffness . . . . . . . . . . . . . . . . . 72
4.2.2 on the system size . . . . . . . . . . . . . . . . . . . . 72
4.3 Number of rattlers at failure . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Weakening process in large systems 77
5.1 Stress-strain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Kinetic energy and vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Volume, injected power and number of contacts . . . . . . . . . . . . . . . 81
5.4 Sliding contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.1 Number of sliding contacts . . . . . . . . . . . . . . . . . . . . . . 81
5.4.2 Strength of Sliding Contacts . . . . . . . . . . . . . . . . . . . . . 85
5.4.3 Understanding the evolution of sliding contacts . . . . . . . . . . . 85
5.5 Local number of sliding contacts . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.1 Definition of boxes . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.2 Number of sliding contacts in boxes . . . . . . . . . . . . . . . . . 89
5.5.3 Local change and disappearance of sliding contacts . . . . . . . . . 89
5.5.4 Spatial distribution of sliding contacts . . . . . . . . . . . . . . . . 93
5.6 Two qualitatively different periods . . . . . . . . . . . . . . . . . . . . . . 93CONTENTS v
5.7 Triangular figures: importance for stability . . . . . . . . . . . . . . . . . . 96
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Precursors of failure 101
6.1 Definition of precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Number of sliding contacts at precursors . . . . . . . . . . . . . . . . . . . 102
6.3 Appearance of an instability . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Localization of the kinetic energy . . . . . . . . . . . . . . . . . . . . . . 105
6.5 of the decrease in sliding contacts . . . . . . . . . . . . . . . 105
6.6 Change in the number of contacts . . . . . . . . . . . . . . . . . . . . . . 105
6.7 Can precursors be observed experimentally? . . . . . . . . . . . . . . . . . 110
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7 Organization of rolling motions 115
7.1 Definition of the rolling velocity . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 Organization of in specific directions . . . . . . . . . . . . . . . . . 116
7.3 Evolution of the organization of rolling . . . . . . . . . . . . . . . . . . . 122
7.4 Length scale of the org . . . . . . . . . . . . . . . . . . . . . . . 122
7.5 Correlation of sliding and rolling motions . . . . . . . . . . . . . . . . . . 126
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8 Comparison of rigid wall and membrane boundaries 129
8.1 Kinetic energy and vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.2 Number of sliding contacts . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.3 Contact status transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.4 Organization of sliding contacts . . . . . . . . . . . . . . . . . . . . . . . 134
8.5 Number of triangular figures: importance for stability . . . . . . . . . . . . 137
8.6 Precursors with membrane boundaries . . . . . . . . . . . . . . . . . . . . 137
8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9 Dependence of failure on particle roughness 143
9.1 Is failure an isostatic trans