Failure of granular assemblies

Abstract

This work investigates granular assemblies subjected to increasing external forces in the quasi-static limit. In this limit, the system’s evolution depends on static properties of the system, but is independent of the particles’ inertia. At the failure, which occurs at a certain value of the external forces, the particles’ motions increase quickly. In this thesis, the properties of granular systems during the weakening process and at the failure are investigated with the Discrete Element Method (DEM). Unlike continuum approaches, in the DEM each particle is represented by one element with certain properties. This method is combined with an analytical description and with numerical simulations of the granular systems. Although many results are obtained for systems with rigid boundaries in two dimensions, several aspects of granular behavior are investigated under different conditions: systems with rigid wall boundaries are compared to systems with membrane boundaries. Two and three dimensional systems with rigid wall boundaries show the dependence of weakening and failure on dimensionality. This enables one to separate the general granular behavior from individual trends. A comparison between two independently developed computer codes shows that the obtained results are a true physical property of our granular model, and are not due to a particular numerical procedure. The dependence of the results on the system size is a further important aspect. Since the size of a grain introduces a characteristic length, small systems and large systems behave differently. For some of the investigated properties, an average over many small systems is equal to the value of one large system, but this is not always the case. Thus the system size dependence of granular properties is nontrivial. This work investigates small systems with only 16 particles but also large systems with 16384 particles. Different aspects of granular behavior are investigated in different chapters. Chapter 1 introduces the reader to granular media, and Chap. 2 provides the theoretical framework for understanding the results in the following chapters. Chap. 3 considers the failure and triggers of failure in small systems with only 16 particles. This size is small enough to identify single particle contact changes and investigate their meaning for weakening and failure. The stability of the systems is calculated from the knowledge of the contact structure, i.e. from position and orientation of the contacts (geometric aspects) and their states (mechanical aspect). From this structure, the stiffness matrix is constructed, and a scalar stiffness is deduced. Changes in the stiffness can in this way be related to changes in the interparticle contact structure. A contact status change always appears at the beginning of the failure, decreasing the stiffness of the system. Thus, this contact status change is at the origin of failure. Through the change, the stiffness either becomes negative (instability), zero (null-mode motion), or very low. In the latter case the transition from a higher stiffness before the change to the low stiffness afterwards is rather slow. During the transition, a subsequent contact status change leads to the failure of the system. The results in Chap. 3 show that the stiffness matrix approach is successful in describing weakening and failure of granular systems. Thus Bagi’s definition of stability, which is based on the stiffness matrix description, is a useful concept for a more precise definition of jamming than the qualitative jamming diagram. In Chap. 4 the failure is investigated in more detail. Several publications show that there is a minimum number of interparticle contacts for which the system can stabilize the external forces at the boundaries. This limit, the so-called isostatic limit, defines the minimum number of contacts required for stability of a frictionless granular system. When friction comes into play, the counting of contacts becomes more complex: closed contacts contribute two stability conditions, and sliding contacts contribute one condition for stability. However, the results in Chap. 4 show that the failure of a granular system does not coincide with the minimum number of constraints required for stability, at least not based on the global comparison of constraints and degrees of freedom presented there. When a system fails, a shear band appears, where the particles’ kinetic energy is higher than elsewhere in the system. This immediately suggests evaluating the number of constraints in that region to be a promising route for refinement. Many aspects of the softening of granular systems on the way to the failure are explained in Chap. 5. The main result therein is that the softening of a granular assembly can be divided in two periods of different granular behavior. In the first period, many contacts become sliding, while in the second period the number of sliding contacts decreases. Also precursors of failure appear in the second period. They can be noticed as plunges in the number of sliding contacts, followed by a fast recovery. The evolution of sliding contacts can be fully understood in terms of contact status transitions. In the beginning of the simulation, most frequently a closed contact becomes a sliding contact. At the transition between the two periods, the number of sliding contacts becomes maximal. At this maximum, the inverse transition from sliding to closed is equally likely. Increasing the force further, another transition comes into play: sliding contacts disappear. This means the total number of contacts decreases. Finally, at the failure, all contact status transitions become equally likely. This behavior is a consequence of the global rearrangement involving a large number of grains. The spatial organization of sliding contacts also changes during the simulation. In the first period, sliding contacts tend to distribute evenly over the system, while they tend to cluster in the second period. This tendency increases until the failure, where the sliding contacts are finally concentrated in a diagonal band. Chapter 6 examines more closely the precursors mentioned in Chap. 5. These are localized instabilities, i.e. a negative local stiffness appears in the system. Consequently, the kinetic energy rises locally very quickly, and decreases again when, after a very short interval of time, stability is recovered. Due to the localization, the kinetic energy is much lower than at the failure. One can more easily identify precursors by the sudden plunge of the number of sliding contacts. This decrease is always temporary, and it can be understood in terms of contact status transitions: when a precursors appears, contacts become nonsliding in the region of the vibration wave, which is initiated by the precursor and radiated outward. When the vibration disappears, these contacts start to slide again. Changes in the contact structure are mainly transitional, therefore the importance of precursors for the internal structure is much smaller than the general trend, which depends on the external force. Chapter 7 examines closely the organization of motions, first observed in Chap. 5. While the system is stable, the two most important relative motions are the sliding of particles and the rolling of grains at their point of contact. This chapter shows that the latter motions do concentrate in bands appearing roughly at the angle pi/4 relative to the direction of increasing force. Such rolling bands were also observed by Kuhn and Bagi. The organization of rolling in bands increases as failure approaches. As the number of bands decreases, the distance between the rolling bands increases. Eventually only very few rolling bands remain, and one of them becomes the shear band which appears at the failure. Thus the rolling motions are concentrated within the shear band, contributing to the process of reorganization. The direction of the shear band found here is similar to the Mohr-Coulomb direction of failure. The latter is also obtained from the principle of least dissipation rate of Toeroek. Another important finding is the correlation of rolling and sliding motions. This correlation is stronger in the simulation beginning and at failure, and it is weaker at the maximum number of sliding contacts. Chapter 8 discusses the influence of the boundary conditions on the results presented in the foregoing chapters. Specifically, results obtained in systems with membrane boundaries are compared to results in systems with rigid wall boundaries. The comparison reveals that many details of the softening process and the failure are qualitatively independent of the boundary conditions. One example is the number of sliding contacts attaining a maximum before the failure, and disemboguing into a sharp maximum at the failure. The maximum always divides the softening process in two periods. On the contrary, the spatial organization of sliding contacts depends on the boundary conditions. Applying membrane boundary conditions leads to a constantly increasing clustering tendency with increasing force, which decreases again only at the failure of the system. When rigid walls boundaries are applied, sliding contacts tend to repel each other in the first period of the simulation, resulting in a more homogeneous distribution of sliding contacts. In the second period until the failure, sliding contacts show an increasing tendency to cluster. In both rigid walls and membrane conditions the sliding contacts concentrate at the failure in the shear band that develops. This suggests that the process of failure might be similar in both cases. A further important finding, explained in Chap. 9, is that the critical force at failure depends on the interparticle friction coefficient µ. When this coefficient is increased, the system fails at a higher value of the external force. A higher µ involves also a stronger decrease in the number of contacts M before failure, while the number of sliding contacts Ms involved in the weakening is lower. A very important finding is that while M and Ms vary considerably, the global hyperstatic number H observed at failure is almost independent of µ. This suggests that failure might be better characterized by µ-independent properties. Chapter 10 shows the behavior of granular systems in three dimensions. Applying similar boundary conditions as in two dimensions (rigid walls on every side of the system), important system characteristics can easily be compared. The comparison reveals that the approaching of failure in three dimensions is in some respects similar to two dimensions, but it has also some different features. One common observation is the initially linear increase in the number of sliding contacts ms. At the maximum ms, the behavior becomes different: ms decreases very quickly in three dimensions, while the decrease in two dimensions is smooth. The spatial organization of sliding contacts is also different: they always tend to cluster in three dimensions, while in two dimensions they only cluster in the second period before failure. Precursors are observed in both two and three dimensions. In both cases, their characteristics are similar, but their number is much higher in three dimensions. At least three main results can be distilled from this thesis. First, organization and clustering of sliding and rolling motions starts a long time before the failure is approached. Thus, in a macroscopic constitutive model, not only the number of sliding motions, but also their organization has to be taken into account. Second, precursors of the failure are always observed, independent of the boundary conditions and the dimensionality. In the literature, precursors of granular avalanches have been reported by several authors. The findings of this thesis show that the precursor phenomenon is more general and applies to rigid wall and membrane boundary conditions as well. Although from the limited number of examples it is impossible to draw a general conclusion, most probably precursors of failure occur for any kind of boundary conditions and for any load protocol. Third, the thesis provides new insight into the nature of failure: in small systems, failure is connected to the appearance of instability and an exponential rise in the kinetic energy. In large systems, vibrations, appearing at precursors, become larger close to failure. Thus the system explores an increasing part of phase space. These vibrations are physical, and not an artifact of the simulation: they appear in the same way after localized instabilities as they also do after the failure, when the system comes back to rest. These vibrations are important for failure in large systems. This importance might trigger future investigations of granular media. What are the perspectives for future work? First, the global hyperstatic number H, defined in Sec. 2.7 and investigated in Sec. 4.1, could be locally defined. The hyperstatic number compares the number of constraints with the number of degrees of freedom. The refinement to a local definition is promising because sliding and rolling motions organize in localized diagonal bands close to failure. Thus failure might be a local happening, and calculating H in these regions might give further insight to what precisely happens at failure. For example, the mystery of failure being approximately an isostatic transition (H = 0) in small systems, but hyperstatic (H > 0) in large systems, could probably be explained by this approach: the isostatic region might be localized. Second, the number of precursors could be counted for each type of boundary conditions, and the rise in this number until failure could be evaluated. Also the position of subsequent precursors could be related to each other. Are the positions correlated? This would be interesting to know. Note that for this purpose it would be advisable to control the boundary motions (i.e. the strain) instead of the external forces (i.e. the stress). This increases the resolution close to the failure, where most precursors occur. Last but not least, a few more words about the jamming diagram, introduced in Fig. 1.1 should be said, linking the results of this thesis with this diagram. Granular systems belong generally to the blue plane in the jamming diagram, defined by the density and the external force, which is the load. Considering the systems here, at least one more parameter must be introduced to fully pin the shaded jamming surface. This parameter is the interparticle friction coefficient µ. When it is reduced, the maximum supported load decreases. As the results in Chap. 9 show, the type of failure transition also changes qualitatively from µ = 0 to µ = infinity (See Fig. 9.1). For small µ, the hyperstatic number H, which reflects the number of stabilizing contact forces, evolves smoothly and continuously at the failure. For µ > 0.3, this number plunges suddenly at the failure. If the simulations are not quasi-static, the parameter load protocol must also be added to the list, because failure appears at a higher load when the external force is quickly increased. In the jamming diagram, the precise meaning of the axis load is also not clear. Most probably the authors mean a deviator stress which is applied to the system. This would be a good specification, because failure is most frequently identified with the point of maximum deviator stress. However, the maximum stress might depend on the dimensionality. Consequently the jamming surface in the diagram proposed by Liu and Nagel is only unique when a specific set of parameters is chosen. But then the same set of parameters must be applied to all systems (i.e. to grains, bubbles, glasses). Further investigation is needed to identify if the jamming diagram in the proposed form relates the granular failure to the glass transition in a favorable and unique way.

Ziel dieser Arbeit ist die Vertiefung des fundamentalen Verstaendnisses von Versagen granularer Anordnungen in zwei und drei Dimensionen. Ein Granulat besteht aus einer typischerweise großen Anzahl von Konstituenten, den Koernern. Aufgrund der Groesse der Koerner (ab ca. 10µm) hat die Temperatur keinen Einfluss auf deren Bewegungen. Die dissipativen Bewegungen werden daher alleine durch Newtonsche Kraefte gesteuert, die dem System von außen aufgepraegt werden. In dieser Arbeit werden sowohl kleine als auch große granulare Systeme untersucht, mit dem Ziel eine Laengenskala zu identifizieren, welche bis zum Versagen so stark anwaechst, dass sie der Systemgroesse entspricht. Versagen ist eine ploetzliche Anderung der Systemabmessungen. Methodisches Kernstueck der Untersuchung ist die Steifigkeitsmatrix, welche aus der Information ueber alle Teilchenkontakte des Systems aufgestellt wird. Diese Matrix bestimmt das dynamische Verhalten eines granularen Systems unter langsam veraenderter Belastung. In einem ersten Schritt werden kleine granulare Systeme in zwei Dimensionen untersucht. In diesen Systemen mit nur 16 Teilchen wurden mit Hilfe der Steifigkeitsmatrix die Ausloeser von Versagen identifiziert und deren relative Haeufigkeit ausgewertet. Wenn Versagen auftritt, geht stets eine Kontaktstatusaenderung voraus, welche eine Aenderung der Systemstabilitaet bewirkt. Es sind drei Folgen moeglich. Erstens kann die Aenderung eine Systeminstabilitaet herbeifuehren, und die kinetische Energie steigt exponentiell. Zweitens kann eine Bewegung mit Stabilitaet Null auftreten. Dieser Mechanismus wird allerdings nur in kleinen Systemen beobachtet. Drittens kann das System so weich bzw. schwach werden, dass schon eine kleine Zunahme der aeusseren Belastung eine hohe kinetische Energie, verbunden mit weiteren Kontaktaenderungen, herbeifuehrt. Auch in diesem Fall versagt das System. Dieser letztgenannte Mechanismus herrscht in großen System mit mehreren zehntausend Teilchen vor. Im Bezug auf das Versagen treten weitere Fragen auf, die bislang nicht hinreichend geklaert wurden. Eine dieser Fragen betrifft die Anzahl der noetigen Stabilitaetsbedingungen. Tritt Versagen genau dann auf, wenn bei Verlust eines einzelnen, beliebig waehlbaren Kontaktes keine Stabilitaet mehr gewaehrleistet werden kann? Diese Untersuchung zeigt, dass dies im Allgemeinen nicht so ist. Das Resultat naehrt den Verdacht, dass Versagen auf einen Teilbereich des Systems beschraenkt ist. Und in der Tat findet man, dass sich sowohl Gleit- als auch Rollbewegungen in einem diagonal verlaufenden Band bei Versagen konzentrieren. Diese Konzentration wird ebenfalls in der vorliegenden Arbeit quantitativ erfasst und charakterisiert. Ein weiteres bemerkenswertes Resultat ist, dass das Versagen im Mittel ueber viele kleine Systeme bei demselben Wert der aeusseren Kraefte auftritt wie in einem großen System. Dies ist ein Beleg dafuer, dass es sinnvoll ist Versagen zunaechst in kleinen Systemen zu charakterisieren, weil sie sich in mancher Hinsicht den großen Systemen aehnlich verhalten, jedoch ihr Verhalten deutlich leichter zu beschreiben und zu verstehen ist. In großen Systemen mit zehntausenden Teilchen fuehren einzelne Kontakt aenderungen nur zu sehr kleinen Aenderungen in der Systemstabilitaet. Auch andere Eigenschaften aendern sich fast kontinuierlich. Die Anzahl an Kontakten, an denen Teilchen aneinander entlanggleiten koennen, nimmt linear mit der Belastung zu bis zu einem Maximalwert. Anschliessend gibt es gleich haeufig Uebergaenge in beide Richtungen (gleitend <-> geschlossen), sodass keine neuen Gleitkontakte entstehen. Jedoch oeffnen sich immer mehr Gleitkontakte, d.h. die Gesamtanzahl der Kontakte sinkt bis zum Versagen ab, wodurch das System weiter geschwaecht wird. Die raeumliche Verteilung der Gleitkontakte ist zu Beginn zufaellig, ein zunehmendes Ordnungsverhalten zeichnet sich jedoch fruehzeitig ab. Nahe dem Versagen organisieren sich Gleitkontakte schließlich in diagonal verlaufenden Baendern. Eines dieser Baender, das sogenannte Scherband, ist bei Versagen besonders deutlich ausgepraegt. Waehrend der Zunahme der Ordnung von Gleitkontakten kommt es manchmal zu ploetzlichen Einbruechen in deren Anzahl. Diese Einbrueche werden mit Vorlaeufern von Versagen identifiziert. Es handelt sich hierbei um lokalisierte Instabilitaeten im System, die zum raeumlich begrenzen Anstieg der kinetischen Energie fuehren. Diese Energie wird, da der Vorlaeufer nur von sehr kurzer Dauer ist, rasch in Form einer Vibrationswelle in umliegende Bereiche dissipiert. Eng mit der Vibrationswelle verbunden ist der Einbruch in der Anzahl der Gleitkontakte im Bereich hoher Energie. Dieser Einbruch entsteht dadurch, dass Kontakte schliessen, d.h. nichtlgleitend werden, und kurz darauf wieder in ihren Ursprungszustand gleitend zurueckkehren. Signifikante bleibende ,,Schaeden”, d.h. Veraenderungen in der Struktur des Granulates, die nach dem Vorlaeufer zurueckbleiben, werden nicht beobachtet. Neben den Gleitkontakten ist das Abrollen von Teilchen an ihren Kontaktstellen ein weiterer Mechanismus, der zur Zunahme der kinetischen Energie fuehrt. Es kann gezeigt werden, dass diese Rollbewegungen raeumlich korreliert sind. Sie organisieren sich, aehnlich wie die Gleitkontakte, in diagonaler Richtung. Diese Organisation startet bereits zu Simulationsbeginn und nimmt kontinuierlich zu. Das Organisationsverhalten unterscheidet sich vom demjenigen der Gleitkontakte, die – zumindest fuer die einfachsten untersuchten Randbedingungen vom Typ zwei- bzw. dreiachsiger Test – nur bei Annaeherung an das Versagen zunehmende Organisation zeigen. Fuer die Organisation der Rollkontakte wurde eine zunehmende Korrelationslaenge bestimmt. Diese wird durch Approximation der Verteilung durch eine endliche Fourierreihe gewonnen. Aus der Approximation werden die Anzahl der Maxima bestimmt. Es zeigt sich, dass die Anzahl der Maxima ab einer bestimmten Anzahl an Fourier-Koeffizienten konvergiert. Diese Anzahl nimmt vom Simulationsbeginn bis zum Versagen um eine Groessenordnung ab (bei Systemen mit 16384 Teilchen). Dies zeigt, dass die Korrelationslaenge im selben Maße zunimmt. Ferner laesst sich zeigen, dass Gleitkontakte und die Bereiche hoher Rollgeschwindigkeiten korreliert sind. Diese Korrelation ist besonders hoch kurz nach Simulationsbeginn und bei Versagen, wo der Korrelationskoeffizient einen Wert von etwa 0.5 annimmt. Bei Untersuchung von Systemen mit andersartigen Randbedingungen, naemlich membranartigen seitlichen Begrenzungen, zeigt sich, dass viele Eigenschaften der Materialschwaechung und des Versagens von den gewaehlten Randbedingungen unabhaengig sind. Die auffaelligste Abhaengigkeit zeigt sich bei der Organisation der Gleitkontakte: Bei membranartigen Raendern nimmt die Konzentration der Gleitkontakte in bestimmten Bereichen schon von Beginn an zu, wohingegen bei starren Waenden zunaechst eine gleichmaessigere Verteilung beobachtet wird. Der letzte Schritt dieser Untersuchung von Versagen ist die Erweiterung der Ergebnisse auf dreidimensionale Systeme und der Vergleich dieser Ergebnisse mit einem weiteren Simulationscode. Dieser Vergleich ist noetig, um moegliche Simulationsartefakte, die keineswegs von vornherein vollstaendig auszuschließen sind, als Ursache bestimmter Ergebnisse auszugrenzen. In dieser Richtung ist das erste Resultat die Beobachtung, dass Gleitkontakte auch in drei Dimensionen maßgeblich an der Materialschwaechung und dem Versagen beteiligt sind. Die Entwicklung ihrer Anzahl ist aehnlich der Evolution in zwei Dimensionen. Das zweite, noch wichtigere Resultat, ist die Beobachtung des Auftretens von Vorlaeufern von Versagen in drei Dimensionen, die auf gleiche Weise wie in zwei Dimensionen zu charakterisieren sind. Dieses Resultat stuetzt die These, dass das Versagen stets durch Voraeufer von Versagen angekuendigt wird. Die Vorlaeufer von Versagen wurden auch bei der vergleichenden Simulation mit einem weiteren Simulationscode beobachtet, sodass Simulationsartefakte als deren Ursache ausgeschlossen werden koennen. Die vorliegende Untersuchung beinhaltet eine Reihe von Resultaten, die das fundamentale Verstaendnis von Versagen granular Andordnungen erweitern. Dieses Wissen ermoeglicht es, die Verbindung zwischen granularem Versagen und dem Glasuebergang, welche zuerst von Liu und Nagel im sogenannten Hemmungsdiagramm aufgestellt wurde, besser zu charakterisieren und zu verstehen.