Analytical and numerical modelling of quaking in tall silos

Although seeming like very simple systems, the flow of granular materials from silos can often be observed to display large, self induced, dynamic pulsations that are known as quakes. These quakes usually take the form of distinct events that occur very rapidly (of the order of milli-seconds), with several seconds (possibly minutes) between each distinct quake. This thesis presents an examination of silo quaking mechanisms from the perspective of its origins in the interaction of the mechanics of the granular material and slip-stick processes. Although the mechanisms for several different types of quaking are given, this study focuses on one particular type of quaking (arguably the most common) - that of a tall mass flow silo. Quaking in tall silos is first examined experimentally using laboratory scale test rigs in which the acceleration of the granular material during quaking was measured. These measurements clearly showed the presence of rarefaction and compression waves as well as many other features of the quakes. Quaking within these tall silos was shown to be governed by the transmission of these slip and stick waves that were found to grow exponentially as they moved up the material bed. A one-dimensional simplification of a hypoplastic constitutive model was developed and used to model the motion of granular materials in the upper section of tall silos. This dynamic version of Janssen's equation was used to understand the behaviour of the rarefaction and compression waves such as those present during quaking, with several interesting results; An asymptotic simplification of the model during loading was found to show both dissipation and dispersion (separately) due to friction, as well as the tendency for compression waves to form shocks. The Riemann invariants for the governing equations of the system were used to define a Hugonoit for the shock waves and this was used as a tool to relate the pressure and velocity fluctuations experienced during a quake. The system in its full complexity was studied numerically and was shown to display the same exponential growth of the disturbances as observed in the experimental work. The correlation between the rate of growth of the numerical solutions and experimental measurements was remarkable, and through this the role of wall friction in the development of severe quaking was laid bare. The significance of these results in the understanding of the quaking phenomenon was discussed.