Wave Propagation in Granular Materials Thesis by Stephen R. Hostler In Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2005 (Defended October 12, 2004) ii °c 2005 Stephen R. Hostler All Rights Reserved iii Acknowledgements I am particularly grateful to my advisor, Prof. Christopher Brennen, and Prof. Melany Hunt. My growth as a researcher and teacher is direct result of their mentorship and support. As a member of the Caltech Granular Flows Group, I had the opportunity to take part in a couple fairly adventurous research projects, namely ¯eld research at the Booming Dunes of Death Valley and the Mojave Desert and weightless research aboard the KC-135 (\vomit comet"). These experiences gave me invaluable research experience and were also quite a bit of fun. I thank Profs. Brennen and Hunt for giving me the chance to be a part of these projects. I would also like to thank the other members of my thesis committee, Prof. Tim Colonius and Prof. John Brady, for their valuable comments on my thesis. I would like to thank those who helped with the setup of the experiments. In particular, I would like to thank Rodney Rojas for help in the machine shop and Raul Relles for help in the vibrations lab. My time at Caltech has also been enriched by all of the other people I have known. I am grateful for discussions with Jim Cory, Gustavo Joseph, Kristo Kriechbaum, Matt Muto, and Steve Alves who were always willing to listen to my (often obscure) questions. More socially, I have had great time as a member of SOPS, the Aero soccer team, and the Thomas Lunch group. Most of all I would like to thank my wife, Rochelle. First of all, I appreciate her willingness to move across the country. Secondly, I appreciate her love and support throughout my studies. I owe much of my success and the retention of the majority of my sanity to her support. iv Abstract Wave propagation is a fundamental property of all physical systems. The wave speed is directly related to the compressibility of the system and determines the rate at which local disturbances are propagated into the bulk of the material. The wave propagation characteristics of conventional forms of matter are well understood and well documented. In contrast, waves in granular materials are more complex due to the heterogeneous nature of these systems. The key element of the mechanics of a granular system is the force chain. It is along these preferentially stressed chains of particles that waves are transmitted. These nonlinear chains are heavily dependent on the geometry of the bed and are prone to rearrangement even by the slightest of forces. Results from both experiments and simulations on wave propagation in granular materials are presented in the current study. The experiments measure the pressures at two points within the granular bed that result from the motion of a piston at one end of the bed. The simulations are a two-dimensional version of the experiments and use a discrete, soft-particle method to detect the wave at both the output of the simulated bed and at any point within it. In addition to examining wave propagation in a granular bed at rest, simulations and experiments are also performed for a granular bed undergoing agitation perpendicular to the direction of the wave input. Imposed agitation increases the granular temperature of the bed and allows for the exploration of the e®ect of granular state changes on the wave propagation characteristics. Such information may provide a means to diagnose the state of a flowing granular material. Measurements of the wave speed and attenuation in the bed reveal the unique properties of waves in granular systems that result from the nonlinearity of the bed and the heterogeneity of the force chains. Sinusoidal waves demonstrate the nondispersive nature of a granular bed and show the transient e®ects of force chain rearrangement. Pulsed waves display a semi-permanent shape qualitatively similar to predictions from nonlinear wave theory. In an agitated granular bed, measurements of the wave characteristics were found to be possible even in the presence of signi¯cant agitation. The prevailing con¯ning pressure, which changes throughout the agitation cycle, was determined to be the system parameter that correlates best with changes to the wave speed. v Contents Acknowledgements iii Abstract iv 1 Introduction 1 1.1 Granular materials and granular flows .......................... 1 1.2 Waves in granular beds .................................. 2 1.2.1 Wave speed ..................................... 3 1.2.2 Dissipation ..................................... 5 1.2.3 Linear waves .................................... 6 1.2.3.1 Dispersion ................................ 6 1.2.4 Nonlinear waves .................................. 7 1.2.4.1 Strongly compressed chain ....................... 9 1.2.4.2 Weakly compressed chain ........................ 10 1.3 Thermodynamic analogy for granular flows ....................... 12 1.3.1 Granular solid ................................... 13 1.3.2 Granular liquid ................................... 13 1.3.3 Granular gas .................................... 14 1.3.4 Agitated granular beds ............................... 14 1.4 Simulation background ................................... 14 2 Experiments and Simulations 16 2.1 Experiments ......................................... 16 2.1.1 Continuous input .................................. 18 2.1.2 Pulsed input .................................... 21 2.2 Simulations ......................................... 21 2.2.1 Contact model ................................... 23 2.2.2 Dimensionless parameters ............................. 26 2.2.3 Preliminary results ................................. 28 vi 2.2.3.1 Settling .................................. 28 2.2.3.2 Pressure scaling with depth ....................... 29 3 Experiments with continuous input 31 3.1 Constant acceleration experiments ............................ 31 3.2 Constant frequency experiments ............................. 37 3.3 Summary .......................................... 40 4 Experiments with pulsed input 41 4.1 General characteristics ................................... 41 4.1.1 Wave shape ..................................... 41 4.1.2 Pulse attenuation .................................. 42 4.2 Constant pulse amplitude experiments .......................... 43 4.2.1 Oscillations ..................................... 47 4.3 Constant pulse width experiments ............................ 50 4.4 Wave speed comparison .................................. 54 4.5 E®ect of consolidation ................................... 55 4.6 Summary .......................................... 59 5 Simulations with continuous input 61 5.1 General characteristics ................................... 61 5.2 Phase and phase speed ................................... 64 5.3 Wave amplitude and attenuation ............................. 65 5.4 Summary .......................................... 68 6 Simulations with pulsed input 69 6.1 General characteristics ................................... 69 6.2 Parameter sensitivity .................................... 74 6.2.1 Poisson's ratio, º .................................. 74 6.2.2 Dissipation factor, A¤ ............................... 76 6.2.3 Pulse width ..................................... 76 6.2.4 Pulse amplitude .................................. 79 6.3 Summary .......................................... 81 7 Waves in an agitated granular bed 83 7.1 Experiments with continuous input ............................ 83 7.2 Simulations with continuous input ............................ 88 7.3 Experiments with pulsed input .............................. 93 vii 7.4 Simulations with pulsed input ............................... 94 7.5 Summary .......................................... 96 8 Discussion 98 8.1 Summary .......................................... 98 8.1.1 Unagitated bed ................................... 98 8.1.2 Agitated bed .................................... 101 8.2 Wave characterization ................................... 102 8.3 Variation with agitation .................................. 103 Bibliography 107 A Additional parameter sensitivity plots 108 viii List of Figures 1.1 Force chains ......................................... 1 1.2 Normal elastic contact force ................................. 7 1.3 Idealized force chain ..................................... 8 1.4 Soliton shape ......................................... 10 1.5 Solitary wave shape ..................................... 11 1.6 Granular states ........................................ 13 2.1 Experimental schematic ................................... 16 2.2 Dynamic response of the piston ............................... 20 2.3 Simulation schematic .................................... 22 2.4 Pulsed piston motion in simulations ............................ 23 2.5 Contact model ........................................ 24 2.6 Concentration wave resulting from particle settling .................... 29 2.7 Pressure with depth ..................................... 30 3.1 Phase vs. frequency ..................................... 32 3.2 Phase speed .......................................... 33 3.3 Group velocity summary .................................. 34 3.4 Attenuation ratio ....................................... 35 3.5 Attenuation ratio summary ................................. 36 3.6 E®ect of consolidation on phase speed ..........................