MEK. Modelling Granular Media and Molecular Dynamics Simulations of Ellipses. K. Nikolaj Berntsen PhD thesis March 2005

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1 Technical University of Denmark Modelling Granular Media and Molecular Dynamics Simulations of Ellipses K. Nikolaj Berntsen PhD thesis March 2005 Department of Mechanical Engineering MEK Coastal, Maritime and Structural Engineering

3 Modelling granular media and molecular dynamics simulations of ellipses K. Nikolaj Berntsen TECHNICAL UNIVERSITY OF DENMARK DEPARTMENT OF MECHANICAL ENGINEERING MARITIME ENGINEERING MARCH 2005

4 Published in Denmark by Technical University of Denmark K. N. Berntsen 2005 All rights reserved Marilir~~e Engineerir~g- Department of Mechanical Engineering Technical University of Denmark Studentertorvet, Building 101E, DK-2800 Kgs. Lyngby, Denmark Phone , Telefax maritime. engineering0mek.dtu.dk WWW: Publication Reference Data Bcrntscn, K. N. Modelling granular media and molecular dynamics simulations of ellipses PhD Thesis Technical University of Denmark, Maritime Engineering. March, 2005 ISBN Keywords: Granular, ellipses, modelling, silo, simulation

5 This thesis is dedicated to my family my son Ask who did not see it begin my wife Marianne for encouraging me my father Aage for artistic inspiration my sister Tinanguak for autistic enthusiasm my mother Martina who did not see it finished.

7 Preface This delayed report is a slightly revised version of the thesis submitted May 30 and defended November 15, 2001, as partial fulfillment of the requirements for the Danish Ph.D. degree. The work has been carried out from 1996 to 2001 at the Department of Structural Engineering and Materials (BKM), Technical University of Denmark (DTU), and (after the reorganization of DTU) at the Section of Maritime Engineering at the Department of Mechanical Engineering. Professor Ove Ditlevsen has supervised the study. Part of the study took place at the James Franck Institute (JFI) at University of Chicago (U of C), under the guidance of Heinrich Jager. Another part of the study took place at the Technical University of Stuttgart, Institute for Computer Applications I, under the guidance of professor Hans H. Hermann. Hellerup, April 18th K. Nikolaj Berntsen, berntsen@efef.dk Acknowledgement: Thanks for all the jsh [Adams, No, sincerely. Thanks to Marianne Dissing for the sacrifices she has made, allowing me to write up this thesis. Thanks to Ove Ditlevsen for the freedom to follow my own ideas, as well as patient supervision. Thanks to Susan Coppersmith for the hospitality at the James Franck Institute at University of Chicago, to Heinrich Jaeger for accepting me and supervising me in his granular group at U of C. Thanks to Tom Witten and Leo Kadanof for illuminating talks, also at U of C, as well as the others there spending time on illuminating me. To Hans Hermann goes many thanks for his hospitality and interest in my work, allowing me to spend some months in his group at Technical University of Stuttgart, Institute for Computer Applications I. During my stay I had many good discussions, too many to list here, but Stephan Luding and Hans-Georg Mattutis deserve special mention. Thanks to Michael Bundgkd Krogh, Jclrgen Flemming Olsen, Christian Veje and Boyan Lazarov for critical comments on early versions of the manuscript.

8 iv Preface

9 Executive Summary Granular matter, powder, is a poorly understood subject of classical mechanics, an area with no general theory except Newtons's laws. In order to try to understand the subject better computer simulations have been used to access variables that are not easily accessible to experiments. Computer simulations of granular matter can be very expensive in computer time and up to now few have made simulations with non-round objects. For this thesis a molecular dynamics programme have been developed allowing the simulations of granular matter consisting of elliptical particles. The computer power available (in 1998) allowed for simulations of up to 3600 particles. The computer programme has been applied to silo filling and the data have been compared to a developed stochastic continuum model for the stress in a silo. The developed stochastic continuum model predicts the mean and variance of the stress and wall stresses in a silo in terms of the statistical properties (mean and variance) a stochastic Janssen constant and the friction distribution. Other geometries of granular configurations have also been studied, and preliminary numerical results have been obtained on the packing, coordination number, particle angle correlation, particle orientation, and contact point for particles filled into a square container. Other preliminary results are on the angle of repose and stresses inside and under a (small) heap. Recently the hypothesis of an exponential tail in the distribution of forces in granular matter have been put forward, and been confirmed by experiments and simulations of circles. We have found support for this hypothesis both from data analysis of previously obtained silo pressures from real silos as well as from the simulations of ellipses. In the appendices we provide background knowledge on geometry, molecular dynamics, statistics and continuum mechanics as well as some description of the developed programme.

10 vi Executive Summary

11 Synopsis Der findes endnu ingen samlet teori for granul~re medier (pulversystemer) udover Newtons ligninger. Et redskab i udviklingen af et generelt teoretisk apparat er computer simuleringer, hvor de enkelte partiklers bev~gelse simuleres. Ved sadanne simulationer kan variable tilgas, som ikke er let tilg~ngelige for eksperimentelle metoder, og herved kan ny inspiration og indsigt opnas. Computer simulationer af granul~rt stof kan vere meget beregningstunge og hidtil er kun fa studier af systemer af ikke-runde partikler gennemfort. I forbindelse med udarbejdelsen af denne afhandling, er et computer program ti1 simulation af granul~rt stof bestaende af ellipser, blevet skrevet. Beregningskraften, der var ti1 radighed i 1998, tillod simulationer af op ti partikler. Det udviklede stykke programmel er blevet anvendt ti1 simulationer af silofyldninger, og data herfra er blevet sammenlignet med en udviklet stokastisk kontinuummodel. Den udviklede model udtrykker middelv~rdi og varians for sp~ndinger i materialet og langs silov~ggene ved stokastiske egenskaber (middelv~rdi og varians) af en stokastisk Janssen konst ant og friktionsfordelingen. Andre geometriske konfigurationer end siloen er ogsa undersogt, og forelobige resultater er opnaet for pakningsgraden, koordinationstallet, partikel-orienterings-korrelationer og kontaktpunkterne for partikler fyldt i kvadratisk beholder. Andre forelobige resultater om hvileoverflade-vinklen (angle of repose) for en dynge granul~rt materiale samt og sp~ndinger i og under er ogsa fremkommet. Nyligt er fremkommet hypotesen om at kontaktkr~fterne i et granul~rt materiale har en eksponentiel hale i fordelingen; en hypotese som er blevet understottet af eksperimenter. Vi har fundet yderligere stotte for hypotesen bade ved analyse af silotryksdata og fra ellipsesimulationerne. Baggrundsinformation om geometri, molekyl~r dynamik simulationer, statistik og kontinuumsmekanik save1 som en overordnet beskrivelse af det udviklede programmel er at finde i appendicerne.

12 viii Synopsis

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154 Coastal, Maritime and Structural Engineering Department of Mechanical Engineering Technical University of Denmark Nils Koppels Allé, Building 403 DK-2800 Kgs. Lyngby Denmark Phone Fax info.mt@mek.dtu.dk ISBN

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