Computational Granular Mechanics and Its Engineering Applications (Springer Tracts in Mechanical Engineering) [1 ed.] 9811533032, 9789811533037

Table of contents :
Preface
Contents
1 Introduction
1.1 Engineering Demands of Granular Mechanics
1.2 Basic Physical and Mechanical Properties of Granular Materials
1.2.1 Friction Law
1.2.2 Grain Silo Effect
1.2.3 Extrusion and Shear Expansion of Granular Materials
1.2.4 The Flow State of Granular Materials
1.3 Computational Analysis Softwares for Computational Granular Mechanics
References
Fundamentals of Computational Granular Mechanics
2 Constructions of Irregular Shaped Particles in the DEM
2.1 Bonding and Clumping Models Based on Spherical Particles
2.1.1 Bonding Models Based on Spheres
2.1.2 Clumping Models Based on Spheres
2.2 Super-Quadric Particles
2.2.1 Super-Quadric Particles
2.2.2 Ellipsoidal Particles Based on Super-Quadric Equation
2.3 Polyhedral and Dilated Polyhedral Particles
2.3.1 Polyhedral Particles
2.3.2 Dilated Polyhedral Particles Based on Minkowski Sum
2.4 Advanced Constructions of Novel Irregular Shaped Particles
2.4.1 Random Star-Shaped Particles
2.4.2 B-Spline Function Models
2.4.3 Combined Geometric Element Method
2.4.4 Potential Particle Model
2.4.5 Poly-superellipsoid Model
2.5 Summary
References
3 Contact Force Models for Granular Materials
3.1 Visco-Elastic Contact Models of Spherical Particles
3.1.1 Linear Contact Model
3.1.2 Nonlinear Contact Model
3.2 Elastic-Plastic Contact Models of Spherical Particles
3.2.1 Normal Elastic-Plastic Contact Model
3.2.2 Tangential Elastic-Plastic Contact Model
3.3 Rolling Friction Models of Spherical Particles
3.3.1 Rolling Friction Law
3.3.2 Rolling Friction Model of Spherical Particles
3.4 Bonding-Breakage Models of Spherical Particles
3.5 Contact Models of Non-spherical Particles
3.5.1 Contact Model Between Super-Quadric Particles
3.5.2 Contact Model of Dilated Polyhedral Particles
3.6 Non-contact Physical Interactions Between Particles
3.6.1 Adhesion Force Between Spherical Particles
3.6.2 Liquid Bridge Force Between Wet Particles
3.6.3 Heat Conduction Between Particles
3.7 Summary
References
4 Macro-Meso Analysis of Stress and Strain Fields of Granular Materials
4.1 Computational Homogenization Method Based on Mean Field Theory
4.1.1 Variational Representation of Frictional Contact Problems
4.1.2 Macro-Meso Two Scale Boundary Value Problems
4.1.3 Macro-Meso Scale Solution Procedures Based on Mean Field Theory
4.2 Meso Analysis of Stress Field of Granular Materials
4.2.1 Average Stress Description of the Micro Topological Structure
4.2.2 Stress Characterization of Particle Aggregates
4.2.3 Description of Macro Stress Based on Virtual Work
4.2.4 The Average Stress of the RVE in a Cosserat Continuum
4.3 Meso Analysis of Strain Field of Granular Materials
4.3.1 Definition of Strain by Bagi
4.3.2 Definition of Strain by Kruyt-Rothenburg
4.3.3 Definition of Strain by Kuhn
4.3.4 Definition of Optimal Fitting Strain by Cundall
4.3.5 Definition of Optimal Fitting Strain by Liao et al.
4.3.6 Definition of Optimal Fitting Strain by Cambou et al.
4.3.7 Definition of Volumetric Strain by Li et al.
4.4 Summary
References
5 Coupled DEM-FEM Analysis of Granular Materials
5.1 Combined DEM-FEM Method for the Transition from Continuum to Granular Materials
5.1.1 Contact Algorithm
5.1.2 Deformation of Element
5.1.3 Failure Model of Materials
5.2 Coupled DEM-FEM Model for the Continua-Discontinua Bridging Domain
5.2.1 Weak Form of Governing Equations for the Bridging Domain
5.2.2 Coupling Interface Force
5.2.3 Coupling Point Search
5.3 Coupled DEM-FEM Method for the Interaction Between Continua and Discontinua
5.3.1 Global Search Detection of Particle-Structure Contacts
5.3.2 Local Search Detection of Particle-Structure Contacts
5.3.3 Transfer of Contact Forces
5.4 Summary
References
6 Fluid-Solid Coupling Analysis of Granular Materials
6.1 DEM-CFD Coupling Method for Granular Materials and Fluid
6.1.1 Basic Governing Equations of Particles
6.1.2 DEM-CFD Coupling Solution Method
6.1.3 Governing Equations of Fluid Domain
6.1.4 Momentum Exchange Between Fluid and Solid Particles
6.1.5 Fluid Volume Fraction
6.1.6 Convection Heat Transfer Term
6.2 DEM-SPH Coupling Method for Granular Materials and Fluid
6.2.1 Integral Representation of Function and Particle Approximation in SPH
6.2.2 SPH Form for Navier-Stokes Equations
6.2.3 The EISPH Method for Incompressible Fluid
6.2.4 DEM-SPH Coupling Model
6.3 DEM-LBM Coupling Method for Granular Materials and Fluid
6.3.1 Lattice Boltzmann Method
6.3.2 DEM-LBM Coupling Immersion Boundary Method
6.3.3 Application of the DEM-LBM Coupling Method
6.4 Summary
References
7 High Performance Algorithm and Computing Analysis Software of DEM Based on GPU Parallel Algorithm
7.1 Developments of Computing and Analysis Software of DEM
7.1.1 GPU Parallel Technology
7.1.2 Development of Discrete Element Computing and Analysis Software
7.2 Numerical Algorithm of DEM Based on CUDA Programming
7.2.1 Hardware and Software Architecture of CUDA
7.2.2 Discrete Element Algorithm in Multi-machine/Multi-GPU Environment
7.3 High Performance Contact Detection Based on GPU
7.3.1 Relationship Between Grid and Particles
7.3.2 Establishment of Neighbor List
7.3.3 Calculation of Contact Force Sequence
7.4 DEM Computing Analysis Software Based on GPU Parallel Algorithm
7.5 Summary
References
Engineering Applications of Computational Granular Mechanics
8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull
8.1 DEM for Sea Ice and Determination of Computational Parameters
8.1.1 Discrete Element Construction of Sea Ice
8.1.2 Discrete Element Analysis of Sea Ice Compression and Flexural Strength
8.1.3 Main Computation Parameters in Sea Ice DEM Simulations
8.1.4 Failure Criteria Between Bonded Particles
8.1.5 DEM Simulations of Sea Ice Strength Influenced by Particle Size
8.2 DEM Analysis for the Interaction Between Sea Ice and Fixed Offshore Platform
8.2.1 DEM Analysis of Ice Load on Cylindrical Leg Structure
8.2.2 DEM Analysis of Ice Load on Conical Offshore Platform Structure
8.2.3 Shadowing Effect of Multi-leg Conical Jacket Platform Structure
8.2.4 DEM Analysis of Ice Load on Jack-Up Offshore Platform Structure
8.3 DEM Analysis for the Interaction Between Sea Ice and Floating Platform and Ship Hull
8.3.1 Analysis of Ice Load on Floating Offshore Platforms
8.3.2 DEM Simulation of the Ship Sailing in the Ice Zone
8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration of Marine Structure
8.4.1 DEM-FEM Coupling Method for Ice-Induced Vibration of Jacket Platform
8.4.2 DEM-FEM Method for Ice-Induced Vibration of Jacket Platform
8.5 Simulation of Ice Load on Marine Structure with Dilated Particles
8.5.1 Interaction Between Broken Ice and Pile Legs, Floating Structures
8.5.2 Interaction Between Level Ice and Ship Hull Structure
8.6 DEM Analysis for Characteristics of Sea Ice Pile-Up on Water Intake of Nuclear Power Plant
8.6.1 Numerical Simulation of Sea Ice Pile-Up at the Water Intake
8.6.2 Analysis of Influencing Factors of Sea Ice Pile-Up Characteristics
8.7 Coarse-Grained DEM Model for Sea Ice Dynamics
8.7.1 The Coarse-Grained DEM of Sea Ice
8.7.2 Numerical Simulation of Sea Ice Dynamic Processes in the Regular Domain
8.7.3 Numerical Simulation of the Dynamic Process of the Bohai Sea Ice
8.8 Summary
References
9 DEM Analysis of Mechanical Behaviors of Railway Ballast
9.1 DEM Analysis of Ballast Particle Breakage
9.1.1 DEM Analysis of Ballast Breakage
9.1.2 The Experimental Verification of Ballast Particle Breakage
9.2 DEM Analysis of Dynamic Response of Railway Ballast
9.2.1 Construction of Irregularly Shaped Ballast
9.2.2 DEM Modelling of Ballast Box Test
9.2.3 Effect of Loading Frequency on Accumulated Settlement of Full-Scale Ballast Bed
9.3 Direct Shear Tests of Fouled Ballast
9.3.1 Direct Shear Tests of Fouled Ballast
9.3.2 Effect of Fine Grains on Shear Strength
9.3.3 Effect of Fine Grains on Dilatancy Behavior of Ballast
9.4 DEM Analysis of Shear Strength of Sand-Fouled Ballast
9.4.1 DEM Simulations of Direct Shear of Sand-Fouled Ballast
9.4.2 Shear Strength and Force Chain Analysis with Different Ballast Contents
9.5 Coupled DEM-FEM Analysis of the Transition Zone Between Ballasted-Ballastless Track
9.5.1 DEM-FEM Algorithm in Railway Ballasted Track
9.5.2 DEM-FEM Model for Ballasted-Ballastless Transition Zone
9.5.3 Settlement Analysis of the Transition Zone
9.5.4 Coupled DEM-FEM Analysis Considering Ballast Embedded in Ballastless Bed
9.6 DEM Simulations of Railway Ballast Bed with Dilated Polyhedral Particles
9.6.1 Ballast Box Test Model with Dilated Polyhedral Particles
9.6.2 Settlement of Ballast Bed Under Various Cyclic Loading Frequencies
9.7 Summary
References
10 DEM Analysis of Vibration Reduction and Buffering Capacity of Granular Materials
10.1 DEM Simulations and Experimental Tests of Vibration Reduction of Granular Materials
10.1.1 Experimental Studies on Particle Damper
10.1.2 Numerical Simulation of Particle Damper
10.2 DEM Analysis of Buffering Capacity of Granular Materials
10.2.1 Experimental Studies on Buffering Capacity of Granular Materials
10.2.2 Numerical Simulations of Buffering Capacity of Granular Materials
10.3 DEM Analysis of Landing Process of Lunar Lander
10.3.1 Discrete Element Model of Landing Buffering System and Lunar Soil
10.3.2 Discrete Element Analysis of Landing Process and Impact Characteristics
10.4 Summary
References

Citation preview

Springer Tracts in Mechanical Engineering

Shunying Ji Lu Liu

Computational Granular Mechanics and Its Engineering Applications

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA

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Shunying Ji Lu Liu •

Computational Granular Mechanics and Its Engineering Applications

123

Shunying Ji Department of Engineering Mechanics Dalian University of Technology Dalian, China

Lu Liu Department of Engineering Mechanics Dalian University of Technology Dalian, China

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-981-15-3303-7 ISBN 978-981-15-3304-4 (eBook) https://doi.org/10.1007/978-981-15-3304-4 Jointly published with Science Press The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Science Press. ISBN of the Co-Publisher’s edition: 978-7-03-064550-0 © Science Press and Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Granular matter exists widely in natural environments, industrial production, daily life, and biological science. Great attention has been attracted because of its peculiar mechanical phenomena in the motion evolution and even in a static state. However, its being a scientific issue, investigated systematically at different scales and from different perspectives, in recent decades. The granular mechanics in research areas of landslide debris flow, desertification, mining, blast shock, agricultural sowing, pharmaceutical process, 3D printing, etc. were investigated in last couple decades. The research scope of granular materials can range from as large as geophysics, even celestial planets, and as small as dust, even molecules. Currently, much more emphasis is placed on problems at the engineering structure scale. Exploring and establishing the basic mechanics and mathematical models of the motion law and mechanical properties of granular materials at different scales is the research task of granular mechanics. Adopting numerical methods to carry out research on the basic mechanical properties of granular materials is inevitable with the rapid development of computer science and technology, especially large-scale and high-performance parallel computing technology. Consequently, the research direction of computational granular mechanics gradually forms. A wide variety of numerical methods are involved in the computational granular mechanics. Among those, the most representative one is the discrete element method (DEM). During the development of this method, the objects of study have been extended from granular media to continua, the study scales have been extended from the single engineering scale to multiple macro- and micro-scale, and materials under investigation have been extended from single medium to multi-media and multi-physical fields. The DEM has solved granular mechanics problems in different fields and faces many challenges as well. Cundall and Strack established the DEM. The paper published in Geotechnique in 1979 has become a milestone in the development of the DEM. The research field of this method has been gradually extended from original rock and soil mechanics to chemical engineering, mechanical processing, transportation, construction, mining, natural disasters, etc. In this process, the shapes of particles have been developed from simple two-dimensional discs to three-dimensional spheres and vii

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some other irregularly shaped particles, and research direction has extended from dry particles to fluid-solid coupling and multi-scale computations. Great changes have taken place in terms of particles constructions, contact models, and computation scale. At the same time, English monographs on DEMs have been published. Contact Mechanics published by Johnson in 1984 is a classic work on the discrete element contact calculations. In recent years, monographs on granular materials and DEMs have emerged and integrated with the corresponding engineering applications closely. However, a comprehensive and systematic introduction to the computational granular mechanics or DEMs has rarely been published. This reflects to a certain extent that computational granular mechanics, as an emerging discipline, is still developing, although it has drove to maturity in recent years. China’s research on granular mechanics began with engineering requirements in geotechnical mechanics, starting in the 1980s. The earliest work on the DEM was the Discrete Element Method and Its Application in Geotechnical Mechanics published by Yongjia Wang and Jibo Xing of Northeastern University in 1991. It focuses on the applications of two-dimensional block elements to solve geotechnical problems such as slope stability and tunnel support. Qun Wei published the Numerical Methods and Procedures for the Fundamental Principles of the Discrete Element Method in 1991, focusing on the applications of two-dimensional block elements and disc elements to solve slope stability and anchor support problem, with providing corresponding calculation procedures. The introduction to Particle Mechanics published by Qicheng Sun and Guangqian Wang of Tsinghua University in 2009 is the first in China to introduce the mechanics of granular materials and related calculation methods. It provides detailed reference for graduate students and technicians in granular mechanics research. In recent years, the incorporation of large-scale and high-performance computing in the DEM has become an effective way to solve related particle engineering problems. The State Key Laboratory of Multiphase Complex Systems of the Institute of Process Engineering, Chinese Academy of Sciences published the GPU-based multi-scale discrete analog parallel computing in 2009 based on its work on GPU parallel computing of granular materials, systematically introducing the CUDA implementation of the GPU high-performance algorithm for granular systems. In recent years, with the applications of computational granular mechanics in different engineering fields such as geotechnical mechanics, road engineering, tunnel engineering, silo materials, cutting processing, and combustion processes, some professional works in corresponding fields have also been published. In addition, with explorations and applications of commercial DEM software such as PFC, DEC, and EDEM, Chinese scholars have also published some reference books for software applications. This has effectively promoted the further development of computational granular mechanics in engineering applications. However, we also notice that graduate students and engineers who have just come into contact with the DEM need to learn the basic theories and numerical calculation methods systematically and get some inspiration from some engineering applications. To this end, this book introduces the basic principles and numerical algorithms in the

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calculations in detail and also explains the relevant engineering applications that the authors are engaged in. This book can be divided into 10 chapters. The first chapter is introduction, which mainly introduces the basic methods, engineering applications, and the current situation of computational analysis software. According to the basic theories of computational granular mechanics, Chaps. 2 through 7 introduce the basics of current computational granular mechanics systematically from aspects of particle shapes, contact models, macro- and micro-analysis, multi-scale analysis, fluid-structure interaction, and analysis software. Chapters 8 through 10 mainly introduce the authors’ work of the DEM in polar ocean engineering, ballasted railway track, particle damper, and buffering capacity characteristics of granular material in recent years. The book aims to provide reference for relevant researchers in solving practical engineering problems in different fields. Contents of this book are mainly based on research achievements in recent years on the DEM and its engineering applications from State Key Laboratory of Structural Analysis for Industrial Equipment in Dalian University of Technology. Meanwhile, the book also cited a large number of related research results at home and abroad in order to introduce the international developments on computational granular mechanics and engineering applications more comprehensively and systematically. The completion of this book also benefits from useful suggestions and works of the cooperators including Ying Yan (Dalian Jiaotong University), Jian Li and Hao Zhang (Northeastern University), Zhengguo Gao (Beihang University), Qicheng Sun (Tsinghua University), Xihua Chu (Wuhan University), Yuanqiang Tan (Huaqiao University), and Yuxin Wang (Dalian University of Technology). Jian Li wrote 3.3 and 10.3 sections of this book. Zhengguo Gao and Hao Zhang wrote 3.4 and 6.3 sections, respectively. Shuai Shao from Inner Mongolia University wrote Chap. 4 with revision and guidance of Xihua Chu from Wuhan University. This book condenses many of the research results of the above scholars and other experts. I am particularly grateful to Professor Hayley Shen and Professor Hung Tao Shen of Clarkson University, USA. During the visit of Clarkson University in 2002, I began to work on the mechanics of granular materials under the guidance of the above two professors, and focused on the computations and analyses using the DEM. In December 2010, I visited ABS in Houston with Professor Hayley Shen and began the development of discrete element software for ice loads on polar ships and offshore engineering. I would like to express my great appreciations to Shewen Liu, Han Yu, Jiancheng Liu, Xiang Liu, Yingying Chen, Wei Xia, Hai Gu, and other experts of American Bureau of Shipping (ABS) for their guidance and assistance. I also appreciate the encouragement of Professors Gengdong Cheng, Xikui Li Li, and Qianjin Yue of Dalian University of Technology to my studies on the DEM and its engineering applications. The research in this book benefits from the following projects. The Major State Basic Research Development Program of China (973 Program) Research on the prediction theory and numerical analysis method of major engineering geological hazards (2010CB731500), which is under the responsibility of Professor Shihai Li,

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Institute of Mechanics of Chinese Academy of Sciences; The Key Program of National Natural Science Foundation of China, Research on Mechanical Behavior, Deterioration Mechanism and Deformation Law of High-speed Railway Bulk Track Bed (U1234209), which is under the responsibility of Professor Wanming Zhai, Southwest Jiaotong University; The Key Program of National Natural Science Foundation of China, Study on Ice-Water-Ship Coupling dynamics (51639004), which is under the responsibility of Professor Duanfeng Han, Harbin Engineering University; The National Key Research and Development Program of China, Development of sea ice model with high precision (2018YFA0605900), which is under the responsibility of Professor Jiping Liu, Institute of Atmospheric Physics of Chinese Academy of Sciences. Meanwhile, the studies included in this book are also supported by the National Marine Environmental Forecasting Center of the State Oceanic Administration (SOA), the Polar Research Institute of China and the National Marine Environmental Monitoring Center. The research achievements of this book have also been funded by the National Key Research and Development Program of China (2018YFA0605902, 2016YCF1401505, 2016YFC1402705, 2016YFC1402706), the National Natural Science Foundation of China (11872136, 11772085, 11672072, 41576179, 11572067, 50808027), the Special Fund of Marine Commonweal Industry (201105016, 201205007, 201505019), and the Fundamental Research Funds for the Central Universities (Grant Nos. DUT19GJ206 and DUT19ZD207). We also received support from the State Key Laboratory of Structural Analysis of Industrial Equipment at Dalian University of Technology. The research results of this book are also benefited from American Bureau of Shipping (ABS), China Classification Society (CCS), CNOOC Information Technology Co., Ltd., China State Shipbuilding Co., Ltd., China Shipbuilding Industry Group Co., Ltd., and China International Marine Containers (Group) Co., Ltd. They provided practical conditions for computational software development and engineering applications of the DEM. It also poses a great challenge to solve practical engineering problems based on computational granular mechanics. I would also like to pay special tribute to the research work of doctoral students Anliang Wang, Shaocheng Di, Shanshan Sun, Shuai Shao, Shuailin Wang, Xue Long, and Xiaodong Chen who graduated from the Group of Computational Granular Mechanics of Dalian University of Technology. The cooperation with Dalian Dizao Science & Technology Co., Ltd. on the discrete element software SDEM also enriches the achievement of this book. Some relevant results can be found on the website www.s-dem.com. Dalian, China December 2019

Shunying Ji

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Engineering Demands of Granular Mechanics . . . . . . . 1.2 Basic Physical and Mechanical Properties of Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Friction Law . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Grain Silo Effect . . . . . . . . . . . . . . . . . . . . . 1.2.3 Extrusion and Shear Expansion of Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The Flow State of Granular Materials . . . . . . 1.3 Computational Analysis Softwares for Computational Granular Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fundamentals of Computational Granular Mechanics

Constructions of Irregular Shaped Particles in the DEM . . . . 2.1 Bonding and Clumping Models Based on Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Bonding Models Based on Spheres . . . . . . . . . . . 2.1.2 Clumping Models Based on Spheres . . . . . . . . . . 2.2 Super-Quadric Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Super-Quadric Particles . . . . . . . . . . . . . . . . . . . . 2.2.2 Ellipsoidal Particles Based on Super-Quadric Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Polyhedral and Dilated Polyhedral Particles . . . . . . . . . . . 2.3.1 Polyhedral Particles . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Dilated Polyhedral Particles Based on Minkowski Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Advanced Constructions of Novel Irregular Shaped Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.1 Random Star-Shaped Particles . . . . . . . 2.4.2 B-Spline Function Models . . . . . . . . . 2.4.3 Combined Geometric Element Method 2.4.4 Potential Particle Model . . . . . . . . . . . 2.4.5 Poly-superellipsoid Model . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contact Force Models for Granular Materials . . . . . . . . . . . 3.1 Visco-Elastic Contact Models of Spherical Particles . . . . 3.1.1 Linear Contact Model . . . . . . . . . . . . . . . . . . . . 3.1.2 Nonlinear Contact Model . . . . . . . . . . . . . . . . . 3.2 Elastic-Plastic Contact Models of Spherical Particles . . . . 3.2.1 Normal Elastic-Plastic Contact Model . . . . . . . . 3.2.2 Tangential Elastic-Plastic Contact Model . . . . . . 3.3 Rolling Friction Models of Spherical Particles . . . . . . . . 3.3.1 Rolling Friction Law . . . . . . . . . . . . . . . . . . . . 3.3.2 Rolling Friction Model of Spherical Particles . . . 3.4 Bonding-Breakage Models of Spherical Particles . . . . . . 3.5 Contact Models of Non-spherical Particles . . . . . . . . . . . 3.5.1 Contact Model Between Super-Quadric Particles 3.5.2 Contact Model of Dilated Polyhedral Particles . . 3.6 Non-contact Physical Interactions Between Particles . . . . 3.6.1 Adhesion Force Between Spherical Particles . . . 3.6.2 Liquid Bridge Force Between Wet Particles . . . . 3.6.3 Heat Conduction Between Particles . . . . . . . . . . 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Macro-Meso Analysis of Stress and Strain Fields of Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Computational Homogenization Method Based on Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Variational Representation of Frictional Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Macro-Meso Two Scale Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Macro-Meso Scale Solution Procedures Based on Mean Field Theory . . . . . . . . . . . . . . . . . . . . 4.2 Meso Analysis of Stress Field of Granular Materials . . . . . 4.2.1 Average Stress Description of the Micro Topological Structure . . . . . . . . . . . . . . . . . . . . . 4.2.2 Stress Characterization of Particle Aggregates . . .

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4.2.3

Description of Macro Stress Based on Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 The Average Stress of the RVE in a Cosserat Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Meso Analysis of Strain Field of Granular Materials . . . . . 4.3.1 Definition of Strain by Bagi . . . . . . . . . . . . . . . . 4.3.2 Definition of Strain by Kruyt-Rothenburg . . . . . . 4.3.3 Definition of Strain by Kuhn . . . . . . . . . . . . . . . . 4.3.4 Definition of Optimal Fitting Strain by Cundall . . 4.3.5 Definition of Optimal Fitting Strain by Liao et al. 4.3.6 Definition of Optimal Fitting Strain by Cambou et al. . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Definition of Volumetric Strain by Li et al. . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

6

Coupled DEM-FEM Analysis of Granular Materials . . . . . . 5.1 Combined DEM-FEM Method for the Transition from Continuum to Granular Materials . . . . . . . . . . . . . . 5.1.1 Contact Algorithm . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Deformation of Element . . . . . . . . . . . . . . . . . . 5.1.3 Failure Model of Materials . . . . . . . . . . . . . . . . 5.2 Coupled DEM-FEM Model for the Continua-Discontinua Bridging Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Weak Form of Governing Equations for the Bridging Domain . . . . . . . . . . . . . . . . . . 5.2.2 Coupling Interface Force . . . . . . . . . . . . . . . . . 5.2.3 Coupling Point Search . . . . . . . . . . . . . . . . . . . 5.3 Coupled DEM-FEM Method for the Interaction Between Continua and Discontinua . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Global Search Detection of Particle-Structure Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Local Search Detection of Particle-Structure Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Transfer of Contact Forces . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid-Solid Coupling Analysis of Granular Materials . . . 6.1 DEM-CFD Coupling Method for Granular Materials and Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Basic Governing Equations of Particles . . . . 6.1.2 DEM-CFD Coupling Solution Method . . . . . 6.1.3 Governing Equations of Fluid Domain . . . .

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6.1.4

Momentum Exchange Between Fluid and Solid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Fluid Volume Fraction . . . . . . . . . . . . . . . . . . . 6.1.6 Convection Heat Transfer Term . . . . . . . . . . . . 6.2 DEM-SPH Coupling Method for Granular Materials and Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Integral Representation of Function and Particle Approximation in SPH . . . . . . . . . . . . . . . . . . . 6.2.2 SPH Form for Navier-Stokes Equations . . . . . . . 6.2.3 The EISPH Method for Incompressible Fluid . . . 6.2.4 DEM-SPH Coupling Model . . . . . . . . . . . . . . . 6.3 DEM-LBM Coupling Method for Granular Materials and Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Lattice Boltzmann Method . . . . . . . . . . . . . . . . 6.3.2 DEM-LBM Coupling Immersion Boundary Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Application of the DEM-LBM Coupling Method 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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High Performance Algorithm and Computing Analysis Software of DEM Based on GPU Parallel Algorithm . . . . . . . . . . . . . . . . . 7.1 Developments of Computing and Analysis Software of DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 GPU Parallel Technology . . . . . . . . . . . . . . . . . . . . 7.1.2 Development of Discrete Element Computing and Analysis Software . . . . . . . . . . . . . . . . . . . . . . 7.2 Numerical Algorithm of DEM Based on CUDA Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Hardware and Software Architecture of CUDA . . . . 7.2.2 Discrete Element Algorithm in Multi-machine/Multi-GPU Environment . . . . . . . . 7.3 High Performance Contact Detection Based on GPU . . . . . . 7.3.1 Relationship Between Grid and Particles . . . . . . . . . 7.3.2 Establishment of Neighbor List . . . . . . . . . . . . . . . . 7.3.3 Calculation of Contact Force Sequence . . . . . . . . . . 7.4 DEM Computing Analysis Software Based on GPU Parallel Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 8

xv

Engineering Applications of Computational Granular Mechanics

DEM Analysis of Ice Loads on Offshore Structures and Ship Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 DEM for Sea Ice and Determination of Computational Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Discrete Element Construction of Sea Ice . . . . . . 8.1.2 Discrete Element Analysis of Sea Ice Compression and Flexural Strength . . . . . . . . . . . 8.1.3 Main Computation Parameters in Sea Ice DEM Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Failure Criteria Between Bonded Particles . . . . . . 8.1.5 DEM Simulations of Sea Ice Strength Influenced by Particle Size . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 DEM Analysis for the Interaction Between Sea Ice and Fixed Offshore Platform . . . . . . . . . . . . . . . . . . . . . . 8.2.1 DEM Analysis of Ice Load on Cylindrical Leg Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 DEM Analysis of Ice Load on Conical Offshore Platform Structure . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Shadowing Effect of Multi-leg Conical Jacket Platform Structure . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 DEM Analysis of Ice Load on Jack-Up Offshore Platform Structure . . . . . . . . . . . . . . . . . . . . . . . 8.3 DEM Analysis for the Interaction Between Sea Ice and Floating Platform and Ship Hull . . . . . . . . . . . . . . . . 8.3.1 Analysis of Ice Load on Floating Offshore Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 DEM Simulation of the Ship Sailing in the Ice Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration of Marine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 DEM-FEM Coupling Method for Ice-Induced Vibration of Jacket Platform . . . . . . . . . . . . . . . . 8.4.2 DEM-FEM Method for Ice-Induced Vibration of Jacket Platform . . . . . . . . . . . . . . . . . . . . . . . 8.5 Simulation of Ice Load on Marine Structure with Dilated Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Interaction Between Broken Ice and Pile Legs, Floating Structures . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Interaction Between Level Ice and Ship Hull Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.6

DEM Analysis for Characteristics of Sea Ice Pile-Up on Water Intake of Nuclear Power Plant . . . . . . . . . . . . . . 8.6.1 Numerical Simulation of Sea Ice Pile-Up at the Water Intake . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Analysis of Influencing Factors of Sea Ice Pile-Up Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Coarse-Grained DEM Model for Sea Ice Dynamics . . . . . . 8.7.1 The Coarse-Grained DEM of Sea Ice . . . . . . . . . . 8.7.2 Numerical Simulation of Sea Ice Dynamic Processes in the Regular Domain . . . . . . . . . . . . . 8.7.3 Numerical Simulation of the Dynamic Process of the Bohai Sea Ice . . . . . . . . . . . . . . . . . . . . . . . 8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

DEM Analysis of Mechanical Behaviors of Railway Ballast . . 9.1 DEM Analysis of Ballast Particle Breakage . . . . . . . . . . . 9.1.1 DEM Analysis of Ballast Breakage . . . . . . . . . . . 9.1.2 The Experimental Verification of Ballast Particle Breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 DEM Analysis of Dynamic Response of Railway Ballast . 9.2.1 Construction of Irregularly Shaped Ballast . . . . . . 9.2.2 DEM Modelling of Ballast Box Test . . . . . . . . . . 9.2.3 Effect of Loading Frequency on Accumulated Settlement of Full-Scale Ballast Bed . . . . . . . . . . 9.3 Direct Shear Tests of Fouled Ballast . . . . . . . . . . . . . . . . 9.3.1 Direct Shear Tests of Fouled Ballast . . . . . . . . . . 9.3.2 Effect of Fine Grains on Shear Strength . . . . . . . . 9.3.3 Effect of Fine Grains on Dilatancy Behavior of Ballast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 DEM Analysis of Shear Strength of Sand-Fouled Ballast . 9.4.1 DEM Simulations of Direct Shear of Sand-Fouled Ballast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Shear Strength and Force Chain Analysis with Different Ballast Contents . . . . . . . . . . . . . . 9.5 Coupled DEM-FEM Analysis of the Transition Zone Between Ballasted-Ballastless Track . . . . . . . . . . . . . . . . . 9.5.1 DEM-FEM Algorithm in Railway Ballasted Track 9.5.2 DEM-FEM Model for Ballasted-Ballastless Transition Zone . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Settlement Analysis of the Transition Zone . . . . . 9.5.4 Coupled DEM-FEM Analysis Considering Ballast Embedded in Ballastless Bed . . . . . . . . . .

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Contents

DEM Simulations of Railway Ballast Bed with Dilated Polyhedral Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Ballast Box Test Model with Dilated Polyhedral Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Settlement of Ballast Bed Under Various Cyclic Loading Frequencies . . . . . . . . . . . . . . . . . . . . . 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 DEM Analysis of Vibration Reduction and Buffering Capacity of Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 DEM Simulations and Experimental Tests of Vibration Reduction of Granular Materials . . . . . . . . . . . . . . . . . . . . 10.1.1 Experimental Studies on Particle Damper . . . . . . . 10.1.2 Numerical Simulation of Particle Damper . . . . . . . 10.2 DEM Analysis of Buffering Capacity of Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Experimental Studies on Buffering Capacity of Granular Materials . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Numerical Simulations of Buffering Capacity of Granular Materials . . . . . . . . . . . . . . . . . . . . . . 10.3 DEM Analysis of Landing Process of Lunar Lander . . . . . . 10.3.1 Discrete Element Model of Landing Buffering System and Lunar Soil . . . . . . . . . . . . . . . . . . . . . 10.3.2 Discrete Element Analysis of Landing Process and Impact Characteristics . . . . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

Generally, granular materials consist of a large number of irregularly-shaped particles and form complex granular systems together with surrounding fluid media and structures. Granular systems exhibit complex multi-scale and multi-medium mechanical properties. Theoretical analysis, numerical calculations and mechanical experiments are utilized comprehensively to investigate the mechanical properties of granular systems. The numerical method can be dated back to the establishment of the discrete element method (DEM) in the 1970s, which is originally oriented towards rock and soil mechanics and gradually extended to the chemical engineering, mechanical processing, transportation, construction, mining, natural disasters, and other fields (Cundall and Strack 1979; Cleary 2009). Currently, the DEM has become a powerful tool for solving the problems of granular materials in different engineering fields. However, it still faces many problems to be solved urgently in terms of the morphology of a real granular particle, granular flow characteristics, multi-medium and multi-scale problems, and large-scale high-performance computing (Wang et al. 2018). In addition, the development of the DEM and its engineering applications have been accompanied by the development of relevant computational analysis softwares. Itasca’s PFC2D and PFC3D and DEM Solutions’ EDEM software have been successfully commercialized (Itasca Consulting Group 2004; DEM Solutions Ltd 2008). In recent years, some open source discrete element softwares have also been rapidly developed and applied to a certain extent due to its unique computing performance (Abe et al. 2011; Goniva et al. 2012; Kozicki and Donze 2009). In China, the development of the DEM software began in the 1980s and has made significant progress since then. In order to solve the problem of computational efficiency and computational scale in the numerical simulations of granular materials, large-scale parallel computation has been developed rapidly in recent years, and the computational scale has reached to be of 107 (ten million level) units (Cleary 2009). Nevertheless, due to the influences of particle morphology, contact models between particles, coupling between particles and fluid, coupling between particles and engineering structures and calculation scale, the DEM and related simulation softwares still face great challenges in terms of calculation accuracy and efficiency when applied © Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_1

1

2

1 Introduction

to engineering applications. Therefore, the DEM involved in computational granular mechanics needs to be constantly developed to better solve different types of granular mechanics problems in engineering practice, whether in the aspect of basic theoretical methods of multi-medium and multi-scale or high-performance computing analysis softwares. This chapter will briefly introduce the basic requirements of computational mechanics of granular materials in engineering applications, the basic physical and mechanical properties of granular materials, and the development of computational analysis softwares of granular materials.

1.1 Engineering Demands of Granular Mechanics Granular materials widely exist in many fields such as natural environment, industrial production and daily life. Granular particles generally have irregular geometric shapes and form complex granular systems with surrounding fluid media and engineering structures. Currently, a cross discipline of multi-scale and multi-medium computational granular mechanics has been preliminarily formed with granular contact mechanics as the bases, fluid mechanics and structural mechanics as the carrier, and granular engineering technology as the application. Theoretical analysis, numerical calculations and mechanical experiments are needed to study the complex mechanical properties of granular materials. The numerical method can be traced back to the DEM developed in the 1970s, which is originally for rock and soil mechanics and gradually extended to the chemical engineering, mechanical processing, transportation, construction, mining, natural disasters, and other fields (Cundall and Strack 1979; Cleary 2009; You and Buttlar 2004). The research on granular mechanics in China began in the 1980s, and has developed rapidly in the past decade in the aspects of basic theories, numerical methods and experimental verifications. Recently, the DEM has been successfully applied in many engineering fields. For example, it has been used to simulate the grinding and sawing process of granite, analyze the dynamic characteristics of rock tunnel, calculate the large deformation and failure process of soil mass in pebble underground engineering, and construct asphalt concrete model to analyze mechanical behaviors of asphalt pavement under different loads in road engineering (Takeda et al. 2018). Moreover, the soil-structure interaction can be analyzed through the coupling model of discrete element and finite element in geotechnical engineering (Wang and Ji 2018; Constant et al. 2019). For the granular flow problems such as landslide, rubble mound foundation and discharging, the computational advantages of the DEM can be given full play. However, the DEM still faces many problems when applied in engineering applications, such as realistic particle morphology construction, multi-medium coupling and large-scale calculation. In daily life and industrial production, granular materials generally have irregular geometric shapes, as shown in Fig. 1.1. Discrete element method usually simplifies irregular granular particles by means of two-dimensional (2D) discs or threedimensional (3D) spheres at the earliest, and describes the complex mechanical

1.1 Engineering Demands of Granular Mechanics

3

(a) Potatos

(b) Tablets

(c) Macadam

(d) Crowd

Fig. 1.1 Irregular granular morphology

behaviors of irregular granular particles by adjusting relevant calculation parameters such as sliding or rolling friction coefficient. However, this kind of simplification faces great limitations in dealing with mechanical properties such as particles’ packing and interlocking dilatancy. Therefore, it is necessary to construct discrete particles with shapes much closer to the realistic particles’ morphologies. At present, different construction methods have been developed to describe morphologies of irregularly shaped particles, such as bonding and clumping model, hyperquadric model, polyhedral and extended polyhedral (Galindo-Torres et al. 2012, 2013). However, the contact models between irregularly shaped particles are more complex than those between regularly shaped particles. The nonlinear mechanical behaviors in different contact modes and how to effectively improve the contact detection efficiency in large-scale calculations should be appropriately considered. Therefore, the realistic construction of irregularly shaped particles, the development of fast contact detection algorithm between irregularly shaped particles, and the establishment of the reasonable contact models between granular particles are of great theoretical importance in solving the complex engineering problems of granular materials. Many practical engineering problems involve interactions between discontinua and continua. For example, the breakage of car windshields, the dynamic behaviors

4

1 Introduction

of railway ballast bed, and the movement of a tire on sand. Coupled discrete and finite element methods have been successively proposed since the 1990s, which can deal with the transformation of the continuum to discrete medium, the continuumdiscontinuum bridging domain from continuum to discrete, and the interaction between discrete materials and continuous structures, respectively (Munjiza et al. 2013). Figure 1.2 shows the phenomenon of breakage of an automobile windshield, in which the complex mechanical process of glass transforming from continuum to discontinuum can be simulated by the coupling method of discrete element and finite element. Figure 1.3 shows the ballasted railway track. The interaction process between ballast bed and subgrade can be simulated by using the discrete and finite coupling method in order to determine the macroscopic settlement characteristics of ballast bed. In the coupling method of discrete element and finite element, the accurate transfer of coupling interface parameters, the unification of time steps and the improvement of computational efficiency are the key problems to be solved (Das and Das 2019). Fluid-solid coupling phenomenon of granular materials widely exists in nature, such as the interaction process between granular materials and fluid in debris flow and floating ice, as shown in Figs. 1.4 and 1.5. When granular materials are in fluid dynamics environment, the fluid-solid coupling characteristics are different under different granular concentrations. The fluid seepage in the pores of granular materials dominates the fluid dynamics process under high granular concentrations; while the drift of granular particles in the fluid dominates under low granular concentrations. The concentration of granular materials changes constantly during the movement process, and seepage and drift occur alternately. As a result, it is necessary to develop

Fig. 1.2 The breakage of car windshields

1.1 Engineering Demands of Granular Mechanics

5

Fig. 1.3 Railway ballast bed

Fig. 1.4 A landslide in Guanling, Guizhou Province in June 28, 2010

fluid-solid coupling models for granular materials under different concentrations. At present, various methods have been developed, including DEM-SPH, DEM-LBM and DEM-CFD, etc. (Jonsen et al. 2014; Galindo-Torres 2013; Washino et al. 2013). All of these methods have their own advantages and disadvantages. It is necessary to select suitable numerical methods in combination with the actual situation to correctly simulate the interaction process between granular materials and fluid media. With the rapid development of the DEM in engineering applications, higher requirements are put forward for the calculation scale and efficiency of the method.

6

1 Introduction

Fig. 1.5 Interaction between sea ice and ship structure

Figures 1.6 and 1.7 show the grain discharging in granary and blown sand phenomenon. It is unrealistic to carry out discrete element calculations for such a huge number of particles at the current stage. The computational scale and computational efficiency are the two issues to be urgently tackled in current engineering applications of the method. Some corresponding solutions are proposed, such as coarse graining, multi-core CPU parallelism, GPU parallelism (Ji and Wang 2018). Coarse graining is the process of integrating several granular assemblies into a computing unit and reasonably establishing the contact model and scale effect analysis between

Fig. 1.6 Granary grain discharging

1.1 Engineering Demands of Granular Mechanics

7

Fig. 1.7 The drift highway buried by wind-blown sand

computing units in order to improve the computing scale and efficiency of the granular system (Hilton and Cleary 2014). In addition, discrete element large-scale GPU parallel computing has made rapid development in recent years, and the computing scale has reached to be of 107 (ten million level) units. However, granular mechanics at engineering scale is generally of 109 (billion level) units or even larger. Currently, the DEM still faces great challenges in terms of calculation scale and calculation efficiency when orientated to engineering applications considering the influences of granular particles’ morphologies, contact models, coupling between particles and fluid, coupling between particles and engineering structures and calculation scales. Numerical methods based on discrete element models have obvious advantages in characterizing the discrete distribution states, granular contacts and movement law of granular materials, which is helpful to reveal the internal mechanisms of mechanical behaviors of granular materials. Moreover, the mechanical problems of complex granular systems in different fields can be effectively solved by establishing coupling models with fluid and structures. However, the theories of the DEM are still in development and improvement. Assumptions are made in both particles’ motions and elastic and plastic deformations, and the calculation parameters are selected with

8

1 Introduction

a certain degree of experience. Therefore, the basic theories, numerical algorithms, error analysis and parameter selections of the DEM should be further investigated. In addition, simulation results needs to be verified by comparisons with experimental results, theoretical solutions and calculation results from other numerical methods in order to improve the calculation accuracy of the DEM.

1.2 Basic Physical and Mechanical Properties of Granular Materials Granular mechanics is a discipline that studies the equilibrium, movement rules and their applications of complex systems formed by interactions of a large number of solid particles. Granular materials are composed of a large number of discrete solid particles. The mechanical characteristics can be summarized as ‘scattering’ and ‘dynamic’. ‘Dispersion’ refers to the dispersion of particles’ physical properties, sizes and shapes; ‘Dynamic’ refers to the transient, fluctuating, colliding granular coagulation state of granular movement and the fracture or fragmentation of agglomeration (Xu et al. 2002). From the point of view of physical state, granular system is not a simple elastic aggregate or a rigid aggregate, but a complex system between them. Therefore, traditional solid theory, fluid theory and gas theory cannot be used to explain various phenomena observed in granular systems. As a material state different from solid and fluid, granular materials exhibit many unique physical and mechanical phenomena at different scales.

1.2.1 Friction Law The research on granular mechanics began in 1773, when French physicist Paul coulomb studied soil mechanics and put forward the law of solid friction. The friction force is proportional to the normal pressure between them, while the coefficient of static friction is greater than that of sliding friction. The theory was mentioned many times in later research, such as in discontinuum mechanics and soil mechanics, including bearing capacity of foundation formed by granular materials, slope stability under the action of weight and external force, action of the bunker wall, action of soil on retaining walls, the motion law of crushed ore, the force state of the storage tower release and its movement rule, soil stress analysis, the effect of pore water in the unsaturated soil and a series of engineering problems. The first study to reveal the constitutive relationship of granular friction was carried out by Bagnold in the 1950s. He conducted the coaxial shear rheometer experiment of wax spheres in glycerol-alcohol-water solution and put forward the famous Bagnold law (Bagnold 1954).

1.2 Basic Physical and Mechanical Properties of Granular Materials

9

Under the enlightening effect of the law of solid friction, the door to granular mechanics research is opened. People gradually try to explain the physical phenomena of granular materials scientifically, and put forward a series of mechanical theories, which provide a strong basis for the study of granular materials.

1.2.2 Grain Silo Effect The granary has a long history as a building for food storage. Although the structure of granary is quite different from that of building materials, its reasonable structure design depends on the understanding and application of granular mechanics. Figure 1.8 shows the damage phenomenon of a steel grain silo. As a classic mechanical phenomenon of granular materials, Roberts, a British scientist, noticed the granary effect as early as 1884. When the height of grain accumulation is greater than twice the diameter of the ground, the pressure on the surface of the granary will not increase with the increase of grain, which is completely different from the behavior of liquid in the container. In 1895, German engineer Janssen used the continuum model to explain the granary effect (Janssen 1895). Due to the interaction between particles, the force in the direction of gravity is decomposed into the horizontal direction, and the granary wall supports part of the weight of grains, making the pressure at the bottom of the granary tend to be saturated. Janssen’s work played a crucial role in the development of granular statics, and the model was then widely accepted (Vanel and Clement 1999).

Fig. 1.8 The silo damaged by the action of grain particles

10

1 Introduction

In addition, the granary does not need to be thickened at the lower part like the trapezoidal dam, because the pressure at the lower part of the granary does not increase linearly with its height. Based on the above theories, granular statics develops rapidly, which plays a key role in people’s understanding of granular materials.

1.2.3 Extrusion and Shear Expansion of Granular Materials In 1885, Renault put forward the Reynolds extrusion expansion principle. When an external force acts on packed particles, as in striking the container storing particles, the packing density changes. If the initial packing density is high, the knock will lower the density, i.e., the volume expands as shown in Fig. 1.9. On the contrary, if the initial packing density is small, the knock will make the density increase. When people walk on the beach where the sand is closely packed, the sand they step on expands and becomes loosened under the pressure and absorbs seawater from surrounding sand grains. As a result, the sand grains around the footprints will become dry as shown in Fig. 1.10. The volume fraction of randomly packed granular materials decreases after shearing, that is, the concentration of particles decreases, which means shearing expansion occurs. (Burman et al. 2018). Shearing dilation is caused by the change of deviatoric stress and strain in granular system, and the volume strain is also affected by the change of deviatoric stress (Dorostkar and Carmeliet 2018). When a stably closed granular sample is subjected to continuous deformation, the densification of the granular sample will be accompanied by shearing expansion (Kozicki and Tejchman 2018). The expansion coefficient depends on the stress level and density of the granular system. Figure 1.11 is a schematic of shearing expansion. When the granular system is subjected to shearing action, the spatial position and volume of the granular system change. The expansion movement of granular materials along the shear direction is inevitable in order to adapt to the change (Forgber and Radl 2017). From the perspective of engineering applications, the normal force generated by the shearing expansion of granular materials is able to bear load effectively, but

(a) Before extrusion Fig. 1.9 Schematic of granular extrusion expansion

(b) After compression

1.2 Basic Physical and Mechanical Properties of Granular Materials

11

Fig. 1.10 Sand around beach footprints

Fig. 1.11 Schematic of shearing expansion of granular materials (Forgber and Radl 2017)

the expansion and contraction phenomenon due to shearing is highly irregular and has many adverse effects on engineering applications.

12

1 Introduction

Fig. 1.12 Wind-blown sand on the Mingshan mountain in Dunhuang

1.2.4 The Flow State of Granular Materials The motion of a single particle obeys Newton’s law. Granular materials, as a whole, may flow and form granular flow when the external force or the internal stress changes (Witt et al. 2018). The phenomenon of granular flow exists widely in nature, such as avalanche, landslide, debris flow and so on. Figure 1.12 shows the wind-blown sand phenomenon. Strictly speaking, granular flow should be considered as a multiphase flow problem because the pores between particles are usually filled with gas or liquid (Chung et al. 2019). The granular flows mentioned above are dense flows, and force chains form between particles due to particles’ mutual contacts. Force chains interconnect with each other to form a spatial network structure supporting the weight and external loads acting on the whole granular system. This is also the source of the overall friction and contact stresses of the granular flow (Constant et al. 2019). Granular flow can be divided into elastic flow and inertial flow according to the existence of relatively stable force chains. As two extreme flow states, the elasticquasi-static flow and inertia-collision flow correspond to quasi-static flow and fast flow, respectively. They are usually treated as continuum and analyzed by friction plastic model and dynamic theory, respectively (Sandlin and Abdel-Khalik 2018). The description of dense granular flow is very difficult (Bartsch and Zunft 2019). Granular particles flow like a fluid and sustain relatively enduring contacts between each other. Obviously, the relatively well developed plastic theory and dynamic theory are not applicable to granular flows. Inspired by the visco-plastic Bingham fluid behaviors, Pouliquen and Forterre proposed the phenomenological model to describe the dense granular flow, thereby reproducing the complex flow of granular

1.2 Basic Physical and Mechanical Properties of Granular Materials

13

materials under different boundary conditions and determining important parameters such as velocity distribution and granular concentration distribution (Pouliquen and Forterre 2002). The model has been applied to analyze natural disasters, such as debris flow and landslide. The flow characteristics predictions of granular materials and the degree of approximation are the two aspects for evaluating the effects of numerical simulations. Walton studied the flow of inelastic balls in the chute (Walton 1992). Karion and Hunt studied the Couette flow of mono-sized particles and binary mixtures (Karion and Hunt 2000). Discharging in a silo is a typical granular flow. Masson and Martinez made an in-depth study on the influence of mechanical properties of the flow (Masson and Martinez 2000). Xu et al. used Thornton’s spherical particles to simulate the influences of material properties on the silo discharging, and explained that the collapse of the silo is attributed to the buckling failure induced by the sudden increase of friction on the silo wall when the dispersion changes from an active state to a passive state at the beginning of discharging (Xu et al. 2002). The solid-liquid transition of granular materials is one of the key issues in granular mechanics. Granular media exhibit mechanical properties similar to those of fluid or solid at different concentrations and shear rates, and solid-liquid phase transition may occur under certain conditions. The flow phase states can be divided into fast flow, slow flow and quasi-static flow. The different flow phase states and their mutual transitions are typical energy dissipation processes. Volfson et al. used a state parameter to describe the proportion of particles showing liquid state flow in the granular system (Volfson et al. 2003). Under different excitation conditions, shear rates and concentrations of granular materials are different, and granular materials may exhibit quasi-solid or quasi-liquid state. Three-dimensional discrete element simulations, as shown in Fig. 1.13, were carried out for the single shear of granular materials using spherical particles, and different flow states were obtained under different shear rates and densities. As the solid-liquid phase transition is an enduring process, the solid-liquid phase transition diagram as shown in Fig. 1.14 can visually Fig. 1.13 Discrete element simulation of a single shear flow

14

1 Introduction

Fig. 1.14 The coordination number in a granular shear flow during solid-liquid transition

characterize the basic law of solid-liquid transition of granular system. In the quasisolid-liquid transition, the macroscopic mechanical behaviors of granular materials are mainly reflected in the corresponding relationship between the average stress and the shear rate. In the solid-like state, the average stress has nothing to do with the shear rate, but is proportional to the contact stiffness; while in the liquid-like state, the average stress is proportional to the square of the shear rate, and has nothing to do with the contact stiffness. The above macroscopic mechanical behaviors are closely related to the interaction characteristics of particles at mesoscale and can be characterized by such parameters as the strength of force chains, contact duration and concentration of granular flow. The development of geotechnical mechanics has been always accompanied by the improvement of granular mechanics and provides an effective way of engineering practice. Soil mechanics researchers have systematically studied the macroscopicmesoscopic mechanical relationship of materials and correlate the micro contact mechanics of granular materials to the stress-strain relationship of continuum mechanics. Strack (Cundall and Strack 1979) adopted graph theory to study the mathematical descriptions of constitutive relationships of granular materials. Oda and Kazama (1998) and Iwashita and Oda (2000) put forward the concept using tensor to express the stress-strain relationship of granular aggregates, and developed the theory of statistical average of granular materials with the correlation of contact forces and the displacement field between particles and macro continuum mechanics’ stress-strain (Oda and Kazama 1998; Iwashita and Oda 2000). This has been further

1.2 Basic Physical and Mechanical Properties of Granular Materials

15

illustrated by discrete element simulations (Thornton and Antony 2000; Thornton and Zhang 2010). Meanwhile, physicists have also contributed to the establishment and improvement of the theoretical foundation of granular mechanics by adopting advanced scientific and technological means, such as vacuum technology, nuclear magnetic resonance (NMR), high speed photography, computer simulations with the nonlinear theory, molecular dynamics, fractal theory, and the non-equilibrium phase transition theory. They have tried to reveal the mystery of granular mechanics in nature and strive to solve particle-related problem in daily life and industrial production. Granular materials is a multi-body system with contact forces as the dominating action. The complex physical and mechanical properties of granular system can be further investigated by analyzing the physical mechanisms at respective scale and establishing correlations between different scales with the consideration of its multi-scale structure characteristics.

1.3 Computational Analysis Softwares for Computational Granular Mechanics Discrete element analysis softwares have been developed continuously since the establishment of the discrete element method in 1970s. The ball discrete element programs were developed earlier. PFC2D and PFC3D , EDEM, LIGGGHTS and ESyS Particle et al. have been widely used in the relevant calculations of granular materials as shown in Figs. 1.15 and 1.16 (Bock and Prusek 2015; Shi et al. 2019). In the development of block discrete element program, 2D UDEC and 3D 3DEC, as well as Rocky, Y2D, Yade and other softwares have been successfully applied in joint rock, mining and grinding industry (Afrasiabi et al. 2019). The development of the DEM and related softwares in China stems from its applications in rock and soil mechanics in the 1980s, namely, the 2D Block and TRUDEC developed by Northeastern University. Recently, CDEM, developed by

(a) PFC3D

(b) EDEM

Fig. 1.15 Typical commercial discrete element softwares

16

1 Introduction

(a) LIGGGHTS

(b) Yade

Fig. 1.16 Typical open source discrete element softwares

Institute of Mechanics of Chinese Academy of Sciences, has been used to carry out a lot of practical research in geological evolution, geological disaster mechanism and other aspects (Stéen et al. 2019). It is a discrete element algorithm based on continuum which has many functions, such as bonding-fracture, fluid-solid coupling, large-scale parallel computing, etc. On the basis of CDEM algorithm, GDEM wa developed jointly by Institute of Mechanics of Chinese Academy of Sciences and JTD Technology Company. In addition, Tsinghua University, Peking University, Institute of Process Engineering, Chinese Academy of Sciences, Lanzhou University, China Agricultural University, Metallurgical University and other research institutes and large enterprises have systematically developed the discrete element method and its engineering applications, and the corresponding analysis softwares. In order to improve the large-scale computing ability of the DEM in engineering applications, it is a trend to use multiple GPU and CPU to work together in a single or multi-machine environment. The high-performance discrete element softwares, such as EDEM, ESyS Particle, LIGGGHTS and Yade et al., have all realized the large-scale calculations of multi-GPU or GPU cluster. With the increasing maturity of graphics processor technology, GPU parallel algorithm has been widely used in the large-scale solutions of fluid dynamics, the finite element method (FEM), the DEM and other mechanical problems. The GPU-based parallel HPC method has been successfully applied into the fluid-solid coupling simulations of granular materials in chemical processes. GPU parallel computing technology can improve discrete element computing efficiency by more than 2 orders of magnitude, and has been applied in geological hazards, marine engineering, civil structure and other fields. In large-scale discrete element computing, the use of irregular particles has become the focus of academic and industrial communities (Kureck, et al. 2019).

1.3 Computational Analysis Softwares for Computational Granular Mechanics

17

Remarkable achievements have been made in the research on discrete element methods and their engineering applications in China in recent years. Nevertheless, with the ever increasing diverse technical requirements and fierce international competition, there is still a lack of a high-performance discrete element analysis software which is easy to operate, user-friendly, rich in pre-processing and post-processing and perfect follow-up services. How to improve the computational efficiency and scale of the DEM to an acceptable level in industry and how to develop simulation software for granular materials that can provide generalized and customized services are urgent problems to be solved.

References Abe S, van Gent H, Urai JL (2011) DEM simulation of normal faults in cohesive materials. Tectonophysics 512:12–21 Afrasiabi N, Rafiee R, Noroozi M (2019) Investigating the effect of discontinuity geometrical parameters on the TBM performance in hard rock. Tunn Undergr Space Technol 84:326–333 Bagnold RA (1954) Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear. Proc R Soc Lond Ser A Math Phys Sci 225(1160):49–63 Bartsch P, Zunft S (2019) Granular flow around the horizontal tubes of a particle heat exchanger: DEM-simulation and experimental validation. Sol Energy 182:48–56 Bock S, Prusek S (2015) Numerical study of pressure on dams in a backfilled mining shaft based on PFC3D code. Comput Geotech 66:230–244 Burman BC, Meng J, Huang J et al (2018) Quasi-Static rheology of granular media using the static DEM. Int J Geomech 18(12):07018015 Chung YC, Wu CW, Kuo CY et al (2019) A rapid granular chute avalanche impinging on a small fixed obstacle: DEM modeling, experimental validation and exploration of granular stress. Appl Math Model 74:540–568 Cleary PW (2009) Industrial particle flow modeling using discrete element method. Eng Comput 26(6):698–743 Constant M, Dubois F, Lambrechts J et al (2019) Implementation of an unresolved stabilised FEM– DEM model to solve immersed granular flows. Comput Part Mech 6(2):213–226 Cundall P, Strack O (1979) A discrete numerical model for granular assemblies. Geotechnique 29:47–65 Das SK, Das A (2019) Influence of quasi-static loading rates on crushable granular materials: A DEM analysis. Powder Technol 344:393–403 DEM Solutions Ltd (2008) EDEM discrete element code. Edinburgh, UK Dorostkar O, Carmeliet J (2018) Potential energy as metric for understanding stick–slip dynamics in sheared granular fault gouge: a coupled CFD–DEM study. Rock Mech Rock Eng 51(10):3281– 3294 Forgber T, Radl S (2017) Heat transfer rates in wall bounded shear flows near the jamming point accompanied by fluid-particle heat exchange. Powder Technol 315:182–193 Galindo-Torres SA, Pedroso DM, Williams DJ et al (2012) Breaking processes in three-dimensional bonded granular materials with general shapes. Comput Phys Commun 183(2):266–277 Galindo-Torres SA (2013) A coupled discrete element Lattice Boltzmann method for the simulation of fluid–solid interaction with particles of general shapes. Comput Methods Appl Mech Eng 265(2):107–119 Goniva C, Kloss C, Deen NG et al (2012) Influence of rolling friction on single spout fluidized bed simulation. Particuology 10:582–591

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Hilton JE, Cleary PW (2014) Comparison of non-cohesive resolved and coarse grain DEM models for gas flow through particle beds. Appl Math Model 38:4197–4214 Itasca Consulting Group (2004) Particle Flow Code in 3 Dimensions. Online Manual Iwashita K, Oda (2000) Micro-deformation mechanism of shear banding process based on modified distinct element method. Powder Technol 109(1–3):192–205 Janssen HA (1895) Test on grain pressure silos. Z. Ver Dtsch Ing 39:1045–1049 Ji S, Wang S (2018) A coupled discrete–finite element method for the ice-induced vibrations of a conical jacket platform with a Gpu-based parallel algorithm. Int J Comput Methods 1850147 Jonsén P, Pålsson BI, Stener JF et al (2014) A novel method for modelling of interactions between pulp, charge and mill structure in tumbling mills. Miner Eng 63(63):65–72 Karion A, Hunt ML (2000) Wall stresses in granular Couette flows of mono-sized particles and binary mixtures. Powder Technol 109(1):145–163 Kozicki J, Donze FV (2009) YADE-OPEN DEM: an open source software using a discrete element method to simulate granular material. Eng Comput 26(7):786–805 Kozicki J, Tejchman J (2018) Relationship between vortex structures and shear localization in 3D granular specimens based on combined DEM and Helmholtz-Hodge decomposition. Granular Matter 20(3):48 Kureck H, Govender N, Siegmann E et al (2019) Industrial scale simulations of tablet coating using GPU based DEM: a validation study. Chem Eng Sci 202:462–480 Masson S, Martinez J (2000) Effect of particle mechanical properties on silo flow and stresses from distinct element simulations. Powder Technol 109(1):164–178 Munjiza A, Lei Z, Divic V et al (2013) Fracture and fragmentation of thin shells using the combined finite–discrete element method. Int J Numer Meth Eng 95(6):478–498 Oda M, Kazama H (1998) Microstructure of shear bands and its relation to the mechanisms of dilatancy and failure of dense granular soils. Géotechnique 48(4):465–481 Pouliquen O, Forterre Y (2002) Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J Fluid Mech 453(453):133–151 Sandlin M, Abdel-Khalik SI (2018) A study of granular flow through horizontal wire mesh screens for concentrated solar power particle heating receiver applications–Part II: parametric model predictions. Sol Energy 174:1252–1262 Shi Y, Sun X, Wang X, et al (2019) Numerical simulation and field tests of minimum-tillage planter with straw smashing and strip laying based on EDEM software. Comput Electr Agric 166:105021 Stéen EJL, Jørgensen JT, Petersen IN et al (2019) Improved radiosynthesis and preliminary in vivo evaluation of the 11C-labeled tetrazine AE-1 for pretargeted PET imaging. Bioorg Med Chem Lett 29(8):986–990 Takeda S, Ogawa K, Tanigaki K et al (2018) DEM/FEM simulation for impact response of binary granular target and projectile. Eur Phys J Spec Top 227(1–2):73–83 Thornton C, Antony SJ (2000) Quasi-static shear deformation of a soft particle system. Powder Technol 109(1–3):179–191 Thornton C, Zhang L (2010) Numerical simulations of the direct shear test. Chem Eng Technol 26(2):153–156 Vanel L, Clément E (1999) Pressure screening and fluctuations at the bottom of a granular column. Eur Phys J B Condens Matter Complex Syst 11(3):525–533 Volfson D, Tsimring LS, Aranson IS (2003) Partially fluidized shear granular flows: continuum theory and molecular dynamics simulations. Phys Rev E 68:021301 Walton OR (1992) Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres. Mech Mater 16(1–2):239–247 Wang S, Fan Y, Ji S (2018) Interaction between super-quadric particles and triangular elements and its application to hopper discharge. Powder Technol 339:534–549 Wang S, Ji S (2018) Coupled DEM–FEM analysis of ice-induced vibrations of a conical jacket platform based on the domain decomposition method. Int J Offshore Polar Eng 28(02):190–199 Washino K, Tan HS, Hounslow MJ et al (2013) A new capillary force model implemented in micro-scale CFD–DEM coupling for wet granulation. Chem Eng Sci 93(4):197–205

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Witt PJ, Sinnott MD, Cleary PW et al (2018) A hierarchical simulation methodology for rotary kilns including granular flow and heat transfer. Miner Eng 119:244–262 Xu Y, Kafui KD, Thornton C et al (2002) Effects of material properties on granular flow in a silo using DEM simulation. Part Sci Technol 20(2):109–124 You ZP, Buttlar WG (2004) Discrete element modelling to predict the modulus of asphalt concrete mixtures. J Mater Civ Eng 16(2):140–144

Part I

Fundamentals of Computational Granular Mechanics

Chapter 2

Constructions of Irregular Shaped Particles in the DEM

The discrete element method was first proposed by Cundall in the 1970s. It has been developed to be a powerful way of investigating behaviors of granular materials numerically. The 2D discs or 3D spheres were originally used to describe the particles’ shapes because of their characteristics of simple calculation and high efficiency. Existing experimental measurements and numerical simulations have focused mainly on spherical particles. However, particle shapes have been demonstrated to strongly affect the dynamics of granular systems in various simulations (Li and Ji 2018; Ji et al. 2019). In fact, granular systems commonly encountered in industry or nature comprise of non-spherical particles, which are of great differences from regular spheres in terms of particle packing, dynamic process and motion morphology. Meanwhile, multi-contact, low flowability and inter-locking between irregularly shaped particles significantly affect the macroscopic mechanical properties of granular materials. The macroscopic and microscopic mechanical properties obtained from spherical particulate systems cannot be simply applied to irregular particulate systems (Lu et al. 2015). With the development of computer technology and higher requirements of the computational accuracy, the influence of particle shape on the macroscopic mechanical properties and dynamic response of granular system has become a main concern of the DEM (Soltanbeigi et al. 2018; Khazeni and Mansourpour 2018). In order to reasonably describe granular materials with irregular shapes, different construction methods have been developed, such as composite approaches based on regular elements, B-spline models based on contour fitting, ellipsoidal models based on quadric equation, super-quadric element based on continuous function representation, polyhedral model based on geometric topology, dilated polyhedral models based on Minkowski sum theory, and star-shaped particles based on X-ray tomography and spherical harmonics (Su and Yan 2018). This chapter mainly introduces basic theories and construction methods of composite approach, ellipsoidal or super-quadric model and polyhedral or dilated polyhedral model.

© Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_2

23

24

2 Constructions of Irregular Shaped Particles in the DEM

2.1 Bonding and Clumping Models Based on Spherical Particles Recently, laser scanners have been used to accurately acquire irregular particle shapes and complex geometric boundaries due to the development of digital scanning technology. Figure 2.1 shows the reconstruction of an irregularly shaped particle. Particle shapes can be described by using complex functions together with the application of these functions into numerical simulations of particles’ collisions and breakage in order to realistically model the interactions between particles. However, this approach will result in a complicated contact detection algorithm and costs a lot of computation time. It is not suitable for large-scale computations due to the disadvantages of low computational efficiency. 2D discs or 3D spheres are commonly employed in the discrete element analysis of granular dynamics with the incorporation of rolling friction to simulate the interlocking between irregular particles (Lim and McDowell 2005; Houlsby 2009; Ngo et al. 2014; Zeng et al. 2015). However, it is difficult for a disc or sphere to effectively simulate mechanical behaviors of particles with irregular shapes (Lim and McDowell 2005; Houlsby 2009; Ngo et al. 2014; Zeng et al. 2015). The shapes of irregular particles can be accurately constructed by bonding or clumping spheres based on the obtained 3D geometric boundaries with laser scanner (Lu and McDowell 2007; Lobo-Guerrero and Vallejo 2006). This approach has been successfully applied in the research on railway ballast and slope stability (Ngo et al. 2014).

2.1.1 Bonding Models Based on Spheres The spherical particle is the most widely used and simplest geometric construction in the 3D DEM. The breakable irregular particle can be constructed by bonding spheres, as shown in Fig. 2.2. Under the external force, the bonded particle breaks according to the corresponding failure criterion. Two bonding models has been established, including the contact bonding model and parallel bonding model (Potyondy and Cundall 2004). Only inter-particle force can be transferred by contact bonding, whereas both the inter-particle force and torque can be transferred by parallel bonding.

Fig. 2.1 Irregularly shaped particle and its high-precision reconstruction

2.1 Bonding and Clumping Models Based on Spherical Particles

25

Fig. 2.2 By bonding spheres (Lim and McDowell 2005)

The material’s fracture characteristics are usually simulated by parallel bonding. The normal and tangential forces of particles are obtained through inter-particle force and torque between bonding spheres. The parallel bonding method can be used to construct irregular particles shapes and simulate the failure process of continuous material. Figure 2.3 shows the compression of a single ballast particle comprised of bonding spheres. Bonding strength between bonding spheres can be used to ensure the overall movement of the particles. When the inter-particle force exceeds the bonding strength, the bonding particle breaks into several pieces. The irregular particles can be more accurately simulated by increasing the number of constitutive spheres, and a sample can even be constructed with ten thousands of spheres. However, as the number of constitutive spheres increases, the number of potential contact pairs increases during the contact detection and fracture determination, which results in a sharp decrease of computational efficiency of the DEM simulation.

Fig. 2.3 Compression of a single ballast particle comprised of bonding spheres (Yan et al. 2015)

26

2 Constructions of Irregular Shaped Particles in the DEM

2.1.2 Clumping Models Based on Spheres The clumping model possesses favorable computational efficiency in describing the geometrical morphology and mechanical behaviors of irregularly shaped particles. Figure 2.4 shows the discrete element models of crushed stones constructed by spheres with different overlaps and combination modes. The particle breakage is not considered in the clumping model, and the number of contact pairs between particles is significantly less than that of the bonding model. In the construction of clumping models, the sizes of constitutive spherical particles are determined according to the size of the crushed stone, the number of spheres and the overlap between spheres. Different spheres’ sizes and overlaps significantly affect the surface smoothness of the crushed stone, consequently affecting the overall macroscopic mechanical properties of granular materials. Figure 2.5 shows the ballast box simulation and the mixing process in which clumping models were adopted. In the DEM simulations of the clumping particles, the mass and the moment of inertia of the particle are the two important parameters. They need to be recalculated due to the significant overlaps between spheres. The calculation method is shown in

Fig. 2.4 Irregularly shaped particles constructed by clumping models (Ngo et al. 2014)

(a) Ballast box simulation [Lim and McDowell, 2005] Fig. 2.5 Applications of clumping models

(b) Mixing process [PFC3D]

2.1 Bonding and Clumping Models Based on Spherical Particles

27

Fig. 2.6 Calculation method of the mass and moment of inertia for a clumping particle

Fig. 2.6.The effects of overlapping regions can be eliminated and the true mass of the particle can be calculated by accumulating the masses of all active grids. The calculation formula for the mass of a clumping particle can be expressed as, Mc =

n 

ρli3

(2.1)

i=1

where Mc is the mass of the clumping particle, n is the number of active grids, ρ is the density of the granular material, li is the grid length. The coordinates of the particle centroid can be calculated as follows, n xc =

ρli3 xi , yc = Mc

i=1

n

ρli3 yi , zc = Mc

i=1

n

ρli3 z i Mc

i=1

(2.2)

where xc ,yc and z c are the coordinates of the particle centroid; xi ,yi and z i are the coordinates of the centroid of the active grid. The moment of inertia of the particle is then obtained as follows, Jxc = Jyc = Jzc =

n 

m i ri2 =

n 

i=1

i=1

n 

n 

i=1 n  i=1

m i ri2 = m i ri2 =

i=1 n 

  ρli3 (yi − yc )2 + (z i − z c )2   ρli3 (xi − xc )2 + (z i − z c )2   ρli3 (xi − xc )2 + (yi − yc )2

(2.3)

i=1

where Jc is the moment of inertia of the particle, m i is the mass of the active grid, ri is the distance between the center of the active grid and the particle centroid.

28

2 Constructions of Irregular Shaped Particles in the DEM

During the calculation, each constitutive sphere within the particle is involved. The whole clumping particle is considered as a rigid body. The quaternion method is used to determine the overall motion of the particle. This method has been widely used in many research fields such as molecular dynamics, solid mechanics and rigid body dynamics.The dynamic components such as the moment and the rotational speed of the particle can be freely converted between the global and local coordinate system by using the quaternion method. Here, a fixed global coordinate system eG and a local coordinate system e B are defined. They can be mutually transformed by the transformation matrix A, e B = A · eG

(2.4)

e G = AT · e B

(2.5)

where the transformation matrix is satisfied with A−1 = AT , and the matrix A can be determined by the quaternion method. In the quaternion method, there are four scalars, namely, Q = (q0 , q1 , q2 , q3 )

(2.6)

where the four scalars should satisfy the following relationship, q02 + q12 + q22 + q32 = 1

(2.7)

The transformation matrix A can be expressed as, ⎤ 2(q1 q3 − q0 q2 ) q02 + q12 − q22 − q32 2(q1 q2 + q0 q3 ) A = ⎣ 2(q1 q2 − q0 q3 ) q02 − q12 + q22 − q32 2(q2 q3 + q0 q1 ) ⎦ 2(q1 q3 + q0 q2 ) 2(q2 q3 − q0 q1 ) q02 − q12 − q22 + q32 ⎡

(2.8)

In local coordinates, the angular acceleration of the clumping particle can be expressed as, 

I yy − Izz MxB ω yB ωzB = + Ix x Ix x 

M yB Izz − Ix x ωzB ωxB ω˙ yB = + I yy I yy 

MzB Ix x − I yy B ωxB ω yB ω˙ z = + Izz Izz ω˙ xB

(2.9)

where Ix x , I yy and Izz are the moment of inertia with respect to the three local coordinates. The moment components in the global coordinate system can be transformed to the local coordinate system by the transformation matrix,

2.1 Bonding and Clumping Models Based on Spherical Particles

⎡

MxB

⎤

⎡

MxG

⎤

⎢ B⎥ ⎢ G⎥ ⎣ M y ⎦ = A⎣ M y ⎦ MzB

29

(2.10)

MzG

where M B and M G are the moments of the clumping particle in the local coordinates and the global coordinates, respectively, which can be calculated by contact forces between particles in the DEM simulations. When the angular velocity of the clumping particle in the local coordinates system is determined, the angular velocity in the global coordinates can be expressed as, ⎡

ωxG

⎤

⎡

ωxB

⎤

⎢ G⎥ T⎢ B ⎥ ⎣ ωy ⎦ = A ⎣ ωy ⎦ ωzG

(2.11)

ωzB

The increment of the quaternion component and the angular velocity of the clumping particle in the local coordinates satisfy the following differential equation, ⎡ ⎤ ⎤ 0 q˙0 ⎢ B ⎢ q˙ ⎥ 1 ⎢ ω ⎥ x⎥ ⎢ 1⎥ ⎥ ⎢ ⎥ = W⎢ ⎢ ⎣ q˙2 ⎦ 2 ⎣ ω yB ⎥ ⎦ B q˙3 ωz ⎡

(2.12)

where W can be expressed as, ⎡

q0 ⎢ q1 W=⎢ ⎣ q2 q3

−q1 q0 q3 −q2

−q2 −q3 q0 q1

⎤ −q3 q2 ⎥ ⎥ −q1 ⎦ q0

(2.13)

In general, the global coordinates and Euler angles of a particle should be defined for its initial position and orientation. Here, the relationship between quaternion and Euler angles is given as follows, ϕ+ψ θ cos 2 2 ϕ−ψ θ q1 = sin cos 2 2 ϕ−ψ θ q2 = sin sin 2 2 ϕ+ψ θ q3 = cos sin 2 2

q0 = cos

(2.14)

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2 Constructions of Irregular Shaped Particles in the DEM

where θ , ϕ and ψ are Euler angles, respectively. At the nth time step, the clumping (n) particle is subjected to the external force F(n) G and the moment MG in the global coordinates. At the (n + 1)th time step, the translational velocity of the particle is updated as follows, (n+1) (n) vG = vG +

F(n) G · t Mc

(2.15)

(n+1) (n) where vG and vG are the translational velocity of the clumping particle at the (n + 1)th and nth time step in the global coordinates, respectively. Meanwhile, the quaternion can be updated by the following equation at the (n + 1)th time step as,

⎡ ⎤(n) ⎤(n+1) ⎡ ⎤(n) 0 q0 q0 ⎢ ⎥ B ⎢q ⎥ ⎢q ⎥ ωx ⎥ t (n) ⎢ ⎢ 1⎥ ⎢ 1⎥ ⎢ =⎢ ⎥ + W ⎢ B⎥ ⎢ ⎥ ⎥ , ⎣ q2 ⎦ ⎣ q2 ⎦ 2 ⎣ ωy ⎦ q3 q3 ωB ⎡

(2.16)

z

The rotational components of the clumping particle can be iteratively solved by using the above transformations between the global and local coordinates.

2.2 Super-Quadric Particles The most common non-spherical particles are ellipsoids in the DEM, which have been used in many fields such as hopper discharging and particles packing. Oblate or prolate ellipsoids with different length-width ratios are obtained based on spheres by changing the semi-axis length of the three major axes. Because of the relatively simple contact detection, ellipsoids are the popular geometric models commonly used in the non-spherical DEM simulations. However, it cannot be used to construct the geometric morphologies with sharp corners or planes. Meanwhile, it cannot reflect the influence of the sharpness of the particle surface on the dynamics of granular systems. The super-quadric method is extended from the quadric equation. It allows the efficient changes of aspect ratio of particles and the smoothness of the surface, and it is a common approach to describe irregular particles’ shapes from the mathematics viewpoint. Statistically, 80% of particle shapes can be represented by super-quadric equations, and the rest can be described by higher-dimensional hyper-quadric equations (Williams and Pentland 1992). Cleary et al. simplified the super-quadratic equation and successfully applied it to ball mills and landslide induced rock avalanche simulations (Cleary et al. 2016, 2017). The super-quadric equations based on continuous function representation (CFR) can accurately describe the irregular particle

2.2 Super-Quadric Particles

31

shapes and effectively solve the problem of contact detection by nonlinear iterative method (Zhao et al. 2017; Ji et al. 2019).

2.2.1 Super-Quadric Particles The super-quadric method is developed by extending the quadric equation, which has been used extensively to describe particles with convex and concave shapes. The super-quadric equation is given as (AH 1981):  x n 2  y n 2 n 1 /n 2  z n 1       +  −1=0   +  a b c

   x 2/n 2  y 2/n 2 n 2 /n 1  z 2/n 1 + + −1=0 a b c

(2.17) (2.18)

where a, b, and c are the semi-axis lengths of the particles along the major axes, n1 and n2 are shape parameters. n1 controls the sharpness of x-z and y-z plane, while n2 controls the sharpness of x-y plane. When n1 = n2 , the above equation can be simplified to,  x m  y m  z m         +  +  −1=0 a b c

(2.19)

This simplified equation may be unable to describe cylinder-shaped particles accurately. Therefore, Eq. (2.17) is usually used as the standard equation for the super-quadric particles. An ellipsoid is obtained if n 1 = n 2 = 2, a cylinder-shaped particle is obtained if n 1  2 and n 2 = 2, and a cubic is obtained if n 1  2 and n 2  2. Figure 2.7 shows the differently shaped particles obtained by varying five parameters in the super-quadric equation. The super-quadric particles constructed by CFR algorithm has been widely used in geotechnical, mining and industrial processing engineering, as shown in Figs. 2.8 and 2.9 (Cleary 2010). Lu et al. studied the relationship between computational efficiency and precision, and shape index and semi-axis length of the super-quadric. It was reported that the closer the shape is to a sphere, the higher the efficiency and precision (Lu et al. 2012). Cleary et al. developed two coupling algorithms of DEM-SPH and DEM-CFD for super-quadric models. The simulations of interaction process of high concentration mud and coarse particles in industrial applications, pneumatic transport and industrial dust were realized by using these models (Cleary et al. 2016). In addition, super-quadric models have also been successfully applied to grinding millers, and the analysis of the influence of particle shape on the mixing characteristics in high shear mixers (Delaney et al. 2015; Sinnott and Cleary 2016). The main advantage of using the CFR algorithm is that the normal direction of particles can be easily obtained,

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2 Constructions of Irregular Shaped Particles in the DEM

(a) Super-quadric particles with different surface smoothness.

(b) Super-quadric particles with different aspect ratios. Fig. 2.7 Super-quadratic particles based on CFR

n 1  x n 2 −1  x n 2  y n 2 n 1 /n 2 −1 sign(x)     +  a a a b      n 1  y n 2 −1  x n 2  y n 2 n 1 /n 2 −1  fy =   sign(y)   +  b b a b n 1  z n 1 −1  fz =   sign(z) c c 

fx =

(2.20)

Based on this method, the problem of contact detection between particles is transformed to an optimization problem solving the minimum distance between particles, and the contact force between particles is calculated by a nonlinear iterative method.

2.2 Super-Quadric Particles

(a) Landslide

33

(b) Granular mixing

Fig. 2.8 Engineering applications of super-quadric particles (Cleary 2010)

(a) Random packing

(b) Grinding and crushing

Fig. 2.9 Crushing process of super-quadric particles and its application (Cleary et al. 2017)

In order to calculate the contact force, it is necessary to establish the target equation and the constraint equation. The target equation is aimed at finding the shortest distance between particles. The constraint equation is to ensure that the contact point is on the surface of the two particles, and contact information can be calculated by Newton iteration method and sequential quadric programing (Wang et al. 2018). However, the CFR algorithm also has certain limitations. Firstly, this method can only be used to produce centro-symmetric shapes like ellipsoids, cylinders, hexahedrons, etc. (Podlozhnyuk et al. 2016); Secondly, the convergence of the iteration should be ensured whether using the common normal method or the minimum geometric potential energy algorithm in terms of contact detection (Wellmann et al. 2008; Lu et al. 2014). In addition, the instability of the algorithm increases as the power of the function increases (i.e., the surface is sharpened). In 1995, Williams and O’Connor proposed a super-quadric detection algorithm using discrete function representation (DFR) (Williams and O’Connor 1995). In this

34

2 Constructions of Irregular Shaped Particles in the DEM

method, the particle surface is discretized to the individual points. If the geometric potential energy of the particle in the local coordinates satisfies f (x, y, z) < 0, the surface point (x, y, z) is inside the particle, and vice versa. Therefore, contact detection is simplified to calculate successively whether the particle surface points are inside the adjacent particles. Although the DFR is considered as a detection algorithm with wide application prospects, a number of critical questions need to be addressed. How many semi-discretization points are required to represent the particle surface with sufficient precision? How should the semi-discretization points be distributed on the surface of a particle? How can the contact point be identified among all semidiscretization points efficiently? Lu et al. (2012) introduced the so-called ‘uniform semi-discretization’ and ‘adaptive semi-discretization’ methods. The ‘uniform semidiscretization’ method is suitable for a perfectly symmetrical particle, i.e. a sphere. Here, the semi-discretization points are of a uniform and equally spaced distribution. On the other hand, the ‘adaptive semi-discretization’ method is suitable for superquadrics, with a larger number of semi-discretization points distributed around sharp vertices and edges, i.e. the spacing between semi-discretization points depends on the local shape of the particles. Lu et al. (2012) compared the computational performance of both DFR and CFR approaches using a number of different test cases. The two approaches gave comparable results in the simulations of dynamic behaviors of granular bed.

2.2.2 Ellipsoidal Particles Based on Super-Quadric Equation The ellipsoidal model is an irregular particle shape described by a quadric equation, and this equation can be simplified by a super-quadric function,  x 2 a

+

 y 2 b

+

 z 2 c

−1=0

x = a cos θ cos ϕ, y = b cos θ sin ϕ, z = c sin θ

(2.21) (2.22)

In a fixed coordinate system, a, b and c are the semi-axis lengths of the particles along the major axes. The value interval of the coefficient θ and ϕ in the parametric equations are taken as −π/2 ≤ θ ≤ π/2 and −π ≤ ϕ ≤ π , respectivley. The oblate and prolate shaped ellipsoids with different aspect ratios can be obtained by changing the proportional relationship between a, b and c, as shown in Fig. 2.10. The geometric potential energy method or the common normal algorithm can be used to solve the collisions of ellipsoids. The geometric potential energy method is to find a point on the surface of the target particle with the lowest geometric potential energy compared with its neighboring particles. If the value is negative, the particles are in contact; otherwise, they are not in contact. The points on the surface of the

2.2 Super-Quadric Particles

35

Fig. 2.10 Ellipsoidal models with different aspect ratios

particle are sequentially searched for the lowest geometric potential energy compared with its neighboring particles. The line connecting the two points is the normal direction of the two particles. The common normal algorithm is to find two points on the surface of the neighboring particles, and the direction of the line connecting the two points should be parallel to the outward normal direction of the neighboring particle. It can be solved by nonlinear iterative algorithm. Compared with the common normal algorithm, the geometric potential approach has higher computational efficiency. However, its disadvantage is that it may not meet the definition of contact mechanics, i.e. the normal force is perpendicular to the surface of two neighboring particles. In DEM simulations, the ellipsoidal model is widely used in many fields, such as pile sinking, packing, and hopper discharge, as shown in Fig. 2.11. Yan et al. (2010) developed a DEM-FEM coupling algorithm with ellipsoids to simulate the penetration depth of piles. Baram and Lind (2012) studied the packing properties of ellipsoids based on geometric potential energy algorithm. Zheng et al. (2013) developed a nonlinear visco-elastic contact model of ellipsoids based on parameter correction. Liu et al. (2014) studied the flow characteristics of ellipsoids in flatbottomed hoppers, and the discharging processes were in good agreement with the experimental results.

(a) Pile

(b) Packing

(c) Discharging

Fig. 2.11 DEM simulations using ellipsoidal particles (Yan et al. 2010; Baram and Lind 2012; Liu et al. 2014)

36

2 Constructions of Irregular Shaped Particles in the DEM

2.3 Polyhedral and Dilated Polyhedral Particles Polyhedral particles can better reflect the geometries of particles and have a wide range of applications in geotechnical and geological simulations. The dilated polyhedral particles using Minkowski sum theory can effectively reflect the geometric morphology of polyhedra with efficient contact detection and contact force models.

2.3.1 Polyhedral Particles The polyhedral particle is composed of several planes, resulting in a geometry mainly composed of vertexes, edges and planes, as shown in Fig. 2.12. The polyhedron may be a convex or a concave one. The contact detection of the convex polyhedron is simpler than that of the concave one, and a concave polyhedron can be made up of convex polyhedra. The convex polyhedron is mainly described here. For a polyhedron composed of N planes, it can be represented by N inequalities, ai x + bi y + ci z ≤ di , i = 1, 2, 3, · · · · ··, N

(2.23)

where ai , bi and ci represent the three components of the outward normal direction of each plane, respectively. di is the distance from the origin to the plane. In fact, the exterior surface of a polyhedron can be expressed in the form of an equation as follows, Fig. 2.12 A polyhedral particle

2.3 Polyhedral and Dilated Polyhedral Particles

f (x, y, z) =

N i=1

37

ai x + bi y + ci z − di  = 0

(2.24)

where  is the Macaulay brackets and can be expressed as x = x, x > 0; x = 0, x ≤ 0. Recently, polyhedra have been widely used in the DEM due to their arbitrary and diverse shapes (Nassauer et al. 2013). Corresponding damage fracture model has also been developed, as shown in Fig. 2.13 (Ma et al. 2016). The main issues of polyhedra are the complicated contact detection and the difficulty in establishing contact force models in different geometric contact forms, as shown in Fig. 2.14 (Nezami et al. 2004, 2006; Boon et al. 2012; Wang et al. 2015). In addition, the discontinuous deformation method based on polyhedral particles plays an important role in geotechnical engineering. It produces good results in the analysis and simulation of mechanical processes such as landslides and fractures of discrete rocks and brittle materials, and gains wide attention in rock and soil mechanics and related engineering applications (Shi 1988; Maclaughlin and Doolin 2006; Zhang et al. 2015).

(a) Hopper flow

(b) Polyhedron coupled with finite element

Fig. 2.13 Discrete element method using polyhedra

(a) Contact detection between polyhedra

(b) Common normal concept

Fig. 2.14 Contact detection algorithm for polyhedra (Nassauer et al. 2013; Ma et al. 2016; Boon et al. 2012; Nezami et al. 2006)

38

2 Constructions of Irregular Shaped Particles in the DEM

2.3.2 Dilated Polyhedral Particles Based on Minkowski Sum Recently, dilated polyhedral particles based on Minkowski sum theory have been rapidly developed in the engineering applications of discrete element methods due to the significant advantages in describing particle shapes, contact detection algorithms and force models. The Minkowski sum method was defined by the German mathematician Herman Minkowski (Varadhan and Manocha 2006). If two sets of points A and B represent closed geometries in space, respectively, their Minkowski sum is defined as, A ⊕ B = { x + y|x ∈ A, y ∈ B}

(2.25)

where x and y are the points in set A and B, respectively, i.e. A ⊕ B is the position vector sum of all points in A and B. The calculation process of Minkowski sum can be summarized as set B sweeping the exterior surface of set A. The produced new geometry is the Minkowski sum of A and B, as shown in Fig. 2.15. The Minkowski sum method has the following properties. If A and B are convex, A⊕ B is also convex;A⊕ B = B ⊕ A; μ(A⊕ B) = μA ⊕ μB; n A = A ⊕ A . . . . . . ⊕ A; A ⊕ (B ∪ C) = (A ⊕ B) ∪ (A ⊕ C). The Minkowski sum method is widely used in complex graphics construction, collision detection of bodies with complicated shapes, motion planning for robots to circumvent spatial obstacles, computer graphics and computer aided design. In DEM simulations, dilated particles constructed by the Minkowski sum of discs and spheres, and of polyhedra and spheres can better solve contact problems of the complex morphological particles and improve search efficiency and precision (Hopkins and Tuhkuri 1999; Hopkins 2004; Galindo-Torres et al. 2012; Liu and Ji 2018). The dilated disc is a particle obtained by performing a Minkowski sum operation of a sphere on a disc, as shown in Fig. 2.16. The center of the sphere moves freely on the surface of the disc, and the envelope of all possible motion trajectories of the sphere is the dilated disc model (Hopkins and Tuhkuri 1999; Hopkins 2004). The particle constructed by this method has a simple geometric description and can accurately calculate contact deformation between particles. The main descriptive parameters of the particle are its central position and the normal direction of the disc. A dilated polyhedron can be constructed by spheres and an arbitrary polyhedron using the Minkowski sum method, as shown in Fig. 2.17 (Galindo-Torres et al. 2012). The diversity of dilated polyhedra can be adjusted by different polyhedral morphologies and spherical sizes (Liu and Ji 2019). The dilated polyhedra can greatly simplify the contact detection between polyhedra, hence effectively improve the calculation efficiency of the discrete element method with the satisfaction of calculation accuracy. Moreover, the sharp vertex contacts between polyhedra can be transformed into smooth spherical or cylindrical surface contacts, resulting in an effective solution of the normal direction and the overlap. Due to the smooth surface of the dilated polyhedra, the traditional contact mechanics models can be used, such as Hertz contact theory.

2.3 Polyhedral and Dilated Polyhedral Particles

39

Fig. 2.15 Constructions of geometric mophologies based on Minkowski sum(Barki et al. 2009; Peternell and Steiner 2007) Fig. 2.16 Construction of the dilated disc

40

2 Constructions of Irregular Shaped Particles in the DEM

(a) Construction of a dilated polyhedron

(b) Dilated polyhedra with different dilation radii Fig. 2.17 Construction of dilated polyhedra with Minkowski sum

2.4 Advanced Constructions of Novel Irregular Shaped Particles In recent years, with the development of large-scale DEM calculations and the higher requirements for construction precision of irregularly shaped particles, the construction of novel irregularly shaped particles has gradually become a research hot topic. The effective and stable contact detection algorithm is the current challenge of the topic (Lu et al. 2015; Zhong et al. 2016).

2.4.1 Random Star-Shaped Particles 3D random star-shaped particles are generally constructed in a spherical coordinate system. The distance from the particle center to the surface is a single-valued function of azimuth and elevation. Therefore, it cannot have ‘cantilever’ and internal cavities. The construction of particles has a large free degree and can generate more complex geometrical shapes. Random star-shaped particles can be used in many fields such as industrial processing, concrete materials, rock, soils, and asteroids in nature, as shown in Fig. 2.18. The construction of random star-shaped particles generally uses a spherical harmonic function method. Firstly, the polar coordinate values of the radial distances of the surface points are obtained from the 3D image of the particle. The particle surface is then represented by the series form of the spherical harmonic function. The surface function of the star-shaped particle can be represented in spherical coordinates, r (θ, ϕ) =

 N n n=0

m=−n

anm Ynm (θ, ϕ)

(2.26)

2.4 Advanced Constructions of Novel Irregular Shaped Particles

41

Fig. 2.18 3D star-shaped random particles of concrete aggregates (Qian et al. 2016; Zhu et al. 2014)

where N is the expansion order and closely related to the accuracy of the particle surface. anm is the expansion factors based on m and n. Ynm is a spherical harmonic function and can be expressed as,  Ynm (θ, ϕ)

=

(2n + 1)(n − m)! m pn cos(θ )eimϕ 4π(n + m)!

(2.27)

where pnm is associated Legendre functions of degree n and order m. Considering cos(θ ) = x, pnm can be expressed as dm pn (x) dx m   n d 1 2 n pn (x) = n (x − 1) 2 n! dx n m

pnm (x) = (−1)m (1 − x 2 ) 2

(2.28) (2.29)

If m is negative, the above equation can be given by, pnm (x) = (−1)−m

(n + m)! −m p (x) (n − m)! n

(2.30)

Figure 2.19 shows the particles’ shapes when N varies between 1 and 30 (anm database is from Garboczi and Bullard (2017).

42

2 Constructions of Irregular Shaped Particles in the DEM

N =1

N =5

N = 10

N = 20

N = 30

Fig. 2.19 Star-shaped random particles when N is equal to 1, 5, 10, 20 and 30

Due to the complexity of the representation of random star-shaped particles, there is no uniform analytical form for its contact detection. Existing methods includes separation axis algorithm and contact function algorithm (Zhu et al. 2014; Garboczi and Bullard 2013). However, these algorithms require more precise function forms due to the precision expectation of complex geometric models, which results in a significant decrease in computational efficiency.

2.4.2 B-Spline Function Models The B-spline model is a non-interpolation method for fitting a target contour through a Bezier curve, which obtains the curve form by interactively determines a set of control points, as shown in Fig. 2.20. This curve equation is based on the Bernstein basis function (Lim and Andrade 2014). If one or several of the control points are changed, the overall shape of the curve is not affected. Various irregular shapes can be obtained since the curve equation is determined by control points. The diversity and mutability of the method provides the possibility of approximation to irregularly shaped particles with functions. The method has the characteristics of the relatively mature Bezier

Fig. 2.20 Constructions of B-spline particles (Lim and Andrade 2014)

2.4 Advanced Constructions of Novel Irregular Shaped Particles

43

Fig. 2.21 Packing of B-spline particles (Lim and Andrade 2014)

surface algorithm, the relatively smooth displayed graphics, and the relatively few control parameters, which makes it suitable for constructing and displaying graphics of irregularly shaped particles. This method is commonly used for the study of the packing and mechanical properties of irregularly shaped particles, as shown in Fig. 2.21. However, the control points of the B-spline model are often selected empirically. In general, when the contour changes drastically and the number of control points should be increased in order to accurately approximate the contour. When the contour changes gently, the control points can be appropriately selected with the satisfaction of precision. In other words, the selection of control points will significantly affect the calculation efficiency.

2.4.3 Combined Geometric Element Method The combined geometric element method is to obtain a set of geometric elements by combining surfaces of any arbitrary geometric shapes, such as plane or smooth curves. This method makes the surface of the particle discontinuous at the vertex or edge. It is suitable for constructing conventional irregular particle shapes with smooth surfaces or edges, such as solid cylinders, pipes, sphero-cylinders and tablets with the convex surfaces at the two ends, as shown in Fig. 2.22. This algorithm is widely used in the fields of chemical industry, geotechnical and minerals. Figure 2.23 shows the static packing process of the elements. This method is different from the traditional spherical clumping algorithm in the sense that it is not limited to the use of spherical particles. A new geometric model can be generated by a combination of elements such as ellipsoids and cylinders.

44

2 Constructions of Irregular Shaped Particles in the DEM

(a) Solid cylinder

(b) Pipeline

(c) Tablet

Fig. 2.22 Combined geometric elements (Guo et al. 2013; Wehinger et al. 2015; Dong et al. 2015)

Fig. 2.23 Static packing of cylinders (Kodam et al. 2012; Dong et al. 2015)

However, the contact detection algorithm must search the vertices, edges, and faces, and determine each contact mode that may exist between elements, including vertexedge, edge-edge, vertex-face and face-face contact. This search process will greatly limit the parallelism ability of large-scale computing (Guo et al. 2013; Wehinger 2015; Dong et al. 2015).

2.4.4 Potential Particle Model Potential particle model is a function representation algorithm that can be used to construct a wide range of particle shapes from circle to polyhedron. By proposing the function concept of surface potential energy, the particle shape, surface curvature and normal direction can be effectively controlled. The exterior dilated spheres can

2.4 Advanced Constructions of Novel Irregular Shaped Particles

45

Fig. 2.24 Construction of potential particle model (Boon et al. 2012)

ensure the strict convex shape of the particle, as shown in Fig. 2.24. For this construction algorithm, the contact detection between particles often depends on finding the lowest geometric potential point on the surface. The numerical iterative algorithm is used to solve the optimization equation, and the contact points and normal direction between particles can be obtained. Meanwhile, the calculation results of the previous DEM time step are used as the initial prediction for the next time step, which can improve the calculation efficiency of the method. In general, this algorithm reduces the complexity of different contact modes of the traditional polyhedral algorithm and the dilated polyhedral algorithm. It is of benefit for large-scale parallel computing. Moreover, this algorithm compensates for the inability to construct a polyhedral particle in quadratic functions and hyper-quadratic functions. Potential particle method is the one that is easy to use widely and convenient for parallelization in the novel construction models of particles (Houlsby 2009).

46

2 Constructions of Irregular Shaped Particles in the DEM

Fig. 2.25 Poly-superellipsoids with random shape parameters (Zhao and Zhao 2019)

2.4.5 Poly-superellipsoid Model The poly-superellipsoid approach is based on the poly-ellipsoid method, which can be used to describe a large number of convex particle shapes and represent major shape characteristics, such as surface roughness and aspect ratio, as shown in Fig. 2.25 (Zhao and Zhao 2019). This novel model is constructed by assembling eight pieces of super-ellipsoids, and the shape of each octant is obtained by super-ellipsoid equations. Hence, one poly-superellipsoid element has 40 unknown parameters. Considering the continuity and smoothness of the surface of the non-spherical particles, only 8 parameters are needed. Moreover, arbitrarily shaped particles can be discretized into a series of point clouds, and the fitted particle shape is obtained by minimizing the relative error of the inside-and-outside function. The contact detection between polysuperellipsoid elements is thus transformed into an optimization problem, which can be established by the common normal method. LM and GJK algorithms are effective approaches to solve the optimization problems and the normal overlap is obtained. This novel model overcomes the shortcomings of the geometric symmetry of the super-ellipsoid elements and can be used to construct geometrically asymmetrical shapes like pebbles.

2.5 Summary

47

2.5 Summary This chapter mainly introduces the bonding and clumping model based on spherical particles, the ellipsoid and super-quadric model based on function envelope, the polyhedral and dilated polyhedral model based on geometric topology and Minkowski sum theory. In addition, the construction methods and applications of novel irregularly shaped particles are briefly described, including 3D random star-shaped model, B-spline function model, combined geometric element model and potential particle model. Compared with the traditional spherical particle model, the irregular particle morphologies can accurately reflect the basic mechanical behaviors of granular materials. Meanwhile, it has a significant effect on the macroscopic and microscopic analysis of granular materials. However, different methods have their unique advantages and limited scope of applications considering the flexibility, numerical stability and computational efficiency of the algorithms. The development of more effective and stable construction methods for irregularly shaped particles is still an important topic in the future. In addition, the contact theory of the traditional spherical particles is mainly extented to the irregular particles. Unfortunately, the contact force is related with the contact area and the equivalent curvature at the overlapping region for irregular particles, in particularly, for those with sharp vertices, sides and faces. Therefore, the development of the contact theory for arbitrarily shaped particles is another important challenge in the future.

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Garboczi EJ, Bullard JW (2017) 3D analytical mathematical models of random star-shape particles via a combination of X-ray computed microtomography and spherical harmonic analysis. Adv Powder Technol 28(2):325–339 Guo Y, Wassgren C, Hancock B et al (2013) Granular shear flows of flat disks and elongated rods without and with friction. Phys Fluids 25(6):876–879 Hopkins MA, Tuhkuri J (1999) Compression of floating ice fields. J Geophys Res Atmos 1041:15815–15826 Hopkins MA (2004) Discrete element modeling with dilated particles. Eng Comput 21(2/3/4):422– 430 Houlsby GT (2009) Potential particles: a method for modelling non-circular particles in DEM. Comput Geotech 36(6):953–959 Ji S, Wang S, Peng Z (2019) Influence of external pressure on granular flow in a cylindrical silo based on discrete element method. Powder Technol 356:702–714 Khazeni A, Mansourpour Z (2018) Influence of non-spherical shape approximation on DEM simulation accuracy by multi-sphere method. Powder Technol 332:265–278 Kodam M, Curtis J, Hancock B et al (2012) Discrete element method modeling of bi-convex pharmaceutical tablets: Contact detection algorithms and validation. Chem Eng Sci 69(1):587– 601 Li Y, Ji S (2018) A geometric algorithm based on the advancing front approach for sequential sphere packing. Granul Matter 20(4):59 Lim K, Andrade J (2014) Granular element method for three-dimensional discrete element calculations. Int J Numer Anal Meth Geomech 38(2):167–188 Lim WL, McDowell GR (2005) Discrete element modeling of railway ballast. Granul Matter 7:19– 29 Liu L, Ji S (2019) Bond and fracture model in dilated polyhedral DEM and its application to simulate breakage of brittle materials. Granul Matter 21(3):41 Liu L, Ji S (2018) Ice load on floating structure simulated with dilated polyhedral discrete element method in broken ice field. Appl Ocean Res 75:53–65 Liu SD, Zhou ZY, Zou RP et al (2014) Flow characteristics and discharge rate of ellipsoidal particles in a flat bottom hopper. Powder Technol 253(253):70–79 Lobo-Guerrero S, Vallejo L (2006) Discrete element method analysis of rail track ballast degradation during cyclic loading. Granular Matter 8(3–4):195–204 Lu G, Third JR, Müller CR (2012) Critical assessment of two approaches for evaluating contacts between super-quadric shaped particles in DEM simulations. Chem Eng Sci 78(34):226–235 Lu G, Third JR, Müller CR (2015) Discrete element models for non-spherical particle systems: From theoretical developments to applications. Chem Eng Sci 127:425–465 Lu G, Third JR, Müller CR (2014) Effect of particle shape on domino wave propagation: a perspective from 3d, anisotropic discrete element simulations. Granul Matter 16(1):107–114 Lu M, McDowell GR (2007) The importance of modelling ballast particle shape in the discrete element method. Granul Matter 9(1–2):69–80 Ma G, Zhou W, Chang XL et al (2016) A hybrid approach for modeling of breakable granular materials using combined finite-discrete element method. Granul Matter 18(1):1–17 Maclaughlin MM, Doolin DM (2006) Review of validation of the discontinuous deformation analysis (DDA) method. Int J Numer Anal Meth Geomech 30(4):271–305 Nassauer B, Liedke T, Kuna M (2013) Polyhedral particles for the discrete element method. Granular Matter 15(15):85–93 Nezami EG, Hashash YMA, Zhao D et al (2004) A fast contact detection algorithm for 3d discrete element method. Comput Geotech 31(7):575–587 Nezami EG, Hashash YMA, Zhao D et al (2006) Shortest link method for contact detection in discrete element method. Int J Numer Anal Meth Geomech 30(8):783–801 Ngo NT, Indraratna B, Rujikiatkamjorn C (2014) DEM simulation of the behaviour of geogrid stabilised ballast fouled with coal. Comput Geotech 55(1):224–231

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Peternell M, Steiner T (2007) Minkowski sum boundary surfaces of 3D-objects. Graph Model 69(3–4):180–190 Podlozhnyuk A, Pirker S, Kloss C (2016) Efficient implementation of superquadric particles in discrete element method within an open-source framework. Comput Part Mech 4(1):101–118 Potyondy DO, Cundall PA (2004) A bonded-particle model for rock. Int J Rock Mech Min Sci 41(8):1329–1364 Qian Z, Garboczi EJ, Ye G et al (2016) Anm: a geometrical model for the composite structure of mortar and concrete using real-shape particles. Mater Struct 49(1–2):149–158 Shi GH (1988) Discontinuous deformation analysis-a new model for the statics and dynamics of block systems. PhD thesis, University of California, Berkeley Sinnott MD, Cleary PW (2016) The effect of particle shape on mixing in a high shear mixer. Comput Part Mech 3(4):477–504 Soltanbeigi B, Podlozhnyuk A, Papanicolopulos S et al (2018) DEM study of mechanical characteristics of multi-spherical and superquadric particles at micro and macro scales. Powder Technol 329:288–303 Su D, Yan WM (2018) 3D characterization of general-shape sand particles using microfocus Xray computed tomography and spherical harmonic functions, and particle regeneration using multivariate random vector. Powder Technol 323:8–23 Varadhan G, Manocha D (2006) Accurate Minkowski sum approximation of polyhedral models. Graph Model 68(4):343–355 Wang J, Li S, Feng C (2015) A shrunken edge algorithm for contact detection between convex polyhedral blocks. Comput Geotech 63(63):315–330 Wang S, Fan Y, Ji S (2018) Interaction between super-quadric particles and triangular elements and its application to hopper discharge. Powder Technol 339:534–549 Wehinger GD, Eppinger T, Kraume M (2015) Evaluating catalytic fixed reactors for dry reforming of methane with detailed CFD. Chem Ing Tec 87(6):734–745 Wellmann C, Lillie C, Wriggers P (2008) A contact detection algorithm for superellipsoids based on the common-normal concept. Eng Comput 25(5):432–442 Williams JR, Pentland AP (1992) Superquadrics and modal dynamics for discrete elements in interactive design. Eng Comput 9(2):115–127 Yan B, Regueiro RA, Sture S (2010) Three-dimensional ellipsoidal discrete element modeling of granular materials and its coupling with finite element facets. Eng Comput 27(4):519–550 Yan Y, Zhao J, Ji S (2015) Discrete element analysis of breakage of irregularly shaped railway ballast. Geomech Geoengin 10(1):1–9 Zeng YW, Jin L, Du X et al (2015) Refined modeling and movement characteristics analyses of irregularly shaped particles. Int J Numer Anal Meth Geomech 39(4):388–408 Zhang Y, Wang J, Xu Q et al (2015) DDA validation of the mobility of earthquake-induced landslides. Eng Geol 194:38–51 Zhao S, Zhang N, Zhou X et al (2017) Particle shape effects on fabric of granular random packing. Powder Technol 310:175–186 Zheng QJ, Zhou ZY, Yu AB (2013) Contact forces between viscoelastic ellipsoidal particles. Powder Technol 248(2):25–33 Zhong W, Yu A, Liu X et al (2016) DEM/CFD-DEM modelling of non-spherical particulate systems: theoretical developments and applications. Powder Technol 302:108–152 Zhu Z, Chen H, Xu W et al (2014) Parking simulation of three-dimensional multi-sized star-shaped particles. Model Simul Mater Sci Eng 22(3):035008 Williams JR, O’Connor R (1995) A linear complexity intersection algorithm for discrete element simulation of arbitrary geometries. Eng Comput 12:185–201 Zhao S, Zhao J (2019) A poly-superellipsoid-based approach on particle morphology for DEM modeling of granular media. Int J Numer Anal Meth Geomech 43:2147–2169

Chapter 3

Contact Force Models for Granular Materials

In the discrete element method (DEM), each particle in the bulk material is considered to be an independent discrete element which has its own physical parameters (shape, density, size and et al.) and mechanical properties (elastic modulus, Poisson’s ratio and et al.). The interconnection and constraints between discrete particles are activated by contacts, which can fully reflect the discontinuity of granular materials. In DEM simulations, particles’ attributes, such as the position, velocity and angular velocity, are computed and stored at each time step. In sum, this method has the advantage of taking full account of the unique properties of each particle. Mechanical descriptions of the static or quasi-static granular contacts are the core of granular dynamics. It is also the basis for the further establishment of viscous collision and dynamic contact model between particles. For the elastic contact of coarse particles, some contact models have been developed. Among these, the linear contact model and nonlinear Hertz-Mindlin contact model are the two used more widely with the consideration of elastic force and viscous force. In order to simulate more complex behaviors of granular materials, the DEM considering elastic-plastic contact theory is gradually developed and improved. For the adhesive contact of fine particles, van der Waals force is the main influencing factor of particle’s motion. In order to more accurately describe the relationship between the force and deformation of the particle, multiple adhesive contact models have been proposed. Friction force also plays an important part in the interaction between particles in contact, which includes rolling friction and sliding friction. Research on the conversion mechanism of the rolling-sliding friction helps better explain the macro dynamic characteristics of granular materials. Finally, in order to simulate the mechanical behaviors of continua with the DEM, the bonding-breakage model based on the beam theory has been developed and applied in many engineering fields. In order to construct complex particles more closer to the actual particles’ geometries, a variety of granular particles with irregularly shapes have been successively proposed, such as hyper-quadric particles and dilated polyhedral particles. However, linear or nonlinear contact models of spherical particles are not applicable to the © Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_3

51

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contact relationship of irregularly shaped particles. It is necessary to develop contact models for irregular shaped particles. The Hertz contact model of spherical particles and the contact model based on the surface curvature of the ellipsoid provide a reference for the calculation of the contact force between irregularly shaped particles. At the molecular or atomic level, the adhesion behavior between particles cannot be neglected. Theories proposed by Bradley, DMT and JKR explain reasonably the phenomenon of inter-particle adhesion. Dry granular matter exhibits specific phenomena and behaviors. When liquid added in, complicated adhesion force is created between particles which makes even more complicated phenomena in granular matter. To explore the cause of macroscopic damage of granular materials during the process from drying to wetting, it is necessary to study the microscopic mechanism of hydrodynamic interactions. A variety of 3D liquid-bridge force models have been proposed. In addition, the heat transfer in granular materials is common in industry, which can be studied by the DEM due to its unique advantages. Based on the heat conductivity equation, the discrete element governing equation is derived, in which the influence of thermal resistance of the contact surface is fully taken into account. The above theories form the basis of the discrete element method.

3.1 Visco-Elastic Contact Models of Spherical Particles The establishment of a reasonable contact model is the foundation for the accuracy of simulation results. However, the realistic interaction between contacting particles is very complicated. A reasonable simplification is necessary for the consideration of computational efficiency and application. Currently, the linear and nonlinear Hertz contact model is commonly used with the consideration of elastic and viscous forces. These models are phenomenological models which adopt springs, dampers and sliders. The contact force can be divided into normal and tangential component, as shown in Fig. 3.1. In the figure, K n and Cn are the normal stiffness and viscous coefficient, respectively. K s and Cs are the tangential stiffness and viscous coefficient, respectively. Contact stiffness is defined as the relationship between the contact force and relative displacement at the contact point of the two particles in contact. At each time step, the normal contact force can be calculated by the normal relative displacement, while the increment of the tangential force can be obtained by the increment of the tangential relative displacement. There are two main types of contact models, that is, the linear contact model and the nonlinear Hertz-Mindlin contact model. In the linear contact model, the contact force and relative displacement exhibit a linear correlation with contact stiffness. It has the advantages of simple calculation and easy to be programmed. In fact, the contact between particles is not a simple face-to-face contact mode because of different shapes of particles. Take the 2D disc as an example. It has an arc-arc contact mode. The larger the interaction between

3.1 Visco-Elastic Contact Models of Spherical Particles

53

Fig. 3.1 The normal and tangential contact model between spherical particles

the contacting particles is, the greater the overlap between the contacting particles is. This inevitably induces the increase of the local contact area. As a result, the contact stiffness changes with the increasing overlap during the deformation of the two contacting particles (Zhou and Zhou 2011; Ji and Shen 2006). Consequently, the nonlinear contact model represented by the Hertz-Mindlin contact model is the model widely used in engineering applications.

3.1.1 Linear Contact Model In the linear contact model of spherical particles, the contact force includes the normal force Fn perpendicular to the contact surface and the tangential force Ft parallel to the direction of the contact surface. The normal force Fn consists of elastic force Fe and viscous force Fv , Fn = Fe + Fv

(3.1)

Fe = K n xn , Fv = −Cn x˙n

(3.2)

with

where xn and x˙n are the normal relative displacement and relative velocity between two contacting particles. K n and Cn are the normal stiffness and damping coefficient.

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The normal stiffness can be expressed by: Kn=

knA knB knA +knB

(3.3)

where superscripts A and B represent the two particles in contact. knA and knB are the normal contact stiffness of particle A and B, respectively. In the linear contact model, the stiffness of the particle can be determined approximately by its size and elastic modulus (Ji and Shen 2008). The normal damping coefficient between the two particles is,  Cn = ξn 2M K n ξn = 

- lne π2

+ (lne)2

(3.4) (3.5)

where ξn is the damping ratio, e is the restitution coefficient of the particle, M is the equivalent mass of the two particles with M=

MAMB MA + MB

(3.6)

The tangential contact force is generally calculated in an incremental form, that is, Fs = K s xs or Fs = K s x˙s t. xs and x˙s are the increment of the tangential displacement and the relative tangential velocity of the two particles at the contact point, respectively. t is time step. K s is the tangential stiffness, and can be calculated by Ks=

ksA ksB ksA +ksB

(3.7)

where ksA and ksB are the tangential contact stiffness of particle A and B, respectively. Based on the assumption that the contact force between particles meets the Coulomb friction law, the tangential force at the contact point can be written as, Ft = K s xs − Cs vs if |Ft | ≤ μ|Fn |

(3.8a)

Ft = μFn if |Ft | > μ|Fn |

(3.8b)

Equations (3.8a) and (3.8b) can be combined into, Ft = min(|K s xs −Cs vs |, |μFn |)

(3.9)

3.1 Visco-Elastic Contact Models of Spherical Particles

55

where Cs is the tangential damping coefficient, μ is the friction coefficient. The tangential stiffness and damping coefficient are proportional to those normal ones. The corresponding relationship is K s = α K n and Cs = βCn . Generally, α = 0.8 or 1.0, β = 0.0 (Babic et al. 1990; Shen and Sankaran 2004; Campbell 2002; Zhang and Rauenzahn 2000). In the linear contact model, the time step size is generally about 1/50 ~ 1/20 of the binary contact time. The duration of binary contact is defined as (Babic et al. 1990), Tbc = 

π   2K n 1 − ξn2 M

(3.10)

where Tbc is the duration of binary contact, which means the contact time of the two spherical particles from collision to separation.

3.1.2 Nonlinear Contact Model The Hertz contact theory is generally used to calculate the normal elastic force between spherical particles. Following assumptions are made. The particle surface is smooth and homogeneous; The contact surface is far less than the particle surface; Only elastic deformation occurs on the contact surface; And the contact force is perpendicular to the contact surface. As the theoretical basis of the elastic contact between particles, the Hertz contact theory is suitable for the elastic contact studies of curved bodies such as spheres, cylinders and ellipsoids. It can also be used in the contact calculation of micro-convex bodies of contacting particles. The tangential force between particles was established by Mindlin and Deresiewicz (1953). Due to the nonlinear relationship between deformation and contact force, the normal and tangential force mix together, and cannot be calculated based on single normal or tangential action. Therefore, deformation and contact force is often calculated in a superposition form in numerical simulations. (1) Nonlinear normal force The normal force between two particles is shown in Fig. 3.2a. R A and R B are the radius of the two contacting particles. The normal force F n can be calculated in terms of the normal overlap δ as follows, 4  1/2 3/2 δ Fn = E ∗ R ∗ 3

(3.11)

where E ∗ and R ∗ are the Young’s modulus and the equivalent radius, respectively, and given by, R∗ =

RA RB EA EB    E∗ =  2 RA + RB 1 − ν A E B + 1 − ν B2 E A

(3.12)

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(a) Deformation under normal force

(b) Deformation under tangential force

Fig. 3.2 The schematic of the two contacting spherical particles

where E A , ν A , E B and ν B are the elastic modulus and Poisson’s ratio of particle A and B, respectively. Based on Eq. (3.11), if the overlap increment between the two contacting particles is δ in a time step, the corresponding increment of the normal contact force is calculated as, Fn = 2a E ∗ δ

(3.13)

 R ∗ 1/3 (Vu-Quoc et al. 2001). where the contact radius a = 3P 4E ∗ Due to the elastic deformation occurring at the contact surface of the two particles, the corresponding pressure can be written as,  p(r ) = pm 1 −

 r 2 1/2 a

(3.14)

where r is the distance from a point on the contact surface to the contact center. pm 3Fn is the maximum pressure when r = 0, and pm = 2πa 2. (2) Nonlinear tangential force In general, the tangential contact force between particles can be calculated by the Mindlin-Deresiewicz contact theory (Mindlin and Deresiewicz 1953; Vu-Quoc et al. 2001). Due to the complexity of the method, only its simplified model is introduced here. When two spherical particles are in contact in the tangential direction, the corresponding deformation is shown in Fig. 3.2b. The tangential force increment

3.1 Visco-Elastic Contact Models of Spherical Particles

57

Fs is defined as a function of relative tangential displacement increment η on the contact surface and expressed as, Fs = 8aG ∗ θk η + (−1)k μ(1 − θk )Fn

(3.15)

where k = 0, 1, 2, which correspond to loading, unloading and unloading-reloading, respectively. If |Fs | < μFn , θk = 1. If |Fs | ≥ μFn , ⎧  1/3 ⎪ n ⎨ 1− Fs +μF ,k=0 μFn θk =  (3.16) 1/3 k ⎪ ⎩ 1− (−1) ( Fs −Fs,k )+2μFn , k = 1, 2 2μFn where μ is the friction coefficient of the particle surface. G ∗ is the shear modulus, which is defined as: 1 2 − νA 2 − νB = + ∗ G GA GB

(3.17)

where G A and G B are the shear modulus of the two particles. Fs,k is the tangential contact force considering unloading or reloading history, which is updated at each time step. Based on Eqs. (3.11)–(3.16), the increment of the tangential force Fs induced by the tangential displacement increment not only depends on the loading history but also is affected by the normal contact force. The tangential overlap between the particles and the contact force at a time step are updated as follows, η N +1 = η N + η

(3.18a)

FsN +1 = FsN + Fs

(3.18b)

where η N and FsN , η N +1 and FsN +1 are the tangential overlap and tangential contact force at time step N and N + 1, respectively. When multiple consecutive contacts are applied on the particles surface, the surface may become harden due to collision, causing the variation of the experiment results. During the process of the increase of the tangential contact force until the relative slip occurring on the surface of the contacting particles, the tangential force will wear away fine unevenness on the surface, resulting in a changed surface performance. Thus, the Hertz normal contact theory and Mindlin-Deresiewicz tangential contact theory are the calculation methods under ideal conditions. When particles are in contact, the contact surface is subjected to an alternating stress. A polarized wave will propagate along the particle surface, which is called the Rayleigh wave. The time step in the Rayleigh wave method is determined by the

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time required by the propagation of contact deformation stress to the particle surface. The velocity of the Rayleigh wave can be calculated as:  v R = (0.163ν + 0.877)

G ρ

(3.19)

The contact force is confined to the two contacting particles and cannot be transmitted to other particles by Rayleigh waves. Then, the time step should be less than the time required for the Rayleigh wave propagating along the hemisphere (Kremmer and Favier 2001a, b): πR πR = t = vR 0.163ν + 0.877



ρ G

(3.20)

For the systems with different particle compositions, the time step is: 

R t = π 0.163ν + 0.877



ρ G

(3.21) min

It should be noted that the appropriate time step is selected to ensure the stability of the numerical calculation based on the complexity of the particle motion.

3.2 Elastic-Plastic Contact Models of Spherical Particles The interaction between granular particles is mainly based on Hertz contact theory, in which the elastic theory is used to predict the collision behavior between particles. Unfortunately, this method is only suitable for low-speed collision problems. When the collision speed is high, the predicted impact force is much higher than the actual impact force. Granular materials are mostly elastic-plastic materials. A significant plastic deformation will be generated between the contacting particles when the impact velocity reaches a certain value. Consequently, the Hertz contact theory is no longer applicable. The elastic-plastic contact model of granular materials is an important part of the discrete element simulations. A detailed contact model can make simulation results more accurate and model more complicated behaviors of granular materials.

3.2.1 Normal Elastic-Plastic Contact Model In the plastic model proposed by Thornton (1997), it is assumed that a plastic stage occurs right after the elastic stage which obeys the Hertz contact pressure distribution.

3.2 Elastic-Plastic Contact Models of Spherical Particles

(a) Contact pressure distribution

59

(b) Load displacement relationship

Fig. 3.3 Ideal elastic-plastic contact model (Thornton 1997)

In the plastic stage, an ultimate pressure is defined approximately as Py ≈ 2.5Y , which is equivalent to the Hertz contact pressure after the despiking of peak pressure (Thornton 1997). As illustrated in Fig. 3.3a, a is the contact radius, a y is the radius when yielding starts on the contact area, a p is the radius of the contact area with an assumed uniform pressure of Py . The normal contact force is the sum of the elastic force acting on the elastic zone and the plastic force acting on the plastic zone. The pressure distribution on the contact area is: p(r ) =

  r 2 1/2 3Fn 1 − 2πa 2 a

(3.22)

It can be seen from the above equation that the contact pressure value reaches its highest at the center of the contact area. Therefore, yielding starts from this area. When the center pressure reaches the yield limit, contacting particles in the range of a p < r < a are in the elastic stage, and those in the range of r ≤ a p are in the plastic stage. Based on the nonlinear Hertz contact theory, the radius of the yielding zone and the overlap can be expressed as, π R ∗ Py 2E ∗   π Py 2 ∗ = R 2E ∗

ay = xn,y

(3.23) (3.24)

During the unloading process, the force-displacement relationship is considered to follow the Hertz theory until the contact force is zero (Fig. 3.3b). For the unloading process of the normal elastic-plastic contact, the normal pressure is a function of the radial position. The comparison between elastic-plastic contact theory and traditional Hertz contact theory is shown in Fig. 3.4a (Mesarovic and Johnson 2000). Although the final contact force tends to be zero, the unloading process is not the same. Wu

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3 Contact Force Models for Granular Materials

Fig. 3.4 Unloading process at the contact of granular materials

(a) Comparison between elastic-plastic and traditional Hertz contact theory [Mesarovic and Johnson, 2000]

(b) Comparison between elastic-plastic contact theory results and finite element results [Wuet al., 2003]

et al. considered the plastic behavior when the particle rebounds, and compared the theoretical results with the finite element analysis results. The two results match well, as seen in Fig. 3.4b (Wu et al. 2003).

3.2.2 Tangential Elastic-Plastic Contact Model Through the Hertz contact theory, Mindlin (1949) obtained the relationship between the tangential force and displacement (Mindlin 1949). When similar spheres are subjected to constant normal force and variable tangential force in elastic-frictional contact, the tangential contact theory so far is the incremental form proposed by

3.2 Elastic-Plastic Contact Models of Spherical Particles

61

Mindlin and Deresiewicz as described in Sect. 3.1.2. The normal force, tangential force and friction force influence each other. Many studies assume that increasing tangential force have no effect on the size and shape of the contact area induced by the normal preload. However, the experimental studies on the evolution of the contact area have proved this is not accurate. For example, the contact area of the steel ball-plate and metallic ball-smooth plate increase with the increasing tangential force by using indirect test methods such as measuring contact force and contact resistance. Based on the experimental results, Tabor (1959) proposed the concept of ‘contact area evolution’ to explain the requirement of remaining the von Mises stress unchanged at the contact point which yields (Tabor 1959). The area of the contact surface already yielded under the normal load is inevitably increased when subjected to a tangential force, thereby reducing the average contact pressure and withstanding additional shear stress. Kogut and Etsion (2003) proposed a semi-analytical approximation method to solve the elastic or elastic-plastic critical sliding problem between a deformable sphere and rigid plane under normal and tangential loads (Kogut and Etsion 2003). According to the Mises criterion, a critical sliding inactivates plastic yielding. However, before the critical sliding, a plastic region other than a point extending to the sphere surface is assumed in the model, as illustrated in Fig. 3.5. The model also assumes that the contact area, overlap and contact pressure distribution induced by the normal preload (under the sliding condition) remain unchanged during the loading process of the additional tangential loads. When the normal preload reaches 14 times the critical load at yielding, the contact surface will not be able to withstand any additional tangential loads. Chang and Zhang (2007) derived the static friction theoretical model for the contact of the elastic-plastic ball and rigid plate (Chang and Zhang 2007). In the model, the evolution of contact area does not exist during elastic contact, but increases after plastic yielding, and reaches to a constant value during fully plastic contact.

(a) Plastic region just reaching the contact interface

(b) Plastic region at sliding inception

Fig. 3.5 Elastic and plastic zones in a sphere under combined normal and tangential loading (Kogut and Etsion 2003)

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3 Contact Force Models for Granular Materials

Although many assumptions have been made, it is still difficult to give an analytical solution for the viscous contact problem due to the behaviors of non-elastic materials and complicated contact conditions. Brizmer et al. studied the contact between the sphere and the rigid plane under the completely viscous and smooth conditions with the consideration of material properties (Brizmer et al. 2006). The elastic-plastic contact were investigated in subsequent studies. Studies show that contact conditions and mechanical properties have little effect on contact parameters. However, the evolution of the plastic zone is essentially different under the fully viscous and smooth contact condition. Olsson and Larsson (2016) proposed an algorithm for calculating the elastic-plastic contact force and contact region between two different spherical particles, in which an implicit function containing the material parameters was included to describe the plastic behavior of the particles(Olsson and Larsson 2016). This model is based on the traditional Brinell hardness test, and the indentation in the contact region is similar to the powder compaction process. The theoretical results are in good agreement with the finite element results. Currently, a perfect theory concerning the influence of the tangential force on the evolution of the contact region has not been yet formed. The acceptable theoretical model is Tabor theory, but the model lacks the exact solution of the stress field in the contact region and is only suitable for high normal preloads. In addition, the related research on the elastic-plastic contact model with respect to relative rolling between two spheres has not been reported.

3.3 Rolling Friction Models of Spherical Particles The microscopic actions between particles determine the macroscopic properties of granular materials. An in-depth study of the particulate behavior on the microscopic scale is an effective way to understand the comprehensive behaviors of granular materials on the macroscopic scale (Wensrich and Katterfeld 2012). The sliding and the rolling friction between particles have a great influence on the microscopic and macroscopic dynamic characteristics of granular materials. In the traditional discrete element models, the spring and damper in both normal and tangential directions are used to characterize the contact constitutive relationship, without considering the rolling effect between particles. However, during the deformation of granular materials, relative rolling or non-slipping rolling may occurs between the contacting particles or between the particles and boundary. The compressive deformation in the contact region hinders the rolling and generates the rolling friction (Iwashita and Oda 2000). In recent years, with the deep research on granular mechanics, the influence of granular rolling behavior on the mechanical properties of granular materials has drawn more and more attention. It has been confirmed that granular rolling cannot be neglected in numerous numerical studies and physical experiments (Jiang et al. 2006).

3.3 Rolling Friction Models of Spherical Particles

63

3.3.1 Rolling Friction Law Bardet and Huang (1992) first considered the rolling constraint in the discrete element model, and found that the discrete element simulation results without considering the rolling constraint are outside the range of theoretical values. Therefore, they speculated that the contact moment may play an important role, and proved that the internal friction angle in the granular system increases if the rolling degree of freedom is restrained (Bardet and Huang 1992). In the shear band test of granular materials, Oda and Iwashita obtained the numerical results which are consistent with the experiment results by introducing rolling resistance into the calculation model, and proposed an improved discrete element model (MDEM) (Oda and Iwashita 2000). After that, the rolling friction model of granular materials has been extensively studied in many fields such as the shear behavior at the granular interface, micro mechanical model of non-isotropic contact of granular materials, shear band in dense granular system and yielding of force chains. Jiang et al. noted the limitations of the MDEM that the empirical trials are needed in the determinations of model parameters. On the basis of contact mechanics, they gave new definitions of pure sliding and pure rolling, and added an additional parameter ‘anti-rotation coefficient β’ into the model, and established a new model with rolling resistance (Jiang et al. 2005). During the movement of granular materials, the rolling resistance is mainly caused by the elastic hysteresis effect and plastic deformation. The micro-slip and adhesion between particles induce little rolling resistance. In general, the deformation caused by an increasing loading is smaller than that caused by a decreasing loading, which results in the phenomenon of an elastic hysteresis ring, as shown in Fig. 3.6a. The shaded area represents the energy loss in the cycle, which induces the asymmetric distribution of the normal force. This is also the main cause of the generation of rolling friction torque, as seen in Fig. 3.6b.

(a) Elastic hysteresis ring Fig. 3.6 Mechanism of rolling friction

(b) Asymmetric distribution of the normal force

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3 Contact Force Models for Granular Materials

3.3.2 Rolling Friction Model of Spherical Particles Both rolling friction and sliding friction belong to contact damping. Rolling friction exists in the form of moment between the contacting particles and is much smaller than sliding friction. Therefore, it is not considered in the early stage of contact models. In the traditional DEM, the rotation of particles is assumed to be determined solely by the tangential contact force. However, some experimental and numerical simulations show that the rolling damping between particles has a significant effect on the micro deformation of the granular matter. Rolling friction increases the contact time between particles. This phenomenon is negligible when the concentration of granular flow is small. However, for high concentration granular flows, especially when the granular flow state tends to be in a solid state, the effect of rolling friction cannot be ignored. As shown in Fig. 3.7, during the rolling process, when the contact stress on the surface reaches a certain value, plastic deformation is first generated at a certain depth away from the surface, which is shown in the figure as a permanent groove. The rolling friction models used in the DEM are mainly divided into three categories: equivalent rolling frictional resistance, viscous rolling frictional resistance and combined rolling frictional resistance. (1) Equivalent rolling frictional resistance The rolling frictional resistance is represented by a constant moment. Its direction is opposite to the direction in which the contact particles roll relative to each other. The moment acts on a pair of contact particles i and j. The rolling resistance moment vector M r on particle i can be expressed as:   M r = −μr  Finj ωi

(3.25)

where μr is the rolling friction coefficient associated with the material, F inj is the 

normal force vector between the contacting particle i and j, ωi is the unit vector of the rolling angular velocity and is defined as,

(a)

(b) z

d

y x

Fig. 3.7 Rolling process of spherical particle on plastic plane

3.3 Rolling Friction Models of Spherical Particles 

ωi =

ωi |ωi |

65

(3.26)

In this model, the constant normal contact force between contacting particles causes the constant frictional resistance, with only the direction being opposite to the rolling direction. Due to the constant rolling friction moment, the resistance moment is not zero when the particle stops rolling. The moment will exist in the form of reciprocating vibration. And a small amount of residual kinetic energy cannot be dissipated. (2) Viscous rolling frictional resistance For this type of model with viscous rolling friction resistance, the rolling frictional resistance is proportional to the tangential velocity of the relative rotation between particles. The moment direction is opposite to the direction of the relative rolling of particles. The moment acts on a pair of contacting particle i and j. And the rolling resistance moment vector M r on particle i can be expressed as,   M r = −μr Vitj  F inj ωi

(3.27)

where Vitj is the relative tangential velocity between the contacting particle i and j, and can be given by:   Vitj = ω j × R j − ωi × Ri 

(3.28)

where Ri and R j are the equivalent radius vectors which are the distance between the centroids of contacting particle i and j and the contact point. If the two contacting particles have identical radii, and their rolling angular velocities are the same in magnitude and opposite in direction, that is, Ri = R j and ωi = −ω j , the relative tangential velocity is zero. As a result, the rolling friction moment calculated is zero. (3) Combined rolling frictional resistance The constitutive model of contact force used in the current discrete element simulation is shown in Fig. 3.8. The rolling frictional resistance model consists of an elastic rotational spring, a rotational dashpot, a non-tensioning node and a slider. Similarly, the moment of the rolling frictional resistance acts on a pair of contact particle i and j. The rolling frictional moment M r on the contacting particles can be written in terms of the rotational spring moment M rk and the rotational viscous moment M rd as follows, M r = M rk + M rd

(3.29)

where rotational spring moment M rk is given by: M rk = −kr θr , M rk ≤ M rmax

(3.30)

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3 Contact Force Models for Granular Materials

Fig. 3.8 Relative rotation angle between contacting particles

where kr is the elastic rotational stiffness, θr is the rotation angle. M rmax is the critical rotational bending moment of the spring and can be expressed as:   M rmax = μr R Finj 

(3.31)

The Rotational viscous moment M rd is obtained by,  M rd =

  −Cr ωr,i j , M id < M rmax − f Cr ωr,i j ,  M id  = M rmax

(3.32)

where Cr is the rolling damping coefficient, ωr,i j is the relative angular velocity, f is a specific coefficient selected based on different problems. When f = 0, energy dissipation in the system is not significant. The viscous resistance moment is mainly to stabilize the vibration of granular particles. When f = 1, the energy dissipation is significant, which can represent the energy dissipation behavior in the system. In general, f can be defined as a function of ωr,i j or M id .

3.4 Bonding-Breakage Models of Spherical Particles The studies on granular breakage mechanism originated from Griffith strength theory. Griffith considered that many intrinsic micro cracks exist in a material. Under the action of the external force, stress concentration occurs at the crack tip, which is always a tensile stress called induced stress. When the induced stress at the crack tip exceeds the tensile strength of the material, micro cracks will expand until the material fails (Mcdowell et al. 1996).

3.4 Bonding-Breakage Models of Spherical Particles

67

The bonding model includes the point-bonding model and the parallel-bonding model. The point-bonding model only activates at the neighborhood of the contact point of adjacent particles and only the normal and tangential force can be applied in the model. It cannot transfer the moment between the contacting particles because of the small bonding region. The parallel-bonding model bonds particles with an additional material filled in the neighborhood of the contact point of two adjacent particles. When a parallel-bonded particle moves, the additional material and the constitutive spheres themselves interact with each other and generate a contact force. Any loads applied on the parallel-bonded particle can be shared by the parallel bond and constitutive spheres. Both the contact force and moment can be transmitted between particles because the contact region is a disc with radius R. If the normal and tangential contact forces are greater than the corresponding bond strength, the bond breaks and the two particles are separated from each other (Fig. 3.9). The parallel-bonding model can be envisioned as a set of linear springs with constant normal and tangential stiffness which distribute uniformly on the disc and center on the contact point. The mechanical properties of the parallel bond are determined by the following five parameters: the normal stiffness K n , the tangential stiffness K s , the normal bond strength σnb , the tangential bond strength σsb , and the radius of bonding disc R, which is usually defined as the smaller radius or average radius of the two bonded particles (Wang and Tonon 2009, 2010; Nitka and Tejchman 2015). The normal and tangential forces and moments associated with the parallel bond are expressed by Fn , Fs , Mn and Ms , respectively.

Fig. 3.9 Parallel-bonding model between particles

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3 Contact Force Models for Granular Materials

According to the beam theory, the maximum tensile stress and shear stress acting on the disc can be written as (Potyondy and Cundall 2004), σmax =

|Ms | |Fs | |Mn | −Fn + Rτmax = + R A I A J

(3.33)

where A is the cross-sectional area of the disc, I is the inertia moment, J is the polar inertia moment. These can be calculated by the following formulae, A = π R2 J =

1 1 π R4 I = π R4. 2 4

(3.34)

If the maximum tensile stress acting on the disc exceeds the bond normal strength or the maximum shear stress exceeds the bond tangential strength, the bond breaks and the bonding effect disappears, which can represent the initiation, propagation and coalescence of micro cracks in the granular material and the fracture failure of the material. Since the fracture energy is released in an instant after the parallel bond ruptures, it is easy to cause the particles splash and affect the stability of the granular system. In order to gradually eliminate the fracture energy generated during the bond breakage process, a softening model with the introduction of damage concept was proposed, as shown in Fig. 3.10 (Onate and Rojek 2004). In the softening model, the normal stiffness K n of the contact point gradually decreases after the tensile force reaches the peak value Fnmax , and the normal bond strength of the material decreases accordingly, which means that the decreasing normal stiffness will lead to a change of the linear relationship between force and displacement during the unloading and reloading. If the normal tensile strength σnb is set to be a constant, the shear strength is determined by the tangential bond strength and the friction caused by the normal stress based on the Mohr-Coulomb friction law. The shear strength τb between the Fig. 3.10 Softening model during the breakage of a parallel bond

3.4 Bonding-Breakage Models of Spherical Particles

69

two bonded particles can be expressed as: τb = σsb + μb σn

(3.35)

where σsb is the tangential strength of the bond, μb is the friction coefficient between two bonded particles, σn is the normal stress of bonded particles. After the bonding particles break, the adhesion force is zero and the friction coefficient is μ. In the previous studies of failure criteria of parallel-bonding, the friction coefficient was set to be zero, which means that the shear strength between particles is independent of the normal stress. Recently, more researchers have noticed the effect of the normal load on the tangential strength (Nitka and Tejchman 2015). The ratio of the tangential bond strength to the normal bond strength is represented by a constant α: α=

σsb σnb

(3.36)

where α can be set to be one or other values (Tarokh and Fakhiimi 2014). The maximum shear force is determined by the normal force, tangential bond strength and friction coefficient (Scholtes and Donze 2012). In the softening failure criterion, the normal stiffness changes with the damage degree between the two bonded particles during their progressive failure process (Tarokh and Fakhiimi 2014). The normal contact force in the bonded particles can be calculated by the following formula, f = K nf u n = (1 − ω)K n u n

(3.37)

where K nf is the positive modulus of elastic damage, ω is the damage variable. u n is the normal overlap between two contacting particles. The scalar damage variable ω is the damage indicator of the material. For a non-damage condition, ω = 0; For a damage condition, 0 < ω ≤ 1. The scalar damage variable ω can be written as: ω=

ϕ(u n ) − 1 ϕ(u n )

(3.38)

where ϕ(u n ) is given by:

ϕ(u n ) =

⎧ ⎪ ⎨ ⎪ ⎩

1, u n ≤ u 0 K n2 u n , u0 (K ns +K n )Fnmax −K ns K n u n

≤ u n ≤ u max

(3.39)

∞, u n ≥ u max

where Fnmax = Aσnb , A is the cross-sectional area of the bonding disc. The contact force-shear displacement law with the incorporation of damage is introduced in the tangential direction. The stiffness and strength in the tangential direction decreases

70

3 Contact Force Models for Granular Materials

Fig. 3.11 Tangential force-displacement relationship

according to the damage state in the normal direction, and λ is defined as: 

σb λ = nb σn

(3.40)



where σnb is the residual tensile strength of the bond, σnb is the initial cohesion force of the bonding particles. When the bond breaks, as shown in Fig. 3.11, considering the Coulomb friction law between the contacting particles, the maximum tangential force Fsmax = Aλσsb . The normal and tangential stiffness can be defined by the disc radius and the elastic modulus as (Azevedo et al. 2015; Wang and Tonon 2010; Nitka and Tejchman 2015), Kn = E

2R A R B Kn β= RA + RB Ks

(3.41)

where the parameter β is closely related to the Poisson’s ratio (Weerasekara 2013). The DEM with the incorporation of the bonding-breakage model has been applied to the numerical simulations of different materials such as sea ice (Long et al. 2019), ceramic (Jiang et al. 2019), soil and rock mass. For example, the blocky sea ice in the form of bonded particles breaks under an external force during the interaction between the icebreaker and the sea ice, as shown in Fig. 3.12. The bonding particles have been widely used in geotechnical mechanics and other fields, and can deal well with the fracture behaviors of continuous materials. Nevertheless, its failure criteria in tension, shear and compression conditions are still key issues which are yet to be further studied (Ji et al. 2017). To sum up, bonding-breakage models with spheres overcome the difficulty in describing the final morphology of a single particle after its breakage. It can realize the process of granular breakage, which has a certain guiding effect on engineering applications.

3.5 Contact Models of Non-spherical Particles

71

Fig. 3.12 Breakage process of bonded sea ice

3.5 Contact Models of Non-spherical Particles In the DEM, the contact models of 2D discs or 3D spheres have been widely developed and applied, which include the linear spring model (Di Renzo and Di Maio 2004) and the nonlinear Hertz-Mindlin model (Zhu et al. 2008). Unfortunately, these models are not suitable for irregularly shaped particles. In recent years, in order to determine the contact forces between irregular particles, the energy conservation theory between particles with arbitrary shapes has been proposed. However, the accurate calculation of the overlapping volume in the theory leads to a lower computational efficiency and is not suitable for the massively parallel computing (Feng et al. 2012). The results obtained by the nonlinear contact theory of spheres with the modifications on the surface curvature are in good agreement with the results by the finite element method considering the characteristics of ellipsoidal particles (Zheng et al. 2013). However, for super-quadratic particles (Wang et al. 2018) and dilated polyhedral particles (Liu and Ji 2019), the method becomes very complicated and cannot accurately calculate the modification parameters under different shapes and smoothness. Fortunately, the Hertz contact model based on spheres and the contact model based on the surface curvature of the ellipsoid provide a good idea for the calculation of the nonlinear contact force of the super-quadratic and the dilated polyhedral particles.

3.5.1 Contact Model Between Super-Quadric Particles During the interaction between super-quadric particles, the normal contact force mainly includes the elastic force, the viscous force, and the additional bending moment induced if the contact force does not pass through the centroid of particles. Figure 3.13 is the contact model that considers the equivalent curvature between super-quadratic particles. The normal forces can be expressed as: Fn = K n δn3/2 + Cn Avn,i j

(3.42)

72

3 Contact Force Models for Granular Materials

Fig. 3.13 Contact model of super-quadratic particles

where K n is the effective normal stiffness of the two contacting particles, δn is the normal overlap, Cn is the normal viscous coefficient, A is the viscous parameter between particles, vn,i j is the relative normal velocity. The effective normal stiffness K n and viscous parameters A are written as: 4  1/2 Kn = E ∗ R∗ 3   1/2 1/2 A = 8m ∗ E ∗ R ∗ δn

(3.43)

i j i j E where E ∗ = 2 1−ν , m ∗ = m i +m , R ∗ = Ri +R . E, ν and R are the elastic modulus, j j ( 2) Poisson’s ratio of the granular material and the average curvature radius at the contact point. The curvature radius is given by:

mm

R R

R = 1/K

(3.44)

where K is the local curvature at the contact point, which can also be written as the average curvature K mean : K mean =

∇ F T H (F)∇ F − |∇ F|2 (FX X + FY Y + FZ Z ) 2|∇ F|3

(3.45)

Or Gauss curvature K gauss : √ K gauss =

AB − C 2 D

(3.46)

with A = FZ (FX X FZ − 2FX FX Z )+ FX2 FZ Z , B = FZ (FY Y FZ − 2FY FY Z )+ FY2 FZ Z ,  2 C = FZ (FX Y FZ − FX Z FY − FX FY Z ) + FX FY FZ Z , D = FZ2 FX2 + FY2 + FZ2 .

3.5 Contact Models of Non-spherical Particles

73

The tangential contact force can be divided into the elastic force Fse and the viscous force Fsd . Based on the Mohr-Coulomb friction law, the tangential elastic force Fse is defined as:      3/2  (3.47) Fse = μs  K n δn3/2  1 − 1 − min δt , δt,max /δt,max where μs is the sliding friction coefficient,δt is the tangential overlap with δt,max = μs (2 − ν)/2(1 − ν) · δn . The tangential viscous force Fsd is expressed as:  1/2    1/2 /δt,max · νt,i j Fsd = Ct 6μs m ∗  K n δn3/2  1 − min δt , δt,max

(3.48)

where Ct is the tangential viscous coefficient, νt,i j is the relative tangential velocity between particles.

3.5.2 Contact Model of Dilated Polyhedral Particles In addition to the resilience caused by the elastic deformation, the energy loss due to the collision between particles is also taken into account in the interactions between dilated polyhedra. Therefore, both the elastic force and the viscous force should be considered simultaneously. For the normal viscous force and tangential force, the contact model of spheres or super-quadric particles is available for reference. Currently, the linear models are commonly used in calculating the normal elastic force of polyhedra and dilated polyhedral particles, namely: Fne = K n δ n

(3.49)

where K n is the normal stiffness coefficient, δ n is the normal overlap. Unfortunately, the normal stiffness coefficient is strongly dependent on empirical equation or needs to be determined by a large number of numerical tests (Behraftar et al. 2017). So the model is currently only suitable for phenomenological simulations and is not in common use. Since a dilated polyhedron is mainly composed of vertices, edges and faces, its effective contact stiffness is closely related to the contact mode between particles. Therefore, the Hertz contact model can be used to solve the contact problem of dilated polyhedral particles. Its simplified form is written by: Fne =

4 ∗  ˜ 23 E Rδn 3

(3.50)

The only variable that needs to be determined in the above equation is the selection ˜ The contact modes between dilated polyhedra can be of the equivalent radius R.

74

3 Contact Force Models for Granular Materials

Fig. 3.14 Sphere-cylinder contact

divided into three categories: sphere-sphere, sphere-cylinder, sphere-face contact; cylinder-cylinder, cylinder-face contact; face-face contact. 1. Sphere-sphere, sphere-face and sphere-cylinder contacts Sphere-sphere and sphere-face contacts can be referred to contact models of spheres, while the sphere-cylinder contacts can be referred to the Hertz contact model of ellipsoids. Figure 3.14b shows two ellipsoids that are in contact along the axial direction. The equivalent radius in the Hertz model can be expressed as:  1 = R˜ j

j

R1i + R1



j

R2i + R2 j

 (3.51)

j

R1i R2i R1 R2

j

where R1i , R2i , R1 and R2 are the curvature radius at the contact point on the two j j contacting ellipsoids, respectively. By setting R1 = R2 = Rspher e , R2i → ∞ and i R1 = Rcylinder , a simplified form of Eq. (3.51) can be written as: 1 1 = Rspher e R˜



1 Rspher e

+

1



Rcylinder

(3.52)

2. Cylinder-cylinder and cylinder-face contact As shown in Fig. 3.15a, b, the cylinder-cylinder contact can be divided into two categories: parallel contact and cross contact. For the cross contact, the equivalent radius can be calculated by the surface distance equation. In Fig. 3.15a, x-O-y plane is perpendicular to the normal contact plane between the two cylinders with contact point O. The distance between these two cylinders in the normal direction can be written as d=

x2 (x cos θ − y sin θ)2 + = ax 2 + 2bx y + cy 2 2R1 2R2

(3.53)

where a = 1/(2R1 ) + cos2 θ/(2R2 ), b = − sin θ cos θ/(2R2 ), c = sin2 θ/(2R2 ). The quadratic form matrix M of the polynomial is expressed as

3.5 Contact Models of Non-spherical Particles

75

Fig. 3.15 Cylinder-cylinder and cylinder-face contact

 M=

ab bc

 (3.54)

Let T = tr(M) = a + c , D = det(M) = ac − b2 . Defining R1 and R2 as the principal curvature radii of the contact point, R1 and R2 can be obtained from eigenvalues of the matrix M: R1 , R2 =

T±

√

2 T 2 − 4D

(3.55)

 where R˜ = R1 R2 . For the parallel contact of two cylinders and the cylinder-face contact, the equivalent radius can be referred to that of two contacting spheres, which is defined as: 1 1 1 = + R1 R2 R˜

(3.56)

3. Face-face contact The normal elastic force of the face-face contact is calculated as (Popov 2010)  Fne

∗

= 2E β

A δn π

(3.57)

where β depends on the shape of the contact area. In general, if the shape is circular, β = 1; if the shape is triangular, β = 1.034; if the shape is rectangular, β = 1.012. For the convenience of calculation, a fixed value can be taken in the actual simulations (Fig. 3.16). These elastic normal force calculations are defined based on different contact modes. They have certain theoretical basis, but the calculation process is relatively complicated. A lot of judgments are needed in the actual programming, which is not suitable for large-scale phenomenological simulations. In fact, the normal contact force can be calculated by a simplified model. Here are the two of them.

76

3 Contact Force Models for Granular Materials

Fig. 3.16 Face-face contact

One is based on the equivalent mass or volume. The mass of a dilated polyhedron is m and the density is ρ. The equivalent radius can be referred to the radius of sphere with the equal mass, and is defined as:  R˜ =

3

3m 4πρ

(3.58)

The other is to calculate the distance from each vertex to the centroid, N 1  R˜ = di N i=1

(3.59)

where di is the distance between vertex number i and the centroid. The above equivalent methods can improve the calculation efficiency and give accurate numerical results in the actual calculations.

3.6 Non-contact Physical Interactions Between Particles The Hertz contact theory is applicable for problems at the macroscopic scale in which the surface interaction force between particles is very weak compared with the elastic force. However, the surface interaction force cannot be neglected at the molecular or atomic scale, and the adhesion of molecules or atoms at the surface of contacting particles should be considered. Bradley’s contact theory regards the particle as a rigid body without considering the surface deformation caused by the adsorption between particles. Johnson-Kendall-Roberts (JKR) contact theory is suitable for low elastic modulus materials with large particle size and low adhesion energy. DMT contact theory is applicable to high elastic modulus materials with small particle size and high adhesion energy. Natural rainfall and various production activities may induce the drying-wetting behaviors of granular materials, which is the key environmental loads and inducing factor leading to landslides, collapses, mudslides and other disasters. The liquid in granular materials affects the strength and deformation properties of the material. For

3.6 Non-contact Physical Interactions Between Particles

77

the heat conduction in granular materials, a simple conduction phenomenon in solid phase particles is insufficient for the understanding of heat conduction. Fortunately, the DEM can provide effective support for the studies of heat conduction in granular materials. The key is the reasonable description of the heat transfer performance between the contacting particles. The studies of heat conduction using the DEM are helpful to understand further the heat transfer in granular materials.

3.6.1 Adhesion Force Between Spherical Particles (1) Van der Waals force Studies have shown that the van der Waals force between particles is proportional to the particle size D p , and the gravity of the particle is proportional to D 3p . That is to say, as the particle size decreases, the van der Waals force becomes significantly important compared with gravity. When there is no electrostatic force and liquidbridge force, particles with a size of 1 mm and above usually do not adhere with each other; While particles with sizes less than 100 μm are just the opposite. The van der Waals force is the main influencing factor of fine grain granular flows. The Hertz theory has been widely recognized for the elastic contact between coarse particles. But for the adhesive contact between fine grains, the van der Waals force between the contacting molecules will cause local deformation of the particles in the contact region. The interaction between molecules usually can be expressed by the empirical Lennard-Jones (as shown in Fig. 3.17): C D U = 12 − 6 = 4U0 z r

Fig. 3.17 Lennard-Jones between molecules

   6  a 12 a − z z

(3.60)

78

3 Contact Force Models for Granular Materials

  where a is the molecule radius, z is the distance between molecules, U0 = C/ 2z 06 , and z 0 = (2D/C)1/6 , C is a constant which is approximately equal to 10−79 Jm6 . Hence, the van der Waals force between particles can be written as: F =−

12D 6c dU = 13 − 7 dz z z

(3.61)

(2) Bradley theory and DMT theory For two solid particles in a van der Waals equilibrium state, the adhesion energy γ can be expressed as: γ = γ1 + γ2 − γ12

(3.62)

where γ1 and γ2 are the free energy on the surface of the two particles, respectively, γ12 is the interface energy. Based on the Lennard-Jones law, the force per unit area between the two surfaces with distance h is (Johnson and Greenwood 1997):  8γ  ε 3  ε 9 σ (h) = − 3ε h h

(3.63)

where ε is the equilibrium spacing of molecules or atoms. When two rigid particles with radius R1 and R2 are in contact, the adhesion force based on Bradley theory can be written as: F0 = 2π R ∗ γ

(3.64)

where R ∗ = R1 R2 /(R1 + R2 ) is the equivalent radius, which is obtained by the potential energy between two spherical particles (Bradley 1932). If the force between atoms is regarded as the interaction force between particles’ surfaces and the paraboloid is used to approximately represent the sphere surface, the gap between the surfaces is h = r 2 /(2R). If σ (h) is the force per unit area, the force between the two spherical particles can be expressed as:  F= 0

∞



∞

2π σ (h)dr = 2π R

σ (h)dh

(3.65)

0

Substituting (3.63) into (3.65), and assuming that the gap between the two spherical particles is h 0 , the force between the spheres can be obtained by:     8  ε 2 ε 2 F(h 0 ) = − π R ∗ γ 4 3 h0 h0

(3.66)

3.6 Non-contact Physical Interactions Between Particles

79

The above formula is called the Bradley equation. It is appropriate for situations with rigid particles, and z  R should be satisfied. However, the scope of its applications is limited. Given the contact between two elastic solid spheres, under the action of the load P, the radius a of the contact region obtained based on the Hertz contact theory is (Landau and Lifshitz 1999), a 3 = R ∗ P/K

(3.67)

  1−ν 2 1−ν 2 where K = 43 ( E1 1 ) + ( E2 2 ) , E 1 and E 2 , and ν1 and ν2 are the Young’s modulus and the Poisson’s ratio of the two spheres, respectively. The elastic compression is δ = a 2 /R ∗ . However, in actual laboratory observations, it was found that when the external load is small, the contact area is much larger than that predicted by the Hertz contact theory. Moreover, the contact area tends to remain a constant as the load gradually decreases to zero. It indicates that adsorption force on the surface exist between the solids, which becomes very important in the process of decreasing load to zero. Considering the adhesion force on the contact surface, Derjaguin et al. (1975) gave a modified Hertz relationship based on Bradley Eq. (3.64) (Derjaguin et al. 1975): a3 K = P + 2π R ∗ γ , δ = a 2 /R ∗ R∗

(3.68)

Equation (3.68) is generally known as DMT theory. It can be seen that when there is no external 1/3 force, a contact circular domain with the radius of a0 =  will be generated under the surface force. 2π R ∗2 γ /K (3) JKR theory Combining the linear superposition of the Boussinesq solution and the Hertz theory, Johnson et al. derived the JKR theoretical model using the principle of minimum energy (Johnson et al. 1971). The JKR contact theory is an extension of the Hertz contact theory. It considers that adhesion only exists on the contact surface. If there is no adhesion between particles, the radius of the contact surface is given by the Hertz contact theory; Otherwise, the radius of the contact surface a > a0 , although the external load is still P. The normal load applied to the particles and the radius of the contact surface meet the following relationship:    3R ∗ ∗ 2 ∗ ∗ F + 3π R γ + (3π R γ ) + 6π R γ P a = 4E ∗ 3

(3.69)

The deformation is induced by the combined actions of the external load F and the adhesion force on the contact surface, and the equivalent load F1 is obtained by:

80

3 Contact Force Models for Granular Materials

Fig. 3.18 Non-dimensional force-overlap curves in Hertz, JKR and DMT contact models

F1 = N + 3π R ∗ γ +



(3π R ∗ γ )2 + 6π R ∗ γ P

(3.70)

In the absence of an applied load, the overlap δ and the radius of the contact surface are both finite values larger than zero. As the tensile load increases, the radius a gradually decreases, and a sudden jump occurs when the critical load is reached. The interface is then out of contact, as shown in Fig. 3.18. Since the controlling tensile load gradually increases, it belongs to hard unloading. For the hard unloading, the a ∼ N relationship given by the JKR does not represent the actual physical process after the pull open point. The pull open point corresponds to the minimum point of the curve. Both δ and a are non-zero finite values, and the corresponding critical load is equal to −1.5π R ∗ γ . If it is a soft unloading, that is, the overlap rate changes slowly instead of gradually increasing after the pull open point is active, the JKR relationship still holds until the slope of the curve N ∼ δ equals to the critical point of infinity where a jump also occurs. The adhesion force exists only in the contact area and the absorption force is ignored outside the contact area in the JKR contact model. If the contact area is large and the effective action distance of the adhesion force is small compared with the overlap ratio, the adhesion force outside the contact zone is smaller than that inside the zone. In this case, the JKR contact model is reasonable.

3.6.2 Liquid Bridge Force Between Wet Particles When particles’ surfaces are wet and approach each other, part of the liquid films gradually coalesces and forms a liquid bridge at and near the contact point. The liquid bridge forces between particles are caused by the pressure difference of the liquid bridge, the surface tension of the liquid and the viscous resistance. For a granular material filled with liquid, the liquid phase can be divided into four states according to the content of liquid, namely, swing state, chain state, capillary state and dipping state, as shown in Fig. 3.19. In a swing state, there is a lenticular or annular liquid

3.6 Non-contact Physical Interactions Between Particles

(a) Swing state

(b) Chain state

81

(c) Capillary state

(d) Dipping state

Fig. 3.19 Different states of liquid in granular materials

phase at the contact point and the liquid phase does not connect to each other. As the liquid content increases, the annular liquid phase grows and connects with each other to form a shape of liquid mesh tissue with air filled in between, this is chain state. In a capillary state, all pores between particles are filled with liquid, only the gas-liquid interface exists at the exterior surface of particles. A dipping state is a state in which particles are immersed in the liquid and a free surface exists. The liquid bridge force is related to the geometry of the liquid bridge. It consists of two parts: surface tension and static pressure. The geometry of the liquid bridge is related to the spacing between the two particles. The geometric features of the 3D liquid bridge projection plane are shown in Fig. 3.20. The gas-liquid interface of the 3D liquid bridge, which is also called the curved surface of the liquid bridge, can be described by the Laplace-Young equation as (Soulié et al. 2010):  p = σsu

1 1 − ρext ρint

 (3.71)

where p is the pressure difference at the interface, σsu is the surface tension of the pore liquid, ρext and ρint are the extrinsic and intrinsic curvature radius of the curved surface of the liquid bridge, respectively, which can be given by: Fig. 3.20 Schematic of the geometry of liquid bridge between particles

82

3 Contact Force Models for Granular Materials  2 3/ 2 [1 + ( y (x)) ] y  (x)   2 = y(x) · 1 + ( y (x))

ρext = ρint

(3.72) (3.73)

The 3D Laplace-Young equation can only be solved in extremely special cases due to its high nonlinearity, hence the numerical method is an effective means. Lian et al. solved the Laplace-Young equation of the 3D liquid bridge by numerical method (Lian et al. 1993). When the contact angle θ < 0.7, the critical fracture distance Dr of the 3D liquid bridge can be expressed in terms of the three-phase contact angle and the volume of liquid bridge: Dr = (1 +

θ ) V 1/ 3 2 w

(3.74)

where θ is the three-phase contact angle, Vw is the liquid volume of the liquid bridge. The liquid bridge force consists of the liquid surface tension and hydrostatic pressure. The calculation method is usually divided into the following two types. Gorge is calculated at the cross section One is the Gorge method, in which the force Fs of the neck of the liquid bridge. In the other method, the resultant force Fsdir of the pore fluid pressure and surface tension acting on the solid surface is calculated directly. The liquid bridge force obtained from the two methods can be written as: FsGorge = π y02 p + 2π y0 σsu

(3.75)

2 Fsdir = π ( R1 sin δ1 ) p + 2π R1 σsu sin δ1 sin(δ1 +θ )

(3.76)

where δ1 is half-filled angle, R1 is the particle radius. The precondition for the above two calculation methods is that the geometric characteristics of the liquid bridge such as the half-filled angle are known. Due to the highly nonlinear 3D Laplace-Young equation, the numerical solution process is extremely complicated. Soulié et al. (2010) and Richefeu et al. (2008) obtained a fitting formula of the 3D liquid bridge force with different forms (Soulié et al. 2010; Richefeu et al. 2008). The 3D liquid bridge force equation obtained by Soulie et al. is (Soulié et al. 2010): Fs = π σsu

√



  D c + ex p a + b R1 R2 R2

(3.77)

where D is the distance between particles, R1 and R2 are particles radii, a, b and c are fitting coefficients, which are written as: 

a = −1.1

Vw R23

 −0.53

3.6 Non-contact Physical Interactions Between Particles

83

    Vw Vw 2 − 0.96) + 0.48 b = (−0.148ln −0.0082 · ln θ R23 R23   Vw c = 0.0018 · ln + 0.078 R23

(3.78)

The 3D liquid bridge force equation proposed by Richefeu et al. is (Richefeu et al. 2008):

Fs =

⎧ ⎨

−2π σsu R cos θ D Dr 0

(3.79)

where λ and R can be expressed as: 0.9 λ = √ Vw1/ 2 R −1/ 2 2



1 1 + R1 R2

1/ 2

 , R = max

R2 R1 , R1 R2

 (3.80)

Furthermore, when there is an interstitial fluid, the relative motion between particles involves primarily the fluid flow in the narrow pores. This physical phenomenon can be described by the Reynolds lubrication equation. For the pendulum-shaped liquid bridge, the influence of liquid viscosity is mainly reflected in the two aspects. One is that the liquid viscosity affects the evolution of the shape of the concave gasliquid surface. The region near the surface is called the non-lubricating region. The other is that the liquid viscosity controls the strength of the liquid bridge, and this region is called the lubrication region. When the distance between the two particles is very small, the contribution of the gas-liquid interface to the dynamic liquid bridge force is negligible. The dynamic viscosity force can be analyzed by the lubrication theory (Briscoe and Adams 1987; Lian et al. 1998, 2001).

3.6.3 Heat Conduction Between Particles Due to the diffusion of granular materials and the complicated interactions between particles, the heat transfer performance of granular material also exhibits nonisotropic characteristics and is affected by particle size, distribution, granular temperature, pore fluid properties, body fraction ratio and boundary conditions (Bala et al. 1989; Hadley 1986). Studies have shown that the change of the granular temperature, which changes the internal structure of the material due to the thermal expansion and contraction of particles, can also affect the macroscopic properties of particles. It provides a means of changing the particles’ properties without any mechanical disturbance applied (Chen et al. 2009, 2012). In engineering applications such as sintering and preparation of powder materials, heat transfer and applications of heat

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3 Contact Force Models for Granular Materials

insulating materials, it is necessary to understand the heat transfer properties of granular materials in order to improve the production technology and formula, hence the products’ performance. However, the heat conduction in granular materials, even for a simple conduction phenomenon through solid-phase particles in contact, is not fully understood. Related theoretical and numerical models of heat conduction in granular materials are an urgent requirement for engineering applications. The DEM has unique advantages in dealing with the microscopic properties of granular materials. In order to simulate the heat transfer behaviors in granular materials, temperature, as a degree of freedom, is introduced to characterize the change of heat in particles. The governing equations and the contact-heat transfer model are the key to the implementation of numerical algorithms. The granular material as the research object mainly has two characteristics. One is that the contact-heat conduction between particles is the only way to conduct heat in granular materials. The particle assembly is in a vacuum state with no filling materials in the pores, and the heat radiation between particles is neglected. The other is that the changing granular temperature does not affect the properties of the granular material and particle size. (1) Governing equations of heat conduction For particle i in the granular material, the change of the temperature meets the heat conduction equation (Bergman et al. 2007): ρ i ci

      ∂ ∂ ∂ ∂ Ti ∂ Ti ∂ Ti ∂ Ti − − − − ρ i q˙ i = 0 (3.81) k1 k2 k3 ∂τ ∂x ∂x ∂y ∂y ∂z ∂z

And the corresponding boundary condition is: P1 : T = T P2 : k1 P3 : k1

∂ Ti ∂ Ti ∂ Ti  n 1 + k2 n 2 + k3 n3 = q ∂x ∂y ∂z

(3.82)

∂ Ti ∂ Ti ∂ Ti n 1 + k2 n 2 + k3 n 3 = −β(T − Ta ) ∂x ∂y ∂z

where ρ i is the particle density; ci denotes the specific heat capacity; q˙ i is the density of the heat source inside particles; n 1 , n 2 and n 3 are the direction cosine of the outward normal of the boundary, respectively. The boundary condition T¯ is the  given temperature at the boundary; q is the heat flux density at the boundary; β is the heat exchange coefficient; k is the heat transfer coefficient. For natural convection conditions, Ta is the ambient temperature. For forced convection conditions, Ta is the adiabatic wall temperature of the boundary layer. If the temperature distribution inside each particle is considered in the discrete element calculation, a huge amount of calculation time is required. To improve the efficiency, each particle has identical temperature. For particle i without heat source

3.6 Non-contact Physical Interactions Between Particles

85

inside, that is, q˙ = 0, all factors which can affect the granular temperature come from external conditions, namely, the sum of heat exchange with other particles is q i . The relationship between granular temperature and heat exchange at the boundary is:    ∂ i ∂ Ti ∂ Ti ∂ Ti ρ i ci v i T = n 1 + k2 n 2 + k3 n 3 da = q i (3.83) k1 ∂τ ∂x ∂y ∂z p

 where vi is the particle volume; p is the integral at particle boundary. By the accutotal heat exchange mulation of the heat exchange q ji over each contact pair, the  with other N particles can be obtained, which is written as q i = Nj=1 q ji . However, q ji is related to the properties and temperature difference of the contacting particles, and can be obtained by: q ji = h i j (T j − T i )

(3.84)

where h i j denotes the contact thermal conductivity between the two contacting particles;T i and T j are the temperature of particle i and j, respectively. The granular temperature can be calculated by the sum of the contact heat exchanges:  ∂ Ti = h i j (T j − T i ) ∂τ j=1 N

ρ i ci v i

(3.85)

In the above equation, the transient heat transfer process of the granular assembly is calculated approximately by a static physical quantity, which requires that the time step is sufficient small so that the heat conduction in each time step can be described in a quasi-static manner. (2) Contact-heat transfer model In the DEM, contact thermal conductivity h i j is an important parameter for calculating the heat exchange between granular contact pairs. The effect of the properties of contact surface is considered here. By introducing the contact thermal resistance model into the DEM, the influence of the contact surface properties on the temperature distribution and heat conductivity of the granular assembly can be analyzed in numerical simulations, which enhances the ability of the DEM to reflect the properties of granular materials. As shown in Fig. 3.21, the contact pair consists of particle i and j; c is the contact point. The contact pair is simplified as one-dimensional heat conduction system composed of the thermal resistance R i , R j and the contact thermal resistance R c in series. The temperature is T i and Tci at the center of particle i and at the contact, j respectively; while T j and Tc are the temperature at the center of particle j and at the contact, respectively.

86

3 Contact Force Models for Granular Materials

Fig. 3.21 Granular contact pair and its heat conduction process

The steady state heat transfer equation can be given by:   ∂T ∂ k =0 ∂z ∂z

(3.86)

where z denotes the direction of heat conduction; the Fourier heat conduction equation can be obtained by integrating the above formula: −k

∂T  =q ∂z

(3.87)



where q is the heat flux density. The heat flow rate q ji of any section A in granular  system can be obtained by integrating the q over the section:  q ji =

k A

∂T dA = ∂z





q dA

(3.88)

A

According to the definition of thermal resistance, the relationship between the heat flow rate on the contact surface of particle i and j and the thermal resistance in the granular system is: j

j

i − Tci T i − Tc Tc − T j Ti −T j q ji = T = = = Ri Rc Rj Ri j

(3.89)

The total thermal resistance R i j of the contact pair, composed of the macroscopic thermal resistance and contact-thermal resistance in series, can be obtained from the heat flow rate q i j and temperature difference T j − T i of the system. The corresponding equation is: Ri j =

1 T j −Ti = qi j hi j

(3.90)

where h i j is the reciprocal of thermal resistance R i j . For the ideal case where the surfaces of the two contacting particles are smooth, based on the Hertz elastic contact theory, the approximate analytical expression of

3.6 Non-contact Physical Interactions Between Particles

87

Fig. 3.22 Schematic of heat flow on the surface of coarse particles ij

the ideal contact thermal conductivity h can be written as (Batchelor and O’Brien 1977):   3 Fn re 1/ 3  h = = 2k 4 Ee R ij

1

(3.91)

where R is the ideal macroscopic thermal resistance of the contact pair when the contact surface is smooth; Fn denotes the contact normal pressure between particles;  k is the effective heat transfer coefficient of the contact pair; re is the effective k, k i and k j can radius; E e is the effective elastic modulus. The relationship between  be given by, −1  k = 2 k i k j (k i + k j )

(3.92)

In fact, the particle surface is rough, and gaps scatter at the contact section, as shown in Fig. 3.22. Since heat conduction needs to be performed through micro contact surfaces and the total area of micro contact surfaces is limited, the rough surface limits the heat transfer capability between contacting particles. Many scholars have studied the thermal conduction model on the contact surface based on the contact problem of elastomer with rough surface (Lambert and Fletcher 1996; Bahrami et al. 2005). It is generally believed that the contact between two rough Gaussian surfaces can be equivalent to a contact between a Gaussian surface containing two rough surface properties and a smooth surface. The equivalent roughness η and equivalent surface slope s are related to the respective attributes of the two rough surface. η and s can be expressed as: η=

  ηi 2 + η j 2 , s = s i 2 + s j 2

(3.93)

The surface slope is a function of surface roughness, Tanner and Faboum gave the conversion relationship (Tanner and Fahoum 1976):

88

3 Contact Force Models for Granular Materials

s = 0.152 η0.4

(3.94)

The contact thermal resistance between the two contacting surfaces Rs is the sum of the contribution of all micro contacts on the contact surface and can be expressed by (Batcgelor and O’Brien 1977):  0.565 H ∗ (η s) Rs =  k Fn

(3.95)

where H ∗ is the equivalent surface micro-hardness coefficient on the surface, which is generally the effective value of Vickers micro-hardness (Sridhar and Yovanovich 1996). The total thermal resistance R i j of the contact pair in series with the macroscopic thermal resistance and the contact thermal resistance can be written approximately as the sum of the macroscopic ideal thermal resistance R and the contact thermal resistance Rs (Lambert and Fletcher 1996; Bahrami et al. 2005): R i j = R + Rs

(3.96)

(3) The effective heat transfer coefficient of granular assembly The temperature boundary conditions are applied on the top and bottom of the granular assembly, as shown in Fig. 3.23. When the granular assembly reaches a thermal equilibrium state, the heat flux flowing into the assembly per unit time is equal to that flowing out the assembly, that is, qin = qout . The heat transfer of a granular assembly in a certain direction can be described by one-dimensional Fourier heat transfer equation: Fig. 3.23 Schematic of heat conduction in a granular assembly

3.6 Non-contact Physical Interactions Between Particles

89

T x

(3.97)

q = − ke A

where ke is the effective heat transfer coefficient of the granular assembly in the conduction direction;A  is the cross sectional area of the granular assembly. The temperature gradient T x can be obtained from the applied temperature boundary conditions and the size of the granular assembly. The heat flow rate q can be obtained by the heat flow on the boundary using the statistics method. Thus, the effective heat transfer coefficient can be directly determined from the above statistics: ke = −q

1 x A T

(3.98)

(4) Effect of granular characteristic parameters on thermal conductivity (a) Effect of contact thermal resistance The contact thermal resistance has an effect on the overall thermal conductivity of granular assembly (Zhang et al. 2011). The granular assembly shown in Fig. 3.24 is composed of 1106 spherical particles. The particle sizes follow a normal distribution with an average of d = 4 × 10−3 m and a standard deviation of 0.25d. The material parameters are listed in Table 3.1. The particles are constrained by a rigid wall in a rectangular region. The rigid wall itself has a mass and can move along the normal direction under the action of the external force and the contact force. The heat is introduced into the assembly by the contact. The heat flux flows into the assembly in each time step can be obtained from the heat flow rate data on the rigid wall, which provides a basis for whether the granular assembly reaches a thermal equilibrium state or not. The application of the external load can be divided into two stages. The first stage is the extrusion phase, where an external force is applied to the rigid wall around the granular assembly until it reaches equilibrium. The second stage is the application of the temperature load, a temperature difference with 1 K is applied in the x direction (The temperatures of rigid walls at both ends of the granular assembly are 273 K and 274 K, respectively). The temperature distribution after heat balance is shown in Fig. 3.24b. The properties of the contact surface will affect the heat transfer properties of the granular assembly. The effect of the material properties of contact surface on the effective conductivity coefficient ke is revealed by changing the equivalent micro-hardness on the surface. The equivalent micro-hardness was taken as H ∗ = 2.3 × 109 Pa, 2.3 × 108 Pa and 2.3 × 107 Pa. The remaining material parameters were kept unchanged. An increasing equivalent micro-hardness will result in an increase of contact thermal resistance. Under the same external load, the effective heat transfer coefficient decreases as the surface equivalent micro-hardness increases. The simulation results with different compression loads show the same trend, as shown in Fig. 3.25. Since the contact thermal resistance is in series with the ideal thermal resistance, when the properties of contact surface are not favorable for heat conduction,

90

3 Contact Force Models for Granular Materials

Fig. 3.24 Schematic of the granular assembly with random packing. a Granular distribution (Dark parts are the granular layers of the thermal propagation path depicted in (e) and (f)); b Relative temperature distribution (Reference temperature 273 K); c Normal force network; d Heat conduction path; e Force chain of dark granular layer in (a); f Heat conduction path of dark granular layer in (a) (Zhang et al. 2011)

3.6 Non-contact Physical Interactions Between Particles Table 3.1 Material parameters of the granular assembly

Parameters

91

Symbol

Unit

Value

Density

ρ

kg·m−3

2.7 × 103

Elastic modulus

E

Pa

6.89 × 1010

Poisson’s ratio

v

Heat transfer coefficient

k

Wm−1 K−1

1.0 × 102

Specific heat capacity

c

J·kg−1 K−1

8 × 102

Friction coefficient

μ

Effective micro-hardness

H∗

Surface roughness

η

0.23

0.5 Pa

2.3 × 108 0.13 × 10−6 m

Fig. 3.25 Schematic of temperature distribution in a granular assembly (reference temperature 273 K) (Zhang et al. 2011)

the contact thermal resistance becomes large. Meanwhile, the effective heat transfer coefficient ke decreases remarkably. When the properties of contact surface are favorable, the contact thermal resistance will be close to zero. The thermal resistance between contact pairs will be close to the ideal thermal resistance. Then the effective heat transfer coefficient ke will remain at a good level. The properties of the contact surface are always determined by the properties of particles’ surfaces. It can be seen that the thermal conductivity properties of the particles themselves and the properties of particles’ surfaces are important for the overall thermal conductivity of the granular assembly. Figure 3.24c shows the spatial network of normal contact forces in the granular assembly, in which the thickness of line segments represents the size of normal forces. In the extrusion stage, the granular assembly reaches an equilibrium state, so

92

3 Contact Force Models for Granular Materials

the particles stop moving before the temperature load is applied; due to the reason that the expansion coefficient of particles is zero, the changing temperature will not affect the contact force and lead to the change of particles’ positions. The topological structure of the contact force network remains the state shown in Fig. 3.24c after the temperature load is applied. The heat conduction path in the granular assembly is shown in Fig. 3.24d. Similar to the force network, the thickness of the line segment represents the heat flux of the contact pair. Figure 3.24e, f show the force and heat conduction paths in the dark granular layer in Fig. 3.24a. The results show that the force and heat conduction paths are identical in the topological structure and different in the thickness of the line segment. In other words, the distribution of heat flux between contacting particles is not identical with that of contact force. The reason is perhaps that although the contact heat flux is a function of the contact force, it is related to the material properties, geometric parameters and the temperature difference between particles and is not completely proportional to the contact force. In addition, duo to the reason that the material is in a vacuum state and the thermal radiation between particles is neglected, the heat conduction path shown in Fig. 3.24d is the only transmission way in the granular assembly. In view of the influence of force chain on the mechanical properties of granular assemblies, the study of heat conduction path will play a positive role in understanding the heat conduction properties of granular assemblies. (b) Effect of compressive load Figure 3.25 shows a granular assembly in which there are 18112 spheroidal particles with different sizes. The average size is 0.25d. The material parameters are given in Table 3.1. A temperature difference of 1 K was applied in the x direction. After the granular assembly arrived at the thermal equilibrium state, an average heat flux density and an average temperature gradient can be obtained. Meanwhile, an effective heat transfer coefficient ke was obtained by Eq. (3.98). Then by keeping parameters of the granular assembly unchanged and changing the external pressure load only, the corresponding discrete element simulations were performed. In the discrete element simulations, the stress in the assembly can be calculated under different pressure loads. Figure 3.26 shows the relationship between the heat transfer coefficient ke and the average stress σ of the assembly. ke increases with increasing σ , which is consistent with the experimental observations given by Weidenfeld et al. (2003, 2004). When the compressive load increases, the contact force between particles increases. If the material parameters are constant, the thermal conductivity h i j of the granular contact pair is determined by the contact force. And h i j increases as the contact force increases. Since the contact pair is the basic unit of heat transfer between particles, if the thermal conductivity of the contact pair is enhanced, the overall thermal conductivity of the granular system is also enhanced accordingly. Therefore, the increasing compressive load plays a positive role in improving the overall heat transfer performance of the contacting particles.

3.7 Summary

93

Fig. 3.26 The effective thermal conductivity ke under different compression stresses (Zhang et al. 2011)

3.7 Summary First of all, the basic principles of the DEM are introduced, including linear and nonlinear contact models between spherical particles, motion equations, damping effects, and the selection of time step. The second, considering the plastic deformation of the material, the radius of the yielding region, the equation of overlap and the radial distribution diagram of the normal pressure between contacting particles are presented in the plastic state next. Then, based on the Hertz contact model of spheres and the surface curvature contact model of ellipsoids, respectively, the contact models between irregular shaped particles such as the super-quadric particles and the dilated polyhedral particles are introduced. And then, for the contact model considering adhesion force, based on van der Waals force, Bradley contact theory, JKR contact theory and DMT theory are introduced, respectively. And then, if the granular surfaces are wet and approach to each other, the fluid bridge model between wet particles can be obtained using the Laplace-Young equation. Finally, the governing equations of the DEM for simulations of the heat conduction in granular materials and the calculation method of heat exchange are discussed.

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3 Contact Force Models for Granular Materials

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Tanner LH, Fahoum M (1976) A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light. Wear 36(3):299–316 Tarokh A, Fakhimi A (2014) Discrete element simulation of the effect of particle size on the size of fracture process zone in quasi-brittle materials. Comput Geotech 62:51–60 Thornton C (1997) Coefficient of restitution for collinear collisions of elastic-perfectly plastic spheres. Trans ASME J Appl Mech 64:383–386 Vu-Quoc L, Zhang X, Lesburg L (2001) Normal and tangential force-displacement relations for frictional elasto-plastic contact of spheres. Int J Solids Struct 38:6455–6489 Wang S, Fan Y, Ji S (2018) Interaction between super-quadric particles and triangular elements and its application to hopper discharge. Powder Technol 339:534–549 Wang Y, Tonon F (2010) Calibration of a discrete element model for intact rock up to its peak strength. Int J Numer Anal Meth Geomech 34(5):447–469 Wang Y, Tonon F (2009) Modeling Lac du Bonnet granite using a discrete element model. Int J Rock Mech Min Sci 46(7):1124–1135 Weerasekara NS (2013) The contribution of DEM to the science of comminution. Powder Technol 248:3–24 Weidenfeld G, Weiss Y, Kalman H (2003) The effect of compression and preconsolidation on the effective thermal conductivity of particulate beds. Powder Technol 133(1):15–22 Weidenfeld G, Weiss Y, Kalman H (2004) A theoretical model for effective thermal conductivity (ETC) of particulate beds under compression. Granular Matter 6(2–3):121–129 Wensrich CM, Katterfeld A (2012) Rolling friction as a technique for modelling particle shape in DEM. Powder Technol 217(2):409–417 Wu CY, Li LY, Thornton C (2003) Rebound behaviour of spheres for plastic impacts. Int J Impact Eng 28(9):929–946 Zhang DZ, Rauenzahn RM (2000) Stress relaxation in dense and slow granular flows. J Rheol 44(5):1019–1041 Zhang HW, Zhou Q, Xing HL et al (2011) A DEM study on the effective thermal conductivity of granular assemblies. Powder Technol 205(1–3):172–183 Zheng QJ, Zhou ZY, Yu AB (2013) Contact forces between viscoelastic ellipsoidal particles. Powder Technol 248:25–33 Zhou Y, Zhou Y (2011) A theoretical model of collision between soft-spheres with Hertz elastic loading and nonlinear plastic unloading. Theor Appl Mech Lett 1(4):34–39 Zhu HP, Zhou ZY, Yang RY et al (2008) Discrete particle simulation of particulate systems: a review of major applications and findings. Chem Eng Sci 63(23):5728–5770

Chapter 4

Macro-Meso Analysis of Stress and Strain Fields of Granular Materials

Compared with typical continuous materials, granular materials have many unique discrete, nonlinear physical and mechanical properties. Originally, granular materials were generally treated as a continuum from a macroscopic perspective, and the mechanical behaviors were analyzed by the finite element method (FEM) or meshless method. The DEM can capture the meso-mechanical information of granular materials at the meso-scale, from which the mechanical behaviors of granular materials can be determined at the macro-scale. However, this method needs a large amount of calculations. The development of multi-scale theoretical models based on macro continuum-meso discrete models is helpful in fully understanding the basic mechanical behaviors of granular materials. The key to the macro-meso analysis is the establishment of the relationship between macro and meso properties of granular materials. The basic approach to determining the macro and micro properties of heterogeneous materials is the physics-based mean field theory and mathematical-based homogenization theory. Mean field theory is a method of averaging the effects of the environment on objects and is used to calculate the most important physical information in a physical model. This method is widely used in the studies of mechanics, condensed matter systems, magnetism and structural phase transitions. Granular materials are composed of discrete solids at the meso scale. When they can be approximately treated as a continuum at the macro scale, the physical and mechanical properties of the quasicontinuous bodies should be obtained by mean field theory or homogenization theory. The relationship between the micro-structural information and macro performance of granular materials can be established by small parameter perturbation method in the homogenization theory, which provides a mathematical framework for multi-scale computing for the discrete element analysis (Xiong et al. 2019; Desrues et al. 2019; Kaneko et al. 2003). Multi-scale computational homogenization method based on

© Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_4

97

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4 Macro-Meso Analysis of Stress and Strain Fields …

mean field theory is an effective means of multi-scale analysis of granular materials since it can analyze the deformation from the two scales and reflect the influence of complex micro-structures on macro mechanical behaviors without macro phenomenological constitutive assumption needed. Due to the discreteness and non-periodicity of granular materials, traditional theories based on heterogeneous continuum are no longer applicable. In addition, the theoretical basis of the DEM is still imperfect compared with the classical continuum mechanics. There is a lot of disagreement on the definition of strain of granular materials based on the discrete model. Although discrete element simulations do not depend on the definitions of stress and strain of granular materials, the establishment of the relationship between external loads and macroscopic mechanical behaviors needs the determination of the stress and strain of granular materials (Bardet and Vardoulakis 2001; Chang and Kuhn 2005; Liu et al. 2019; Garcia and Bray 2019). Although granular materials have strong discrete characteristics, its macroscopic stress state is still an important basis for describing its mechanical behaviors, which is also an important research topic in the DEM.

4.1 Computational Homogenization Method Based on Mean Field Theory Computational homogenization is a multi-scale computational method based on representative volume elements (RVE) under the framework of mean field theory. The most important issue in computational homogenization method is to find the relationship between the meso and macro scale variables. Figure 4.1 shows that if the constitutive relation of the particle contact is known, it is necessary to establish the relationship between the particle response and the macro quantities of the RVE in order to obtain the macroscopic constitutive relation of the equivalent continuum of the granular material based on meso-mechanics, that is, localization analysis. Fig. 4.1 Flow chart of homogenization of granular materials

4.1 Computational Homogenization Method Based on Mean Field …

99

4.1.1 Variational Representation of Frictional Contact Problems The force acting on the granular assembly under quasi-static conditions is shown in Fig. 4.2a (Kaneko et al. 2003). The granular assembly is composed of a large number of meso-cell structure with periodic arrangement, where the cell structure is composed of a plurality of elastic particles and pores, as shown in Fig. 4.2b. It is assumed that this is a plane strain problem, and the size of each cell is small enough compared with the overall structure and is represented by symbol ε. Suppose Ω ε is an arbitrary open domain in the granular assembly. The boundary of the assembly is ∂Ω ε . Ω ε consists of three parts, as shown in Fig. 4.2b, Ω ε = Ω Pε ∪ ΩVε ∪ C ε

(4.1)

where Ω Pε is the open domain of particles, ΩVε is the open domain of pores, and C ε is the sum of the surfaces of the internal contacting particles. In addition, ΩCε is defined as the domain in Ω ε excluding C ε , that is, ΩCε = Ω ε \C ε = Ω Pε ∪ ΩVε . Hence, this problem can also be described as the equilibrium problem in C ε of the contact force and frictional force in ΩCε within the boundary ∂Ω ε . The stress equilibrium equation in ΩCε and the boundary conditions of the granular assembly are expresses as: divσε (x) + bε (x) = 0 in ΩCε

(4.2)

uε = 0 on ∂u Ω ε and σε · n = t on ∂t Ω ε

(4.3)

where uε is the displacement vector, bε (x) is the body force, t is the external force vector on ∂t Ω ε , and n is the outward normal unit vector of ∂t Ω ε . It should be pointed out that the superscript ε indicates microscopic heterogeneity.

(a) Macro-structure

(b) Meso-structure

(c)Particle contact pair

Fig. 4.2 Granular assembly composed of discs (Kaneko et al. 2003)

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4 Macro-Meso Analysis of Stress and Strain Fields …

The mechanical behaviors of particles and pores follow the elastic constitutive equation: σε (x) = Dε (x) : eε (x)

(4.4)

where eε is the strain; Dε is the elastic tensor. The strain can be expressed in terms of displacement as: eε (x) = ∇ (S) uε

(4.5)

where ∇ (S) is the gradient operator. A second-order symmetric tensor can be produced by the gradient operator. Certain constraints are also required for solving the above equations. The contact friction equation in C ε is shown in Fig. 4.2c. The outward normal unit vector of particle a at the contact point is nC , the tangential unit vector is tC , and (nC , tC ) follows the right-hand rule. The normal and tangential components of the displacement vector uε and the stress vector Tε in C ε are,   uε = {unε , utε }T · · · · · · unε = uε · nC , utε = uε · tC

(4.6)

  Tε = {Tεn , Tεt }T · · · · · · Tεn = Tε · nC , Tεt = Tε · tC

(4.7)

The contact state in C ε can be obtained by the above components: −Tεn ≥ 0, [[unε ]] ≥ 0, −Tεn [[unε ]] = 0 in C ε

(4.8)

Mohr-Coulomb law of friction can be expressed as:   −μTεn + c ≥ Tεt  in C ε

(4.9)

where μ and c are the friction angle and cohesion, respectively. So far, the quasi-static problem of the granular assembly can be described by the equilibrium Eqs. (4.2)–(4.5) and the constraint Eqs. (4.8)–(4.9), respectively. The original problem is converted into the following variational inequality:   uε ∈ κ ε ΩCε : α(uε , vε − uε ) + j(uε − vε ) − j(uε , uε ) ≥ l(vε − uε )   ∀vε ∈ κ ε ΩCε

(4.10)

where the bilinear form α(·, ·) represents the virtual work done by the internal force; the linear form l(·) represents the external force; j(·, ·) represents the friction force. Hence,  ∇vε : Dε : ∇uε dx (4.11) α(uε , vε ) = ΩCε

4.1 Computational Homogenization Method Based on Mean Field …

l(vε ) = ε

 ∂t Ω ε ε

j(u − v ) =

t · vε ds +  Cε



101

b·vε dx

(4.12)

   μTεn (uε ) vtε ds

(4.13)

ΩCε

The permissible displacement in C ε is expressed as:      κε ΩCε = vε |viε ∈ γ ΩCε ; [[vnε ]] ≥ 0 in C ε

(4.14)

     γε ΩCε = vε |viε ∈ H l ΩCε ; viε = 0 on ∂t Ω ε

(4.15)

  where H l ΩCε is the first-order Sobolev space.

4.1.2 Macro-Meso Two Scale Boundary Value Problems An important idea in the mean field theory is to establish the relationship of (x, y) at macro-meso two scale. The conversion relationship between x and y is y = x/ε. According to Fig. 4.2, domain ΩCε formed by particles and pores can be divided into domain Ω at the macro scale x and domain YC = Y \C at the meso scale y, as shown in Fig. 4.3. ΩCε = Ω × YC = {(x, y)|x ∈ Ω ⊂ R 2 , y = x/ε ∈ YC ⊂ R 2 }

(4.16)

where Y is the domain at the meso scale; C is the contact domain. In this way, the superscript of the variable can be redefined at the macro and meso scale:

uε (x) = u(x, y), eε (x) = e(x, y), σε (x) = σ(x, y) bε (x) = b(x, y), Dε (x) = D(x, y)

(4.17)

The above equation has a periodic nature at the meso scale y. Terada and Kikuchi demonstrated the boundary value convergence conditions for anisotropic materials with periodic meso-structures at macro and meso scales (Terada et al. 2000; Terada and Kikuchi 2001). Equation (4.10) should also satisfy the similar convergence conditions:  Ω

    ∇x v0 − u0 : D : ∇x u0 dx +



+

Ω



     ∇x v0 − u0 : D : ∇ y u1 dx +



Ω



    ∇ y v1 − u1 : D : ∇x u0 dx 

Ω

    ∇ y v1 − u1 : D : ∇ y u1 dx

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4 Macro-Meso Analysis of Stress and Strain Fields …

Fig. 4.3 Decomposition to macro and meso scales (Kaneko et al. 2003)

 + C

   μTn (u1 )[[vt1 ]]ds −  −

∂t Ω

 Ω

   μTn (u1 )[[ut1 ]]ds −

 Ω

  b · v0 − u0 dx

  t · v0 − u0 ds ≥ 0, ∀v0 ∈ γ , ∀v1 ∈ kYC

(4.18)

where · represents the average volume of the unit cells in Y; ∇x and ∇ y represent the gradients at the macro and meso scale, respectively. u0 (x) and v0 (x) are independent of each other at meso scale and can be expressed as:  γ = v0 (x)|vi0 ∈ H l (Ω); vi0 = 0 on ∂u Ω

(4.19)

On the other hand, the displacement u 1 (x, y) at meso scale and its variation v (x, y) have meso-periodic properties, which can be expressed as: 1

 γYC = v1 (x, y)|vi1 ∈ H l (Ω × YC ); Y - Periodic

(4.20)

 kYC = v1 (x, y)|v1 ∈ γYC ; [[vn1 ]] ≥ 0 in C

(4.21)

In (4.18), the trial function v1 can be selected as v1 = u1 . Since γ is a linear space, v can be chosen as v0 = u0 = αw0 . Here, w0 is an arbitrary function in space γ, and α is an arbitrary real number. The inequality (4.18) should satisfy the following equality: 0

4.1 Computational Homogenization Method Based on Mean Field …

 Ω

   ∇x (w0 ) : D : ∇x (u0 ) + ∇ y (u1 ) dx =

103

 Ω

 b · w0 dx +

∂t Ω

t · w0 ds, ∀w0 ∈ γ (4.22)

In (4.18), v0 can also be selected as v0 = u0 , and v1 can be selected as v1 = u1 = αw1 . w1 is an arbitrary function in κYC , and α is an arbitrary positive real number. u1 (x, y) satisfies the following equation in a cell: 

 ∇ y (w1 ) : D : ∇ y (u1 )dy + YC

C

 ≥−

   μTn (u1 )[[wt1 ]]ds −

∇ y (w1 ) : Ddy



   μTn (u1 )[[ut1 ]]ds

 : ∇x (u0 ), ∀w1 ∈ κYC

(4.23)

YC

The existence condition of an unique solution for Eq. (4.23) in κYC is:   κ˜ YC = v1 (x, y)|v1 ∈ κYC ; v1 = 0

(4.24)

Equation (4.23) describes the mechanical behaviors of the meso-structure. Under the constraint Eq. (4.23), Eq. (4.22) gives the averaged mechanical behaviors of the macro-structure. Therefore, as long as there is a solution to the meso-scale problem, the macro-scale problem can be solved, and vice versa. The strong form of Eq. (4.23) is defined as: divσ0 = 0 in YC  −Tn ≥ 0, [[un1 ]] ≥ 0 in C −μTn + c ≥ |Tt |

(4.25)

(4.26)

The constitutive equation at the meso scale is:     σ0 (x, y) = D : ∇x u0 + ∇ y u1 = D : E + ∇ y u1

(4.27)

Here, the macroscopic strain E is defined as:   E = ∇x(s) u0

(4.28)

Hence, the mesoscopic displacement vector u(x, y) can be expressed by the macroscopic (uniform) displacement vector u0 and the mesoscopic periodic displacement u1 (x, y): u(x, y) = E · y + u1 (x, y)

(4.29)

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4 Macro-Meso Analysis of Stress and Strain Fields …

Equation (4.22) can be equivalently written as the partial form of a macroscopic boundary value problem: div



u0 = 0, · · ·

+B = 0 in Ω 

·n = t on ∂t Ω

(4.30) (4.31)

 where and B are macroscopic stresses and body forces, respectively, which can be determined by mesoscopic uniform variables in the unit cell, that is,      (x) = σ0 (x, y) = D : E + ∇ y u1

(4.32)

B(x) = b(x, y)

(4.33)

4.1.3 Macro-Meso Scale Solution Procedures Based on Mean Field Theory A single cell composed of hard spherical particles is shown in Fig. 4.4a. The average stress at the macro scale can be written as:   1 1  E E σ0 dy = ≈ T y (4.34) |Y | Y |Y | E where E is the sum of the contacts between the cell and the exterior boundary ∂Yc ; T E is the force of the exterior particles to the interior particles; yE is the position vector of the contact point between the exterior particles and the interior particles;

(a) Unit cell

(b) Stress state between unit cell and exterior particles

Fig. 4.4 Unit cell composed of spherical particles (Kaneko et al. 2003)

4.1 Computational Homogenization Method Based on Mean Field …

105

|Y | is the volume of the unit cell. This equation is also used to define the average stress in the meso mechanical analysis of granular materials (Oda and Iwashita 2000). Although there is a certain error in the above equation, its approximation to the stress is satisfactory. Equation (4.22) at the macro scale can be analyzed by the FEM, and its mathematical form can be expressed as:   G( ) = {Fext } in Ω

(4.35)

  where {Fext } is the nodal  external force. G( ) is the internal nodal load associated with the macro stress , and its value at each node can be calculated by Eq. (4.34). The meso scale stiffness equation in the DEM is expressed as: {H} = [K]{u} in Yc (at meso scale)

(4.36)

The constraints are expressed as: [[uni j ]] = 0 in C i j ∈ C (contact conditions)

(4.37)

   ij Tt  ≤ −Tinj tan φ in C i j (friction law)

(4.38)

⎧ ⎨ Tit j ≤ −cn   in C i j (cohesive conditions) ⎩ Tit j  ≤ −Tinj tan φ + ct

(4.39)

where {H} is the external force increment caused by the macroscopic strain increment E; [K] is the overall stiffness matrix; {u} is the displacement increment of ij the particle, including two horizontal displacements and one rotational angle; [[un ]] ij is the normal component of the relative displacement of the contact point C of ij ij particle i and j; Tn and Tt are the normal and tangential components of the contact force on contact point C i j ; φ is the friction angle; cn and ct are the normal and tangential components of the cohesive force, respectively. In the micro scale analysis, all particles in the cell should satisfy the following equilibrium equation: α  

 Ri j Ti j = {0} in Yc (i = 1, . . . , n)

(4.40)

j=1

where n is the number of particles in the unit cell; α is the number of particles in  contact with particle i; Ri j is the coordinate transformation matrix at the contact point C i j . For more details of meso scale Eqs. (4.36)–(4.40), please refer to Kishino (1989) (Kishino 1989).

106

4 Macro-Meso Analysis of Stress and Strain Fields …

The macro model based on the mean field theory is shown in Fig. 4.5a. The upper and lower surfaces of the calculation model have a width of 50 mm and a height of 100 mm. Uniformly distributed normal load is applied to the upper, left and right sides, and the vertical displacement constraint is applied to the bottom. A schematic of each cell in the macroscopic model is shown in Fig. 4.5b. Each cell contains approximately 200 particles with different diameters. When the confining pressure is 0.3 MPa, the volumetric strain of the structure is shown in Fig. 4.6. It can be found that when the axial strain is about 2.0%, the structure shows a significant expansion effect around its center; when the axial strain is about 2.4%, the expansion ratio of the overall structure reach 0.5%. It means that particles inside the unit cell, namely, volume of the meso-structure, have expanded.

(a) Macro-structure in finite element

(b) Meso-structure in discrete element

Fig. 4.5 Macro-meso model of granular material (Kaneko et al. 2003)

(a) 1.2%

(b) 1.6%

(c) 2.0%

(d) 2.4%

Fig. 4.6 Volumetric strain nephogram (data represents axial strain) (Kaneko et al. 2003)

4.2 Meso Analysis of Stress Field of Granular Materials

107

4.2 Meso Analysis of Stress Field of Granular Materials The definition of the stress of granular materials began with the investigation of the equilibrium state of the internal contact force and the external force of the granular assembly. The change of the RVE will result in a different descriptive approach for the stress of granular materials. Currently, the mathematical expression of the stress of granular materials tends to be uniform, but there is still some controversy in the symmetry of stress and the calculation method of the high-order stress. Because granular materials are usually heterogeneous and non-periodic, traditional homogeneous material-based theories are no longer applicable. According to the concept of continuum mechanics, stress is defined as the internal force per unit area at a point. Due to the strong non-uniform nature of the granular material and existence of both solid particles and pores, the stress defined in infinitesimal regions is not only highly non-uniform but also discontinuous at the boundaries of solid particles. In order to introduce the concept of stress into granular materials, it is necessary to examine it at the macro scale. Generally, the definition of the stress of granular materials is closely related to the way of characterizing the RVE, and the representation of the internal average stress of the RVE is also different according to different representations of the RVE. Drescher and De Josselin (1972) used a assembly of spherical particles with an arbitrary shape and a total volume of V as RVE, and defined the stress σij as the average stress within the set of spheres (Drescher and De Josselin 1972): σij =

Ne 1  xk f ke V k=1 i j

(4.41)

where k is the symbol of contact within the assembly; xik is the coordinate of the particle on the RVE boundary; f ke j is the external force acting on the boundary points; N e is the total number of external force acting on the boundary. The definition of Eq. (4.41) requires an accurate description of the location of the boundary particles, and the contacts within the assembly are not taken into account. The average stress calculation consistent with Eq. (4.41) is given by considering the microscopic topological structure described by different RVEs’ forms and corresponding action modes of contact forces. Among them, Bagi (2006) gave an equivalent continuum description of the granular assembly based on the proposed spatial cell model, and derived the mathematical expression of the stress and strain (The strain will be introduced in Sect. 4.3.) as well (Bagi 2006). The effect of boundary and volume forces on the stress was considered in the subsequent research. Ehlers et al. used the equilibrium condition of the granular assembly and gave the expressions of the average stress and couple stress essentially the same as Eq. (4.41) (Ehlers et al. 2003). The effect of the particle’s rotation was not considered in the above studies. Subsequently, relevant scholars introduced the contact couple and the particle’s rotation

108

4 Macro-Meso Analysis of Stress and Strain Fields …

into the stress calculation, and obtained a more accurate stress expression (Bardet and Vardoulakis 2001). Based on this, Bardet and Vardoulakis further analyzed the symmetry of the average stress (Bardet and Vardoulakis 2001): σ i j − σ ji =

 1   ae e e xi f j − xae j fi V e

(4.42)

The above equation shows that if there is no external couple at the macro scale, the moment caused by the external force about the centroid of the RVE may not be zero. This results in asymmetry of the stress. The analysis of the stress symmetry of granular materials by Bardet and Vardoulakis has led to widespread debate. Consequently, Researchers realized the stress of granular materials requires uniform specifications. Chang and Kuhn (2005) suggested that the description of the macro and average stress should satisfy three conditions (Chang and Kuhn 2005). The description of the macro or average stress does not depend on the selection of the reference point for determining the branch vector; For the same particulate assembly, the stress measured by different observers should be the same through appropriate tensor transformation, that is, the stress objectivity requirement; The representation of the macro or average stress does not depend on the choice of the center of granular materials.

4.2.1 Average Stress Description of the Micro Topological Structure Based on the concept of Cauchy stress tensor in continuum mechanics, the average stress is constructed based on the exterior contact force of the RVE (Fortin et al. 2002, 2003; Saxcé et al. 2004; Nicot et al. 2013). The Cauchy stress in the 3D subspace V is expressed as: σ(x) = σt (x)

(4.43)

The basic surface of the 3D subspace is defined as dS = dx1 × dx2 . The outward normal vector is n = −dx1 × dx2 /dS, as shown in Fig. 4.7. The stress tensor is p = dF/ds or p=σ(x)n. The 3D space shown in Fig. 4.7 can be divided into several tetrahedrons, as shown in Fig. 4.8. The volume of a tetrahedron can be expressed as: dV =

1 dx1 · (dx2 × dx3 ) 6

(4.44)

The linear operator can be expressed by: I=

dx1 · (dx2 × dx3 )t + dx2 · (dx3 × dx1 )t + dx3 · (dx1 × dx2 )t dx1 · (dx2 × dx3 )

(4.45)

4.2 Meso Analysis of Stress Field of Granular Materials

109

Fig. 4.7 3D subspace in continuum mechanics

Fig. 4.8 Schematic of the tetrahedron after segmentation (Fortin et al. 2003)

The four outward surface vectors of the tetrahedron are expressed as: 1 1 1 dx2 × dx3 , dS2 = dx3 × dx1 , dS3 = dx1 × dx2 2 2 2 1 dS4 = (dx3 − dx1 ) × (dx2 − dx1 ) 2

dS1 =

(4.46)

Four faces will form a closed space, hence 1 dxi dSit 3 i=1 3

IdV =

(4.47)

The original position vector becomes the following equations through the rotational tensor δx, as shown in Fig. 4.9. dxi → dxi = dxi − δx

(4.48)

110

4 Macro-Meso Analysis of Stress and Strain Fields …

Fig. 4.9 Schematic of the rotation of three coordinate axes (Fortin et al. 2003)

3IdV =

3 

dxi dSit

=

3 

i=0

dxi dSit

+ δx

i=0

3 

dSit

(4.49)

i=0

Substituting Eq. (4.49) into Eq. (4.47) and (4.48) gives, 3IdV =

3 

dxi dSit − δxdSt4

(4.50)

i=0

1   t dx dS 3 i=1 i i 4

IdV =

(4.51)

The force acting on the fourth face can be expressed as: dFit = −σ(x)dSi

(4.52)

Considering the symmetrical nature of the stress tensor, Eq. (4.52) can be expressed as: dFit = −dSit σ(x)

(4.53)

The following formula can be derived by combining Eqs. (4.51) and (4.53): σ(x) = Iσ (x) =

4 1  dxi dFit 3dV i=1

(4.54)

The corresponding stress is expressed as: σ(x) = −

4 1  dxi dFit 3dV i=1

(4.55)

4.2 Meso Analysis of Stress Field of Granular Materials

111

Fig. 4.10 The RVE composed of several spherical particles (Fortin et al. 2003)

So far, the stress tensor based on continuous tetrahedral elements can be solved by the above equations. The average stress tensor of the elements can be obtained by the force acting on each face of the tetrahedral element according to Eq. (4.55). In the following, the calculation of the average stress tensor in a RVE will be introduced in detail. The form of the RVE composed of spherical particles should be defined first at the meso scale, which falls in between the micro and macro scale, as shown in Fig. 4.10. The RVE is represented by symbol v, and its volume is given by: V =



v p (1 + e)

(4.56)

p∈V

where e is the porosity, which is obtained by:  V − p∈v v p e=  p∈v v p The outward surface force of the RVE is calculated as follows:  δxa ra dF =

(4.57)

(4.58)

a

where ra and δxa represent contact displacement and contact stiffness, respectively. The following satisfies in an arbitrary trial domain wi ,  wdFt dS = ∂v p

 a

w(xa )rat

(4.59)

112

4 Macro-Meso Analysis of Stress and Strain Fields …

The equation for the calculation of the average stress of the RVE is:

σv p

⎛ ⎞   1⎜ ⎟ = ⎝ xa rat + xf t dV ⎠ vp

(4.60)

vp

where f indicates the body forces of particles which can be calculated by gravity and internal forces between particles. The average stress tensor within the RVE is obtained by: ij

σV =

1  i j r d V a∈K a a

(4.61)

v

where V represents the RVE volume; N represents the number of particles contained j in the RVE; rai represents the contact force; da is the distance between the centroid of the two contacting particles; K v is the number of all contact pairs in the RVE. In Eq. (4.61), the summation formula traverses all contact pairs, including both the particle-particle contacts and the contact between particles and boundaries.

4.2.2 Stress Characterization of Particle Aggregates The contact force in the particle aggregates is mainly distributed at the contact points between particles. This distribution is not uniform, which brings great difficulty to the stress calculation. Christoffersen et al. (1981) proposed a method for calculating the macro stress by applying the principle of virtual work with the consideration of the contact force and particle size (Christoffersen et al. 1981). The spatial position relationship between the two contacting particles A and B is shown in Fig. 4.11. Their centroid coordinates are x A and x B , respectively, and the coordinates of the contact point are x AB . fiAB and fiB A are the force of particle Fig. 4.11 Spatial position relationship between contacting particles A and B

4.2 Meso Analysis of Stress Field of Granular Materials

113

A on particle B and the force of particle B on particle A, respectively. According to Newton’s third law of motion, the contact forces satisfy the following: fiAB + fiB A = 0

(4.62)

Since particle A is in force balance condition, fiAB is expressed as: κ 

Aβ

fi

=0

(4.63)

β=1

where κ is the number of contact points on particle A. According to the equilibrium of moment, fiAB is expressed as: κ 

Aβ

fi

κ      Aβ Aβ Aβ x j − x Aj = fi xi − xiA

β=1

(4.64)

β=1

The summation is carried out over all particles inside the aggregate and the moment at each contact point is expressed as:     A B A AB fiAB x AB x j − x Bj j − x j + fi

(4.65)

Hence, N 

fiα dαj

=

α=1

N 

f αj diα

(4.66)

α=1

where α represents the contact point; N is the number of contact points; diAB = xiB − xiA is a vector connecting the centroid of particle A and B. The relationship between contact force and stress is established by the principle of virtual work. It is assumed that the particles are subjected to the traction force Ti on the boundary S, that is: Ti = σi j v j On face S

(4.67)

where v is the outward normal unit vector of face S. Assume a virtual displacement ui on the exterior surface of the particle. This virtual displacement leads to virtual displacement iα at the contact point α. Assume that the traction Ti is equal to the contact force fiα , the following can be obtained based on the principle of virtual work: N  α=1

fiα iα

1 = V

 Ti ui dS S

(4.68)

114

4 Macro-Meso Analysis of Stress and Strain Fields …

Let the virtual displacement ui be a linear function, then: ui = φi, j x j + ci On face S

(4.69)

where φi j is an arbitrary constant second-order tensor; c is a constant vector, hence iα = φi, j d aj

(4.70)

Substitute Eqs. (4.70) and (4.69) into Eq. (4.68) with the consideration of the divergence theorem,  φi j σ¯ i j −

N 

fiα dαj

=0

(4.71)

α=1

! where σ¯ i j = 1v v σi j dV is the average volume stress. φi j is an arbitrary tensor, according to Eq. (4.71): σ¯ i j =

N   1 α α fi d j + f αj diα 2 α=1

(4.72)

This equation is applicable for both 2D and 3D conditions.

4.2.3 Description of Macro Stress Based on Virtual Work In order to satisfy the three conditions of the previous section, Chang and Kuhn (2005) proposed a generalized virtual work expression. This expression takes into account the force balance conditions within the particulate assembly and the displacement field is described in a polynomial form. The expressions of σi j , σ jki , Ti and Ti j are given based on the principle of virtual work, representing macro stress, macro high-order stress, macro internal torque and macro torque stress, respectively (Kuhn and Chang 2006; Alonso-Marroquín 2011). The representative volume of the granular material is shown in Fig. 4.12. It is typically a sub-domain of the particulate assembly. The average stress in the representative volume can be obtained by statistically calculating the forces and moments within the representative volume. The representative volume is V, including interior particles, boundary particles, and pores. Two loads are generated. One is from the internal forces and moments, and the other is the interaction between particles in the volume and the boundary particles. The centroid position of the representative volume in the global coordinate system X is denoted by Xi0 . The stress at the centroid is used to replace the average stress in the representative volume. A local coordinate system x is established in the representative

4.2 Meso Analysis of Stress Field of Granular Materials

115

Fig. 4.12 Representative volume of granular material (Chang and Kuhn 2005)

volume with origin Xi0 . Therefore, the coordinates of any arbitrary point in the volume can be expressed as: xi = Xi − Xi0

(4.73)

In the local coordinate system x, the centroid coordinate of the nth particle is xin . The internal contact force and moment between particle m and n are denoted as finm and minm , respectively, as shown in Fig. 4.13. The external contact force and moment between boundary b and particle n are denoted as finb and minb , respectively, as shown in Fig. 4.14. Static equilibrium analysis of particle n can be expressed as: fin +

 b∈B

Fig. 4.13 The internal contacts of the representative volume (Chang and Kuhn 2005)

finb +

 m∈V

finm = 0

(4.74)

116

4 Macro-Meso Analysis of Stress and Strain Fields …

Fig. 4.14 Boundary particles and exterior contacts (Chang and Kuhn 2005)

min +

    nm  nb nm minb + ei jk rnb + mi + ei jk rnm =0 j fk j fk b∈B

(4.75)

m∈V

where rnm j represents the vector of the centroid of the nth particle to the internal mth contact point, as shown in Fig. 4.13; rnb j represents the vector of the centroid of the nth particle to the exterior bth contact point, as shown in Fig. 4.14; ei jk is the arrangement tensor. The effects of body force and body torque are ignored, i.e., fin = 0 and min = 0. Hence, Eq. (4.74) and (4.75) can be simplified to: finm = −fimn

(4.76)

minm = −mimn

(4.77)

Therefore, all the internal contact forces are self-balancing and cancel each other out. According to the principle of energy conservation, the virtual work in the representative volume can be obtained by: δW

d,1

 1   nb  nm = f + fi δuin V n b∈B i m∈V     nm  1    nb nb nm mi + ei jk rnb + mi + ei jk rnm δωin = 0 + j fk j fk V n b∈B m∈V (4.78)

The effects of body force and body torque are not considered in the above equation. Assume a virtual displacement δuin and a virtual rotational angle δωin at the centroid of the nth particle, a ‘discrete’ virtual work δW d,1 can be obtained by the summation over all the particles in the representative volume. For any arbitrary δuin and δωin , the virtual work is always equal to zero. The superscript d in the virtual work δW d,1 represents the virtual work of the discrete system, and the superscript 1 represents the first of the two virtual work equations to be discussed. Because the displacement and rotation of the boundary particles in the representative volume may be different from

4.2 Meso Analysis of Stress Field of Granular Materials

117

those of the internal particles, the virtual work equation can be further expressed as: δW

d,2

 1   nb  nm = f + fi δuin V n b∈B i m∈V     nm  1    nb nb nm mi + ei jk rnb + mi + ei jk rnm δωin + j fk j fk V n b∈B m∈V    1  b 1 f − fib δuin + mb − mib δωin = 0 + (4.79) V b∈B i V b∈B i

The external force finb and the moment minb in the last two items on the right side of the above equation are replaced by fib and mib , then are multiplied by the boundary virtual displacements δuin and δωin , respectively. The virtual work δW d,1 and δW d,2 are equivalent, because the last two items do not do work. Equation (4.79) can be divided into two parts, that is, the virtual work δW Ed,1 caused by the boundary displacement and the virtual work δW Id,1 caused by the internal particle displacement. δW Ed,1 and δW Id,1 satisfy the following: δW d,1 = δW Ed,1 − δW Id,1 = 0

(4.80)

The virtual work caused by the boundary displacement and internal force in unit volume are expressed as: δW Ed,1 = δW Id,1 = −

 n 1   nb n 1    nb nb mi + ei jk rnb fi δui + j fk δωi V n b∈B V n b∈B

(4.81)

 n 1   nm n 1    nm nm mi + ei jk rnm δωi fi δui − j fk V n m∈V V n m∈V

(4.82)

Similarly, the second virtual work equation can also be divided into the virtual work caused by the boundary displacement and by the internal particle displacement, respectively, i.e., δW Ed,2 =

1  b b 1  b b fi δui + m δω V b∈B V b∈B i i

(4.83)

 n 1   nm n 1    nm nm mi + ei jk rnm δωi fi δui − j fk V n m∈V V n m∈V  1  b b  1  1  b b n b + fi δui − δuin + mi δωi − δωin − ei jk rnb j fk δωi V b∈B V b∈B V b∈B

δW Id,2 = −

(4.84)

118

4 Macro-Meso Analysis of Stress and Strain Fields …

where the virtual work δW Ed,2 done by the external force is caused by the boundary virtual displacement, and the virtual work δW Id,2 done by the internal force is caused by the virtual displacement between particles. The virtual work done by the internal force can also be expressed as:   1    nm  n n n nm f δui − ei jk rnm j δωi + mi δωi V n m∈V i    1   b b  n n + fi δui − δui − ei jk rnb + mib δωib − δωin j δωi V b∈B

δW Id,2 = −

(4.85)

n where the displacement of the nth particle is calculated by δuin − ei jk rnb j δωi . The internal virtual work is calculated by the contact force and moment caused by the relative displacement and rotation between two contacting particles. For boundary particles, the relative motion between particles generated by the external contact n and δωib − δωin . Meanwhile, the forces is calculated by δuib − δuin − ei jk rnb j δωi displacement caused by the relative motion between boundary particles is also contributed to the virtual work in the representative volume. Therefore, the second virtual work equation δW Id,2 is more accurate than the first virtual work equation δW Id,1 . The virtual displacement can be expressed in the form of a polynomial as:

1 δui (x) = δu i + δu i j x j + δu i jk x j xk 2

(4.86)

δωi (x) = δωi + δωi j x j

(4.87)

Substituting Eqs. (4.86) and (4.87) into Eqs. (4.83) and (4.84) gives: 1  b 1  b b 1  b b b fi + δu i j fi x j + δu i jk f x x V b∈B V b∈B 2V b∈B i j k 1  b 1  b b + δωi mi + δωi j m x V b∈B V b∈B i j

δW Ed,2 = δu i

(4.88)

1  cc 1  c c 1  fi l j + δu i jk fi J jk − δωi ei jk fkc lcj V c∈V ∪B V c∈V ∪B V c∈V ∪B   1   cc mi l j + eikl fkc Jlcj − xlc lcj (4.89) + δωi j V c∈V ∪B

δW Id,2 = δu i j

The macro stress is thus expressed as: σi j =

1  b b 1  cc f x = f l V b∈B i j V c∈V ∪B i j

(4.90)

4.2 Meso Analysis of Stress Field of Granular Materials

σ jki = Ti = T ji =

1  b b b 1  c c fi x j xk = f J 2V b∈B 2V c∈V ∪B i jk

1  b 1  mi = − ei jk fkc lcj V b∈B V c∈V ∪B

  1  b b 1   cc m l + eikl fkc Jlcj − xlc lcj mi x j = V b∈B V c∈V ∪B i j

119

(4.91) (4.92) (4.93)

where B is the contact point between particles and the boundary; b is the contact point between particles; xbj is the coordinates of the contact point; fib , fic , mib and mic represent the force and moment acting on the boundary point or the particle-particle contact point, respectively; ei jk is the permutation symbol; lcj and Jcjk are related to the position of the particle, and lcj is defined as: " lic

=

xim − xin , c ∈ I xib − xin , c ∈ B

(4.94)

where xim and xin represent the coordinates of the particles forming the contact point c, respectively. Chang and Kuhn (2005) further introduced the expression of the macro couple stress μ ji : μ ji = T ji − ekli σl jk

(4.95)

Based on the principle of energy equivalence, Chang and Kuhn (2005) distinguished the equivalent continuum of granular materials into classical continuum, micro-polar Cosserat continuum and high-order Cosserat continuum. The corresponding average stress expressions were obtained based on the principle of virtual work. It has been shown that the average Cauchy stress is consistent with the macro Cauchy stress; It is symmetrical; However, the high-order average stress is different from the corresponding high-order macro stress, and it is generally asymmetrical. In addition, the particle assembly has very complicated mechanical properties due to its discrete properties. When the internal force network is stable, it exhibits solid-like mechanical properties; When the rapid flow occurs, it exhibits fluid-like mechanical properties. The general form of the internal stress state of the particle assembly is expressed as: σi j = pδi j +

Np  1  ˜ ˜ 1 p m x˙ l x˙ J + σi j dV 2V 1 V

(4.96)

where V is the volume of the particle assembly; N p is the number of particles conp tained in the assembly; m is the particle mass; # x˙ is the pulsating flow rate; σi j is the average stress state in a particle.

120

4 Macro-Meso Analysis of Stress and Strain Fields …

The first term on the right hand of the above equation is obtained by the fluid pressure p, and an isotropic stress tensor component is also obtained. The second term is the density of the pulsation energy, also called as the Reynolds stress which is usually anisotropic. The third term is the Cauchy stress term produced by the interaction between particles. In a quasi-statically loaded dense granular system, the contact force between particles is usually long-lasting with small velocity change. Hence, the second item is very small and negligible. Therefore, the effective stress σij in the assembly is defined as:  1 p  σi j = σi j − ρδi j = (4.97) σi j dV V p

where σi j is the average stress tensor in a particle and can be expressed as: 1 = p V

p σi j

 σi j dV p

(4.98)

where V p is the volume of a particle. According to the divergence theorem, p

σi j =

1 Vp

 xi t j dS

(4.99)

Considering that the integral term in the above equation can be replaced by the p summation of the n contact forces acting on the particle, σi j can be rewritten as, p

σi j =

n 1  xi F j Vp 1

(4.100)

Due to the fact that the effective stress defined by Eq. (4.97) is not continuously distributed within the particle assembly (σij = 0 at the pores between particles), Eq. (4.97) can be rewritten as: σij =

Np Np n 1  p p 1  σi j V = xi F j V 1 V 1 1

(4.101)

For discs or spherical particles, the centroid coordinates of each particle can be expressed in the form of vector as xi = Rn i . Similarly, the contact force F can be resolved into normal and tangential components Fn and Ft , which are denoted by Fni = Fn ni and Fti = Ft ti . Here, ni is the normal vector at the contact point, and ti is orthogonal to ni . Therefore, Eq. (4.101) can also be expressed as:

4.2 Meso Analysis of Stress Field of Granular Materials

σij =

121

C C   1  A 1  A R + R B Fn ni n j + R + R B Ft ni t j V 1 V 1

(4.102)

where R A + R B is the distance between two contacting particles; C is the number of contact points.

4.2.4 The Average Stress of the RVE in a Cosserat Continuum The macro and meso homogeneous model in a Cosserat continuum can be derived from the Hill theorem in the classical Cauchy continuum (Li and Tang 2005; Liu et al. 2014; Tang et al. 2019). According to the definition of the average stress, the following equation can be obtained: σJ1 =

1 V

 σ ji dV V

1 = V

1 V

  σki δ jk dV = V

V

 $ %   σki x j ,k − σki,k x j dV

1 V

 σki x j,k dV V

(4.103)

V

According to the Gauss theorem and the equilibrium equation, εi j can be expressed as: εi j = u i, j − ek ji wk

(4.104)

Equation (4.103) can thus be rewritten as: (4.105) This equation expresses the average stress of the RVE. ti and x j are the traction vector and position vector at any point on the boundary S of the RVE, respectively. The average value k ji of the microscopic curvature of the RVE k¯ ji of the Cosserat continuum is defined as:     1 1 1 1 k ji dV = wi, j dV = wi n j dS or k¯ = n ⊗ wdS k¯ ji = V V V V V S V s (4.106) where wi and ni are mesoscopic rotational vectors and unit normal vectors at any point on the boundary S of the RVE, respectively.

122

4 Macro-Meso Analysis of Stress and Strain Fields …

The granular material is now introduced into the RVE. The RVE should be considered as a granular assembly. The internal particles are treated as spherical rigid bodies with the assumption that the particle displacement caused by the contact force between particles is negligible. The particles within the RVE are divided into two parts, i.e., interior particles and peripheral particles. The peripheral particles are assumed to be in contact with the medium outside the RVE. In fact, the boundary S of the RVE is the closed boundary of peripheral particles of the assembly, as shown in Fig. 4.15. The number of particles located on the boundary of the RVE is definite. The number of all particles in the RVE is N g with N p and Ni being the number of particles located on the boundary and inside the RVE, respectively, i.e., N g = Ni + N p . Take a 2D case as an example. The number of contact points between the exterior particles and the boundary is equal to N p or N p + 4. For the circular and rectangular boundaries, in order to simplify the transfer of boundary conditions from exterior particles to interior particles, Nc = N p is assumed temporarily. Therefore, each exterior particle in the RVE has only one contact point with the boundary, as shown in Fig. 4.16. The integral formulae defined in Eqs. (4.105) and (4.106) can be discretized as: σ¯ =

1 V

 x ⊗ tds = s

Fig. 4.15 Macro and meso homogenization of granular materials: discrete particle assembly-Cosserat continuum

Nc Nc 1  1  xic ⊗ tic Si = xc ⊗ fic V i=1 V i=1 i

(4.107)

4.2 Meso Analysis of Stress Field of Granular Materials

123

Fig. 4.16 The meso structure inside the RVE

1 k¯ = V

 n ⊗ wdS = s

Nc 1  nc ⊗ wic Si V i=1 i

(4.108)

where xic is the position vector of the ith contact point between the exterior particle and the boundary; fic is the external force vector applied to the exterior particle through the ith contact point; nic is the outward normal unit vector of the boundary at the ith contact point; wic is the rotational vector of the exterior particles at the ith contact point forming the boundary. In the current homogenization process, the macro stress assumption at the integration point is equal to the meso stress represented by the RVE. According to Eq. (4.107) and (4.108), the macro stress can be expressed as: σ˙ = σ˙¯ = d  ˙ k˙ = k¯ = d

1 V

1 V

 Nc

 Nc i=1

i=1

xic

⊗

fic

Nc Nc 1  1  c c ˙ x˙ c ⊗ fic + x ⊗ fi + V i=1 i V i=1 i

nic ⊗ wic Si +

(4.109)

Nc Nc 1  1  ˙ ic Si + n˙ c ⊗ wic Si nic ⊗ w V i=1 V i=1 i

(4.110) It can be found that the average stress is related not only to changes in contact force and rotation, but also to the geometrical configuration of the RVE.

4.3 Meso Analysis of Strain Field of Granular Materials Compared with the stress of granular materials, there is a great debate about the definition of the strain of granular materials. Bagi (1996, 2006) summarized the relevant strain definitions and broadly divided them into two categories. One is the strain of an equivalent continuum based on the analysis of the internal micro structure

124

4 Macro-Meso Analysis of Stress and Strain Fields …

of the granular material; The other is the optimal fitting strain through the statistical average of the position and displacement of particles (Bagi 1996, 2006). Several typical strain definitions of granular materials have been described based on Bagi’s classification method.

4.3.1 Definition of Strain by Bagi Bagi (1996, 2006) defined strains for 2D and 3D granular materials based on equivalent continuum models of a particle assembly (Bagi 1996, 2006). The spatial cells required to establish the equivalent continuum model are shown in Fig. 4.17. The spatial cells interconnect the centers of adjacent particles to form different triangles or tetrahedrons. The corresponding continuous displacement field is defined as follows. The nodal displacements are compatible with those of the corresponding particle center, and the displacement of any point in the spatial cell can be obtained by the interpolation of nodal displacements. The definition of strain is illustrated by taking the 2D particle assembly shown in Fig. 4.17 as an example. Each triangle is a spatial cell, and the central points of the boundary particles are connected to form the boundary, as shown by the thick solid line in Fig. 4.17. The translational gradient in the Lth cell is expressed as: d a¯ iLj =

Fig. 4.17 Spatial cells defined by Bagi (1996)

1 AL

& du j ni dl lL

(4.111)

4.3 Meso Analysis of Strain Field of Granular Materials

125

where A L represents the area of cell L; du j represents the displacement vector of the boundary point; ni represents the outward normal direction at the boundary point. The area average of the overall granular material is: d a¯ i j = q

1  L L A d a¯ i j A L

(4.112)

p

With ucj = du j − du j , the translational gradient of the assembly can be expressed as: d a¯ i j =

1  ucj dic A c

(4.113)

 where A = c 21 lic dic . dic is the area auxiliary vector at the contact point c; lic is the  edge vector; A = c 21 lic dic . The symmetrical part of Eq. (4.112) is the strain of the granular material.

4.3.2 Definition of Strain by Kruyt-Rothenburg Kruyt and Rothenburg (1996, 2014) and Kruyt et al. (2010, 2014) defined the strain of the granular material based on the equivalent continuum model, as shown in Fig. 4.18. The equivalent continuum model is suitable for 2D particle assemblies, so the strain definition by Kruyt and Rothenburg is only valid for 2D particle assemblies (Kruyt and Rothenburg 1996, 2014; Kruyt et al. 2010, 2014). The equivalent continuum model of Kruyt and Rothenburg is described by the polygons, which is formed by the lines connecting the centers of the contacting

(a) Polygon formed by contact pairs

(b) Granular assembly boundary

Fig. 4.18 Equivalent continuum of particle assembly defined by Kruyt-Rothenburg (Bagi 1996)

126

4 Macro-Meso Analysis of Stress and Strain Fields …

(b) Contact pair is the boundary of

(a) Contact pair is the common edge

the particle assembly

of adjacent polygons

Fig. 4.19 Rotational polygon vector (Bagi 1996)

particles, as shown in Fig. 4.18a. The assembly boundary is shown in Fig. 4.18b. The definition of the continuous displacement field can be summarized as follows. The displacement vector at any node is the same as the displacement vector at the central point of the corresponding particle, and is linearly distributed along the line connecting the centers of the contacting particles (Kruyt and Rothenburg 2004). The average displacement gradient of an arbitrary polygon inside the particle assembly can be represented by the displacement of the particles on the boundary of the polygon. To establish the relationship between the displacement gradient and pq the relative displacement of the particles, a rotational polygon vector gi defined by pq contacting particle p and q is introduced. If the vector l connecting the centers of particles p and q is the common edge of adjacent polygons, the vector connecting pq the center of the two polygons is defined as gi , as shown in Fig. 4.19a; If l pq only belongs to one polygon, the vector connecting the center of the polygons and the pq pq center of l pq is defined as gi , as shown in Fig. 4.19b. Rotate gi by 90° clockwise pq pq to obtain a polygon vector hi = −ei j g j (e12 =+1, e21 = −1, e11 = e22 = 0). The contact point between particle p and q is c, which is used to replace the superscript pq in the above equation. The total area of the equivalent continuum is: A=

1 c

2

lic hic

(4.114)

The relative displacement of the two contacting particles is represented by ducj , and the average displacement gradient of the particles is defined as: da i j =

1  ducj hic A c

(4.115)

The symmetrical part of the above equation is the strain of the granular assembly.

4.3 Meso Analysis of Strain Field of Granular Materials

127

4.3.3 Definition of Strain by Kuhn The strain model developed by Kuhn is suitable for 2D granular systems (Kuhn 1999, 2005, 2010). The equivalent displacement field is similar to that defined by Kruyt and Rothenburg. Considering the particle assembly shown in Fig. 4.20, the displacement gradient of a polygon L is expressed as: da¯ iLj =

1  Q k1 k2 ukj1 bik2 A L (k ,k ) 6 1

(4.116)

2

where ukj1 is the relative displacement of the two contacting particles on edge k1 of the polygon; bik2 is the outward normal vector of edge k2 , and its length is equal to that of k2 ; A L is the area of the polygon; Q k1 k2 is a correction factor. The average displacement gradient of the equivalent continuum can be expressed as the average of the entire polygon displacement gradient with the area as a weight function on the particle assembly, which can be expressed as: da¯ i j =



A L da¯ iLj =

L

1 6

L

⎛ ⎝



⎞ Q k1 k2 ukj1 bik2 ⎠

(4.117)

(k1 ,k2 )

The symmetrical part of the above equation is the average micro structure strain of the granular system defined by Kuhn. Kruyt and Rothenburg (1996) introduced the rotational degree of freedom of particles based on the equivalent continuum model and defined the Cosserat strain of the 2D particle assembly (Kruyt and Rothenburg 1996). Cambou et al. also developed a corresponding equivalent continuum model and described the strain of a 2D particle assembly, which was substantially the same as the strain defined by Bagi (Cambou et al. 2000). The main difference between the above various strain definitions based

Fig. 4.20 Kuhn polygon (Bagi 1996)

128

4 Macro-Meso Analysis of Stress and Strain Fields …

on the equivalent continuum model is the spatial cells of the particle assembly, in other words, the RVEs are defined differently, which results in different representations of the displacement gradient and different forms of strain expressions.

4.3.4 Definition of Optimal Fitting Strain by Cundall The strain model defined by Cundall is suitable for particle assembly composed of particles with any shapes (Cundall and Strack 1979). Consider a particle assembly p composed of N particles. xi is the initial position of any particle p (center position), p dui represents the displacement of the particle center. The average value of the position vector of particle center xi0 and the center displacement dui0 can be respectively expressed by: xi0 =

dui0 =

N 1  p x N p=1 i

(4.118)

N 1  p du N p=1 i

(4.119)

Taking xi0 as the reference point, the relative position of the center of particle p p is denoted as xi . Taking dui0 as the reference value, the relative displacement of the p center of particle p is denoted as dui , then: p

p

# xi = xi − xi0 p

p

d# ui = dui − dui0

(4.120) (4.121)

If the deformation gradient tensor ai j in the particle assembly is equal everywhere, p xi , which the displacement of each particle center corresponds exactly to ai j and # should be satisfied: p

p

xj d# ui = a ji#

(4.122)

However, because of the complexity of the motion in the granular assembly, p p x j = 0 usually works. The objective function constructed by the sum of d# ui − a ji# the squares of Eq. (4.122) is expressed as: Z=

N     p p p p d# ui − a ji# ui − a ji# x j d# xj P=1

(4.123)

4.3 Meso Analysis of Strain Field of Granular Materials

129

When the objective function described by Eq. (4.123) satisfies the extreme conditions, i.e., ∂Z = 0(i, j = 1, 2, 3) ∂a ji

(4.124)

An equation about the deformation gradient tensor ai j can be obtained. For a 2D case, the equation can be expressed as: ⎡

N 

⎢ ⎢ p=1 ⎢ ⎣ N p=1

p p # x1 # x1 p p

# x1 # x2

N  p=1 N  p=1

⎧ ⎫ N  ⎪ p p⎪ ⎪ ⎪ ⎪ d# u1 # x1 ⎪ ⎨ ⎬

⎤ p p # x2 # x1

⎥" ⎥ a1i p=1 = ⎥ N  ⎪ a ⎦ p p p p⎪ 2i ⎪ ⎪ ⎪ # x2 # x2 d# u1 # x2 ⎪ ⎩ ⎭

(4.125)

p=1

The coefficient matrix on the left hand of the equation is a positive definite matrix, which is also a necessary and sufficient condition for the existence of the Cundall optimal fitting strain. This condition requires N ≥ 3 in the granular assembly and there are at least three particles with central non-collinearity. If zi j is used to represent the inverse of the coefficient matrix on the left hand of Eq. (4.125), the optimal fitting deformation gradient # ai j solved by Eq. (4.122) can be expressed as: # ai j = Zik

N 

p p

d# uj# xk

(4.126)

p=1

where the symmetrical part of # ai j is the Cundall optimal fitting strain. The necessary and sufficient condition for the existence of the Cundall optimal fitting strain is N ≥ 4, and there exists at least four particles with their centers being non-coplanar. The parameters required in the Cundall strain can be obtained in the DEM simulations without adding additional calculations. However, this definition only considers the translation of the particle center and does not take into account the effect of particle rotation.

4.3.5 Definition of Optimal Fitting Strain by Liao et al. Liao et al. (1997) defined the best fitting strain using the same method as Cundall, but further increased a characteristic displacement which is the contact deformation determined by the translation and rotation of the particles (Liao et al. 1997). In the particle assembly, the contact point c is formed by particle p and q, and the vector lcj is directed from the center of particle p toward that of particle q. If the deformation gradient tensors ai j within the particle assembly are equal everywhere, each contact displacement duic corresponds exactly to ai j , and it can be expressed as:

130

4 Macro-Meso Analysis of Stress and Strain Fields …

duic = ai j lcj

(4.127)

Based on the idea of Cundall’s optimal fitting strain, the above equation is not valid under normal conditions, and the optimal fitting deformation gradient is required. If there are totally N c contacts in the granular system, the objective function is defined as: Z=

Nc   c   ai j l j − duic ai j lcj − duic

(4.128)

c=1

2 From the extreme condition ∂Z ∂ai j = 0 of Z, the equations for calculating the deformation gradient can be obtained, i.e., N c 

lnc lmc ani =

c=1

Nc 

duic lmc

(4.129)

c=1

The coefficient matrix on the left hand of the above equation being positive determinate is the necessary and sufficient condition for the existence of strain. The optimal fitting deformation gradient a˜ i j of the particle assembly can be obtained by solving the above equation, i.e., a˜ i j = Zik

Nc 

ducj lkc

(4.130)

c=1

where Zi j represents the inverse of the left hand coefficient matrix of Eq. (4.129). The symmetrical part of the above equation is the optimal fitting strain defined by Liao et al. The significant difference from the strain defined by Cundall is the consideration of the effect of particle rotation.

4.3.6 Definition of Optimal Fitting Strain by Cambou et al. On the basis of Liao and other research work, Cambou et al. (2000) obtained two different forms of strain expression by selecting different characteristic displacements (Cambou 1998; Cambou et al. 2000; Nguyen et al. 2009; Dedecker et al. 2015). In the first form, the relative displacement uic of the center of the two contacting particles p and q are selected as the characteristic displacement. The method of finding the best fitting deformation gradient a˜ i j is consistent with the corresponding process of Liao’s strain, and a˜ i j is expressed as:

4.3 Meso Analysis of Strain Field of Granular Materials

a˜ i j = Zik

Nc 

cj lkc

131

(4.131)

c=1

where the matrices Zik and lkc are the same as those in Eq. (4.130). The symmetrical part of Eq. (4.131) is the optimal fitting strain. The second form further considers the effect of particle contact on the displacement based on the contact point displacement being the characteristic displacement. The expression of the best fitting deformation gradient a˜ i j is: a˜ i j = Zik

M 

cj lkc

(4.132)

c=1

where M is the total number of contact pairs. The basic definition of Zik is the same as that in Eq. (4.131), but the specific values are different. The symmetrical part of Eq. (4.132) is the optimal fitting strain. It can be seen from the above discussion that the focus of the optimal fitting strain is to obtain the optimal fitting deformation gradient, which depends on the selection of the characteristic displacement. The accuracy of the strain definition is determined by whether the characteristic displacement can reflect the characteristics of the movement and deformation of the granular assembly.

4.3.7 Definition of Volumetric Strain by Li et al. The relationship between the relative positions of the two particles A and B is shown in Fig. 4.21. The centroid coordinates of particle A and B at different time tn and n+1 tn+1 are expressed as XnA , XnB and Xn+1 A , X B . For different time tn and tn+1 , the Fig. 4.21 Position relationship between particle A and its adjacent particles (Li and Tang 2005)

132

4 Macro-Meso Analysis of Stress and Strain Fields …

relative positions of particle A and B in the global coordinates are expressed as (Li et al. 2010, 2018): XnB A = XnB − XnA

(4.133)

n+1 Xn+1 − Xn+1 B A = XB A

(4.134)

According to the local coordinate system (xln ,yln ) shown in Fig. 4.21, the relative positional relationship between particle A and B are expressed as: xnB A = xnB − xnA

(4.135)

n+1 xn+1 − xn+1 B A = xB A

(4.136)

n+1 where xnA , xnB and xn+1 A , x B represent the local coordinates of the centroids of particle A and particle B at time tn and tn+1 , respectively. The corresponding coordinate transformation relationship can be expressed as:

xnB A = TXnB A

(4.137)

n+1 xn+1 B A = TX B A

(4.138)

3

cosα1 sinα1 T= −sinα1 cosα1

4 (4.139)

Under the local coordinate system, the relative position changes of particle A and B at time tn and tn+1 can be expressed in terms of the deformation gradient Fn : Fn =

xn+1 BA = Rn Nn xnB A

(4.140)

where Rn can be expressed as: 3

cos(α2 − α1 ) − sin(α2 − α1 ) Rn = sin(α2 − α1 ) cos(α2 − α1 ) 3 4 λ AB 0 Un = 0 1 λ AB =

4

 n  n+1  n+1  l n+1 AB n     n , l AB = x B A , l AB = x B A l AB

(4.141) (4.142) (4.143)

where α1 and α2 are the angles between the global X-axis and the local x-axis at different times, respectively, as shown in Fig. 4.21. According to Eq. (4.140), Xn+1 BA

4.3 Meso Analysis of Strain Field of Granular Materials

133

can be obtained by: n Xn+1 B A = FX B A

(4.144)

F = TT fn T

(4.145)

where F can be expressed as:

The displacement derivative matrix can be further expressed as: D=F−I

(4.146)

where I is the unit matrix, thus: 3

γ AB

2 Di j Di j = 3

4 21 (4.147)

According to the relative position relationship between particle A and its surrounding particles and the continuum theory, the effective strain γ A at the centroid of particle A can be expressed as: γA =

nA 1  γ n A B=1 AB

(4.148)

The volumetric strain at the centroid of particle A can be expressed as: γ Av

nA 1  = γ v , γ v = λ AB − 1 n A B=1 AB AB

(4.149)

where n A is the number of particles adjacent to particle A. The mechanical response of the rectangular particle assembly was analyzed according to the definition of volumetric strain, as shown in Fig. 4.22a. The sample size was 86.7 cm × 50 cm with a total number of 4950 spherical particles. A displacement-controlled normal load was applied to the upper and lower surfaces of the sample. The left and right side of the sample were free surfaces. The nephogram of the volumetric strain is shown in Fig. 4.22b. It can be found that the volumetric strain defined above can describe the distribution of shear bands inside the material as well as the mode of failure.

134

4 Macro-Meso Analysis of Stress and Strain Fields …

(a) The sample

(b) Volumetric strain at 3.6 cm normal displacement

Fig. 4.22 Volumetric strain distribution of a rectangular particle assembly (Li and Tang 2005)

4.4 Summary In this chapter, the macroscopic behaviors of granular materials are described by means of the concept of stress and strain based on the multi-scale calculation method in solid mechanics, considering the characteristics that granular materials are often regarded as continuum models at the macro scale and discrete particle models are adopted at the micro scale. In the framework of the mean field theory, in order to find the relationship between variables at the macro-meso scale, the macroscopic constitutive relation of the equivalent continuum of granular materials is obtained using the computational homogenization algorithm based on the multi-scale calculation method of RVEs. In addition, the concept of stress is introduced into granular materials. The average stress, the average stress of the particle assembly in the Cosserat continuum, and the macro stress based on the virtual work principle are introduced. Finally, several calculation methods of strain of granular materials are introduced. The above multi-scale methods have been successfully used to solve many complex problems, such as heterogeneous, multi-phase analysis of mechanical behaviors, local high gradient problems, crack propagation problems, and so on. However, whether the computational homogenization method of granular materials or the application of stress and strain in engineering fields are still at the exploratory stage. Further research and its applications in engineering fields are of important theoretical and practical significance.

References

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References Alonso-Marroquín F (2011) Static equations of the Cosserat continuum derived from intra-granular stresses. Granular Matter 13(3):189–196 Bagi K (1996) Stress and strain in granular assemblies. Mech Mater 22:165–177 Bagi K (2006) Analysis of microstructural strain tensors for granular assemblies. Int J Solids Struct 43(10):3166–3184 Bardet JP, Vardoulakis I (2001) The asymmetry of stress in granular media. Int J Solids Struct 38(2):353–367 Cambou B (1998) Behaviour of granular materials. International Centre for Mechanical Sciences, 385 Cambou B, Chaze M, Dedecher F (2000) Chang of scale in granular materials. Eur J Mech A/Solids 19(6):999–1014 Chang CS, Kuhn MR (2005) On virtual work and stress in granular media. Int J Solids Struct 42(13):3773–3793 Christoffersen J, Mehrabadi MM, Nematnasser S (1981) A micromechanical description of granular material behavior. J Appl Mech 48(2):339 Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnque 29(1):47–65 Dedecker F, Chaze M, Dubujet P et al (2015) Specific features of strain in granular materials. Int J Numer Anal Meth Geomech 5(3):173–193 Desrues J, Argilaga A, Caillerie D et al (2019) From discrete to continuum modelling of boundary value problems in geomechanics: an integrated FEM-DEM approach. Int J Numer Anal Meth Geomech 43:919–955 Drescher A, De Josselin de Jong G (1972) Photoelastic verification of a mechanical model for the flow of a granular material. J Mech Phys Solids 20(5):337–340 Ehlers W, Ramm E, Diebels S et al (2003) From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int J Solids Struct 40(24):6681–6702 Fortin J, Millet O, Saxcé GD (2002) Mean stress in a granular medium in dynamics. Mech Res Commun 29(4):235–240 Fortin J, Millet O, Saxcé GD (2003) Construction of an averaged stress tensor for a granular medium. Eur J Mech 22(4):567–582 Garcia FE, Bray JD (2019) Modeling the shear response of granular materials with discrete element assemblages of sphere-clusters. Comput Geotech 106:99–107 Kaneko K, Terada K, Kyoya T (2003) Global-local analysis of granular media in quasi-static equilibrium. Int J Solids Struct 40(5):4043–4069 Kishino Y (1989) Computer analysis of dissipation mechanism in granular media. Powders and grains. Balkema, Rotterdam, pp 323–330 Kruyt NP, Rothenburg L (1996) Micromechanical definition of the strain tensor for granular materials. J Appl Mech 63(3):706–711 Kruyt NP, Rothenburg L (2004) Kinematic and static assumptions for homogenization in micromechanics of granular materials. Mech Mater 36(12):1157–1173 Kruyt NP, Rothenburg L (2014) On micromechanical characteristics of the critical state of twodimensional granular materials. Acta Mech 225(8):2301–2318 Kruyt NP, Agnolin I, Luding S et al (2010) Micromechanical study of elastic moduli of loose granular materials. J Mech Phys Solids 58(9):1286–1301 Kruyt NP, Millet O, Nicot F (2014) Macroscopic strains in granular materials accounting for grain rotations. Granular Matter 16(6):933–944 Kuhn MR (1999) Structured deformation in granular materials. Mech Mater 31(6):407–429 Kuhn MR (2005) Are granular materials simple? An experimental study of strain gradient effects and localization. Mech Mater 37(5):607–627

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Kuhn MR (2010) An experimental method for determining the effects of strain gradients in a granular material. Int J Numer Methods Biomed Eng 19(8):573–580 Kuhn MR, Chang CS (2006) Stability, bifurcation, and softening in discrete systems: a conceptual approach for granular materials. Int J Solids Struct 43(20):6026–6051 Li X, Tang H (2005) A consistent return mapping algorithm for pressure-dependent elastoplastic Cosserat continua and modelling of strain localisation. Comput Struct 83(1):1–10 Li XK, Liu QP, Zhang JB (2010) A micro-macro homogenization approach for discrete particle assembly Cosserat continuum modeling of granular materials. Int J Solids Struct 47(2):291–303 Li X, Wang Z, Zhang S et al (2018) Multiscale modeling and characterization of coupled damagehealing-plasticity for granular materials in concurrent computational homogenization approach. Comput Methods Appl Mech Eng 342:354–383 Liao CL, Chang TP, Young DH et al (1997) Stress-strain relationship for granular materials based on the hypothesis of best fit. Int J Solids Struct 34(31–32):4087–4100 Liu Q, Liu X, Li X et al (2014) Micro–macro homogenization of granular materials based on the average-field theory of Cosserat continuum. Adv Powder Technol 25(1):436–449 Liu J, Bosco E, Suiker ASJ (2019) Multi-scale modelling of granular materials: numerical framework and study on micro-structural features. Comput Mech 63:409–427 Nguyen NS, Magoariec H, Cambou B et al (2009) Analysis of structure and strain at the meso-scale in 2D granular materials. Int J Solids Struct 46(17):3257–3271 Nicot F, Hadda N, Guessasma M et al (2013) On the definition of the stress tensor in granular media. Int J Solids Struct 50(14–15):2508–2517 Oda M, Iwashita K (2000) Study on couple stress and shear band development in granular media based on numerical simulation analyses. Int J Eng Sci 38(15):1713–1740 Saxcé GD, Fortin J, Millet O (2004) About the numerical simulation of the dynamics of granular media and the definition of the mean stress tensor. Mech Mater 36(12):1175–1184 Tang H, Dong Y, Wang T et al (2019) Simulation of strain localization with discrete elementCosserat continuum finite element two scale method for granular materials. J Mech Phys Solids 122:450–471 Terada K, Kikuchi N (2001) A class of general algorithms for multi-scale analysis of heterogeneous media. Comput Methods Appl Mech Eng 90(40–41):5427–5464 Terada K, Hori M, Kyoya T et al (2000) Simulation of the multi-scale convergence in computational homogenization approaches. Int J Solids Struct 37(16):2285–2311 Xiong H, Yin Z, Nicot F (2019) A multiscale work-analysis approach for geotechnical structures. Int J Numer Anal Methods Geomechan 43:1230–1250

Chapter 5

Coupled DEM-FEM Analysis of Granular Materials

The finite element method (FEM) is mature in theory. It has the advantage of being suitable for solving continuum problems with the characteristics of accurate and reliable numerical results and high efficiency when modelling small deformation problems. However, the FEM simulations of large deformation problems is not ideal. Discrete element method (DEM), on the contrary, is suitable for dealing with discontinuous media, material damage during the transition from continuum to discontinuum, and large deformation problems. Unfortunately, it has the disadvantages of large amount of computations, time-consuming and low efficiency. Since the 1990s, DEM-FEM coupling models have been developed and improved. It is a multi-scale numerical method which can effectively simulate the coupling interactions between granular materials and engineering structures, and between granular materials and continua. Specific coupling algorithm should be proposed according to different engineering requirements. Currently, there are mainly three kinds of DEM-FEM coupling models, namely, the combined finite-discrete element model (combined FEM-DEM) to simulate the fracture of brittle materials, the coupling model of the DEM-FEM bridging domain (bridging scale method), and the interaction model between discrete media and continua. The combined FEM-DEM for the fracture of continuum was pioneered by Munjiza (Munjiza et al. 2013), which simulated the crack propagation through the fracture of joint elements and has been widely used (Smoljanovi et al. 2013). In the bridging scale DEM-FEM model, the DEM is adopted for the local micro scale analysis, and the FEM is used in the remaining domain for the macro scale simulation. The transfer of the mechanical and physical characteristics can be achieved by introducing a transitional layer (Haddad et al. 2016) or imposing the displacement constraint conditions by the penalty function method (Xu and Zang 2014). The last one is adopted for the interaction between granular media and continuous structure, realizing the transfer of the boundary conditions of the two media (Chung et al. 2016). This method has been used in the simulation of the tyre-pavement interaction (Michael et al. 2014), the ice-inducted vibrations of conical jacket platform (Wang et al. 2018), impact of © Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_5

137

138

5 Coupled DEM-FEM Analysis of Granular Materials

granular materials (Liu et al. 2013), the effect of earthquake on pipeline (Rahman and Taniyama 2015), and so on. The interaction between granular materials and engineering structures is a typical DEM-FEM coupling problem at different scales. The DEM is used to simulate the movement of granular materials, and the FEM is used to calculate the structural deformation and vibration. By this way, the coupled DEM-FEM is established. The displacement at the interface, the force transfer and the difference in time step should be precisely treated in order to couple the dynamic process of the two computational methods.

5.1 Combined DEM-FEM Method for the Transition from Continuum to Granular Materials The combined DEM-FEM method for the transition from continuum to discrete materials was proposed by Munjiza in the 1990s. In this method, the system is discretized into a group of discrete elements, and each discrete element is divided into several finite elements, as shown in Fig. 5.1. The basic idea is as follows. The DEM is adopted to solve the discontinuum mechanics problems such as contact between particles, and the FEM is used to solve continuum mechanics problems such as deformation (Munjiza and Latham 2004). In this way, the advantages of the FEM in deformation problems and the DEM in contact problems are both remained. The method can be used to solve large displacement of structures (Owen et al. 2007; Karami and Stead 2008; Choi and Gethin 2009) and highly nonlinear mechanical problems of contact and dynamic failure of materials (Lewis et al. 2005; Gethin et al. 2006; Frenning 2008). The algorithm includes contact algorithm, deformation calculation, explicit integral algorithm and failure model. Fig. 5.1 Schematic of the combined DEM-FEM model by Munjiza (Munjiza 2004)

5.1 Combined DEM-FEM Method for the Transition …

139

5.1.1 Contact Algorithm Contact algorithm of the combined DEM-FEM method mainly includes two aspects: contact search and contact detection. The contact search is to determine the condition of adjacent elements to each element and to prepare for the contact force calculation. The contact algorithm is time-consuming and its computation amount is much large. Therefore, the contact search algorithm requires not only good robustness, but also high computational efficiency and low memory consumption. Munjiza proposed a linear basic algorithm (NBS algorithm) (Munjiza and Andrews 1998). NBS algorithm is a method based on spatial discretization (Han et al. 2007), that is, the computational domain is divided into independent cells with size d.   Each cell can be represented by a set of numbers i x , i y , i x = 1, 2, 3 . . . n x , i y = 1, 2, 3 . . . n y . n x and n y are the sum of grids in x and y directions, and can be respectively expressed as: nx =

xmax − xmin d

(5.1)

ny =

ymax − ymin d

(5.2)

Under the condition that each discrete element can only be in one cell, the connection between elements and cells can be established by the coordinates of elements and the location of grids, that is:   x − xmin i x = Int d   y − ymin i y = Int d

(5.3) (5.4)

where i x , i y is the cell number of the discrete element, (x, y) is the elemental coordinates. Then, the neighboring elements of each cell are determined by comparing the target cell and its neighboring cell. Because the algorithm limits the searching area of each cell to a small range, as shown in Fig. 5.2, this ensures that the algorithm has high computational efficiency. On the other hand, the NBS algorithm uses the linked-list storage technology. Its storage process is similar to the ‘stack’ process, which can be effectively used to arrange and store disorderly repetitive finite integer sequences. The linked-list storage technology ensures that the memory occupancy of the NBS algorithm is only proportional to the number of cells in the system, and independent of the computational space and cell size. After the determination of the neighboring elements of each element, it is necessary to carry out contact detection based on the geometry and spatial position of the element. Contact detection is mainly carried out in two steps, namely, estimating

140

5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.2 Cell search area

whether the target element is in contact with the neighboring element, and calculating the contact force between elements if they are in contact. Munjiza proposed a penalty function method based on potential function to calculate the contact force (Munjiza and Andrews 2000). The basic idea is that the free movement of bodies in space and the existence of small penetration between bodies in contact are allowed. The non-penetration between bodies in contact is approximately satisfied by the penalty mechanism, that is, once the penetration occurs between bodies in contact, the repulsive force is generated in the contact area through the penalty mechanism, and the penetration between bodies will gradually decrease to zero under the action of the repulsive force. The overlapping area of the two contacting elements is S, the boundary is , as shown in Fig. 5.3. Assuming that the infinitesimal area of any point in the overlapping region is d A, the repulsive force generated is as follows:   d f = gradϕc (Pc ) − gradϕt (Pt ) dA

(5.5)

where ϕc and ϕt are some potential functions defined on the master and the slave contact body, respectively. Pc and Pt are the corresponding points at the infinitesimal element on the master and the slave contact body, respectively. By integrating over Fig. 5.3 Contact force due to an infinitesimal overlap around point Pc and Pt (Munjiza 2004)

5.1 Combined DEM-FEM Method for the Transition …

141

the overall penetration region, the repulsive force between two bodies in contact can be obtained as:    fc = gradϕc − gradϕt dA (5.6) S=βt ∩βc

Applying the Gauss formula, the above formula can be rewritten as: fc =

βt ∩βc

n  (ϕc − ϕt )dA

(5.7)

where βt ∩ βc is the boundary surface of the penetration region, n  is the outward normal of boundary . In the combined DEM-FEM, individual discrete elements are discretized into finite elements, and each discrete element can be expressed in terms of its finite elements, namely, βc = βc1 ∪ βc2 . . . βci . . . βcn

(5.8)

βt = βt1 ∪ βt2 . . . βt j . . . βcm

(5.9)

where β c and β t are the contactor and target discrete elements, respectively, m and n are the total number of corresponding finite elements. The potentials φ c and φ t can also be associated with corresponding finite elements: ϕc = ϕc1 ∪ ϕc2 · · · ϕci · · · ϕcn

(5.10)

ϕt = ϕt1 ∪ ϕt2 · · · ϕt j · · · ϕcm

(5.11)

Integrating over finite elements in the overlapping region, the repulsive force can be obtained: fc =

m  n

βci ∩βt j

i=1 j=1



 gradϕci − gradϕt j dA

(5.12)

In a similar way, the above equation can be transferred to the integration over the boundaries of finite elements through Gauss formula, fc =

m  n

i=1 j=1

βci ∩βt j

  n βci ∩βt j ϕci − ϕt j dA

(5.13)

where the contact between two discrete elements can be obtained through relevant boundaries of finite elements involved.

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5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.4 Tetrahedron volume coordinates

To preserve the energy conservation during contact, the potential function of tetrahedron element can be expressed as:       (P) = min 4V1 V, 4V2 V,4V3 V,4V4 V

(5.14)

where Vi is the volume coordinates of point P, as shown in Fig. 5.4, which can be expressed as: V1 =

V (P234) V (1234)

V2 =

V (P341) V (1234)

(5.15)

V3 =

V (P412) V (1234)

V4 =

V (P123) V (1234)

(5.16)

It can be proved that the contact force calculated by the potential function and the work due to the contact force are independent of the selection of coordinates system and the rigid body motion.

5.1.2 Deformation of Element Each discrete element represents a single deformable body, which occupies a region of space at any instance of time as shown in Fig. 5.5. For regions with special meanings, designate the sub-region B as the initial or reference configuration, that is, p∈B

(5.17)

5.1 Combined DEM-FEM Method for the Transition …

143

Fig. 5.5 Deformation of a discrete element (Munjiza 2004)

where p is the material point. The region bounded by the body and boundary is called parts. Body deforms via mapping: x = f ( p)

(5.18)

where f is a smooth mapping function. For any point p, it satisfies that: det∇ f ( p) > 0

(5.19)

In the global coordinates, the deformation gradient tensor can be written using the following matrix: ∂u x + u(x, y, z) 1 + ∂u ∂u ∂x ∂y ∂z ∂v ∂v ∂v F = ∇f = ∇ y + v(x, y, z) = ∂ x 1 + ∂ y ∂z ∂w ∂w ∂w z + w(x, y, z) 1 + ∂z ∂x ∂y

(5.20)

The deformation of a body usually consists of rigid displacement and deformation with constant deformation gradient. Rigid displacement will not produce deformation. Hence, the deformation with constant deformation gradient is F(p) = const

(5.21)

which is referred to as homogeneous, and can be expressed as: f(p) = f(q) + F(p − q)

(5.22)

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5 Coupled DEM-FEM Analysis of Granular Materials

Two important examples of homogeneous deformation are stretch from q and rotation about q. Stretch from q can be written as follows: f(p) = q + U(p − q)

(5.23)

where U is a symmetric and positive definite tensor, can be written as follows: s1 0 0 U = 0 s2 0 0 0 s 3

(5.24)

where si are eigenvalues corresponding to three coordinates. For homogeneous isotropic materials, the constitutive relations are as follows: T=

1 (|detF|)2/ 3

 E  E  Ed + Es + 2μD (1 + v) (1 − 2v)

(5.25)

where E and v are Young’s modulus and Poisson’s ratio of materials, respectively. μ is viscous damping coefficient.   FFT 1  − I Ed = 2 (detF)2/ 3   2/ 3 − 1 ([detF])  Es = I 2

(5.26)

(5.27)

are the Green–St. Venant strain tensor due to shape changes and volume changes, respectively. D is the deformation rate: ⎛ ∂vxc ∂ xi 1 ⎜ ∂v D = ⎝ ∂ xyci 2 ∂vzc ∂ xi

∂vxc ∂vxc ∂ yi ∂z i ∂v yc ∂v yc ∂ yi ∂z i ∂vzc ∂vzc ∂ yi ∂z i

∂v xc ∂ xi ∂vxc + ∂ xi ∂v xc ∂ xi

∂v yc ∂vzc ∂ yi ∂z i ∂v yc ∂vzc ∂ yi ∂z i ∂v yc ∂vzc ∂ yi ∂z i

⎞ ⎟ ⎠

(5.28)

where vxc , v yc , vzc are the coordinate of the material point in the current coordinate system, xi , yi , z i are the coordinates of the material point in the initial coordinate system.

5.1.3 Failure Model of Materials In order to fast simulate the failure of brittle materials, a single/smeared failure model based on experimental stress-strain relationship was proposed (Munjiza et al. 1999). The stress-strain curve of a typical brittle material in uniaxial tension is shown in

5.1 Combined DEM-FEM Method for the Transition …

145

Fig. 5.6. The curve can be classified into two stages, that is, the strain-hardening stage and the strain-softening stage. In the strain-hardening stage, failure does not occur and the standard FEM can be used to solve the mechanical behaviors; While in the strain-softening stage, cracks occur in the material, the stresses decreases with the increase of strain, and the single-crack model is used. As shown in Fig. 5.7, the separation vector δ at any point on the crack can be divided into two parts: δ = δn n + δt t

(5.29)

where n and t are the unit vector in the normal and tangential direction of the point, δn = δ·n, δt = |δ − δn · n| are the normal and tangential component of δ, respectively. Meanwhile, as shown in Fig. 5.8, the tensile force P at the point can be resolved into p = σn + τt

(5.30)

where σ and τ are the normal and tangential stresses, and can be determined by: Fig. 5.6 Stress-strain curve of a typical brittle material in uniaxial tension (Munjiza 2004)

Fig. 5.7 Separation vector divided into two components at any point on the crack (Munjiza and Latham 2004)

146

5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.8 Traction vector divided into two components at any point on the crack (Munjiza and Latham 2004)

σ = z ft

(5.31)

τ = z fs

(5.32)

where f t and f s are the tensile and shear strength of the material. z is a homogenous variable depending on both material constants and the separation vectors, that is, 

  a + b − 1 d(a+cb/ ((a+b)(1−a−b))) × a(1 − d) + b(1 − d)c e z = 1− a+b

(5.33)

where a, b, c are material parameters chosen to fit a particular experimental curve. d is the amount of the homogenous crack opening and can be calculated through the separation vector components:   d=

δn δcn

2

 +

δt δct

2 (5.34)

where δcn and δct are the crack opening limit and slippage limit determined by the energy release rate Gf and the strength. Due to the complexity of fracture mechanism, the shape of the strain-stress curve of mode II loaded cracks is simply assumed the same as that of mode I loaded cracks, thus δcn and δct are δcn = 2.7488G I δct = 2.7488G II

 

ft

(5.35)

fs

(5.36)

where G I and G II are the energy release rates required for the failure of mode I and mode II loaded cracks, respectively.

5.1 Combined DEM-FEM Method for the Transition …

147

In the combined DEM-FEM, another two assumptions are made. Cracks always form at the boundaries of finite elements; All common faces of adjacent finite elements can have cracks. For computational convenience, a special joint element is inserted between the two adjacent finite elements initially. At the initial moment of time, the thickness of the joint element is zero and its geometric shape is in accordance with the boundaries of finite elements. The strain-hardening and strain-softening stages can be simulated through the establishment of the constitutive relationship of the joint element. Figure 5.9 shows the schematic of the joint elements corresponding to linear tetrahedron element (6-node) and nonlinear quadric tetrahedron element (10-node) and their finite elements together. Figure 5.10 shows the joint element in the linear tetrahedron element. The upper and lower surface are defined in the joint element, respectively. The upper surface is determined by nodes from 1 to 6, while the lower one is determined by nodes from 7 to 12. The coordinates of any point on both surfaces can be interpolated by corresponding nodes. Thus, the coordinates x u of any point on the upper surface and the coordinate vector x l ψ of any point on the lower surface are:

(a) Linear tetrahedron element

(b) Nonlinear tetrahedron element

Fig. 5.9 The joint elements and its adjacent finite elements (Munjiza 2004)

Fig. 5.10 The joint elements inserted in the linear tetrahedron elements (Munjiza 2004)

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5 Coupled DEM-FEM Analysis of Granular Materials

xu =

6

Ni xi

(5.37)

Ni xi

(5.38)

i=1

xl =

12

i=7

where xi is the coordinate vector of node i. Ni is the standard shape function, which is a function with respect to the natural coordinates, N1 = (1 − 2ε − 2η)(1 − ε − η) N2 = (2ε − 1)ε N3 = (2η − 1)η N4 = 4ε(1 − ε − η) N5 = 4εη N6 = 4η(1 − ε − η)

(5.39)

and Ni = N(i−6) , if i > 6. In order to calculate the separation vector components, another surface named the base surface is defined:  1

1 u x + xl = Ni (xi + xi+6 ) 2 2 i=1 6

xb =

(5.40)

The unit vector normal to the base surface at any point is v1 × v2 |v1 × v2 |

(5.41)

v1 =

 1 ∂ Ni  ∂ xb = x i + x (i+6) ∂ε 2 i=1 ∂ε

(5.42)

v2 =

 1 ∂ Ni  ∂xb = xi + x(i+6) ∂η 2 i=1 ∂η

(5.43)

n= where v1 and v2 can be expressed as

6

6

The separation vector δ at any point is δ = xu − xl =

6

i=1

  Ni xi − x(i+6)

(5.44)

5.1 Combined DEM-FEM Method for the Transition …

149

For computational convenience, the joint element is inserted into the boundary of finite element before calculation starts. If the material does not fail, the continuity of the material is guaranteed by the penalty function method, namely, two variables δ pn and δ pt need to be introduced to determine whether the material is in the strain hardening stage or not, δ pn =

2h f t p0

(5.45)

δ pt =

2h f s p0

(5.46)

where δ pn and δ pt are the normal and tangential separation allowed before the material reaches the tensile strength f t and shear strength f s , respectively. h is the element size. p0 is the penalty factor. Due to the continuity of the material, both δ pn and δ pt should theoretically be zero. This requirement can be achieved by increasing the penalty factor. When the penalty factor is infinite, it satisfies: lim δ pn = 0

(5.47)

lim δ pt = 0

(5.48)

po →∞

po →∞

Increasing the penalty factor can ensure the continuity of adjacent finite element boundaries before the material reaches its tensile strength and shear strength. When the separation variables satisfy both δn < δ pn and δt < δ pt , the material is in the stage of strain hardening, and the bonding stresses between the upper and lower joint elements are as follows: 

   δn 2 δn − σ = 2 ft δ pn δ pn   2  δt δt τ= 2 − fs δ pt δ pt

(5.49)

(5.50)

Otherwise, the material is in the stage of strain softening, the bonding stresses should follow Eqs. (5.31) and (5.32). After the determination of bonding stresses p, the equivalent nodal forces of the joint element are  pNi |v1 × v2 |ds (1 ≤ i ≤ 6)

fi = − s

(5.51)

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5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.11 Impact failure process of sphere based on the combined block DEM-FEM method (Guo et al. 2016)

 pN j |v1 × v2 |ds (7 ≤ i ≤ 12)

fj = −

(5.52)

s

Guo et al. (2016) developed a 3D failure model for brittle materials through the combination of the block DEM and FEM, and simulated the impact failure of a ball (Guo et al. 2016). The simulation process is shown in Fig. 5.11. The spherical element is composed of 4-node tetrahedral elements. The material failure criterion is applicable to joint elements. If failure occurs, a number of tetrahedral elements will be generated at the failure site. Mohr-Coulomb failure criterion is used to determine the failure state of joint elements. In the combined DEM-FEM model, FEM and DEM are separated in terms of space and time during calculation. In terms of space, the deformation of finite elements is continuous, while discrete elements describe the discontinuity at the contact of elements. In terms of time, the continuity between tetrahedral elements is only constrained by the joint elements before crack initiation. Then, after crack formation, the interaction between tetrahedral elements on the two sides of the crack is simulated by contact search and interaction algorithm (Guo et al. 2016). Munjiza applied the combined DEM-FEM method to anisotropic material deformation. Figure 5.12 is a 2D simulation of the propagation of explosion in anisotropic geological structure. The combined model used 6-node composite triangular element, and the detailed description of its constitutive model is referred to Lei et al. (2016).

5.2 Coupled DEM-FEM Model for the Continua-Discontinua Bridging Domain For many engineering problems, the region where large deformation occurs is usually the local region of the whole computational domain. In order to give full play to the advantages of each computational method, the DEM is used to simulate large deformation regions which are local and need to be focused on, and the FEM is

5.2 Coupled DEM-FEM Model for the Continua-Discontinua …

(a) Anisotropic model of typical geological structures

151

(b) Process of explosion propagation in geology

Fig. 5.12 Simulation of explosion propagation in anisotropic geological structure based on the combined block DEM-FEM method (Lei et al. 2016)

used to simulate the remaining small deformation regions. Hence, coupling DEMFEM models for the continua-discontinua bridging domain have been established by effectively combining the FEM and DEM (Li and Wan 2011). This kind of coupling model not only has a good ability to simulate large deformation, but also maintains good computational efficiency (Oñate and Rojek 2004).

5.2.1 Weak Form of Governing Equations for the Bridging Domain The bridging scale DEM-FEM model puts forward requirements of additional displacement constraint conditions at the DEM-FEM interface. It transforms the variational principle with additional conditions to the one without additional conditions by adopting constrained variational principle. The algorithm starts from the minimum potential principle and decomposes the calculation domain into discrete element region and finite element region. As shown in Fig. 5.13, the spatial domain is divided Fig. 5.13 The schematic of domain division (Xu and Zang 2014)

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5 Coupled DEM-FEM Analysis of Granular Materials

into subregion a and b represented by Ωa and Ωb , respectively. The surface of subregion a is further divided into Sσ a and Sua , while the surface of subregion b is divided into Sσ b and Sub . Here, Sσ a and Sσ b are the surfaces on which the external forces are prescribed; Sua and Sub are the surfaces on which the displacements are prescribed. The coupling interface between subregion a and b is Sab . The displacement functions in subregions a and b are u ia and u ib , respectively, and they are independent from each other. Generally, additional constraint conditions are introduced into the functional, and a modified functional is reconstructed as, ∗ 

=



p

+

u

 cp

=



+

pa



+

pb



(5.53)

cp

   where u is the functional without considering the interface. pa and pb are a b the  functionals that are related to the displacement field u i and u i , respectively. cp is the functional with the incorporation of additional constraint conditions. The constrained condition is the compatibility of the displacement on the interface Sab (Lei and Zang 2010): u ia − u ib = 0

(5.54)

Currently, there are two common methods to construct modified functional by introducing additional constraint conditions, namely, Lagrange multiplier and  penalty function method. For the Lagrange multiplier method, cp can be expressed as:     (5.55) = λ u ia − u ib ds Sab

cp

where λ is the function vector of a set of coordinates on interface Sab , Lagrange multiplier.  For the penalty function method, cp can be expressed as:  cp

 = Sab

2  α u ia − u ib ds

 cp

is called

(5.56)

where α is the penalty coefficient. Although the Lagrange multiplier method can precisely satisfy the additional displacement constraint conditions, it has some disadvantages. The increase of the order of the equation set with increasing additional conditions and having zero entries on the principle diagonal of the stiffness matrix bring in computational inconvenience  and increase computational cost. The modified functional ∗p no longer maintains the same extremal property as that of the original functional and need to be further determined. If explicit central difference is adopted when solving, the inertial term

5.2 Coupled DEM-FEM Model for the Continua-Discontinua …

153

is incompatible with explicit computational scheme due to the increasing degree of freedom of the equation. Compared with the Lagrange multiplier method, the penalty function method has just the reverse advantages and disadvantages. Its advantage is as follows. The degree of freedom does not increase. The modified functional still maintains the same extremal property as that of the original functional. If explicit central difference is adopted when solving, the inertial term is compatible with explicit computational scheme. As a result, the penalty function method is widely applied in the coupling problems of different media, connections of different type structures or elements. Its disadvantage is that additional constraint conditions can only be approximately satisfied (Hunˇek 1993).  The first order variation of cp in Eq. (5.56) is: δ



 = Sab

cp

   2αi u ia − u ib δu ia − δu ib ds

(5.57)

Hence, the first order variation of Eq. (5.53) with the combination of Eq. (5.57) gives, δ

∗  p

=δ

 u

 + Sab

   2αi u ia − u ib δu ia − δu ib ds

(5.58)

Based on the principle of virtual work, the interface coupling force is calculated as:   FiD = −2αi u ia − u ib

(5.59)

Based on the above equations, two conclusions can be drawn. No combined methods are needed when Ωa = 0 or Ωb = 0; The solutions in different subregions can be obtained by using different methods, as long as the interface stress is calculated by a proper method. In the coupling algorithm, governing equations of DEM and FEM are independent with increasing only the coupling interface force in the loading term and other terms remaining unchanged. Hence, no modifications are needed in structural linear elastic analysis or material nonlinear problems. When using explicit algorithm to solve the governing equations, the coupling interface force can be calculated based on the displacement state variable at the end of previous time step. In this way, the coupling interface force becomes known quantity, and the iteration is avoided in the implicit algorithm.

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5 Coupled DEM-FEM Analysis of Granular Materials

5.2.2 Coupling Interface Force The solving of the interface force is the key of the model, and the interface force is related to the displacement of the points on the interface. The displacement in the FEM region can be obtained by nodal interpolation. However, no mathematical expression exists describing the displacement of any point in the DEM region. Therefore, the coupling is divided into the coupling of discrete elements with finite element nodes, face center of finite elements and any position on the finite element surface, respectively (Gao et al. 2014). (1) Coupling with finite element nodes Figure 5.14 shows the initial relative position of discrete elements and finite elements on the interface. D1 , D2 , D3 and D4 are the four discrete elements. N1 , N2 , N3 and N4 are the four nodes on the coupling surface of finite elements. L 1 , L 2 , L 3 and L 4 are the points on discrete elements who are coincident with finite element nodes at initial moment of time. The interface to the finite element side can be obtained by the shape function interpolation of the four nodes N1 , N2 , N3 and N4 of the finite element; While the interface to the discrete element side can be obtained by the interpolation of L 1 , L 2 , L 3 and L 4 with same shape functions. At the initial moment of time, the coincidence of N1 , N2 , N3 and N4 with L 1 , L 2 , L 3 and L 4 , and the same shape functions used result in exact coincidence of the interfaces on both sides. It indicates that the coupling interface satisfies the constraint of u d = u f at the initial moment of time. In the coupling method, the penalty function is used to guarantee the satisfaction of constraint conditions. Theoretically, the bigger the penalty coefficient, the more the constraint condition approaches satisfaction. If the penalty coefficient is selected to be infinity, the constraint conditions are precisely satisfied. In fact, the value of the penalty coefficient is under certain restrictions. The selection of excessive values will cause computational instability and too small time step. Therefore, the constraint conditions in the penalty function method can only be approximately satisfied in actual computations. The displacement field expressed by nodal displacement interpolation, Fig. 5.14 The schematic of domain division (Gao et al. 2014)

5.2 Coupled DEM-FEM Model for the Continua-Discontinua …

 Sd f

   f α u id − u i δu id ds =

 Sd f

fN

u dI iB − u I i

 α N I N J δu dJ i ds

is converted to matrix form:   

 T f δu e K be u e α u id − u i δu id ds = Sd f

155

(5.60)

(5.61)

e

where K be can be expressed as,  Kbe

=

Sde f

NT αNds

(5.62)

It is a 12 × 12 stiffness matrix of a single coupling interface; is a 12 × 3 shape function matrix; α = αE is a 3 × 3 penalty coefficient diagonal matrix. u e = u d B − u f N is the relative displacement matrix of the coupling points, in which u f N and u d B are the displacement matrices of the finite element nodes and discrete element coupling points, respectively. u d B can be obtained by using the motion equation of the discrete element. K be in Eq. (5.62) is converted to a diagonal matrix K beDia for simplification. For densely packed discrete elements, the diagonal stiffness matrix K beDia of the interface at the initial moment of time is (Lei and Zang 2010), KbeDia =

αA E 4

(5.63)

√ where A = 2 2r 2 is the area of rhombus whose corners are N1 ,N2 ,N3 and N4 , r is the radius of each discrete element and E is a 3-order unit matrix. The penalty coefficient α is selected as, α =γK

(5.64)

where γ is a scaling = min(K   coefficient.K   f , K d ) is a material related parameter. Here, K f = E f 1 − 2μ f and K d = E d (1 − 2μd ), where E f and E d , μ f and μd are the Young’s modulus and the Poisson’s ratio of the materials in the finite element and discrete element subregions, respectively. (2) Coupling with face center of finite elements Figure 5.15 shows the relative positions of spherical discrete element and hexahedral finite element at the initial moment of time. N1 , N2 , N3 and N4 are four nodes of finite element on the separating interface. Suppose the centers of N1 , N2 , N3 and N4 , N1 , N4 , N5 and N6 , N7 , N5 , N1 and N6 , N8 , N9 , N2 and N1 are C1 , C2 , C3 C4 , respectively. The coincident points with C1 , C2 , C3 and C4 on the initial discrete elements are B1 , B2 , B3 and B4 , respectively. Here, C1 , C2 , C3 and C4 are treated as the virtual finite element nodes. The finite element interface is generated with C1 , C2 , C3 and C4 as corners by using shape function interpolation. The discrete element interface

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5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.15 Schematic of the face-centroid coupling (Lei and Zang 2010)

is interpolated with B1 , B2 , B3 and B4 with the same shape functions. Similarly, penalty function method is used to guarantee the coincidence of the two interfaces in the calculation process. The stiffness matrix of the coupling surface is computed at the initial moment and converted to a diagonal matrix, so the penalty force on the discrete element is, Fp =

γKA e u s = γ K r 2 u es 4

(5.65)

where u es = u d B − u f G is the relative displacement of the coupling points between the virtual nodes of finite elements and the corresponding discrete elements.  of the coupling points of discrete elements. u f G =   u d B is the displacement 1 4 u N1 + u N2 + u N3 + u N4 is the displacement of the coupling points of finite elements. u N1 , u N2 , u N3 and u N4 are the displacement of four nodes on the coupling surface. Based on Newton’s third law, the penalty force on virtual nodes of the finite element coupling surface is—F p . (3) Coupling with any position on finite element surface In the above two kinds of coupling methods, the positions of coupling points on finite element surface are special, and finite element mesh is restricted by the arrangement and size of discrete elements. This limits the application of DEM-FEM coupling method in engineering. To overcome these, the coupling with any position on finite element surface can be adopted. Figure 5.16 shows the relative positions of spherical discrete element and hexahedral finite element at the initial moment of time. N1 , N2 , N3 and N4 are four nodes of finite element on the interface. Suppose that the coupling points on the discrete element are B1 , B2 , B3 and B4 . The corresponding four coupling points on the surface of the finite element are C1 , C2 , C3 and C4 . The interpolation of C1 , C2 , C3 and C4 gives the spatial surface Si . Virtual work done by the discrete element penalty force can be expressed as:

5.2 Coupled DEM-FEM Model for the Continua-Discontinua …

157

Fig. 5.16 Schematic of arbitrary coupled spatial position on the DEM-FEM interface (Lei and Zang 2010)

 Sd f

 

 T f δu e K be u e α u id − u i δu id ds =

(5.66)

e

where Sd f is the interface between the discrete element subdomain and the finite element subdomain, which is made up of Si . For a cubic packing of discrete elements, the initial diagonal stiffness matrix of the interface can be expressed as: KbeDia =

αA E 4

(5.67)

where A = 4r 2 is the area of the quadrilateral formed by C1 , C2 , C3 and C4 at the initial moment of time. The penalty force F p acting on the discrete element can be expressed as: Fp =

γKA e u s 4

(5.68)

where u es is the relative displacement between the coupling point of the discrete element and the corresponding virtual node of the finite element.

5.2.3 Coupling Point Search In the DEM-FEM contact search, a spatial region is generated consisting of discrete element centroids and finite element nodes at the interface. The space is divided into a number of cubic grids (Hallquist et al. 1985). The nodes in each grid need to be

158

5 Coupled DEM-FEM Analysis of Granular Materials

searched and stored. In order to reduce memory usage, linked-list technology can be used for storage (Gao et al. 2014). During the algorithm realization, the finite element coupling surface is first input as known information. Each finite element coupling surface includes four nodes information. The finite element coupling surface where each node is located needs to be searched in the program. Then, loop over all the discrete elements. The lattice is calculated where the discrete element centroid is located, all the nodes in the lattice and the neighboring lattices are searched by using the node table, and the nodes are found with the shortest distance from the discrete element. The brittle fracture of laminated glass under impact was simulated by the coupling DEM-FEM method for the discrete material and continua bridging domain. The calculation model is shown in Fig. 5.17. The laminated glass is supported by four supports. The top surface of the upper support and lower surface of the lower support are fully constrained. In the combined model, the glass is discretized into 28000 cubic finite elements of size 1 mm and 12000 discrete elements with radius 0.5 mm. The impactor is just on the top of the glass beam, and impacts vertically the beam with an initial velocity of 3.13 m/s. All the materials in the model are linear elastic due to the low velocity impact. The simulation results of impact failure process of glass beams are shown in Fig. 5.18.

Fig. 5.17 The DEM-FEM coupling model for the laminated glass beam under impact load (Xu and Zang 2014)

5.3 Coupled DEM-FEM Method for the Interaction Between ...

159

Fig. 5.18 Cracks profiles of laminated glass with different element sizes (Xu and Zang 2014)

5.3 Coupled DEM-FEM Method for the Interaction Between Continua and Discontinua In order to simulate the interaction between discrete media and continua and realize the transfer of boundary conditions of the two media, a DEM-FEM coupling model has been established and applied in many fields (Jonsén et al. 2011; Rahman and Taniyama 2015; Nishiyama et al. 2016). The numerical simulations of tire-pavement interaction, submarine pipeline impact, tumbling mill and fluidized bed using the coupling method are shown in Fig. 5.19. In the DEM-FEM coupling method for the interaction between discrete materials and continuum materials, the key is to search contacts fast, and to solve and transfer the contact forces between discrete elements and finite elements. These two aspects will be introduced in detail next.

5.3.1 Global Search Detection of Particle-Structure Contacts The contact detection of granular discrete element and continuous finite element should be performed in the global coordinate system. The aim is to find out all the potential contact pairs (Zhong, 1993; Williams et al. 2004). It includes following steps: (1) Space decomposition The contact domain is divided into cells as shown in Fig. 5.20. The numbers of cells n x , n y and n z in the x, y and z directions, can be determined by the following equations:  n x = int(xmax − xmin ) L c + 1

(5.69)

 n y = int(ymax − ymin ) L c + 1

(5.70)

 n z = int(z max − z min ) L c + 1

(5.71)

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5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.19 Applications of the interaction between discrete medium and continuum

(a) Interaction between tire and pavement [Nishiyama et al., 2016]

(b) Interaction between granulesand tumbling mill [Jonsén et al., 2011]

where xmin , ymin and z min are the minimum boundary values of the contact domain in the x, y and z directions, whereas xmax , ymax and z max are the maximum boundary values. L c is the cell size, which can be given by: L c = λL

(5.72)

where λ represents the effect of cell size, L is a characteristic length of the model determined by:

5.3 Coupled DEM-FEM Method for the Interaction Between ...

161

Fig. 5.20 Schematic of the spatial cell division

L=

D S 1

1

di + sj D i=1 S j=1

(5.73)

where D and S are the total numbers of discrete elements and finite segments, respectively; di is the diameter of discrete element i and s j is the diagonal length of finite segment j. Each cell is numbered uniquely, that is,   t = i x + i y − 1 n x + (i z − 1)n x n y

(5.74)

where t is the cell number, ix , iy , iz are the cell number in the x, y, z direction, correspondingly. The lower and upper boundaries of cell t are expressed as:  c = xmin + (i x − 1)(xmax − xmin ) n x xtmin  c xtmax = xmin + i x (xmax − xmin ) n x    c ytmin = ymin + i y − 1 (ymax − ymin ) n y  c ytmax = ymin + i y (ymax − ymin ) n y  c z tmin = z min + (i z − 1)(z max − z min ) n z  c z tmax = z min + i z (z max − z min ) n z

(5.75)

c c c , ytmin and z tmin are the minimum boundary values of cell t in the three where xtmin c c c , ytmax and z tmax are the maximum boundary values. directions, whereas xtmax

162

5 Coupled DEM-FEM Analysis of Granular Materials

Fig. 5.21 Bounding boxes: a discrete element, b finite element segment (Zheng et al. 2017)

(2) Bounding boxes Because the global search is time consuming, the bounding boxes are constructed to avoid performing it every time step (Benson and Hallquist 1990). For the discrete element as shown in Fig. 5.21a, the positions of the bounding box are given by: d = x i − r i − γ1 ximin d ximax = x i + r i + γ1 d yimin = yi − ri − γ1 d yimax = yi + ri + γ1 d z imin = z i − r i − γ1 d z imax = z i + r i + γ1

(5.76)

and γ1 is given by: γ1 = η

D 1

di D i=1

(5.77)

where η represents the effect of boundary size; ri is the radius of discrete element i; d d d , yimin and z imin are xi , yi and z i are the central coordinates of discrete element i; ximin d , the minimum value of the bounding box in the x, y and z directions, whereas ximax d d and z imax are the maximum value. yimax Similarly, for a linear hexahedral element containing four nodes of N1 , N2 , N3 and N4 , as shown in Fig. 5.21b, the positions of its bounding box are given by: f

x jmin = min[n 1x , n 2x , n 3x , n 4x ] − γ2 f

x jmax = max[n 1x , n 2x , n 3x , n 4x ] + γ2   f y jmin = min n 1y , n 2y , n 3y , n 4y − γ2   f y jmax = max n 1y , n 2y , n 3y , n 4y + γ2

5.3 Coupled DEM-FEM Method for the Interaction Between ...

  f z jmin = min n 1z , n 2z , n 3z , n 4z − γ2   f z jmax = max n 1z , n 2z , n 3z , n 4z + γ2

163

(5.78)

where γ2 is given by: γ2 = η

S 1

sj S j=1

(5.79)

here, n kx , n ky and n kz are the coordinates of node k (k = 1, 2, 3, 4) in the x, y and z f f f directions; x jmin , y jmin and z jmin are the minimum value of the bounding box in the f

f

f

x, y and z directions, whereas x jmax , y jmax and z jmax are the maximum value. Because of the construction of bounding boxes for discrete elements and finite elements, the global search process does not need to be performed every time step. Hence, the frequency to perform the global search can be expressed as: m  

f d vimax + v jmax · ti ≥ (γ1 + γ2 )

(5.80)

i=n

where n is the time step at which the global search process is performed last time d and and m is the current time step; ti denotes the size of the ith time step; vimax f v jmax denote the maximum velocity of the discrete elements and finite nodes at ith time step, respectively. (3) Element positioning All the discrete elements and finite segments are mapped into cells. Take the discrete element shown in Fig. 5.22 as an example determining whether D E 1 and D E 2 are located in cell Ct . If the relationship between discrete element D E i and cell Ct satisfy Eq. (5.81), the element is in cell Ct : Fig. 5.22 The mapping between discrete element and cell (Zheng et al. 2017)

164

5 Coupled DEM-FEM Analysis of Granular Materials d c ximin ≤ xtmax d c ximax ≥ xtmin d c yimin ≤ ytmax d c yimax ≥ ytmin d c z imin ≤ z tmax d c z imax ≥ z tmin

(5.81)

(4) Determination of the potential contact pairs If the discrete elements and finite element segments are in the same cell, they are determined as potential contact pairs. A preliminary check based on Eq. (5.82) is performed to reduce the storing memory of potential contact pair lists as shown in Fig. 5.23. D E 1 and F E 2 located in cell C 2 are a potential contact pair, and f

d x jmin ≤ ximax f

d x jmax ≥ ximin f

d y jmin ≤ yimax f

d y jmax ≥ yimin f

d z jmin ≤ z imax f

d z jmax ≥ z imin

(5.82)

Generally, a potential contact pair may appear in two or more cells at the same time, which causes unnecessary repetitive judgments. Consequently, a simple method is adopted. If the maximum lower boundaries of a contact pair are in a cell, this cell is chosen to be judged, that is, Fig. 5.23 The schematic of preliminary check on the potential contact pairs (Zheng et al. 2017)

5.3 Coupled DEM-FEM Method for the Interaction Between ...

165

 f d c max ximin , x jmin ≥ xtmin  f d c max yimin , y jmin ≥ ytmin  f d c max z imin , z jmin ≥ z tmin

(5.83)

5.3.2 Local Search Detection of Particle-Structure Contacts The further local search after the global search is to identify the contact type, contact point and calculate the contact force (Belytschko and Neal 1991; Chaudhary and Bathe 1986). Here, the local search algorithm based on the inside-outside approach of the finite element was adopted (Wang and Nakamachi 1997; Gao et al. 2016). For a discrete element and a finite segment, the contacts between them can be classified into three types, i.e. point-face, point-edge and point-point. As shown in Fig. 5.24, four nodes of finite segments A, B, C, D and its surroundings can be roughly divided into three parts, PTF, PTE and PTN, which are three potential contact regions between the discrete element DE and the segment. Following steps show how to determine the relationship between the discrete element-finite segment contact and the three parts. Firstly, the unit normal vector ns of the finite segment is calculated: 4 !

ns =

i=1 4 !



i=1

Fig. 5.24 Three potential contact regions between a discrete element and a segment (Wang and Nakamachi 1997)

ni ni

(5.84)

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5 Coupled DEM-FEM Analysis of Granular Materials

where ni is the cross product of two neighboring edge vectors. Take n1 as an example: n1 = ab × ad

(5.85)

Then, calculate the judgment value ϕi (i = 1, 2, 3, 4). Take ϕ1 as an example: ϕ1 = nabp · ns

(5.86)

nabp = ab × ap

(5.87)

where nabp can be determined by:

According to the sign of ϕi (i = 1, 2, 3, 4), the relative position between the discrete element DE and the corresponding edge ei is determined. That is to say, if ϕi > 0, the discrete element DE is defined to be inside the edge ei , otherwise it is outside the edge ei . Then, given the combinations of the signs of ϕ1 , ϕ2 , ϕ3 and ϕ4 , one can know which potential contact region the discrete element is located in. (1) PTF contact In this case, the position of O can be calculated as: xof =

4

Ni x i

(5.88)

i=1 f

where xo is the position vector of the projection of the discrete sphere center p on the finite segment; xi is the position vector of the four nodes onthe segment; Ni is the shape function at node i. Here, Ni is defined as: N1 = ϕ2 ϕ3 ϕ, N2 = ϕ3 ϕ4 ϕ,  N3 = ϕ4 ϕ1 ϕ, N4 = ϕ1 ϕ2 ϕ, ϕ = (ϕ1 + ϕ3 )(ϕ2 + ϕ4 ). Then the penetration depth of the discrete element into the finite segment is calculated by:   gnf = ns · x p − xo − r f

(5.89) f

where x p is the position vector of node p. gn ≤ 0 means penetration, while gn > 0 means non-penetration. (2) PTE contact Taking PTE1 as an example, the discrete element may be in contact with edge ab. The unit vector of edge ab is:  e = (xb − xa ) |xb − xa |

(5.90)

The projection point on ab is: xoe = xo + t · (xb − xa )

(5.91)

5.3 Coupled DEM-FEM Method for the Interaction Between ...

167

Fig. 5.25 Edge E belongs to two segments, FE1 and FE2 (Zheng et al. 2017)

 where t = (xb − xa ) · e x p − xa is a direction vector. Then the penetration depth gne is given by: gne = x p − xo − r

(5.92)

where gne ≤ 0 means penetration, while gne > 0 means non-penetration. Since one edge may be shared by two segments, we check the IDs of the segments and choose the segment with a smaller ID for this kind of contact judgment, as shown in Fig. 5.25. (3) PTN contact Take PTN1 as an example. The discrete element may be in contact with node a. Then we can calculate the penetration depth gnn : gnn = x p − xa − r

(5.93)

Similarly, gnn ≤ 0 means penetration, while gnn > 0 means non-penetration. Since one node may be shared by several segments, we check the IDs of the segments and choose the segment with the smallest ID for this kind of contact judgment, as shown in Fig. 5.26. Fig. 5.26 Node N belongs to four segments (Zheng et al. 2017)

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5 Coupled DEM-FEM Analysis of Granular Materials

5.3.3 Transfer of Contact Forces The forces acting on the boundary finite elements are from the DEM contact forces arising from many particle contacts. In finite element analysis, the equivalent concentrated loads are generally applied at the nodal degrees of freedom. DEM-FEM contact points may be chosen as nodal points, but this requires to refine the finite element mesh, which dramatically influences the computational efficiency. In addition, a single finite element may be in contact with multiple discrete elements during the loading processes. As a result, it is not realistic to describe the equivalent nodal force through the refinement of the finite element mesh. The proposed linking technique overcomes these difficulties and allows the finite element mesh to be designed independently of the DEM mesh to satisfy the mesh convergence requirement for the finite element analysis (Chung and Ooi 2012). A boundary surface is generally represented using many triangular elements and there may be many particle contacts on each triangular element. Consider a 3-node triangular plate element and the contact forces resulting from M particle contacts on the triangular plate element, as shown in Fig. 5.27. The local coordinate system (x,y,z) is at the centroid P where plane x−y is coincident with the plane of the plate element. The unit normal vector n of the triangular plate element is given by, ui j × uik n= ui j × uik

(5.94)

Fig. 5.27 Contact force between triangular element and discrete element (Chung and Ooi 2012)

5.3 Coupled DEM-FEM Method for the Interaction Between ...

169

where ui j = vi × v j denotes the unit vector along the line joining point i and j. uik = vi × vk denotes the unit vector along the line joining point i and k.vi , v j , vk are the unit vector of the triangular plate element at nodes i, j and k. Here, x direction is coincident with ui j . The three unit vectors ux , u y and uz of the local coordinate system can be expressed as: ux = ui j u y = n · ux uz = n

(5.95)

The relationship between the local coordinate system (x, y, z) and the global coordinate system (X, Y, Z) can be expressed as: |x, y, z|T = Ttran,1 |X, Y, Z |T

(5.96)

where Ttran,1 is a 3 × 3 transformation matrix and is given by:  T Ttran,1 = ux , u y , uz

(5.97)

A total of 6 degrees of freedom are needed at each node to model shell and plate bending response, three displacements u x , u y , u z and three rotations θx , θ y , θz . Rotation θz can be assumed to be negligible when the bending moment is much larger than the in-plane torque. The displacement field U can be related to the nodal displacement A via the shape interpolation function N, and expressed as: U=N·A

(5.98)

where N is the shape function matrix for a 3-node plate element. External virtual work δW induced by contact forces is given by δW =

M

UmT · fcon,m

(5.99)

m=1

where δ is variation operator, fcon,m is the contact force vector applied at the contact point m in the local coordinate system. Subscript m refers to the contact point, m is the number of particle contacts on the triangular element. Substituting Eq. (5.96) into (5.99) gives, δW = δAT ·

M

m=1

NmT · fcon,m

(5.100)

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5 Coupled DEM-FEM Analysis of Granular Materials

where Nm is the value of shape function associated with the contact point m. The equivalent nodal forces in the local coordinate system fcon,nodal can be expressed as: fcon,nodal =

M

NmT · fcon,m

(5.101)

m=1

Using the coordinate transformation, the contact forces in the local coordinate system can be transformed to those in the global coordinate system Fcon,m , that is, Fcon,m = Ttran,2 · fcon,m

(5.102)

where Ttran,2 is a 6 × 6 transformation matrix and is given by: T 0 Ttran,2 = tran,1 0 Ttran,1

(5.103)

Similarly, the equivalent nodal forces in the global coordinate system Fcon,nodal can be transformed through the equivalent nodal forces in the local coordinate system fcon,nodal as: Fcon,nodal = Ttran,3 · fcon,nodal

(5.104)

where Ttran,3 is a 18 × 18 transformation matrix and is given by,

Ttran,3

Ttran,2 0 0 = 0 Ttran,2 0 0 0 Ttran,2

(5.105)

The equivalent nodal forces in the global coordinate system is thus obtained, Fcon,nodal =

M

Ttran,3 · NmT · Ttran,2 · fcon,m

(5.106)

m=1

Oñate and Rojek (2004) analyzed the cutting process and impact failure process of rock mass by using the coupled DEM-FEM method for the contact of discrete medium and continuous medium. In the rock cutting simulation, DEM was adopt to simulate the rock, and triangular or tetrahedral finite element was used to simulate the tool, as shown in Fig. 5.28a. In the impact failure process of rock, the FEM was used to analyze the impact body, as shown in Fig. 5.28b. The coupling method is suitable for the analysis of the interaction between brittle materials or bulk materials and other continuum. Generally, the mechanical behaviors of brittle materials and bulk materials are solved by the DEM, and the mechanical behaviors of other materials are analyzed by the FEM.

5.4 Summary

171

(a) Cutting process of rock

(b) Impacting process of rock Fig. 5.28 Cutting and impacting process of rock based on the coupled DEM-FEM method (Oñate and Rojek 2004)

5.4 Summary In this chapter, three types of the coupled DEM-FEM method, the combined DEMFEM method for the transition from continuum to granular material, the coupling DEM-FEM model for the continua-discontinua bridging domain (the bridging scale method), and the DEM-FEM coupling method for the interaction between continuum and granular material, are introduced respectively. The combined DEM-FEM method for the transition from continuum to granular material is mainly introduced from three aspects: contact algorithm, element deformation and material failure model. The method can be used to solve high-degree non-linear mechanical problems such

172

5 Coupled DEM-FEM Analysis of Granular Materials

as large displacement, strong contact and material dynamic failure. For the bridging scale method coupling multiscale DEM-FEM analysis for continua-discontinua domain, the basic theory of the DEM-FEM coupling method and three coupling modes are introduced. Then, the surface arbitrary position coupling algorithm is discussed by the global search, local search and coupling force calculation. Finally, for the DEM-FEM coupling method for the interaction between continuum and granular material, the global and local search detection of granular material and structure and the transfer mode of contact force on the coupling interface are introduced.

References Belytschko T, Neal MO (1991) Contact-impact by the pinball algorithm with penalty and Lagrangian methods. Int J Numer Meth Eng 31(3):547–572 Benson DJ, Hallquist JO (1990) A single surface contact algorithm for the post-buckling analysis of shell structures. Comput Methods Appl Mech Eng 78(2):141–163 Chaudhary AB, Bathe K-J (1986) A solution method for static and dynamic analysis of threedimensional contact problems with friction. Comput Struct 24(6):855–873 Choi JL, Gethin DT (2009) A discrete finite element modelling and measurements for powder compaction. Modell Simul Mater Sci Eng 17:035005 Chung YC, Lin CK, Chou PH et al (2016) Mechanical behavior of a granular solid and its contacting deformable structure under uni-axial compression–Part1: joint DEM-FEM modelling and experimental validation. Chem Eng Sci 144:404–420 Chung YC, Ooi JY (2012) Linking of discrete element modelling with finite element analysis for analysing structures in contact with particulate solid. Powder Technol 217(2):107–120 Frenning G (2008) An efficient finite/discrete element procedure for simulating compression of 3D particle assemblies. Comput Methods Appl Mech Eng 197(49–50):4266–4272 Gao W, Tan Y, Jiang S et al (2016) A virtual-surface contact algorithm for the interaction between FE and spherical DE. Finite Elem Anal Des 108:32–40 Gao Wei, Zang M, Xu W (2014) An approach to freely combining 3d discrete and finite element methods. Int J Comput Method 11(01):1350051 Gethin DT, Yang XS, Lewis RW (2006) A two dimensional combined discrete and finite el-ement scheme for simulating the flow and compaction of systems comprising irregu-lar particulates. Comput Methods Appl Mech Eng 195(41–43):5552–5565 Guo LW, Xiang JS, Latham JP et al (2016) A numerical investigation of mesh sensitivity for a new three-dimensional fracture model within the combined finite-discrete element method. Eng Fract Mech 151:70–91 Haddad H, Guessasma M, Fortin J (2016) A DEM-FEM coupling based approach simulating thermomechanical behaviour of frictional bodies with interface layer. Int J Solids Struct 81:203–218 Hallquist JO, Goudreau GL, Benson DJ (1985) Sliding interfaces with contact-impact in large-scale Lagrangian computations. Comput Methods Appl Mech Eng 51(1–3):107–137 Han K, Feng YT, Owen DRJ (2007) Performance comparisons of tree-based and cell-based contact detection algorithms. Eng Comput 24(2):165–181 Hunˇek I (1993) On a penalty formulation for contact-impact problems. Comput Struct 48(2):193– 203 Karami A, Stead D (2008) Asperity degradation and damage in the direct shear test: a hybrid FEM/DEM approach. Rock Mech Rock Eng 41(2):229–266 Lei Z, Rougier E, Knight EE et al (2016) A generalized anisotropic deformation formulation for geomaterials. Comput Part Mech 3(2):215–228

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Lei Z, Zang M (2010) An approach to combining 3D discrete and finite element methods based on penalty function method. Comput Mech 46(4):609–619 Lewis RW, Gethin DT, Yang XS et al (2005) A combined finite-discrete element method for simulating pharmaceutical powder tableting. Int J Numer Meth Eng 62(7):853–869 Li X, Wan K (2011) A bridging scale method for granular materials with discrete particle assembly– Cosserat continuum modeling. Comput Geotech 38(8):1052–1068 Liu T, Fleck NA, Wadley HNG et al (2013) The impact of sand slugs against beams and plates: coupled discrete particle/finite element simulations. J Mech Phys Solids 61(8):1798–1821 Michael M, Vogel F, Peters B (2014) DEM-FEM coupling simulations of the interactions between a tire tread and granular terrain. Comput Methods Appl Mech Eng 289:227–248 Munjiza A (2004) The combined finite-discrete element method. Wiley Munjiza A, Andrews KRF (1998) NBS contact detection algorithm for bodies of similar size. Int J Numer Meth Eng 43(1):131–149 Munjiza A, Andrews KRF (2000) Penalty function method for combined finite–discrete ele-ment systems comprising large number of separate bodies. Int J Numer Meth Eng 49(11):1377–1396 Munjiza A, Andrews KRF, White JK (1999) Combined single and smeared crack model incombined finite-discrete element analysis. Int J Numer Meth Eng 44(1):41–57 Munjiza A, Latham JP (2004) Some computational and algorithmic developments in computational mechanics of discontinua. Philoso Trans R Soc Lond Ser A Math Phys Eng Sci 362(1822):1817 Munjiza A, Lei Z, Divic V et al (2013) Fracture and fragmentation of thin shells using the combined finite-discrete element method. Int J Numer Meth Eng 95:479–498 Nishiyama K, Nakashima H, Yoshida T et al (2016) 2D FE-DEM analysis of tractive performance of an elastic wheel for planetary rovers. J Terrramech 64:23–35 Oñate E, Rojek J (2004) Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput Methods Appl Mech Eng 193(27–29):3087–3128 Owen DRJ, Feng YT, Cottrell MG et al (2007) Computational issues in the simulation of blast and impact problems: an industrial perspective. Extreme man-made and natural hazards in dynamics of structures. Springer, Netherlands Jonsén P, Pälsson BI, Tano K et al (2011) Prediction of mill structure behaviour in a tumbling mill. Miner Eng 24(3):236–244 Rahman MA, Taniyama H (2015) Analysis of a buried pipeline subjected to fault displacement: A DEM and FEM study. Soil Dyn Earthq Eng 71:49–62 Smoljanovi´c H, Živalji´c N, Željana N (2013) A combined finite-discrete element analysis of dry stone masonry structures. Eng Struct 52(9):89–100 Wang S, Ji et al (2018) Coupled DEM-FEM analysis of ice-induced vibrations of a conical jacket platform based on the domain decomposition method. Int J Offshore Polar Eng 28(2):190–199 Wang SP, Nakamachi E (1997) The inside-outside contact search algorithm for finite element analysis. Int J Numer Meth Eng 40(19):3665–3685 Williams JR, Perkins E, Cook B (2004) A contact algorithm for partitioning N arbitrary sized objects. Eng Comput 21(2/3/4):235–248 Xu W, Zang M (2014) Four-point combined DE/FE algorithm for brittle fracture analysis of laminated glass. Int J Solids Struct 51(10):1890–1900 Zheng Z, Zang M, Chen S et al (2017) An improved 3D DEM-FEM contact detection algorithm for the interaction simulations between particles and structures. Powder Technol 305:308–322 Zhong ZH (1993) Finite element procedures for contact-impact problems. Oxford University Press

Chapter 6

Fluid-Solid Coupling Analysis of Granular Materials

The coupling between granular materials and fluid widely exist in nature and human activities, such as sediment in the river course, seepage of liquid water in sand and gravel soil, and interactive movement in the gas-solid two-phase flow in chemical equipment. In order to study the fluid-solid coupling problem of granular materials, the discrete element method (DEM) is usually adopted to simulate solid particles, while various computational methods can be used to simulate fluid. According to the traditional computational fluid dynamics (CFD) method, the mesh-based DEMCFD model (Washino et al. 2013; Zhao and Shan 2013), the DEM-SPH based on the mesh-free particle method (Jonsen et al. 2014; Sun et al. 2013), and other methods have been developed. It should be noted that the lattice Boltzmann method (LBM) for the simulation of fluid dynamics based on mesoscopic dynamic theory has obtained great attention recently, and the DEM-LBM coupling method was also developed (Galindo-torres 2013; Tran et al. 2016). In recent years, SPH and LBM are becoming mature numerical methods for fluid simulations. The numerical coupling methods of DEM-CFD, DEM-SPH and DEM-LBM are mainly introduced here, while the applications for specific problems are also introduced in details from the aspects of core fundamentals and coupling approach.

6.1 DEM-CFD Coupling Method for Granular Materials and Fluid In fluid-solid system of granular materials, the coupling method should be developed to deal with the interaction between fluid and particles. Kloss et al. (2012) made a detailed introduction of the DEM-CFD coupling theory method based on open source software (Kloss et al. 2012). Norouzi et al. (2016) elaborated the basic theory of DEM-CFD, specific implementations and applications (Norouzi et al. 2016). In the DEM-CFD coupling method, the fluid motion is described by two-phase coupled Navier-Stokes (N–S) equation, in which the influence of solid phase is considered © Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_6

175

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6 Fluid-Solid Coupling Analysis of Granular Materials

and the void fraction and energy exchange term are added. Particle motion depends on two aspects, one is the force generated by the collision between particles, and the other is the influence of fluid on particle motion. It should be noted that the CFD method means the traditional mesh-grid method here.

6.1.1 Basic Governing Equations of Particles The DEM is used to calculate the particle trajectory. The motion equation of particle i is: m i x¨ i = Fi,n + Fi,t + Fi, f + Fi,b

(6.1)

Ii ω˙ i = ri,c × Fi,t + Ti,r

(6.2)

where, Fi,n is the normal contact force between particles; Fi,t is the tangential contact force between particles; Fi, f is the force exerted by the fluid around the particle; Fi,b represent gravity, electromagnetic force and other forces; Ti,r is the moment that the fluid impedes the particle’s rotation, including the rotational resistance torque acting on the particle and an unsteady term in the linear acceleration.

6.1.2 DEM-CFD Coupling Solution Method According to the size of the particle and grid, the DEM-CFD coupling method can be divided into the unresolved approach and the resolved approach based on the immersed boundary method. The unresolved approach is applicable to the cases that the size of solid particles is smaller than the computational grid, i.e., a grid contains many particles. The N–S equations for volume averaging can compute the fluid phase by local averaging of solid phase properties. The average properties of solids are fluid-particle interactions and solid volume fractions. The resolved approach is applicable to the cases that the size of solid particles is larger than the computational grid. Thus, solid particles may occupy multiple computational grids. The particle phase is represented by an imaginary domain method, in which there is only one velocity field and pressure field in the fluid phase and particle phase, and the covered area of the particle is the same as the velocity of the particle itself. Since the size of the fluid grid is much smaller than the particle size, the fluid flow can be completely solved on the entire surface of the particle, and therefore the specific flow field distribution can be obtained around all the particles. In the resolved approach, since the size of the fluid element is much smaller than the size of the particle, the fluid flow can be completely solved on the whole

6.1 DEM-CFD Coupling Method for Granular Materials and Fluid

177

surface of the particle, the detailed flow field distribution around the particle can be obtained, and the force acting on a single particle can be obtained at the same time. However, due to the high computational cost of solving the fluid phase equation of the entire grid element, this treatment is only applicable to the system with no more than a few hundred of particles, so the unresolved approach is used more widely in practical problems. This section mainly introduces the basic theory of the unresolved approach.

6.1.3 Governing Equations of Fluid Domain In the CFD method, the continuous fluid domain is discretized into elements in grid. In each element, the velocity, mass and pressure of the fluid are all local average quantities. In the presence of particle phase, the motion of incompressible fluid phase is described by the continuity equation and the N–S equation based on volume averaging, and the governing equations are:   ∂α f + ∇ · αfuf = 0 ∂t

    ∂ αfuf p + ∇ · α f u f u f = −∇ − Rpf + ∇ · τ + α f g ∂t ρf

(6.3) (6.4)

where, α f is the volume fraction occupied by the fluid, α f = 1 means the element are full of fluid; ρ f is the density of the fluid; u f is the velocity of the fluid; R p f represents the momentum exchange between fluid phase and particle phase; τ = ν f ∇u f is the stress tensor of the fluid phase; g stands for gravitational acceleration. Due to the requirements of particle collision dynamics and particle collision model, the DEM time step should be at least one order of magnitude smaller than the CFD time step. For convenience, the time steps of the DEM and CFD can be independent of each other, and the coupling interval can be set to adapt to the physical coupling between them.

6.1.4 Momentum Exchange Between Fluid and Solid Particles There are several approaches for momentum exchange between fluid and particles. Most of them estimate the interaction force between two phases through the summation of the resistance of particles in the fluid elements (Kafui et al. 2002; Zhu et al. 2007; Tsuji et al. 2008; Kloss et al. 2012; Zhao and Shan 2013). Here are two commonly used approaches.

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6 Fluid-Solid Coupling Analysis of Granular Materials

(1) Momentum exchange of Kloss et al. (2012) The momentum exchange term can be divided into and explicit terms. Mean implicit  while, according to the overall average velocity u p of particles in the fluid unit, the momentum exchange term R p f can be defined as:    Rpf = Kpf u f − up

(6.5)

where, K p f is the interaction factor between fluid and solid, and can be defined as: Kpf =

Vcell

 F  i d    · u f − up 

(6.6)

 where, Vcell is the velocity of the fluid element; i Fd is the resultant drag force of the fluid on all solid particles in the element. The fluid-solid interaction factor K p f adopted here is defined as follows. When α f > 0.8, K p f can be defined as:    3 α f 1 − α f u f − u p  −2.65 αf K p f = Cd 4 dp  0.687 24  Cd = 1 + 0.15 α f Re p α f Re p   u f − u p  dp Re p = vf

(6.7) (6.8) (6.9)

When α f ≤ 0.8, K p f can be defined as:   2  1 − αf vf 1 − α f u f − u p  = 150 + 1.75 α f d 2p dp 

Kpf

(6.10)

(2) Momentum exchange of Zhao and Shan (2013) The interaction between fluid and particles is divided into resistance and buoyancy, and the resistance is expressed as: Fd =

 1−χ   1 Cd ρ f π d 2p u f − u p u f − u p α f 8

(6.11)

where, Cd is the drag coefficient related to Reynolds number Re p ; d p is the particle diameter. Here, Cd and χ are denoted as:

4.8 Cd = 0.63 +  Re p

2 (6.12)

6.1 DEM-CFD Coupling Method for Granular Materials and Fluid

  u f − u p  Re p = α f dp vf

 2  1.5 − log10 Re p χ = 3.7 − 0.65ex p − 2

179

(6.13)

(6.14)

where, χ is the correction function considering the influence of other particles in the system on this particle; v f is the coefficient of kinematic viscosity. The buoyancy F b of particle can be expressed as: Fb =

1 πρ f d 3p g 6

(6.15)

6.1.5 Fluid Volume Fraction The volume fraction of the fluid is also known as the void fraction. The governing equation of the DEM-CFD contains the derivative term of volume fraction with respect to time, so the selection of this parameter directly affects the final calculation result. In two consecutive time steps of CFD and DEM, the abrupt change of the volume fraction will lead to the singularity of the pressure in the result, which makes the accurate calculation of the volume fraction of the fluid quite important. In the computational element, the fluid volume fraction α f can be estimated as: αf = 1 −

kv 1  ϕi Vi Vcell i=1

(6.16)

where, Vcell is the volume of the computing unit; kv is the number of particles in the same fluid element; ϕi is the estimated volume fraction of a particle between the partial volume in the fluid element and the overall particle volume; Vi is the volume of the particle. Virk et al. (2012) proposed a new computation algorithm for the void fraction of particles located at the grid boundary, which significantly improved the computational accuracy without increasing the complexity (Virk et al. 2012).

6.1.6 Convection Heat Transfer Term In the DEM-CFD coupling, there is heat transfer between fluid and particle phase, which can be solved by temperature scalar transport equation, that is:

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6 Fluid-Solid Coupling Analysis of Granular Materials

    ∂Tf + ∇ · T f · u f = ∇ · κe f f ∇T f + ST ∂t

(6.17)

The source term in the equation can be calculated by the Nusselt number N u p , which can be expressed as a function of Reynolds number Re p and Prandtl number Pr. When Re p < 200, N u p can be expressed as: 1/3

N u p = 2 + 0.6r np Re1/2 p Pr

(6.18)

When 200 < Re p < 1500, N u p can be expressed as: 1/3

1/3

n 0.8 N u p = 2 + 0.5r np Re1/2 p Pr + 0.02r p Re p Pr

(6.19)

When 1500 < Re p , N u p can be expressed as: N u p = 2 + 0.000045r np Re1.8 p

(6.20)

where, r p represents the particle radius; Re p represents Reynolds number of particles; Pr denotes Prandtl number. The exponent n is generally 3.5. The heat transfer coefficient h can be utilized to calculate q p as follows: λN u p dp   q p = h A p T f − Tp h=

(6.21) (6.22)

where, λ is the thermal conductivity. Accordingly, the source term ST can be expressed as: ST = −

qp ρ f C Vcell

(6.23)

Based on the basic theoretical method of the DEM-CFD coupling, many scholars have done a lot of extensibility work. Chu et al. (2009) proposed a new numerical model to simulate the typical flow phenomenon in the heavy medium cyclone (also called as dense medium cyclones, DMCs), and explained the so-called surging phenomenon in the typical flow by employing the force between particle-particle, particle-fluid and particle-wall (Chu et al. 2009). Figure 6.1 shows the distribution of the flow field. There is a hollow area filled with air around the central axis. Zhao and Shan (2013) adopted the DEM-CFD method to simulate the characteristics of sand heaps formed by sand entering the water through a funnel (Zhao and Shan 2013). It considered the influence of the rolling friction and the polydispersity of the water. During the entire simulation process, the motion characteristics of solid particles generated by fluid-particle interaction can be well captured. As shown in

6.1 DEM-CFD Coupling Method for Granular Materials and Fluid

181

Fig. 6.1 Flow field distribution in the cyclone (Chu et al. 2009)

Fig. 6.2, sand, poured from a funnel into a container filled with water, forms a coneshaped sand pile on a circular plate placed at the bottom of the container. Figure 6.3 is the comparison of local void fraction. Figure 6.3a, b represent the cloud maps of void fraction in the monosized case and the polydisperse case, respectively.

Fig. 6.2 Formation process of sand accumulation in water (Zhao and Shan 2013)

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6 Fluid-Solid Coupling Analysis of Granular Materials

Fig. 6.3 Comparison of void fraction between dry and wet sand: a the monosized case, b the polydisperse case (Zhao and Shan 2013)

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid In the fluid-solid coupling between granular materials and fluid, since the discrete particle assumption in the meshfree particle method (MPM) and the geometric shape of DEM particles have highly consistent data structure and numerical algorithm in the practical programming. The MPM are more suitable to be used in the fluid-solid coupling simulations of granular materials simulated by the DEM. Comparing with the traditional mesh-grid CFD method, the smoothed particle hydrodynamics (SPH) has advantages in the simulations of free liquid surface and surface tension, etc. It is widely used in the fluid-solid coupling problem of granular materials. It should be noticed that some other particle methods, such as the molecular dynamics (MD), the dissipative particle dynamics (DPD), etc., have the advantages of the MPM, but they are at such microscopic scales that the computational cost is huge. As a result, these methods are seldom used on a large scale fluid-solid coupling simulations, and usually used only for solving some special problems. The SPH is an adaptive meshfree Lagrangian particle method to simulate fluid flow (Monaghan 1988; Koshizuka and Oka 1996). It employs the integral representation to approximate the field function, and the particle approximation to represent the relevant physical quantities. Consequently, it converts a series of partial differential equations into time-dependent ordinary differential equations, and the change of each particle’s field variables with time can be obtained by using the time integral. At present, the SPH has been extensively developed in various fields, and widely adopted and acknowledged in engineering (Yang et al. 2016; Sun et al. 2017). The DEM-SPH coupling is developed by coupling at the interface between fluid and particles, i.e., the force between DEM particle and SPH particle satisfies the Newton’s third law.

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid

183

6.2.1 Integral Representation of Function and Particle Approximation in SPH In the domain Ω, the continuous function f (x) can be defined as the integral form:  f (x) =

Ω

    f x  δ x − x  dx 

(6.24)

  where, δ x − x  is the Dirac delta function, and can be written as: 

δ x−x





 =

1, x = x  0, x = x 

(6.25)

In the SPH, a kernel function is adopted to replace the Dirac function, so f (x) can be approximately expressed as:  f (x) ≈

Ω

    f x  W x − x  , h dx 

(6.26)

where, W is called as the smoothing kernel function, smoothing function, or kernel function. h is the smoothing length that defines the scope of the kernel function W . The typical form of a kernel function is shown in Fig. 6.4, which should satisfy the following conditions.

Fig. 6.4 Kernel function in the SPH method

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6 Fluid-Solid Coupling Analysis of Granular Materials

(1) Normalization condition (unity condition):  Ω

  W x − x  , h dx  = 1

(6.27)

(2) Compact condition. We have W equals to zero outside the supporting region, and can be written as:     W x − x  , h = 0,  x − x   > h

(6.28)

(3) When the smoothing length approaches zero, the kernel function should satisfy the Dirac function property:     lim W x − x  , h = δ x − x 

h→0

(6.29)

6.2.2 SPH Form for Navier-Stokes Equations The basic governing equations of fluid dynamics are based on three basic laws of conservation of physics, namely, conservation of mass, conservation of momentum and conservation of energy. The fluid governing equations described by Lagrangian method can be expressed as a series of partial differential equations. These equations are the famous Navier-Stokes equations, which include the conservation of mass, momentum and energy. If the coordinate direction is denoted by the Greek letters α and β, the index method can be used to express the superposition of equations. The derivative of motion with respect to time is introduced in the Lagrangian framework. Therefore, N–S equations include the following series of equations (Liu and Liu 2004). The mass conservation equation, or the continuity equation is: ∂vβ dρ = −ρ β dt ∂x

(6.30)

The momentum conservation equation is: dvβ 1 ∂σ αβ = dt ρ ∂ xβ

(6.31)

The energy conservation equation is: de σ αβ ∂vα = dt ρ ∂ xβ

(6.32)

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid

185

where, v is the velocity vector; x is the position vector; σ is the stress tensor, which is composed of the isotropic pressure p and the viscous stress τ , that is: σ αβ = − pδ αβ + τ αβ

(6.33)

where δ αβ is the Dirac delta function. In Newtonian fluids, the viscous shear stress is proportional to the shear strain ε, and the scale factor is the viscosity coefficient μ, that is, τ αβ = μεαβ

(6.34)

where εαβ can be expressed as: εαβ =

∂vβ ∂vα 2 + − (∇ · v)δ αβ ∂ xα ∂ xβ 3

(6.35)

If the stress tensor is decomposed into the isotropic pressure and the viscous stress, the energy equation can be rewritten as: de p ∂vβ μ αβ αβ =− ε ε + β dt ρ ∂x 2ρ

(6.36)

According to the integral representation of function and the particle approximation, N–S equations, named as the density, momentum and energy conservation equation, can be transformed into particle forms in the SPH. (1) Particle approximation of density Since the distribution of particles and the change of smoothing length mainly depend on the density, the density approximation method is very important in the SPH method. There are two ways to solve the density in the SPH. The first one is to apply the particle approximation directly to the density, which is called the density summation method. For any particle i, with density summation method the density can expressed as the following two forms: ρi =

N 

m j Wi j

(6.37)

j=1

N

j=1

ρi =  N

m j Wi j

mj j=1 ρ j

Wi j

(6.38)

where m represents the mass of the particle. The other one is to perform the particle approximation transformation of the continuity equation, which is called the continuity density method. It can mainly be written as the following two forms:

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6 Fluid-Solid Coupling Analysis of Granular Materials

 m j β ∂ Wi j dρi v = ρi dt ρ j i j ∂ x iβ j=1

(6.39)

 ∂ Wi j dρi β = m j vi j · β dt ∂ xi j j=1

(6.40)

N

N

(2) Particle approximation of momentum The momentum equation approximation in the differential form can be derived by differential transformation. The SPH method is commonly used to solve the momentum equation in two forms: αβ

αβ

N  σi + σ j ∂ Wi j dviα = mj β dt ρi ρ j ∂ xi j j=1

αβ αβ N  σj σi dviα ∂ Wi j = mj + 2 β 2 dt ρ ρ ∂ xi j i j j=1

(6.41a)

(6.41b)

From Eqs. (6.33) and (6.34), the momentum equation can be written as: αβ αβ N N  dviα pi + p j ∂ Wi j  μi εi + μ j ε j ∂ Wi j =− mj + m j β dt ρi ρ j ∂ x iαj ρi ρ j ∂ xi j j=1 j=1



αβ N N αβ  μjεj μi εi dviα ∂ Wi j p j ∂ Wi j  pi =− mj + 2 mj + α + β 2 2 2 dt ρi ρ j ∂ xi j ρi ρj ∂ xi j j=1 j=1

(6.42a)

(6.42b)

The formula for the shear strain ε in the SPH is: ⎛ ⎞ N N N    m ∂ W m ∂ W m 2 j β ij j α ij j αβ v ji v ji −⎝ v ji · ∇i Wi j ⎠δ αβ (6.43) εi = α + β ρ ∂ x ρ 3 ρ j j j ∂ x i i j=1 j=1 j=1 (3) Particle approximation of energy The two main forms of the energy are:

 N pi + p j β ∂ Wi j dei 1 μi αβ αβ = v + ε ε mj dt 2 j=1 ρi ρ j i j ∂ x iβ 2ρi i j



 N p j β ∂ Wi j pi 1 dei μi αβ αβ = + 2 vi j + ε ε mj β dt 2 j=1 2ρi i j ρi2 ρj ∂ xi

(6.44a)

(6.44b)

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid

187

(4) Artificial viscosity When the SPH is used to deal with the dissipation problem, the artificial viscosity term can significantly improve the stability of the simulation. The artificial viscosity converts kinetic energy into heat energy, providing essential dissipation of the system and preventing non-physical penetration of particles as SPH particles approach each other. The specific expression of artificial viscosity is as follows:  i j =

−α ci j φi j +β φi2j ρi j

0,

vi j · xi j < 0; vi j · xi j < 0;

(6.45)

where φi j can be expressed as: h i j vi j · x i j φi j =  2 xi j  + ϕ2

(6.46)

where ϕ = 0.1h i j . With the artificial viscosity, the momentum and energy equations can be modified as follows:

αβ αβ N  σi + σ j dviα ∂ Wi j = mj − i j (6.47) β dt ρi ρ j ∂ xi j j=0

   N pi + p j 1 dei μi αβ αβ β ∂ Wi j = − i j vi j + ε ε mj (6.48) β dt 2 j=1 ρi ρ j 2ρi i j ∂ xi As shown in Fig. 6.5, the dam-break flow can be simulated with the SPH. The validation with other classical results illustrates good agreement (Ji et al. 2019). Fig. 6.5 The SPH simulations of dam-break flow (Ji et al. 2019)

(a) t = 0.1 s

(b) t = 0.3 s

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6 Fluid-Solid Coupling Analysis of Granular Materials

6.2.3 The EISPH Method for Incompressible Fluid In the SPH simulations, the particle pressure is the key factor to reasonably solve the particle motion. In the traditional SPH method, also known as the weakly compressible SPH (WCSPH), pressure is solved through the equation of state (EOS) related to density. The EOS can be written as (Sun et al. 2013; Robinson et al. 2014):   c02 ρ0 ρ γ [ − 1] p= γ ρ0

(6.49)

where, ρ0 is the density of the fluid; γ is a constant coefficient. In order to ensure the stability of the density in the simulation, normally γ = 7 so that ρ ≈ ρ0 (Sun et al. 2013). Taking the derivative of density in Eq. (6.49), the following equation can be obtained:  γ −1 ρ dp = c02 dρ ρ0

(6.50)

= c02 can be yielded when γ = 1, i.e., the change of pressure with density is directly related to the square of sound speed. It can avoid the influence of other parameters on the pressure. Therefore, γ = 1 is also used in some literatures (Eitzlmayr et al. 2014). According to Eq. (6.49), particle pressure can be directly obtained from density, and then particle motion can be solved from the momentum equation. This method has good intuitiveness and feasibility. It is also suitable for the large-scale parallel algorithm. However, it is difficult to guarantee the stability and uniformity of the fluid pressure field in the practical simulation. On the other hand, the velocity v∗ in an “intermediate state” can be adopted in the difference form of the momentum equation, and thus the pressure Poisson equation (PPE) can be yielded to solve the pressure. The difference form of momentum equation can be written as: dp dρ

1 vn+1 − vn = − ∇ p n+1 + v∇ 2 v + g t ρ

(6.51)

where n denotes the time step. Accordingly the following equation can be obtained: vn+1 = −

t ∇ p n+1 + vn + (v∇ 2 vn + g)t ρ

(6.52)

The last two terms on the right hand side can be expressed as the velocity of “intermediate state” v∗ between vn and vn+1 , that is: v∗ = vn + (v∇ 2 vn + g)t

(6.53)

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189

Thus, Eq. (6.53) can be written as: vn+1 = −

t ∇ p n+1 + v∗ ρ

(6.54)

Taking the spatial derivative of the above equation, and ∇ · vn+1 = 0, the PPE can be written as: ∇ 2 p n+1 =

ρ ∇ · v∗ t

(6.55)

The left hand side of the above formula is the Laplacian term of pressure, which can be expressed by the particle approximation method (Rafiee and Thiagarajan 2009). Thus the matrix formula of pressure on each particle can be obtained and solved. v∗ can be directly calculated on the right hand side of the equation. Or it can be rewritten according to the continuity equation (Lee et al. 2008): ∇ 2 p n+1 =

ρ0 − ρ ∗ 1 1 dρ ∗ (ρ0 ∇ · v∗ ) = − = t t dt t 2

(6.56)

Equation (6.55) or (6.56) can be written as a matrix equation about the pressure of each particle, and the matrix equation needs to be assembled and solved at each time step. Besides, an explicit approach can be developed to explicitly solve the pressure without solving the matrix equation (Rafiee and Thiagarajan 2009). This method is named as explicit incompressible SPH (EISPH). This equation which uses the PPE to solve the particle pressure is also known as incompressible SPH (ISPH), and it is abbreviated as the strong form SPH. This method is more accurate than the weak compressible SPH in solving pressure, which can obtain stable fluid pressure field. However, the solution procedure of SPH in strong form is complex and cumbersome. Each time step requires the solution of the matrix equation whose dimension is the same as the number of particles. As a result, the storage of matrix and the solution for the equation would greatly decrease the computational efficiency (Lee et al. 2008). Therefore, this method is not conducive to the large-scale simulations for engineering.

6.2.4 DEM-SPH Coupling Model The SPH described by Lagrangian method does not need to deal with complex free surface and moving solid boundary problems, and it is more effective for the simulation of moving boundary and free surface flow. According to different practical problems and physical models, the coupling of DEM-SPH is usually carried out in different ways. The DEM-SPH generally has two categories to achieve the coupling. One is direct numerical simulation (DNS); The other is local averaging approach (Anderson and

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6 Fluid-Solid Coupling Analysis of Granular Materials

Jackson 1967). In the DNS method, hydrodynamic force acting on solid particles is obtained by solving N–S equations directly. The significant disadvantage of this approach is the requirement for extremely fine particle (SPH or MPS particle) resolution to accurately simulate the fluid flow (Potapov et al. 2001; Gotoh and Sakai 2006). Therefore, DNS method is difficult to be applied to large-scale particle system, while the hydrodynamic force in this method is usually calculated directly by empirical model. The mesoscopic scale model is usually adopted for solid-liquid and gas-solid multiphase flow problems in atmospheric science and process engineering. The interaction between DEM particles and SPH particles is not refined, only considering the local averaging of liquid to solid particles (Robinson et al. 2014; Deng et al. 2013; Bokkers 2005), as shown in Fig. 6.6. Meanwhile, the volume occupied by solids, named as the influence of fluid porosity on the fluid, is employed to consider the interaction between fluid and particles (Sun et al. 2013). In the local averaging approach, Anderson and Jackson (1967) defined a smoothing operator similar with that in the SPH, and used this operator to calculate the smoothing variable and local porosity field (Anderson and Jackson 1967). Fig. 6.6 SPH simulation of fluidized bed (Deen et al. 2007)

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid

191

For the numerical simulation of fluid-particle coupling problems, different coupling calculation methods are required according to the scale difference. For problems with very small scales it is easier to solve the interaction between fluid and particles by DNS. For many practical problems, however, the dynamics occurs on a scale larger than the particle size. In this instance, the computational cost for fully solving the mechanical problem inside the particle void is huge. So it is necessary to use the unresolved approach for the simulation of the meso scale fluid. For the macro scale simulation, it is very difficult to simulate the behaviors of particles in the particle-fluid coupling simulation with the DEM, so the discrete particle system should be regarded as a continuous system such as the dual-fluid model. Since the theoretical basis of the local averaging approach is kernel interpolation integral, it is advantageous to employ SPH method to solve the local average N–S equations (AVNS). The DEM-SPH coupling model based on local mean method is a coupling method suitable for large and meso scale problems. The model can be physically decomposed weakly into the motion of the fluid at a larger scale and the motion of the particles and interstitial fluid at a smaller scale. But the decomposition between them cannot be completely decoupled and separated like the DNS. In this method, the SPH is utilized to simulate the fluid motion at a larger scale while the DEM is utilized to simulate the particle motion at a smaller scale. Meanwhile, parametric resistance model is employed to solve the influence of interstitial fluid on particles. Therefore, this method is required to meet the following assumptions. The scale between solid and liquid phases is similar, but they can still be separated from each other. The coupling of phases (the effect of interstitial fluids on solid particles) needs to be adequately described by the local value of porosity and fluid-dependent variables. The DEM-SPH coupling model based on the local averaging method inherits the advantages of the two mesh-free particle methods, and is particularly suitable for solid-liquid flows containing free surfaces, such as mixing problems. In the local averaging approach, a smoothing function g(r ) is defined, whose value is greater than 0. The function g(r ) decreases monotonically with the increase of r. This function also has an arbitrary nth derivative, g n (r ) theoretically satisfies the normalization condition g(r )dV = 1. Any field variable a  defined in the fluid domain can be obtained by the convolution formula of the smoothing function: 

a  (y)g(x − y)dVy

ε(x)a(x) =

(6.60)

vf

where, x and y are the position coordinates, and the integral traverses the volume v f of the interstitial fluid; ε(x) is the local porosity. vs is the volume of solid particles, which can be expressed as:  ε(x) = 1 −

g(x − y)dVy vs

(6.61)

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6 Fluid-Solid Coupling Analysis of Granular Materials

Similarly, the local average of any field quantity a  (x) defined in the solid boundary is given by the following equation:  (1 − ε(x))a(x) =

a  (y)g(x − y)dVy

(6.62)

vs

Anderson and Jackson (1967) applied the local averaging approach to the N–S equations (AVNS), and obtained the incompressible fluid continuity equation according to the local average variable:     ∂ ερ f + ∇ · ερ f u = 0 ∂t

(6.63)

where, ρ f is the density of the fluid; u is the velocity of the fluid. Correspondingly, the momentum equation is:  ερ f

 ∂u + u · ∇u = −∇ P + ∇ · τ − nf + ερ f g ∂t

(6.64)

where, P is the fluid pressure; τ is the viscous stress tensor; nf is the fluid-particle coupling term. The fluid is assumed to be an incompressible Newtonian fluid, in which  of  the viscous stress tensor are given by the formula τi j =  the components μ ∂u i ∂ x j + ∂u j ∂ xi . Note that turbulence is not considered here. In the similar way, the SPH adopts the local averaging approach to AVNS equation. The kernel function W (r, h) in the SPH is naturally used to replace the smoothing function g(r ) in the local averaging approach for the kernel interpolation integration. To calculate the porosity at the positions of SPH/DEM particles, the integral Eq. (6.61) can be transformed into the summation of DEM particles within the kernel interpolation radius, that is: εa = 1 −



Wa j (h c )V j

(6.65)

j

  where, V j is the volume of solid particle j; Wa j (h c ) = W ra − r j , h c is the length of the coupling smooth h c , generally setting the coupling smoothing length equal to the smoothing length of the SPH. The longer the coupling smoothing length, the smoother the porosity field is. If the coupling smoothing length is small enough, important local details of porosity can be captured. The continuity equation and momentum equation of N–S equations are deduced based on local averaging approach, as shown in Eqs. (6.63) and (6.64). Before converting these two equations into SPH equations, a surface density is defined as ρ = ερ f . The surface density is substituted into the continuity and momentum equation, and the normal N–S equation is obtained. The change rate of surface density is calculated by the continuous equation, and the discrete form of the continuous equation can be expressed as:

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid

1  Dρa = m b uab · ∇a Wab (h a ) Dt Ωa b

193

(6.66)

where, uab = ua −ub , subscripts a and b respectively represent SPH particle a and its neighboring particle b. For the sake of convenience, the subscripts in the following follow this convention. Ωa can be expressed as: Ωa = 1 −

∂h a  ∂ Wab (h a ) mb ∂ρa b ∂h a

(6.67)

The momentum equation, if the gravity acceleration is ignored, can be written in the discretized form:    Pa du a mb + ab ∇a Wab (h a ) =− dt Ωa ρa2 b    Pb + fa /m a ∇ + +  W (6.68) (h ) ab a ab b Ωb ρb2 where, fa is the coupling resultant term, which is the reaction force of DEM particles on the SPH particles of the liquid phase; ab is the divergence of the viscous stress tensor (Monaghan 1997). For the pressure term in the pressure gradient in the above equation, the pressure is calculated explicitly in the SPH by the EOS:  Pa = B

ρa εa ρ0

γ

 −1

(6.69)

The computational parameters in the EOS can be referred to Monaghan’s work (Monaghan et al. 2005). When the local average N–S equations are solved by SPH, the surface density is closely related to the local porosity. Therefore, in order to satisfy the accuracy of the simulation, the smoothing length needs to be updated in time so that there are enough particles around. The smoothing length h a can be calculated by the following formula:  ha = σ

ma ρa

1/d (6.70)

where, d is the dimension of the simulation (d = 1, 2, 3). σ determines the resolution of kernel interpolation integral. The higher the value is, the higher the precision is. Usually the value of σ is 1.3. In the coupling simulation based on local average DEM-SPH, the DEM (The default is based on spherical particles) is relatively simple to be solved, i.e., only a solid-liquid coupling force needs to be added to the right side of the DEM governing equations. The key lies in the accurate calculation of the coupling force fi . According

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6 Fluid-Solid Coupling Analysis of Granular Materials

to the derivation of Anderson and Jackson’s local averaging (Anderson and Jackson 1967), the coupling force fi on each particle can be expressed as: fi = Vi (−∇ P + ∇ · τ )i + fd (εi , us )

(6.71)

The first two terms in the above equation represent the forces of fluid, buoyancy and shear stress, on particles, respectively. The shearing stress divergence is considered to ensure that the motion of particles can move along the streamline. fd is determined by local porosity and apparent flow velocity (Van der Hoef et al. 2005), where parameters in different resistance models need to be determined by experiments. Robinson et al. (2014) successfully simulated the bidirectional coupling between particles and liquid using the local average DEM-SPH, and compared it with the standard test. The comparison shows that the simulation error can be kept below 1% under a wide range of Reynolds number fluctuations. It illustrates that the coupling method of local average DEM-SPH can achieve remarkable result, as shown in Fig. 6.7 (Robinson et al. 2014). The local averaging approach is widely used in the numerical analysis of fluidsolid coupling problems of granular materials. This is due to the fact that most solid particles are small and mostly characterized as spheres. However, the local averaging cannot be well applied to the complex shaped and large discrete elements, especially in the coupling of SPH and polyhedral particles. Therefore, the interaction between each SPH particle and solid particle should be considered specifically, i.e., the force between fluid and solid should be directly calculated. The detailed calculation of fluid-solid coupling with SPH can be referred to some applications in other fields (Marrone et al. 2012). Some application examples can be found in marine engineering, as shown in Fig. 6.8.

Fig. 6.7 Local average fluid-solid coupling of granular materials (Robinson et al. 2014)

6.2 DEM-SPH Coupling Method for Granular Materials and Fluid

195

Fig. 6.8 The force between fluid and solid by direct method (Marrone et al. 2012)

(a) t = 0 s

(b) t = 1.4 s

(c) t = 1.4 s

(d) t = 2.3 s

Fig. 6.9 Dam-break flow impacts on three cubes simulated with the DEM-SPH method (Ji et al. 2019)

The interaction between polyhedral particles and fluid can be simulated by the repulsive force model, as shown in Fig. 6.9. The movement of the solid cube can be validated with experimental results, which shows good agreement (Ji et al. 2019).

6.3 DEM-LBM Coupling Method for Granular Materials and Fluid Lattice Boltzmann Method (LBM), derived from Lattice gas automata, is a computational fluid dynamics method developed in the mid-1980s (McNamara and Zanetti 1988). Compared with the traditional CFD method for solving nonlinear partial differential equations, LBM conducts numerical simulation based on linear equations, which fundamentally simplifies the modeling process. At present, LBM has become

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6 Fluid-Solid Coupling Analysis of Granular Materials

a very important numerical method. Based on the characteristics of LBM itself, this section attempts to clarify the unique treatment method in the coupling process of DEM and LBM method and its successful applications in the particle-fluid system.

6.3.1 Lattice Boltzmann Method In the traditional LBM simulation, the computational domain is dispersed into a series of grids and computing time are modeled as a series of time steps. The fluid density distribution function is calculated on the grid points and moves in accordance with the specified grid lines, i.e., collision and migration. The spatial and temporal discretization of LBM has a high correlation. During the iteration of each time step, the fluid density distribution function inside the flow field only allows migration from the current lattice point to the neighboring lattice point or not, while the fluid density distribution function on the solid boundary lattice point only needs to be intuitionistic “rebound”. Therefore, from the perspective of algorithm, LBM calculation has a strong “locality”, which can identify different phases on the grid points easily and describe the interaction between phases directly. This method is suitable for dealing with multiphase flows problems and complex geometric boundary problems, and it is considered to have a natural parallelism. Theoretically, LB equation can be seen as a special discrete form of simplified Boltzmann equation (BGK) (He and Luo 1997). The similarity between LBM and DEM gives them a natural advantage in coupling. Firstly, both governing equations are discretized with the explicit difference scheme, DEM-LBM coupling algorithm facilitates processing of dynamic fluid-solid coupling problems. Secondly, the calculation and interaction of the two algorithms are of high “locality”, which facilitates the implementation of parallel acceleration calculation of the coupling algorithm by splitting the computing regions. Thirdly, LBM can process all important data (velocity, force, torque and heat flow, etc.) generated by the DEM when processing complex granular materials (especially irregular granular morphology) and accurately feedback the interaction between particles and fluid, which means the respective advantages can be remained to the maximum extent in the coupling model. Therefore, the DEM-LBM method is often used to study the mechanism of complex multi-physical field, multi-phase flow and multi-scale problems. At the same time, it must be pointed out that both the DEM-LBM coupling method and the respective methods themselves are still severely constrained by the computational resources, which makes it difficult to achieve real-time simulations on the engineering scale. The application of parallel acceleration technology such as graphics processor (GPU) in the field engineering computing provides an effective way to improve the computational efficiency and calculation scale of the DEM-LBM (Xu et al. 2012). As mentioned above, the LBM recognizes the existence of other phases by judging lattice points. In other words, the numerical description of the physical shape of particles in the flow field is often composed of several grids, as shown in Fig. 6.10. The smooth curve is the real physical boundary of particles, the serrated shadow

6.3 DEM-LBM Coupling Method for Granular Materials and Fluid

197

Fig. 6.10 Description of solid particles in the LBM flow field

grid is the numerical boundary of non-slip rebound, and the points on the curve are the numerical boundary of particles. In this way, the flow pattern and fluid pressure distribution around the particle can be roughly obtained by applying non-slip bounces or other high-order boundary conditions on the representative particle lattice points. It is not difficult to find that the rough description of solid particles in the flow field using lattice points will bring some problems. Firstly, there may be a large error between the saw-tooth description of the particle and its smooth physical shape, which will inevitably affect the accuracy of the calculation. In addition, when solid particles move in the flow field, the lattice representing particles will be updated accordingly. The difference of numerical boundary morphology before and after particle movement will also produce additional errors. Certainly, it is possible to improve these differences by densifying the fluid grid, but it also increases the amount of calculation. Ladd (1994a) corrected the traditional non-slip bounce-back boundary by forcibly arranging the intersection point between particle physical boundary and lattice line at the midpoint representing the lattice point and its adjacent flow frame point, considering the effect caused by the density distribution function on the points of the translational and rotational convection frame of solid particles (Ladd 1994a, b). Momentum exchange of fluid density distribution function near the boundary is applied to calculate the coupling force and torque. The correction method aims to properly “polish” the serrated edges of the numerical boundary of particles and narrow the difference between them and the real physical boundary. Ladd successfully simulated several typical examples such as particle suspension with this method, which was also the earliest application of the LBM in fluid-particle system. The inadequacy of the Ladd model is that even the shape of the numerical boundary is improved, the edges are still jagged. The bigger defect is that when the solid particles move in the lattice at

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a higher speed, the resultant coupling force calculated by this method has obvious numerical instability. These problems were gradually solved with the introduction of various highprecision boundary methods, such as IMB (Noble and Torczynski 1998) and IBM (Peskin 1997). IMB can be regarded as a further modification of Ladd’s model, which introduces the concept of weight in the calculation process of fluid-solid coupling force, and further makes clear the contribution of relevant lattice points to control the area covered by solid particles and the free fluid area in the body. Such treatment can ensure that the smooth and excessive fluid-solid coupling force and torque are still obtained when the solid particles rapidly cross different lattice points. Han et al. (2007) made a clear comparison and description of the modified model of IMB and Ladd’s (Han et al. 2007). Cook et al. (2004) first constructed the DEM-LBM coupling model based on IMB, and studied the fluid corrosion problem through numerical simulations (Cook et al. 2004). Subsequently, the construction of the DEM-LBM coupling model framework and the discussion of some calculation details by Swansea University in the UK are very helpful for beginners to understand and master (Feng et al. 2007; Owen et al. 2011). At about the same time, Feng and Michaelides from Tulane university in the United States realized the coupling of DEM and LBM based on IBM, and studied the sedimentation process of solid particles in a large system in a fluid (Feng and Michaelides 2004). Different from the way that IMB determines the position of representative particle lattice points by obtaining the physical boundary of particles and the intersection point of grid lines, IBM uses two sets of independent grids to store particle information in Lagrangian grid and flow field information in Euler coordinates, respectively. The information between the two sets of grids is transmitted through numerical interpolation. Once a Lagrangian point is generated in IBM, it is only necessary to update it according to the particle’s position and deflection angle during calculation. Therefore, compared with IMB, IBM not only describes the numerical boundary of particles more accurately, but also does not need to pay much attention to the specific location of the intersection of particles and grid lines in the implementation of the calculation program, which is much simpler than IMB. Compared with the previous solution of fluid-solid coupling problem based on direct numerical simulation (Hu et al. 2001), the introduction of IBM saves the generation of extremely time-consuming body-fitted grid and significantly improves the computational efficiency. Therefore, the following focuses on the IBM-based DEM-LBM method. Currently, there have been several reviews (Chen and Doolen 1998) and monographs (Wolf-Gladrow 2004; Succi 2001) on LBM. The common BGK model (Qian et al. 1992) and the double distribution function model (He et al. 1998) are taken as the starting point to expand the narration in the following description. The D3Q15 model is selected here, and its velocity distribution is shown in Fig. 6.11. The governing equations of the LBM dual-distribution function model can be expressed as:

6.3 DEM-LBM Coupling Method for Granular Materials and Fluid

199

Fig. 6.11 Velocity distribution diagram of D3Q15 model in LBM

⎧ eq f α (r, t) − f α (r, t) ⎪ ⎪ f δ , t + δ + Fα (r, t)δt = f t) − ⎪ (r+e ) (r, α t t α ⎨ α τ f

eq ⎪ gα (r, t) − gα (r, t) ⎪ ⎪ + G α (r, t)δt ⎩ gα (r+eα δt , t + δt ) = gα (r, t) − τg

(6.72)

where, f α (r, t) and gα (r, t) are the internal energy distribution function, respectively, α = 0 ∼ 14 (see Fig. 6.11). r And t represent spatial position and time, respectively, δt is the time step. In the D3Q15 model, the fluid velocity eα is defined as: ⎧ α=0 ⎨ (0, 0, 0), eα = (±c, 0, 0), (0, ±c, 0), (0, 0, ±c), α = 1 ∼ 6 ⎩ α = 7 ∼ 14 (±c, ±c, ±c),

(6.73)

where, c is the lattice velocity. In addition, the superscript eq in Eq. (6.72) represents the equilibrium distribution function:   ⎧ 9 3 2 2 ⎪ ⎪ |u| 1 + 3(e · u) + · u) − t) = ρω f (r, (e a α α ⎨ α 2 2   (6.74) ⎪ 9 3 ⎪ ⎩ gα (r, t) = T ωa 1 + 3(eα · u) + (eα · u)2 − |u|2 2 2 14 = 1/9, ω7∼14 = 1/72; ρ = where, ω0 = 2/9, ω1∼6 !" α=0 f α , u = 14 14 1 1 ρ and T = α=0 f α eα + 2 (FB + Fe )δt α=0 gα eα + 2 Q B δt represent the macroscopic density, velocity and temperature of the fluid, respectively. Several

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source terms Fe (r, t), FB (r, t), Fα (r, t), Q B (r, t), G α (r, t) and G α (r, t) are derived from the influence of solid particles on the fluid. Obviously, these items should be zero when no  solid particles are present in the flow field. The buoyancy lift Gr u c T , with u c as the characteristic velocity. The specific forms of Fe (r, t) = Re 2 c other source terms will be introduced in the next section. Dimensionless relaxation factor τ f and τg can be defined as: 

τf = τg =

L c uc + 0.5 Recs2 δt L c uc + 0.5 Pr Recs2 δt

(6.75)

"√ where, L c is the characteristic length, cs = c 3 is the lattice sound velocity. The fluid pressure and kinematic viscosity coefficients meet the following: p = cs2 ρ υ=

  1 1 h2 τf − 3 2 δt

(6.76) (6.77)

where, h = cδt is the space step length of LBM.

6.3.2 DEM-LBM Coupling Immersion Boundary Method In the DEM-LBM coupling method, the influence of solid particles on the flow field is not in the form of a solid boundary, but is introduced into the governing equation in the form of source terms, which is the essence of the IBM. The IBM has grown rapidly since it was proposed. The source terms (i.e., force density f α (r, t) and heat flux gα (r, t)) are calculated in slightly different ways. For example, in the calculation of fluid-solid coupling force, Peskin (1997), Feng and Michaelides (2004) introduced an artificial parameter k as the elastic coefficient (Peskin 1997; Feng and Michaelides 2004, 2005; Uhlmann 2005) proposed a direct calculation method for fluid-solid coupling force without artificial parameters (Uhlmann 2005; Feng and Michaelides 2005). This section mainly introduces the IBM based on momentum exchange proposed by Niu et al. (2006). This method combines Ladd’s modified model and advantages of the IBM. While using two independent grids to describe the flow field and particle, it is proposed to calculate the fluid-solid coupling force based on momentum exchange without artificial parameters. Compared with other IBMs of the same type, the implementation of the model algorithm of Niu et al. is relatively simple. Subsequently, the IBM method based on momentum exchange was further improved, making it one of the most commonly used IBM-LBM methods

6.3 DEM-LBM Coupling Method for Granular Materials and Fluid

201

(Wu and Shu 2009, 2010). Zhang et al. first adopted the coupling proposed by Niu et al. to realize the coupling of the DEM and LBM, and verified the feasibility of applying the DEM-LBM model in complex fluid-solid coupling problems (Zhang et al. 2014a, b). First, a very important interpolation tool, the Dirac Delta function (Peskin 1997), is introduced:         xi jk − X yi jk − Y z i jk − Z 1 δh δh (6.78) Di jk r i jk − X l = 3 δh h h h h where, X l is the position of solid particles, and the subscript l represents the information of Lagrangian point. δh can be defined as:  δh (a) =

1 4

1+ cos

π|a| 2

!!

0,

, |a| ≤ 2 |a| > 2

(6.79)

The fluid parameters at the boundary of solid particles can be obtained by using the Dirac Delta function with numerical interpolation. For example, the fluid velocity and temperature at the solid boundary can be obtained by the following equation: ⎧    ⎪ u f (X l , t) = u f (r, t)Di jk r i jk − X l h 3 ⎪ ⎪ ⎨ i jk    ⎪ ⎪ T f (X l , t) = T f (r, t)Di jk r i jk − X l h 3 ⎪ ⎩

(6.80)

i jk

Here, in order to reflect the interaction between fluid and solid, subscript f is used to represent fluid and subscript s is used to represent solid particles. In this way, the force density and heat flux of the fluid and solid phase on the particle surface can be estimated, and then the source term of the equation can be calculated. Based on the momentum transfer criterion, the source term in Eq. (6.74) can be calculated as follows: ⎧ !   ⎨ Fα (r, t) = 1 − 1 ωα 3 eα −u +9 eαc4·u eα · (FB (r, t)+Fe (r, t)) 2τ f ! c2 (6.81) ⎩ G α (r, t) = 1 − 1 ωα Q B (r, t) 2τg where, FB (r, t) and Q B (r, t) can be expressed as: FB (r, t) =

 l

  F(X l , t)Di jk r i jk − X l sl

   F(X l , t) = 2ρ(X l , t) u s (X l , t) − u f (X l , t) h δt    Q B (r, t) = Q(X l , t)Di jk r i jk − X l sl l

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6 Fluid-Solid Coupling Analysis of Granular Materials

   Q(X l , t) = 2 Ts (X l , t) − T f (X l , t) h δt  where, sl is the unit area represented by a single Lagrangian point, and l represents all the Lagrangian points traversed on the particle surface. The source item calculation actually consists of three steps. The force density F(X l , t) and heat flux Q(X l , t) at the surface of the particles are first calculated; The force density and heat flux of the particle boundary are distributed to the lattice points of Euclidean LBM to obtain the force density FB (r, t) and heat flux Q B (r, t); Finally, the source term Fα (r, t) and G α (r, t) are calculated. Then, according to Newton’s third law, the resultant fluid-solid coupling force, torque and heat flux applied by the fluid on the solid particles can be obtained by integrating the particle surface area as: F f pi = −



F(X l , t)sl

l

T f pi = −



(X l − s) × F(X l , t)sl

l

q f pi = −



Q(X l , t)sl

(6.82)

l

where, s is the center coordinate of particles. The general calculation flow of the DEM-LBM algorithm is shown in Fig. 6.12. The DEM-LBM coupling model above requires the size of solid particles to be larger than that of a single LBM lattice, which is typically several or dozens of times. This method provides as much fluid-solid coupling information as possible, and the requirements for computing resources are also greatly improved. As mentioned above, due to the huge amount of calculation and the imperfection of relevant models, the DEM-LBM method can only be used for mechanism research at present, which is difficult to be applied to the actual simulations of industrial scale processes. In many industrial processes with dense gas-solid systems, such as circulating fluidized bed and pneumatic conveying, people are more interested in the complex multi-scale structure and meso scale mechanism of the formation of a large number of particles. The simulations of these systems can adopt coarse-mesh in fluid calculation, allowing a single particle size smaller than a single CFD computational grid, and the coupling force between fluid and solid can be estimated by an appropriate traction model, named as the DEM-CFD method, to improve the computational efficiency (Zhu et al. 2007). According to Professor Aibing Yu, academician of Australian Academy of Science and Academy of Engineering, the main difficulty in the current numerical simulation algorithm of fluid-particle system is the particle side rather than the fluid side (Yu and Xu 2003). Because of the advantages of the DEM-CFD algorithm in computational efficiency, it is widely used in engineering. The team led by Professor Yu (SIMPAS) has done a lot of pioneering work on the proposal, development and application of the DEM-CFD method. SIMPAS was invited to give a comprehensive

6.3 DEM-LBM Coupling Method for Granular Materials and Fluid

Fig. 6.12 The DEM-LBM algorithm flow chart (Zhang et al. 2014)

203

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6 Fluid-Solid Coupling Analysis of Granular Materials

review on the theories and applications of the DEM-CFD method, which has exerted a great influence in the field (Zhu et al. 2007, 2008). In recent years, in order to improve the computational efficiency of the DEMLBM, the DEM-LBM model with solid particle size smaller than a single LBM lattice size has been proposed. The calculation of fluid-solid coupling force is either based on the EMMS drag force model (Wang et al. 2013), momentum exchange model (Zhang et al. 2015a), or Stokes drag force model (Habte and Wu 2017). When the LBM is introduced into such a coupling model, it is only one of many CFD solvers, and its coupling mode is the same as other DEM-CFD methods.

6.3.3 Application of the DEM-LBM Coupling Method Due to the great computational cost of the DEM-LBM method, most of the existing literatures reported 2D results. Zhang et al. (2015a) put forward the particle immersion boundary method (PIBM), regarding a single Lagrangian point as a solid particle, in which the particle size is allowed to be smaller than the grid size, and the coupling force between fluid-solid is still calculated based on momentum exchange. The PIBM method can then be implemented to achieve the DEM-LBM coupling. The 3D numerical simulation by this method was developed to study the settlement behavior of 8125 solid particles in the square box. The results are shown in Fig. 6.13 (Zhang et al. 2015a). The effect of initial void fraction on particle deposition velocity was also studied. Zhang et al. (2016) also implemented temperature simulations to study the settlement process of 3D high-temperature particles in the flow field by the PIBM model. The temperature change of the fluid in the box during the settlement process is shown in Fig. 6.14 (Zhang et al. 2016). The heating process of the surrounding fluid by hot particles can be observed obviously. Galindo-torres further developed the IMB and simulated the diving process of seabirds, and obtained very interesting numerical simulation results, as shown in Fig. 6.15 (Galindo-Torres 2013). In this coupling model, the DEM adopts the extended polyhedral particles based on Minkowski Sum, the LBM adopts the D3Q15 format, and the immersion method is adopted for the interaction between fluid and solid. This method can achieve the fluid-solid coupling simulation of complex shaped solid particles. Since the DEM-LBM method can accurately calculate the drag force and heat transfer information in the fluid-solid coupling process, it is often used to study the basic experimental research to obtain the drag force coefficient and average Nusselt number under different simulation conditions. These parameters can be used to improve the calculation of the coupling force and heat transfer in DEM-LBM simulations. Figure 6.16 shows the instantaneous flow field distribution when fluid passes through a particle group composed of two shapes of ellipsoids by the DEMLBM method (Rong et al. 2015). Through a lot of simulations, the more accurate

6.3 DEM-LBM Coupling Method for Granular Materials and Fluid

205

Fig. 6.13 Particle location distribution diagram of particles at different moments. a 2.5 s; b 5.0 s; c 7.5 s; d 10.0 s (Zhang et al. 2015a)

drag coefficient can be obtained. These formulae provide important tools for the fluid-solid coupling simulations of the dense non-spherical particle systems. At present, the DEM-LBM method is mostly used in the fundamental research of multi-physical field and multi-phase flow. But the applications in industrial systems are few. With the rapid development of computer software and hardware and the further improvement of the DEM-LBM theoretical system, the DEM-LBM will have greater development in the field of science and engineering.

6.4 Summary This chapter introduces three basic fluid simulation methods for the fluid-solid coupling of granular materials, namely, the DEM-CFD, the DEM-SPH and the DEM-LBM. In the DEM-CFD coupling, the incomplete solution scheme is mainly introduced. The interaction between fluid and particle is introduced in the governing equations for fluid simulation. In the DEM-SPH coupling, the basic method of the SPH and coupling method is introduced, including the traditional WCSPH and ISPH,

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6 Fluid-Solid Coupling Analysis of Granular Materials

Fig. 6.14 Distribution of temperature field in the square box at different times. a 2.5 s; b 5.0 s; c 7.5 s; d 10.0 s (Zhang et al. 2016)

and the local averaging approach in the DEM-SPH coupling. In the DEM-LBM coupling, the origin and development of this method are briefly introduced, the coupling theory of the DEM-LBM method based on the IBM is expounded, and some typical examples are given at last. The CFD and SPH are numerical methods of Navier-Stokes equations based on macroscopic computational fluid theory. The LBM is a new fluid simulation method based on mesoscopic statistics theory. The traditional CFD method, with its solid theoretical foundation, has been used in engineering applications, but the treatment of complex boundary grids is more complicated. The SPH is adaptive to the simulations of free surface of liquid, and there is no grid problem, but it is complex to deal with the interaction between fluid and solid. The DEM-LBM method is widely used in fundamental research such as multi-physical field and multi-phase flow, but the

6.4 Summary

207

Fig. 6.15 The diving process simulated by the DEM-LBM method (Galindo-Torres 2013)

(a) Flattened particles

(b) Slender particles

Fig. 6.16 The DEM-LBM simulation of flow field distribution diagram (Rong et al. 2015)

application of industrial system simulation is quite few. With the rapid development of computer software and hardware and the further improvement of relevant theoretical system, the fluid-solid coupling simulations of granular materials will play a greater role in scientific research and practical applications.

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Chapter 7

High Performance Algorithm and Computing Analysis Software of DEM Based on GPU Parallel Algorithm

Taking the DEM as the main approach, computational granular mechanics has been widely applied in geotechnical engineering, chemical process, geological disaster, mining engineering, marine engineering, agricultural engineering, mechanical engineering, atmospheric science, medical pharmacy, life science and other fields, and obtained many important research results. With the increasing particle types and combination with traditional numerical methods such as computational solid mechanics and computational fluid dynamics, the DEM has been gradually extended to the material damage and fracture, fluid-structure interaction and other fields. Meanwhile, it also provides a powerful approach to solve the granular mechanics problems in the natural ecology, industrial engineering and life sciences. With the development of computational mechanics of granular materials, the corresponding simulation software have been gradually developed and applied to different engineering fields, which have become an effective means to solve the granular materials problems.

7.1 Developments of Computing and Analysis Software of DEM 7.1.1 GPU Parallel Technology With the rapid development of computer hardware, high-performance parallel computing has become an important means to improve the efficiency of computer simulation and expand the scale of computing. Parallel computing is to decompose an application into multiple sub-tasks and assign them to different processors. The processors cooperate with each other to execute sub-tasks in parallel so that the solution speed is improved or the solution scale is expanded. Traditionally, there are MPI, OpenMP, fork, thread and so on. Among them, message passing interface (MPI) is a messaging programming environment, which is suitable for massively parallel © Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_7

211

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processors and cluster programming. In contrast, open multi-processing (OpenMP), based on the shared memory model, is more suitable for stand-alone programming. It adopts simple pseudo-instruction, which can quickly realize the parallelization of the original serial code. With the development of Central Processing Unit (CPU) multiple nuclear technologies, parallel computing has attracted more and more attention in the field of computing. In recent years, the parallel technology, which adopts Graphics Processing Unit (GPU) as the main hardware carrier, has been developed rapidly. It derives a cross-platform common parallel environment with CPU-GPU collaboration, that is, CPU controls task allocation and main program flow, while GPU performs parallel computing tasks. At present, the parallel environment based on CPU-GPU collaboration mainly includes CUDA, OpenCL, OpenACC and so on. CUDA was launched by NVIDIA company in November of 2006. It is used to play the general computing power of the NVIDA GPU, mainly using the CUDA C computing language for program compilation and development. Currently, CUDA only supports NVIDIA GPUs on the hardware, so it must use OpenCL if users want to use AMD or Intel graphics cards. OpenCL is an open standard for heterogeneous parallel programming maintained by the Khronos Group, a non-profit organization. It supports single and collaborative parallelism of multiple accelerators, as well as data and task parallelism. It can achieve greater computing scale through built-in multi-GPU parallelism. Therefore, the application of OpenCL is wider than CUDA. However, because CUDA was launched early and strongly supported by NVIDA company, more programmers are using this environment. Besides, CUDA is more stable and has a strong library of scientific computing functions. In addition to CUDA and OpenCL, OpenACC uses compiled guidance statements to build high-level heterogeneous programs, without explicitly initializing the accelerator and data transfer, but only by compiling guidance statements to specify. This pattern is similar to OpenMP in that it enables rapid parallelization of serial code and allows programmers to focus more on specific research rather than parallel environment and code construction (Table 7.1).

7.1.2 Development of Discrete Element Computing and Analysis Software Due to the high parallelism of discrete element algorithm, high performance parallel computing technology can be widely used in numerical simulations of largescale physical models consistent with objective conditions. With the deepening of scientific research and engineering applications in the field of discrete media, high-performance computational granular mechanics analysis software based on discrete element has gained great attention, and various related computational analysis software has been developed rapidly.

7.1 Developments of Computing and Analysis Software of DEM

213

Table 7.1 Comparisons of main GPU parallel environments Parallel environment

Developer

Hardware support

Portability

Efficiency

Notes

CUDA

NVIDIA

NVIDA GPU

Low

High

Highest utilization rate, high stability, strong extension function library

OpenCL

Khronos Group

CPU, NVIDA/AMD GPU

High

High

Lack of maintenance, poor stability, low utilization rate

OpenACC

Cray, NV, AMD and so on

NVIDA/AMD/INTEL GPU

High

Low

Fast parallelization can be achieved

(a) PFC 2D

(d) EDEM

(b) PFC 3D

(e) ROCKY

(c) 3DEC

(f) MercuryDPM

Fig. 7.1 Main commercial software for discrete element calculation and analysis

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7 High Performance Algorithm and Computing Analysis Software …

ITASCA, the company joined by Dr. Cundall, the founder of the DEM, was the first to develop commercial PFC2D and PFC3D software developed primarily from the 2D BALL and 3D TRUBAL programs. As shown in Fig. 7.1a, b, PFC2D and PFC3D are based on 2D discs and 3D spherical particles, respectively. The software can be used to simulate the flow of a large number of discrete media, material mixing, and the fracture and dynamic failure of rock mass. In addition, the company also developed the 2D and 3D block discrete element analysis software, called UDEC and 3DEC, respectively, which is suitable for simulating static and dynamic problems of jointed rock system or discontinuous block system, as shown in Fig. 7.1c. The software provides Visual C++ self-development interface in which the function modules are relatively complete. At present, it has a wide range of influence in the field of engineering and scientific research. EDEM is the world’s first general CAE simulation analysis software designed by modern discrete element model technology, which is widely used in simulations of behavior characteristics of particulate system, as shown in Fig. 7.1d. The pre- and post-processing function of the software is strong, with good secondary development ability. It also has an excellent compatibility with other software. EDEM is becoming the leading product in the field of discrete element commercial software. Rocky is a powerful DEM package developed jointly by Granular Dynamics International, LLC and Engineering Simulation and Scientific Software Company (ESSS), as shown in Fig. 7.1e. The software supports GPU parallelism, non-spherical granular units, and integrated computation with ANSYS. At present, there is a large number of open source software using the DEM. BALL&TRUBAL was the first one; LAMMPS based on molecular dynamics can also be used for discrete element simulations and GPU parallel computation. LIGGGHTS is an open source software developed based on LAMMPS with strong parallel computing power, as shown in Fig. 7.2a. Using multiple parallel modes such as MPI and GPU, it can be used for simulations of particle flow motion and heat transfer. Meanwhile, this open source code has a powerful fluid calculation package, which can be applied to fluid-structure coupling analysis with OpenFOAM, SPH, LBM and other methods. It has been developed as an independent community project and widely used in the research field of computational granular mechanics. Yade and ESyS-Particle are mainly used in the field of geotechnical engineering. It has the application structure supporting Python language with strong scalability, as shown in Fig. 7.2b. Both of them can be used to conduct a fluid-structure coupling analysis with OpenFOAM. Meanwhile, they also support parallel acceleration of OpenMP and MPI. Unfortunately, Yade can only run on LINUX system. Woo based on Yade supports Windows system and has a good user interface. Originally developed by Chinese scholar Chen Feng, ESyS-Particle now supports Windows and is developing rapidly under the Aussie Earth System Computing Center’s funding. MechSys is able to support extended polygonal/polyhedral particles and can adopt the LBM method for fluid-structure coupling. In addition, dp3D, SDEC, LMGC90, PASIMODO and other important open source software also have widely applications at home and abroad.

7.1 Developments of Computing and Analysis Software of DEM

215

(a) LIGHTS

(b) Yade Fig. 7.2 Main open source software currently for discrete element calculation and analysis

The studies on the DEM in China began in 1986. Professor Wang Yongjia of Northeastern University first introduced Cundall’s DEM to simulate rock mechanics and particulate system. He systematically introduced the basic principles and calculation examples of the DEM in the first national discussions on numerical calculations and model tests of rock mechanics. Since then, the DEM has been continuously applied in rock and soil mechanics, mechanical engineering, warehouse analysis, process engineering and other fields. As a result, it promotes the development of computational granular mechanics in China. For the development of discrete element software, Li Shihai’s research team from the institute of mechanics, Chinese Academy of Sciences, developed discrete element computing software CDEM for continuity-discontinuity-analysis based on GPU parallel algorithm, which has been successfully used in geotechnical engineering, geological hazards, water conservancy engineering and other research fields, as shown in Fig. 7.3a (Li et al. 2007; Feng et al. 2014). DEMLab, developed by Zhao Yongzhi’s group of Zhejiang University is a large-scale software based

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7 High Performance Algorithm and Computing Analysis Software …

(a)

(b)

(c)

Fig. 7.3 Main discrete element calculation and analysis software in China a CDEM software developed by Institute of Mechanics, Chinese Academy of Sciences. b DEMSLab software developed by Zhejiang University. c Software developed by the Institute of Processes, Chinese Academy of Sciences (Qiu et al. 2017)

7.1 Developments of Computing and Analysis Software of DEM

217

on GPU parallel computational algorithm, which has excellent computing capacity in chemical process, soil mechanics and other fields, as shown in Fig. 7.3b. The institute of processes, Chinese academy of sciences has also successfully applied GPU-based parallel high-performance computing to the gas-solid coupling process of granular materials in chemical processes, and achieved remarkable research results, as shown in Fig. 7.3c (Qin and Zhan 2012). Professor Yu jianqun from Jilin University applied the DEM to the field of agricultural machinery, made a detailed analysis of the interaction between the mechanical parts and particles, and developed the corresponding discrete element software combined with commercial CAD software. Professor Jin Feng from Tsinghua University used block particles to simulate the stability of slope, arch dam, etc., and established a relatively complete discrete element analysis method based on polyhedral particles, including contact models, bonding models and fracture models, as well as coupling with finite element and boundary element methods. Zang Mengyan, professor of South China University of Technology, introduced the DEM-FEM coupling method into the analysis of automobile safety, and carried out a systematic discrete element simulations on the interaction between automobile tires and bulk media and the breakage process of automobile glass. Zhou Wei, professor of Wuhan University, made an in-depth analysis of the failure of rockfill dam and other problems by using the cementing block particles, and provided a very important method for the study of related problems in the field of rock mechanics and geotechnical engineering. For the large deformation and failure problems in the field of geology and geotechnical engineering, Dr. Liu Chun from Nanjing University independently developed the large-scale discrete element software MatDEM for geotechnical mass, which integrated the functions of pre-processing, discrete element calculations based on GPU, post-processing and secondary development. The software has been successfully used in the numerical simulations of conventional triaxial tests, consolidation tests and uniaxial compression tests, and can also be used for discrete element analysis of geological disasters, tunnel excavation and other geological and geotechnical engineering problems. In addition, China Agricultural University, Zhejiang University, Lanzhou University, Tongji University, Hohai University, Xiang Tan University, University of Science and Technology of China, the Chinese Academy of Sciences Institute of Physics and other colleges and universities and research institutes are deeply engaged in the basic theories and engineering applications of the DEM in order to effectively promote the development of the computational granular mechanics in China. SDEM, developed by the computational mechanics of granular materials research group of Dalian University of Technology, is a discrete element computing and analysis software based on GPU parallel computing, as shown in Fig. 7.4. At present, the dynamic and thermal simulation of the interaction between sea ice and structure in ocean engineering of cold regions is mainly based on the spherical particles. However, for the construction method of discrete particles, the software can also be compatible with the DEM-FEM, dilated polyhedral particles and super-quadric particles, and can realize fluid-structure coupling analysis based on the SPH method.

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7 High Performance Algorithm and Computing Analysis Software …

(a) Navigation of ships in ice areas

(b) Construction of irregular particles with superquadric equation Fig. 7.4 SDEM based on GPU parallel discrete element computing analysis software

In practical engineering applications, in addition to sea ice dynamics in marine engineering in cold regions, it can also provide the mechanical performance analysis of ballasted/ballastless railway track, landing process of space probe, basic mechanical behaviors of rock and soil material and many other fields.

7.2 Numerical Algorithm of DEM Based on CUDA Programming

219

7.2 Numerical Algorithm of DEM Based on CUDA Programming The discrete element algorithm based on CUDA parallel environment is very different from that based on serial environment. The CUDA parallel environment is also different from the traditional parallel setting based on OpenMP or MPI. This section mainly discusses the realization of the discrete element algorithm including the CUDA parallel architecture and discrete element algorithm in Multi-Machine or Multi-GPU environment.

7.2.1 Hardware and Software Architecture of CUDA Under the CUDA environment, the C-Like language CUDA C is adopted to complete code compilation (Storti and Yurtoglu 2015). In fact, CUDA also supports Fortran language under the PGI compiler. However, using C/C++ to implement the underlying software code has become a common method for many software developers. Fortran, traditionally widely used in computing, is not suitable for large-scale software development, maintenance and upgrades in future. Well-known computing software developers such as ANSYS and MSC have spent a decade or so converting the underlying code from Fortran to C/C++. C/C++ has become the most widely used language in the field of numerical computations. A complete CUDA program consists of a CPU mainline and several GPU functions that need to be executed in parallel. The function executed on the GPU is called kernel function, which is the execution unit for each thread. In CUDA, the parallel parts are not only allocated by threads, but also multiple threads are divided into several thread blocks. As shown in Fig. 7.5a, the CUDA program is executed according to the flow path of the main thread when it runs. Then, the main thread allocates tasks to the GPU according to the program design where it needs to be parallel. GPU calls in relevant data for calculation according to the definition of kernel function and sends the calculation results back to CPU through fixed interface, and then continues the execution according to the original thread. CUDA divides memorizer into different types at multiple levels, as shown in Fig. 7.5b. It mainly has the six kinds of memories. Register, the cache on the GPU chip, is the fastest access memory; Local memory, located in graphics memory, has a slow access speed; Shared memory, high-speed memory on GPU chip with fast access speed, is not a private memory for each thread; Global memory, the main part of graphics memory, can be accessed by both CPU and GPU; Constant memory, a read-only memory, is located in graphics; Texture memory is developed from a GPU dedicated graphics unit for texture rendering and rarely used in general calculations. Register is a high-speed memory at the thread level. It can be seen that the memory hierarchy of GPU registers is related to the allocation of threads and thread blocks. In practice, it is necessary to closely cooperate with the allocation of threads in order to

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C Program Sequential Execution

Grid0 Block(0,0)

Block(1,0)

Shared Memory

Shared Memory

Serial code Registers

Parallel kernel

Thread(0,0)

.. .. ..

Thread(1,0)

Local memory

Serial code Parallel kernel

Registers

Local memory

Registers

Thread(0,0)

Local memory

Registers

Thread(1,0)

Local memory

Global Memory

Host

(a) CUDA programming model

Constant Memory Texture Memory

(b) CUDA memory hierarchy

Fig. 7.5 CUDA parallel architecture

achieve the mechanism of accessing high-speed memory as much as possible (Cook 2014). It should be noted that the register memory is allocated equally among the thread blocks according to the number of thread blocks. That is to say, the amount of register memory that can be called by each thread block is the same. At the thread level, if each thread in a kernel function requires too much register memory, the computation core of the GPU will be limited by register. As a result, the activity of thread blocks reduces, resulting in weakened computing performance. The other type of high-speed memory on the GPU is shared memory, whose access speed is seven times faster than that of global memory, but only 1/10 as fast as register memory. In the early NVIDIA graphics architecture, each thread block only allocated shared memory with 16 K, but now it is possible to implement dynamic shared memory. That is, the size of a thread block or a shared memory array can be defined manually. The specifications of the utilization method can refer to the relevant information in NVIDIA official website. The aim of setting shared memory is to mitigate the large gap between registers with high speed and global memory with low speed. It is more appropriate only when data is reused, global memory is merged, or there is shared data between threads. Otherwise, the method of loading data from global memory into registers will be better. CUDA adopts Single Instruction, Multiple threads (SIMT) execution model. In short, CUDA parallelism is multiple executions of a single instruction. Unlike OpenMP and MPI, which can parallelize different instructions, CUDA parallelism only occurs at the loop computing position of the program, such as “for…” in C language or Fortran’s “ do…” loop statement. Therefore, the parallel of CUDA can

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take the loop body as the function body of kernal function, modify the global and local parameters in combination with the memory mechanism of CUDA, and match the corresponding CUDA C syntax to achieve simple single GPU parallel. Further optimization should be carried out gradually in combination with specific programs and detailed understanding of developers to parallelism. In CUDA programming, optimization of program branches is helpful to improve computational efficiency. A program branch is a change in the sequence of program execution caused by program logic. In simple terms, it is a program logic event caused by a program’s if statement. During the compilation of a program, two different instructions are generated depending on whether the “if” statement is true or false. In fact, before the “if” statement is executed, there is a stop-logic-resume procedure. The CPU avoids program stop by setting the branch predictor, that is, it predicts the true or false of “if” statement in advance by means of prediction, and switches to another instruction when the prediction is wrong. At present, most of the current CPU branch predictors have a prediction accuracy of more than 80%, some of which even have a prediction accuracy of more than 90%. However, there is no such branch predictor in the calculation core of GPU, so “if” statement should try to avoid in the core function of CUDA, especially in loop body, such as the following program segment: a[10] = {0,1,7,2,6,3,4,5,8,9,10}; float sum = 0; for (int i=0; i5) sum +=a[i] }

The purpose of this section of the program is to obtain the sum of number which is greater than 5 in array a. In actual programming, the program can be optimized by using logical operators, such as: a[10] = {0,1,7,2,6,3,4,5,8,9,10}; float sum = 0; for (int i=0; i5) }

Since the “if” statement is removed, the sum operation is added, and the overall computation is also increased. But by reducing the program branch, the instruction delay caused by the branch is greatly reduced. It actually improves the execution efficiency of the program.

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7.2.2 Discrete Element Algorithm in Multi-machine/Multi-GPU Environment When the number of particles is in the millions, the single GPU programming can achieve a higher acceleration ratio. However, when the number is increased to tens of millions or even higher, the parallel performance of single GPU programming will be sharply reduced due to memory limitations, access delays between threads and other factors. Therefore, the development of multi-GPU discrete element program is becoming more and more important. For multi-GPU programming, multiple threads can be opened up on the host through MPI or OpenMP. Each thread corresponds to one GPU’s memory and instruction management. In order to ensure the integrity of discrete element system, data between parallel GPUs need to communicate. Among them, MPI supports Ethernet data communication between different nodes, which is suitable for cluster computing in large server machine. For multi-GPU programming on a single node, namely, a single machine, the communication between GPUs is through the PCI-E bus. So OpenMP or POSIX thread is more appropriate. In recent years, each new generation of NVIDIA GPUs and CUDA platform upgrades have been accompanied by improvements in parallel programming across multiple GPUs. Starting from CUDA 4.0, the GPU direct technology developed by NVIDIA allows the third party to access directly. That is, the data exchange between multiple GPUs no longer needs the transfer process of physical memory at the host. The point-to-point memory access can be directly realized, effectively improving the communication efficiency of multi-GPU programming. After CUDA 5.0, GPU direct further supports the direct access of remote memory, which is significantly increased the GPU memory access speed between nodes. Compared with large multi-node server clusters, simple workstations or single-node servers realize multi-GPU computing with the advantages of convenient construction, low maintenance cost and more convenience and flexibility. Meanwhile, achieving a fast parallel computing using 2–8 GPUs on a single computer is the goal of most developers in order to greatly improve the computing scale and efficiency. The main problem of multi-GPU programming is how to divide computing tasks and data into relatively independent parts, so that each GPU can operate relatively independently and the massive data communication can also be avoided in the computing process. Generally speaking, the multi-GPU parallel of particle algorithm can adopt three methods, namely, particle decomposition method, force decomposition method and region decomposition method. The particle decomposition method distributes the particles evenly among the GPUs to update the force, space position and speed. This method can ensure that the computation amount of each GPU is roughly the same. But each GPU needs to obtain all the information of particles. There is a large amount of communications between GPUs at each time step. A large amount of memory is occupied. The force decomposition method allocates operations according to contact pairs. Since the contact calculation is the most computationally intensive part of the discrete element, an insightful distribution approach of contact force

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Fig. 7.6 Multi-GPU region decomposition method

can greatly improve the computational efficiency of the contact pair. However, this approach also requires a lot of memory swapping and storage. The regional decomposition method divides the simulated geometric space and distributes the particle units in each region to each GPU for calculation, so that the corresponding particles of each GPU are constantly changing. However, the internal data of each GPU is relatively independent, which avoids the communications between GPUs. Usually, only one time step is needed for data exchange between GPUs. In general, the method of region decomposition has the advantages of relative independence and is more suitable for discrete element programming with multiple GPUs. In the regional decomposition method, in order to achieve the data in each GPU to remain independent in the calculation and avoid the communication between GPUs in the specific calculation, the actual particle dimensions in each GPU are slightly larger than the actual area. Although it slightly increases the amount of data storage and calculation in each GPU, it ensures the data independence in each GPU. As shown in Fig. 7.6. Take the area decomposition method with 3GPU parallel as an example. Each GPU needs to accept more data from a part of adjacent areas. There is overlapping area of data between GPUs. In this way, data communication with other GPUs is no longer needed in the calculation within each GPU. Independent calculations can be achieved.

7.3 High Performance Contact Detection Based on GPU As with serial programs, the establishment of neighbor list between particles is conducive to improving the computational efficiency of the program (Kalms 2015; Nyland et al. 2007; Müller et al. 2003; Harada 2007). In the process of building a list of neighbors, it is obviously unreasonable to directly traverse all the particles to

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check whether other particles are in the adjacent domain. At present, there are many contact detection methods in 3D, such as spatial grid method (Munjiza and Andrews 1998; Munjiza 2004), octree method (Vemuri et al. 1998), boundary box method (Hart and Cundall 1988; Cleary et al. 1997), etc. Various search methods in parallel environment need to be adjusted according to the specific parallel mechanisms. Due to the complexity of octree and bounding box algorithms and the large amount of memory, this section will mainly focus on the list search method based on CUDA parallel.

7.3.1 Relationship Between Grid and Particles In the list search method, the first step is to divide the cube search grid within the computational domain. The grid is different from Lagrangian grid or Euler grid. It is only a background auxiliary grid, which has nothing to do with contact force calculations. In some serial algorithms, the size of the grid may be limited to ensure only one particle exists in per grid for computational effort. However, the parallel algorithm has no such limitation. In fact, if there is only one particle in each grid, the number of grids will increase significantly, which will lead to a large amount of video memory and reduce the access speed of memory. However, the grid with large size will increase the amount of computation per thread, which is also not conducive to improving computational efficiency. It is recommended to use 2 times of average particle size or maximum particle size to define the grid size dgrid , that is: dgrid = 2Dmean or dgrid = 2Dmax

(7.1)

where, Dmean and Dmax represents the average and maximum particle size, respectively. In addition to the geometric spatial coordinates of the particle, the grid coordinates of the particle also need to be defined to represent its grid position. If the spatial coordinates (x, y, z) of a particle are known, then its grid coordinates (x, y, z) can be obtained by integer operation, that is: i x = int((x − xmin )/dgrid )

(7.2)

i y = int((y − ymin )/dgrid ) i z = int((z − z min )/dgrid ) In the serial programming, in order to establish the relationship between the spatial position of a particle and its grid position, one or more array sequences with the number of grids in length are usually used to record the number of particles in each grid. This approach is also possible in GPU parallel environments. However, this method takes up a lot of free memory and does not make full use of the high-speed

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Fig. 7.7 Space and grid for each particle

memory mechanism of GPU to improve efficiency. This section introduces a method of using high speed memory, shared memory in GPU, to establish the corresponding relationship between spatial location and grid location. Take 2D space as an example. It is assumed that the spatial distribution of each particle is shown in Fig. 7.7. First, an array of grid Particle Hash is created to record the grid number where each particle resides. The length of the array is the number of particles. If the number of particles in a discrete system is N, the scale of divided grid is N x × N y × Nz = Ngrid . In a 3D or 2D grid, the grid number is recorded in a one-dimensional manner, that is: grid Particle Hash [i] = i z · N x · N y + i y · N x + i x , i = 1, 2, . . . , N

(7.3)

In the formula, the calculation on the right side of the equal sign can adopt other similar methods, as long as the relationship between each spatial grid and its ordinal number is guaranteed. According to Eq. (7.3), the grid Particle Hash sequence can be established. Considering its particle number, the second column can be formed in Table 7.2, that is, grid number + particle number. The current sequence is arranged by particle number. Next, the grid ordinal number is arranged, and the particle ordinal number changes with the grid ordinal number, namely, the third column in Table 7.2. In this case, the order of the third column is arranged according to the grid ordinal number. When generating the third column in Table 7.2, a sorted array of particle numbers can be formed taking into account the position of particle numbers in the sequence, that is, the particle sequences in the first and third columns of Table 7.2. The array establishes an association between the new and old ordinal numbers of the particles. For programmers who are not computer professionals, it is difficult and timeconsuming to write efficient sorting programs in CUDA parallel environment. The

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Table 7.2 Sorting processing of grid location of a particle Sequence Number

Unsorted list (cell ID, particle ID)

List sorted by cell ID

New particle ID

0

(9, 0)

(4, 3)

(0, 3)

1

(6, 1)

(4, 5)

(1, 5)

2

(6, 2)

(6, 1)

(2, 1)

3

(4, 3)

(6, 2)

(3, 2)

4

(6, 4)

(6, 4)

(4, 4)

5

(4, 5)

(9, 0)

(5, 0)

library Thrust, included in CUDA, is recommended here, containing the follow sort functions required, which is convenient to use. The next step is to create a shared memory array, shareHash. The array is used to record the grid number in its previous thread, which forms the array of the fourth column, as shown in Fig. 7.8. Here, take the 4th row as an example. If it is not equal, it indicates that the thread number (i.e. the new particle number, here 2) is the first particle number in the 6th grid and the last particle number in the 4th grid. Here the particle number is the new number. After the whole process is completed, the first particle number recorded by array cell-start and the last particle number recorded by cell-end in each grid are formed, as shown in Table 7.3. Cell-start and cell-end will not change for other meshes without particles.

Fig. 7.8 Record granular information within each grid in Shared memory

Table 7.3 Grid cell-start and cell-end

List sorted by cell ID

Cell-start

Cell-end

4

0

2

6

2

5

9

6

~

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It should be noted here that cell-start and cell-end computations need to occur after the sharedHash is fully generated, so the program needs to be synchronized after sharedHash is generated. In addition, the algorithm does not require sharedHash to synchronize all threads, but only within the thread block, that is, the __syncthreads() operation in CUDA. This ensures that the shared memory array within each thread block is fully generated. The above process completes the information recording of the first and last particle in each grid. All the particle information in the grid can be accessed by traversing the first particle to the last particle. In addition, through the array of old and new particle numbers, the old particle number information in each grid can be further obtained.

7.3.2 Establishment of Neighbor List After generating the particle number information in each grid, it is necessary to traverse all the particles and search other particles in the adjacent grid where the particle is located to obtain the surrounding particle information of each particle. As the interaction of particles is mutual, the forward or backward search method is usually adopted to avoid the secondary calculation of the same contact pair, that is, each particle only records the adjacent particles larger or smaller than the number of the particles. The general method is to build a neighbor array which is called NeighbourParticle. It is supposed that the maximum number of neighbors of the particle is Nmax , then the length of the array is N × Nmax . Thus, each particle has a series of location with Nmax to store the serial number of neighbor particles. It should be noted that when setting up the neighbor array, the stored neighbor particle number can be either the new one or the old one. It needs to be noted that the corresponding adjustment should be made in the following specific calculations. If we directly use the adjacent neighbor array for parallel, each thread needs to calculate multiple contact pairs, which increases the granularity of parallel and reduces the overall parallel performance. The optimal approach is that only one contact pair exists in per thread and parallelism granularity is minimized. This approach requires the establishment of relationship between contacts pairs. When the neighbor array Neighbour Particle is established, the neighbor number of each particle is recorded and deposited in the array neighbour Number. The array is then prefix summed to provide the third column named Prefix-Sum in Table 7.4. The process can be achieved using the Thrust library function. Then, the particle neighbor’s contact pair ParticleList need to be built, which is the number of the two particles in contact. Here it is recommended to use two sequences such as ParticleListA and ParticleListB to record separately. In fact, the Prefix-Sum sequence determines the location of all neighbors of each particle throughout the contact pair sequence. Here, the particle with number 1 in Table 7.4 is taken as an example, the location of its three neighbors in ParticleList is 2–5. If the number of three neighbors are 5, 6, and 7, then ParticleLis [2] = (1, 5), ParticleLis

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Table 7.4 Processing of particle neighbor sequences

Particle ID

Neighbor number

Prefix-Sum

0

2

2

1

3

5

2

2

7

3

5

12

4

4

16

5

3

19

6

1

20

7

3

23

[3] = (1, 6), and ParticleLis [4] = (1, 7) in ParticleList. In this way, the contact pairs list of the whole discrete system can be established completely. The last number of the Prefix-Sum sequence is the number of all the contact pairs. The contact pair list can be parallelized in the subsequent calculation of contact forces. Only one contact pair per thread is needed to achieve parallel granularity minimization.

7.3.3 Calculation of Contact Force Sequence In the calculations of contact forces, each thread only needs to calculate the contact force between one contact pair in order to achieve a high degree of parallelism. However, in discrete element parallel algorithms, a simple problem in serial algorithms perhaps become complicated due to different memory operations or designs. Since the force between particles is mutual, the contact force applied to the two contacting particles calculated by each thread can satisfy Newton’s third law. Moment between contacting particles also follows a similar pattern. In each thread of a contact pair, the two particles can be respectively accumulated by the contact force. However, in the parallel environment of CUDA, the storage conflict problem of global memory will occur, that is, when multiple threads store data in the same location in the global memory, it will cause unpredictable errors in the data calculation at the location. Since multiple threads of global memory access are merged within CUDA, there is no similar problem with multiple threads accessing the same location. In fact, all parallel computing platforms have similar problems, which is one of the difficulties in parallel algorithm design. To solve this problem, CUDA provides atomic operation functions. Atomic operations essentially lock a global memory location while it is accessed by a thread, thus preventing other threads from storing it. This design effectively solves the above problems.

7.3 High Performance Contact Detection Based on GPU Table 7.5 Parallelism of contact pairs

229

Pair ID

Particle ID

Force summation

0

(0, 4)

Force[0]

Force[4]

1

(0, 6)

Force[0]

Force[6]

2

(1, 3)

Force[1]

Force[3]

3

(1, 5)

Force[1]

Force[5]

4

(1, 6)

Force[1]

Force[6]

5

(2, 3)

Force[2]

Force[3]

6

(2, 6)

Force[2]

Force[6]

7

(3, 7)

Force[3]

Force[7]

Given the contact conditions in a discrete system as shown in Table 7.5, it is possible for threads 0 and 1 to store Force [0] at the same time, for threads 1, 4, and 6 to store Force [6] at the same time, and so on. At this point the atomic operation can be used to accumulate the contact forces. The atomic addition function of singleprecision floating-point number is atomicADD. The atomic addition function of double-precision floating-point number needs to be built by itself. The NVIDIA website provides a specific function called atomicAdd_double. In the actual computations, due to a series of reasons such as hardware itself and instruction scheduling, the computation speed of each thread will be slightly different. In the comparison between threads, these differences are relatively obvious. The atomic operation adopts the “who comes first, who locks first” approach, that is, the thread that counts to the accumulative position first takes priority in the atomic operation. After that, the subsequent thread executes the queuing mechanism. This will inevitably lead to different accumulative order of contact forces, and eventually lead to different results of multiple calculations of the same example. In Table 7.5, for example, threads 1, 4, and 6 access Force[6] by atomic function, since threads 1, 4, and 6 execute at different speeds over multiple computations. In the first calculation, the cumulative order can be 1, 4, 6; However, in the second time, it can be 4, 6, 1, and so on. Multiple computations here refer to multiple executions of the program, rather than different time steps in the same execution. The above operations are the same seemingly, but there are numerical errors in floating-point calculations. Different accumulative orders can cause different numerical errors. Since the discrete element does not have the same convergence problem as the finite element, its calculation results completely depend on the deduction from the previous step to the current step, and will not converge to the same state. After the iteration of multiple time steps, the difference of the results of multiple calculations will be obvious. It is worth mentioning that the calculation is not wrong because each calculation result is different. The difference is only caused by computer error. In fact, the difference in the calculation results does not make the difference in the order of magnitude. The calculation results are still stable. In fact, this difference can be used to calculate the results of many times into an operation library. Statistical calculations are performed to obtain more general rules. In addition, there will be

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Table 7.6 Particle No. 4’ neighbour particle 5

6

7

Table 7.7 The assumed information of the array ParticleList

3

List ID

1

Particle ID

… 4

(1, 4)

… 9

(3, 4)

… 13

(4, 5)

14

(4, 6)

15

(4, 7)

…

doubts for the robustness of the calculation method due to the different results of multiple calculations. In this way, the corresponding algorithm can be designed to avoid the problem. If the order of each sum is sorted every time, this problem can be avoided directly. However, the sorting algorithm is time-consuming and will greatly reduce the computational efficiency in the actual operation. This problem can be solved by building the relationship between each particle and the contact sequence (Nishiura and Sakaguchi 2011). First, the array Reference Series is created and declared, whose length is the same with the neighbor array neighbour Particle. When calculating the neighbor array neighbour Particle, each particle only records particles whose numbers are bigger or smaller than that of the particle. Here, the array neighbour Particle also records the information (the numbers included) of particles whose number are smaller than that of the particle. Records are in the reverse order. Suppose that the contact particles of particle No. 4 is No. 1, No. 2, No. 3, No. 5, No. 6, and No. 7 in a discrete system, Nmax = 10. So particle No. 4 in array neighbour Particle on the storage location is shown in Table 7.6. Reference Series are built when the particle neighbor’s contact list ParticleList sequence is generated. Take particle No. 4 above as an example. It is assumed that the relevant information of all the contact pairs is shown in Table 7.7, that is, the location of particle No. 5, No. 6, and No. 7 in the ParticleList is 13, 14, and 15. The location of particle No. 1, No. 2, and No. 3 in the ParticleList is 4, 7, and 9, respectively. Since the number of contact particles for each particle is known, including the contacting particles whose numbers are bigger or smaller than the particle number, the corresponding relationship between the contact number and each particle can be established in the Reference Series. Table 7.8 lists the storage of particle No. 4 in the Reference Series. When calculating the contact forces between particles, the contact forces need only be stored in an array with the same size of ParticleList. Then, the Reference Series can be used to find the corresponding ParticleList number for each particle

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Table 7.8 Reference Series of particle No. 4

13

14

15

4

9

+

+

+

-

-

and the corresponding contact forces. By the comparison of the number of contact particles with larger or smaller number, the contact force can be determined as a positive or negative action, as shown in the second row of Table 7.8. Through developing the contact search algorithm based on GPU parallel environment and considering the GPU computing power and data structure, the grid method within the GPU can be used in sorting and then building the neighbor list between particles. Based on the minimization principle of parallel particle size, the transformation between the subscript of contact pair and the coordinates of particles can be completed, resulting in the realization of efficient computing of particle contact forces.

7.4 DEM Computing Analysis Software Based on GPU Parallel Algorithm In recent years, the computational granular mechanics team of the state key laboratory for structural analysis of industrial equipment in Dalian University of Technology has developed the Smoothed Discrete Element Method computing analysis software, SDEM, based on GPU parallelism. It is composed of a series of professional software, which can be used to calculate and analyze ice load of marine engineering in cold regions (IceDEM), dynamics of ballasted railway track (BallastDEM), and progressive evolution of geohazards (PDEM-EDG). Currently, the software SDEM has developed a new interface and software architecture, which can integrate various particles unit types and the solving methods, and consider the coupling of granular particles with engineering structures and fluid. Orientated developments can be carried out according to specific industrial problems. The software has the advantages of friendly operation interface, efficient numerical algorithms and computations at engineering-scale. SDEM includes spherical particles, bonding and clumping particles, dilated polyhedral particles, super-quadric particles and so on, which can be called as “smooth discrete element” because it has smooth surface without sharp edges. SDEM employs GPU parallel algorithm with high performance and has great advantage in discrete element computing efficiency. In the numerical methods, the DEM as the main line, SDEM has developed coupled DEM-FEM method and coupled DEM-SPH method based on finite element method and the smoothed particle hydrodynamics. For further

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(a) Submarine floating process under the ice

(b) Numerical simulation of railway ballast-ballastless transition zone Fig. 7.9 Engineering applications of SDEM software

developments, the FEM-DEM-SPH algorithm provides the research foundation of the continuum-discrete-fluid coupling, and can achieve more practical applications in the field of engineering. In ocean engineering in cold regions, SDEM-IceDEM software can simulate the interaction between ocean platforms, ships and sea ice, and provide an effective

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engineering numerical analysis method for China polar ocean engineering development. Figure 7.9a shows the floating process of a submarine under the ice. In the field of geotechnical engineering, SDEM can also be applied to the dynamic analysis of railway ballast bed, which provides a basis for the degradation mechanisms and reasonable maintenance of railway ballasted track. Figure 7.9b shows the ballast-ballastless transition zone simulated by the DEM-FEM coupling algorithm.

7.5 Summary This chapter introduces the program design of the DEM from the perspective of CUDA parallel programming based on GPU algorithm, and briefly summarizes the basic methods from CPU parallel to GPU parallel of MPI and OpenMP, as well as the research and development of high-performance computing software for the DEM at home and abroad. The programming design of CUDA is different from serial programs and CPU-based parallel architecture. This chapter introduces the CUDA hardware and software system, and describes the basic knowledge in the CUDA programming; Then, the discrete element program design based on GPU is introduced in detail, including the establishment of neighbor list and the processing and calculations of contact forces. Finally, SDEM and its engineering applications are introduced. In recent years, with the rapid development of artificial intelligence, people pay more and more attention to the calculations based on the GPU. The software and hardware innovations in the field are becoming more and more fast, such as Tensor Processing Unit based on Tensor operation (TPU). NVIDIA GPU adds to a series of revolutionary computing architectures and platforms with the unified addressing and tensor operation function. Therefore, based on these new technologies, algorithms and platforms, modifications on the proposed algorithms and improvement on the computational efficiency and accuracy will be an important research direction in the field of discrete element simulations in future.

References Cleary PW, Metcalfe G, Liffman K (1997) How well do discrete element granular flow models capture the essentials of mixing processes? Appl Math Model 22(12):995–1008 Cook S (Write) (2014) Su Tonghua etc. (Translation). CUDA parallel programming: a guide to GPU programming. China Machine Press Feng C, Li SH, Liu XY et al (2014) A semi-spring and semi-edge combined contact model in CDEM and its application to analysis of Jiweishan landslide. J Rock Mech Geotechn Eng 6:26–35 Harada T (2007) Real-time rigid body simulation on GPUs. GPU Gems 3. Addison Wesley Hart R, Cundall PA, Lemos J (1988) Formulation of a three-dimensional distinct element model— Part II. Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks. Int J Rock Mech Min Sci 25(3):117–125

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Kalms M (2015) High-performance particle simulation using CUDA. Linköping University Li SH, Wang JG, Liu BS et al (2007) Analysis of critical excavation depth for a jointed rock slope by face-to-face discrete element method. Rock Mech Rock Eng 40(4):331–348 Müller M, Charypar D, Gross M (2003) Particle-based fluid simulation for interactive applications. In: Proceedings of 2003 ACM SIGGRAPH symposium on computer animation, 2003, pp 154–159 Munjiza A, Andrews KRF (1998) NBS contact detection algorithm for bodies of similar size. Int J Numer Meth Eng 43(1):131–149 Munjiza A (2004) The combined finite-discrete element. Wiley, Chichester Nishiura D, Sakaguchi H (2011) Parallel-vector algorithms for particle simulations on sharedmemory multiprocessors. J Comput Phys 230:1923–1938 Nyland L, Harris M, Prins J (2007) Fast N-Body simulation with CUDA. GPU Gems 3. Addison Wesley Qin CZ, Zhan L (2012) Parallelizing flow-accumulation calculations on graphics processing unitsfrom iterative DEM preprocessing algorithm to recursive multiple-flow-direction algorithm. Comput Geosci 43:7–16 Qiu X, Wang L, Yang N et al (2017) A simplified two-fluid model coupled with EMMS drag for gas-solid flows. Powder Technol 314:299–314 Storti D, Yurtoglu M (2015) CUDA for engineers: an introduction to high-performance parallel computing. Addison-Wesley Professional Vemuri BC, Cao Y, Chen L (1998) Fast collision detection algorithms with applications to particle flow. Comput Graph Forum 17(2):121–134

Part II

Engineering Applications of Computational Granular Mechanics

Chapter 8

DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

With the development of offshore oil and gas industry in the cold region, the ice load and ice failure modes during the interaction between sea ice and offshore platform structures in cold regions play important roles in the structural design and fatigue vibration analysis of offshore platforms (Yue and Bi 2000; Yan et al. 2009; Ji and Wang 2018). There are various types of offshore structures, such as jacket platforms, caisson structures, lighthouses, floating platforms and ship structures (Qu et al. 2006; Timco et al. 2006; Brown and Määttänen 2009). Compared with the field measurements and model experiments for ice loads, numerical simulations have the characteristics of low cost and short cycle. The DEM has obvious computational advantages in determining the ice load on offshore structures as the interaction between sea ice and offshore structures presents a failure process from continuum to discrete blocks (Paavilainen and Tuhkuri 2013; Hopkins 1997; Lau 2001). In addition, the introduction of GPU-based high performance computing methods into DEM numerical simulation can significantly improve the computational scale and computational efficiency (Ji et al. 2015). In this chapter, spherical particles, dilated discs and dilated polyhedral particles are used to simulate the sea ice (Ji et al. 2015; Sun and Shen 2012; Liu and Ji 2018). In the spherical particle calculations, the parallel bonding model is applied to bond the discrete elements of the spherical particle, while the bond strength is preset considering the influence of ice temperature and ice salinity. The DEM is used to calculate and analyze the fracture process of sea ice interacting with the offshore platform and ship hull, and determine the ice load characteristics under the influence of parameters such as ice thickness, ice speed and ice strength. Besides, the coarse-grained DEM is used to analyze the dynamic-thermal process of sea ice at the geophysical scale.

© Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_8

237

238

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

8.1 DEM for Sea Ice and Determination of Computational Parameters Considering the material properties and discrete characteristics of sea ice, the discrete element models of sea ice can be constructed by spherical particles with a bondingbreakage parallel bonding model, 3D dilated discs and dilated polyhedral particles. For the spherical particles, the sizes of the particles and the failure criteria among particles have significant influence on the ice failure mode and the ice load. In DEM simulations to analyze the mechanical properties of sea ice, it is difficult to obtain the simulation parameters directly from the mechanical tests Therefore, the macromechanical properties of sea ice is used to calibrate the meso-mechanical parameters in DEM simulations.

8.1.1 Discrete Element Construction of Sea Ice (1) Spherical particle of sea ice Under natural conditions, sea ice will form different types during its generation, dissipation and movement, which are mainly level ice, crushed ice, ice ridge and accumulated ice, as shown in Fig. 8.1. In the DEM simulations of sea ice, the geometric structure and mechanical properties of different types of sea ice should be constructed. The bonding model of spherical particles can be used to effectively construct the continuous and discrete distribution characteristics of sea ice, in order to describe the breakage phenomenon of sea ice in the dynamic process. The different types of sea ice constructed by spherical particles are shown in Fig. 8.2. In addition to the regular packing of spherical particles, the random arrangement of spherical particles can also be used in order to avoid the anisotropic mechanical behaviors of sea ice in DEM simulations. Figure 8.3 shows an ice ridge constructed by a regular packing and a random packing of spherical particles. Due to the influence of computational scale and efficiency, the DEM can only be performed in the local domain of an infinite ice area. Therefore, it is necessary to develop reasonable boundary conditions. For the level ice area, considering the conditions of displacement and force transfer between the simulation domain and the distant field, the corresponding horizontal and vertical stiffness are set for each particle on the boundary, as shown in Fig. 8.4. If the overall motion of sea ice is further considered and the corresponding horizontal velocity is set for boundary particles. The normal and tangential bonding strength between bonded spheres can transfer the force and moment between two spheres, as shown in Fig. 8.5. The normal and tangential stresses between bonding spheres can be obtained by the tension, torsion and bending of elastic beams under combined loads. When the normal or tangential stress reaches its corresponding strength, the bond breaks. The failure process and failure criterion have a significant influence on the simulation results, which is also an important issue of the current DEM research.

8.1 DEM for Sea Ice and Determination of Computational Parameters

(a) Level ice

(c) Ice ridge

239

(b) Crushed ice

(d) Rubble ice

Fig. 8.1 Different types of sea ice in the field

(2) The dilated polyhedral particles of sea ice In view of the irregular polygon geometry of sea ice in the broken ice filed (as shown in Fig. 8.6), the dilated polyhedral particles based on Minkowski Sum method can be used to calculate the interaction between broken ice and offshore structures. In order to construct broken ice elements with random distribution and irregular geometry, the 2D-Voronoi tessellation algorithm is used to randomly generate several polygons with different shapes and sizes in the calculation domain. For the generated polygons, the Minkowski Sum method is used to construct the corresponding dilated polyhedral sea ice particle, which is to construct smooth dilated polyhedral particle by superimposing the spherical particles on the polyhedral particles, as shown in Fig. 8.7. The size and concentration of ice can be determined in the 2D-Voronoi tessellation algorithm. For ice fragments, collision between ice fragments, buoyancy and dragging force of sea water should be taken into account. The quaternion method is used to calculate the dynamic forces of ice fragments at six degrees of freedom.

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(a) Level ice

(b) Crushed ice

(c) Ice ridge

(d) Rubble ice

Fig. 8.2 Discrete element constructions of different types of sea ice

(3) The 3D dilated disc particles of sea ice When the level ice is broken, the polyhedral ice blocks tend to be smooth in multiple collisions, forming pancake ice floes, as shown in Fig. 8.8. Hopkins initially established a dilated disc to effectively describe the broken ice geometry of the ice field (Hopkins and Shen 2001; Hopkins 2004). Subsequently, this method is used to calculate the ice load in the interaction between pancake ice and offshore platform piles and hull structures (Sun and Shen 2012; Liu et al. 2017; Ji et al. 2013). The 3D dilated disc of pancake ice consists of a circle disc and a dilated sphere with radius R and r, respectively, as shown in Fig. 8.9. Each point of the circular plane is dilated by a sphere with radius r, thus a 3D dilated disc particle with outer diameter and central ice thickness is constructed from a circular plane. In this way, the interaction between discs can be transformed into the contact detection and force calculation of spherical particles.

8.1 DEM for Sea Ice and Determination of Computational Parameters

241

(a) Regular packing of spherical particles

(b) Random packing of spherical particles Fig. 8.3 Ice ridges constructed by the regular and random packing of spherical particles

Fig. 8.4 Level ice constructed by bonded spherical particles

A

B

A

B L

Fig. 8.5 Schematic of beam element at the contact point

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Ship sailing in the broken ice filed Fig. 8.6 Ice floes with irregular polygon geometry

Fig. 8.7 Broken ice filed generated by the 2D-Voronoi tessellation algorithm

Fig. 8.8 Pancake ice field around an oil platform in the Bohai

8.1 DEM for Sea Ice and Determination of Computational Parameters

(a) Calculating parameters R and r of sea ice

243

(b) The 3D dilated discs of sea ice

Fig. 8.9 The dilated disk element of broken ice

8.1.2 Discrete Element Analysis of Sea Ice Compression and Flexural Strength Sea ice is a kind of natural composite material with complex mechanical properties. Its mechanical properties include compressive strength, tensile strength, flexural strength, shear strength, modulus of elasticity, Poisson’s ratio and fracture toughness. The mechanical properties of sea ice are affected by salinity and temperature, internal structure (ice crystal size and orientation, porosity) and external loading environment (loading rate, loading direction, test size). The uniaxial compressive strength and three-point flexural strength tests are used to calibrate the discrete element models. In the uniaxial compression test of sea ice, a rectangular ice specimen of size 70 mm × 70 mm × 175 mm was used and a vertical load was applied. The loading diagram is shown in Fig. 8.10a. The three-point bending test is used to obtain the flexural strength of sea ice, and the sea ice sample was cut as a 75 mm × 75 mm × 700 mm cuboid sample, the loading diagram is shown in Fig. 8.10b. The typical stress-time curves in uniaxial compression and three-point bending tests of sea ice are shown in Fig. 8.11, from which the uniaxial compressive strength and flexural strength of sea ice can be determined. The quality of initial sea ice sample has a great influence on the numerical simulation results. There are two ways of particle arrangement in the initial sample, i.e. regular arrangement and random arrangement. Among them, the method of generating samples by regular arrangement has the characteristics of simple structure, single particle size, irregular surface of sample, and anisotropy of sample. Common rules are Face-Centred Cubic (FCC) and Hexagonal Close Packed (HCP). The initial samples of sea ice are generated by HCP regular arrangement. The uniaxial compression test samples and the three-point bending test samples are shown in Fig. 8.12. In the uniaxial compression of sea ice, the upper loading plate is loaded downward at a fixed rate, and all boundary conditions are assumed to be rigid. In the bending of sea ice, rigid cylinders are used as the supporting points. The two bottom cylinders are fixed, and the top cylinder is loaded downward at a constant rate. The main computation parameters are listed in Table 8.1. The breaking process of sea ice is shown

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

P

a

H

a

(a) Uniaxial compression test

P h L0

b L

(b) Three-point bending test Fig. 8.10 Uniaxial compression and three-point bending test of sea ice

in Figs. 8.13 and 8.14. The calculated curves of stress-strain under compression and stress-deflection under bending of sea ice are shown in Fig. 8.15 and Fig. 8.16, respectively. Consequently, the uniaxial compressive strength σc = 5.64 MPa and flexural strength σf = 1.25 MPa of sea ice can be obtained. The uniaxial compression and flexural strength of sea ice are affected by factors such as sea ice brine volume (a function of temperature and salinity) and loading rate. Considering the influence of the volume of brine, the bond strength between sea ice granular elements can be expressed by the maximum bond strength σbmax and expressed as σbn = β(vb )σbmax

(8.1)

where, β(vb ) is the reduction coefficient of sea ice strength under the influence of brine volume vb ; The maximum bond strength σbmax between sea ice elements can be determined by sensitivity analysis of sea ice uniaxial compression strength. Considering that the compression and flexural strength of sea ice have a similar relationship with the volume of brine (Ji et al. 2011) √

β = e−4.29

vb

(8.2)

8.1 DEM for Sea Ice and Determination of Computational Parameters

245

6.0 5.0

σ (MPa)

4.0 3.0 2.0 1.0 0.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

t (s) (a) Uniaxial compression test 1.4

σf

1.2

σ (MPa)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

1.0

2.0

3.0

4.0

5.0

t (s) (b) Bending test Fig. 8.11 Typical stress-time curves in uniaxial compression and bending tests of sea ice

(b) The three-point bending test samples (a) The uniaxial compression test samples Fig. 8.12 DEM for uniaxial compression and three-point bending of sea ice

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Table 8.1 Main computational parameters in sea ice numerical simulations Definitions

Symbols

Units

Values

Ice density

ρ

kg/m3

920

Granular size

D

mm

10

Elastic modulus

Ec

GPa

1.0

Shear stiffness/Normal stiffness

ks /kn

–

0.5

Coefficient of friction

μp

–

0.2

Shear_damping/Normal_damping

ξ

–

0.36

Maximum bond strength

σbmax

MPa

1.0

Sea ice temperature

T

°C

−10

Sea ice salinity

S

‰

1

Number of elements

Np

–

3,300 (compression) 5,040 (bending)

Loading rate

vL

m/s

0.02

Fig. 8.13 DEM simulations of compression test

Fig. 8.14 DEM simulations of three-point bending test

where, vb can be set as a function of sea ice temperature and salinity as Frankenstein and Garner (1967),

8.1 DEM for Sea Ice and Determination of Computational Parameters

247

6.0

Fig. 8.15 Stress-strain curves of sea ice under uniaxial compression simulated by the DEM

5.0

B

σ (MPa)

4.0 3.0

A

2.0 1.0 0.0

0.0

0.1

0.2

0.3

0.4

ε (%)

Fig. 8.16 Three-point bending stress deflection curve of sea ice simulated by the DEM

1.4 1.2

σ (MPa)

1.0 0.8 0.6 0.4 0.2 0.0

0.0

0.5

1.0

1.5

2.0

u (mm)

 49.185 vb = S 0.532 + (−22.9 ◦ C ≤ T ≤ −0.5 ◦ C) |T | 

(8.3)

where, T is the ice temperature (°C), S is the ice salinity (%). In order to analyze the influence of sea ice temperature and salinity on uniaxial compressive strength and three-point flexural strength, the maximum bond strength σbmax between sea ice granular particles is set at 0.7 MPa. The bond strength σbn under different brine volumes is calculated by determining the strength reduction coefficients β at different temperatures and salinities. By adjusting temperature and salinity to make brine volume vary uniformly in the domain [0.0, 0.4], the relationship between compressive strength, flexural strength and brine volume is calculated as shown in Fig. 8.17 and 8.18. Fitting curves between compressive strength and flexural strength and brine volume are given. It can be found that σc and σf have a strong negative exponential relationship with (vb )0.5 , which is consistent with the exponential relationship which is measured in the field.

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull 7.0

5.0

5.0

4.0 3.0 2.0 1.0 0.0 0.0

0.5

σc = 6.26e-4.13 (υb)

6.0

σc (MPa)

σc (MPa)

7.0

= 6.12 = 5.50 = 5.03

6.0

4.0 3.0 2.0 1.0

0.1

0.2

0.3

0.0 0.0

0.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(vb)0.5

(vb)0.5

(a) DEM simulation results

(b) Field measurement results

Fig. 8.17 Effect of brine volume on compressive strength of sea ice 2.5

2.5

= = 1.69

0.5

= 1.37

1.5 1.0 0.5 0.0 0.0

σf = 2.41e-4.29 (υb)

2.0

σf (MPa)

σf (MPa)

2.0

1.5 1.0 0.5

0.1

0.2

0.3

0.4

0.0

0.0

0.1

(a) DEM simulation results

0.2

0.3

0.4

0.5

0.6

0.7

(υb)0.5

(vb)0.5

(b) Field measurement results

Fig. 8.18 Effect of brine volume on flexural strength of sea ice

8.1.3 Main Computation Parameters in Sea Ice DEM Simulations The purpose of the calibration of DEM computation parameters is to establish the relationship between model parameters and macroscopic mechanical properties by using mesoscopic parameters. Through the comparisons between the simulation results and the macroscopic test results, the accuracy of the adopted model parameters in simulating the macroscopic mechanical properties is determined, and the model parameters are finally determined. The objective of simulating the mechanical properties of sea ice by using the meso-mechanical model is to make the macro-mechanical properties of the numerical samples approach the macro-mechanical properties of the real sea ice by constantly changing the meso-mechanical parameters of the particles in the samples. Therefore, the process of determining the meso-mechanical parameters is a process of parameter optimization, i.e., constant modifications to obtain the final parameters of simulations. The macroscopic mechanical properties of sea ice mainly include density ρi , Poisson’s ratio ν, elastic modulus E, compressive strength σc , bending strength σf , and loading rate vL .

8.1 DEM for Sea Ice and Determination of Computational Parameters

249

The meso-mechanical parameters of the model include granular size D, granular density ρ, interparticle friction coefficient μ, normal contact stiffness kn and tangential contact stiffness ks between particles, normal bond strength σbn and tangential bond strength σbs in the parallel bonding model, etc. Through the dimensional analysis method, the corresponding relationship between the micromechanical parameters and the macroscopic mechanical properties of sea ice can be obtained. When the discrete element parallel bonding model is used to simulate the mechanical properties of sea ice, the relationship between the response of particles and the meso-mechanical parameters in the numerical samples can be expressed as follows (Huang 1999)   RS = f H, a, D, ρ, μ, kn , ks , σbn , σbs , vL

(8.4)

The height H and width a of the sample are consistent with those of the sea ice compression sample, and the element density ρ can be obtained from the material density ρi of the sea ice. The above equation becomes   RS = f D, μ, kn , ks , σbn , σbs , vL

(8.5)

The parameters in this equation have three independent dimensions, m, s, and kg. The responses of the sample can be represented by five independent dimensionless parameters as  Rs = f

ks σ s D vL , μ, , bn , √ a kn σb kn /ρ

 (8.6)

When the simulations √ of the mechanical properties of sea ice are carried out under quasi-static loads (vL / kn /ρ  1) and sufficient degrees of freedom (D/a  1), the elastic modulus, Poisson’s ratio, compressive strength and flexural strength of the sea ice can be determined based on the friction coefficient, the ratio between normal and the tangential stiffness, the ratio between normal and tangential bond strength.

8.1.4 Failure Criteria Between Bonded Particles When using the bonding model to simulate the breaking process of sea ice from crack initiation, expansion, penetration to final failure, it is necessary to determine the reasonable failure criterion to detect the crack initiation of the pores inside the sea ice first. There are many forms of failure criteria, among which the most commonly used one is the maximum principal tensile stress criterion. When the maximum tensile stress at a certain point inside the sea ice exceeds the tensile strength σbn of the material, the crack inside the material begins to sprout, and the sea ice will further damage. Here, the tension-shear zoning fracture criterion is adopted, i.e., the normal tensile strength and tangential shear strength are set as the fracture strength between

250

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.19 Tension-shear fracture criteria for parallel bonding particles

the normal and tangential direction of the particles, respectively (Hentz 2004; Wang and Tonon 2010). The schematic diagram of the tension-shear fracture criterion is given in Fig. 8.19. The fracture function is defined as F(σn ) = σn − σbn

(8.7)

F(τ ) = τ − σbs

(8.8)

where, σn is the normal stress experienced by the interparticle bonding part; τ is tangential stress; σbn and σbs are tensile and shear strengths. When the stress combination of a point makes the fracture function F < 0, the point is in the linear elastic stage; When the stress state of a point makes F ≥ 0, the bond breaks. It can be concluded that the fracture is mainly caused by two kinds of fracture modes: tension fracture and shear fracture. It is assumed that at a certain time t, the stress state of the bonding part is in the linear elastic stage, located in region ➀ of the figure, the normal stress and tangential stress of the bonding part are calculated according to the linear elastic contact model. At time t + t, if the calculated stress makes the stress state of a point outside the cracking surface, that is, F(σn ) ≥ 0 or F(τ ) ≥ 0 or both are greater than 0, the bond begins to fracture. When the stress state at one point is in region ➁ of the figure, tensile fracture occurs without shear fracture. The normal component of the fracture stress is the tensile strength σbn . When the stress state of a point is located in region ➂, both tensile fracture and shear fracture occur. The stress at which the fracture occurs is the intersection point of two fracture functions. The normal and tangential components of the cracking stress are σ0 = σbn

(8.9)

τ0 = σbs

(8.10)

when the stress state of a point is located in region ➃, shear fracture occurs, but no tensile fracture occurs. The shear component of the fracture stress is shear strength σbs .

8.1 DEM for Sea Ice and Determination of Computational Parameters

251

The stress (σ0 , τ0 ) at the time of the fracture of the bonded particles under the condition of tensile shear stress can be established; When the bond is in the compressive shear stress state (σn < 0), the yielding function of the compression-shear fracture is defined as F(σn , τ ) = |τ | + σn tan φ − σbs

(8.11)

where, φ is the friction angle. When the stress state is located in region ➄, F(σn , τ ) < 0, it is in the linear elastic state of compression-shear; When the stress state is located in region ➅, F(σn , τ ) ≥ 0, it is the compression-shear fracture. After the fracture criterion of the bonding model is developed, the sea ice breaking process can be simulated, in addition to the uniaxial compressive strength and threepoint bending strength of sea ice. It is found that the normal tensile bond strength, tangential bond strength and interparticle friction angle will affect the macroscopic strength of the sample. When the interparticle friction coefficient μ = tan φ = 0.2 is used, the values of normal bond strength and tangential bond strength vary in the range of 0.1–2.0 MPa, and the uniaxial compressive strength, flexural strength and their ratio R under different bond strength can be obtained. The contour diagram is shown in Fig. 8.20. By comparing with the results of sea ice physical and mechanical tests, a reasonable coefficient of friction between the bonded particles can be determined.

8.1.5 DEM Simulations of Sea Ice Strength Influenced by Particle Size In the discrete element models, particle size is an important parameter. In the ice discrete element simulations, the particle size has an important influence on the simulation results, and the sample size also has impacts on the simulation results. Here, the size effect of the macroscopic mechanical properties of ice is studied by changing the particle size and sample size, respectively. Six groups of sea ice samples with different sizes are set up for uniaxial compression tests and four groups for threepoint bending tests, and the particle size are changed at the same time. The sample size is listed in Table 8.2. In order to study the effect of particle size and sample size on the mechanical properties of sea ice, a dimensionless parameter D = D/L is introduced here, where D is the particle size, L is the sample size. L is a in the uniaxial compression test, while L is h or b in bending text. With compressive strength and bending strength as ordinates and D as abscissa, the relative strength-particle size relationship is shown in Fig. 8.21. It can be seen that for different sizes of sea ice samples, as long as the ratio of particle size D to sample size L is equal, the compressive strength and flexural strength of the sample are the same, and can be expressed as:

252

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(a) Uniaxial compression strength

(b) Flexural strength

(c) Ration between compression and flexural strength Fig. 8.20 Contour of uniaxial compressive strength, flexural strength and their ratio of ice when μ = 0.2 Table 8.2 Size of ice sample for uniaxial compression and three-point bending tests

Uniaxial compression test (a × a × H) (mm3 ) 70 × 70 × 175; 100 × 100 × 250; 150 × 150 × 375; 200 × 200 × 500; 1 000 × 1000 × 2500; 2000 × 2 000 × 5000 Three-point bending test (L 0 × h × b) (mm3 ) 250 × 37.5 × 37.5; 500 × 75 × 75; 750 × 112.5 × 112.5; 1000 × 150 × 150

8.1 DEM for Sea Ice and Determination of Computational Parameters

253

7.0 6.0

σc (MPa)

5.0

Sample size (cm) 7×7×17.5 10×10×25 15×15×37.5 20×20×50 100×100×250 200×200×500

4.0 3.0 2.0 1.0 0.0

0

σc = − 11.36(D/L) + 5.96

0.03

0.06

0.09

0.12

0.15

D/L

(a) Uniaxial compression strength 2.5

σf (MPa)

2.0

σf = − 3.47(D/L) + 1.94

1.5 1.0 0.5 0.0 0.00

Sample size(cm) 25×3.75×3.75 50×7.5×7.5 75×11.25×11.25 100×15×15 0.04

0.08

0.12

0.16

0.20

0.24

D/L

(b) Flexural strength Fig. 8.21 Relation between mechanical properties of sea ice and relative particle size D at different sample sizes

σc = −11.36(D/L) + 5.96

(8.12)

σf = −3.47(D/L) + 1.94

(8.13)

The intersection point of the two fitting straight lines and the ordinate is the uniaxial compressive strength σc = 5.96 MPa and the flexural strength σf = 1.94 MPa of the sample, and the corresponding interparticle bonding strength σb = 0.73 MPa.

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

8.2 DEM Analysis for the Interaction Between Sea Ice and Fixed Offshore Platform The fixed offshore platform structure is widely used for marine engineering in the cold regions. The base of fixed offshore structure is fixed on the seabed, generally adopting pile foundation or gravity foundation. The natural frequency of fixed structures is normally higher than that of ice load, while the vibration period is close to the range of the ice load period. In many cases, strong ice-induced vibration of the structure under alternating ice load (Yan et al. 2012). The DEM analysis and experimental verification of the ice load on the flexible fixed jacket platform structure is introduced here. Due to the different structural forms of the legs, the jacket platform can be divided into the cylindrical leg structure and conical structure. Since sea ice is different from the fracture forms of these two types of structures, its ice load characteristics are also different. Based on this, the DEM is used to calculate and analyze the interaction process between sea ice and offshore platform structures.

8.2.1 DEM Analysis of Ice Load on Cylindrical Leg Structure When sea ice moves rapidly and interacts with cylindrical structures, the sea ice will show brittle crushing and the structure vibrates under random excitation. During brittle extrusion on the structure, sea ice breaks into powdery particles, as shown in Fig. 8.22.

Fig. 8.22 The interaction process of sea ice and cylindrical leg structure

8.2 DEM Analysis for the Interaction Between Sea Ice … Table 8.3 Main computational parameters for DEM simulations of interaction between sea ice and cylindrical leg structure

Definition

255

Symbols

Values

Initical ice region

L×W ×H

20 m × 10 m × 0.18 m

Particle size

D

4.48 cm

Normal stiffness

kn

1.72 × 107 N/m

Shear stiffness

ks

1.72 × 106 N/m

Particles-particle friction

μp

0.2

Normal bond strength

σbn

0.5 MPa

Shear bond strength

σbs

0.5 MPa

Ice velocity

Vi

0.2 m/s

Drag coefficient

Cd

0.05

Number of particles

Np

608,790

In order to simulate the crushing process of sea ice interacting with the cylindrical platform structure, level ice is constructed by parallel bonding model of spherical particles. According to the physical and mechanical properties of sea ice, the bond strength between two bonding particles can be set as a function of sea ice temperature, salinity and loading rate. When the shear or tensile stress between bonding particles is higher than the preset shear or tensile strength, the bond will fail (Ji et al. 2013). The main calculation parameters are listed in Table 8.3. The interaction process between level ice and cylindrical leg structure is simulated by the DEM. The animations at six different times are shown in Fig. 8.23. During the interaction between sea ice and the vertical leg structure, sea ice is mainly crushed. The main morphological feature is that the sea ice is powdery after crushing, and the DEM results can reproduce this characteristic. The ice load time history in the x direction is shown in Fig. 8.24a. The time history curve shows strong randomness. The circle mark in the figure is the peak of ice load. The maximum peak value is 56.2 kN and the minimum is 28.2 kN, with an average of 40.9 kN. The statistical results of the peak ice force are shown in Fig. 8.24b. It can be seen that the peak ice load shows a normal distribution with a mean of 26.4 kN and a standard deviation of 8.3 kN. The vertical leg structure and ice breaking process are shown in Fig. 8.25, and the typical ice breaking time history of brittle extrusion measured by the pressure box is shown in Fig. 8.26 (Yan et al. 2012). The ice thickness is 0.18 m and ice speed is 0.2 m/s in this case. The maximum and peak mean values of ice load simulated by the DEM are 56.2 kN and 40.9 kN, respectively, which are close to the corresponding 57.8 and 45.2 kN measured in situ.

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(a) t = 0s

(d) t = 27s

(b) t = 9s

(e) t = 36s

(c) t = 18s

(f) t = 45s

Fig. 8.23 Simulation of the interaction between sea ice and vertical structures by the DEM

8.2.2 DEM Analysis of Ice Load on Conical Offshore Platform Structure Firstly, a conical structural element is used to construct a single conical structure on the JZ20-2 MUQ platform. The DEM of the cone part is shown in Fig. 8.27. The diameter of the leg is 1.708 m before the cone is added, the height of the cone is 2.5 m, the maximum diameter of the cone is 4.0 m, and the diameter at the water line is 2.85 m. JZ20-2 MUQ platform has four pile legs, but only one of them is equipped with ice pressure box. Therefore, only one pile leg is simulated to interact with sea ice. The pile leg is regarded as a three-degree-of-freedom system, i.e., the leg can only move in horizontal and rotate around the vertical direction. The interaction between the sea ice and the inclined structure is mainly affected by the bending failure of the ice sheet in front of the inclined structure. The distance between the structure model and the ice sheet lateral limit is more than four times larger than the diameter of the structure model. It can be concluded that the simulated ice load is not affected by the boundary. Since the diameter of the cone at the waterline is 2.85 m, the width and length of the ice sheet are set at 18 and 40 m. The meso-parameters of the DEM used in the calculation are listed in Table 8.4. The numerical simulation of the interaction process between the level ice and the cone structure is shown in Fig. 8.28, and the total simulation time is 90 s. The figure shows the interactions at three different moments. During the interaction between

8.2 DEM Analysis for the Interaction Between Sea Ice …

257

60 50

Fx (kN)

40 30 20 10 0

0

5

10

15

20

25

30

35

40

45

t (s)

(a) DEM simulation of dynamic ice load on the vertical leg structure

(b) Normal probability distribution of ice load peaks Fig. 8.24 The statistical characteristics of the dynamic ice load and its peak value of the sea ice on the vertical leg structure simulated by the DEM

sea ice and cone structure, sea ice is mainly caused by bending failure. The size of the broken ice is about 3–7 times of the ice thickness in the field measurements, which can be reproduced in the DEM simulation results. After the ice cover is broken, the ice force will be unloaded to zero until the subsequent ice cover interacts with the cone structure again. The simulated ice load time history curve of cone in x direction is shown in Fig. 8.29. It can be seen that the peak of ice load presents a strong periodicity. The circle in the figure is the peak of ice load in each cycle. The maximum value of ice load is 109 kN, the minimum value is 70.6 kN, and the average value is 91.2 kN. According to the statistics of the peak ice load, the peak ice intensity is mainly between 75 and 110 kN, with the most near 95 kN, and the time interval of force peak is mainly concentrated between 1.5 and 4.5 s. The ice speed here is 0.43 m/s. It can be calculated that the size of ice floes is about 1.1 m, and the ice thickness

258

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(b) Vertical pile leg structure

(a) The JZ9-3MDP-1vertical jacket platform in the Bohai Sea, China

Fig. 8.25 Interaction process between sea ice and vertical leg structure 60 50

Fx (kN)

40 30 20 10 0

0

5

10

15

20

25

30

35

40

45

t (s) Fig. 8.26 The ice load history curve on the JZ9-3MDP-1 platform by the pressure box 1.708m

2.85m 4.0m

2.5m

Fig. 8.27 Conical platform of JZ20-2 MUQ platform in Liaodong Bay and its pile leg in DEM simulations

8.2 DEM Analysis for the Interaction Between Sea Ice … Table 8.4 Main calculation parameters in DEM simulations of interaction between sea ice and conical structure

259

Definition

Symbol

Value

Particle diameter

D

8.735 cm

Normal stiffness

kn

1.72 × 107 N/m

Shear stiffness

ks

1.72 × 106 N/m

Friction coefficient

μp

0.2

Normal bond strength

σbn

0.5 MPa

Shear bond strength

σbs

0.5 MPa

Number of particles

Np

326,304

(a) t = 13.5s

(b) t = 31.5s

(c) t = 90.0s

Fig. 8.28 Interaction process between sea ice and conical structure simulated by the DEM 120 100

Fx (kN)

80 60 40 20 0

0

10

20

30

40

t (s)

50

60

70

80

90

Fig. 8.29 Dynamic ice load on conical structures simulated by the DEM

is 0.23 m. It can be concluded that the size of ice floes is about 5 times of the ice thickness, which is consistent with the field observation results. JZ20-2 MUQ jacket platform in Liaodong Bay, China, is a four-legged jacket platform. Similar ice-breaking cones are installed on each leg. The ice-breaking cones installed and the ice-breaking process are shown in Fig. 8.30. When sea ice

260

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.30 JZ20-2 MUQ conical pile structure in Liaodong Bay, China

acts on a narrow cone structure, radial and circumferential cracks will appear from the boundary of the cone to the inside of the ice sheet. Finally, the ice sheet will bend and form many wedge-shaped ice sheets. The time history of ice load on the cone, measured by the pressure box, is shown in Fig. 8.31. It can be seen that the ice load is completely unloaded after each bending and crushing, and it will decrease to zero. At the same time, the ice load acting on the conical structure has obvious periodicity, and the period is about 2 s. In addition, the failure modes of sea ice obtained by DEM simulations and field measurements are both bending and fragmentation, while the ice loads have periodic changes. The maximum and peak mean values of ice load 140 120

Fx (kN)

100 80 60 40 20 0

0

10

20

30

40

50

60

70

80

90

t (s) Fig. 8.31 Ice load history curve measured by the cone pressure box of JZ20-2 MUQ platform

800

800

600

600

PSD (kN2/Hz)

PSD (kN2/Hz)

8.2 DEM Analysis for the Interaction Between Sea Ice …

400 200 0

400 200 0

0

2

4

6

8

10

261

0

2

4

6

Frequency (Hz)

Frequency (Hz)

(a) Field results

(b) DEM results

8

10

Fig. 8.32 Comparison of power spectrum density curves between field results and DEM simulation of ice load

simulated by DEM are 109.0 kN and 91.2 kN, respectively, which are close to 127.8 and 86.1 kN in the field measurements. The dynamic ice load acting on the conical structure belongs to random load, and the power spectrum density function can be used to analyze the characteristics of the random process. The ice load power spectrum density curves in field measurements and DEM simulations are shown in Fig. 8.32. It can be seen that both the spectrum density curve peak and the peak frequency are close. Therefore, the DEM can effectively simulate the interaction process between sea ice and conical structure and determine the ice load.

8.2.3 Shadowing Effect of Multi-leg Conical Jacket Platform Structure For the shadowing effect of multi-leg platform structure, Kato (1990) and Kato et al. (1994) first determined the shadowing coefficients of vertical legs and conical platforms under different ice directions through model tests (Kato 1990; Kato et al. 1994). The shadowing effect of ice load is an important issue in the field monitoring and model tests of sea ice in the Bohai Sea (Timco et al. 1992; Yue et al. 2007). In this section, the shadowing effect of different ice direction is analyzed for the interaction process between sea ice and multi-pile cone offshore platform structure. (1) Ice load of multiple pile leg structures with different ice direction During the interaction between sea ice and multi-legged offshore platform, the ice load on each leg is greatly different under different ice moving direction. The ice load will show a significant ice-shadowing effect of multi-pile structure. Based on the Bohai JZ20-2 MUQ platform, the DEM is employed to analyze the ice-shadowing effect between the legs. The JZ20-2 MUQ platform in the DEM simulation are shown in Fig. 8.33.

262

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.33 Bohai JZ20-2 MUQ four-cone platform and discrete element model

The distance between the legs of the platform is about 20 m. In order to avoid the influence of the boundary effect on the calculation results, the size of the ice cover is set to 50 m × 40 m, and the ice thickness is 0.23 m. The DEM parameters of the sea ice are consistent with the previous ones listed in Table 8.3. In order to analyze the interaction between sea ice and four-legged conical structure under different ice directions, the ice directions are set to 0°, 5°, 10°, 15°, 20°, 25°, 30°, 35°, 40° and 45°, respectively. The interaction processes between sea ice and four-legged conical structure with ice angles of 0°, 15°, 30°, and 45° are shown in Fig. 8.34. In the process of interaction between the front cone and sea ice, sea ice will break up and form a waterway, and the ice force of the back pile leg will be sheltered. The channel formed by the interaction of sea ice and cone will be quite different under different ice directions. When the ice direction θ = 0◦ (Fig. 8.34 a), the rear cone is located in the channel formed by the front cone. Only the crushed ice act on the rear cone so that the ice force is smaller. When the ice direction θ = 15◦ (Fig. 8.34b), part of the rear leg is in the water channel and is affected by the ice edge, and the ice load is also significantly reduced. When the ice direction θ = 30◦ (Fig. 8.34c), the two sides of the ice sheet contacting with pile leg No. 2 are free edges, which makes the period of ice load longer when the sea ice acts with the pile leg. When the ice direction θ = 45◦ (Fig. 8.34d), leg No. 2 is just inside the water channel of leg No. 4. The time history of ice load on each leg of the platform when the ice direction is 45° is shown in Fig. 8.35. It can be found that the ice load of each leg shows a very random pulse load, which is consistent with the field monitoring of sea ice and the model test results (Tian and Huang 2013). The average value of the corresponding ice load peak is shown in Fig. 8.36. It can be seen that the ice load on No. 1 and No. 4 remains stable in different ice directions as the legs of No. 1 and No. 4 are not affected by the shadowing effect. When the ice direction is 45°, leg No. 2 is completely covered by leg No. 4. Therefore, the average peak value of the ice load increases first and then decreases with the increase of the ice direction angle. It can be seen that the changing trend of ice load on each leg is determined by the shadowing effect of each leg. When the number of legs and the spatial layout of the platform changes, the shadowing effect of each leg changes with the ice direction. Changes

8.2 DEM Analysis for the Interaction Between Sea Ice …

263

Fig. 8.34 Interaction process between different ice direction and four-leg conical structure

occur and further affect the distribution characteristics of the overall ice force of each leg and platform. (2) Ice shadowing effect of multi-pile cone structure The attenuation expression of ice load of single pile in multi-pile structure established by Kato (1990), Kato et al. (1994) is as follows: F = ks F0

(8.14)

where, F is the ice load of single pile in multi-pile structure; F0 is the ice load on the independent pile; ks is the attenuation coefficient of single pile in multi-pile structure. The dimensionless distance s ∗ = s/D is defined here, where s is the distance from the pile to the free boundary and D is the diameter of the pile. The attenuation

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

120

60

100

50

80

40

Fx (kN)

Fx (kN)

264

60 40 20

20 10

0

20

40

60

80

0

100

0

20

40

60

t (s)

t (s)

(a) Leg No.1

(b) Leg No.2

120

100

100

80

80

Fx (kN)

Fx (kN)

0

30

60 40

80

100

80

100

60 40 20

20 0

0 0

20

40

60

80

0

100

20

40

60

t (s)

t (s)

(c) Leg No.3

(d) Leg No.4

Fig. 8.35 Dynamic ice load on each leg when the ice direction θ = 45◦

80

Fig. 8.36 Average ice peak load on each leg under different ice direction

1#

70

2#

3#

4#

Fx (kN)

60 50 40 30 20 10 0

0

5

10

15

20

25

30

35

40

45

θ (º)

coefficient of the pile at different free boundary distances can be obtained by the empirical formula. The attenuation coefficient can be divided into two segments, as  ks =

if s ∗ < 1.0 s∗, ∗ min(0.89 + 0.02s , 1.0), if s ∗ ≥ 1.0

(8.15)

8.2 DEM Analysis for the Interaction Between Sea Ice …

265

In calculating the total ice load of multi-pile platform structure, the factors such as non-simultaneous destruction of sea ice, shadowing effect and pile-up of sea ice should be taken into account simultaneously. The total ice load of multi-pile platform structure is as follows (ISO 19906 2010) FT = K s K n K j FA

(8.16)

where, FT is the total ice load of the multi-pile structure; FA is the total ice load received by multiple independent piles; K s is the shadowing coefficient; K n is a factor considering non-simultaneous destruction, generally chosen as 0.9; K j is a factor considering ice pile-up, for level ice K j = 1.0. The shadowing coefficient K s is related to the layout of pile legs, ice direction, pile diameter and other factors. For the four-pile conical platform structure, the following is an analysis of the shadowing effect of each leg. Figure 8.37 is a schematic diagram of a waterway formed by cutting pile No. 1 and No. 4 with different ice directions. The waterway forms a free boundary between pile No. 2 and No. 3. When 0 ≤ θ ≤ 45◦ , the geometric parameters in the graph have the following relations: 

√ L 31 (θ ) = L 2 + B 2 sin(45◦ − θ ) L 21 (θ ) = L 43 (θ ) = L × sin θ

(8.17)

The distance between piles and legs is L = B = 20 m. The width of the channel formed by the cutting of pile legs on sea ice is calculated by the DEM, where w = D/4 is taken. Thus, the dimensionless distances between piles No. 2 and No. 3 from the free boundary are:

Fig. 8.37 Calculation of the distance between the leg and the free boundary of the channel

266

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.38 Ice load attenuation coefficient of pile No. 2 and No. 3 under the shadowing influence in multi-leg structure

⎧ ∗ ⎪ ⎨ s21 = s21 /D = (L 21 (θ ) − w)/D ∗ = s31 /D = (L 31 (θ ) − w)/D s31 ⎪ ⎩ ∗ s43 = s43 /D = (L 43 (θ ) − w)/D

(8.18)

where, s21 is the distance from the waterway boundary formed by pile No. 2 to pile No. 1; s31 and s43 are the distance from pile No. 3 to the waterway boundary formed by pile No. 1 and No. 4 pile, respectively. The attenuation coefficients of pile No. 2 and No. 3 in Fig. 8.37 are calculated by using Eqs. (8.15), (8.17), and (8.18). The attenuation coefficient of the two legs obtained is shown in Fig. 8.38. The comparison between the DEM results and the empirical formula of Eq. (8.15) is given. It can be seen that the ice load attenuation coefficient has a good consistency in trend. However, the result obtained by Eq. (8.15) is higher than the discrete element result. This is mainly due to the calculation of the ice load attenuation using the Eq. (8.15) does not consider the early damage caused by the action of the ice cover and the front leg. Because pile No. 1 and No. 4 have no ice-shadowing effect, ks1 = ks4 = 1.0. Pile ∗ can be No. 2 is mainly affected by the water channel formed by pile No. 1, and s21 determined. For pile No. 3, it is necessary to consider the water channel formed by  ∗ ∗  ∗  ∗ , s43 = ks s31 · ks s41 . If the pile No. 1 and No. 4, which can be written as: ks3 s31 ice load attenuation caused by the shadowing effect of the four legs is considered at the same time, the shadowing coefficient of the total ice load is:  ∗  ∗ ∗   + k3 s31 , s43 /4 K s = ks1 + ks4 + ks2 s21

(8.19)

By substituting the attenuation coefficients of each leg under different ice directions into Eq. (8.19), the total ice shadowing coefficients of the four-pile conical platform structure can be obtained, as shown in Fig. 8.39. The figure also gives the total ice shadowing coefficient calculated by the DEM. It can be found that the ISO results are in good agreement with the DEM results. However, the pre-damage caused by the interaction between sea ice and front leg is not considered in ISO, which leads

8.2 DEM Analysis for the Interaction Between Sea Ice …

267

Fig. 8.39 Shadowing coefficient of total ice load for four-pile cone platform structure

to that the total ice load shadowing coefficient is slightly higher than the result by the DEM.

8.2.4 DEM Analysis of Ice Load on Jack-Up Offshore Platform Structure The jack-up offshore platform structure is shown in Fig. 8.40a, and truss structures for platform deck lifting are installed on each pile leg. In order to simplify the calculation of the interaction between sea ice and jack-up platform, the influence of truss structure on sea ice damage is not considered. The calculation model of single leg of jack-up platform is shown in Fig. 8.40b. During the interaction process between sea ice and offshore platform structure, the ice breaks up continuously, while the platform structure vibrates under the ice load. In the process of interaction between sea ice and jack-up platform, brittle extrusion is the main failure mode of

Fig. 8.40 DEM model of jack-up offshore platform and pile structure in Bohai

268

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Table 8.5 Main calculation parameters of DEM simulation of interaction between sea ice and jack-up platform

Definition

Symbol

Date

Water velocity

V

1.0 m/s

Ice thickness

hi

0.5 m

Ice cover area

L×W

100 m × 90 m

Particle diameter

D

0.275 m

Number of particles

Np

332,000

sea ice, and random dynamic ice load is generated on the platform structure. In this section, the DEM numerical analysis of the random failure process of sea ice is carried out. The simulation parameters of sea ice are given in Table 8.3, and the other parameters are listed in Table 8.5. The interaction process between sea ice and platform structure is simulated by the DEM as shown in Fig. 8.41. The figure shows the process of sea ice breaking during the interaction between the pile legs and sea ice, and the pile-up of sea ice in the pile truss. The ice load time histories of the front leg in the x, y and z directions are shown in Fig. 8.42. It can be found that the sea ice load is mainly concentrated in the x direction which is the direction of sea ice movement. The ice loads F y and F z in the y and z directions are mainly caused by the action of oblique rods and the friction between the surface of piles and columns. Although the whole structure and single leg structure of jack-up platform are symmetrical, due to the randomness of

(a) t = 0.0s

(c) t = 59.4s

(b) t = 15.0s

(d) t = 74.2s

Fig. 8.41 DEM simulation of sea ice interacting with jack-up platform

1600 1400 1200 1000 800 600 400 200 0

(a) Leg_1

269

(b) Leg_1

1200 800

Fy (kN)

Fx (kN)

8.2 DEM Analysis for the Interaction Between Sea Ice …

400 0 -400 -800

0

15

30

t (s)

45

60

75

-1200

0

15

30

45

60

75

t (s)

Fz (kN)

(c) Leg_1

200 150 100 50 0 -50 -100 -150 -200 0

15

30

45

60

75

t (s)

Fig. 8.42 DEM simulation of ice load time history on front pile legs

sea ice breakage, F y and F z on the three legs show alternating positive and negative fluctuations.

8.3 DEM Analysis for the Interaction Between Sea Ice and Floating Platform and Ship Hull The floating platform structure usually consists of an upper floating body and a lower mooring line, and the floating body is submerged in water and supported by buoyancy to its own weight. Floating platforms are generally positioned by mooring systems, usually consisting of more than eight radially arranged anchors, which are connected with the platform by catenary-shaped anchors (Zhou et al. 2012).

8.3.1 Analysis of Ice Load on Floating Offshore Platforms The Kulluk offshore platform is an anchor floating drilling platform. It served in Beaufort Sea of Canada from the mid-1970s to the early 1990s. The water depth of the offshore area is 20–80 m. Kulluk is in round shape and has a circular design that resists waves or sea ice in any direction, with a deck diameter of 81 m and a waterline diameter of 70 m, as shown in Fig. 8.43. The Kulluk platform is inverted cone-shaped hull structure which can cause the sea ice to bend and break when it

270

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.43 The Kulluk offshore platform

collides with sea ice, reducing the ice load of the hull. The inclination angle of the inclination is 31.4°. The structure of the bottom opening outwards makes the broken ice block bypass. The hull does not gather at the bottom of the hull. Kulluk is fixed to the seabed by 12 cables with a diameter of 0.09 m, and the ice load is obtained by measuring the cable. The discrete element model will be used to simulate the interaction of sea ice with the Kulluk offshore platform. The Kulluk structure in the discrete element model is a mosaic of triangular elements. The ice floes around Kulluk are dominated by annual ice floes. With the help of an icebreaker, larger sea ice breaks into pieces that are 10–50 m in size. In the DEM simulations, the average area of crushed ice is 400 m2 and the thickness of ice floes is 1.5 m. The original shape of the ice floes is obtained by the 2DVoronoi tessellation algorithm, and four different concentrations, 20, 50, 80, 90%, are simulated by the DEM. The initial arrangement is shown in Fig. 8.44. The ice floes move to the right at a speed of 0.5 m/s under the push of the fixed boundary on the left side. The main calculation parameters in the DEM simulations are listed in Table 8.6. The DEM analysis of the interaction process between the four different density ice floes and the Kulluk floating platform is carried out. The interaction process between the ice floes and the Kulluk with DEM simulation intensity of 80% is shown in Fig. 8.45. The total simulation time is 600 s. The figure shows the process at three different moments. Under the action of current dragging, the frontal ice first

8.3 DEM Analysis for the Interaction Between Sea Ice …

(a) C = 20%

271

(b) C = 50%

(c) C = 80%

Fig. 8.44 Initial arrangement of ice floes at different concentrations

Table 8.6 Main calculation parameters in Kulluk ice load DEM simulations

(a) t = 20 s

Definition

Symbols

Values

Sea ice density

ρi

920 kg/m3

Sea water density

ρw

1035 kg/m3

Ice thickness

hi

1.5 m

Particle-particle friction

μp

0.1

Wall-particle friction

μw

0.3

Normal bond strength

σbn

0.65 MPa

Tangential bond strength

σbn

0.65 MPa

Ice velocity

Vi

0.5 m/s

Seawater drag coefficient

Cd

0.05

(b) t = 240 s

(c) t = 420 s

Fig. 8.45 The process of interaction between crushed ice and Kulluk simulated by the DEM

contacts Kulluk, and the frontal ice at the front of Kulluk will be blocked and slowed down and accumulated in the front of the platform. The frozen ice on both sides will continue to move around the platform, thus creating a channel equal to the width of the platform behind the platform. As the subsequent ice floes continue to push

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull 600

200

500

150 100 50 0

Fglobal (kN)

Fglobal (kN)

250

0

100

200

300

400

500

400 300 200 100 0

600

100

200

300

400

500

600

400

500

600

t (s)

(a) 20%

(b) 50%

2500

3000

2000

2500

1500 1000 500 0

0

t (s)

Fglobal (kN)

Fglobal (kN)

272

2000 1500 1000 500

0

100

200

300

400

500

600

0

0

100

200

300

t (s)

t (s)

(c) 80%

(d) 90%

Fig. 8.46 Ice load history of the Kulluk floating offshore platform at different concentrations

forward, the ice floes will re-aggregate behind the platform, and the ice floes at the front end of the platform will accumulate further. The ice load time-history curves of the ice floes on the platform under different concentrations are shown in Fig. 8.46. For the case of 20%, the load on the platform shows strong dispersion due to less ice breakage, while other concentrations of ice crushing have a sustained load on platforms. The ice load tends to increase with the increase of ice fragmentation. The comparison between the field measurement and simulated results, as shown in Fig. 8.47, shows that the simulated results are in good agreement with the measured results.

8.3.2 DEM Simulation of the Ship Sailing in the Ice Zone The “Xuelong” ship is the first polar scientific research vessel of China for navigation in the north and south polar ice regions. It has a maximum speed of 17.9 knots and can continuously break 1.2 m thick ice (including 20 cm of snow) at a speed of 1.5–2 knots (Dawei et al. 2017). The “Xuelong” ship is 167 m long, with a ship width of 22.6 m, a depth of 13.5 m, a draft of 9 m and a full-load displacement of 21,025 t, as shown in Fig. 8.48a. In order to simulate the navigation of the “Xuelong” ship in the ice area with the DEM, the hull model should be established first. Since only the hull contacts with sea ice during the voyage, the triangular discretization of the hull is considered to construct the ship structure, as shown in Fig. 8.48b. The hull model

8.3 DEM Analysis for the Interaction Between Sea Ice …

273

DEM results Field data

Fig. 8.47 Comparison of field measurement and DEM results of ice load on Kulluk platform

Fig. 8.48 “Xuelong” Polar expedition ship

is composed of 1430 triangular elements. In the DEM simulation of the interaction process between “Xuelong” ship and level ice, the motion of the ship hull in the ice zone at a given speed or driving power can be implemented. When the ice thickness is 0.6 m, the results of DEM simulations of the straight line navigation of the “Xuelong” ship in the level ice area are shown in Fig. 8.49. Under this navigational condition, sea ice mainly acts on the ship stern. The maximum and instantaneous ice pressure distribution on the ship hull is shown in Fig. 8.50. Figure 8.51 shows the time history of ice load on the ship hull.

274

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.49 The interaction process between the “Xuelong” ship and the level ice simulated by the DEM

8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration of Marine Structure Installation of ice-breaking cone in the pipeline part of jacket offshore platform leg can change the form of sea ice breakage, hence reducing the ice load and its impact on platform structure. However, field observations indicate that there is still significant ice-induced vibration on the offshore platform where the ice-breaking cone is installed (Ji et al. 2002). In order to calculate and analyze the ice-induced vibration phenomenon of the offshore platform structure, the DEM is used to analyze the sea ice breaking process, and the FEM is used to analyze the vibration characteristics of the offshore platform structure, thus establishing the DEM-FEM coupling method for ice-induced structural vibration, as shown in Fig. 8.52 (Polojärvi and Tuhkuri 2013).

8.4.1 DEM-FEM Coupling Method for Ice-Induced Vibration of Jacket Platform When using the DEM-FEM coupling model to analyze the platform ice-induced vibration, the parameter transfer between the two calculation methods is particularly important. Here, the ice loads on jacket offshore platforms are transferred as force boundary conditions of the FEM, and the dynamic response of offshore platforms

8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration …

275

(a) Maximum ice pressure distribution

(b) Instantaneous ice pressure analysis Fig. 8.50 The DEM simulation of the “Xuelong” ship ice pressure distribution 1800

Fig. 8.51 The ice load on the “Xuelong” ship simulated by the DEM Fx (kN)

1500 1200 900 600 300 0

0

100

200

300

t (s)

400

500

600

276

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.52 DEM-FEM coupling model of ice-induced jacket platform

Fig. 8.53 DEM-FEM coupling analysis process

is calculated. Then the updated platform displacement is taken as the displacement boundary conditions of the DEM as shown in Fig. 8.53. Sea ice is constructed by spherical particles, and jacket offshore platforms are established by beam elements. The contact force between the spherical particles and the beam element and the coordinates of the contact points can be obtained in the local coordinate system. The contact force calculation is the same as that of the DEM of the sphere. According to the Hertz-Mindlin theory, the contact force between the discrete sphere and the beam element consists of two parts, normal force and tangential force: f i  = f n · ei  + f s · ei 

(8.20)

where, f i  is the contact force of the beam element i direction in the local coordinate system; f n and f s are the normal and tangential contact forces obtained by DEM simulations. The normal contact force consists of elastic and viscous forces as:   f n = kn · ξnt + cn · ξ˙nt · n

(8.21)

8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration …

277

where, n is the normal unit vector of the contact between the sphere and beam element; kn is the normal stiffness coefficient between the sphere and beam element; cn is the damping coefficient between the sphere and beam element; ξnt is the normal overlap of the sphere and beam element at time t, which can be expressed as: ξnt = (R − x t − ut ) · n

(8.22)

where, R is the radius of the sphere; x t is the position vector of the sphere at time t; ut is the position vector of the beam element at time t. For the calculation of the tangential force between the sphere and beam element, the Mohr-Coulomb friction criterion is considered here f s∗ = ks · (x˙ts − u˙ st ) · t

(8.23)

     f s = sign f s∗ min f s∗ , μw · f n · s

(8.24)

⎧ ⎨ −1, x < 0 sign(x) = 0, x = 0 ⎩ 1, x > 0

(8.25)

where, s is a tangential unit vector between the sphere and beam element; f s∗ is the tangential force of the current spherical particle to the beam element and cannot exceed the maximum static friction; ks is the tangential stiffness between the sphere and beam element; μw is the coefficient of friction between the sphere and beam element; x˙ts and u˙ st are the tangential velocity at the contact point; t is the calculated time interval; sign(x) is the sign function. During the action of sea ice and offshore platform structure, the contact position between spherical particle and beam element is random, so it is necessary to determine the equivalent node load of offshore platform under ice load. In the local coordinate system, assuming that the two ends of the beam element are fixed as shown in Fig. 8.54, the equivalent nodal force {Fe } of the beam element in the local coordinate system can be obtained from the static equilibrium: Fig. 8.54 Effective calculation model of the beam element

278

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

 T {Fe }T = [N ]T · f i  , f j  , f k 

(8.26)

where,  ] is the equivalent nodal force transformation matrix composed of  A B[N N , N , which can be expressed as: 

NiAj = NiBj =

(3a+b)·b2 L3 (3b+a)·a 2 L3

· δi j (i, j = 1, 2, 3) · δi j (i, j = 1, 2, 3)

(8.27)

where, L is the length of the beam element; a and b are the distances from the contact point to the nodes at both ends; δi j is the Kroneck symbol. In the finite element calculation of the offshore platform structure, the structural vibration response is calculated by the Newmark method in the stepwise integration method. On the basis of guaranteeing the transfer of DEM and FEM boundary conditions, the selection of time step directly affects the results of DEM-FEM coupling algorithm. The time step of a general DEM is much smaller than that of the FEM. In order to obtain accurate calculation results, this section uses the DEM time steps for calculation.

8.4.2 DEM-FEM Method for Ice-Induced Vibration of Jacket Platform The DEM-FEM coupling model of ice-induced structural vibration is verified by comparison with the ISO 19906, JTS144-1-2010 standards and the measured vibration acceleration data of JZ20-2 MUQ cone platform in Bohai Sea. Concrete sea ice discrete element calculation parameters are listed in Table 8.7. The interaction between sea ice and conical offshore platforms is simulated under different ice velocities and ice thickness with parameters listed in Table 8.8. The processes of interaction between sea ice and conical jacket offshore platform at different times are obtained. It is found that sea ice is bent and destroyed under Table 8.7 Calculation parameters in sea ice DEM simulation

Definition

Symbols

Values

Elastic modulus

E

109 Pa

Normal stiffness

kn

1.6 × 108 N/m

Shear stiffness

ks

0.8 × 108 N/m

Normal damping

cn

2.4 × 105 N/m

Shear damping

cs

1.2 × 105 N/m

Particle-particle friction

μp

0.2

Wall-particle friction

μw

0.1

8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration …

279

Table 8.8 Values of ice velocity and thickness for the simulation Definition

Values

Ice velocity (m/s)

0.2

0.3

0.4

0.5

Ice thickness (m)

0.1

0.15

0.2

0.25

Number of particles

160,173

99,000

63,083

40,640

(a) t = 0.0s

(b) t = 60.0s

(c) t = 120.0s

(d) t = 180.0s

Fig. 8.55 DEM-FEM simulation of sea ice interaction with JZ20-2 MUQ platform

the action of pile legs and forms obvious water channel, as shown in Fig. 8.55. Figure 8.56 gives the comparison of sea ice failure modes between field observations and numerical simulations, which shows that the simulation results are close to the field observation results. In addition, from the simulation results, we can get the breakage of the level ice in front of the cone, that is, the ice sheet presents the process of flexural failure, climbing and clearing, which causes the alternating ice force. Considering the ice load on the platform structure, the two legs facing the ice direction are the main force-bearing parts. Because of the symmetry of the structure, Fig. 8.57 gives only the ice load time history along the horizontal direction of a single ice-facing pile leg (vi = 0.2 m/s and Hi = 0.2 m), in which the circular point represents the peak load in the ice load time history. From the ice load time history, it can be found that the ice load on the leg of the pile has multiple peaks and presents obvious randomness, which is consistent with the field monitoring of sea ice and the indoor model test results.

280

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(a) On-site monitoring of sea ice damage before cones

(b) Failure mode of ice cone in DEM-FEM simulation

Fig. 8.56 Fracture when sea ice interacts with cones

Fig. 8.57 The time history of ice load on the piles by the DEM (vi = 0.2 m/s, Hi = 0.2 m)

Firstly, the vibration data of the field measured under the same ice conditions, and the time history of the ice-induced platform vibration acceleration obtained by numerical simulation are given, as shown in Fig. 8.58. It can be observed from the figure that the measured and numerical results vary from −0.04–0.04 m/s2 . The maximum values of vibration acceleration obtained under different ice velocities and ice thickness are compared with the measured data in the Bohai Sea, as shown in Fig. 8.59a and b. The circle points in the figure represent the maximum vibration of the structure in 5 min measured in the field; the maximum vibration acceleration obtained by the DEM-FEM coupling method is represented by a triangle; the two dotted lines represent the trend of the measured data and numerical results, respectively. Since the actual measured ice conditions are complex and involve various factors

8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration …

(a) Field data

281

(b) DEM-FEM data

Fig. 8.58 The vibration acceleration of the platform under vi = 0.2 m/s, Hi = 0.2 m

such as ice thickness, ice speed, ice direction, temperature, etc., when comparing the simulation results with the measured data, it is more common to select the measured data points corresponding to the numerical simulation conditions. Figure 8.59 shows that the numerical results of the DEM-FEM are basically consistent with the trend of field measurement results, that is, the ice speed and the vibration acceleration increase linearly, and the ice thickness and vibration acceleration have a quadratic nonlinear relationship. In order to illustrate the influence of ice velocity and ice thickness on the vibration acceleration of structures, the energy input of sea ice discrete elements per unit time step is written according to the momentum theorem: E k = Fmean · tDEM · vi

(8.28)

where, E k is the energy of the input structure; Fmean is the mean ice load calculated by the DEM-FEM coupling model; tDEM is the DEM time step; vi is the velocity of sea ice. According to Formula (8.28), the trend charts of the structure input sea ice energy under different ice conditions can be obtained, as shown in Fig. 8.60. Figure 8.60a shows the input energy of the structure at different ice velocities when the ice thickness is 0.2 m, and the dot represents the energy value of the input structure. Figure 8.60b shows the energy transferred to the structure by different ice thickness when the ice velocity is 0.5 m/s, in which the triangle represents the energy value of the input structure. The above results show that ice speed increases linearly with energy, while ice thickness increases nonlinearly with energy. According to the statistics of the filed data of the ice-induced vibration acceleration of the JZ20-2 MUQ platform on the Bohai Sea, it is found that the acceleration of the ice-induced vibration is linear with the product of the ice speed and the square of the ice thickness: amax = γ · Hi2 · vi

(8.29)

282

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(a) Simulated results and field data under different ice velocities

(b) Simulated results and field data under different ice thicknesses

Fig. 8.59 Comparison between numerical results and field data of ice-induced vibration acceleration of platform structure under different ice conditions

8.4 DEM-FEM Coupling Analysis for Ice-Induced Vibration …

283

(a) Different ice velocities ( H i = 0.2m ) (b) Different ice thickness ( vi = 0.5m/s ) Fig. 8.60 Sea ice energy input from cone platform structure under different ice conditions

where, amax is the maximum value of vibration acceleration; Hi is the thickness of sea ice; vi is the velocity of sea ice; γ is a linear coefficient. According to the statistical rule shown in Formula (8.29), the simulated 16 kinds of working conditions are recombined according to the above parameters, as shown in Fig. 8.61. In the figure, the abscissa represents the product of the ice speed and the square of the ice thickness, and the ordinate is the maximum value of the ice-induced vibration acceleration. The data points represent the field data and the numerical simulation results. It can be seen that the relationship between the field data and the

Fig. 8.61 Vibration accelerations calculated by the proposed method and field measurements

284

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

numerical simulation results is close and consistent. The DEM-FEM coupling model proposed can reveal the general law of ice-induced vibration of the offshore platform of the Bohai, China. In addition, from the perspective of input structure energy, it can be found that the square of ice speed and ice thickness are linear with the sea ice energy of the input structure. It is assumed that ice velocity and ice thickness are independent variables, so the product of ice velocity and ice thickness square should also be linear with the ice energy input to the structure. The relationship between ice-induced vibration acceleration and ice speed and ice thickness is explained from the structural energy.

8.5 Simulation of Ice Load on Marine Structure with Dilated Particles The DEM has been applied to the dynamic simulation of sea ice. Among them, the spherical particles (Ji et al. 2016), the dilated discs (Hopkins and Jukka 1999; Sun and Shen 2012) and the dilated polyhedral particles (Liu and Ji 2018) are widely used to simulate the interaction between different types of sea ice and structures, to determine the magnitude and distribution characteristics of the ice load on the structure, and to analyze the vibration of the structure. Meanwhile, the development of the bonding-breakage model in the DEM provides an effective approach for simulating the breaking process of sea ice. In this section, the Minkowski Sum is adopted to generate dilated polyhedral ice particles to simulate the interaction between ice and structure.

8.5.1 Interaction Between Broken Ice and Pile Legs, Floating Structures (1) Construction of dilated ice particles based on Minkowski sum According to the Minkowski Sum method, the dilated disc is constructed by sphere and disc, as shown in Fig. 8.62. Similarly, dilated polyhedral sea ice particles can be constructed, as shown in Fig. 8.63.

Fig. 8.62 The dilated disc particle of ice

8.5 Simulation of Ice Load on Marine Structure with Dilated Particles

285

Fig. 8.63 The dilated polyhedral particles of ice

(a) t = 0s

(b) t = 30.0s

(c) t = 60.0s

Fig. 8.64 DEM simulation of the process of crushing ice and ship hull

(2) The interaction between dilated disc particle and ship hull Broken ice consists of 3D dilated discs. The buoyancy, drag force on the ship hull structure, in addition to the additional mass under the action of sea currents, are considered. The length and width of the computational domain are L (x direction) and B (y direction), and the z axis is perpendicular to the water surface. Periodic boundary conditions are used in the horizontal direction of the computational domain. The diameter of sea ice particles ranges from 4.0–6.0 m, while the concentration is 50% and the total number of sea ice particles is 1380. The initial distribution of the hull and floating ice is shown in Fig. 8.64a. The seawater flows in the x direction at a flow rate of 0.4 m/s, while the ship travels in the opposite direction of x at a speed of 4.0 m/s. The above DEM model is used to simulate the dynamic process of the interaction between hull and sea ice in 60 s. Figure 8.64b and c show the state of interaction between broken ice and the hull at t = 30 and 60 s. During the voyage, the sea ice particles contact with the hull and stack at the bow, as shown in Fig. 8.65. The time course curve of the total resistance of sea ice to the hull is shown in Fig. 8.66. It can be found that the ice load shows a strong random fluctuation. In the x direction, which is the ship’s navigation direction, the ice load is positive; while in the y direction, the ice load mean is close to zero and is generally symmetrically distributed. (3) The interaction between dilated polyhedral particles and Marine structures The dilated polyhedral particles are used to simulate the interaction between floating ice and cylindrical piles. The distribution between floating ice and cylindrical piles

286

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.65 Collision between crushed ice and ship hull structure 400

1000

200

600

Fy (kN)

F x(kN)

800

400

-200

200 0

0

0

10

20

30

40

50

60

-400

0

10

20

30

40

50

t (s)

t (s)

(a) Dynamic ice load in x direction

(b) Dynamic ice load in y direction

60

Fig. 8.66 Dynamic ice loads on the ship hull in x and y directions

is shown in Fig. 8.67. 2D-Voronoi tessellation algorithm is used to generate 200 randomly distributed polyhedral ice floes. The average size of the ice floes is 2.8 m, and the initial concentration is about 64%. The flow velocity is 0.3 m/s along the x direction. On both sides of the water flow direction are the periodic boundaries, and the others are free boundaries. The DEM simulation of the dynamic process of the Fig. 8.67 Schematic of the interaction between ice floes and cylindrical piles

8.5 Simulation of Ice Load on Marine Structure with Dilated Particles

287

Fig. 8.68 Interaction process between ice floes and cylindrical piles simulated by dilated polyhedral particles

interaction between floe ice and cylindrical legs is carried out as shown in Fig. 8.68. It can be found that the floating ice drift under the drag of the water flow and collide with the cylindrical legs. Under the blocking effect of the pile, the floating ice region has obvious water channels on the back side of the pile. Using the dilated polyhedral particles to simulate ice floes, the results of the ice load on the piles are shown in Fig. 8.69. It can be seen that the ice load caused by the impact of the ice floes on the pile in the x direction is significantly higher than those on other directions, and the ice load has significant impulse characteristics under the collision of the discontinuous ice. In the y direction, ice blocks also collide on both sides of the cylindrical piles, and the ice load appears positive and negative. From the simulated ice load time history, the impact of sea ice on the cylindrical pile structure is also quite random. The dilated polyhedral particles can also simulate the interaction between ice floes and the floating platform. The simulation results are shown in Fig. 8.70 (ship hull) and Fig. 8.71 (floater Kulluk), respectively. 6

20

4 2

Fy / kN

Fx / kN

15 10

0 -2

5 -4 0

20

40

60

80

100

-6

20

40

60

80

t/s

t/s

(a) Ice load in x direction

(b) Ice load in y direction

Fig. 8.69 Load of ice floes on cylindrical piles

100

288

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.70 Interaction between ice and ship hull simulated with dilated polyhedral particles

(a) Ice floes concentration 40%.

(b) Ice floes concentration 80%.

Fig. 8.71 Interaction between ice and floater simulated with dilated polyhedral particles

Generally, the crushed ice exerts relatively small force on the ship hull, and the ice load on the structure is mainly reflected in the interaction with smooth ice, ice ridge and other continuous floating ice. On the other hand, the load of ice floes on the floater Kulluk is small. The ice load on the hull structure is mainly reflected in the interaction with large ice floes, or level ice, ice ridges.

8.5.2 Interaction Between Level Ice and Ship Hull Structure The icebreaker is an important equipment in the scientific investigation of the polar region, and oil and gas exploration and development. Usually, the icebreaker uses the gravity of the hull to cut the ice from top to bottom through a spoon-shaped ship bow. In the design process of the icebreaker, the overall and local ice load on the

8.5 Simulation of Ice Load on Marine Structure with Dilated Particles

289

ship hull structure is an important indicator for design in different ice conditions. The bonded dilated polyhedral particle simulates the process of smoothing the interaction of ice with the hull, as shown in Fig. 8.72. In addition, considering the 6-degree-offreedom motion model of the hull, the rigid body vibration of the hull can be analyzed. Figure 8.73 shows the dynamic ice load on the hull in three directions. In order to rationally design floating offshore platform structures operating in polar regions, it is very important to study the ice load on floating structures when interacting with level ice. The bonded dilated polyhedral particle simulates the process of smoothing the interaction of ice with the floating structure, as shown in Fig. 8.74. Figure 8.75 shows the dynamic ice load of the hull in the x and z directions.

Fig. 8.72 Dilated polyhedral particle simulation of interaction between level ice and hull structure

Fy (kN)

Fx (kN)

1200 800 400 0

6000

1000

4000

800

2000

Fz (kN)

1600

0 -2000

25

50

t (s)

75

100

-6000

400 200

-4000 0

600

0

25

50

t (s)

75

100

0

0

25

50

t (s)

75

Fig. 8.73 The dynamic ice load on the hull in three directions

Fig. 8.74 Interaction between level ice and floating structure with dilated polyhedral particles

100

290

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

1200

2000

1000 1500

Fz (kN)

Fx (kN)

800 600 400

1000 500

200 0

0

100

200

300

400

0

0

100

t (s)

200

300

400

t (s)

Fig. 8.75 Ice load of the floating structure in the x and z directions

8.6 DEM Analysis for Characteristics of Sea Ice Pile-Up on Water Intake of Nuclear Power Plant The broken ice accumulated in front of the structure will not only affect the ice load on the structure, but also cause blockage of the multi-leg structure or the water intake structure of the nuclear power plant, and cause potential safety hazards to the stability and normal operation of the structure (Wang and Ji 2018). Taking the water intake of Hongyanhe Nuclear Power Station in Liaodong Bay as an example, the pile-up process of broken ice at the water intake is simulated by the DEM. The influence of flow velocity, sea ice concentration and average size of the crushed ice on the occurrence and development of broken ice pile-up in the water intake is analyzed. The dangerous range of blocking caused by these factors is determined.

8.6.1 Numerical Simulation of Sea Ice Pile-Up at the Water Intake During the ice period of the Liaodong Bay, China, there is a large amount of broken ice covered on the sea surface. When the broken ice enters the sea area near the intake, it will flow into the open channel under the action of the intake flow field, resulting in the blockage or damage of the intake. In order to effectively prevent the damage caused by sea ice blockage, the DEM is used to accurately predict the pile-up of broken ice in front of the water intake, according to the characteristics of ice, wind and flow fields in the water intake of nuclear power plant. The structure of the water intake is mainly composed of two parts: the diversion dike and the water intake structure, which is simplified as a schematic of the plane structure as shown in Fig. 8.76. The AB section is the water intake building, the BC section, the CD section and the DE section together constitute the diversion dike. According to the actual size of the intake structure, L AB = 232.6 m, L BC = 82.9 m, L CD = 93.9 m and L DE = 43.8 m are set here. The water intake building

8.6 DEM Analysis for Characteristics of Sea Ice …

291

(a) Schematic of water intake structure

(b) DEM model

Fig. 8.76 Water intake of Hongyanhe Nuclear Power Plant in Liaoning, China

has a length of 118 m and a vertical height of 16 m. The DEM model established with the above structural parameters is shown in Fig. 8.76b. The horizontal region is first divided into several polygonal particles by 2D-Voronoi tessellation algorithm. These irregular polygons are extended along the ice thickness direction to generate polyhedral particles with a certain thickness. The polyhedral particles are then filled with spherical particles with bonding and breakage functions to form the ice breaking area shown in Fig. 8.76b. The concentration of broken ice formed by 2D-Voronoi tessellation algorithm can be controlled and the average size of broken ice can be adjusted, so that the DEM can be used to simulate the pile-up of crushed ice in different shapes, sizes and densities. In the model shown in Fig. 8.76, the thickness of sea ice is 0.3 m, the average size of broken ice is 20 m2 , and the concentration is 60%. The specific discrete element parameters are shown in Table 8.9. Due to the requirements to simulate the moving process of broken ice under the combined drag of wind and water flow, the wind and Table 8.9 Main parameters of discrete element simulations

Definition

Symbols

Unit

Sea ice density

ρ

kg/m3

Values 920

Elastic modulus

E

GPa

1.0

Particle size

D

m

0.3

Normal stiffness

kn

N/m

2.5 × 106

Shear stiffness

ks

N/m

2.5 × 105

MPa

0.6

Particle bond strength

σb

Particle-particle friction

μp

0.2

Coefficient of restitution

ep

0.3

Wall-particle friction

μw

0.15

292

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

wave effects are considered in the calculation, while the wind speed is 15 m/s and the wind drag coefficient is 0.1; The flow rate is 0.5 m/s, the drag coefficient of the water flow is 0.3, the wave period is 20 s, and the wave height is 0.5 m. Figure 8.77 shows the pile-up of sea ice above and below the water surface in front of the structure. Compared with the initial state of broken ice in Fig. 8.76b, broken ice keeps approaching the intake under the combined action of wind and wave, hence the concentration of broken ice increases continuously (Fig. 8.77a). Finally, broken ice overlaps and piles in front of the structure (Fig. 8.77b). Assuming that the height of the water line is zero, the pile-up height of broken ice above and below the water surface in front of the intake structure changes with time as shown in Fig. 8.78. It can be seen that when the pile-up begins, the pile-up height under the water surface increases with time, and the pile-up process on the water surface does not change significantly, i.e., the pile-up height under the water surface is significantly larger than that on the water surface. When t = 60 s, the pile-up of broken ice tends to be stable. Although the amount of broken ice increases with time, the pile-up height before the intake structure does not change much. Under this condition, the maximum height of broken ice pile-up in water is 1.2 m, and the maximum height of underwater pile-up is 4.1 m. The water intake is 5.7–9.7 m below the waterline, i.e., when the pile-up height of the broken ice under the water surface exceeds 5.7 m, it will affect the routine operation of the water intake. When the pile-up height under the water surface exceeds

(a) Pile-up on the water

(b) Pile-up under the water

Fig. 8.77 Pile-up of broken ice before the water intake structure (t = 80 s)

Fig. 8.78 Curve of the pile-up height of the broken ice on the water surface with time

8.6 DEM Analysis for Characteristics of Sea Ice …

293

9.7 m, the water intake will be completely blocked by the broken ice, posing a great threat to the safe operation of nuclear power facilities. If the water level in the sea area is below −2 m under this condition, the pile-up height of broken ice 4 m below the water surface will affect the routine operation of the intake. It is necessary to systematically consider the influence of the concentration, size and flow velocity of the broken ice on the pile-up height, in order to better study the pile-up law of broken ice at the water intake, consequently, to make a better prediction of the blockage of the water intake caused by the broken ice pile-up.

8.6.2 Analysis of Influencing Factors of Sea Ice Pile-Up Characteristics (1) Effect of initial ice concentration on floe ice In order to study the influence of the initial concentration of sea ice on the pile-up of sea ice, the distance between the polygons generated by the 2D-Voronoi tessellation algorithm was changed to generate a discrete element model of ice floe with different concentrations. The ice floe concentration set here is 20%, 40%, 60% and 80%, respectively, as shown in Fig. 8.79. The average size of the ice floe is 20 m2 and the ice speed is 0.5 m/s. Other parameters are the same as those in the previous section. From the sea ice model, it can be found that the greater the concentration is, the more the number of floe ice pieces of sea ice is. The total number of sphere particles also increases. In order to illustrate the change law of sea ice pile-up height above and below the water surface under different concentrations, after the stabilization of the pileup height of floe ice in front of the structure, the comparison results show that the pile-up height of floe ice increases with the increase of concentration as shown in Fig. 8.80. The pile-up height above the water surface is obviously larger than that below the water surface, and the influence of concentration is nonlinearly increasing. The greater the concentration, the more obvious the variation of the pile-up height. The reason is that the greater the concentration of floe ice, the greater the number of ice cubes. As a result, the ice cubes overlap each other in front of the structure and are more likely to pile. (2) The effect of the average size of ice floe The average size of floe ice refers to the average area of all ice blocks, with a concentration of 60%, and an average size distribution of 15, 20, 25 and 30 m2 . The distribution of ice blocks is shown in Fig. 8.81. The ice blocks differ in size, but the ice concentration remains unchanged. It can be seen from Fig. 8.82 that the average size of the floe ice has no obvious effect on the pile-up height, and the height variation of the upper and lower pile-up is not obvious. When the average size of the floe ice increases, the number of ice blocks generated in the same sea area decreases, while the total number of sphere particles does not substantially change. The amount of sea ice accumulated during the same period of time as the structure acts increases with

294

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

(a) The concentration is 20%

(b) The concentration is 40%

(c) The concentration is 60%

(d) The concentration is 80%

Fig. 8.79 Distribution of ice floe under different initial concentrations

Pile-up height (m)

8

Above waterline Below waterline

6 4 2 0

20

30

40

50

60

70

Ice concentration (%) Fig. 8.80 Effect of initial concentration of ice floe on pile-up height

80

8.6 DEM Analysis for Characteristics of Sea Ice …

(a) The average size is 15 m2

295

(b) The average size is 25 m2

Fig. 8.81 Distribution of ice floe under different average sizes

8

Pile-up height (m)

Fig. 8.82 Effect of average size of ice floe on pile-up height

Above waterline Below waterline

6 4 2 0

15

20

25

30

Mean size (m2)

the decrease of the average size. The larger the average size is, the more obvious the pile-up of ice is. The effects of the two on the pile-up process cancel each other out, so the impact on the average size of sea ice during sea ice pile-up is small.

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics The sea ice discrete element model can be used not only for the collision between ice at small scales, but also for the evolution of ice ridges and ice gaps at the meso scale, and for numerical simulation of sea ice dynamic processes at geophysical scales in the polar regions, which has clear physical meaning and high accuracy (Hopkins et al. 2004; Herman 2012). At large and meso scales, the distribution characteristics of sea ice can be well presented from its satellite remote sensing images, as shown in Fig. 8.83. Because the DEM uses explicit calculations in the sea ice dynamic equations, the time step is rather small, which cannot meet the time limit requirements for the large scale sea ice numerical prediction and long-term sea ice dynamic simulations (Feng and Owen 2014; Sakai et al. 2012). Therefore, based on

296

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.83 Satellite remote sensing image of Bohai sea at large scale (February 16, 2001)

the DEM models of sea ice, this section constructs the sea ice elements and develops the coarse-grained DEM of sea ice dynamic process by integrating the idea of sea ice smoothed particle hydrodynamics. The viscous-elastic-plastic constitutive model is used to calculate the internal forces and the elastic-plastic deformation between elements, and the drift, fracture, pile-up and divergence of sea ice are then simulated.

8.7.1 The Coarse-Grained DEM of Sea Ice In the improved discrete element models of sea ice dynamics, the sea ice element is set as a collection of many ice blocks, and the contact and collision effects between them are affected by parameters such as average thickness, concentration and element size. When the normal stresses between elements exceed their compressive strength, the sea ice undergoes plastic deformation, which leads to the change of element size. The movement law of the sea ice element depends on the drag force of the wind and current, the sea ice pressure gradient force caused by the sea surface tilt, and the force between ice. (1) Equation of motion of sea ice element The movement of sea ice mainly depends on the force between elements, the drag of wind and current, the Coriolis force and the pressure gradient force caused by the sea surface tilt. The dynamic equation is described by Newton’s law

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics

M

DV = −MfK × V + τa + τw − Mg∇ξw + ∇ · σ Dt

297

(8.30)

where, M is sea ice mass per unit area with M = Nρi Hi ; V is the velocity vector of sea ice; f is a Coriolis parameter with f = 2ωe sin φ; ωe is the rotation speed of the earth; φ is geographic latitude; K is the unit vector perpendicular to the sea surface; ∇ξw is the instantaneous sea level gradient. The drag force per unit area of wind and current is τa = ρa Ca |Vai |Vai and τw = ρw Cw |Vwi |Vwi , where Ca and Cw is the drag coefficient of wind and current; Vai and Vwi are wind velocity and velocity vectors relative to sea ice, and ∇ · σ are internal force vectors per unit area caused by the interaction between sea ice elements. (2) Contact model between sea ice elements The force between sea ice elements mainly includes the elastic force caused by the mutual overlap between elements in the normal and tangential directions and the viscous force due to the relative velocity. In addition, the sliding friction based on the Mohr-Coulomb criterion needs to be considered in the tangential direction (Shen et al. 2001). In the normal direction, the contact force between sea ice elements is   Fn = min Fe + Fv , Fp

(8.31)

where, Fe and Fv are the elastic and viscous forces between sea ice elements; Fp is the force when sea ice elements deform plastically, which can be written as ˜ P Fp = h i Ni Dσ

(8.32)

where, h i and Ni are the thickness and concentration of sea ice; D˜ is the effective contact length, which is set to the average diameter of the two sea ice elements; σP is the plastic stress. The normal elastic and viscous forces between sea ice elements can be expressed as Fe = K n xn , Fv = Cn x˙n

(8.33)

where, K n and Cn are the normal stiffness and viscous coefficients, while xn and x˙n are the normal relative displacement and relative velocity between ice elements. In the tangential direction, based on the Coulomb friction law, the frictional forces between sea ice elements are   Ft = min K t xt , sign(K t xt )μp Fn − Ct x˙t

(8.34)

where, K t and Ct are the tangential stiffness and viscous coefficients, xt and x˙t are the relative displacement and velocity, and μp is the sliding friction coefficient between sea ice elements.

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

The normal and tangential stiffness of sea ice elements are closely related to its concentration. According to the corresponding relationship between the mechanical properties of sea ice and its concentration, the following conclusions can be drawn as  j Ni K n = E Hi (8.35) Nmax where, E is the elastic modulus; Hi is the thickness; Ni is the concentration; Nmax is the maximum density, where Nmax = 100%; j is usually 15 (Shen et al. 1990). In the discrete element model, the tangential stiffness of the element is generally set as K t = 0.5K n . The viscous coefficient Cn of sea ice element can be determined by its mass, normal stiffness and dimensionless viscous coefficient as:  (8.36) Cn = ξn 2M K n  where, ξn is dimensionless viscous coefficient with ξn = − ln e/ π 2 + ln2 e, where e is the restitution coefficient; M is the effective mass. Here, the tangential viscous coefficient is set as Ct = 0.5Cn . (3) Plastic deformation of sea ice element When sea ice elements are extruded under external forces, the normal stress between can be determined by the normal force Fn and the contact area S, i.e. σn = Fn /S. In the improved discrete element model of sea ice dynamics, sea ice elements are aggregates of many ice blocks. The normal compressive strength can be determined by the hydrostatic pressure of the ice fragments in the vertical direction and the MohrCoulomb strength criterion. This method has been successfully applied to the SPH simulation of sea ice dynamics. When the normal stress of the ice element exceeds its compressive strength, plastic deformation will occur, and the ice element will shrink and the concentration and thickness of the element will change accordingly. According to the plastic failure criterion of granular materials and considering hydrostatic pressure effect in ice, the compressive strength is  σc = K c P0 + 2c K d

(8.37)

  where, K c = tan2 π4 + ϕ2 with ϕ as the internal friction angle of the sea ice. c is the cohesive force between ice blocks, and c = 0 for non-frozen crushed ice. P0 is the vertical average pressure in the ice layer, i.e., hydrostatic pressure, which is calculated as Shen et al. (2001)   j  Ni ρi ρi g Hi P0 = 1 − ρw 2 Nmax

(8.38)

where, ρi and ρw are densities of sea ice and sea water, respectively; g is gravitational acceleration.

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics

Convergence

Rafting and ridging

299

Divergence

Fig. 8.84 Sea ice convergence-diverging process and changes in the corresponding area and concentration of sea ice elements

When the plastic deformation occurs, the area of sea ice element decreases correspondingly, which leads to the increase of concentration and convergence of sea ice. When the sea ice concentration reaches a maximum value Nmax , if the internal force of the sea ice further increases, the ice blocks inside the element will pile up. On the contrary, when the stress between sea ice elements is less than its compressive strength, the sea ice elements divergence under hydrostatic pressure, which leads to the increase of the area of sea ice elements and the decrease of the concentration. The convergence and divergence of the sea ice under internal force is shown in Fig. 8.84. During this convergence and divergence, the mass of the sea ice element remains constant, and the area, concentration and thickness change accordingly.

8.7.2 Numerical Simulation of Sea Ice Dynamic Processes in the Regular Domain In order to verify the reliability and accuracy of the coarse-grained discrete element model, the following two numerical analysis of the typical sea ice dynamic process in the regular domain are carried out. One is the drift of sea ice under wind and current towing in variable width waterway and the pile-up process; The other is the dynamic process of sea ice in the rectangular domain under the action of the rotating wind field. (1) Drift and pile-up of sea ice in variable width channels The variable width channel is shown in Fig. 8.85 with a triangular obstacle placed in the channel. The total length L = 170 km, the width of left side and right side is W 1 = 30 km and W 2 = 14 km, respectively. The initial thickness of sea ice is H i0 = 1.0 m, the initial density is N i0 = 100%, the initial length of ice area is L 1 = 60 km, and the wind velocity is 20 m/s (Table 8.10). L1=60km

W1=30km

L2=60km

Initial ice field Wind velocity 20m/s

Fig. 8.85 Initial distribution of sea ice in variable width channel

L3=50km

W2=14km

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Table 8.10 Main computational parameters in the DEM simulation of sea ice dynamic processes Parameters

Definitions

Value

E

Young modulus of sea ice (MPa)

100

Ca

Drag coefficient of wind

0.0015

Va

Wind velocity (m/s)

20.0

D0

Initial element diameter (km)

1.0

ϕ

Sea ice internal friction angle

46°

ρi

Sea ice density (kg/m3 )

917

μp

Coefficient of friction between sea ice elements

0.5

Cw

Drag coefficient of water

0.0045

Vw

Wind velocity (m/s)

0.0

H i0

Initial sea ice thickness (m)

1.0

N0

Initial sea ice concentration

100%

ρw

Sea water density (kg/m3 )

1006

Under the action of wind and current, sea ice drifts around the triangular obstacle to the right and accumulates at the right boundary. The pile-up height increases with the passage of time, which leads to the increasing internal force of sea ice. When the internal force of the sea ice and the drag force of the wind and current reach a balance, the sea ice velocity tends to zero, and the pile-up height reaches a stable state. During the drift and pile-up of sea ice, its velocity, size and thickness change. The average ice thickness on the 1st, 2nd, 3rd and 7th day are shown in Fig. 8.86. Influenced by the boundary friction, the ice velocity near the boundary decreases significantly, and the average ice thickness is smaller than that in the middle of the channel. To further verify the accuracy of the model for ice thickness simulation, the average thickness of the sea ice after stabilization is compared with the analytical solution. When considering boundary friction, the steady-state pile-up height of sea ice can be calculated from the classical ice dam theory. This theory applies the force balance equation in terms of the wind and current drag force, the sea ice internal force and the boundary friction in the length direction of the waterway to determine the pile-up height of the ice dam. The stress in the water channel width direction is averaged, and the pile-up height in the longitudinal direction is:    Hi =  Hi02 +

     B ρa Ca Va2 + ρw Cw Vw2 2(1 − sin ϕ) tan ϕ  1 − exp −  x B ρi g tan ϕ(1 + sin ϕ) 1 − ρρwi (8.39)

The results of discrete element simulation and the analytical solution of Eq. (8.39) are shown together in Fig. 8.87. It can be found that the simulated average ice thickness has a high fitting degree with the analytical solution, which verifies the accuracy of the improved discrete element simulation in sea ice dynamics.

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics

301

Sea ice thickness (m)

Fig. 8.86 Average sea ice thickness distribution at different times

(2) Dynamic process of sea ice under rotating wind field The simulation of sea ice under rotating wind field can effectively verify the effectiveness of the numerical method of sea ice dynamics. This method will be used to verify the effectiveness of the improved DEM. The boundary of the cyclone study is a square computational area of 500 km × 500 km. The center of the wind field is located at (250, 220 km), and the sea ice is initially distributed in the upper half of the computational domain, as shown in Fig. 8.88.

302

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull 3.5 3.0

Hi (m)

2.5

Analytical Solution DEM results

2.0 1.5 1.0 0.5 0.0 120.0

130.0

140.0

x (km)

150.0

160.0

170.0

Fig. 8.87 Comparison of DEM results and analytical solution of sea ice average thickness

Initial ice field N0 =80%, Hi0 =1.0m

Fig. 8.88 Initial distribution of rotating wind field and sea ice

The initial thickness of sea ice Hi0 = 1.0 m, the initial concentration Ni0 = 80%, and the initial diameter of sea ice element Di0 = 4.0 km. The wind velocity at position r in the calculation domain is:   r λ K× W(r ) = min ωr, r r

(8.40)

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics

303

Fig. 8.89 Sea ice velocity and average ice thickness distribution under rotating wind field

where, W is the wind velocity vector; r and r are the position vector of any point in the computational domain and the distance to the center of the wind field; wind velocity parameters ω = 0.5 × 10−3 m/s, λ = 8 × 102 m2 /s. When the sea ice freezing is not considered, the bonding force between the elements is zero. The improved discrete element model is used to calculate the sea ice dynamic process under the rotating wind field for 10 days. The ice velocity and average ice thickness distribution on the 2nd, 5th, and 10th day are shown in Fig. 8.89. It can be found that the sea ice vortexes around the center of the wind field in the rotating wind field under the drag of wind and current. In the process of sea ice movement, the average ice thickness increases due to the compression and pile-up of the ice elements near the left boundary and decreases due to the drift of the ice elements near the right boundary. In addition, it can be seen from the contour distribution of the average ice thickness that two distinct vortices are formed on both sides of the vortex center. They move and deform during the sea ice drift. The above calculation results show that the improved discrete element model can accurately simulate the dynamic evolution process of sea ice under the action of rotating wind field.

8.7.3 Numerical Simulation of the Dynamic Process of the Bohai Sea Ice In order to further verify the applicability of the improved discrete element model in the simulation of sea ice dynamics, the model is applied to the numerical analysis of

304

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

the dynamic evolution process of sea ice in the Bohai Sea. The results are verified by sea ice satellite remote sensing data and sea ice monitoring data in oil and gas fields. (1) Extraction and numerical simulation of ice parameters distribution in Bohai Sea In this section, the parameters of sea ice are extracted from the satellite remote sensing image at 10:50 on January 22, 2010, which mainly includes ice thickness and concentration. The initial ice velocity field is approximately set as tidal current field, and the maximum initial diameter of sea ice is 2 km. Satellite remote sensing images and digitized distribution of sea ice elements, ice thickness and concentration are shown in Fig. 8.90. The 48 h numerical simulation of the dynamic process of sea ice is carried out by using the improved discrete element model. The location of sea ice elements, ice concentration, ice thickness and ice velocity distribution are shown in Fig. 8.91 and Fig. 8.92, respectively. Initial distribution of sea ice 22 jan, 2010, 10:50

(a) Satellite remote sensing image

(b) Sea ice elements distribution

(c) Sea ice thickness distribution

(d) Initial distribution of sea ice concentration

Fig. 8.90 Sea ice satellite remote sensing image and initial field of sea ice parameters in the Liaodong Bay of the Bohai Sea

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics

305

Fig. 8.91 Sea ice satellite remote sensing image and discrete element simulation of sea ice location

From the simulation results, it can be found that the location of sea ice element and the distribution law of ice margin line are basically consistent with satellite remote sensing data. Within 48 h of numerical simulation, the wind velocity is between 2 and 8 m/s, and the temperature is between −5 and −8 °C. The change of meteorological conditions is gentle. The sea ice movement is mainly controlled by tidal current, and the ice thickness is less affected by thermal factors. The coarse-grained DEM can accurately describe the process of sea ice movement through analyzing the transport, collision and overlap characteristics of sea ice elements in the Lagrangian coordinates, and changing the location, size, concentration and ice thickness of the elements as well. In particular, the numerical simulation of the drift velocity and the motion trajectory of the sea ice elements can accurately determine the position of the ice edge line without the numerical expansion phenomenon under the Euler coordinates.

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8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull

Fig. 8.92 Sea ice concentration, thickness and velocity distribution simulated by the DEM

(2) Evolution of sea ice parameters in the oil and gas area of JZ20-2 in Liaodong Bay In the dynamic simulation of the Bohai sea ice, the determination of sea ice parameters in the oil and gas operation area has important engineering significance for ensuring the safe operation of oil and gas, and is also an important part of the sea ice management in the glacial oil and gas operation. In the above improved discrete element simulation of sea ice dynamic process in Liaodong Bay, sea ice parameters of JZ20-2 oil and gas operation area (121° 21 , 40° 30 ) can be extracted. The Gauss function of the SPH method is used to interpolate the sea ice elements of the oil and gas operation area to obtain the sea ice thickness, concentration and velocity in the sea area. Figure 8.93 shows the drift trajectory of a sea ice element near JZ20-2 oil and gas operation area. It can be found that it drifts southwestward during the reciprocating movement due to the dragging effect of reciprocating tidal current and northwest wind. Figures 8.94 and 8.95 show the variation of sea ice velocity and thickness within 48 h. The measured values are obtained by the on-site monitoring and digital image processing system of sea ice on the oil and gas platform. The results

8.7 Coarse-Grained DEM Model for Sea Ice Dynamics

307

40.60

Latitude ( ° )

Fig. 8.93 Drift trajectory of sea ice element in JZ20-2 sea area of the Liaodong Bay

40.55 40.50 40.45 40.40 121.2

121.3

121.4

121.5

121.6

Longitude ( ° ) 1.0

Ux (m/s)

0.5 0.0 -0.5

DEM simulated -1.0

0

6

12

18

24

30

Filed measured 36

42

48

t (hr)

(a) Ice velocity in the east-west direction 1.0

Uy (m/s)

0.5 0.0 -0.5

DEM simulated -1.0

0

6

12

18

24

30

Field measured 36

42

48

t (hr)

(b) Ice velocity in the north-south direction Fig. 8.94 48 h simulated and measured ice velocity in the JZ20-2 sea area of the Liaodong Bay

show that the simulation results of the coarse-grained DEM for sea ice parameters in the sea area are basically consistent with the field monitoring values.

308

8 DEM Analysis of Ice Loads on Offshore Structures and Ship Hull 0.10

Hi (m)

0.08

0.06

Field measured DEM simulated 0.04

0

6

12

18

24

30

36

42

48

t (hr)

Fig. 8.95 48 h simulated and measured ice thickness in the JZ202 sea area of the Liaodong Bay

8.8 Summary This chapter mainly introduces the wide application of the DEM in sea ice in ocean engineering. Aiming at the characteristics of crushing and bending failure of sea ice interacting with the offshore platform structure and ship hull, the DEM is used to numerically analyze the failure mode of sea ice and the ice load on structures under different ice conditions. Accordingly, the ice-induced vibration of different types of offshore platforms are further analyzed by the DEM-FEM coupling method. The corresponding characteristics of ice loads are determined and compared with standards and criteria. The comparison results are in good agreements. The coarsegrained DEM for the sea ice dynamic process is developed for the discrete distribution characteristics of sea ice and the deformation characteristics of sea ice drift and overlap at the geophysical scale, and accurately calculates the plastic deformation of sea ice during the pile-up process. It can effectively simulate and study the large and medium scale sea ice formation and dispersion. Meanwhile, considering the fragmentation and pile-up of sea ice under waves, it is possible to accurately predict the impact of sea ice pile-up and blockage on the water intake of nuclear power plants. In addition, for the geometric characteristics and distribution of ice in the icecutting area, the 2D-Voronoi tessellation algorithm is used to generate the randomly distributed dilated polyhedral sea ice particles. It can be seen that the DEM simulation not only effectively improves the computational scale and efficiency, but also provides a strong basis for the design of marine structures. The DEM approach demonstrates its application prospect and potential in marine engineering.

References

309

References Brown TG, Määttänen M (2009) Comparison of Kemi-I and Confederation Bridge cone ice load measurement results. Cold Reg Sci Technol 55(1):3–13 Dawei G, Xiaoping P, Quan S et al (2017) GPS-based “Snow Dragon” ship impact icebreaking pattern recognition research. Polar Res 29(3) Feng YT, Owen DRJ (2014) Discrete element modelling of large scale particle systems-I: exact scaling laws. Comput Part Mech 1:159–168 Frankenstein G, Garner R (1967) Equations for determining the brine volume of sea ice from −0.5 °C to −22.9 °C. J Glaciol 6:943–944 Hentz S (2004) Identification and validation of a discrete element model for concrete. J Eng Mech 130(6):709–719 Herman A (2012) Influence of ice concentration and floe-size distribution on cluster formation in sea-ice floes. Cent Eur J Phys 10(3):715–722 Hopkins MA (1997) Onshore ice pile-up: a comparison between experiments and simulations. Cold Reg Sci Technol 26(3):205–214 Hopkins MA (2004) Discrete element modeling with dilated particles. Eng Comput 21:422–430 Hopkins MA, Jukka T (1999) Compression of floating ice fields. J Geophys Res Atmos 1041(C7):15815–15826 Hopkins MA, Shen HH (2001) Simulation of pancake-ice dynamics in wave field. Ann Glaciol 33:355–360 Hopkins MA, Susan F, Thorndike AS (2004) Formation of an aggregate scale in Arctic sea ice. J Geophys Res Oceans 109(109):1032 Huang HY (1999) Discrete element modeling of tool-rock interaction. University of Minnesota, Minnesota ISO 19906 (2010) Petroleum and natural gas industries-Arctic offshore structures. International Organization for Standardization Ji S, Wang S (2018) A coupled discrete-finite element method for the ice-induced vibrations of a conical jacket platform with a GPU-based parallel algorithm. Int J Comput Meth 17(4):1850147 Ji S, Di S, Liu S (2015) Analysis of ice load on conical structure with discrete element method. Eng Comput 32(4):1121–1134 Ji S, Di S, Long X (2016) DEM simulation of uniaxial compressive and flexural strength of sea ice: parametric study. J Eng Mech 143(1):C4016010 Ji S, Yue Q, Bi X (2002) Probability distribution of sea ice fatigue parameters in JZ20-2 sea area of the Liaodong Bay. Ocean Eng 20(3):39–43 Ji S, Li Z, Li C, Shang J (2013) Discrete element modeling of ice loads on ship hulls in broken ice fields. Acta Oceaologica Sinica 32(11):50–58 Ji S, Wang A, Jie S, Yue Q (2011) Experimental studies on elastic modulus and flexural strength of sea ice in the Bohai Sea. J Cold Reg Eng 25(4):182–195 Kato K (1990) Total ice force on multi legged structures. In: Proceedings of 10th international symposium on ice. Espoo, Finland, vol 2, pp 974–983 Kato K, Adachi M, Kishimoto H et al (1994) Model experiments for ice forces on multi conical legged structures. In: Proceedings of the 4th ISOPE conference, Osaka, Japan, vol 2, pp 526–534 Lau M (2001) A three dimensional discrete element simulation of ice sheet impacting a 60° conical structure. In: Proceedings of the 16th interactional conference on port and ocean engineering under Arctic conditions, Ottawa, Canada, 2001 Liu L, Ji S (2018) Ice load on floating structure simulated with dilated polyhedral discrete element method in broken ice field. Appl Ocean Res 75:53–65 Liu L, Sun S, Ji S (2017) Interaction between floater and sea ice simulated with dilated polyhedral DEM. In: International conference on discrete element methods. Springer Singapore Paavilainen J, Tuhkuri J (2013) Pressure distributions and force chains during simulated ice rubbling against sloped structures. Cold Reg Sci Technol 85:157–174

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Polojärvi A, Tuhkuri J (2013) On modeling cohesive ridge keel punch through tests with a combined finite-discrete element method. Cold Reg Sci Technol 85:191–205 Qu Y, Yue Q, Bi X et al (2006) A random ice force model for narrow conical structures. Cold Reg Sci Technol 45:148–157 Sakai M, Takahashi H, Pain CC et al (2012) Study on a large-scale discrete element model for fine particles in a fluidized bed. Adv Powder Technol 23(5):673–681 Shen HT, Shen HH, Tsai SM (1990) Dynamic transport of river ice. J Hydraul Res 28(6):659–671 Shen HT, Liu L, Chen YC (2001) River ice dynamics and ice jam modeling. In: IUTAM symposium on scaling laws in ice mechanics and ice dynamics. Springer, Netherlands Sun S, Shen HH (2012) Simulation of pancake ice load on a circular cylinder in a wave and current field. Cold Reg Sci Technol 78:31–39 Tian Y, Huang Y (2013) The dynamic ice loads on conical structures. Ocean Eng 59:37–46 Timco GW, Irani MB, Tseng J et al (1992) Model tests of dynamic ice loading on the Chinese JZ-20-2 jacket platform. Can J Civ Eng 19:819–832 Timco GW, Wright BD, Barker A et al (2006) Ice damage zone around the Molikpaq: implications for evacuation systems. Cold Reg Sci Technol 44(1):67–85 Wang S, Ji S (2018) Coupled DEM-FEM analysis of ice-induced vibrations of a conical jacket platform based on the domain decomposition method. In: International society of offshore and polar engineers, 1 June 2018. ISOPE-18-28-2-190 Wang Y, Tonon F (2010) Calibration of a discrete element model for intact rock up to its peak strength. Int J Numer Anal Meth Geomech 34(5):447–469 Yan H, Qing-zeng S, An S (2009) Model test study on ice-induced vibrations of compliant multicone. China Ocean Eng 23(2):317–328 Yan Q, Qianjin Y, Xiangjun B et al (2012) A random ice force model for narrow conical structures. Cold Reg Sci Technol 45(3):148–157 Yue Q, Bi X (2000) Ice-induced jacket structure vibrations in Bohai Sea. J Cold Reg Eng 14(2):81–92 Yue Q, Qu Y, Bi X, Tuomo K (2007) Ice force spectrum on narrow conical structures. Cold Reg Sci Technol 49:161–169 Zhou L, Su B, Riska K et al (2012) Numerical simulation of moored structure station keeping in level ice. Cold Reg Sci Technol 71:54–66

Chapter 9

DEM Analysis of Mechanical Behaviors of Railway Ballast

The ballasted railway track, as one of the most important ways of transportation, has a wide distribution in China. The main functions of the railway structure are to withstand and transfer the load from the sleeper, to maintain the stability of the track geometry, and to mitigate and absorb the impact and vibration from the wheel-rail interaction. As a kind of typical granular materials, railway ballast exhibits strong nonlinear properties such as non-uniformity, discontinuity and anisotropy. The irregular shapes of ballast particles play an important role in the stability of the ballast bed (Lu and McDowell 2007). Currently, the discrete element method (DEM) has been widely used to analyze the mechanical behaviors of ballast since it is an effective numerical method with great advantages in simulating granular materials (Ngo et al. 2014; Lu et al. 2010; Huang and Tutumluer 2011). The DEM can accurately model the microscopic characteristics, such as the sizes, shapes, gradation and porosity of ballast and ballast particle breakage as well (Huang and Tutumluer 2011; Tutumluer et al. 2013; Liu et al. 2019). The method has been used to analyze the track settlement under external load, the distribution of force chains and the rearrangement of ballast particles on the microscopic scale (Hossain et al. 2007; Bian et al. 2019; Lu et al. 2010; Stahl and Konietzky 2011). With the rapid development of China’s high-speed railway technology, the ballastless track is widely distributed on high-speed railway. However, the ballasted track is still the main structural form in China railway. Consequently, the connection of ballasted and ballastless track inevitably exists. Because of the differences in the strength, stiffness, and settlement of the ballasted and ballastless track, transition zones need to be installed. How to realize a gradual change in stiffness of the transition zone is an urgent problem to be solved (Indraratna et al. 2019). This chapter summarizes numerical simulations of the mechanical behaviors of railway ballast related research.

© Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_9

311

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

9.1 DEM Analysis of Ballast Particle Breakage 9.1.1 DEM Analysis of Ballast Breakage The breakage mechanism of ballast is complicated. It can mainly be classified as particle splitting and abrasion. A lot of experiments have been done on the breakage behaviors of crushed stones, which mainly focus on carrying out compressive tests of crushed stones to determine the equivalent compressive strength of stones with different sizes. The DEM is an effective way to numerically simulate the breakage of granular materials. Irregularly shaped ballast particles can be constructed based on the bonding-breaking effect between spheres. The breakage law between bonded particles can then be characterized by the development of corresponding failure criteria (Potyondy and Cundall 2004; Lim and McDowell 2005; Indraratna et al. 2010). The breakage of ballast is mainly related to its tensile strength, which can be determined through compressive tests (McDowell and Bolton 1998). The laboratory tests and the DEM simulations show that the tensile strength of ballast meets the Weibull distribution for the given sizes of particles in the experiments (Lim et al. 2004; Yan et al. 2015). The bonding model is adopted to build irregular shaped ballast particles. The model parameters are listed in Table 9.1. Take a ballast particle with an effective size of 45.14 mm as an example. It was generated with 293 bonded spheres and 449 bonding pairs. Figure 9.1 shows the progressive failure modes of the ballast particle. It can be observed that at the initial stage of loading, a few bonds break at Table 9.1 Main parameters in discrete element simulations

Parameter

Unit

Density ρ

kg/m3

2530.0

Size Dav

mm

20–75

Young’s modulus E c

GPa

60

Normal stiffness k n

MN/m

660

–

0.2

Normal to tangential stiffness ratio k s /k n Normal strength of bond

σbn

Tangential strength of bond σbs

( a ) ε = 0.035

Value

(b) ε = 0.044

( c ) ε = 0.050

Fig. 9.1 The breakage of ballast during compression

MPa

191.7

MPa

191.7

( d ) ε = 0.058

9.1 DEM Analysis of Ballast Particle Breakage

313

Fig. 9.2 The effective stress and bond breakage numbers with respect to the effective strain

the ballast particle’ contacts with the platen due to the stress concentration; Breakage then aggravates, and the internal crack forms gradually; When the radial penetration occurs, the ballast splits into two smaller pieces. Figure 9.2 shows the relationship between the effective stress σe f f and the effective strain εe f f , and the total number of bond breakage during the process of the ballast particle breakage. When the effective strain is small (εe f f < 0.035), the effective stress is low; When εe f f > 0.035, the effective stress increases nonlinearly with the increase of the effective strain, and a peak appears at εe f f = 0.044; The maximum value reaches at εe f f = 0.058, then the effective stress value declines sharply. It indicates that the ballast breakage process consists of two parts. The first breakage occurs at the first peak of the effective stress, and the overall splitting occurs at the maximum effective stress. It also shows the number of bond breakage of bonded spheres reaches its maximum at εe f f = 0.058. Figure 9.3 shows the evolution of the force chain during compression which is another micro indication of the ballast particle breakage. The strong force chains concentrate near the upper and lower loading plates, and then extend to the entire ballast particle. When the effective strain εe f f = 0.058, the broken force chains run through the particle, resulting in an overall failure of the ballast particle. In order to analyze the statistical properties of the equivalent compressive strength of ballast, 65 ballast particles were constructed using the 3D scanning technology. The sizes of ballast are in the range of [20 mm, 75 mm], which is consistent with those in the experiment. Figure 9.4 shows the effective compressive strength of the ballast at the different equivalent sizes. It indicates that the compressive strength decreases with the increase of the equivalent size and exhibits significant dispersion. The equivalent compressive strength is statistically analyzed, and the results follow the Weibull distribution. The simulated Weibull residual probability distribution is shown in Fig. 9.5. When the residual probability of ballast is 37%, the tensile stress σ = 9.44 MPa, which is in good agreement with the experiment results. This

314

9 DEM Analysis of Mechanical Behaviors of Railway Ballast

( a ) ε = 0.035

(b) ε = 0.044

( c ) ε = 0.050

( d ) ε = 0.058

Fig. 9.3 Distribution of the force chain under different equivalent strains

Fig. 9.4 The plot of compressive strength and equivalent sizes

indicates the reasonability of using the DEM. In addition, the simulated result m = 1.93 is lower than that obtained from the experiment which is m = 2.75. It indicates that the dispersion degree of the compressive strength of ballast is relatively lower than that obtained from the DEM.

9.1 DEM Analysis of Ballast Particle Breakage

315

Fig. 9.5 Weibull residual probability distribution of the effective compressive strength

9.1.2 The Experimental Verification of Ballast Particle Breakage The material of the selected ballast is granite. The selected samples have similar geometric shapes of an approximate hemisphere. Owing to the irregularity of ballast, it is difficult to select ballast with identical sizes. Therefore, the samples within a certain size range are selected in the experiment. The sample size is measured by a vernier caliper. Define the maximum length direction of the ballast particle as one of the 3D coordinates, and three sizes perpendicular to each other are measured and averaged. The average size is in the range of [20 mm, 75 mm]. The test setup is shown in Fig. 9.6. The machine indenter and bottom plate are made of high quality steel. During the test, the working surface of the top indenter is parallel to the bottom plate. The selected contact point between the ballast top and the machine indenter should be located at the ballast center. The loading is realized by using a displacement-control method. The loading rate is l mm/min. The force and displacement data are recorded by the data acquisition system. The typical force-displacement curve is shown in Fig. 9.7. The strength of ballast is determined by the empirical formula proposed by Jaeger (1967), σe f f =

Ff dav

(9.1)

where σe f f is the effective strength; F f is the normal force; dav is the average of the maximum and minimum diameter of the ballast particle. In order to explore the relationship between the ballast strength and its size, the experiment results are plotted in Fig. 9.8. The abscissa is the average size and the ordinate is the compressive strength. It shows that the strength of ballast decreases as the size increases. Based on the Griffith strength theory, more micro-cracks exist in ballast particles with the increase of ballast particle size. The possibility of splitting

316

9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Fig. 9.6 Ballast compression test setup Fig. 9.7 Force-displacement curve in the crushing test

Fig. 9.8 The effective strength of ballast with different effective sizes

9.1 DEM Analysis of Ballast Particle Breakage

317

induced by the crack coalescence is greater under the radial load. As a result, the effective compression is lower. The brittle materials have greater dispersion of strength compared with elasticplastic materials. The Swedish scientist Weibull proposed an empirical formula for the stress and failure probability of brittle materials under tension in 1951, which has been widely adopted (Weibull 1951). Jayatllake studied the distribution characteristics of materials with microcracks, deduced a general expression of the failure probability, and gave the specific physical meanings of the parameters in the formula (Jayatilaka and Trustrum 1977). The Weibull empirical formula is based on two assumptions. The material is isotropic; The propagation of a certain crack in the material will cause the final failure of the material. The residual probability formula of the ballast breakage under the Weibull probability distribution is defined as,     σ − σu m Ps = 1 − P f = exp − σ0

(9.2)

where Ps is the residual probability; P f is the granular breakage probability; σ is the applied stress; σu is the stress when the failure probability is 0 (For brittle materials, σu = 0); σ0 is the tensile stress when the residual probability of ballast is 37%, which is also defined as the material characteristic strength based on Weibull statistics; m is a constant related to material properties also called Weibull modulus. The range of strength increases with the decreasing the value of m. The internal stress distributions of ballast particles are very complicated under external loading due to the diversity of ballast particles’ sizes and shapes. The multilevel fracture may occur. Therefore, two basic assumptions need to be made if the Weibull empirical formula is used to analyze the strength of ballast. Ballast particles should have similar geometries so that the internal stress distributions induced are the same when particles are subjected to external loadings; the particle’s breakage is caused by the internal tensile stress rather than surface crack propagation. The fracture probability and the tensile strength of the sample can be determined from the obtained experiment results using the probability statistics. The residual probability can also be written as,   m  σ Ps = exp − σ0

(9.3)

 Rewriting Eq. (9.3), the linear relationship between ln(ln(1 Ps )) and ln σ can be obtained as,   1 = m ln σ − m ln σ0 ln (9.4) Ps

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Fig. 9.9 Weibull residual probability of effective compressive strength of ballast

 where ln σ is the x-axis coordinate, ln(ln(1 Ps )) is the y-axis coordinate. σ0 is the tensile stress when residual probability is 37%, denoting the y-intercept. m represents the slope of the curve, which is also called Weibull modulus. The residual probability of the ballast is calculated by the average order method as follow, Ps = 1 −

i N +1

(9.5)

where Ps is the residual probability, i is the number of particle breakage under a certain stress, N is the total number of particles. Figure 9.9 shows the Weibull residual probability distribution of 147 samples. The calculated tensile stress σ = 9.39 MPa when the residual probability of ballast is 37%, and the Weibull modulus m is equal to 2.75.

9.2 DEM Analysis of Dynamic Response of Railway Ballast The elasticity of ballast bed is significantly influenced by its accumulated deformation. The discrete element analysis of dynamic process of ballast is helpful in revealing the deformation mechanism of ballast bed and providing a basis for degradation of railway ballast. Here, the accumulated settlement and elastic characteristics of railway ballast under cyclic loading is analyzed by the DEM with the irregularly shaped ballast constructed by clumping spheres.

9.2 DEM Analysis of Dynamic Response of Railway Ballast

319

Fig. 9.10 Ballast particles with irregular shapes

9.2.1 Construction of Irregularly Shaped Ballast The ballast particles are generally of irregular polyhedral shapes with corners and edges as shown in Fig. 9.10. Currently, the morphological description of irregular ballast particles mainly includes parallel-bonding (Ergenzinger et al. 2012; Lu and McDowell 2007; Lobo-Guerrero and Vollejo 2006), clumping (Wang et al. 2015; Laryea et al. 2014; Ferellec and McDowell 2010; Peters and Džiugys 2002), polyhedral and dilated polyhedral particles (Galindotorres et al. 2010; Alonso-Marroquín and Wang 2008). The clumping method is used here to the construct the irregular ballast given the complex configurations of ballast. Firstly, the corresponding polyhedral structure is generated according to the particle size and geometry. Secondly, spheres with smaller sizes are filled into polyhedra. Finally, the spheres inside are expanded to the designed sizes and form irregularly shaped ballast. Here, six ballast particles with different shapes are selected as shown in Fig. 9.11. The ballast particles are made up of several spheres with different overlaps and orientations. The volume, centroid and moment of inertia of particles can be determined by the finite partition method (Yan and Ji 2009). In view of the randomness of the effective size of ballast, the size ranges from 37 to 54 mm.

9.2.2 DEM Modelling of Ballast Box Test The size of the ballast box is set to be 700 mm × 300 mm × 450 mm. The main parameters are listed in Table 9.2. In the discrete element simulations, the ballast particles are randomly placed in the box. In order to obtain a compact state, the particle sizes are reduced by 2/5 times of their original sizes first, then gradually increased with time steps until their true

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Fig. 9.11 Discrete element models of railway ballast

Table 9.2 Main parameters in ballast box test Parameter

Unit

Value

Size of box

mm3

700 × 300 × 450

Size of sleeper

mm3

295 × 250 × 170

Size of ballast

mm

37–54

Stiffness of ballast

N/m

7.25 × 107

Density of ballast

kg/m3

2545.0

Friction coefficient between ballast

–

0.1–1.0

Friction coefficient between ballast and wall

–

0.2

Maximum applied force

kN

30.5

Minimum applied force

kN

5.5

sizes are reached. After the initial packing, the sleeper is slowly placed on the surface of the ballast material as shown in Fig. 9.12. The sleeper is slowly loaded to Fmean . Then a repeated sine loading is applied, with the maximum value Fmax = 30.5 kN, the minimum value Fmin = 5.5 kN and the average value Fmean = 18.0 kN, and the loading frequency f 0 = 3.0 Hz. The calculated results with 35 loading cycles are shown in Fig. 9.13. In the first loading cycle, the settlement is large. As the number of loading cycles increases, the settlement curve becomes more and more dense. In other words, the settlement of the sleeper tends to be stable. Figure 9.13c shows the evolution of accumulated settlement of sleepers with the loading numbers. It shows that the settlement of the sleeper changes periodically with the cyclic loading. The accumulated settlement increases with the increasing number of cycles, but showing a gentle increasing trend. This is mainly due to the fact that at the initial stage of loading, the pores

9.2 DEM Analysis of Dynamic Response of Railway Ballast

321

40

35

35

30

30

25

25

P (kN)

P (KN)

Fig. 9.12 The DEM model of the ballast box test

20 15

20 15 10

10

5

5 0

0 0

1

2

3

4

5

6

0

7

1

2

3

4

5

t (s)

u (mm)

(a) Cyclic sine loading

(b) Relationship between force P and settlement

7

u = 2.969N 0.239

6

u (mm)

5 4 3 2 1 0

0

5

10

15

20

25

30

35

N (cyclic loading number)

(c) Accumulated settlement u with cyclic loading number N Fig. 9.13 P − u curve of ballast during cyclic loading

6

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

between ballast particles are relatively large. Under the action of cyclic loadings, ballast particles move with each other, resulting in smaller pores and denser packing. As the number of loading cycles increases, the internal packing becomes continuously dense, and the accumulated settlement becomes stable. By the exponential fitting with cyclic number N, the accumulated settlement u can be expressed as, u = 2.969N 0.239

(9.6)

where the unit of settlement is mm. In order to further analyze the elastic deformation of ballast under cyclic loading, the force P on the sleeper is linearly fitted with its displacement, and the deformation modulus can be defined as, M=

P u

(9.7)

where the unit of deformation modulus M is N/m. Taking the second loading cycle as an example (as shown in Fig. 9.14a), the relationship between the load P and the sleeper settlement shows a fine linearity. The deformation modulus M = 12.34 × 106 N/m. Thus, the deformation modulus of 35 loading cycles is determined and the result is shown in Fig. 9.14b. During the initial cycles, the deformation modulus increases significantly. As the cycle number increases, the deformation modulus tends to be stable. Both the deformation modulus and accumulated settlement effectively reflect the compaction process of ballast under cyclic loading, which characterize the evolution law of the effective elastic deformation and permanent deformation of the ballast bed, respectively. The force chain is the major transmittal mode of contact forces between particles in granular materials. The microscopic analysis of the spatial structure and strength distribution of force chains helps revealing the basic mechanical behaviors of ballast. Here, the force chain distribution in a loading cycle is shown in Fig. 9.15. When 35

20

P (kN)

M (kN/mm)

N =2 P = 13.338u - 92.254

30 25 20 15 10 5

0

0.5

1

1.5

2

2.5

3

3.5

15 10 5 0

0

5

10

15

20

25

30

u (mm)

N

(a) Linear fitting between P-u

(b) Deformation modulus with cyclic number

Fig. 9.14 Relationship between force and settlement and evolution of deformation modulus

35

9.2 DEM Analysis of Dynamic Response of Railway Ballast

(a) t = 1.65s

(c) t = 1.81s

323

(b) t = 1.73s

(d) t = 1.98s

Fig. 9.15 Force chain distributions under cyclic loading

t = 1.65 s, the minimum force is applied on the sleeper. The resulted force chain is evenly distributed and its strength is weak due to the sufficient contacts between particles. When t = 1.81 s, the sleeper force applied on the ballast increases. The maximum contact force between ballast particles also increases. During this process, force chains break obviously and reconstruct, and the strong force chains concentrate radially under the sleeper. When t = 1.98 s, the force applied on the sleeper is unloaded to the minimum value. The contacts change again, resulting in the redistribution of force chains and the breakage of strong force chains. It is concluded that in the process of loading-unloading, the movement and rearrangement between ballast particles result in corresponding changes of the force chain structure.

9.2.3 Effect of Loading Frequency on Accumulated Settlement of Full-Scale Ballast Bed Railway ballast is currently subjected to dynamic train loadings with higher frequency components due to the continuous increase of train speed. The loading frequency has significant influence on ballast permanent deformation. It is of significant importance to take the effect of loading frequency into account when investigating the permanent

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

deformation of high-speed railway ballast under cyclic loadings (Zhang et al. 2019). The clumping model is also used here to construct the irregular ballast. Meanwhile, based on GPU parallel algorithm, a full scale discrete element model of the sleeperballast bed is established. The effects of the loading frequency on the mechanical properties of ballast bed are studied. The settlement of the ballast bed is obtained and its internal mechanism is explored. Figure 9.16 shows the DEM model of the ballast bed containing eight sleepers. The sleeper is simplified to be a rigid rectangular structure with a dimension of 2.6 m × 0.24 m × 0.174 m and a weight of 2 kN. The spacing between adjacent sleepers is 0.6 m. The total thickness of the bed is 0.35 m. The width of the upper ballast bed surface is 3.6 m, and the slope of is 1:1.75. A sine cyclic loading is applied vertically to the sleeper. The two ends of the ballast bed are constrained by rigid boundaries, which mean no interlocking effect on neighboring ballast particles. Hence, boundaries have a weaker restriction on the ballast rotation. In order to reduce the influence of the rigid boundary on the accumulated settlement, only the settlements of No. 3–No. 6 sleepers are considered and the movement of ballast in the corresponding zone is analyzed. Figure 9.17 shows the simulated deformation of the ballast bed under different loading frequencies. It indicates that the accumulated settlements are basically the same when the loading frequency is less than 25 Hz. When the loading frequency is between 25 and 60 Hz, the settlement increases steadily as the increasing loading frequency. When the frequency exceeds 60 Hz, the settlement increases sharply with the increase of loading frequency and reaches a maximum value at 80 Hz. Thereafter, when the loading frequency is further increased, the vertical displacement of the sleeper gradually decreases. In order to visually analyze the effect of loading frequency, the velocity vectors of ballast are plotted and represented by black arrows. Figure 9.18 shows the velocity distribution of ballast in the vertical section with a thickness of 10 cm in the middle of bed when the loading frequency is 60 Hz and 80 Hz, respectively. The maximum

Fig. 9.16 The full scale DEM model of ballasted railway bed

9.2 DEM Analysis of Dynamic Response of Railway Ballast

325

Fig. 9.17 The accumulated settlement of sleepers under different loading frequencies

(a) 60Hz loading frequency

(b) 80Hz loading frequency Fig. 9.18 The velocity distribution of ballast bed under different loading frequencies

loading is applied on sleeper No. 5. The velocity of ballast under the sleeper has a fan-shaped distribution obviously, showing a pronounced trend of vertical downward movement. The closer the ballast is to the bottom of the sleeper, the greater the vertical velocity is. Besides, when the loading frequency is 80 Hz, the velocity response is more intense than at 60 Hz. The movement of the ballast under other sleepers is also intense, and there are no obvious rules. However, the velocities of crib ballast are small. It indicated that the train loading transferred by the sleeper is mainly absorbed by the ballast under sleepers. In order to further analyze the vibration response of ballast particles at different locations, five observation points are selected and the maximum vertical and lateral acceleration of the ballast nearest to the observation points are calculated. As shown

326

9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Fig. 9.19 The locations of observation points

in Fig. 9.19, observation point No. 1, No. 2, and No. 3 are located under the bottom of sleeper No. 5, whose depths are 50 mm, 150 mm, and 300 mm, respectively. No. 4 and No. 5 are at the same depth as that of No. 2, but with a different lateral position. Figure 9.20 is the histogram of the vertical and lateral vibration acceleration of ballast at different loading frequencies. The results show the amplitudes of the vertical and lateral vibration acceleration at observation point No. 1–No. 3 decrease as the depth of the points increases at loading frequencies of 20–60 Hz. In addition, for all observation points, the amplitudes of the vertical and lateral vibration accelerations reach a maximum at loading frequency of 80 Hz, which indicates that the external loading frequency of 80 Hz is closer to the natural frequency of the discrete element model of the ballast bed.

9.3 Direct Shear Tests of Fouled Ballast Significant pores exist between ballast particles. Under the long-term operation of the track line, ballast particles break. Broken particles fill in the pores between ballast particles and make ballast bed fouled. Researchers have found not only broken ballast particles but also fine grains, such as fines from the subgrade, coal, sand grains and other contaminants, may infiltrate into the pores, which reduce the porosity of the ballast bed, lower the shear strength, produce large plastic deformation and reduce the drainage capacity of the track (Indraratna et al. 2009, 2014; Danesh et al. 2018). Different fouling materials, the proportion of fouling and the water content in the ballast bed can seriously affect the stability and performance of the ballast layer.

9.3.1 Direct Shear Tests of Fouled Ballast Direct shear tests of ballast containing different fine grains were carried out in order to determine the shear strength, maximum dilated displacement and the frictional angle of ballast at different normal loadings using a large direct shear apparatus. Figure 9.21 is the schematic of a large multi-functional direct shear apparatus in which the normal and horizontal actuator are used to provide a constant normal load and a horizontal tensile force, respectively. The normal displacement meter is installed to obtain the dilated displacement of ballast during the shearing process

9.3 Direct Shear Tests of Fouled Ballast

327

(a) Vertical vibration acceleration

(b) Lateral vibration acceleration Fig. 9.20 The histogram of the vertical and lateral vibration acceleration of ballast at different loading frequencies

by measuring the displacement of the actuator head. Meanwhile, the horizontal displacement meter is used to measure the horizontal shear displacement. The upper cover is used to transfer the constant load provided by the normal actuator to the ballast in the shear box. The linear guide at the bottom is used to limit the movement

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Fig. 9.21 The schematic of direct shear apparatus for ballast

of the lower shear box in the vertical direction and reduce the friction generated by the lower shear box in the process of sliding. The sizes of the upper and lower shear boxes are 400 mm × 400 mm × 200 mm and 500 mm × 400 mm × 200 mm, respectively. The lower shear box is 100 mm larger in length than the upper one, which ensures the area of the shear plane remains constant during the shear process. The direct shear test of the ballast material is shown in Fig. 9.22. The ballast material used here meets the requirements of special grade ballast in terms of lithology and texture. Its density is 2545 kg/m3 . Before the tests, the ballast particles were sieved according to Table 9.3. Ballast in different size ranges are then selected and thoroughly mixed to obtain a sample which meets the graduation requirement of special grade ballast. In order to explore the influence of different fine grains on the shear performance of the ballast bed, two typical fine grains, coal powder and sand grain, are selected. The density of the coal powder and sand grain are 1500 kg/m3 and 1650 kg/m3 , respectively. The coal powder and sand grains are mixed in each layer of the ballast. Fouled ballast with different content of fine grains are obtained by adding different masses of coal powder or fine sand. The PVC method is used to calculate the content α of fine grains in the ballast material, namely (Ebrahimi et al. 2014),

9.3 Direct Shear Tests of Fouled Ballast

329

Fig. 9.22 The laboratory direct shear test of ballast fouled by fine grains

Table 9.3 The ballast gradation in direct shear tests

Sieve size/mm

Mass percent passing sieve/% Gradation range of special grade ballast

Gradation in the experiment

22.4

0–3

2

31.5

1–25

18

40

30–65

55

50

70–99

85

63

100

α=

m s /ρs V − m b /ρb

100

(9.8)

where m s and ρs are the mass and density of the fine grain, m b and ρb are the total mass and density of ballast, respectively, V is the volume of the part located at the bottom of the upper cover.

9.3.2 Effect of Fine Grains on Shear Strength In order to reduce the influence of ballast fracture on the test results, new ballast with the same content of fine grains should be used in the new tests. Figure 9.23 shows the shear stress-strain curve of both sand-fouled and coal-fouled ballast when the normal stress σn = 50 kPa. The shear stress decreases as the content of fine

330

9 DEM Analysis of Mechanical Behaviors of Railway Ballast

(a) Coal-fouled ballast

(b) Sand-fouled ballast

Fig. 9.23 Shear stress-strain curve of fouled ballast with different contents of fine grains

grains increases. In addition, all shear stresses with different contents of fine grains fluctuate obviously, the stresses of clean ballast fluctuate the largest, and the shear stress fluctuates slightly after the shear strain reaches 9%. Besides, the fluctuation of curves in the shearing process is mainly caused by the displacement rearrangement and partial fragmentation between ballast particles (Ngo et al. 2014). The interval between shear-stress curves of the coal-fouled ballast is relatively larger than those of the sand-fouled ballast. The reason perhaps is that the coal powder is easy to break and adhere to the surface of ballast during the shear process, thereby reducing the sliding friction between ballast particles. With the increasing content of coal powder, its influence on the friction is more significant, resulting in lower shear strength of ballast. On the contrary, the strength of sand grain is larger and not easy to break and adhere to the surface of ballast. Therefore, the effect of sand grain on the shear strength of ballast is less than that of coal powder. The maximum shear stress in the direct shear process is used as the shear strength of the coal-fouled and sand-fouled ballast with different fine contents. According to the nonlinear strength envelope, the regression analysis is done by power function. The shear strength of ballast under different normal loads and fine grain contents is plotted, as shown in Fig. 9.24. The power function fitted curves all pass through the

(a) Coal-fouled ballast

(b) Sand-fouled ballast

Fig. 9.24 Shear strength of fouled ballast under different normal stresses

9.3 Direct Shear Tests of Fouled Ballast

331

origin point, and the shear strength of ballast increases with the increasing normal load at the same fine grain content. Meanwhile, both the coal powder and sand grain reduce the shear strength of ballast, and the effect of coal powder is significantly stronger than that of sand grain. When the normal load σn is 100 kPa, the shear strength of the clean ballast is 302.5 kPa; the shear strength of coal-fouled ballast is 162.5 kPa with 40% of coal; the strength of sand-fouled ballast is 209.25 kPa with 40% of sand. The shear strength is reduced about 46.3% and 30.8%, respectively, compared with the clean ballast.

9.3.3 Effect of Fine Grains on Dilatancy Behavior of Ballast Figure 9.25 is the normal displacement-shear strain curves of fouled ballast when the normal load σn = 50 kPa with different grain contents. When the normal displacement is positive, it indicates that dilatancy occurs in ballast material; on the contrary, the normal displacement is negative, it contracts. In the initial stage of the direct shear, the fouled ballast with different grain contents have undergone contraction. Compared with the sand-fouled ballast, the shear strain of the coal-fouled ballast is larger when the contraction between ballast particles changes to dilatancy. At this time, due to the low strength of coal powder, the internal pores between ballast containing coal powder are more easily squeezed to produce a larger volume compression. As the test continues, the normal displacement is all greater than zero when the shear strain reaches 15%. It shows that at the normal load of σn = 50 kPa, the ballast material exhibits obvious dilatancy characteristics. The normal displacement, when the shear strain is 15%, is used as the maximum dilated displacement of the fouled ballast mixture. The maximum dilated displacement under three different normal loads is recorded, as shown in Fig. 9.26. It shows the smaller the normal load is, the larger the dilated displacement is. Meanwhile, the effect of coal powder and fine sand on the dilated displacement of ballast is similar.

(a) Coal-fouled ballast

(b) Sand-fouled ballast

Fig. 9.25 Normal displacement-shear strain curves of fouled ballast under the normal load of 50 kPa

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

(a) Coal-fouled ballast

(b) Sand-fouled ballast

Fig. 9.26 The maximum dilatation of ballast under different grain contents

The maximum dilated displacement decreases with the increase of fine grain content. But the effect of coal powder on the dilatancy of ballast is more significant. When the normal load σn = 50 kPa, the dilated displacement of the clean ballast is 9.55 mm. The dilated displacement of ballast is 2.9 mm when the content of coal powder is 40%, while the dilated displacement of ballast with the same content of sand is 5.98 mm. The addition of coal powder or fine sand in the ballast material reduces the friction coefficient between ballast particles, thereby reducing the dilated tendency of the ballast during the direct shear process.

9.4 DEM Analysis of Shear Strength of Sand-Fouled Ballast In the wind-blown sand areas, sand accumulation in the ballast bed is very common. Fine sand particles may penetrate into the interior of the ballast bed. The sand grains in ballast bed will weaken the interlocking ability between ballast, resulting in the lower shear strength and large plastic deformation of the ballast bed (Indraratna et al. 2014). As a powerful numerical method for mechanical behaviors of discrete particles, the DEM can be effectively used to simulate the distribution of the internal force chain of sand-fouled ballast and the particles’ displacements. The deformation and the shear strength of the ballast bed can also be analyzed on the meso scale, hence, the macroscopic mechanical behaviors and shear deformation mechanism can be revealed.

9.4.1 DEM Simulations of Direct Shear of Sand-Fouled Ballast The mass of sand and ballast are determined by the normal probability distribution, thereafter their sizes are determined. The dimension of the shear box is 35 cm × 35 cm × 13 cm. The ballast and sand particles are randomly generated in a cubic

9.4 DEM Analysis of Shear Strength of Sand-Fouled Ballast

333

domain enclosed by the surfaces of the upper and lower shear boxes. The initial packing of the sand-rock mixture is shown in Fig. 9.27 with different contents of ballast in the box. After the particles are stabilized under gravity, a certain normal load is applied at the top wall of the upper shear box. At the same time, the lower box is moved horizontally to realize shear deformation. The maximum shear displacement is set to be 15% of the box length. The total normal force on the shear band between the upper and lower shear boxes consists of the weight W B of the upper shear box, the internal particles’ mass W P and the applied normal force P. The shear force on the shear plane is determined by the static equilibrium equation of the shear box in the horizontal direction. Thus, the normal and tangential forces on the shear band are given by, FN = P + W P + W B Fs =

N 

(9.9)

(Nwi + Swi )

(9.10)

i=1

where Nwi is the normal force between the particles and the left and right side walls of the upper shear box. Swi is the tangential friction between the particles and the front and rear walls of the upper shear box. The number of total walls is N = 5. Assuming that the length of the shear box is L, the width is W and the shear rate is V; the area of the shear band is W (L − Vt ) at time t. Then, the normal and shear stress on the shear band are defined as,

(a) Clean sand

(d) 60% ballast

(b) 20% ballast

(e) 80% ballast

(c) 40% ballast

(f) Clean ballast

Fig. 9.27 Initial packing of the sand-rock mixture with different ballast contents

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

σzz =

FN W (L − V t)

(9.11)

τzx =

Fs W (L − V t)

(9.12)

With the main calculation parameters listed in Table 9.4, the effects of ballast content on the shear strength and deformation of the ballast bed with sand are analyzed. When the ballast content γ = 60%, the process of the direct shear test is shown in Fig. 9.28. The shear deformation of the sample can be clearly seen during the shearing process. Different colors are adopted in order to identify the movements of particles. In the initial stage of the test, the intervals of different colors are clear. As the shear displacement increases, the particles near the shear band are mixed with each other. It indicates that these particles move more significantly compared with those far away from the shear band. Therefore, the deformation of the sample mainly concentrates near the shear band. The vertical displacement and the shear stress of the top wall at different shear displacements are shown in Fig. 9.29. When the shear displacement is less than 40 mm, the vertical displacement increases significantly and reaches its maximum at 40 mm shear displacement as the shear displacement increases; The vertical displacement remains substantially unchanged as the shear displacement continues to increase. The reason is that the particles in the box are closely packed at the initial stage of the test. During the shearing process, the particles move and exhibit shear dilatation. During the late stage of shearing, the vertical displacement of the sample top tends to be stable because the sufficient movements of the particles near the shear band causes the sample enter into a critical state. The sliding and rolling effects of the irregularly shaped ballast make the shear dilatation more pronounced during shearing. Based on the variation of shear stress, the shearing process can be roughly divided into three stages. (1) Linear elastic stage. In the initial stage of shearing, the relationship between shear stress and shear displacement is approximately linear. The Table 9.4 Main calculation parameters in DEM simulations of direct shear tests Parameter

Symbol

Unit

Value

Density of sand

ρ

kg/m3

1800.0

Density of ballast

ρ

kg/m3

2650.0

Elastic modulus of ballast

Eb

GPa

58.0

Elastic modulus of sand

Es

GPa

0.58

Friction between particles

μp

–

0.7

Friction between particles and side wall

μ

–

0.2

Shear rate

ν

m/min

1.6 × 10−2

Content of ballast

–

%

0.0–100

Sand size

r

mm

13.3–14.3

Ballast size

R

mm

37.4–54.4

9.4 DEM Analysis of Shear Strength of Sand-Fouled Ballast

(a) t = 0.30s

335

(b) t = 0.34s

(c) t = 0.35s

(d) t = 0.36s

1800

18

1500

15

1200

12

900

9

600

6 τ zz

300 0

z /mm

τ / kPa

Fig. 9.28 Shear deformation of ballast sample with γ = 60% at different times

0

10

20

30

40

50

3 0 60

x /mm Fig. 9.29 Shear stress and vertical displacement of the top wall at different shear displacements

shear stress increases rapidly. It indicates that the particles in the box are closely packed and undergo elastic deformation. Meanwhile, the relative motion between particles is not obvious. (2) Elastic-plastic deformation stage. As the shear displacement increases, the shear stress continues to increase, but with a slow increasing rate. At this stage, the contacts between ballast-ballast, ballast-sand and sand-sand

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

are sufficient, and the shear strength reaches a maximum value. (3) Residual stage. As the shear displacement continues to increase, the shear stress decreases and tends to stabilize. At this stage, the shear band in the box develops completely and enters a critical stable state.

9.4.2 Shear Strength and Force Chain Analysis with Different Ballast Contents The ballast content γ is an important factor affecting the shear strength of the sandfouled ballast. When the normal stress is 1.2 MPa, discrete element simulations of direct shear tests are carried out under the conditions of the ballast content γ = 0%, 20%, 40%, 60%, 80%, 100%, respectively. Figure 9.30a, b show the vertical displacement z and shear stress τ with shear displacement x with different ballast contents. Figure 9.30a shows that the movement of ballast results in extensive dilatancy with different ballast contents, and the dilatancy are more obvious with the increase of ballast content. This reflects that the influence of ballast on shear deformation becomes more and more dominant as the content of ballast increases. In other 2000

20

100%

15

80% 60%

10

1000

20

40

500 0

60

0

10

20

(a) Vertical displacement

40

(b) Shear stress 25

2000 剪切强度 max

20

竖向位移

1500

τ max / kPa

30

x / mm

x /mm

15 1000 10 500 0

z / mm

0

γ

100% 80% 60% 40% 20% 0%

1500

40% 20% 0%

5 0

τ / kPa

z /mm

25

5 0

20

40

60

80

0 100

γ/%

(c) Shear strength and maximum vertical displacement

Fig. 9.30 Effect of ballast content on vertical displacement and shear strength

50

60

9.4 DEM Analysis of Shear Strength of Sand-Fouled Ballast

337

words, the rotation and sliding of ballast during the shearing process lead to a rapid increasing vertical displacement. In addition, due to the relative motion of ballast particles, the vertical displacement exhibits a significant fluctuation as the ballast content increases. The shear stress curve also shows strong fluctuation as shown in Fig. 9.30b. It indicates that the shear stress of the sand-fouled ballast increases with the increasing ballast content. The possible reason is that the interlocking among ballast particles is pronounced with high ballast contents. When the ballast content γ < 40%, the shear stress does not change a lot. This is mainly because the pores between ballast particles are filled by fine sand, and ballast particles no longer act as the ‘skeleton’ of the mixture. In other words, the current sample presents a ‘sandy’ characteristic. In addition, the corresponding shear displacement is smaller as the maximum shear stress is reached with the increasing ballast content. Figure 9.30c shows the vertical displacement and shear strength with different ballast contents. Both the vertical displacement and the shear strength increase nonlinearly with the increase of the ballast content. When the ballast content is less than 40%, the sand grains play an important role in shear capacity of the specimen, and the shear strength increases slowly; when the ballast content is larger than 40%, the ballast particles play a leading role in the shear capacity due to the larger contact area of ballast near the shear band, and the shear strength increases rapidly. The force chain is the major mode of force transmission between particles in a granular system. It is helpful to reveal the effect of ballast content on the strength of sand fouled ballast by analyzing the spatial structure and intensity probability distribution of the force chain between sand-ballast contacts at microscopic scale. During the shearing process, the internal ballast particles collide and move with each other. The breakage and restruction of the weak force chain between sandsand contacts and the strong force chain between ballast-ballast contacts should be considered simultaneously during the direct shearing process. The distributions of the force chain at the maximum shear stress are shown in Fig. 9.31 when the ballast content is 0%, 20%, 40%, 60%, 80% and 100%, respectively. The results show that the lines representing the values of force chain become thicker and the color of the lines become darker with the increasing ballast content, which denotes a higher intensity of the force chain. With low ballast contents, the force chains are mainly formed by sand grains with less strength and dense distribution. With medium and high ballast contents, the force chains are mainly formed by ballast particles with larger strength and sparse distribution.

9.5 Coupled DEM-FEM Analysis of the Transition Zone Between Ballasted-Ballastless Track Bridges, culverts, tunnels and crossroads are widely applied in railway lines, resulting in foundations with different mechanical properties under the sleeper (Sayeed and Shahin 2016; Paixão et al. 2014; Song et al. 2019). When the train passes through

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

(a) γ = 0%

(d) γ = 60%

(b) γ = 20%

(e) γ = 80%

(c) γ = 40%

(f) γ = 100%

Fig. 9.31 Distributions of force chain with different ballast contents

the joints of different foundations, the train may jump slightly due to the sudden change of the stiffness of the foundation. Simultaneously, the corresponding track substructure suffers from huge impact loads (Sañudo et al. 2016). Moreover, the increase of train speed also increases the wheel-rail interaction. Under the longterm action of high-speed train load, the additional impact load will cause local irregularity of the track bed and increase the maintenance cost (Paixão et al. 2016; Lei and Mao 2004). In order to reduce the wheel-rail impact load caused by different foundation stiffnesses, it is necessary to set up a transition zone at the joint of the two different foundations to reduce the stiffness difference of the foundation at both ends of the transition zone. Therefore, a coupled discrete element-finite element model is introduced in order to analyze the mechanical behaviors of the transition zone between ballasted and ballastless tracks. The irregularly shaped ballast particles are constructed by clumping method. The bonding effect of polyurethane (ballast glue) reinforced in between ballast particles is analyzed by incorporating parallel bonding between clumps.

9.5.1 DEM-FEM Algorithm in Railway Ballasted Track 1. Coupled DEM-FEM algorithm For the coupled DEM and FEM, the key is the transmission mode of mechanical parameters in the discrete element and finite element domains. Using the DEM, the contact force and contact position can be obtained between discrete particles and the

9.5 Coupled DEM-FEM Analysis of the Transition …

339

exterior surface of the finite element. The contact forces are then applied to the nodes of the finite elements as force boundary conditions of the finite element domain, and the deformation of finite element can be solved. After that, the motion of the finite element surface will be used as the displacement boundary condition of the discrete element domain in the next time step. As a result, a coupling discrete element and finite element model is established. Unfortunately, most of the contact forces on the coupling surface are usually not located on the nodes of finite elements. Therefore, the interpolation method of shape function is used. According to the principle of conservation of energy, the work done by contact forces of discrete elements on the contact surface is, δW = δU T Fcon

(9.13)

where Fcon is the contact force vector on the coupled contact surface. U is the displacement vector of the contact point, which can be interpolated by nodal displacement and shape function. The expression of the displacement is, U = Ni20 u i , i = 1 ∼ 20

(9.14)

where Ni20 is the shape function of the 20-node iso-parametric element at the contact point. u i is the displacement of node i. Substituting Eq. (9.14) into (9.13), we get,  T δW = δ u iT Ni20 Fcon , i = 1 ∼ 20

(9.15)

The virtual work on a node of the finite element is given by, δW = δu iT Fnodal,i , i = 1 ∼ 20

(9.16)

Thus, the relationship between the equivalent nodal force and the contact force can be expressed as,  T Fnodal,i , i = Ni20 Fcon , i = 1 ∼ 20

(9.17)

with Ni20 =

(1 + ε0 )(1 + η0 )(1 + ζ0 )(ε0 + η0 + ζ0 − 2) ,i = 1 ∼ 8 8

 1 − ε2 (1 + η0 )(1 + ζ0 ) 20 Ni = , i = 17 ∼ 20 4 

1 − η2 (1 + ζ0 )(1 + ε0 ) 20 , i = 9, 11, 13, 15 Ni = 4

(9.18) (9.19) (9.20)

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Ni20

 1 − ζ 2 (1 + ε0 )(1 + η0 ) = , i = 10, 12, 14, 16 4 ε0 = εi ε, η0 = ηi η, ζ0 = ζi ζ

(9.21) (9.22)

where (ε, η, ζ ) (−1 ≤ ε, η, ζ ≤ 1) is the coordinates of the contact point in the local coordinate system. In the process of interaction between the granular material and the continuum, the contact points will always be located on the exterior surface of the finite element since the breakage of the continuum is not considered. Here the discrete particles are assumed to act on the upper surface of the finite element mesh. Thus, ζ = 1. The values of ε and η can be solved by the Newton iterative method using the shape function of 8-node iso-parametric planar element, as shown in Fig. 9.32. Ux =

8 

Ni8 u i x , U y =

i=1

8 

Ni8 u i y

(9.23)

i=1

where Ni8 satisfies the following equations, Ni8 = −

(1 + εi ε)(1 + ηi η)(1 − εi ε − ηi η) ,i = 1 ∼ 4 4 

1 − ε2 (1 + ηi η) 8 , i = 5, 7 Ni = 2 

1 − η2 (1 + εi ε) 8 , i = 6, 8 Ni = 2

(9.24) (9.25) (9.26)

where (x, y) is the coordinate of the contact point in the global coordinate system. (x i , yi ) is the node coordinate of the finite element in the global coordinate system. Ni8 is the shape function of the 8-node iso-parametric planar element. Fig. 9.32 The diagram of 8-node iso-parametric element for solving ε and η

9.5 Coupled DEM-FEM Analysis of the Transition …

341

8 8 Let f (ε, η) = x − i=1 Ni8 xi = 0 and g(ε, η) = y − i=1 Ni8 yi = 0, then ε and η can be solved by the Newton iteration method, ε = εk +

f (εk , ηk )gη (εk , ηk ) − g(εk , ηk ) f η (εk , ηk ) gε (εk , ηk ) f η (εk , ηk ) − f ε (εk , ηk )gη (εk , ηk )

(9.27)

η = ηk +

g(εk , ηk ) f ε (εk , ηk ) − f (εk , ηk )gε (εk , ηk ) gε (εk , ηk ) f η (εk , ηk ) − f ε (εk , ηk )gη (εk , ηk )

(9.28)

∂ where εk and ηk are the initial values. f ε (εk , ηk ) = ∂ε f (εk , ηk ), f η (εk , ηk ) = ∂ ∂ ∂ f , η , η g(ε , η , η g(ε , ηk ). g = g = (ε ), (ε ) ), (ε ) k k ε k k k k η k k k ∂η ∂ε ∂η

2. Coupled DEM-FEM in case of discrete particles embedded in the finite element Under the long-term action of cyclic loadings of the train, ballast particles are gradually embedded into the foundation of the ballast bed which in turn restrains the movement of ballast particles. In order to analyze the embedding process of ballast particles in the ballasted track, a new coupled DEM-FEM method considering discrete particles embedded in the finite element is developed. The calculation model is shown in Fig. 9.33. A number of embedded discrete particles, called embedded particles, are placed at the edges of the finite element mesh. The displacements of the embedded particles are not controlled by the discrete element, but the resultant force of the free particles in the discrete element acting on the embedded element can be calculated by the contact model between the particles. This resultant force is the force boundary condition for the finite element calculations. For the embedded particles, the force boundary condition of the finite element can be obtained by static equilibrium analysis,

Fig. 9.33 Schematic of the coupled DEM-FEM for discrete particles embedded in the finite element

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

FF E M =

n 

FiD E M

(9.29)

i=1

where FF E M is the finite element force applied by the discrete particle, i is the number of embedded particle, n is the total number of embedded particles, FiD E M is the resultant force of discrete free particle on each embedded particle. After the completion of the discrete element calculations at each time step, FiD E M can be determined and sequentially added to the node of the finite element as the external load to obtain the equivalent nodal load. The schematic of the coupling process between a single embedded particle and a finite element is shown in Fig. 9.34. According to the principle of energy conservation, the relationship between the equivalent nodal force of the finite element and the resultant force of the embedded particle can be obtained by, Fnodal,i =

n 

[Ni ]T F jD E M , i = 1 ∼ 8

(9.30)

j=1

where Ni is the shape function at the centroid of embedded particle and can be calculated as follows, Ni =

(1 + εεi )(1 + ηηi )(1 + ζ ζi ) 8

(9.31)

In order to obtain the local coordinates of the embedded particle, the following equations are to be solved, Ux =

8  i=1

Ni u i x , U y =

8 

Ni u i y , U z =

i=1

8 

Ni u i z

i=1

Let f (ε, η, ζ ), g(ε, η, ζ ) and h(ε, η, ζ ) satisfy the follow equations,

Fig. 9.34 Mapping between global and local coordinate systems

(9.32)

9.5 Coupled DEM-FEM Analysis of the Transition …

f (ε, η, ζ ) = x −

8 

343

Ni x i = 0

i=1

g(ε, η, ζ ) = y −

8 

Ni yi = 0

i=1

h(ε, η, ζ ) = z −

8 

Ni z i = 0

(9.33)

i=1

Assuming that f (ε, η, ζ ) is continuous at the neighborhood of (ε0 , η0 , ζ0 ) with a second order continuous partial derivative, and (ε0 + l, η0 + m, ζ0 + n) is an arbitrary point in the neighborhood, thus, f (ε0 + l, η0 + m, ζ0 + n) ≈ f (ε0 , η0 , ζ0 )   ∂ ∂ ∂ f (ε, η, ζ )|ε=ε0 + m f (ε, η, ζ )|η=η0 + n f (ε, η, ζ )|ζ =ζ0 + l ∂ε ∂η ∂ζ

(9.34) Equation (9.34) can then be approximately expressed as,   ∂ ∂ ∂ f (ε, η, ζ )|ε=εk + m f (ε, η, ζ )|η=ηk + n f (ε, η, ζ )|ζ =ζk = 0 f (εk , ηk , ζk ) + l ∂ε ∂η ∂ζ

(9.35)

That is, f (εk , ηk , ζk ) + (ε − εk ) f ε (εk , ηk , ζk ) + (η − ηk ) f η (εk , ηk , ζk ) + (ζ − ζk ) f ζ (εk , ηk , ζk ) = 0

(9.36)

Similarly, we can get, g(εk , ηk , ζk ) + (ε − εk )gε (εk , ηk , ζk ) + (η − ηk )gη (εk , ηk , ζk ) + (ζ − ζk )gζ (εk , ηk , ζk ) = 0

(9.37)

h(εk , ηk , ζk ) + (ε − εk )h ε (εk , ηk , ζk ) + (η − ηk )h η (εk , ηk , ζk ) + (ζ − ζk )h ζ (εk , ηk , ζk ) = 0

(9.38)

 



When f ζ gε − gζ f ε (gη h ε −h η gε )−(gζ h ε −h ζ gε ) f η gε − gη f ε = 0, combining Eqs. (9.36)–(9.38), 





 f gζ − g f ζ gη h ζ − h η gζ − gh ζ − hgζ f η gζ − gη f ζ 





 ε = εk +

f ζ gε − gζ f ε gη h ε − h η gε − gζ h ε − h ζ gε f η gε − gη f ε  



( f gε − g f ε ) gζ h ε − h ζ gε − (gh ε − hgε ) f ζ gε − gζ f ε 





 η = ηk +

f ζ gε − gζ f ε gη h ε − h η gε − gζ h ε − h ζ gε f η gε − gη f ε

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

 



(gh ε − hgε ) f η gε − gη f ε − ( f gε − g f ε ) gη h ε − h η gε 





 ζ = ζk +

f ζ gε − gζ f ε gη h ε − h η gε − gζ h ε − h ζ gε f η gε − gη f ε

(9.39)

The deformation of the finite element mesh can be obtained according to the equivalent nodal force of the finite element. The coordinates of the embedded particle’s centroid are then updated by the shape function and the nodal displacement of the finite element. The motion of the embedded particle is used as a displacement boundary condition in the subsequent discrete element calculations. Finally, the bi-directional coupling process between the discrete element and finite element is completed.

9.5.2 DEM-FEM Model for Ballasted-Ballastless Transition Zone The clumping method is adopted to construct the irregularly shaped ballast particles. The size of the generated ballast particles ranges from 21 to 63 mm with an average of 40 mm. The finite element model of the ballastless bed is shown in Fig. 9.35, in which a 20-node iso-parametric element is used including 775 elements and 4018 nodes. The linear elastic model is used for dynamic calculation. The boundary conditions in the coupled DEM-FEM model are as follows. The upper surface of the ballastless bed is free, the horizontal displacement constraints is applied on the boundary to the right; in contrast, the boundary to the left is free, both the front and rear boundaries are subjected to the displacement constraints in y direction, the bottom of the model is constrained by vertical displacement. In order to reduce the difference in stiffness between the ballasted and ballastless track, the ballast glue is used to reinforce the strength of the ballasted track, as shown in Fig. 9.36. Fig. 9.35 The finite element model of the ballastless bed

9.5 Coupled DEM-FEM Analysis of the Transition …

345

Fig. 9.36 Ballast reinforced by polyurethane material

The transitional area between ballasted and ballastless track are shown in Fig. 9.37. The polyurethane material can provide additional cohesion between ballast particles, resulting in an increase in the stiffness of the ballast bed (Coelho et al. 2011). In order to ensure an uniformly reduced structural rigidity from the ballastless to the ballasted track, the consumption volume of polyurethane material needs to be gradually reduced. In order to analyze the influence of polyurethane volume on the mechanical properties of the ballast bed, four parts are selected in the transition zone, as shown in Fig. 9.37b, that is, the unbounded part, semi-bonded part, completely bonded part and the coupling mode of the ballast and ballastless track. In the DEM-FEM coupling method, the ballast bed and the ballastless bed are simulated by the DEM and the FEM, respectively. The coupling model is used at the interface of the ballasted and ballastless track. In the numerical model as shown in Fig. 9.38, the dimensions of both the ballast and ballastless bed are 0.45 m × 0.23 m × 0.3 m. The generated ballast bed consists of 690 clumps containing 4027 constitutive spherical particles. In order to raise the computational efficiency, some local parts in the transition zone are selected and analyzed. Actually, the ballast region and the ballastless region are not separated and connected to each other by a certain means. Although the established numerical models are different from the actual railway track bed, they can still provide a basis for analyzing the effect of the polyurethane on the ballast bed.

9.5.3 Settlement Analysis of the Transition Zone In order to analyze the effect of polyurethane content on the settlement of the ballast bed, five observation points were selected as illustrated in Fig. 9.37, with the 5th observation point being under the sleeper in the ballastless track. The settlements

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

(a) Ballasted-ballastless transition zone on site

(b) Schematic of ballasted-ballastless transition zone Fig. 9.37 Ballasted-ballastless transition zone on site and its schematic

of the five observation points are shown in Fig. 9.39. It clearly shows that the polyurethane has a significant effect on the settlement. The settlement of the ballast bed without polyurethane is the largest; With the increasing polyurethane content, the strength of the ballast bed becomes higher. In the immediate transition from the ballasted to the ballastless tracks, the difference of settlement is not obvious. The simulated results show the bonding effect of the polyurethane can reduce the settlement of the sleepers. In addition, the settlement decreases with the increasing injection depth of the polyurethane (Jing et al. 2012). It indicates that polyurethane material is useful to reduce the settlement of the ballast bed. It should be noted that the tamping process of ballast is not taken into account in the DEM-FEM simulation, which result in larger settlement compared with the actual field measurements. In order to further investigate the reinforcement effect of the polyurethane material, the force chain distributions of both unbonded and completely bonded ballast are shown in Fig. 9.40. In the unbonded ballast, the force chain exhibits a trapezoidal distribution, and the ballast bed can transfer most of the train load to the foundation. In the completely bonded ballast, the force chain distribution is more uniform. Due

9.5 Coupled DEM-FEM Analysis of the Transition …

(a) Unbonded ballast bed

(c) Completely bonded ballast bed

347

(b) Semi-bonded ballast bed

(d) Transitional area between ballast track and ballastless track

Fig. 9.38 Numerical models of ballasted and ballastless track bed

Fig. 9.39 Settlement at observation points of the transition zone

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

(a) Unbonded ballast bed

(b) Completely bonded ballast bed

Fig. 9.40 Force chain distributions of unbonded and completely bonded ballast bed

to the bonding action of the polyurethane material, the left and right sides of the upper ballast bed can also withstand part of the train load. The polyurethane can provide additional constraint force between ballast particles, thus improving the bearing capacity and self-locking effect of the railway ballasted track (Wang et al. 2013).

9.5.4 Coupled DEM-FEM Analysis Considering Ballast Embedded in Ballastless Bed A coupled DEM-FEM model of the transition zone considering discrete particles embedded in the finite element structure is established, as shown in Fig. 9.41. The dimension of the ballast bed is 0.4 m × 0.38 m × 0.23 m. The total ballast bed consists of 736 clumps and 4291 spherical particles. The coupled region is composed of 46

Fig. 9.41 Coupled DEM-FEM model of the transition zone considering ballast particles embedded in ballastless bed

9.5 Coupled DEM-FEM Analysis of the Transition …

349

Fig. 9.42 Force chain distribution of ballast bed and stress nephogram of ballastless bed in z direction

clumps and 264 spherical particles. The finite element model of the ballastless bed consists of the track bed and a sleeper. The 3D 8-node iso-parametric element is used to model the ballastless track containing 775 elements and 4018 nodes. In the ballastless bed, the elastic modulus is set to be 3.5 × 104 MPa and the Poisson’s ratio is 0.17. The left surface of the model is free, which is only constrained by the ballast in the bed. The right surface is subjected to the displacement constraint in the x direction. The front and rear surfaces are subjected to the displacement constraint in the y direction. The lower surface is applied with the displacement constraint in the z direction. Finally, the cyclic train load is applied on the upper side of the sleeper in an in-plane form. When the cyclic loading reaches the peak, the force chain distribution and the zdirection stress nephogram in the ballastless bed are shown in Fig. 9.42. It shows that the internal strong force chain concentrates under the sleeper, showing a trapezoid shape. The stress concentration is mainly distributed in the lower part of the track bed, which is consistent with the distribution of the strong force chain in the ballast bed. Meanwhile, it can be seen from the force chain image that the strong force chain is not bilateral symmetric, in which the strong force chain distribution on the right side is higher than that on the left side. The reason is that the particles are embedded in the finite element mesh in the coupling region and its displacement is not affected by the finite element. Namely, it increases the roughness on the interface between the ballast bed and the ballastless bed, resulting in a pronounced interlocking between ballast particles in the coupling region, which can effectively limit the displacement of ballast particles near the interface and reduce the settlement of the ballast bed. The displacement nephogram of the ballastless bed at the peak load is shown in Figs. 9.43 and 9.44. The displacement deformation in the ballastless bed corresponds to the stress distribution in the ballast bed. It can be seen that the displacement deformation of the ballastless bed in the x direction is mainly concentrated in the bottom left of the

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Fig. 9.43 Force chain distribution of ballast bed and displacement nephogram of ballastless bed in x direction

Fig. 9.44 Force chain distribution of ballast bed and displacement nephogram of ballastless bed in z direction

ballastless bed due to the load resulting from the ballast particles on the right surface of the ballast bed. The displacement is large. The displacement of the ballastless bed in the z direction is mainly located under the sleeper and the displacement on the left side is slightly larger than that on the right side.

9.6 DEM Simulations of Railway Ballast Bed …

351

9.6 DEM Simulations of Railway Ballast Bed with Dilated Polyhedral Particles In this section, the dilated polyhedral particles are used to model the irregularly shaped ballast. The settlement and effective stiffness of the ballast bed are investigated through ballast box tests under cyclic loadings.

9.6.1 Ballast Box Test Model with Dilated Polyhedral Particles The inter-locking effect between ballast particles can be better simulated considering the irregular shapes of ballast. The dilated polyhedral particles are adopted here in order to resemble the irregular shapes of ballast particles, as shown in Fig. 9.45.

Fig. 9.45 Generated dilated polyhedral particles

Fig. 9.46 Gradation of ballast used in the DEM simulations

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

In addition, the settlement of ballast bed is found to be sensitive to its aggregate gradation (Tutumluer et al. 2013). The current railway industry standard TB/T 2140-2008 of China is adopted in the generation of ballast particles. Figure 9.46 shows the corresponding particle size distribution (PSD) curve of ballast. The ballast sample is obtained by calculating the mass of ballast passing each sieve size, generating ballast particles based the masses of particles and mixing the generated ballast particles thoroughly. The computational domain, as shown in Fig. 9.47, is based on the work of Lim and McDowell (McDowell et al. 2004). The ballast samples is first dropped into a box of length 700 mm and width 300 mm under the action of gravity. Next, a wall is applied and loaded with a sinusoidal loading on the top to make the sample achieve

Fig. 9.47 Computational domain in ballast box test

Fig. 9.48 The DEM model of the ballast box test

9.6 DEM Simulations of Railway Ballast Bed …

353

the initial compacted state. Finally, the top wall is removed and a sleeper of length 300 mm, width 300 mm and mass 34 kg is placed on the top of the ballast until ballast comes to static state. After the initial preparation, a cyclic loading is applied on the sleeper. In order to model the dynamic behaviors of ballast bed on large scale, periodic boundaries are applied in the y direction of the computational domain. The generated DEM model of the ballast box test is shown in Fig. 9.48. In order to check the influence of domain width, two additional width of 1.4 and 2.1 m are used, resulting in two and three sleepers resting on ballast as shown in Fig. 9.49a, b. In the above simulations, the dilating sphere of radius is 4 mm, the amplitude of the sinusoidal loading is from 3 to 40 kN, the frequency is 6 Hz, and

(a) Domain width L = 1.4m

(b) Domain width L = 2.1m

(c) Variation of cyclic loading with axial strain at various domain widths Fig. 9.49 Influence of domain width

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

Table 9.5 Main parameters used in the simulations

Parameters

Symbols

Unit

Ballast density

ρ

kg/m3

2600

Shear modulus

G

GPa

20

Values

Poisson’s ratio

ν

0.3

Coefficient of restitution

ε

0.8

Coefficient of sliding friction

μ

0.5

Time step

dt

S

2.1 × 10−6

Total mass of ballast

M

Kg

110

simulation time is 2 s. The main computational parameters are listed in Table 9.5. Figure 9.49c plots the variation of cyclic load with axial strain at various domain widths. The curves are very close and the influence of the domain width can be ignored. Hence, the domain width of 0.7 m is used in the following studies.

9.6.2 Settlement of Ballast Bed Under Various Cyclic Loading Frequencies Because of the increasing demand of high speed trains, five loading frequencies of 3, 6, 10, 20 and 30 Hz are chosen here. Figure 9.50 plots the axial strain of the sleeper which is defined as the ratio of the sleeper settlement to the initial thickness of ballast bed under different cyclic loading frequencies. It shows that the axial strain increases continuously with fluctuation. During the same loading period, the higher the loading frequency is, the larger the axial strain is. Fig. 9.50 The axial strain under various cyclic loading frequencies

9.6 DEM Simulations of Railway Ballast Bed …

355

Fig. 9.51 The effective stiffness versus simulation time

The effective stiffness is an important parameter for ballast bed, which is defined as the slope of the strain-stress curve in the process of loading. Figure 9.51 shows the effective stiffness as a function of simulation time, which generally increases with simulation time. In initial cyclic loading, the effective stiffness of ballast bed fluctuates greatly due to the unstable spatial distribution of the ballast bed. As the cyclic number increases, the ballast bed structure tends to be stable and the effective stiffness increases. The higher the frequency, the bigger the effective stiffness is. The effective stiffness and accumulated settlement reflect the compacting process of the ballast bed under the cyclic load and characterize the elastic and permanent deformation of the ballast bed.

9.7 Summary This chapter mainly introduces the application of the DEM and coupled DEM-FEM method in simulations of mechanical behaviors of railway ballast and ballast related issues. The settlement, degradation mechanism and deformation of the ballast bed under cyclic loading are studied. Based on the bonding of spheres, the irregular ballast particles were constructed. The radial loading method was used to simulate the breakage process of a single ballast particle. The evolution process of failure was obtained, and the compressive strength of ballast was determined. The numerical results of the ballast box test show that the settlement of the ballast bed is closely related to its elasticity. On one hand, the more easily the ballast bed settles, the smaller the deformation modulus of the ballast bed is; On the other hand, the accumulated deformation will seriously affect the elasticity of the ballast bed in turn. A full scale DEM model of ballast bed was established based on GPU parallel algorithm, the settlement and vibration behaviors under different loading frequencies were analyzed systematically.

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9 DEM Analysis of Mechanical Behaviors of Railway Ballast

The shear performance of fouled ballast was studied by a large direct shear device and DEM numerical simulations as well. On the microscopic scale, the shear strength of ballasts is analyzed based on force chain distribution, revealing the shear strength and deformation mechanism of ballast particles under macroscopic conditions. The coupled DEM-FEM was used to analyze the dynamic characteristics of the transition zone between ballasted and ballastless track. Based on the study of settlement of ballast bed, the internal mechanism of non-uniform settlement of the transition zone was explored. Finally, dilated polyhedral particles were constructed based on the Minkowski sum operator in order to resemble the irregular shapes of ballast particles. The relationship between the settlement of the sleeper and the applied load was obtained by numerical simulation. By linear fitting method, the effective stiffness of the ballast bed and the accumulated settlement of the sleeper were obtained with number of cycles. Meanwhile, the dilated polyhedral particles can reasonably simulate the dynamic behaviors of ballast and reveal the corresponding macroscopic evolution law by its mesoscopic characteristics.

References Alonso-Marroquín F, Wang Y (2008) An efficient algorithm for granular dynamics simulation with complex-shaped objects. Granul Matter 11(5):317–329 Bian X, Li W, Qian Y et al (2019) Analysing the effect of principal stress rotation on railway track settlement by discrete element method. Géotechnique, 1–19 Coelho B, Hölscher P, Priest J et al (2011) An assessment of transition zone performance. J Rail Rapid Transit 225(1):1–11 Danesh A, Palassi M, Mirghasemi AA (2018) Effect of sand and clay fouling on the shear strength of railway ballast for different ballast gradations. Granul Matter 20(3):51 Ergenzinger C, Seifried R, Eberhard P (2012) A discrete element model predicting the strength of ballast stones. Comput Struct 108:3–13 Ebrahimi A, Tinjum JM, Edil TB (2014) Deformational behavior of fouled railway ballast. Can Geotech J 52(3):1–12 Ferellec J, McDowell G (2010) Modelling realistic shape and particle inertia in DEM. Geotechnique 3:227–232 Galindotorres SA, Muñoz JD, Alonsomarroquín F (2010) Minkowski-Voronoi diagrams as a method to generate random packings of spheropolygons for the simulation of soils. Phys Rev E Stat Nonlinear Soft Matter Phys 82(2):056713 Hossain Z, Indraratna B, Darve F et al (2007) DEM analysis of angular ballast breakage under cyclic loading. Geomech Geoengin 2(3):175–182 Huang H, Tutumluer E (2011) Discrete element modeling for fouled railroad ballast. Constr Build Mater 25:3306–3312 Indraratna B, Vinod JS, Lackenby J (2009) Influence of particle breakage on the resilient modulus of railway ballast. Geotechnique 59(7):643–646 Indraratna B, Thakur PK, Vinod JS (2010) Experimental and numerical study of railway ballast behavior under cyclic loading. Int J Geomech 10(4):136–144 Indraratna B, Nimbalkar S, Coop M et al (2014) A constitutive model for coal-fouled ballast capturing the effects of particle degradation. Comput Geotech 61:96–107

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Indraratna B, Sajjad MB, Ngo T et al (2019) Improved performance of ballasted tracks at transition zones: a review of experimental and modelling approaches. Transp Geotech, 100260 Jing GQ, Shao L, Zhu YD et al (2012) Micro-analysis of railway ballast bond effects. Adv Mater Res 446–449:2492–2496 Jaeger JC (1967) Failure of rocks under tensile conditions. Int J Rock Mech Min Sci Geomech Abstracts Pergamon. Pergamon 4(2):219–227 Jayatilaka AS, Trustrum K (1977) Statistical approach to brittle fracture. J Mater Sci 12(7):1426– 1430 Laryea S, Baghsorkhi MS, Ferellec JF, Well GR, Chen C (2014) Comparison of performance of concrete and steel sleepers using experimental and discrete element methods. Transp Geotech 1:225–240 Lei XY, Mao LJ (2004) Dynamic response analyses of vehicle and track coupled system on track transition of conventional high speed railway. J Sound Vib 271(s3–5):1133–1146 Lim WL, Well GR, Collop AC (2004) The application of Weibull statistics to the strength of railway ballast. Granul Matter 6(4):229–237 Lim W, McDowell G (2005) Discrete element modelling of railway ballast. Granul Matter 7:19–29 Liu GY, Xu WJ, Sun QC et al (2019) Study on the particle breakage of ballast based on a GPU accelerated discrete element method. Geosci Front Lobo-Guerrero S, Vallejo LE (2006) Discrete element method analysis of rail track ballast degradation during cyclic loading. Granul Matter 8:195–204 Lu LY, Gu ZL, Lei KB et al (2010) An efficient algorithm for detecting particle contact in nonuniform size particulate system. Particuology 8(2):127–132 Lu M, McDowell GR (2007) The importance of modelling ballast particle shape in the discrete element method. Granul Matter 1(1–2):69–80 McDowell GR, Bolton MD (1998) On the micromechanics of crushable aggregates. Geotechnique 48(5):667–679 McDowell GR, Lim WL, Collop AC, Armitage R, Thom NH (2004) Comparison of ballast index tests for railway trackbeds. Geotech Eng 157:151–161 Ngo NT, Indraratna B, Rujikiatkamjorn C (2014) DEM simulation of the behavior of geogrid stabilized ballast fouled with coal. Comput Geotech 55:224–231 Paixão A, Fortunato E, Calçada R (2016) A contribution for integrated analysis of railway track performance at transition zones and other discontinuities. Constr Build Mater 111:699–709 Paixão A, Fortunato E, Calçada R (2014) Transition zones to railway bridges: track measurements and numerical modeling. Eng Struct 80:435–443 Peters B, Džiugys A (2002) Numerical simulation of the motion of granular material using objectoriented techniques. Comput Methods Appl Mech Eng 191:1983–2007 Potyondy DO, Cundall PA (2004) A bonded-particle model for rock. Int J Rock Mech Min Sci 41(8):1329–1364 Sañudo R, dell’Olio L, Casado JA et al (2016) Track transitions in railways: a review. Constr Build Mater 112:140–157 Sayeed MA, Shahin MA (2016) Three-dimensional numerical modelling of ballasted railway track foundations for high-speed trains with special reference to critical speed. Transp Geotech 6:55–65 Song W, Huang B, Shu X et al (2019) Interaction between railroad ballast and sleeper: a DEM-FEM approach. Int J Geomech 19(5):04019030 Stahl M, Konietzky H (2011) Discrete element simulation of ballast and gravel under special consideration of grain-shape, grain-size and relative density. Granul Matter 13:417–428 Tutumluer E, Qian Y, Hashash Y et al (2013) Discrete element modeling of ballasted track deformation behavior. Int J Rail Transp 1:57–73 Wang ZJ, Jing GQ, Liu GX (2013) Analysis on railway ballast-glue micro-characteristics. Appl Mech Mater 477–478:535–538 Wang Z, Jing G, Yu Q et al (2015) Analysis of ballast direct shear tests by discrete element method under different normal stress. Measurement 63:17–24

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Weibull W (1951) A statistical distribution functions of wide applicability. J Appl Mech 18(3):293– 297 Yan Y, Ji S (2009) Discrete element modeling of direct shear tests for a granular material. Int J Numer Anal Meth Geomech 34(9):978–990 Yan Y, Zhao J, Ji S (2015) Discrete element analysis of breakage of irregularly shaped railway ballast. Geomech Geoengin 10(1):1–9 Zhang X, Zhao C, Zhai W (2019) Importance of load frequency in applying cyclic loads to investigate ballast deformation under high-speed train loads. Soil Dyn Earthq Eng 120:28–38

Chapter 10

DEM Analysis of Vibration Reduction and Buffering Capacity of Granular Materials

Granular materials is a complex system, and the energy of the system is mainly dissipated through inelastic collisions and sliding friction between particles (Royer et al. 2007; Ramírez et al. 1999). Under the external load, strong compression and collision between particles have been observed. The force chain structure within the granular system is broken and reorganized, thereby consuming the energy of the system (Brzinski et al. 2013; Kim et al. 2018; Kondic et al. 2012). When the collision between particles is inelastic, there is viscous and plastic deformation between particles, and a part of the energy is absorbed through the plastic deformation (Du et al. 2010). In granular systems, the contact force is transmitted through the force chain, and the local impact load is continuously expanded in the 3D space. As a result, the impact force is reduced (Omidvar et al. 2014; Umbanhowar and Goldman 2010). The evolution of the force chain can reflect the inter-particle forces, which makes the instantaneous impact load delayed and reflects a certain buffering performance. Therefore, granular materials have excellent energy dissipation characteristics (Sakamura and Komaki 2012) and can be designed as a granular damper (Fraternali et al. 2009). In this chapter, the vibration reduction of granular materials and its application in particle dampers are first introduced. Next, the buffering capacity of granular materials will be investigated using the DEM and the mechanical experiments as well. Finally, the theory of the dissipated energy of granular materials is used to analyze the landing impact process of the lunar lander.

10.1 DEM Simulations and Experimental Tests of Vibration Reduction of Granular Materials The structural vibrations are common in the daily life and engineering production, which brings serious harm to people’s life and industrial production (Bai et al. 2009). Therefore, the structural vibration should be reduced to decrease its adverse effects. The damping technique is an effective approach to reduce structure vibrations and © Science Press and Springer Nature Singapore Pte Ltd. 2020 S. Ji and L. Liu, Computational Granular Mechanics and Its Engineering Applications, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3304-4_10

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can be applied to vibrating bodies in closed spaces filled with frictional granular materials (Shah et al. 2011). Lieber and Nin applied the concept of single-degree-offreedom particle damping to aircraft vibration reduction in 1945, and achieved well results (Ikago et al. 2012). The particle damping technology has many advantages such as broad application environment scope, less modification to the original structure, small additional mass and a significant vibration reduction effect. As a new type of damper, the particle damper has the advantages of wide vibration reduction frequency band, low impact force, low noise, and strong temperature adaptability, and can be well used for structural vibration reduction (Nayeri et al. 2007). At present, the particle damping technique has been successfully applied to different structures, such as splitter blades, rib structures, gear shaft structures, and drum brakes of aircraft engines. However, the mechanism of vibration energy dissipation in particle damping technique is very complex, including the basic mechanical behaviors of granular materials, and the coupling between granular materials and engineering structure. Therefore, the engineering applications of particle damping technique are limited when its mechanism is not well understood (Chen and Georgakis 2013). Recently, with the development of the fundamental theory of granular materials and the improvement of high-performance computing technology, it is expected to promote the development of particle damping technique by studying the vibration energy dissipation mechanism of granular materials at the micro scale (Li and Darby 2008). In this section, the principle of granular damping and its engineering applications are introduced based on mechanical experiments and DEM numerical simulations.

10.1.1 Experimental Studies on Particle Damper The particle damper works as follows. The granular material is placed in a built-in or external closed structural cavity at a given filling rate. During collisions, friction and momentum exchange between particles or between the particles and the inner wall of the cavity, the kinetic energy of the system can be consumed to reduce the structural vibration (Saeki 2005). Experiments can be performed with the excitation generated by using the vibration table or the exciter, and the motion can be observed through transparent cavity, thereby achieving the energy dissipation and vibration reduction of particle damping system (Lu et al. 2012). The shape and size of the cavity of the external particle damper influence the effect of vibration reduction. Here, experimental and theoretical studies of the rectangular and cylindrical particle damper are performed, respectively, as shown in Fig. 10.1. The results show that the geometry of the cavity has an important effect on the distribution of the contact force between particles and the dynamic characteristics of the particle damper. The vibration reduction effect of particle damper always increases first, and then decreases as the cavity size increases. The particle size of the external particle damper is another important factor that affects the performance of vibration reduction of the external particle damper. Figure 10.2 is the dynamic diagram of particles with different diameters. Steel

10.1 DEM Simulations and Experimental Tests …

(a) Rectangular particle damper

361

(b) Cylindrical particle damper

Fig. 10.1 Experimental device for external particle damper

(a) Small-sized particles

(b) Medium-sized particles

(c) Large-sized particles

Fig. 10.2 Vibration reduction tests of particle damper with different sizes

spherical particle damper has diameters of 4.32 mm (small), 12.34 mm (middle) and 16.32 mm (large), and accelerometer is installed. The experiment was performed under the same random vibration excitation, and each experiment was repeated 3 times independently. The experimental results show that the medium-sized particle damper has the lowest energy response and the largest energy loss compared with other two types of steel balls, which indicates that medium-sized steel balls have the best vibration reduction effect. Moreover, different filling rates produce different vibration reduction effect. When the filling rate is 60%, the energy responses between steel balls of small-sized and large-sized particle damper are almost equal. When the filling rate is less than 60%, the value of energy response between steel balls of large-sized damper is lower than that of small-sized damper. When the filling rate is between 60% and 80%, the value of energy response between steel balls of smallsized particle damper is lower than that of large-sized particle damper, and equal to that of medium-sized damper. It can be concluded that medium-sized damper has the best vibration reduction performance; The vibration reduction performance of large-sized and small-sized damper alternates with the increase of filling rate. In addition, when the filling rate is too small or too large, the particle damper exhibits quasi-liquid and quasi-solid according to the structural vibrations. The damper cannot exert the vibration reduction performance, and it is only equivalent to an additional mass. The experimental results with same free masses of steel balls indicate that there is an optimal filling rate for the filling granular materials.

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For different granular materials, cavity sizes and structures, appropriate filling rate should be adopted based on the specific situations to achieve the best performance of vibration reduction. The working efficiency of the particle damper is affected by many factors. It is found that particle size, filling rate of particles and cavity shape have great influence on the working efficiency. In order to achieve the best performance of vibration reduction, particles can be selected from different materials, such as stainless steel, polymer, ceramics, plexiglass, etc.. When particles with high density, large internal friction coefficient, small restitution coefficient are selected, the performance of vibration reduction is better. The selection of granular material and cavity material also directly affect the mechanical parameters between particle and cavity, such as restitution coefficient and friction coefficient. The working principle of particle damper tends to be clear, which will provide a strong theoretical basis for its applications in industrial practice and better service in vibration reduction. Many experiments have been conducted on the energy dissipation process of particle damper (Zalewski and Szmidt 2014; Bajkowski et al. 2016). For example, the vibration process of the steel frame connected to the multi-degree-of-freedom system damper was also studied experimentally under dynamic load. Figure 10.3 shows the vibration table of a three-layer steel frame with particle damper (Lu et al. 2012). The particle damper is placed on the top layer of the structural model, and the vibration reduction performance of the damper can be detected and evaluated when the structure moves violently. The bottom of the high-rise steel frame structure is bolted to the floor beam, and the additional mass is applied to each layer of the structure, which enables a change in free mass. Fig. 10.3 Vibration table of a three-layer steel frame with particle damper (Lu et al. 2012)

10.1 DEM Simulations and Experimental Tests …

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10.1.2 Numerical Simulation of Particle Damper In order to systematically study the working principle and influencing factors of particle damper, the working process of particle damper can be numerically simulated by the DEM (Mao et al. 2004). According to the design principle schematic shown in Fig. 10.4, the particle damper model is established for numerical simulation, as shown in Fig. 10.5. The energy dissipation mechanism of particle damper is divided into friction energy dissipation and impact energy dissipation. The friction energy dissipation and impact energy dissipation can be calculated in the DEM calculation, respectively. The results show that at the beginning, as the excitation amplitude (acceleration, velocity, displacement) increases, the proportion of impact energy dissipation increases, and the proportion of friction energy dissipation decreases. Then they gradually stabilizes. At a low excitation level, the relative motion between particles is at a low level, most Fig. 10.4 A design schematic of a tuned particle damper

Fig. 10.5 Discrete element model of granular system

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particles cluster together, and the collisions between particles are weak. The energy of the system is dissipated mainly through particle rotation induced sliding friction. However, at a large excitation level, the relative motion between particles intensify, particles separate from each other, and the collision between particles are enhanced. In this case, system energy is dissipated mainly through inelastic collisions between particles (Duan et al. 2013). The relationship between the frequencies of different excitation forces, coordination number, number of free particles and energy dissipation rate per unit volume indicates that and the particle flow is quasi-solid before reaching the resonance region if the coordination number is large. After entering the resonance region, the coordination number and the number of free particles both change suddenly, indicating that the quasi-solid-liquid phase transition occurs in the granular flow. At this time, the energy dissipation rate per unit volume increases suddenly, which indicates that the energy dissipation of granular media is the largest in the quasi-solid-liquid phase transition. When the excitation frequency exceeds the resonance region, the coordination number decreases again. The number of free particles and the energy dissipation rate per unit volume return to the pre-resonance level. These studies show that the parameters characterizing the quasi-solid-liquid phase transitions of granular flow, such as coordination number and the number of free particle, can be used as a criterion for the extreme of the energy dissipation of the particle damping system. Particle damper has a good vibration reduction effect, and it is widely used in engineering. For example, a new type of tuned mass damper can be used to detect seismic responses. Based on the principle of equal cavity volume and equal particle mass in the particle damper before and after equivalence, the multi-particle damper is equivalent to single particle damper, and the numerical model of particle-tuned mass dampers is established. The particle-tuned mass damper connects the cavity with the main structure through the pendulum rope. The numerical simulation results show that the method proposed in this section can be used to calculate the response of the damper system with additional particle-tuned mass under the actual seismic excitation. However, further research is still needed to more accurately evaluate the performance of the tuned mass damping system with additional particles under seismic excitation, especially considering the impact of particle collisions on the vibration reduction performance (Ikago et al. 2012). Particle damper is widely used in viaduct vibration control because vibration reducton is an important part of protecting the performance of the viaduct and extending its service life. Considering the highly nonlinear characteristics of particle damping, a simplified mechanical model of the damper is established and simulated with the FEM based on the experimental results of a tuned particle damper (TPD) used in a scale model of an elevated continuous beam bridge. The model bridge is a 4 m × 4 m four-span continuous viaduct. The intermediate piers and girders are connected by a fixed bracket, and the rest are connected by unidirectional sliding support. According to the similarity theory of the bridge, its tuned particle damper is mainly composed of three parts, the damper cavity, the connector between the damper and controlled structure and damping particles, as shown in Fig. 10.6.

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Fig. 10.6 Overall layout of the model bridge

Under the action of low-frequency excitation, the umbrella resonance of the wheel body structure is easily excited. Compared with other dampers, particle damping technology overcomes the difficulties of easy aging and poor stability in harsh temperature environment. This section presents a particle damper designed for wheel bodies. It is called 2D particle damping model. This model is composed of two parts, the cavity model and particle model, as shown in Fig. 10.7. Here, “2D” means that the movement of the cavity is 2D, while the movement of particles inside the cavity is 3D. Figure 10.8 shows a solid model of wheel structure with particle damper. This structure is equipped with particle damper in five grooves of the hub, and the mounting holes of the grooves are connected to the work table. Therefore, the axial, radial, and circumferential displacements of the five mounting holes are constrained to zero in the finite element model. Fig. 10.7 2D model of particle damper

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Fig. 10.8 Solid model of wheel structure with particle damper

10.2 DEM Analysis of Buffering Capacity of Granular Materials Energy dissipation of granular system is mainly based on inelastic collision and sliding friction between particles (Duan et al. 2013). Under the action of external load, intense extrusion and friction between particles will be observed. The complex force chain structure will break and reconstruct, thus consuming a lot of energy. When the collision between particles is non-linear, there is viscous action and plastic deformation. Meanwhile, the deformation of particles and the interaction between them are irreversible, and the energy conversion will be not observed (Geng et al. 2001). For the collision of two spherical particles, the granular materials with low yielding strength, high elastic modulus and high density have stronger energy dissipation (Stefan 1997, 2005). Granular buffering has been widely used in the anti-collision energy absorption of aerospace return devices, automobiles, trains and high-speed trains. This section will focus on granular buffering through experiments and numerical simulations.

10.2.1 Experimental Studies on Buffering Capacity of Granular Materials In order to study the buffering effect of granular materials, the impact tests of the spheres onto a granular layer were performed, as shown in Fig. 10.9. The height, outer diameter and wall thickness of the plexiglass cylinder are 30 cm, 20 cm and 0.5 cm, respectively. Three CL-YD series piezoelectric force sensors are installed on the

10.2 DEM Analysis of Buffering Capacity of Granular Materials

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Fig. 10.9 The test device for the buffering effect

bottom of the steel cylinder to measure the impact force. The three force transducers with an angle of 120° between each other are installed on the cylinder with a distance of 6.5 m away from the center of cylinder bottom. With simultaneous measurements of the three impact loads, the resultant impact load can be obtained. During tests, regular glass particles and irregular sand particles with different sizes are selected as granular materials, as shown in Fig. 10.10. The parameters used in the tests are listed in Table 10.1.

Fig. 10.10 Four granular materials used in the experiments

Table 10.1 The parameters used in the experiments Particle size

Parameters

Fine glass

Coarse glass

Fine sand

Coarse sand

Minimum size

Dmin

0.4

4.0

0.4

1.5

Maximal size

Dmax D

0.6

5.0

0.7

2.5

0.5

4.5

0.5

2.0

Average size

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In the experiment, the stone sphere was used as the impact weight with a mass of m = 167 g and a diameter of d = 5 cm. The diameter of the container is 4 times of that of the impact sphere and more than 8 times of that of the particle. Smooth plexiglass side wall is adopted to eliminate the influence of the boundary on the simulation results (Seguin et al. 2011; Su et al. 2019). Glass particles or sand particles with different thicknesses are placed in the cylinder, and the thickness range is between 0 and 9 cm. The impactor dropped at a height of 50 cm to the surface of the granular bed. The impact force at the bottom of the cylinder is measured by a force sensor with a collection frequency of 50 kHz. Firstly, glass particles with average size D = 0.5 mm are used. When the granular layer thickness H is 1 cm and 6 cm, respectively, the relationship between impact force and time is shown in Fig. 10.11. When H = 1 cm, the impact force at the bottom of the cylinder has multiple peaks and show a regular decreasing trend. When H = 6 cm, the impact force presents a major peak, and several smaller peaks appear randomly until the impact contact force approaches zero. These two typical time-historical curves represent the changing characteristics of the impact force of buffering particles in a thin layer and a thick layer, respectively. The first peak value most directly reflects the buffering effect of the granular system. In order to better characterize the impact law of the same material, the influence of the different thickness on the first peak superposition was investigated. Then, The thickness of the granular bed is used to analyze the first peak. Firstly, the average size of fine glass is D = 0.5 mm. The thickness of the particles are 0, 0.5, 1.0, 2.0, 3.0, 4.0, and 5.0 cm, respectively. The effect of the thickness on the first peak is shown in Fig. 10.12a. It can be found that as the filling thickness of the granular bed increases, the corresponding peak value of the impact force decreases gradually, which indicates that the impact force transmitted to the bottom plate decreases gradually during the impact process, and the energy dissipated by the granular system gradually increases. Then, fine sand with an average particle size of D = 0.5 mm is used for experiment, as shown in Fig. 10.12b. It can be seen that the change of the impact peak with the thickness of the granular system is similar to that of fine glass, that is, as the particle thickness increases, the impact peak gradually

(a) H 1cm

(b) H

6cm

Fig. 10.11 Relationship between the impact force and the time with different granular thickness

10.2 DEM Analysis of Buffering Capacity of Granular Materials

(a) Fine glass

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(b) Fine sand

Fig. 10.12 The impact force under different granular thickness

decreases. However, the overall curve fluctuate much with weak regularity. This is mainly because the irregularity of fine particles makes the energy dissipation of inter-particle friction more significant, hence the randomness of this effect is more obvious. Numerical results were verified by experiments, and the results showed that the parameters of granular materials have a great influence on the buffering performance, such as granular size, friction coefficient between particles and granular concentration. To study the influence of the thickness of the granular system, the experiments with different granular thickness H were repeated for three times, and the relationship between the peak value and the granular thickness is shown in Fig. 10.13. It can be found that as the thickness increases, the impact value gradually decreases. Meanwhile, there is a critical value of thickness for the glass and sand. When the granular thickness is greater than the critical value, the value of impact force tends to be stable, and the factors such as particle size and shape have no effect on the buffering capacity of granular system. In this case, the granular thickness become the major factor. When the granular thickness is less than the critical thickness, the buffering performance of sand is significantly better than that of glass, which is mainly due to the different morphology of the two materials. The irregular sand have rough surfaces and larger friction coefficient. As a result, the sand has a better buffering capacity. Fig. 10.13 Relationship between the peak value of the impact force and the granular thickness

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10.2.2 Numerical Simulations of Buffering Capacity of Granular Materials The buffering capacity of granular system is studied by using the DEM. Mesoscopic analysis of the particle-impactor and particle-particle interaction are performed to obtain the macroscopic mechanical properties, such as energy dissipation of the granular material, the force chain structure, crater depth and impactor velocity and force during the impact process. Consequently, the internal mechanism of the buffering performance of granular materials can be revealed at the meso scale. In addition, the DEM analysis can effectively analyze the effect of particle shape, size, impact velocity and angle, particle density, friction coefficient, boundary constraints and other factors on the buffering performance of granular materials. In view of the structural characteristics of granular materials and their buffering characteristics against impact load, the contact force between particles is calculated by using spherical nonlinear force model, and the forces between the impactor and the granular bed during the impacting process are simulated. To analyze the buffering performance of granular materials, spherical glass particles with a thickness of 10 mm are dropped in a rigid cylinder, and spherical impactor is placed on the top of the granular bed. According to the conditions of granular buffering performance, the particles are randomly generated within the cylindrical container. The total number of particles is determined by the granular thickness, and all particles remain motionless under gravity, as shown in Fig. 10.14. In the DEM simulation, the diameters of Fig. 10.14 DEM model of buffering capacity of granular materials

10.2 DEM Analysis of Buffering Capacity of Granular Materials

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spherical particles are randomly distributed within [4.0 mm, 5.0 mm] according to uniform probability density function, and the average diameter is 4.5 mm. The diameter of the spherical impactor is donated as D, the initial velocity is donated as V0 , and the position of its center of mass is donated as H1 . In order to eliminate the effect of the wall on the impact process, the inner diameter R of the cylinder is about 4 times of that of the impactor. The main calculation parameters of the DEM simulations are listed in Table 10.2. It is found that the thickness of granular material is an important factor affecting the buffering performance. The experimental study on the buffering performance of granular materials in a cylindrical container shows that the impact force acting on the bottom of the cylinder decreases with the increase of the granular thickness, and stabilizes when the thickness reaches a critical value Hc . In order to obtain a better impact process and buffering performance, the initial velocity V0 = 5.0 m/s, mass m = 0.3 kg and initial height of the impactor is H1 = 0.30m. Other calculation parameters are the same as in Table 10.2. The impacting processes of granular materials within the range of H = 0 − 8.0 cm are analyzed by the DEM. When the granular thickness is H = 0, there is no particle, and the relationship between the impact force and time is shown in Fig. 10.15. It is showed that the peak value is Pa0 = 13.46 kN. When the granular thickness is H = 0.8, 1.0, 3.0, 4.5, 6.0 and 8.0 cm, respectively, the relationship between the impacting force and time are shown in Fig. 10.16. It can be found that the impact force has one or two peaks for different granular thicknesses. When the granular thickness is H = 0.8 cm, the peak value is Pa = 6.73 kN. If the buffering rate of granular material is defined as λ = ( Pa0 − Pa )/ Pa0 , the buffer rate increases with the increase of granular thickness. The peak value of the impact force Pa decreases gradually, and a new peak value Pb gradually generates before Pa , Table 10.2 The main computational parameters in DEM simulations Parameters

Symbol

Unit

Cylinder bore

R

cm

Impactor height

H1

Impactor diameter

Value

Parameters

Symbol

Unit

Value

19.0

Granular diameter

d

mm

4.0–5.0

cm

50.0

Granular elasticity modulus

E

GPa

5.0

D

cm

5.0

Poisson’s ratio of particle

v

0.22

Impactor mass

M

kg

0.167

Granular restitution coefficient

e

0.80

Impact velocity

V0

m/s

0.0

Granular friction coefficient

μ pp

0.50

Granular density

ρ

kg/m3

2650.0

Cylinder friction coefficient

μ pw

0.15

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Fig. 10.15 Relationship between the impact force and time when there are no particles within the container

Fig. 10.16 Relationship between the impact force and time under different granular thickness. a H = 0.8 cm; b H = 1.0 cm; c H = 3.0 cm; d H = 4.5 cm; e H = 6.0 cm; f H = 8.0 cm

10.2 DEM Analysis of Buffering Capacity of Granular Materials

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which indicates that as the granular thickness increases, the time period between the two peaks becomes longer. When the granular thickness is H = 8.0cm, Pa gradually disappears, and Pb becomes the only peak. The density, shape and diameter of particles are important factors affecting the buffering performance of granular materials. The smaller the elastic modulus, the higher the density and the more irregular the granular shape, the better the energy dissipation performance. In addition, the mixing ratio, diameter ratio, and friction coefficient all affect the buffering performance of granular materials. Particle friction coefficient and density have significant influences on energy dissipation and flow characteristics of granular materials. The relationships between the impact force and depth and different friction coefficients are shown in Fig. 10.17, and the change of impact force with time is shown in Fig. 10.18. It can be found that as the friction coefficient increases, the impact force increases significantly, and the impact depth decreases and tends to be stable. Especially for completely smooth granular materials (μ pp = 0), the duration of the impact process is obviously longer, thereby reducing the impact force. In conclusions, the smooth particles have better buffering performance, and the impactor is easier to move in the smooth granular system.

(a) Peak value

(b) Impact depth

Fig. 10.17 Variation of peak value and impact depth with different friction coefficients

Fig. 10.18 The impact force over time under different friction coefficients

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The movements of granular materials are closely related to their concentration. When the concentration is low, the air effect during particle impact cannot be ignored, which causes the formation of jet flow of fine particles. As the concentration increases, the air effect decreases. The shape of impact pit is significantly affected by the medium concentration. The buffering performance of dense granular materials is further analyzed. The relationships between the peak value and impact depth and the packing fraction are shown in Fig. 10.19. It can be found from Fig. 10.19a that there are many pores between particles when the initial concentration is C0 < 0.56. Under the disturbance of external force, there is a large space for free movement of particles, and P is not sensitive to C0 . When C0 > 0.562, the particles with dense arrangement have a strong resistance to external impact, and P increases with the increase of C0 . Therefore, the more dense the particle arrangement is, the weaker the buffering performance the granular system has. Moreover, it can be found from Fig. 10.19b that d decreases with the increase of C0 . In other word, the more dense the particle arrangement is, the shallower the impact depth of the impactor is. The relationship between the impact force and time under different concentrations are shown in Fig. 10.20. It can be found that the denser the granular material is, the higher

(a) Peak value

(b) Impact depth

Fig. 10.19 Peak value of impact force and impact depth as a function of packing fraction

Fig. 10.20 Relationship between the impact force and the time

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375

the peak value of the impact force is, and the longer the duration of impact force is. This is mainly because the movement of particles in the dense granular materials is strongly restrained, which increases the resistance of the impactor and shortens the duration of the impact force. As a result, a weak buffering performance is obtained. The force chain structure is the main force transmission way in granular materials, and it is also the basic framework for the granular system to withstand external forces. The analysis of the force chain structure of granular materials during the impact process is helpful to reveal the internal mechanism of the buffering performance of granular materials. When the granular systems are subjected to an impact load, the force is transmitted through the force chain and exhibits a pronounced arching structure. The calculation results are shown in Fig. 10.21. To clearly observe the force chain structure of the granular system, only forces larger than the average contact force are shown. When t = 55.8ms, the impactor reaches the surface of the granular material. The force chain expands from the contact point to the surroundings, and the number of force chains gradually increases and the strength decreases, as shown in Fig. 10.21a. Furthermore, the force chain gradually extends to the surrounding space. The length of the force chain increases, and the number and strength of the force chain increase, as shown in Fig. 10.21a–e. When t = 57.2ms, the peak value of the impact force reaches its maximum. Both the number and strength of force chains decreases, and the impactor remains motionless. The above process shows the change of force chain structure corresponding to the evolution of impact force.

Fig. 10.21 Evolution of force chain structure during impact process

376

(a) H 3.0cm

10 DEM Analysis of Vibration Reduction and Buffering Capacity …

(b) H 5.0cm

(c) H 7.0cm

Fig. 10.22 The force chain structure and the force distribution on the bottom plate under different granular thickness

The process of initiation, expansion and disappearance of the force chain can be used to characterize the change law of inter-particle force, and it is also the process of the transmission and dissipation of the elastic potential energy between particles. In order to analyze the influence of granular thickness on the buffering performance, the force chain structure and the contact force distribution at the bottom of the container under different granular layer thicknesses H = 3.0, 5.0 and 7.0 cm are shown in Fig. 10.22. The force chain between particles is mainly concentrated at the bottom of the impactor, and the strength of the force chains decreases as the particle depth increases. In addition, the impact force of granular materials on the bottom plate is non-uniform. When the thickness of the granular bed is small, the impact force on the bottom plate is significantly higher than that acting on the granular bed with a larger thickness. With the increase of the thickness of granular bed, the inter-particle force chain becomes denser. As a result, the pressure on the bottom plate is obviously weakened. In conclusion, the buffering performance of granular materials is analyzed by experiments and numerical simulations, which provides a theoretical support for the applications of buffering performance of granular materials. In recent years, the buffering performance of granular material has been widely used in the anti-collision energy absorption of aerospace return devices, cars, trains and high-speed trains.

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10.3 DEM Analysis of Landing Process of Lunar Lander The lunar lander produces dramatic impact during its landing process, which poses a serious threat to the safety of the exploration equipment and lunar landing personnel (Sutoh et al. 2017; Kulchitsky et al. 2016; Smith et al. 2013). The research on impact force and buffering devices during the landing process is crucial to ensure the integrity of the internal precision instruments and achieve a safe soft landing of the lander (Knuth et al. 2012; Metzger et al. 2018; Arslantas and Theil 2017). Figure 10.23 shows the first soft landing lander successfully launched by the United States on May 30, 1966. Figure 10.24 shows the design of the Chang ‘e-3 lander designed by China. It has a certain buffering equipment that is used to mitigate the impact on the lander (Li et al. 2016). Existing studies mainly focus on the buffering performance of the lander, including the airbag buffering with a simple structure and good buffering performance. However, it is easy to rebound and roll and difficult to control (Jeong and Shin et al. 2017; Kariya et al. 2018; Bahrani et al. 2014). The spring ratchet buffer has the advantages of strong space adaptability, simple structure and self-repairing of landing attitude. However, it is poor in stability and difficult Fig. 10.23 The first soft landing lander launched by the United States on the moon on May 30, 1966

Fig. 10.24 Schematic of Chang’e-3 lander design

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to control. Wang et al.(2004) carried out high and low temperature impact tests on the buffering materials, such as aluminum honeycomb, metal rubber, eddy current magnetic damping, and analyzed the landing performance of the buffer (Wang et al. 2004). So far, research is mainly centered on the buffer facilities of the lander. Strict assumptions have been made on the lunar surface, and most research is based on the FEM. The importance of lunar soil in the landing of landers is ignored. The basic properties of lunar soil was first studied by using an astronomical telescope, and the surface of the moon was observed. With the development of human exploration of the moon, 382.0 g lunar soil was brought back to the earth by the probe, which provided valuable data for the further study of lunar soil. In recent years, continuous model and discrete model of lunar soil have been established by the FEM and DEM, respectively (Arslantas et al. 2016). The impact process during the landing of aircraft has been simulated using the DEM (Hou et al. 2018). In the DEM, the lunar soil is regarded as a large number of discrete particles. As for the buffering performance, the calculation of the force between particles can be realized, and the buffering energy absorption of lunar soil during landing can be simulated more effectively.

10.3.1 Discrete Element Model of Landing Buffering System and Lunar Soil For the configuration of the lander, the five-legged inverted triangles, four-legged cantilevers, and three pushing triangle have been widely applied to simulate the lander. Figure 10.25 shows the stable polygon of the landing structure, that is, the regular polygon with the center of the foot pad as the vertex is called the stable polygon. The more landing legs and the larger the area of the stable polygon, the higher the landing stability. Due to the improvement of the landing pod, the five-legged inverted triangle was abandoned because it was no longer suitable for the installation. Compared with the inverted triangle type, the cantilever type can effectively shorten the length of the

(a) Three legged

(b) Four legged

Fig. 10.25 Stable polygon of landing structure

(c) Five legged

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buffer, and effectively reduce the mass of each landing leg. It also has higher spatial practicability, and is widely used in future research. The lander configuration with four-leg cantilever type is introduced in this section. Considering the discrete characteristics of lunar soil, the dynamic process can be numerically analyzed by the DEM. The contact force between particles can be calculated based on the nonlinear contact model of spherical particles. The contact forces between particles include elastic force and viscous force based on the MohrCoulomb criterion. Therefore, the Hertz-Mindlin nonlinear contact model is used here. The energy of the lander in the vertical direction during landing can be expressed as Wh = 21 m vh2 +m gl h. Here, m is the total mass of the lander; vh is vertical landing speed; gl is the gravity acceleration on the moon; h is the change in the height of the center of mass. The horizontal energy during the landing process of the landing module can be expressed as Wv = 21 m vv2 . Here, vv is the horizontal landing velocity. The lander has a certain buffering performance, that is, the lander is equipped with buffering devices (Udupa et al. 2018; Wang et al. 2019; Zheng et al. 2018). During the landing process, the energy absorption of the device in the vertical direction is denoted as Ph , and that in the horizontal direction is Pv . The energy absorbed vertically and horizontally in the buffering process of the lunar soil granular system can be denoted as Q h and Q v , respectively. When the speed of the lander becomes zero, the impact process ends and the energy of the whole process no longer changes, i.e. Wv + Wh = Pv + Ph + Q v + Q h . Figure 10.26 shows the footprints of Armstrong, the first astronaut landing on the moon on July 16, 1967. Figure 10.27 shows the Fig. 10.26 Footprints left by Apollo astronaut Neil Armstrong on the surface of the moon on July 16, 1967

Fig. 10.27 The surface of a plateau near the moon’s South Pole (Forest et al. 2012)

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Fig. 10.28 Discrete element model of lunar soil

Table 10.3 Calculation parameters of lunar soil Parameters density/kg·m−3

Symbols

Values

ρ

1300

Average granular diameter/mm

D

65.5

Elasticity modulus/MPa

E

48.8

Friction coefficient

f

0.62

Restitution coefficient

e

0.1

Granular

Poisson’s ratio

μ

0.32

Number of particles

n

64046

Calculating range scale of lunar soil/m3

LBH

4×4×1

Notes

0.58–0.64 0.31–0.34

plateau surface near the South Pole of the moon (Forest et al. 2012), which reflects the basic morphology of the moon. According to the physical properties of the lunar soil (Jiang et al. 2013; Berger et al. 2013), discrete element model of lunar soil materials was used to simulate the lunar soil particles, and the spherical granular systems were generated by irregular arrangement method, as shown in Fig. 10.28. The relevant calculation parameters are listed in Table 10.3. When the lander is in contact with the lunar soil, the lunar soil can rapidly absorb the energy of the lander through the interaction between particles, thus functioning as a buffer.

10.3.2 Discrete Element Analysis of Landing Process and Impact Characteristics The lander impacts the surface of lunar soil at a certain speed. During the landing process, particles play a buffering role through collision and friction. Finally, the speed of the lander becomes zero, in other words, the landing process ends. The landing process of the lander can be simplified as a combination of vertical impact and horizontal glide. The interaction between the foot pad and the lunar soil during the vertical impact is mainly investigated in this section. Momentum and energy

10.3 DEM Analysis of Landing Process of Lunar Lander

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Fig. 10.29 The landing impact of the lander

transfer at the instant of contact of the two. The specific simulation process is shown in Fig. 10.29. In DEM simulations, the depth of the lander in the lunar soil, the lander acceleration and the impact force between the lander and the lunar soil were analyzed by changing the mass of the lander and the landing speed of the lander. The landing process of the lander mainly includes three stages: (1) impacting stage after contact between the foot pad and lunar soil. (2) gliding stage of foot pad in lunar soil. (3) stable stage. Because the gliding stage is mainly horizontal movement, the scale of the lunar soil model is very important, which will greatly affect the computational efficiency of DEM simulations. In addition, the impact process is mainly studied in this section, that is, the motion of the lander in the vertical direction. the magnitude of impact force directly affects the dynamic process of soft landing. In the landing process of the lander, the relationship between the vertical force and the time is shown in Fig. 10.30. The above results show that the maximum force occurs at the initial contact with the particles. During this process, the kinetic energy is transferred from the lander to the lunar soil. Thereofore, the lander’s energy is reduced and the buffering performance is quantified by the interaction between the lunar soil particles and the lander and between lunar soil particles. Figure 10.31 Fig. 10.30 The force-time curve of the lander in vertical direction

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Fig. 10.31 The velocity change of the lander during landing

shows the change of lander velocity during the whole landing process. The velocity and acceleration of the lander are large at the initial moment. That is, the velocity of the lander changes rapidly at the beginning stage, then with small fluctuations, and becomes zero finally. The lander reaches a stable state. It is also found that one of the factors affecting the landing effect is the vertical impacting velocity, which plays a key role in the buffering performance of lunar soil. During the impacting process, the landing speed is 2 m/s, 2.5 m/s, 3.0 m/s, 3.5 m/s, 4.0 m/s, 4.5 m/s, 5.0 m/s, 6.0 m/s, respectively, and the mass of the lander is 9,000 kg. The change of the peak value of impact force with the impact velocity is shown in Fig. 10.32. The peak value increases linearly with the increase of impact velocity, that is, the greater the impact velocity is, the greater the peak value is. The changes of the acceleration during the impacting process of the lander is shown in Fig. 10.33. It can be seen that the peak acceleration of the lander during impact increases as the landing velocity increases, and this relationship is approximately linear. Figure 10.34 shows the relationship between impact depth and impact velocity. The kinetic energy of the lander increases as the impact velocity increases. The energy absorbed by the lunar soil during the impact process is shown in Fig. 10.35. The results show that Fig. 10.32 The relationship between impact peak and impact velocity

10.3 DEM Analysis of Landing Process of Lunar Lander

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Fig. 10.33 The relationship between peak acceleration and impact velocity

Fig. 10.34 The relationship between impact depth and impact velocity

Fig. 10.35 The relationship between energy absorption and impact velocity

the contact time between the lander and the particles becomes longer and the energy absorbed by the particles becomes greater as the impact velocity increases. The above research shows that the impact velocity is an important factor affecting the landing process. The experimental results show that the impact velocity, soil compactness and the quality of the lander have a significant effect on the peak value

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of impact force and acceleration of the lander (Goldenberg and Goldhirsch 2004). The mass of the lander is 7000, 7500, 8000, 8500, 9000 and 9500 kg, respectively. The influence of the mass on the buffering performance is studied, and the impact velocity is 4 m/s. The influence of the mass of the lander on the peak value of impact force of the lander is further studied, as shown in Fig. 10.36. Obviously, the peak value of impact force increases with the increase of the lander mass. Figure 10.37 shows the relationship between the acceleration and the lander mass. When the mass of the lander is relatively small, the acceleration increases rapidly; when the mass reaches 8500 kg, the mass of lander has no effect on the acceleration. In addition, the relationship between impact depth and quality is studied, as shown in Fig. 10.38. It can be seen that the linear relationship between impact velocity and mass is very prominent. The change of energy absorbed by particles with mass is obtained, as shown in Fig. 10.39. The relationship between the absorbed energy and the mass is linear. The change in impact velocity and lander mass is essentially a change in impact energy. In conclusion, the peak value of impact force increases with the increase of impact energy. Fig. 10.36 The relationship between impact peak and lander mass

Fig. 10.37 The relationship between acceleration of lander and the mass

10.3 DEM Analysis of Landing Process of Lunar Lander

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Fig. 10.38 The relationship between impact depth and lander mass

Fig. 10.39 The relationship between particle energy absorption and lander mass

Although the contact model is relatively simple and the influencing factors considered are not comprehensive, the landing process of the lander can be used to verify the buffering performance of granular materials and provide a new way for the engineering applications of the buffering effect of granular materials.

10.4 Summary The particle damper and buffering performance fully reflect the vibration reduction characteristics of granular materials, which is also crucial for studying the basic mechanical properties of granular materials. This chapter mainly discusses the performance of energy dissipation and vibration reduction of granular materials and its applications in industrial production and practice. Experimental studies and DEM simulations on the particle damper and buffering capacity are introduced in detail,

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focusing on the underlying mechanism of particle damping and buffering performance and its influencing factors. This can provide theoretical basis for the studies on particle dampers and buffering performance. Finally, the buffing effect of lunar soil on the lunar lander is introduced during landing process of the lander.

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