Delft University of Technology
FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING
Department Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl
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Specialization: Transport Engineering and Logistics
Report number: 2014.TEL.7896
Title:
Contact models for modeling
cohesive materials
Author:
R.W. Toetenel
Title (in Dutch) Contact modellen voor het modelleren van cohesieve materialen
Assignment: literature
Confidential: no
Initiator (university): prof.dr.ir. G. Lodewijks Supervisor: dr. ir. D.L. Schott
Delft University of Technology
FACULTY OF MECHANICAL, MARITIME AND MATERIALS ENGINEERING
Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl
Student: Rogier Toetenel Assignment type: Literature
Mentor: Dr.ir. Dingena Schott Report number: 2014.TEL.7896
Specialization: TEL Confidential: No
Creditpoints (EC): 10
Subject: Contact models for modeling cohesive materials
Within the research theme Dynamics and Interaction of Material and Equipment the simulation tool Discrete Element Method (DEM) is used. DEM is a particle based method to model interaction of particulate material (powders or bulk material) with equipment. Until now within our group mainly free flowing materials have been modeled. However a lot of materials show cohesive behavior especially if the material consist of small particles and contains moist. Therefore, this research focusses on the modeling of cohesive behavior of bulk materials and powders.
Your assignment is to investigate and make an overview of the contact models that are used in literature to model cohesive materials. An example of such a contact model is Johnson-Kendall-Roberts [1].
This assignment comprises amongst others (but is not limited to) the following: • Find relevant literature and research groups
• Describe and explain the different contact models used by researchers
• Describe in which context the contact models are used (kind of material, kind of application, etc.)
• Describe to which extent the simulation results found are validated with experiments • Classify and make an overview of the available contact models and compare them
It is expected that you conclude with a recommendation for further research based on the results of this study.
The report should comply with the guidelines of the section. Details can be found on blackboard.
The supervisor,
Dr. ir. Dingena Schott
[1] K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc. London A 324 (1971) 301-313
S
UMMARY
Over the last few decades The Discrete Element Method (DEM) is used more and more to simulate numer-ous bulk solid materials and is also called the Distinct Element Method. The main goal of these simulations within the Transport Engineering department at the University of Technology is to get a better understanding of the interaction between the bulk solid materials such as grain or iron ore and its handling equipment. The closer a simulation gets to reality, the better several parameters involved in the transport of the materials can be predicted. Interesting parameters to look into before designing the equipment are for example the wear rate, the forces and the (dis)charge capacity. These parameters can be essential when designing the capacity, throughput time and the financial consequences of certain design choices. It was only in the last decade that researchers also tried to incorporate the cohesive behaviour of bulk solid materials in the DEM. This litera-ture research aims to bundle and consolidate the worldwide research on the different contact models and its applications for modelling cohesive materials with DEM.This report starts with an introduction to the basic principles of DEM. The working principle, different ap-proaches and the interaction forces between individual particles are discussed. In the third chapter, the most used contact models are explained and compared with each other. The contact models are divided by their main acting direction, namely the normal direction or the tangential direction to the contact with another particle. The popular and well known Johnson-Kendall-Roberts (JKR) and Derjaguin-Muller-Toporov (DMT) contact models are included since they act as foundation for many other contact models created after these two. According to the Tabor parameter, every contact model has its own range in which it is most accurate compared to the other contact models. Since the cohesive behaviour changes significantly by adding mois-ture to the bulk solid, this influence is explained. The Edinburgh Elasto-Plasto Adhesive (EEPA) model gets special attention since it is a new model specifically focussed on modelling with cohesive materials. This model is created by researchers at the University of Edinburgh in 2013 and it is expected that it will gain pop-ularity in the next years.. The contact models can be adjusted with many parameters and each of them has its own influence on the results of the simulation. The parameters are separated in three categories namely model characteristics, material characteristics and parameters for contact model setup and their influence is discussed. Finally, the application of the contact models in the simulations discussed.
It can be concluded that despite the effort of researchers to model the cohesive behaviour of bulk solid ma-terials in the last decade, there is still a lot unknown. The outcomes of several simulations are based on a lot of assumptions of which their influence is not yet known. For example, most contact models assume the particles to be perfect shaped spheres which is obviously not the case in the real life. This can encourage researchers to carry on with their current work because there is still a lot to explore.
Secondly, a lack of a uniform DEM simulation testing environment makes it hard to compare different re-searches. It is recommended that at some point in time, a standard is made to be able to compare the different contact models and their parameters on an quantitative scale instead of the current qualitative comparisons. Researchers are now primarily validating their own simulations with experiments which is hard to reproduce by others. This is due to the fact that most papers do not explicitly describe which parameters are adjusted and which contact models are used in their DEM simulations. It is therefore recommended that a more open culture is created in which researchers are willing to share all the details of their simulations instead of fo-cussing on validating it by themselves.
L
IST OF
S
YMBOLS
Symbol Namea Contact radius
B og Cohesive granular bond number
E∗ Equivalent Young’s modulus
e Inter-particle coefficient of restitution
Fa Attractive inter-particle force
Fi Particle-particle collision force
G Shear modulus
g Gravitational constant
G∗ Equivalent shear modulus
h Distance between two particles
Ii Moment of inertia of particle i
kn Inter-particle normal stiffness coefficient
kt Inter-particle tangential stiffness coefficient
Mi Total moment acting on particle i
mi Mass of particle i
P Pull force
Pc Critical pull-off force
R∗ Equivalent contact radius
S Saturation level
Tc Peeling force
t Time
V Volume
w Adhesive work
x50 Mass median of particle size distribution (PSD)
xi Position of particle i
βn Particle-wall normal damping coefficient
βt Particle-wall tangential damping coefficient
γ Surface energy
γn Inter-particle normal damping coefficient
γt Inter-particle tangential damping coefficient
δ Overlap of two colliding particles
² Material porosity
θi Angular position of particle i
λ Maugis parameter
µD Dynamic particle friction coefficient
µPW,n Particle-wall normal friction coefficient
µPW,t Particle-wall tangential friction coefficient
µS Static particle friction coefficient (Coulomb)
µT Tabor parameter
ν Poisson ratio
ρ Density
σ1 Pre-defined stress level
σc Contact bond normal strength
τ Contact bond shear strength
φi Angle of internal friction
C
ONTENTS
Summary i
1 Introduction 1
2 The Discrete Element Method 3
2.1 History & developments . . . 3
2.2 Soft- and hard-particle approach . . . 3
2.3 The working principle. . . 4
2.3.1 DEM Algorithm . . . 4 2.3.2 Contact detection . . . 4 2.3.3 Equations of motions . . . 5 2.4 Interaction forces . . . 5 2.4.1 Contact forces . . . 6 2.4.2 Non-contact forces. . . 6
2.4.3 Overview of the non-contact forces . . . 8
2.5 Cohesive and non-cohesive materials. . . 8
2.6 Summary . . . 9
3 Contact models for DEM 11 3.1 Linear spring . . . 13
3.2 Normal direction models . . . 13
3.2.1 Johnson-Kendall-Roberts (JKR) . . . 13
3.2.2 Derjaguin, Muller and Toporov (DMT) . . . 15
3.2.3 Maugis-Dugdale . . . 15
3.3 Tangential direction models . . . 16
3.3.1 Hertz-Mindlin and Deresiewicz (H-MD) . . . 16
3.3.2 Mindlin, Savkoor and Thornton . . . 17
3.3.3 Vu-Quoc . . . 17
3.4 Edinburgh Elasto-Plastic Adhesion (EEPA) . . . 18
3.4.1 Proposed model . . . 18
3.4.2 Model Overview . . . 18
3.4.3 The Edinburgh Powder Test (EPT) . . . 20
3.4.4 Verification . . . 20
3.5 Influence of adding moisture . . . 20
3.6 Comparison of the different models. . . 21
3.7 Summary . . . 22
4 Parameters used in contact models 23 4.1 Parameters in DEM . . . 23
4.1.1 Model characteristics . . . 23
4.1.2 Material characteristics . . . 24
4.1.3 Parameters for contact model setup . . . 25
4.2 Particle Size Distribution . . . 25
4.3 Summary . . . 26
5 Implementation of contact models in simulations 27 5.1 Verification and validation . . . 27
5.2 Simulations with the EEPA contact model. . . 27
5.3 Simulations with other contact models . . . 28
5.3.1 Rojek et al. - influence of several parameters. . . 28
vi CONTENTS
5.3.3 Tsuji et al. - bulldozer blade . . . 30
5.3.4 Gröger et al. - tensile and shear test . . . 30
5.3.5 Moreno-Atanasio et al. - flowability of cohesive powders . . . 31
5.3.6 Richefeu et al. - theoretical DEM box . . . 31
5.3.7 Anand et al. - hopper discharge . . . 31
5.3.8 Remy et al. - bladed mixer . . . 32
5.3.9 Cleary and Robinson - falling stream cutters . . . 33
5.3.10 Asmar et al. - stepwise simulation . . . 33
5.3.11 Hou et al. - screw feeder . . . 33
5.4 Overview . . . 34
5.5 Summary . . . 34
6 Conclusions & Recommendations 35
1
I
NTRODUCTION
Over the last few decades the Discrete Element Method (DEM) is used more and more to simulate numer-ous bulk solid materials and is also called the Distinct Element Method. The main goal of these simulations within the Transport Engineering department at the University of Technology is to get a better understanding of the interaction between the bulk solid materials such as grain or iron ore and its handling equipment. The closer a simulation gets to reality, the better several parameters involved in the transport of the materials can be predicted. Interesting parameters to look into before designing the equipment are for example the wear rate, the forces and the (dis)charge capacity. These parameters can be essential when designing the capacity, throughput time and the financial consequences of certain design choices.
The Discrete Element Method was first described by Cundall and Strack [1979]. Earlier studies and researches were mainly based on laboratory experiments and real life experience with the interaction of materials and equipment. Several contact models were created to describe the behaviour of the colliding particles. The most powerful way to simulate the bulk solid materials is by using DEM. Therefore, the breakthrough of Cun-dall and Strack created immense opportunities for researchers to increase the reliability and accuracy of the simulations.
Nowadays DEM is not only used by researchers but also by companies involved in the handling of bulk solid materials. Several companies are offering a wide scale of DEM-software that is easy accessible and easy in use. Some of these programs are EDEM, Newton, Rocky and AppliedDEM™. Research into the fields of DEM, has focussed on the non-cohesive or cohesionless materials such as sand or gravel primarily. Over the last years, however, the interest in cohesive materials used in DEM simulations increased. Several researchers started to create contact models to implement the cohesive behaviour of bulk solid materials in DEM simulations. The individual particles of a cohesive material wants to stick together and a specific force is needed to separate the particles. On the other hand, non-cohesive materials are free flowing on its own and no separation force has to be overcome to separate the particles. For example, clay shows cohesive behaviour while grind can be classified as a non-cohesive material. In many literature, the term adhesion is also used for almost identical purposes as cohesion. The difference between both is that cohesion acts between two particles of the same material while adhesion describes the forces between two different materials. This research will only focus on the cohesive behaviour of materials and the adhesive part is left out of this report. A more detailed expla-nation and calculation of cohesion can be found in Section 2.5.
One of the research groups that are investigating this cohesive behaviour is situated at the University of Ed-inburgh. Mr. John P. Morrissey promoted on a closely related subject in 2013 at this university within the section Institute for Infrastructure and Environment School of Engineering. Therefore his PhD-thesis and its accompanying publications were of great value for this literature study.
This literature research aims to bundle and consolidate the worldwide research on the different contact mod-els for modelling cohesive materials with DEM. This can be formulated in one main question for this research:
"Which contact models for modelling cohesive behaviour of bulk solid materials in DEM simulations are now used, what are their applications and on which points do they differ from each other?"
2 1.INTRODUCTION
Throughout the report, the next sub questions will be answered to answer the main question with the literature that has been found:
• Which different contact models are used in literature and how does these models work?
• In which context are these contact models used (kind of material, kind of application, etc.)?
• To which extent are the simulation results found in literature validated with experiments?
• What are the similarities and differences between the different contact models?
Chapter 2 starts with an introduction to the basic principles of DEM. The working principle, different ap-proaches and the interaction forces between individual particles are discussed. In the third chapter, the most used contact models are explained and compared with each other. The contact models are divided by their main acting direction, namely the normal direction or the tangential direction to the collision with another particle. The popular and well known Johnson-Kendall-Roberts (JKR) and Derjaguin-Muller-Toporov (DMT) contact models are included as well in this chapter. The chapter also covers the Edinburgh Elasto-Plastic Adhesive contact model since it is a new model specifically focussed on modelling with cohesive materials and it is expected that this model will be used more in the future since it is developed in 2013. Chapter 4 will describe many parameters that can be adjusted in DEM to model different materials, environments and parameters. The parameters are separated in three categories, namely model characteristics, material char-acteristics and the contact model charchar-acteristics and their influence is discussed to a certain extent. Chapter 5 will explain the applications of different contact models in DEM.
Finally, the conclusion and recommendations that are drawn from the report are discussed and the main question formulated in this introduction is answered.
2
T
HE
D
ISCRETE
E
LEMENT
M
ETHOD
Starting with the first findings in 1979 by Cundall and Strack, the Discrete Element Method (DEM) has devel-oped enormously since then and has become a compulsory part in the design procedures in many industries. DEM can generate important information at any specific moment during the experiment which is often not possible with the conventional experimental setups. It overcomes some of the disadvantages of Continuum Mechanics Method (CMM) such as Finite Element Analysis, which ignores individual unit characteristics, and relies on the highly simplified mechanical equations excessively [Zhang and Li, 2006]. This chapter briefly touches on the history of DEM and will continue with a more detailed explanation of the working principle of DEM. The equations of motions and the forces acting on every single particle will be explained as well.
2.1.
H
ISTORY
&
DEVELOPMENTS
Researchers are interested in the behaviour of bulk solid materials on a macroscopic level for a long time. One of the first examples of this interest is described in the paper of Bradley [1932], where he explains the theory of cohesive forces between solid surfaces present in bulk solid materials. Journals from that period of time were mainly based on numerous experiments with the granular materials. One of the downsides of experiments is the non uniformity of the surrounding- and time dependant circumstances influencing the results of the experiments. The need for a numerical method to describe and simulate the behaviour of granular materials was obvious. Numerical modelling creates more flexibility in modelling the materials over the earlier used analytical methods. Besides, the numerical method can be stopped at any time during a simulation to investigate the data of a specific stage off the test.
It was up until 1979 that Cundall and Strack came up with a numerical way to model the contact forces of the individual particles of granular materials. With the technological improvements of the 1970’s, it was now possible to create a numerical method for calculation of the contact forces of all the individual particles and start the numerical studies in the DEM.
2.2.
S
OFT
-
AND HARD
-
PARTICLE APPROACH
As mentioned in the introduction, multiple software programs are available to model DEM simulations. These different programs or methods can be divided into two different types of DEM’s namely the soft-particle and the hard-soft-particle approach.
The soft particle approach was first developed and explained by Cundall and Strack [1979] and is used in most DEM simulations. In this approach, particles are allowed to deform during a collision with another par-ticle and multi-parpar-ticle collisions are allowed simultaneously [Bertrand et al., 2005],[Di Renzo and Di Maio, 2004]. These deformations are minute but affect the forces on the particles which are calculated by the well known Newton’s laws of motion. The newly calculated deformations are used in their turn to calculate the elastic, plastic and frictional forces between the particles. Because of smaller integration steps to model these deformations, the soft-particle approach is 10-100 times slower due to the computational force than the hard-particle approach [Luding, 2008]. On the other hand, the hard-hard-particle approach only allows the hard-particles to have an instantaneous contact with another particle. No deformation is allowed and the forces between par-ticles are not calculated explicitly. This method is used more in rapid granular flowing models [Zhu et al., 2007].
4 2.THEDISCRETEELEMENTMETHOD
2.3.
T
HE WORKING PRINCIPLE
The principle working behind the DEM is simple yet very effective. The goal of every time step in the simu-lation is to determine the trajectory of a finite number of particles within the system and to determine their positions after this certain time step. These particles will interact with each other and the boundaries of the system with both contact and non-contact forces. Newton’s equation of motion is describing the translational and rotational movements of every discrete particle. Some of the forces that are encountered by the simu-lation are gravitational forces, Van der Waals and cohesive forces and will be described later on in Section 2.4. The chosen time step should be small enough to prevent a particle colliding through a nearby particle without being noticed by the DEM. The time step should allow the collisions to be noticeable by the DEM.
2.3.1.
DEM A
LGORITHMDEM simulations are always finite in time and thus in time steps. After a certain amount of time steps the simulation will stop and start to analyse the results. This subsection will describe the general DEM algorithm step by step between two time steps. These five steps are listed chronologically below [Schott, 2013], [Asmar et al., 2002]:
1. Determine initial conditions.This step is only needed at the start of the model (t = 0). Particle, environ-ment and simulation properties are all impleenviron-mented in the model. Besides, a contact model is chosen in this step.
2. Detect contacts. The simulation program will detect particle-particle and particle-environment con-tacts. The particles are either checked one by one (direct) or from a grid-based environment where all cells are checked for contacts. For further explanation, see Subsection 2.3.2.
3. Calculate interaction forces. This step will actually calculate all the contact forces and non-contact forces (explanation in Section 2.4). The outcome strongly depends on the chosen contact model in step 1. Besides, the external forces such as gravity are implemented.
4. Calculate particle positions.The acceleration of all individual particles is calculated what results in new particle velocities. The model computes the new position of the particle by integrating the obtained particle velocities.
5. Determine t = tend. If the last time step is reached, the simulation stops. Otherwise, the steps are
repeated from step 2 to step 5 until the last time step is reached.
2.3.2.
C
ONTACT DETECTIONThe contact detection used by different DEM simulations methods is described very clearly by Morrissey [2013, p. 40-41] and is therefore cited and duplicated from this source in this whole subsection.
"Contact detection is one of the most time consuming parts of a DEM simulation. The amount of time used for this task is proportional to the number of particles in a system and therefore an efficient method of de-tecting particles in contact is required. It is very inefficient to simply loop through every individual particle and check the separation distance between two particles at every timestep, particularly for a large number of particles. To improve the efficiency through either reducing the number of times all particles are checked or by reducing the number of neighbours checked, three main types of contact detection schemes are used:
• Verlet Neighbour Lists.Verlet neighbours list construct a list of particles within a certain search radius, typically 2-3 particle radii, and only this list is searched for contact, rather than every individual particle in the system. The neighbour list is not updated every timestep, but instead is updated every 20-50 timesteps or if displacements are large.
• Link/Grid Cells. Link or Grid cells divide the simulation domain into a number of equally sized cells which are larger than a couple of particle diameters. A list is maintained of all the particles in each cell, and their respective positions. Contacts are only checked for particles within the same cell and the neighbouring cells.
• Lattices.The lattice method divides the simulation domain into a number of equally sized cells, where every cell is given the size of a particle (each cell can contain only one particle). If a polydisperse particle size distribution is being used the cell size will be related to the size of the smallest particle. Each
2.4.INTERACTION FORCES 5
particle is indexed in relation to a grid point and a neighbour list is created for all cells within a particle diameter d .
The contact detection method employed by the EDEM code is a hybrid between the lattice method and the link/grid cells method. The user specifies the size of the grid to be used with the recommended range being 2-3R, where R is the radius of the smallest particle. At a value of 2R this is the lattice method, while at values larger than 3-4R it become the link/grid cell method. Each cell is checked for more than one particle, and if found these are checked first for contacts. The EDEM Simulator module is most efficient at a grid radius of 2R, but this can lead to large amounts of memory being used."
2.3.3.
E
QUATIONS OF MOTIONSSubsection 2.3.1 gives a very global and textual way of describing the DEM algorithm. This subsection aims to clarify these steps by giving the equations of motion defined by Newton’s second law. The calculations of step 4 are divided in two groups based on the equations of motion namely in the translational/normal direction and the rotational/tangential direction. For a certain particle i , these equations are listed in Equation 2.1 and 2.2.
Fi+ mig = mix¨i (2.1)
where mi is the mass of particle m, xi is its position, Fi=Pcficis the force acting on the particle due to the
particle-particle collisions and g is the gravitational constant.
Mi= ¨θiIi (2.2)
where Ii is the moment of inertia of particle i ,θi is the angular position of this particle and Mi is the total
moment acting on the particle due to the rotational accelerations [Schott, 2013],[Morrissey, 2013].
Figure 2.1: Schematic illustration of the forces action on a particle i from contacting particle j and non-contacting particle k [Zhu et al., 2007, p 3380]
2.4.
I
NTERACTION FORCES
When a bulk solid material consists of more then just one particle, the particles will share the interaction forces amongst each other. All these forces together are forming the material characteristics such as Yield modulus, coefficient of friction and fracture toughness. Numerous other characteristics can be summarized but that is not of main concern for this section. This section will explain the different interaction forces and their relevance to DEM. The interaction forces can be divided into the contact forces and the non-contact forces and are described in the next two subsections.
6 2.THEDISCRETEELEMENTMETHOD
2.4.1.
C
ONTACT FORCESThe contact between two particles might look like a single point collision but is divided over a certain finite area of these two particles due to the elastic behaviour of the materials. These forces are directly following from the equations of motions described in Subsection 2.3.3. In general, these forces are separated in two directions of the collision. The first are the forces directing into the contact plane (tangential forces) and the other one directs in normal direction to the plane. This separation of forces can also be found in the contact models used for DEM. Some of them are mainly describing one of the directions while others are taken the two directions with equal attention into account. The specific forces and accompanying equations are therefore discussed in more detail in Chapter 3.
2.4.2.
N
ON-
CONTACT FORCESWhen particles are together and not on their own in a fluid, they exert interaction forces on each other. How-ever, contact is not necessary for the non-contact forces. These non-contact forces influence the flow and packing behaviour of a material and is therefore of main concern when using DEM as simulation method. The non-contact forces are often a combination of the three fundamental forces explained below [Zhu et al., 2007]. While the van der Waals forces are most dominant at small inter particle distance, the capillary and electrostatic forces become more dominant when the distance is increased [Morrissey, 2013].
Figure 2.2: Comparison of the magnitude of interaction forces [Zhu et al., 2007, p 3382].
VAN DERWAALS FORCES
The van der Waals forces may be the best known of the three non-contact forces. The van der Waals forces are acting between particles or molecules with closed shells and can be either repulsive or attractive. Its funda-mental working principle is based on the difference of electrostatic load of the particles. As two surfaces come in contact, the transfer of electrons creates positive and negative charges on both surfaces. These oppositely charges will lead to a strong adhesive bond. The van der Waals force between molecules is proportional to
h−6, where h is the distance between the particles. The Hamaker theory [Hamaker, 1937] is used often to im-plement the van der Waals forces in DEM simulations. The Hamaker theory, however, shows that the van der Waals forces become infinite when the particles are in contact. Therefore, a cut-off distance is implemented, the forces will only be calculated when the particles are outside this cut-off distance which is typically ranging between 0.165 to 1nm according to Zhu et al. [2007].
2.4.INTERACTION FORCES 7
CAPILLARY FORCES
Capillary forces are caused by condensed moisture on the surface of particles which creates surface tension at solid/liquid/gas interfaces. Dry granular materials are therefore not affected by capillary forces since their magnitude is negligible to the other two interaction forces. The scope of this research is, however, not limited to dry granular materials and the capillary forces should therefore be taken into account as well. There are many types of capillary forces but the most dominant for cohesive materials is the formation of liquid bridges between the particles. When a certain amount of moisture is added to a granular material, the liquid will form bridges between the particles that were first dry. Different stages of the formation of these liquid bridges and its capillarity depicted in Figure 2.3. These liquid bridges are causing the material to become cohesive to a certain extent. Richefeu et al. [2008] explains that the maximal capillary force of a millimetre size particle is around 4 ·10−4N independent of the volume of the capillary bond. These forces are almost four times as high
as the weight of a millimetre size grain. A more detailed calculation method for the capillary forces is made by Lambert et al. [2008], where the Laplace approach is compared with the energetic approach [Castellanos, 2005], [Zhu et al., 2007], [Zhu et al., 2008], [He et al., 2014]. The specific influence of adding moisture to a bulk solid can be found in section 3.5. The most used theory to calculate the capillary forces is made by Mikami et al. [1998]. Nearly all of the above listed references are using this model as a base point and are implementing their own theory into this base model. This model describes the behaviour of capillary forces between particles but also between particle and wall.
Figure 2.3: Illustration of moisture content for various stages of capillarity [Morrissey, 2013, p 19].
ELECTROSTATIC FORCES
Electrostatic forces are present between particles with a difference in electro-negativity and can be divided in three types: Coulomb forces, image-charged forces and space charge forces. As two surfaces come into contact, the transfer of electrons between the two materials creates the exchange of positive and negative charges which creates a strong cohesive bond. The electrostatic forces in DEM simulations are often imple-mented with the Coulomb equation, created in the late 18t hcentury. The electrostatic forces are typically an order one or two lower then the van der Waals and capillary forces as can be seen in Figure 2.2. However, the electrostatic forces should still be implemented in DEM while they contribute for a part in the calculation of the total interaction forces and become more dominant when the inter particle distance is increased [Tomas, 2007].
8 2.THEDISCRETEELEMENTMETHOD
2.4.3.
O
VERVIEW OF THE NON-
CONTACT FORCESThe grand total of all the non-contact forces acting on a particle can now easily be calculated as can be seen in Equation 2.3. The equations for calculating Fv, Feand Fccan be found in Table 2.1. Several parameters in this table are not declared in the list of symbols or at another place in this report and can therefore be found in the original source of these equations [Zhu et al., 2007].
Ft ot= Fv+ Fe+ Fc (2.3)
Table 2.1: List of three non-contact forces [Zhu et al., 2007, p. 3382]
Cohesive force Origin Formula
Van der Waals force Molecular dipole interaction Fv= −Adp
24h2nˆ1,2
Electrostatic force Coulomb force Fe= −£2πR sinφsin(θ + φ) + πR2∆p sin2φ¤ ˆn 1,2
Capillary force Surface tension (capillary pressure and contact line force)
Fc= −16πqQ2 oh2 µ 1 −p h (R2+h2) ¶ ˆ n1,2
Several other non-contact forces are described by Schulze [2008] and Tomas [2007]. Since their influence on the contact models and the non-contact forces are expected and sometimes tested as being limited, the forces are only summarized below and are not taken into account any further in this research.
• Visco-plastic or plastic deformation at particle contacts, which leads to an increase in adhesive forces through approach of the particles (distance is reduced) and enlargement of contact areas.
• Solid bridges due to solid crystallizing when drying moist bulk solids, where the moisture is a solution of a solid and a solvent (e.g., sand and salt water).
• Solid bridges from the particle material itself, e.g., after some material at the contact points has been dissolved by moisture (e.g., adsorbed moisture from the ambient atmosphere), and later the moisture has been removed (e.g., crystal sugars with slight dampness).
• Bridges due to sintering during storage of the bulk solid at temperatures not much lower than the melt-ing temperature. This can appear, for example, at ambient temperature durmelt-ing the storage of plastics with low melting points.
• Chemical processes (chemical reactions at particle contacts).
• Biological processes (e.g., due to fungal growth on biologically active ingredients).
2.5.
C
OHESIVE AND NON
-
COHESIVE MATERIALS
Since this research mainly focusses on the influence of cohesive bonds between particles, this section will explain the definition of cohesion. Besides, it will describe the different categorization of cohesion and how this is measured.
In many literature, two terms are used for almost the same characteristic namely adhesion and cohesion. Whereas adhesion is termed as the bonding of two different materials, cohesion describes the bonding of two of the same materials. For example, the process of a water droplet falling on concrete is adhesion and when this water droplet falls into a lake the process can be categorized in cohesion. This research will only focus on the influence of cohesive materials used in DEM-simulations. This means that only the interactions between the particles are investigated and the interaction with the boundaries or walls of the simulation are neglected since these are typically made of a different material. Besides, the materials that shown no cohe-sion between the particles, the free flowing materials, are not investigated in this research.
2.6.SUMMARY 9
But the main question is now, when is a material called cohesive? In general, the separation between co-hesive or non-coco-hesive materials is made on the ability of cohesion forces to act between individual particles. In order to separate or crack a cohesive materials, these forces need to be overcome. For example, grind or sand are non-cohesive materials while on the other hand clay shows cohesive behaviour. However, a material can not be classified into just one category under all circumstances. When moisture is added to sand, it will be less free flowing but will show cohesive behaviour as well. Therefore it is always necessary to clearly define the material properties that are implemented in DEM-simulations. Castellanos [2005] is the most cited paper (148 times according to Google on 2014-10-01) to categorize materials quantitatively into either cohesive or non-cohesive materials. He uses the "Cohesive Granular Bond Number" which is explained in Equation 2.4 and relates the inter particle force Fa to the weight m of the particle. The calculation of Fais different for
every contact model and can be found in Chapter 3.
B og=
Fa
mg (2.4)
When B og< 1, the maximal cohesive force on a particle is lower than the particle’s weight and one generally
speaks of a non-cohesive material. On the other hand, when B og≥ 1, the maximum cohesive force is greater
than or equal to the particle’s weight and the material is called cohesive.
2.6.
S
UMMARY
This chapter has given general information regarding the context of the contact models in the DEM. Besides, both the contact as well as the non-contact forces are explained. The next chapter will use the information from this chapter to describe the different contact models that are mostly often used in DEM simulations.
3
C
ONTACT MODELS FOR
DEM
One of the main focusses of this research is describing the similarities and differences between different con-tact models. This chapter explain the different particle concon-tact models that are created by researchers in more detail. It will turn out that some models are tested extensively with experiments while others are mainly based on theoretical knowledge. When a particle-particle or particle-surrounding collision occurs, the con-tact forces are usually decomposed into one tangential to the concon-tact plane and one normal to the concon-tact plane. This separation in contact models can also be seen in literature where certain papers or researchers are focusing on one of the two directions of the contact forces. Some models are mainly describing the influence of tangential or rotational forces while others look more into the normal direction of the colliding particles. The contact model for tangential directions have been less studied by researchers in the past decades. Figure 2.1 shows a schematic illustration of the forces acting on a particle. The forces between particle i and k are the cohesive forces, which are discussed in Chapter 2.
Section 3.2 will explain the available contact models for the normal directions while Section 3.3 will focus on the tangential direction. Some models are describing the forces in both directions to a certain extent.
Figure 3.1 on the next page shows a schematic overview of the contact models. Some of them can be traced back to a specific paper and the number of citations of that paper are listed as well. Interestingly enough is that some papers gained more than 10% of its citations in the two months that this report took to write. Especially the fundamental theories of Cundall and Strack and Johnson et al. gained a lot in popularity. This indicates that the research in the field of DEM simulations is a hot topic at this moment and a lot of new papers are written over this subject.
12 3.CONTACT MODELS FORDEM
3.1.LINEAR SPRING 13
3.1.
L
INEAR SPRING
One of the most used and earliest investigated contact models explicitly for the DEM is clearly the linear spring (dashpot) model (LSD-model). Cundall and Strack [1979] proposed a linear spring-dash-pot model for both the normal and the tangential direction. The linear spring obeying Hooke’s law, which is cut-off at the Coulomb friction force, imitates the elastic behaviour of materials while the dash-pot accounts for the dissipation due to plastic deformations. An schematic illustration of this model is depicted in Figure 3.2, where it can also be observed that the spring and dash-pot are placed in a parallel orientation to each other. The LSD-model can be used for particle-particle collisions as well as for particle-wall interaction. When the model is used for the calculation of the forces during a particle-wall collision, they elastic spring constant will be set to zero to represent the wall. This assumption is does not always results in accurate outcomes since some walls also are a little bit elastic (e.g. a plastic wall). However, the LSD-model does not account for cohesive materials and is therefore not further discussed in this research besides for the explanation of some of the other contact models, which implemented a cohesive force element [Di Renzo and Di Maio, 2004], [Gröger et al., 2003], [Morrissey, 2013].
Figure 3.2: Schematic diagram of the linear spring-dashpot model (LSD) showing the contact forces during a collision of two particles [Asmar et al., 2002, p. 788]
3.2.
N
ORMAL DIRECTION MODELS
The first research in the behaviour of elastic contact between two spheres was conducted by Hertz in 1886. He made a basic model that demonstrated that the size and shape of the zone of contact was directly caused by the elastic deformation of both spheres [Johnson et al., 1971],[Zhu et al., 2007]. Hertz only looked at the normal direction and considered that the relationship the normal force and its displacement between two spheres was non-linear. He used an optical microscope to measure the contact between glass spheres to verify his experiments and therefore his theory. Over the past decades his theory was either fine-tuned or partially falsified by several researchers. However, nobody could neglect the findings of Hertz for their own research and some of the theories in the next subsections even used his theory as the base and expanded it with more accurate experiments or simulations. The theories are listed in chronological sequence.
3.2.1.
J
OHNSON-K
ENDALL-R
OBERTS( JKR)
The Johnson-Kendall-Roberts (JKR) theory is based on the non linear contact model first formulated by Hertz in 1886. According to Johnson et al. [1971], the prior attempts to relate different measurements describing the mechanical force needed to separate two bodies had not been very successfully until 1971. It was noted that at low loads the contact areas of two bodies were much larger than predicted in the model of Hertz. Be-sides, the contact area tended to go towards a constant value as the load was almost reduced to zero while at high loads the results closely fitted the Hertz theory. This strong adhesion was observed if the surfaces of the spheres were clean and dry. These observations strongly proved that contact forces were operating between the particles and while these forces were relatively unimportant at high loads, they became increasingly im-portant as the load was reduced to zero [Johnson et al., 1971],[Morrissey, 2013]. At this point, they came up with their own contact model which is known as the ’Johnson Kendall Roberts-model (JKR-model)’, which tried to estimate the contact forces more accurate. Their model is supported by numerous experiments and is in a certain way an addition to Hertz’ theory. The JKR model describes that the contact radius of two collid-ing particles depends on both tensile and compressive interactions. The JKR model states that the adhesion force acting between two spheres, Fa, is not dependant on the elastic modulus of the materials in contact.
It depends, however, on both tensile and compressive interactions [Zhu et al., 2007]. The elastic modulus influences the contact radius a as can be seen later on. Mechanical work has to be applied to separate the
14 3.CONTACT MODELS FORDEM
two bodies and thus overcome the adhesive forces. This work creates new surfaces energy on its own and is defined as the free energy of the solid. The overlap caused by the additional surface force can be described by Equation 3.1, where a is the contact radius, R∗is the equivalent radius and E∗is the equivalent Youngs modulus,∆γ is the surface energy of the contact which is defined by Equation 3.2 [Savkoor and Briggs, 1977]. Equations 3.1, 3.5 and 3.7 are from Morrissey [2013] and Barthel [2008].
Figure 3.3: The contact between two convex bodies of radii R1and R2under a normal load P and whereδ is the elastic displacement.
The presence contact radius is a1and the absence radius is a0[Johnson et al., 1971, p 303].
δJ K R= a2 R∗− r 2π∆γa E∗ (3.1) ∆γ = γ1+ γ2− γ12 (3.2)
The equivalent Young’s modulus E∗is defined in Equation 3.3, whereν represents Poisson’s ratio [Wikipedia, a]. E∗=" 1 − ν 2 1 E1 + 1 − ν22 E2 #−1 (3.3)
The equivalent radius R∗used in Equation 3.1, is defined in Equation 3.4 [Di Renzo and Di Maio, 2005].
R∗=· 1 R1+ 1 R2 ¸−1 (3.4)
The Hertz equation modified to include surface energy is given by Equation 3.5.
FJ K R=
4E∗a3 3R∗ − 4
q
π∆γE∗a3 (3.5)
The contact radius a is given by Equation 3.6, where P is the pull force.
aJ K R= 3 r 3R∗ 4E∗[P + 3π∆γR∗+ q 6π∆γR∗P + (3π∆γR∗)2] (3.6)
The force that enables the two particles to separate is called the critical pull-off force and can be found in Equation 3.7.
PJ K R= −
3 2π∆γR
∗ (3.7)
When the surface energy is zero, the JKR model is simplified to the Hertz-Mindlin theory as described in Sub-section 3.3.1. This generalized Hertz equation for the contact area under zero load can be found in Equation 3.8 and K is defined in Equation 3.9.
a30=6π∆γR ∗2
K (3.8)
K = 4
3E∗ (3.9)
The JKR model is a commonly used contact model for DEM software whereas it is a default cohesion model in EDEM software [DEMSolutions]. However, the Van der Waals part is not included in the EDEM software. The JKR model is especially used when modelling fine cohesive powders.
3.2.NORMAL DIRECTION MODELS 15
3.2.2.
D
ERJAGUIN, M
ULLER ANDT
OPOROV(DMT )
Whereas the JKR model only takes forces within the contact area into account, Derjaguin et al. [1983] created a model that considers the (adhesive) forces outside this area as well. This model is commonly known as the DMT model, named after the first letter of the surnames of the authors. The theory of Hertz acts as the base for the DMT model and it tries to implement the adhesive forces in a different way as the JKR model. The overlap of spheres caused by the additional surface force is described in Equation 3.10, the contact radius is given in Equation 3.11 and the critical pull-off force can be found in Equation 3.12 [Derjaguin et al., 1983], [Morrissey, 2013]. δD M T= a2 R∗ (3.10) aD M T= 3 r 3R∗ 4E∗[P + 2π∆γR∗ (3.11) PD M T= −2πγR∗ (3.12)
The two theories of JMT and DKR appeared to be contradictory until Tabor [1977] showed that the theories are not contradictory but only act on the other extreme side of a spectrum of the Tabor parameter showed in Equation 3.13. µT= µ R∗∆γ2 (E∗)2Z3 ¶ (3.13)
This parameter is commonly known as the Tabor parameter and is used to draw adhesion maps (see Figure 3.4) to determine where certain contact models are better than others [Johnson and Greenwood, 1997], [Mor-rissey, 2013], [Tabor, 1977]. In 1997, Johnson and Greenwood stated that in general, whenµT << 1, the DMT
theory more appropriate while the JKR theory is more applicable whenµT >> 1. Later on, Xu et al. [2007]
changed these boundaries by also including the Maugis-Dugdale theory, which is explained in the next sub-section.
3.2.3.
M
AUGIS-D
UGDALEAdhesion forces outside the area of contact are neglected and elastic stresses at the edge of the contact are infinite according to the JKR approximation as was explained in Subsection 3.2.1. On the other hand, the DMT approximation described in subsection 3.2.2 takes the adhesion forces into account but the profile is assumed to be Hertzian. This means that the adhesive forces can not deform the surfaces of the colliding particles. Maugis [1992] extended the analytical DMT and JKR solutions to include a constant adhesion po-tential between both particles outside their contact area. This profile outside the contact area is given by elliptic integrals which can be found in Maugis [1992] and will not be explained in detail in this report. The Maugis-Dugdale model shows the transition from DMT to JKR behaviour as two opposite ends of a continu-ous spectrum based on the Maugis-Dugdale-parameter which can be found in Equation 3.14.
λ = 2σ0 µ R
πwK2 ¶1/3
(3.14)
In case of a solid-solid interaction, the Maugis parameter can be coupled to the Tabor parameter byλ = 1.16µT. Based on the adhesion map depicted in Figure 3.4, the JKR-analysis becomes most appropriate
whenλ or µT> 5; when λ or µT < 0.1, the DMT model will give the best solution and the intermediate range
16 3.CONTACT MODELS FORDEM
Figure 3.4: The adhesion map defining the optimal working regions for each contact model [Johnson and Greenwood, 1997, p 329].
3.3.
T
ANGENTIAL DIRECTION MODELS
Besides, the normal forces, particle interaction within a DEM simulation involve oblique or tangential forces as well. Yet relatively little information is described in literature regarding this tangential direction contact models and the oblique forces created by this impact. Criteria for tangential separation have been proposed by different researchers, including that a tangential force is capable of causing normal separation of the sur-face, which is called “peeling” [Cheng et al., 2002], [Savkoor and Briggs, 1977]. Figure 3.5 shows an illustration of the tangential displacement of a particle.
Figure 3.5: Illustration of tangential displacement [Di Renzo and Di Maio, 2004, p. 527].
3.3.1.
H
ERTZ-M
INDLIN ANDD
ERESIEWICZ(H-MD)
After the introduction of the JKR-model, researchers mainly focussed on the influence of the normal forces in the contact models. Mindlin and Deresiewicz [1953] mainly focussed on the normal direction but were one of the first that also explained the behaviour in tangential direction. After the theories of Hertz in 1883, limited research was done in the field of the elastic contact between two spheres for a long period of time. One of the possible explanations of this long period can be be found in the First and Second World War that took place. A precise reason, however, could not be found in literature. It was only in 1953 that Mindlin and Deresiewicz picked up the research from Hertz where he basically stopped. They used the theories of Hertz as their only reference points since this described everything that was yet known about this subject. Their model became known as the Hertz-Mindlin and Deresiewicz (H-MD) contact model and is used numerous times. It was very strange that the specific paper [Mindlin and Deresiewicz, 1953] could not be find after an extensive search and requests to different institutions. Therefore, the H-MD model is described by means of citations of other researchers that explained the model for their own paper. According to Di Renzo and Di Maio [2005], the H-MD model identified a series of conditions and rules for the generalization from simple
3.3.TANGENTIAL DIRECTION MODELS 17
cases to the oblique impact problem from an incremental procedure. Their work demonstrates that, due to the presence of tangential slip, the general force–displacement relation depends on the whole loading history and on the instantaneous rate of change of the normal and tangential force or displacement. The complete explanation of the H-MD model including all the governing equations, can be found in Di Renzo and Di Maio [2005] and Vu-Quoc et al. [2001].
3.3.2.
M
INDLIN, S
AVKOOR ANDT
HORNTONMindlin and Deresiewicz [1953] described both the normal and the tangential direction in the H-MD model but their focus was in the tangential direction. Savkoor and Briggs [1977] continued this focus on the tangen-tial direction because they stated that the forces in the tangentangen-tial direction during a impact played a bigger role than was thought of so far. They described that the contact area between two particles decreases with an increasing value of the tangential force in a stable manner until a critical value of the force is reached. They proved this with experiments in which they kept the normal force constant and therefore verified their results. The calculation of the shear tractions have not been conducted. Instead, an energy balance approach is used to prove that a reduction of the contact area can be caused by a peeling action. This peeling failure force is listed in Equation 3.15. The only parameter that has not been defined yet, is the equivalent shear modulus
G∗, which can be found in Equation 3.16.
Tc= 4 " (P PJ K R+ P2J K R)G∗ E∗ #1/2 (3.15) G∗=· 2 − ν1 G1 + 2 − ν2 G2 ¸−1 (3.16)
After the exploration work done by Savkoor and Briggs [1977], Thornton and Yin [1991] were the first to con-tinue this work and the JKR-model including adhesive behaviour in the tangential direction. However, they do not write about tangential forces but calls them oblique forces instead. The use of the term oblique forces is a more general approach to the involvement of all the forces that do not act in normal direction, as ear-lier discussed in section 3.2. Thornton and Yin [1991] combines the previous work of Savkoor and Briggs [1977] and Mindlin and Deresiewicz [1953] to create a new theory to describe the oblique behaviour. The new theory differs from the previous models, which are all based on Hertz’ theor. A new sliding criterion to account for the displacement is proposed. Since their model is a very complicated and mathematical model, the explanation of the model can be found in the original paper.
3.3.3.
V
U-Q
UOCThe latest improvement in the tangential direction dates from 2004, when Vu-Quoc et al. [2004] used the model of Thornton and Yin [1991] and expanded this with new insights and some novelties, which are listed below:
1. The additive decomposition of the elasto-plastic contact area radius into an elastic part and a plastic part where the effect of plastic deformation is now accounted included.
2. The correction of the particles’ radii at the contact point which represent a permanent indentation after the impact.
3. The correction of the particles’ elastic moduli. This part represents the correction of the elastic moduli due to a softening of the material due to plastic flow.
Vu-Quoc et al. [2004] extensively describes the use of his new tangential force-displacement (TFD) model and how his model accounts for the novelties listed above. Just like his predecessors, he uses the models of Mindlin and Deresiewicz [1953] as starting point for his TFD and tests the outcomes of his model with these previous models. However, in the opinion of Di Renzo and Di Maio [2005], the TFD model created by Vu-Quoc et al. [2004] needs more experimental validation. On the other hand, the verification of this model is sufficient for using it in other researches as well.
18 3.CONTACT MODELS FORDEM
3.4.
E
DINBURGH
E
LASTO
-P
LASTIC
A
DHESION
(EEPA)
The most cited source in this research so far is the PhD-thesis of Morrissey [2013]. The subject of his pro-motion was "Discrete Element Modelling of Iron Ore Pellets to Include the Effects of Moisture and Fines" in which a new contact model for the modelling of cohesive materials in DEM is created. Since he is work-ing at the University of Edinburgh, the new model is called the Edinburgh Elasto-Plastic Adhesion model (EEPA-model). This new model includes some novelties which are able to describe the behaviour of cohesive materials in DEM simulations. Since Morrissey promoted just in 2013 on this subject, EEPA has not yet been used or tested by others outside the research group of the University of Edinburgh. It is expected that the EEPA contact model will gain popularity in the coming years due to the extensive research and experiments and its focus on cohesive materials. Since this extensive research is one of the first primarily focussing on cohesive behaviour of bulk solids, this section is dedicated to the EEPA model and will elaborate further on the proposed model and its simulation results.
3.4.1.
P
ROPOSED MODELAll the models that are discussed so far are using single particles for the calculation of the equations of motion, the pull-off force and the relative approach. The summation of all these equations results in a time step in the DEM simulation environment. Since individual particles are calculated, the computation of a time step is very time consuming and complex. Morrissey created a new way to reduce this computational time while still maintaining the accuracy and reliability of the DEM simulation. A new elasto-plastic adhesive contact model is created that will consider the behaviour of bulk solids on a meso-scale rather than on a micro-scale and will be referred to as the Edinburgh Elasto-Plastic Adhesive contact model (EEPA). This methodology means that each DEM particle will represent the meso-structure of the bulk material. This meso-structure of each DEM particle will represent the associated local network, which will consist of a significant number of real individual particles or agglomerates that exist at a certain length-scale and the contacts and interactions between them [Morrissey, 2013]. The researchers of EEPA listed three main reasons and advantages to use the meso-scale. First of all, DEM simulations are computationally very intense and any way to reduce the computational work, is beneficial for the simulation. With the meso-scale, less particles have to be modelled to simulate the same amount of particles so the computational time will reduce as well. Secondly, particles rarely exist naturally as single particles but tend to form agglomerations, which replicate the meso-structure modelled in the EEPA. Finally, the researchers state that it should be easier to calibrate a model on a meso-scale rather than a model on a micro-meso-scale. They argue that the micro-meso-scale calibration requires more detailed information such as contact stiffness, pull-off force and moisture. One can discuss the last argument used by the researchers. Literature shows indeed that it is difficult to obtain reliable and consistent data from different experiments but the argument that this data should be more detailed on a micro-scale over the meso-scale is debatable.
3.4.2.
M
ODELO
VERVIEWIn the previous section, the proposed model was discussed and the explanation of using the meso particle-scale over the micro particle-particle-scale. The model is based on the hysteretic linear spring model originally pro-posed by Walton [1986]. The EEPA model aims to replicate the behaviour of two agglomerates in contact where during contact they are pressed together and undergo elastic and plastic deformations. A schematic force-displacement overview of the implemented hysteretic spring model can be found in Figure 3.6. The trajectories of the lines can be defined in equations and can be found in Equation 3.17 where k1represents a virgin loading branch, k2declares the unloading/reloading branch and the adhesive branch is referred to as
kad h. The component n in the equations accounts for the possible non-linear behaviour of k1and k2. In the EEPA model, k2is modelled as a constant and does not vary with the applied load. It should be noted that when n = 1, the model follows the linear spring model and when n = 1.5, the model replicates the Hertzian spring model.δpdescribes the plastic overlap when the two particles collide, and f0is the constant pull-off force. This constant pull-off force is according to the model of Thakur et al. [2013].
fh y s= f0+ k1δn f1 ⇒ if k2(δn− δnp) ≥ k1δn f0+ k2(δn− δnp) f2 ⇒ if k1δn> k2(δn− δnp) > −kad hδx f0− kxad h fad h ⇒ if − kad hδx> k2(δn− δnp) (3.17) k1= 4 3 p R∗E∗ (3.18)
3.4.EDINBURGHELASTO-PLASTICADHESION(EEPA) 19
Figure 3.6: Non Linear version of the Edinburgh Elasto-Plasto Adhesion (EEPA-NL) model [Morrissey, 2013, p. 68]
δp= µ 1 −k1 k2 ¶n1 δ (3.19) INITIAL CALCULATIONS
The most used governing equations of different contact models were described Chapter 3. This subsection will explain the calculations for these equations within the EEPA contact model. These equations will be the same even when the parameters are changed and are performed at every time step again. The plastic contact path radius a atδpis calculated for each contact to allow the contact area, due to plastic deformation of the
contact, to be calculated from Equation 3.20. In this equation, d represents the distance between particle centres and Riand Rjare the particle radii.
aE E P A= 1 2d q 4d2R2 i − (d2− R 2 j+ R 2 i)2 (3.20) where d = ½ d 1, for d1< d2 d2, for d2≥ d1 (3.21)
The separation distance d1while along k1and the separation distance for other branches, k2and kad h, are
defined in Equation 3.22 and 3.23 respectively.
d1= (Ri+ Rj) − δi j (3.22)
d2= (Ri+ Rj) − δp (3.23)
Now that the calculations of the plastic contact radius and the separation distances are explained, the mini-mum adhesive force fmi n(see Figure 3.6 can be defined by Equation 3.24. The new parameterψ is introduced
in this equation and stands for the adhesion constant similar in form used in the JKR or DMT theory.
fmi n= π∆γψaE E P A (3.24)
The final calculation for the pull-off force, the force at which the particles will start to separate, is defined in 3.25.
fmi n= PE E P A=
3
2π∆γaE E P A (3.25)
DAMPING MODEL
Whereas a lot of prior contact models either described the tangential or the normal direction of the particle collision, the EEPA model accounts for both. First the calculations for the normal direction are done. After these are finished, the tangential direction calculations will ’correct’ the model mainly with the difference of damping models in both directions. The tangential direction is influenced by the sliding frictionµ and the tangential stiffness coefficient kt. Detailed calculations of the order of calculations and the damping models
20 3.CONTACT MODELS FORDEM
LINEARMODE
One of the main novelties in the EEPA model is the implementation of the non-linearity behaviour of bulk solids, especially when these are cohesive. Figure 3.6 shows the non-linear track of the normal direction model with the factor n in Equation 3.17 accounting for the non-linearity when n 6= 1. However, it may also be interesting to look into the behaviour of the EEPA model when n = 1, in other words, when the model describes linear behaviour. The trajectory of the normal direction is depicted in Figure 3.7. When this linear mode is modelled, the contact model is similar to that of Luding [2008] but still accounts for both the normal and the tangential direction.
Figure 3.7: Linear version of the Edinburgh Elasto-Plasto Adhesion (EEPA-L) model [Morrissey, 2013, p. 80]
3.4.3.
T
HEE
DINBURGHP
OWDERT
EST(EPT )
The use of uniaxial compression tests for fine grained, cohesive bulk solids is problematic due to various rea-sons. First of all, the obtained yield strength values appears to be too low. Besides, the preparation of the hollow cylinder to obtain frictionless walls is very time-consuming [Schulze, 2008]. The Edinburgh research group created their own powder tester, the Edinburgh Powder Tester (EPT), to test different materials on its cohesive behaviour. Other then other common test methods such as Jenike’s Sheartest or the uniaxial com-pression test, the EPT can endure materials to an unconfined comcom-pression test. The EPT is a semi-automated uniaxial tester in which the cohesive strength of a bulk solid is evaluated from an unconfined compression test following a period of consolidation to a pre-defined stress level (σ1) [Morrissey, 2013, p. 107].
One of the main advantages of the EPT over the traditional testing methods is the speed at which the test can be repeated without the loss of its reliability. The outcomes of the EPT have been extensively verified and can be found in Morrissey [2013, chapter 5]. The EPT is verified with the aspect ratio of the samples, the crushing strain rate and its repeatability and focuses on the impact of the test on iron ore fines.
3.4.4.
V
ERIFICATIONWhile the EDEM software package has been tested numerous of times for different purposes, it is still nec-essary to verify the software again when implementing a new contact model. The contact model needs to be checked whether it reproduces the right behaviour and that the numerical implementations corresponds with the real life behaviour of materials determined in experiments. More information regarding verification in general can be found in Section 5.1.
3.5.
I
NFLUENCE OF ADDING MOISTURE
As earlier mentioned, a material is not either cohesive or non-cohesive under all circumstances. Whether a material is called cohesive or non-cohesive depends on its Bond number in which the weight of a particle does have a great influence. In industrial processes a material is sometimes starting to show cohesive be-haviour when moisture is added to this material, changing its particle weight and therefore changes the Bond number. This section will elaborate on the influence of adding moisture to a material and how the character-istics are changed by this. The main reference for this section is [Schulze, 2008, section 7.2.3.].
3.6.COMPARISON OF THE DIFFERENT MODELS 21
Figure 3.8: Flowability as a function of moisture content (in principle) [Schulze, 2008, p. 216].
narrow gaps between particles and starts to form liquid bridges (see Subsection 2.4.2). The liquid bridges are strengthening the adhesive forces between the particles which results in an increase of the material’s strength and a decrease of flowability (see Figure 3.8). When enough liquid is added, the material gets saturated which means that the voids within the particles are totally filled with liquid. This results in decreasing surface ten-sion and adhesive forces between particles. As soon as saturation is reached, the flowability starts to increase very rapidly. The moisture content at a saturated level is not a fixed value but can differ and can be explained with Equation 3.26, where S represents the saturation level, V the volume and² the porosity of the material which can be found in Equation 3.27.
S =VLi qui d VV oi d s = Vl i qui d ² ·V (3.26) ² =VV oi d s VTot al (3.27)
The density of the bulk solid material, and thus the porosity² can change. For example, a very moist but not yet saturated bulk solid can be transformed into a saturated state just by compaction, without adding more liquid. This effect can be demonstrated, for example, on a beach during falling tide: when one starts to tap his foot on a section of moist sand, the initially stable sand below the foot begins to compact (² decreases) and becomes a suspension (saturation). Thus it becomes clear that saturation can be attained at different values of moisture content and thus influence the flowability of a material. Handling of saturated bulk solid materials might cause unexpected problems when it initially appears to be good flowing but after a small amount of liquid has drained off, a state just below saturation is reached where flowability is very poor again [Schulze, 2008, section 7.2.3.].
3.6.
C
OMPARISON OF THE DIFFERENT MODELS
Most of the contact models that have been discussed so far are a reproduction with new insights of Hertz’ contact model. This means that the models show some similarities but differ in certain opinions. Besides, the models are not created simultaneously so some of them can make use of some technological improve-ments or breakthroughs that happened in between the creation of the different models. Figure 3.9 shows the equations of the four contact models that were discussed in this chapter in the normal direction. This figure should act as a summary and to provide a clear overview of the different calculations used for determining the contact radius a, the relative approachδ and the pull-off force Pc.
According to Morrissey [2013], the results from the DEM simulations show that while the JKR model can be used successfully for replicating situations such as powder packing, it fails to capture the stress history de-pendent behaviour that is so important for the storage and handling systems for cohesive granular materials. It should be noted that not all contact models found in literature were explained in this chapter. For example, Walton and Braun [1986] also created a contact model in for the normal direction that is used and cited by other researchers. This model is based on a linear spring dashpot (LSD) model but now including hysteretic behaviour. This means that the spring is not only dependant on the current and previous position but the stress history is also of importance. However, this model can be considered at an intermediate level between the full Hertz-Mindlin-Deresiewicz (H-MD) model and a simplified no-slip model (H-MDns).
