State-of-the-art in Discrete and Finite Element Modeling for bulk handling/Het modernste op het gebied van Discrete en Eindige Elementen Modellering voor de afhandeling van bulkgoederen

(1)

Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Maritime and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

Specialization: Transport Engineering and Logistics

Report number: 2014.TEL.7837

Title:

State-of-the-art in Discrete and Finite Element Modeling for bulk handling

Author:

Mark Erkens

Title (in Dutch) Het modernste op het gebied van Discrete en Eindige Elementen Modellering voor de afhandeling van bulkgoederen

Assignment: Literature

Confidential: no

Initiator (university): Dr. ir. Dingena Schott

(2)

Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Maritime 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: Mark Erkens Assignment type: Literature

Mentor: Dr.ir. Dingena Schott Report number: 2014.TEL.7837

Specialization: TEL Confidential: No

Credit points (EC): 10

Subject: State-of-the-art in Discrete and Finite Element Modelling for bulk handling

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. A well-known tool to model structures on the other hand is Finite Element Method, e.g. Ansys. This assignment focusses on the combination or co-simulation of DEM and FEM.

This research comprises not only the dry bulk industry. Every industry where the particulate materials are in contact with the equipment should be investigated, e.g. pharmaceutical industry, chemical industry, powder technology, food industry, agricultural industry.

Your assignment is to investigate and make an overview of the literature where DEM and FEM are combined or used together. That comprises amongst others (but is not limited to) the following:

 Find relevant literature and research groups

 Explain the methodology and theory of the combination of DEM and FEM

 Describe for which purposes the tool was used

 Describe details of the simulation tools used, e.g. software, contact models, etc.

 Describe advantages and disadvantages of the method and tools

 Make an overview of the available methods/tools and compare them.

 Classify and compare the literature found.

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

(3)

Summary

This literature report is used to provide an overview of the use of the combination of the finite and discrete element methods. The finite element method (FEM) is a method developed to help design and engineer products, by simulating the behavior of the material under the desired conditions. It is based on the principle of modelling a continuum by dividing it into a number of small elements called finite elements. The discrete element method (DEM) is a numerical model to describe the behavior of assemblies of discs and spheres. Unlike the FEM, with the DEM every particle is modelled as one element. Every particle is modelled as a disk in 2D, or a sphere in 3D, which can make contact with other particles. The contact forces are calculated at each time-step and then the motions of each particle calculated from there. The DEM is very useful for modelling granular materials for their behavior, like dry bulk materials. Dry bulk however is always handled by equipment like conveyor belts, grabs, chutes and silos. Usually these parts of equipment are modelled with FEM to get the optimal designs, but for the forces from the bulk material are not exactly known. Therefore the discrete element model of the particulate material has to be combined with the finite element model of the equipment.

In order to find the best method for the combination of the FEM and DEM the following research question:

“Which applications are simulated using a combination of the finite element method and the discrete element method, what principles are used to realize this combined?”

To answer this question many researches have been investigated and divided into two groups: - Contact between separate finite and discrete element structures.

- A single structure with both finite and discrete elements

In the first group separate parts exist in the model which are modelled with either a finite element mesh or as a discrete element. The advantage of a combined model with this kind of set-up is that both the behavior of the granular material modelled by the discrete elements can be simulated, as can the impact of this particle behavior on the finite element structure. This can be used for many

different applications such as shot peening, earth reinforced by a geosynthetic sheet, a tumbling mill, the deflection of a conveyor belt or rock cutting.

The second group uses the combination of FEM and DEM together in one single structure. This can be implemented in various ways. These can be used to model the failure of structures. For example dry stone masonry or a sheet of ice collapsing against a wall.

The researches for each of these different applications are investigated and explained in this report. Each of these researched methods are compared to each other based on four main topics: the contact models, simulation software and procedures, validation and verification principles that are used.

(4)

The overall conclusion of this report is that based on all these researches definitely a good combined DEM and FEM model can be created. However the exact principles that are most suited depend strongly on the application of the model and the data is needed from the simulation. This needs to be determined before the contact models or interaction between the discrete and finite elements can be chosen. Also the need for a one or two way coupling needs to be determined. In addition more research can be done on the best simulation software and process for the combined methods.

Because the verification and validation is insufficient is most researches it is recommended that before any method is used for simulations and design of large scale products, the model has to be verified and validated properly.

(5)

List of symbols

µ Coefficient of friction

C Middle point of contact line A-B cn Viscous damping coefficient

d Distance between penetrating object and center of discrete element E Young's modulus

FN Normal contact force

Fnd Damping normal contact force Fne Elastic normal contact force Fp Plastic force limit

FT Tangential contact force G Center of overlap area I Moment of inertia kn Coefficient of elasticity L Buckling length n unit normal vector

R Radius of discrete element S Overlap area

vrn Relative normal velocity w length contact line A-B W Virtual work

γ scaling factor

δ Maximum overlap distance σt Tensile strength

(6)

List of abbreviations

CAD Computer Aided Design DE Discrete Element

DEM Discrete Element Method

EDEM Experts in Discrete Element Modeling FE Finite Element

FEM Finite Element Method GE General Electric PFC Particle Flow Code

(7)

Introduction

This literature report provides an overview of the use of the combination of the finite and discrete element methods. There are many different applications and methods to apply these two principles. First the basis of these principles will be explained briefly and then why there is demand for the combination of these two principles. Then the advantages of combining these method will be discussed and then the research question and the structure of the report is explained.

1.1 Finite Element Method (FEM)

The finite element method (FEM) is a method developed to help design and engineer products, by simulating the behavior of the material under the desired conditions. This method is based on the works of Lord Rayleigh (1870) and W. Ritz (1909). The real development started in the early 1950’s along with the development of the computer. Later the finite element method has been the

researched intensely, hundreds of articles and books can be found on this subject. It is based on the principle of modelling a continuum by dividing it into a number of small elements called finite elements. An example of this is shown in Fig. 1.

In this example a cantilever beam is subjected to forces. The method divides a solid structure, in this case the cantilever beam, in a finite number of smaller elements in order to calculate the deformations and stresses in each part of the structure. This is still an approximation of the reality, but with a very large number of small elements, the results can be very accurate.

1.2 Discrete/Distinct Element Method (DEM)

The discrete element method (DEM) is a numerical model to describe the behavior of assemblies of discs and spheres. This method is more recently developed than FEM and can be used to model materials that consist of multiple granular particles, like dry bulk. The use of discrete elements for modelling granular materials was originally developed by Cundall [1] and later he published more papers which extended this method. Unlike the FEM, with the DEM every particle is modelled as one element. Every particle is modelled as a disk in 2D, or a sphere in 3D, which can make contact with other particles. In order to calculate the contact forces the soft disk model developed by Cundall and Hart [4] is used. This means that all particles are modelled as rigid disks and that the contact forces

(10)

between these disks are calculated using the overlap distance of these disks, denoted as δ. This is shown schematically in Fig. 2. Fig. 3 shows the case when there is contact between a rigid wall and a discrete element.

For both the cases in Fig. 2 and Fig. 3 then the overlap distance can be calculated according to equation (1). To calculate δ first the point C has to be determined. In Fig. 3 this is simply the middle of line A-B.

R d



 

(1)

Where R is the radius of the particle and d is the distance between the penetrating object and the center of the particle. From the overlap distance δ the contact forces can be determined according to different contact models such as linear elastic, Hertz, Winkler or power law. Which of these contact models is used will be explained in chapter 2 and chapter 3 for each individual application.

The contact forces are calculated at each time-step and then the motions of each particle calculated from there. This is a major difference between the finite element method, where the calculations are time independent and the discrete element method, which uses time-steps. This use of time-steps makes it possible to model the behavior of for example granular materials in a silo.

1.3 Combination of FEM and DEM

The DEM is very useful for modelling granular materials for their behavior, like dry bulk materials. A downside however of the discrete element method is that the objects other than the particulate material are modelled as a rigid material. Dry bulk however is always handled by equipment like conveyor belts, grabs, chutes and silos. These types of equipment need to be designed, which is impossible with the DEM only, because this cannot calculate the stresses and deformations in the equipment. Normally these types of equipment are designed using finite element simulations. However because the forces on the finite element mesh of the equipment are caused by particulate materials, this is better to simulate with discrete elements. So in order to keep the advantages of the DEM for the simulation of the bulk material and the ability to calculate the stresses and deformations in the equipment of the FEM, the two methods have to be combined.

(11)

In 1995 Munjiza [2] started with the combination of both FEM and DEM and he extended this in later researches. He wrote a book about the combined finite-discrete element method in 2004 [3] and many of the researches discussed in this report, refer to this work.

Because there are various ways to combine the two methods and everyone uses different principles to obtain this combined simulation method this report is written. The main goal of this report is to provide a complete overview of every application of the combination of both the FEM and DEM, along with the principles that are used to combine these two methods. There are various researches that use this combination for a specific application. These will all be compared in order to give an idea of which principles are best for a specific application. To accomplish this, the following research question will be answered in this report:

“Which applications are simulated using a combination of the finite element method and the discrete element method, what principles are used to realize this combined?”

The first issue when combining the FEM and DEM together is that they are both simulated in different software programs. Therefore a coupling has to be made between these two software packages. Because the DEM software calculates behavior of particles in small time steps and FEM software calculates deformations caused by forces at a specific moment, this can be realized in different manners. For example the contact forces from a DEM analysis can be transferred to the FEM software to calculate the resulting deformed shape of the finite element structure caused by the discrete elements only. This will be referred as a one way coupling and it is schematically shown in Fig. 4.

FEM

Software

DEM

Software

Contact forces

Fig. 4: One way coupling between DEM and FEM software.

However this way the DEM analysis can be performed for a certain amount of time steps and afterwards all contact forces are transferred to the FEM software. This can be useful when the expected deformations of the finite element structure are very small and do not affect the behavior of the discrete element particles in the DEM analysis. However if this is not the case, then a two way coupling between the two software packages is desirable. This means that after a few time steps the contact forces are transferred to the FEM software and a deformed shape of the finite element structure is calculated. This deformed shape is then transferred back to the DEM software as new geometry for the next time steps. These steps can then be repeated as long as necessary. This is shown schematically in Fig. 5.

(12)

FEM

Software

DEM

Software

Contact forces Deformed shape

Fig. 5: Two way coupling between DEM and FEM software.

1.4 Structure of the report

In order to compare all researches on the combination of FEM and DEM the papers are divided into two groups based on the application used in the research:

- Contact between separate finite and discrete element structures (Fig. 6). - A single structure with both finite and discrete elements (Fig. 7).

In the first group separate parts exist in the model, as shown in Fig. 6. These different parts are modelled with either a finite element mesh or as a discrete element. In this example the rigid disk is a discrete element and it bounces against a finite element structure. In this type of combination

between the FEM and DEM the contact and forces between the finite and discrete parts are the most difficult part. The researches on this type of combination using both FEM and DEM are explained in chapter 2.

The second group uses the combination of FEM and DEM for a different purpose. Here both FEM and DEM are used together in one single structure, as shown in Fig. 7. Normally these type of

constructions are modelled with just FEM, but here the DEM particles are used to get a better model for the failure of the structure. This can be used for example to model the failure of the structure. This group is the topic of chapter 3.

Each chapter is then divided in paragraphs based on the specific application of a research. In order to compare all researches even though the applications can be different, this comparison is done

according to three main topics.

Fig. 6: Separate discrete element and structure of finite element mesh [4]

Fig. 7: Single structure with both finite and discrete elements [19].

(13)

First the method of modelling the contact or interaction between the finite and discrete elements will be explained, then the simulation software and procedures are listed and after this the validation and finally the validation and verification principles that are used come will be discussed.

(14)

Contact between separate finite and discrete element

structures

This chapter focusses on the researches based on the principle that multiple structures exist in a model which are each modelled with either finite or discrete elements. This type of modelling has been used for several different applications. All different applications that were found on the combination of FEM and DEM with two separate parts of the model are explained in chronological order. For each research the contact models, software package that was used will be described for each research. Finally it will be discussed whether the method used is verified or validated in the research.

The applications of the researches in this chapter are shot peening in paragraph 2.1, earth structures reinforced by geosynthetic in paragraph 2.3, a tumbling mill in paragraph 2.4 and rock cutting in paragraph 2.6. Finally in paragraph 2.7 the methods found in all these different researches and applications will be compared, in order to discuss the similarities and differences.

2.1 Shot peening

Shot peening is shooting projectiles at a work piece with high velocities to harden the surface by creating stresses in the work piece. These projectiles can be made from cast iron, glass or steel [4]. To predict these stresses, earlier finite elements were used for both the work piece as the shot [5]. This works very well for a single shot, however shot peening is a process which uses a lot of shots during a substantial period. On the topic of shot peening with the combination of FEM and DEM a number of different papers are published, [4] [5] [6] [7] [8] and [9]. Because they all use the same applications and use each other’s works in their own researches these are all described in this paragraph.

First the contact models used in these researches on shot peening are explained in paragraph 2.1.1, then the different software and modelling procedures will be discussed in paragraph 2.1.2, Then paragraph 2.1.3 describes the verification and validation methods that are used.

2.1.1 Contact models

In 2000 Han and Perić et al. [2] introduced the use of discrete elements to model the shot, for the work piece still finite element are used. The most difficult part of the use of both the finite and discrete element method is the transfer of forces between the different type elements. The discrete element is modelled as a rigid circular disc in 2D models or a rigid sphere in 3D models. Because in reality the particle that has been sot at the work piece can also deform, even though it has a much higher surface hardness than the work piece, an overlap between the disk and the work piece is allowed, see paragraph 1.2.

(15)

The set-up of the model is shown in Fig. 8a. When the finite elements are larger in size than the discrete element shots the contact is practically the same as with the normal DEM, because one discrete element only contacts one finite element, as shown in Fig. 8b. However when the finite element size is much smaller, it is very likely that contact with multiple finite elements simultaneously will occur. In this case the contact can be as shown in Fig. 9.

Because the force has to be calculated for each of the contacting finite elements, it is harder to determine the actual overlap and the direction of the force of one single finite element. Han and Perić et al. [4] solved this problem by extending the line of the contact surface of one finite element to assume full contact, this is shown in Fig. 10a. Here contact line A-B is extended as line I-J. This way the overlap can be calculated for the case of full contact.

Fig. 8: Discrete element shot and finite element work piece [4].

Fig. 9: Contact with multiple finite elements [4].

(16)

Because the actual contact area is smaller than with full contact, the calculated force has to be scaled down with a certain ratio γ to estimate the force on line A-B only, according to equation (2).

, ,

N partial N full

F





F

(2)

Han and Perić et al. [4] considered two methods of scaling down the force and determining the direction of the force. The first method is shown in Fig. 10a and equation (3). This uses the ratio between the overlap area of full contact (dotted area, Sf) and partial contact (latticed area, Sp). The intersection point between the line from the middle of the partial overlap area (G) and the middle of the discrete shot (O) is chosen as point C. The direction of the force is then along line C-O.

p s f

S





(3)

The second method is shown in Fig. 10b and equation (4). This uses the ratio between the length of line I-J (wf) and line A-B (wp) to scale the force. To determine the direction of the force, point C is simply chosen as the middle of line A-B, then the force is again along line C-O.

p w f

w





(4)

The second method is easier and requires less computations. Therefore it reduces the simulation time. About the results Han and Perić et al. [4] write:

“Theoretically, it seems that the first approach may give a more accurate solution but, taking into consideration the interactive model as a whole, no noticeable difference in terms of solution accuracy has been observed between the two approaches in shot peening simulations.”

When the forces on the discrete shot from each finite element are all combined together this results in the total force on the shot, Fd, this is shown in Fig. 11.

(17)

From the overlap the force can be calculated Han and Perić et al. [4] considered four contact models in their work: Linear/Hooke, Hertz, Winkler and Power Law. When generalized for the specific case with a rigid disk and under de assumption of small deformations, according to equation (5), all these contact models can be represented with the power law model.

2R





(5)

Table 1 shows all these models and their form under the assumption of small deformations. Then the last column can be taken as the coefficient kn and the fourth column as the power m in the general form of the power law model.

The conclusion of Han and Perić et al. [4] is that under the assumption of only small deformations, all these models can be written in the form of the power law model. As the shot is assumed

undeformable all the contact models are penalty based, in other words a penalty value can be chosen which influences the penetration of the particles and finite elements. This can affect the stability and accuracy of the model.

Now the normal forces can be calculated. However there are also tangential forces that have to be taken into account. In this research Han and Perić et al. [4] made the assumption that:

“The normal impact force is independent of the friction, i.e. the normal and tangential contacts are uncoupled in the sense that the presence of friction in the impact will not affect the magnitude of the normal force”

The tangential forces are obtained following the standard elastic-plastic theory of friction. This means that the tangential force increases with the tangential displacement δt according to slope kt. The tangential force will linearly increase to a maximum value according to the Coulomb law, with µ the coefficient of friction, shown in equation (6). Then only the displacement increases, meaning the discrete shot sliding across the finite element.

(18)

max

( )

F

_t





F

_N (6)

When the relative displacement decreases, the force will unload with the same slope as the loading, until it reaches the same maximum value only in the opposite direction. This loading and unloading of the tangential force is shown in Fig. 12.

Bhuvaraghan et al. [6] and [7] refers to the described contact models in Han and Perić et al. [4] and [8]. It is however not mentioned which of these four is eventually used in the simulation. The only clear difference in this research are and interesting assumptions made by Bhuvaraghan et al. [7] with respect to the contact of the shots with the finite element mesh:

“The shot is assumed to impact the surface in normal direction. The effect of friction is ignored as it does not have much influence on the residual compressive stress.”

This is surprising, especially because earlier Han and Perić et al. [8] stated in their research that:

“Friction plays a significant role in the contact model. The resulting residual stress is increased with increase of the frictional coefficient µ.”

This last statement is backed through the results shown in Fig. 13. This indicates that the residual stresses are definitely influenced by the friction between the shot and the work piece, because the residual stress in the work piece changes with a changing coefficient of friction.

(19)

2.1.2 Simulation software and procedures

In the two papers written by Han and Perić et al. [4] and [8] show examples of their simulations of shot peening with a single shot. Nothing is mentioned in both papers what kind of software is used for either the finite element analysis or the discrete element analysis.

In 2008, Hong [9] used FEM to model the impact of the shot on the work piece, both parts of the model are a finite element mesh. Then they used DEM to model the behavior of the shots, after hitting the work piece. For the finite element model the commercial software package Abaqus/Explicit and for the discrete element model the software EDEM were used. This particular research uses both FEM and DEM, but the results of EDEM are manually transferred to the finite element software. This means that the contact forces from the DEM analysis are written down and later used as an input for the FEM analysis. This is not desirable for a simulation with multiple shots, which is usually the case with shot peening.

A research group from General Electric (GE), Bhuvaraghan et al. [6] and [7] did further research on the modelling of shot peening. The basis of this research were the two models used by Hong [9], only now a one-way coupling between the two models was established. The model was made in a CAD environment, then transferred to EDEM for the modelling of the shot peening process. The contact forces are the transferred to ANSYS in order to perform a finite element analysis to calculate the residual stresses in the work piece. In Fig. 14 the process of the combined models is shown.

Then the simulation is extended to a multi-ball simulation. In EDEM the shots are simulated for a certain period, as shown in Fig. 14, until there is sufficient coverage. EDEM calculates the contact forces of every shot impact and the locations of these contact forces. These contact forces are then

(20)

loaded into ANSYS, on a fine model mesh. There the contact forces can be translated into pressures on the elements. From there the stresses in the work piece can be calculated.

The process of data transfer from DEM to FEM that is used is shown in Fig. 15. All contact coordinates and associated contact forces are taken from the DEM software. Then the nearest nodes in the FE model are determined and the contact force translated into a pressure distribution on these nodes.

Fig. 14: Process of the combined model using EDEM and ANSYS [6]

(21)

2.1.3 Validation and verification

The researches of Han and Perić et al. [4] and [8] have not shown any validation or verification in their researches. Their methods however are used as a basis of many researches about the combination of FEM and DEM in later years.

Bhuvaraghan et al. [6] and [7] used the already common used finite element method to verify their simulation. They first modelled a single shot case is in EDEM and compared the results with the results of a simulation in Abaqus/Explicit in order to verify the EDEM results. The results of these simulations shown in Fig. 16, are very similar. The velocities before and after impact are practically the same. This shows that the same amount of energy is transferred from the shot to the work piece in both simulations. There is no verification for the residual stress in the work piece between the finite element simulation and the combined finite and discrete element simulation.

The multi ball analysis however is not compared with other methods or a real experiment in this research. The simulation described by Bhuvaraghan et al. [6] and [7] seems to work for shot peening. Because there is only a one way coupling from DEM to FEM, the deformations of the work piece have to be very small. For applications with larger deformations it is desirable to have a two way coupling between the DE and FE simulations. This way the deformed shape after one time step can be again used for new calculations with the DE model.

2.2 Various geomechanical applications

In 2004 Oñate et al. [10] used the combination of FEM and DEM for the simulation of different

geomechanical problems, such as rock cutting. However they explain their model using the example of a unconfined compression test on a rock specimen.

The contact models used in this research will be explained in paragraph 2.2.1, then the simulation software and procedures will be discussed in paragraph 2.2.2 and finally in paragraph 2.2.3 the methods used to verify and validate the method are shown.

(22)

The normal contact force between the discrete elements and the surface of a finite element is calculated according to equation (7).

n ne nd

F



F



F

(7)

Where the Fne is the elastic normal force, calculated in the same manner as the earlier described linear elastic model explained in paragraph 2.1.1, as shown in equation

ne n

F



k



(8)

Where kn is the normal stiffness and δ the overlap distance. This method of Fne is the same as

described earlier by Han et al. [4], however in this research also a damping force is taken into account in the form of Fnd :

nd n r n

F



c v

(9)

There cn is a damping coefficient and vr n the normal component of the relative velocity at the contact point.

The tangential forces are calculated using the regularized Coulomb law of friction, as described in paragraph 2.1.1 and shown in Fig. 12.

Several examples are provided in this particular research of simulations of different types of problems. However, the type of simulation software is not mentioned at all, so it is not known which software packages or packages are used for these simulations.

An unconfined compression test is performed in a model and in a laboratory experiment. The geometry of the model is shown in Fig. 17. The failure mode of the numerical model, shown in Fig. 18, is similar to the failure mode of the rock specimen after the same test performed in a laboratory, shown in Fig. 19.

(23)

There are more examples in this research with other applications, however there is not really validation or verification performed by comparing actual values of stresses or forces. There is simply the statement that the model and experiment are in fair agreement.

2.3 Earth structures reinforced by geosynthetic

In 2009 Villard et al. [11] used both finite and discrete elements in the modelling of a geotechnical problem. When building road or rail infrastructure a geosynthetic sheet is used to reinforce the soil, see Fig. 20.

Fig. 17: Numerical model of unconfined compression test

Fig. 18: Failure mode obtained from numerical simulation [10].

Fig. 19: Rock specimen after laboratory unconfined compression test [10].

(24)

A granular material like gravel or sand rests on top of the sheet. When a sinkhole appears underneath the sheet, large deformations occur in this structure. It needs to be calculated whether the sheet can hold the granular material and therefore this is simulated. A schematic image of a geosynthetic sheet covered with granular soil with a sinkhole underneath the sheet is shown in .

The contact models used in this research will be explained in paragraph 2.3.1, then the simulation software and procedures will be discussed in paragraph 2.2.2 and finally in paragraph 2.3.3 the methods used to verify and validate the method are shown.

However not the entire weight of this material rests on the sheet. The arching effect will result in a transfer of the load. In order to calculate the actual load on the sheet, not only the FEM can be used. The DEM is used to model the behavior of the granular material. The soil underneath the sheet is assumed to be undeformable. For the sheet, finite elements are used. Because the sheet is very thin, it is modelled as a membrane. This assumption means that the sheet cannot have any bending stresses, only forces in one direction. Therefore the most important transition of forces between the granular material and the sheet are frictional forces. For the finite element three-node elements are chosen. The interaction between the discrete elements and the finite elements is realized using the linear elastic contact model, shown in Table 1 Linear/Hooke. Because with the sheet no bending stresses can occur, the friction forces are the most important forces in the model. The behavior in friction is independent of the normal contact stiffness kn, therefore this value is chosen as large value

in order to minimize the overlap between the discrete elements and the finite elements. For the tangential forces the same method is chosen as described by Han and Perić et al. [4] in equation (6) together with Fig. 12.

Fig. 20: Geosynthetic sheet used to reinforce the soil for rail or road infrastructure.

(25)

In this research the used software for the simulation of the DEM and FEM parts are not mentioned.

In order to verify and validate the model, the results of the numerical model are compared with analytical methods and a true-scale experiment. Analytical methods exist to describe the behavior of reinforced embankments submitted to sinkholes. Examples of these methods are provided by Villard et al. [11], these will not be explained in this report. The model was validated using a true-scale experiment, with geometry as shown in Fig. 22. On the position of the sinkhole balloons were fitted. These balloons were slowly emptied to simulate the forming of a sinkhole in reality. The strain was measured in the geosynthetic sheet using Bragg Gratings sensors [12] continuously during the experiment. These sensors were fixed on the sheet at many points with intermediate distance of 1m. Displacements of the sheet were measured manually.

These measurements are compared with the results of the numerical model. The geometry of the numerical model is shown in Fig. 23. The result of the model after the sinkhole is shown in Fig. 24.

Fig. 22: Geometry of the true-scale experiment of reinforces embankment with sinkhole [11].

(26)

For comparing the numerical model, analytical method and true-scale experiment, the measured strains and vertical displacements of the sheet over the sinkhole are compared to the calculated and simulated values. In Fig. 25 the results of the measured strains in the experiment are compared to the analytically calculated values and the results of the numerical model. Fig. 26 shows the vertical displacements of the geosynthetic sheet over the sinkhole.

According to Villard et al. [11] these results show that:

Fig. 24: Views of result of the numerical model after the sinkhole [11].

Fig. 25: Comparison between strain results of numerical model, analytical method and true-scale experiment [11].

Fig. 26: Comparison between vertical displacement of the geosynthetic sheet between numerical model, analytical method and true-scale experiment [11].

(27)

“The numerical model reproduces the main mechanisms involved during the subsidence of the reinforced embankment in a satisfactory manner: membrane effect, stretching, friction, and sliding of the geosynthetic sheet in the anchorage areas, increase in strain (and tensile force) at the edges of the cavity and the behavior of the granular soil layer under large displacements.”

The small differences are attributed to the simplifications made in the model, such as the undeformable soil underneath the sheet.

The final conclusions of Villard et al. [11] are that the combined discrete and finite element model reproduce the behavior of the with geosynthetic sheet reinforced soil with localized sinkhole in a satisfactory manner. The discrete element can realistically model the granular soil and the finite element can take the fibrous structure of the sheet into account. It still has to be researched whether the results of this model are still satisfactory when it is used for a three-dimensional application.

2.4 Granular materials in a tumbling mill

Jonsén et al. [13] and [14] did research on the combination of DEM and FEM in the modelling of a grinding mill with a grinding medium. Research on the use of DEM on the simulation of particle flow in grinding mills has already been done extensively. In this research the mill structure is modelled using finite elements and the grinding medium with discrete elements. This way DEM can be used to predict the behavior of the grinding medium and FEM to predict the stresses in the mill structure.

Paragraph 2.4.1 explains the contact models used in this research, paragraph 2.4.2 then shows the simulation software and procedures that are used and finally in paragraph 2.4.3 the methods used to verify and validate the method will be discussed.

There is barely any explanation of the contact model that is used. The only thing mentioned is that for the contact between the particles and the structure a ‘‘nodes to surface’’ contact is used in LS-Dyna v971. The support webpage of LS-Dyna [15] has a very limited explanation on this contact method. Only a simple example of a geometry with nodes in shown in Fig. 27. The relation between contact forces of the surfaces and nodes is not explained.

(28)

The non-linear finite element code LS-Dyna v971 has been used for the modelling and simulation of the mill. The structure of the mill is and consists of a mantel made of steel and lifters and liners made of rubber. The steel mantel is modelled as a rigid material and the rubber parts as hyper-elastic material, both eight-node solid elements. The geometry of the model is shown in Fig. 28 and Fig. 29.

The ball charge is modelled as DEM particles. Two cases are modelled, one with ball charge of diameter 20mm and the other with equally distributed diameters between 15mm and 25mm.

To verify the results of the model, an experiment has been executed. The setup of the experiment is shown in Fig. 30 and Fig. 31. During the rotation of the mill, a force acts on the lifter bar caused by the charge. This causes a deflection of the strain gauge sensor which converts this into an electrical signal. This way the force on the lifter bar can be measured during operation. The ball charges used in the experiment have diameters between 10mm and 30mm.

Fig. 27: Contact nodes to surface [15].

(29)

During a rotation the force on the lifter bar has a certain pattern. This pattern is compared between the results of the experiment and the model. Fig. 32 shows the results of both cases of the model and the experiment. The graph represents the displacement of the lifter bar during a part of the rotation. According to Jonsén et al. [14] the results of the deflection of the model show an acceptable

agreement compared to the experimental results.

It would have been more logical if the ball charge size of the model would have been chosen the same size as the ones used in real experiment. Because now the ball charges are different, it is hard to say that the model can be validated with this experiment.

2.5 Conveyor belt deflection

A research conducted in 2013 by Dratt et al. [16] uses the combined FEM and DEM approach for the simulation of the deflection of a conveyor belt transporting dry bulk material. Because a belt is only

Fig. 30: Cross section of a mill [14]. Fig. 31: Strain gauge sensor embedded inside a lifter bar [14].

(30)

supported by idler rolls which are placed a certain distance apart from each other, the belt can sag under the weight of the bulk material it transports. For modelling the bulk material, the DEM is very suited. For modelling the deformation of the belt however, it would be desirable to model this with finite elements. This research proposes a method of combining these two methods for this particular application.

Because the belt is modelled with finite elements and the bulk material with discrete elements, the interaction between the two methods must be modelled as well. In this research an elaborate explanation is provided on the deformation of the finite elements. How this is done can be found in their paper [16]. There is no mentioning of the calculation of the contact forces between the discrete and finite elements. In the previous researches often the relation between the overlap of the finite and discrete elements is explained, but this is not mentioned in this research. Here there is explained how the geometry of the FE model is prepared for use in the DE model and the DEM contact laws. What kind of DEM contact law is not mentioned, which seems strange because DEM software can use multiple contact laws.

Because the finite element calculations are made in a different software environment than the discrete element model, the data has to be transferred from one program to another. The finite element mesh is transferred into an STL-format (Standard Tessellation Language). An STL file renders surfaces of the model as a mesh of triangles. Because the finite elements can deform and de STL-format inly uses planar triangles, the number and size of the triangles determines how accurately curved surfaces are printed. In this research each rectangular finite element is divided into two triangles. This step is shown in Fig. 33.

Then the geometry in STL-format is used as an input for the DEM software, here the contact forces can be calculated and then transferred to the FEM software for the calculation of the deformations. In this research for the FEM analysis ANSYS is used PFC for the DEM analysis. For simple applications with bulk handling like transfer chutes or silos, usually a one way coupling between the two modelling

(31)

methods is sufficient. However because a conveyor belt has large deflections between idler sets, a two way coupling is required. In Fig. 34 a schematic view of the model of the belt conveyor is shown.

First the model of the belt is tensioned and loaded with particles. When the model is settled the belt is started for the dynamic analysis. When the particles are settled again and a constant load condition is reached, the loads on the belt calculated by PFC are transferred to ANSYS and the positions and velocities of every particle is saved. Then in ANSYS the forces on the belt from PFC are used to calculate the deflection of the belt. This new geometry is then translated into STL-format and used for the next time steps in PFC. These steps are then repeated until a steady belt deflection is reached.

For verification of the model, the results of the forces on the bearings of the idlers from a simulation with the proposed model are compared to the analytical values following the theory of Krause and Hettler [17]. A simulation with the model shown in Fig. 34 provided the numerical results.

According to Dratt et al. [16] there is a good correlation between the theoretical approach and the coupled simulations.

Also the proposed method is compared to a simulation with only a FE model. The setup is a simple finite element beam supported on both sides is loaded with discrete elements. This example is shown in Fig. 35. The finite element mesh of the beam is loaded with randomly generated discrete particles. In the FE model the force of the particles is simulated as a constant distributed force.

Fig. 34: Conveyor belt model with three idler stations, colors indicates particle velocity [16].

(32)

It is simply stated that the difference between the results of the two simulations is less than 1% and that this difference is due to the irregular distribution of the particle bed. It is strange to compare these results, because combined model uses a two way coupling, which means that the direction of the forces on the belt changed during the simulation due to deflection of the belt. Therefore it seems likely that there is a difference between the contact forces from the DEM software compared to a simple uniform distribution. The proposed combined model is designed because a uniform distribution in a FEM model is not realistic.

A great advantage of this method compared to the earlier described ones, is the two way coupling by means of the STL-format. This makes it possible to simulate structures with large deflections or deformations, which is otherwise not possible.

2.6 Rock cutting

In 2013 Fang et al. [18] used the combination of FEM and DEM to simulate the cutter of the earth pressure balance shield machine (EPBSM) in 3D.

Paragraph 2.6.1 explains the contact models used in this research, paragraph 2.6.2 then shows the simulation software and procedures that are used and after that in paragraph 2.6.3 the methods used to verify and validate the method are shown.

The contact models used in this research are derived from the earlier described works of Han and Perić et al. [4] and [8]. Here they chose the linear elastic model is used because it is the simplest model to calculate. The overlap distance δ is defined in the same way as in equation (1). Only the normal force Fn is now calculated according to equation (10).

n n iw

F



K



n

(10)

Where niw is the unit normal vector of the plane and Kn is the secant modulus determined using

equation (11), where Kni and Knw are the secant modulus of the particle and the plane.

(33)

i w n n n i w n n

K K

K





(11)

The tangential force Ft is calculated according to equation (12).

if

sgn(

) if

s s t n t n s t n

K

U

F

U

F











 









(12)

Here ΔUs is the relative shear-displacement increment between the particle and the plane and µ the coefficient of friction. The tangent modulus Ks is determined with equation (13), where Ksi and Ksw are the tangent modulus of the particle and the plane. Equation (12) shows that the tangential force increases with a relative displacement between plane and particle, until a maximum value of µFn.

i w s s s i w s s

K K

K





(13)

The position of the point of contact between the particle and the plane is defined according to equation (14), with Xb the position of the center of the particle, d the distance between the center of the particle and the plane and niw the normal unit vector of the plane.

c b iw

X



X



dn

(14)

Fang et al. [18] then use this contact model in a combined FEM and DEM model of the earlier described cutter head. The spherical discrete elements are used to model the soil area. This soil consists out of thousands of particles with radii between 0.01 m and 0.02 m. For this assembly a program written in Visual C++ 6.0 and on the OpenGL platform is used by Fang et al. [18]. The cutters and cutter head are modelled with finite elements. The design of these parts in made in a CAD environment and discretized into a tetrahedral mesh with FEMAP or PATRAN. This is then converted to the OpenGL format. The triangles surface meshes are used for the contact detection during the simulation and the tetrahedral mesh is used for the visualization of forces and stresses on the cutters and cutter head. First one single cutter is simulated, Fig. 36 shows the stress distribution on this cutter at a certain the 15 000th_{time step. It seems logical that the stress reaches its maximum values}

near the cutting edge, but also at the points where the cutter is fixed to the cutter head. There the moment caused by the force on the cutting edge reaches its maximum value, so the bending stress is there the largest.

(34)

In Fig. 37 the simulation of the entire cutter head is shown. The cutter head is simplified, the number of cutters on the head is reduced from 204 in reality to 21 in the model to minimize calculation time. The results show, that the stresses increase when the cutter head reaches a greater depth, which seems logical. Also the stress in the cutters further from the center of the head is larger than in the cutters closer the center. This is because the velocity of the cutters increases when the distance to the center of the head increases.

In the research by Fang et al. [18] there is no verification or validation performed. They simply show their results and give conclusions.

Fang et al. [18] conclude that coupling the FEM and DEM is good to simulate the interaction between a continuous structure and discrete medium. The downside of this method is that the interaction between a large number of particles and a fine finite element mesh is limited by the capabilities of

Fig. 36: Stress distribution on the cutter on the 15 000th time step [MPa] [18].

(35)

current computers. So simplifications have to be made still, like larger particles, larger finite elements, smaller overall model or splitting the model in smaller parts.

2.7 Comparison between the found methods

First the different parts of each research will be compared, starting with the used contact models in paragraph 2.7.1. Then the different types of simulation software and strategies are compared in paragraph 2.7.2. Later paragraph 2.7.3 provides an overview of the validation en verification methods used in all researches and finally paragraph 2.7.4 summarizes the advantages and disadvantages of the different researches over the other ones.

In this chapter there is contact between different structures. These structures can be either discrete elements of structure that consist of finite elements. Because there is interaction between these parts contact models have to be used.

Because the DEM also uses contact models for interaction between the different discrete elements, this can almost be used directly in the combined model. The contact between discrete elements in the combined model remains the same. In case of contact of one single finite element with a discrete element this is also still the same, only the forces have to be divided over the nodes of this particular element in the FEM software. However in case of contact of one discrete element with multiple finite elements, the principle described by Han and Perić et al. [4] in paragraph 2.1.1 can be used to divide the total force between the number of finite elements. This method is only used in later research by Fang et al. [18] and not mentioned in all other researches while it seems necessary for most models.

All researches which explain their contact model use the overlap distance δ to calculate the contact forces. The only difference is that some mention which contact model they use to relate the overlap distance δ to the normal contact force. Han and Perić et al. [4] and [8] only describe the possible contact models but do not mention which one is used in their simulations. Villard et al. [11] and Fang et al. [18] use the linear elastic contact model because it is the easiest to calculate. Oñate et al. [10] also use linear elastic, but they extend this with a damping factor which seems legit. Finally Dratt et al. [16] simply state that the “DEM contact laws” are used and Jonsén et al. [13] and [14] state that the “nodes to surface” method is used, which is more a method for contact detection.

For the frictional forces all the found researches use practically the same method. Some use different terms for it, but it always boils down to the same. The friction increases linearly with the relative tangential displacement of two contacting surfaces. This continues until the force reaches a maximum value equal to the Coulomb friction, calculated using the normal force and the coefficient of friction of the two materials. When this value is reached, the two surfaces slide across each other. The only surprising statement on the topic of friction is made by Bhuvaraghan et al. [7], according to their research the frictional forces can be neglected because they have no influence on the results of a shot

(36)

peening process while Han and Perić et al. [8] prove in their research that friction plays a significant role in the outcome.

Simulation process and the software packages that are used, for both FEM and DEM analysis is something that is very interesting to know about a research in order to reproduce it and review the quality of the method. Therefore it is strange that a lot of researches do not even mention the type of simulation software that is used, they simply provide pictures from models without revealing the source. Han and Perić et al. [4] and [8], Oñate et al. [10] and Villard et al. [11] do not mention anything about the simulation software or process they use.

The most used software in the other researches for the FEM models is ANSYS, but also

Abaqus/Explicit is used by Bhuvaraghan et al. [6] and [7]. For the DEM model mostly EDEM or PFC is used. The main difference in the simulation methods the coupling between the DEM and FEM

software. Jonsén et al. [13] and [14] state that they use the non-linear finite element code LS-Dyna v971 for the combined FEM and DEM model. It is however unknown how this would work and if this is even possible with this software package.

Bhuvaraghan et al. [6] and [7] use a one way coupling between ANSYS and EDEM. In this process the contact forces from EDEM are transferred to the FEM model in ANSYS.

Dratt et al. [16] use PFC for the DEM simulation and ANSYS for the FEM simulation. This research however introduces a two way coupling between the two programs. To realize this the geometry of the (deformed) FEM model is translated to an STL-format for use in PFC, to calculate the contact forces which then are transferred again to ANSYS again. For applications with large deflections a two way coupling is very desirable.

Some researches do not even verify or validate their proposed methods. However sometimes the validation of verification seems to be not very convincing. All researches are very quick to say the results agree in an acceptable manner, without backing this up with statistics. Before accepting results for the validation or verification of a method or model it seems that a more scientific or statistic support of this statement is needed. With the validation and verification in these researches, none of the proposed models can be guaranteed to provide reliable results and therefore cannot be used for the construction of a product.

2.7.4 Advantages and disadvantages

In this chapter all researches propose a model with separate discrete elements and structures of finite elements. The advantage of a combined model with this kind of set-up is that both the behavior of the granular material modelled by the discrete elements can be simulated, as can the impact of this particle behavior on the finite element structure. To sum up the reasons to use a combined model

(37)

with DEM and FEM for applications with both particulate materials and solid structures the advantages and disadvantages of the different model types are shown in Table 3.

Table 3: Comparison of different modelling methods for application with both particulate materials and solid structures.

Model type Advantage Disadvantage

FEM a. Calculation of deformations and stresses in a structure.

No accurate force distribution (uniform) on the structure, because behavior of the particles is

unknown.

DEM b. Behavior of particulate materials. No data on the effect of contact forces caused by the particulate materials on the structure

Combined DEM/FEM one way Both a. and b.

Effect of the deformation of the solid structure on the behavior of the particulate material not

taken into account.

Combined DEM/FEM two way

Both a. and b.

Also take the effect of the deformation of the solid structure on the behavior of the particulate

material into account.

Longer computation time and more complex procedure by transferring deformed shape with

nodes to DEM software.

2.7.5 Overview of used researches

This paragraph simply provides an overview of all the used researches and their properties shown in Table 4. Then Table 5 shows the advantages and disadvantages of these different researches over each other.

Table 4: Overview of researches, empty means not mentioned in this particular research.

Research Year Contact model Friction model 2D or 3D FEM software

DEM

software Verified Validated Paragraph

[4] 2000 Linear elastic, Hertz, Winkler, power law Elastic-plastic friction / Modified classic Coulomb friction model 2D Yes, compared to FEM model 2.1 [8] 2000 Linear/elastic, Hertz, Winkler, power law Elastic-plastic friction / Modified classic Coulomb friction model 3D Yes, compared to FEM model 2.1 [6] [7] 2008 2010 Friction neglected 3D ANSYS and Abaqus / Explicit EDEM Yes, compared to FEM model 2.1 [10] 2004 Linear elastic and viscous damping Elastic-plastic friction / Modified classic Coulomb friction model 2D Yes, compared to true-scale experiment 2.2

(38)

[11] 2009 Linear elastic Elastic-plastic friction / Modified classic Coulomb friction model 2D Yes, compared to analytical method Yes, compared to true-scale experiment 2.3 [13] [14] 2009 2010 Nodes to Surface, LS DYNA 2D LS-Dyna v971 LS-Dyna v971 Yes, compared with true-scale experiment 2.4 [16] 2012 3D ANSYS PFC Yes, compared to FEM model and analytical method 2.5 [18] 2013 Linear elastic Elastic-plastic friction / Modified classic Coulomb friction model 3D FEMAP / PATRAN Visual C++ 6.0 on OpenGL platform 2.6

(39)

Table 5: Advantages and disadvantages of the different researches compared to each other.

Research Advantages Disadvantages

[4] [8]

 Different contact models explained.

 Contact with multiple finite elements considered.

 Proof that friction cannot be neglected.

 Model in both 2D and 3D

 No decision on the best contact model.

 No validation or verification.

 No mentioning of the used software for the simulation.

[6] [7]

 Coupling between FEM and DEM software.

 Model in 3D.

 Only single shot case verified.

 No mentioning of the used contact model

 Coupling between software packages only one way.

 Friction neglected which results in an error.

[10]  Viscous damping considered.

 Validation of the method is poorly executed.

 Model in 2D

[11]  Model validated and verified in a proper manner.

 Linear elastic contact with high stiffness seems unrealistic.

 Model in 2D

[13] [14]

 Used software is mentioned.

 Validated using a real experiment.

 No explanation of the simulation process. Simply a software package is stated.

 No explanation of the used contact model.

 Validation poorly executed: differences in the setup and results do not agree.

 Model in 2D

[16]

 Two way coupling between ANSYS and PFC using STL-format

 Possible simulating problems with large deflections.

 Model in 3D

 Verification method using FE model seems unreliable.

 Verification using analytical method is not proven statistically.

[18]  Model in 3D

 Using linear elastic law because it is the simplest seems strange.

 No validation or verification.

(40)

Single structure with both finite and discrete

elements

In this chapter the combination of discrete and finite element in used in one structure. For example a beam is modelled using discrete elements, connected by finite elements. This is used in order to model the behavior of the different parts of the beam when it fractures in multiple parts. There are different types of methods that combine the FEM and DEM in one structure. These different methods are depicted in this chapter.

3.1 Multi scale analysis

In 2007 Rojek et al. [19] extend their earlier work described in paragraph 2.2 for allowing the use of both the FEM and DEM in different subdomains of the same body. Paragraph 3.1.1 explains the contact models used in this research, paragraph 3.1.2 then shows the simulation software and procedures that are used and after that in paragraph 3.1.3 the methods used to verify and validate the method are shown.

Unlike the research of Paavilainen et al. [20], the combination is not along the entire model. Here the different methods are used to simulate different parts of the model for different scale calculations. The DEM is used for the parts where the interest lies in the microscopic scale and the FEM for the macroscopic scale. The coupling allows partial overlap between the two different domains. The idea in this method is to model different parts of the structure with different elements. Fig. 38a shows an example of a deformable body with domain Ω. Normally in a FE model, the entire body is meshed using finite elements. In this method the body is divided in three parts, a subdomain with only finite elements ΩF, a subdomain with only discrete elements ΩD and a subdomain where the two methods overlap ΩF-D. This is shown in Fig. 38b.

(41)

The coupling between the two types of elements in the subdomain with the overlapping elements is done through the principle of virtual work. The virtual work in domain W can be calculated according to equation

(1

)

F D

W







 

 

(15)

With α equals zero in ΩD and one in ΩF. In the mixed domain:

( )

g

L



x



x

(16)

The vector x represents the coordinate of the point in the domain on coordinate system shown in Fig. 38. In Fig. 39 the parameters g(x) and L(x) are shown graphically.

There is nothing mentioned in this research about the used simulation software and procedures.

An application that is provided in this research is rock cutting. This is usually simulated with the DEM only as shown in Fig. 40. The method described in this paragraph is now used to model this using a hybrid finite/discrete element model as shown in Fig. 41. In this model the cutter is still modelled as a rigid object.

(42)

The model used in Fig. 40 section has been modified by replacing discrete elements far away from the fracturing region by finite elements. The DEM and FEM subdomains partially overlap, there the

coupling is performed according to equation (15). In order to verify the new modelling method, the results of the cutting forces are compared between the new hybrid model and the DEM model. These cutting forces are shown in Fig. 42 for both models.

About the comparison between the two results Rojek et al. [19] state:

“Both curves show oscillations typical for cutting of brittle rock. In both cases similar values of amplitudes are observed. Mean values of cutting forces agree very well. This shows that combined DEM/FEM simulation gives similar results to a DEM analysis.”

Therefore the conclusion is that the model can be used for this specific application.

Fig. 40: Simulation of rock cutting with discrete element model [19].

Fig. 41: Simulation of rock cutting with hybrid discrete/finite element model [19].

(43)

In order to validate the proposed method a different application is used. The results of a numerical simulation are compared to laboratory scale experiment of a Brazilian test. This is a test that is used to measure the tensile strength of are rock, the setup and failure mode are shown in Fig. 43.

The same test has been simulated with the numerical model. The resulting failure mode and stress distributions in both the direction normal and parallel to the load are shown in Fig. 44 and Fig. 45. These results seem to be in good agreement with theoretical results.

Fig. 43: Brazilian test, rock sample before (a) and after failure (b) [19].

Fig. 44: Stress results of simulation in direction along loading [19].

Fig. 45: Stress results of simulation in direction normal to loading [19].

State-of-the-art in Discrete and Finite Element Modeling for bulk handling/Het modernste op het gebied van Discrete en Eindige Elementen Modellering voor de afhandeling van bulkgoederen

Specialization: Transport Engineering and Logistics

Report number: 2014.TEL.7837

Title:

Author:

Mark Erkens

Summary

List of symbols

List of abbreviations

Contents

Introduction

1.1 Finite Element Method (FEM)

1.2 Discrete/Distinct Element Method (DEM)

R d



 

1.3 Combination of FEM and DEM

FEM

Software

DEM

Software

FEM

Software

DEM

Software

1.4 Structure of the report

Contact between separate finite and discrete element

structures

2.1 Shot peening

F





F

S

S





w

w





2R





( )

F





F

2.2 Various geomechanical applications

F



F



F

F



k



F



c v

2.3 Earth structures reinforced by geosynthetic

2.4 Granular materials in a tumbling mill

2.5 Conveyor belt deflection

2.6 Rock cutting

F



K



n

K K

K

K

K





if

sgn(

) if