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Discrete element simulation of elasto-plastic shockwaves in high-velocity compaction

Muhammad Shoaib

Doctoral Thesis

Stockholm March 2011Royal Institute of TechnologySchool of Engineering Sciences

Department of Aeronautical and Vehicle EngineeringThe Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL)

Postal address Visiting address Contact

Royal Institute of Technology Teknikringen 8 Tel: +46 8 790 8904Aeronautical and Stockholm Email: [email protected] EngineeringSE-100 44 StockholmSweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholmframläggs till o�entlig granskning för avläggande av teknologie doktorsexamen tisdagden 29 mars 2011, 10:00 i sal F3, Lindstedtsvägen 26, KTH, Stockholm.

TRITA-AVE-2011:17ISSN-1651-7660ISBN-978-91-7415-905-9

© Muhammad Shoaib, mars 2011

Abstract

Elasto-plastic shock waves in high-velocity compaction of spherical metal particles arethe focus of this doctoral thesis consisting of an introduction and four appended papers(A-D). The compaction process is modeled by a discrete element method while usingelastic and plastic loading, elastic unloading as well as adhesion and friction at contacts.Paper A investigates the dynamic compaction of a one-dimensional chain of homoge-nous aluminum particles. The development of the elasto-plastic shock waves, theirpropagation and in�uence on the compaction process are examined. Simulations yieldinformation on the contact behavior, velocity of the particle and its deformation duringdynamic compaction. Furthermore, e�ects of changing loading parameters on thecompaction process are discussed.Paper B addresses the non-homogeneity in a chain having particles of di�erent sizesand materials, voids between the particles and particles with/without adhesion betweenthem. Simulations show transmission and re�ection of elasto-plastic shock wave duringthe compaction process. The particle deformation during incident and re�ected shocksand particle velocity �uctuations due to voids between particles are simulated. Finally,the e�ects of adhesion on particles separation during unloading stage are also discussed.Paper C develops a simulation model for a high-velocity compaction process withauxiliary pistons, named relaxation assists, in a compaction assembly. The simulationresults reveal that the relaxation assists o�er smooth compaction during loading stage,prevention of the particle separation during unloading stage and conversion of higherkinetic energy of hammer into particles deformation. Furthermore, the in�uence ofvarious loading elements on compaction process is investigates. These results supportthe �ndings of experimental work.Paper D further extends the one-dimensional case of Paper A and B into a two-dimensional assembly of particles while adding friction between particles and betweenparticles and container walls. Three special cases are investigated including closelypacked hexagonal, loosely packed random and a non-homogenous assembly of particles ofvarious sizes and materials. Consistent with the one-dimensional case, primary interestis the linking of particle deformation with the elasto-plastic shock wave propagation.Simulations yield information on particle deformation during shock propagation andoverall particles compaction with the velocity of the hammer. Furthermore, the forceexerted by particles on the container walls and rearrangement of the loosely packedparticles during dynamic loading are investigated. Finally, the e�ects of presence offriction and adhesion on both overall particles deformation and compaction process aresimulated.The developed models in this doctoral thesis provide insights to understand variousaspects of high-velocity compaction.

Dissertation

The doctoral thesis consists of this summary and four appended papers listed below andreferred to as Paper A to Paper D. Leif Kari has acted as a supervisor for this thesisand Bruska Azhdar is co-supervisor.

Paper A. M. Shoaib and L. Kari, "Discrete element simulation of elasto-plastic shockwave propagation in spherical particles", Submitted for publication in Advances in

Acoustics and Vibration.

Paper B. M. Shoaib and L. Kari, "Simulating the dynamic loading of non-homogenousspherical particles using discrete element method", Submitted for publication in Acta

Acustica united with Acustica.

Paper C. M. Shoaib, L. Kari and B. Azhdar,"Simulation of high-velocity compactionprocess with relaxation assists using the discrete element method", Submitted forpublication in Powder Technology.

Paper D. M. Shoaib and L. Kari, "High-velocity compaction simulation of atwo-dimensional assembly of spherical particles using the discrete element method",Submitted for publication in Advanced Powder Technology.

Parts of work has been presented at the following two conferences:

M. Shoaib, L. Kari 2010 23rd Nordic Seminar on Computational Mechanics"High-velocity compaction simulation using the discrete element method"M. Shoaib, L. Kari 2008 International Symposium on Non-Linear Acoustics"Elasto-plastic wave front propagation in non-linear particle systems"

iii

iv M. Shoaib

Acknowledgements

The work presented in this thesis was carried out at The Marcus Wallenberg Laboratoryfor Sound and Vibration Research (MWL) part of The Department of Aeronautical andVehicle Engineering at the Royal Institute of Technology, Sweden. The funding for thisresearch is provided by The Higher Education Commission (HEC) of Pakistan and theSwedish Institute (SI). I gratefully acknowledge this support.

I owe my gratitude to many people for the success of this thesis. First and foremost,I gratefully acknowledge the unparalleled support, guidance and appreciation from mysupervisor professor Leif Kari without whom this thesis would not exist. Thank you,Leif for your belief in me and accepting me as a PhD student. I learned a lot from yourknowledge, discussions and ideas. I really appreciate your care during my whole stayin the division. Indeed, you have been a great source of inspiration during the wholejourney towards my graduation, not only a supervisor but a wonderful person in my life.

I would also like to thank my co-supervisor Bruska Azhdar for help and guidance. I amgrateful for all the help from my colleagues, especially thanks to those together withwhom I have shared o�ce.

Finally, it is almost too easy to forget something that has always been taken for granted,such as the cordial support that I have always been extremely fortunate to receive frommy family. Its importance to me expressed in words, would easily outsize this thesis.Thank you!

Muhammad Shoaib

Stockholm, March 2011

Contents

I Overview and Summary 1

1 Introduction 3

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Particles contact behavior 7

3 Model 10

3.1 Discrete element method . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Compaction model for one-dimensional chains of particles . . . . . . . . . 113.3 Compaction model for piston-die assembly with relaxation assists . . . . 133.4 Compaction model for two-dimensional assembly of particles . . . . . . . 14

4 Dynamic compaction of spherical particles 17

4.1 Compaction of a one-dimensional chain of particles . . . . . . . . . . . . 174.2 Incorporating the relaxation assists in the piston-die assembly . . . . . . 214.3 Compaction of a two-dimensional assembly of particles . . . . . . . . . . 28

5 Conclusions and outlook 33

6 Summary of Appended Papers 35

6.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.3 Paper C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.4 Paper D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Bibliography 38

v

Part I

Overview and Summary

1

Chapter 1

Introduction

High-velocity compaction (HVC) of particulate material is a rapid production techniquewith the capacity to signi�cantly improve the mechanical properties of the compact.HVC �nds its applications in powder material structural parts and soft magneticcomposites. In conventional compaction, powder is compacted by pressure in about onesecond, while in HVC particles are compacted with shock waves in less than 0.01 seconds.These shock waves are generated by giving the projectile or hammer a certain initialvelocity, usually by a hydraulic press in the case of manufacturing process. This initialvelocity along with the mass of the hammer results in a certain kinetic energy whichcontrols the dynamic compaction process. During last years, high-velocity compactionof ferrous, ceramic and polymer powders have been investigated. The majority of thesestudies are restricted to the experimental work while mainly focusing on the gain incompact density and process parameters. In previous work at Department of Fibre andPolymer Technology at Royal Institute of Technology (KTH) [1, 2], various aspects ofHVC process have been investigated including the pull-out phenomena, spring back anddelay time. In order to minimize these problems, new technique has been developed byintroducing the auxiliary pistons, named relaxation assists, in a compaction assembly.These experiments were performed on polymer powder.

The present work is on numerical simulation of high-velocity compaction process. Sincethe elasto-plastic shock waves are the main source of particle compaction during HVC,shock waves have been investigated in details in the present work. The main focusis to understand the generation, propagation and re�ection of elasto-plastic shockwaves and their e�ects on particle deformation and their movement during dynamiccompaction. To investigate these and other issues, compaction model is reduced toa one and two-dimensional assemblies consist of one hundred to about one thousandparticles. The reason for selecting spherical metal particles instead of polymer particlesis the availability of contact models during compaction process. In the present work,well-established contact models for elastic and plastic loading and elastic unloading withand without adhesion and friction between contacts are used. When primary aspectsof elasto-plastic shock wave are understood; then it is straightforward to extend thecompaction model to any number of particles with 3-dimensions to address a more

3

4 M. Shoaib

realistic practical situation. The discrete element method (DEM) is used to model thecompaction process. This numerical method has been widely used for simulating thelarge number of particles while studying granular mater and powder compaction.

The present work starts from the dynamic compaction of a one-dimensional chainof homogenous particles. Elasto-plastic shock wave generation, its propagation andin�uence on the compaction process are studied. Then non-homogeneity is introducedby considering the chain having particles of di�erent sizes and materials, voids betweenthe particles and particles with/without adhesion between them. Simulations revealtransmission and re�ection of elasto-plastic shock wave; particle deformation duringincident and re�ected shocks and particle velocity �uctuations due to voids betweenparticles. In addition, the e�ects of adhesion on particles separation during unloadingstage are discussed. Furthermore, the e�ects of the presence of auxiliary pistons,known as relaxation assists, in a compaction assembly are investigated. It is shownthat relaxation assists o�er an increased locking of the particles during loading andunloading stages. These results support the �ndings of above mentioned experimentalwork. Finally, two-dimensional assembly of particles with friction between particles andbetween particles and container walls is investigated. The main focus is on the linkingof particle deformation with the elasto-plastic shock wave.

1.1 Background

Considerable interest in the dynamic response of particulate and granular materialsexists in the powder compaction, geomechanics and other branches of engineering.Quasi-static and dynamic compaction modeling of particulate material is studied invarious domains. During compaction of particulate material; load is transferred almostequally to all the particles in quasi-static compaction, while during dynamic loadingit propagates as shock wave. This shock wave propagation in discrete media di�ersconsiderably from the wave propagation in continues media. The granular mattershows discrete behavior when subjected to dynamic loading [3−5]. The dynamic wavepropagation in granular media shows distinct behavior from the wave propagation incontinues media [3]. Shukla and Damania [6] discuss the wave velocity in granular matterand shown experimentally that it depends upon elastic properties of the material andon geometric structure. Similarly, Shukla and Zhu [7] investigate explosive loading of adiscs assembly and found that the force propagation through granular media depends onimpact duration, arrangement of the discs and the diameter of discs. Tanaka et al. [8]investigate numerically and experimentally the dynamic behavior of a two-dimensionalgranular matter subjected to the impact of a spherical projectile.To investigate dynamic response, many researchers [9−13] have modeled the granularmatter as spherical particles using the micromechanical modeling of contact betweenparticles. These studies focused on equivalent macro elastic constitutive constantsduring dynamic loading. Similarly, experimental work using dynamic photoelasticityand strain gage are performed to investigate contact loads between particles both under

DE simulation of elasto-plastic shock waves in high-velocity compaction 5

static and dynamic loading [14, 15]. Saad et al. [14] perform numerical simulationsto investigate the e�ects of the contact laws on wave propagation in granular matter.Similarly Saad et al. [15] use the discrete element method (DEM) to simulate wavepropagation in granular materials. Results of that study show wave propagation speedand amplitude attenuation for two-dimensional assembly of spherical particles. However,that study is restricted to the elastic range only while the material sti�ness and dampingconstants used in the model are determined by photoelasticity. The DEM was initiallydeveloped by Cundall and Strack [16] and this numerical method has been widelyused for granular material simulations [17−19]. Di�erent engineering approaches arediscussed in Refs. [20, 21] to model the behavior of granular matter using DEM. Dynamiccompaction of metal powder is also reported in the literature [22−26] and studies thedistribution of stress, strain and wave propagation. However, these studies treat thepowder as a continuum and determine the material constants experimentally.Force transmission in spherical particles occurs in a chain of contacts, which isusually referred as the force chain. Force chains in granular matter have been widelyinvestigated, experimentally [27, 28] and in simulations [29, 30]. However these studiesdo not consider wave propagation velocity during loading. Liu and Nagel [31] and Jia etal. [32] found experimentally that sound propagates in granular media along strong forcechains. Somfai et al. [33] investigate the sound waves propagation in a con�ned granularsystem. Recently, Abd-Elhady et al. [34, 35] studied contact time and force transferreddue to an incident particle impact while using the Thornton and Ning approach [36]and found a good agreement between DEM simulation and experimental measurements.However, these studies are restricted to the particle collision and are not modeling theshock wave propagation during dynamic loading of particles. Micromechanical modelingof powder compaction process is now well established. The pioneer work performedby Wilkinson and Ashby [37−40] and after that theoretical and experimental studies[41−44] resulted in a complete theory for static compaction. A detailed model of normalcontact between viscoplastic particles is presented by Refs. [45] which relies on powerlaw creep [46, 47], plastic �ow theory [48, 49] and general viscoplasticity [48, 50]. Themodel proposes analytical equations which show the e�ects of the material and processparameters during powder compaction. The technique is used to analyze monolithicand composite powders compaction [51, 52].Further progress in micromechanical modeling is achieved by numerical simulationsbased on the discrete element method. Recently, to investigate di�erent aspects ofpowder compaction, DEM is used by Skrinjar and Larsson [53, 54] and also by Martinand Bouvard [55, 56]. Martin [57] models the powder compaction by introducing plasticloading, elastic unloading and decohesion at contacts using the formulation of Mesarovicand Johnson [58]. However, in these investigations, dynamic e�ects are included fornumerical reasons only and their goal is to attain static equilibrium solutions duringisostatic and die compaction. Quasi-static compaction models include the mechanicaldeformation mechanism of particles and do not analyze the mechanical energy transferduring shock propagation. During the last few years; high velocity compaction offerrous, ceramic and polymers powders has been investigated in various domains [59−66].

6 M. Shoaib

However, most of these studies are restricted to the experimental work while mainlyfocusing on the gain in compact density. Wang et al. [59] determines experimentally thee�ects of impact velocity on green density and on the properties of the compacted body.Similarly, Refs. [60, 61] compare the conventional compaction and HVC to determinethe conditions for attaining the maximum compact density. Doremus et al. [67] compareHVC and conventional compaction of metallic powders for di�erent process parametersincluding the material densi�cation and tooling strength. To understand the di�erentaspects of powder compaction, comparison between numerical and experimental resultsis also reported in the literature [68−70].Recently, Bruska et al [1, 2] investigated the di�erent aspects of HVC process anddeveloped a new technique by introducing the projectile supports, known as relaxationassists. These metal pieces are of the same diameter and material as those of the piston.It is found that incorporation of the relaxation assists increases the material densi�cationand compact properties including the reduction of pull-out phenomena − the top surfaceof the compact becomes uneven and the spring back − expansion of the compact duringunloading stage. It has been suggested that above mentioned improvements are mainlydue to the better locking of the powder during compaction process. Furthermore, theresults of these studies suggest that HVC is not a continuous process but rather itincludes several discontinuities which may e�ect di�erently the compact properties.In the present work, discrete element method is used to simulate elasto-plastic shockwave propagation in a homogeneous, Paper A, and in non-homogeneous, Paper B,chains of spherical particles. The e�ects of shock wave on particle deformation and theirmovement and on the loading parameters are investigated. Paper C examines thesigni�cance of relaxation assists in a compaction assembly. It is shown that relaxationassists provide an increased locking of the particles during loading and unloading stages.Finally, high-velocity compaction of a two-dimensional assembly of spherical particlesis examined in Paper D. Tightly packed, random and non-homogeneous particleassemblies are investigated while the main interest is to link particle deformation withelasto-plastic shock waves.

Chapter 2

Particles contact behavior

This section summaries the theories that describe the contact behavior between particlesand between particles and container walls during loading-unloading-reloading stages.Here, the compact is modeled as a one and two-dimensional assembly of particles thatindent each other. Both homogeneous and non-homogeneous particles assemblies areconsidered. Non-homogeneity is introduced by changing the size and material of theparticles. The materials are assumed to be elastic, perfectly plastic while having nostrain hardening. For two contacting particles p and q with radius Rp and Rq; e�ectiveradius R0, equivalent elastic modulus E∗ and yield stress at contact σy are

R0 =RpRq

Rp +Rq

, (2.1)

E∗ =

[1− ν2

p

Ep

+1− ν2

q

Eq

]−1

(2.2)

andσy = min (σp, σq), (2.3)

where Ep and Eq are the Young's modulus; νp and νq are the Poisson's ratio; σp and σqare the yielding stress; for the particles p and q, respectively. The indentation or overlapbetween particles is denoted h. The elastic normal contact force follows the Hertzianlaw [71] in the beginning of loading process

Fe =4

3E∗√R0h

32 . (2.4)

In the plastic regime, normal contact force is [52, 57]

Fp = 6πc2σyR0h, (2.5)

where the constant c2 = 1.43 for ideally plastic material behavior. The contact radiusa is given by

a2 = 2c2R0h (2.6)

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8 M. Shoaib

Figure 2.1: Schematic of two particles in contact, here h is the overlap between particles.

and the contact sti�ness is de�ned as

k = 6πc2σyR0. (2.7)

The parameters F0, a0 and h0 denote normal contact force, contact radius and overlap,respectively, at the end of plastic loading stage. The uniform pressure at the contact isp0 = 3σy and w is the work of adhesion. In the present case, it is set to w = 0.5 J/m2.During unloading of the contact, the distance between the centers of the particles hpqcan be written as

hpq = R1 +R2 − h0 + hu (2.8)

and the overlap ish = h0 − hu, (2.9)

where hu denotes the indentation recovered during unloading of the contact, reading [56,71]

hu =2p0a0

E∗

√1−

( aa0

)2

. (2.10)

The contact radius a0 during unloading of the contact can readily be determined fromEq. (2.10). The normal contact force Fu during unloading is [56, 71]

Fu = 2p0a20

[arcsin

( aa0

)− a

a0

√1−

( aa0

)2]

(2.11)

−2√

2πwE∗a32 .

In Eq. (2.11), the term −2√

2πwE∗a32 is the force due to adhesion traction between

particles [57, 58]. This term should be included in loading equations when adhesionbetween particles is considered. Furthermore, the friction between particles and between


particles and container walls is considered in this work. Thus, a Coulomb-type frictionlaw is incorporated as

Ff = µfFn, (2.12)

where Ff is the friction force perpendicular to the normal force Fn. Friction forceopposes the relative motion between the contacting surfaces. In the present case, thecoe�cient of friction is set to µf = 0.1.

Chapter 3

Model

3.1 Discrete element method

The dynamic compaction process is simulated by DEM. This numerical method is usedby Martin and Bouvard [56] and other researchers to simulate static compaction. Inthe present work, DEM is extended to simulate shock wave propagation in a chainof spherical particles using established contact models. Here each particle is modeledindependently and the interaction between neighboring particles is governed by contactlaws as described in the previous section. This contact response plays an important rolein the use of DEM to simulate shock wave propagation through the particles. Duringcalculations at time t + ∆t, where t is previous time and ∆t is the time step, contactforce between particles is calculated which determines the net force or compaction forceF acting on each particle. By using Newton's second law, these resultant forces enablenew acceleration, velocity and position of each particle. At time t = 0, force, velocityand position of each particle are known because it is the moment of the �rst hit. The

velocity v(t+∆t)i of a particle i at a time t + ∆t, is determined by adopting a central

di�erence scheme as

v(t+∆t)i = v

(t+∆t/2)i +

Fi

mi

∆t

2, (3.1)

where the position xi is given by

x(t+∆t)i = xti + v

(t+∆t/2)i ∆t. (3.2)

During iterative calculations, the size of time step ∆t plays an important role toensure numerical stability. For problems of a similar nature, Cundall and Strack [16]have proposed a relationship to calculate the time step which is further developed byO'Sullivan and Bray [72] for the central di�erence time integration scheme as

max (∆t) = ft

√m

k, (3.3)

where correction factor ft = 0.01 for the present case, m is the mass of the lightestparticle and k is the approximate contact sti�ness given by expression Eq. (2.7). This

10


Figure 3.1: One-dimensional dynamic particle compaction model.

value of the time step is shown to be su�cient to ensure numerical stability duringcalculations.During the compaction process, particle contact goes through several loading, unloadingand reloading sequences. In the beginning of compaction, contact force is initially elasticfor small values of contact radius a and it is given by Eq. (2.4). The contact forcefollows the same curve during unloading and reloading in the elastic regime. At largercontact radius, the contact becomes plastic and contact force follows Eq. (2.5). Theterm relating the adhesion −2

√2πwE∗a

32 is added in elastic and plastic equations when

considering adhesion traction. If the contact is unloaded, normal force follows elasticunloading Eq. (2.11). When contact is reloaded during unloading then it follows thesame equation up to the value of the contact radius on which it was unloaded. Beyondthis point, plasticity is reactivated and Eq. (2.5) applies.The compaction models used in the dynamic loading of particles in Papers A-D areexplained in the following sections.

3.2 Compaction model for one-dimensional chains of

particles

The dynamic compaction models for homogeneous (in Paper A) and non-homogeneous(in Paper B) chains of particles are shown in the Figure 3.1 and 3.2, respectively. InFigure 3.1, there is a chain of micron-sized identical, spherical particles aligned in acontainer with one end open and the other blocked. At the open end, these particles arein contact with a compaction tool which has the same diameter as those of the particles.Friction between the particles and the container walls is not considered. In Figure 3.2,there are chains of spherical particles with di�erent arrangements. These particles arein contact with compaction tool at one end and with the rigid wall on the opposite end.The model is con�ned to the one-dimensional motion of the particles. In both cases,hammer and compaction tool form the dynamic load which is given a certain initialvelocity by a hydraulic pressure supply. The impact velocity along with the loadingmass determines the compaction energy.

12 M. Shoaib

Figure 3.2: One-dimensional compaction model. (a) Homogenous particles, (b) particles ofdi�erent materials, (c) particles with di�erent sizes, (d) particles with voidsbetween them, (e) considering adhesion/no adhesion between the particles, (f)combination of particles with di�erent materials, sizes and gaps.


Figure 3.3: Drawing of the compaction equipment used in the experimental work of Refs. [1,2].

3.3 Compaction model for piston-die assembly with

relaxation assists

The equipment used in experimental work of Bruska et al. [1, 2] is shown in theFigure 3.3. In those investigations, two new metal pieces, named relaxation assistsare introduced in the piston-die assembly. These parts are of identical material anddiameter as those of the piston but of di�erent lengths. By using the setup of Figure3.3, they performed experiments for high-velocity compaction and compared the resultsobtained with and without relaxation assists. The one-dimensional compaction modelused for numerical simulation in Paper C is shown in the Figure 3.4. There is a chainof spherical particles aligned in a container. These particles are in contact with oneof the loading elements − hammer, piston or relaxation assist − at front end and withrelax assist RA2 or die wall at the rear end. Friction between the particles and containerwalls is assumed small and, therefore, not taken into account. The number of particles isidentical in all the cases. The case B and E correspond to the experimental work [1, 2].However, more cases i.e., A to E, are simulated in this paper to explain the e�ects ofindividual elements in detail. To start the compaction process, the hammer is given acertain initial velocity. The impact velocity along with hammer mass determines thecompaction energy. The �rst case uses the hammer only as excitation and then pistonand relaxation assists are incorporated into the model, as illustrated in Figure 3.4.

The variables describing the dynamic loading process are explained subsequently. Thesum of ends displacement (SED) is the sum of net displacement of the compaction endsand the sum of particle deformation (SPD) is de�ned as the sum of the deformations of

14 M. Shoaib

Figure 3.4: One dimensional compaction model with various loading element con�gurations.(A) Hammer only, (B) hammer and piston, (C) hammer, piston and RA2, (D)hammer, piston and RA1, (E) hammer, piston and both RA.

all the particles, see Figure 3.5. The net displacement of a loading element is the changeof its position during compaction process. As compaction proceeds, front compactionend moves forward with the loading element and particles are deformed resulting in anincrease of SED and SPD. These two quantities change almost identically during theloading stage. However, this is not generally the case during the unloading stage. Thereason for the di�erence between SED and SPD is that the loading elements move backto a longer distance compared to the total expansion of the particles (due to elasticunloading). This situation creates some space for particles to move freely and becomeseparated. Here it is referred as room for separation and it can be interpreted as: Roomfor separation = SPD − SED.

3.4 Compaction model for two-dimensional assembly

of particles

The two-dimensional dynamic compaction model used in Paper D is shown inthe Figure 3.6. Three particular assemblies of micron-sized spherical particles are


Figure 3.5: Sum of ends displacement (SED) and sum of particle deformation (SPD) are usedto explain the compaction process.

investigated. These include a closely packed hexagonal assembly (case A), loosely packedrandom assembly (case B) and a non-homogenous assembly of particles with di�erentsizes and material (case C). To start the compaction process, the hammer is given acertain initial velocity applying a hydraulic pressure supply. The impact velocity alongwith the hammer mass mainly determine the compaction energy. During the compactionprocess, particles deform, change their position and have velocities both in x and y-

directions. Particle rotation is neglected in this study. A particle may make severalcontacts with other particles and container walls at a time and its contacting partnersmay change as compaction proceeds. When a new contact between two particles ismade, it is assumed to be in virgin state and then its history is maintained. If particlesare separated and then make contact again, contact is resumed from the last unloadingstage.

16 M. Shoaib

Figure 3.6: Two-dimensional compaction model. (A) Tightly packed hexagonal, (B) looselypacked random, (C) non-homogeneous assembly. In model A and C, containerhaving width of 32 normal particle radii is used.

Chapter 4

Dynamic compaction of spherical

particles

This section describes the main results of simulation of dynamic loading of one andtwo-dimensional assemblies of spherical particles and e�ects of relaxation assists in acompaction assembly. The time step used is ∆t = 2.3 ns which is estimated as explainedin the Section 3.1. The most frequent particles are of aluminum of density 2700 kg/m3,yielding stress 146 MPa, Young's modulus 70 GPa and Poissons's ratio 0.30. Thematerial properties of the steel particles are: density 7800 kg/m3, yielding stress 400MPa, Young's modulus 210 GPa and Poissons's ratio 0.35.

4.1 Compaction of a one-dimensional chain of parti-

cles

Dynamic loading of a homogenous and non-homogenous particles are simulated inPaper A and B respectively. In both cases, one hundred particles are used. In �rst case,particles are of aluminum having diameter 100 µm while in second case heterogeneityis introduced by changing the material and size of the particles, voids between theparticles and considering adhesion/no adhesion between them. Dynamic compressionload is applied on the particles by supplying hydraulic pressure of 13.5 MPa which givesthe hammer an impact velocity of 10 m/s. This impact velocity along with the di�erentchoices of loading mass results in a compaction energy of 1 J/g to 6.5 J/g. These loadingparameters correspond to a typical high-velocity compaction process.Dynamic e�ects during particle compaction like elasto-plastic shock wave propagation,particle contact behavior and their velocity along with loading parameters are investi-gated in Paper A. The dynamic load transferred in particles is described using elasto-plastic shock wave propagation variables like shock wave front velocity. The shock wavefront is interpreted as the maximum absolute compaction force at a particular time whileshock wave velocity is de�ned as the velocity of wave front. The movement of the shockwave from compaction to dead end and then back to the compaction end is describedas one compaction cycle. The shock completes its one compaction cycle in the following

17

18 M. Shoaib

2 4 6 8 10 12

x 10−5

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time (s)

Net

forc

e (N

)

P 8

P 20

P 32

P 44

P 56

P 68

P 80

P 92

2nd Cycle

(a)

2 4 6 8 10

x 10−5

0

2

4

6

8

10

Time (s)

Par

ticle

vel

ocity

(m

/s)

P 8

P 20

P 32

P 44

P 56

P 68

P 80

P 92

(b)

2nd Cycle

2 4 6 8 10

x 10−5

0

2

4

6

x 10−6

Time (s)

Par

ticle

def

orm

atio

n (m

)

P 8

P 20

P 32

P 44

P 56

P 68

P 80

P 92

(c)

2nd Cycle

Figure 4.1: Dynamic loading of homogenous particles. (a) Propagation and re�ection ofelasto-plastic shock wave, (b) particle velocity, (c) particle deformation.


way; it starts from the compaction end, passes through the particles, re�ected from thedead end and then moves back from the last particle to the �rst particle. During forwardpart of the cycle, as shock passes through a particle, net force on the particle increaseswhich results in the particle motion. This net force decreases to zero as particle gainsapproximately the hammer velocity. The shock amplitude becomes negative as wavemoves back from the last particle to the �rst particle. It decreases the particle velocitywhich eventually becomes zero. Shock wave propagation and velocity of particles areshown in Figure 4.1a and b, repectively. As shock wave passes through the chain, itresults in particle plastic deformation. During one compaction cycle, deformation of allthe particles remains almost the same as in Figure 4.1c.E�ects of non-homogeneity on elasto-plastic shock propagation are examined in PaperB. The results of the compaction of a mixture in which half of the aluminum particles(particles 51 to 100) are replaced by steel particles are shown in the Figure 4.2. In thebeginning of the compaction process, aluminum particles are compacted in a normalway. They are compacted and start moving approximately with the hammer velocityas can be seen in Figure 4.2a. When the shock hits the steel particles, they are lesscompacted and gain less velocity, due to their large contact sti�ness and more mass.It results in lowering the velocity of the aluminum particles while creating a re�ectedshock in the aluminum particles. As incident shock wave proceeds in steel particles(particles 51 to 100), a re�ected wave moves back in aluminum particles towards thecompaction end. As the shock propagates in steel particles, they start moving whilevelocity of aluminum particles decreases by re�ected wave. Due to symmetric particlesarrangement, incident shock and re�ected shock hit the compaction end and dead end,respectively, almost at the same time. At this moment all the particles have almostthe same velocity which is nearly half the velocity of hammer. The incident shock isthen re�ected back from the dead end while a re�ected shock from the compactionend goes through the aluminum particles. In this way, during one compaction cycle,there are two extra re�ected waves in the aluminum particles. Beside the incidentshock, aluminum particles are also compacted during these re�ected waves as shown inFigure 4.2b. This result in more deformation of aluminum particles compared to thedeformation of aluminum particles in a normal chain while having the same loadingconditions. Wave propagation velocity of the re�ected shock is approximately equal tothat of incident shock in aluminum but the shock amplitude decreases considerably. Theaverage shock propagation velocity during �rst compaction cycle; for a chain consistingof only aluminum particles is approximately 827 m/s, chain of only steel particles 803m/s and for the symmetric case of aluminum and steel is 816 m/s. During the simulationof these velocities, adhesion between particles is not taken into account. The wavepropagation velocity reduces due to adhesion between particles; for instance, in onlyaluminum case it becomes approximately 763 m/s. In a random population of steelparticles in an aluminum chain, it is hard to distinguish between incident and re�ectedshocks. However, it is possible to visualize easily the velocity of particles and theirdeformation. Like a di�erent material particle, changing the size of a particle e�ectsthe shock propagation and deformation of particles during the compaction process.

20 M. Shoaib

2 4 6 8

x 10−5

0

2

4

6

8

10

Time (s)

Par

ticle

vel

ocity

(m

/s)

P 8

P 20

P 32

P 44

P 56

P 68

P 80

P 92

(a) 2nd Cycle

2 4 6 8

x 10−5

2

4

6

8

x 10−6

Time (s)

Par

ticle

def

orm

atio

n (m

)

P 8

P 20

P 32

P 44

P 56

P 68

P 80

P 92

Aluminum

2nd Cycle

R

F

Steel

R = Def. during reflected shockF = Def. during incident shock (b)

Figure 4.2: Compaction of di�erent material particles, Particles 1-50 are of aluminum and51-100 are of steel. (a) Particle velocity, (b) particle deformation.


Generally, mixtures of particles with di�erent sizes and materials depict similar behaviorduring the compaction.In a chain of particles, voids or gap between particles e�ect shock propagation and

their deformation during compaction. The results of a chain of homogenous aluminumparticles with two voids before particles 56 and 57 are shown in the Figure 4.3. At thestart of the compaction, shock travels from particle 1 to 55 in a normal way and theparticles start to move with almost the hammer velocity. Due to void after particle 55,net force on it increases and results in a higher particles velocity, See Figure 4.3a. Particle56 undergoes the same motion. When shock hits the particle 57, which is stationarybefore voids are covered, it starts to move. However, velocity of the particles behindit reduces. This �uctuation, caused by the sudden decrease in velocity of particlesbefore voids, goes back to hammer, meanwhile the velocity of the particles after thevoids increases. Fluctuation is re�ected from the hammer and it moves in particlesjust before the main shock passes a particle. This �uctuation is re�ected from thecompaction ends and travels in particles distorting the motion of main shock. Sincethere is no continuous source behind this wave, it decreases and disappears practicallyafter the third cycle. Particles are also compacted when this distortion passes throughthem. After the �uctuation dies out, particles are compacted in the normal way whichis shown in Figure 4.3.Heterogeneity due to change in sizes or material of particles have di�erent in�uence thanvoids between particles. Disturbance caused by the voids between the particles decreasesin strength and dies out after a few compaction cycles. However, disturbance due tochange in size or material remains through out the compaction process. It is possibleto visualize and trace the e�ects of a single disturbance, even of a small magnitude, innormal chain of particles. However in the case of more disturbances, usually the e�ectof disturbances with lower strength is suppressed by those of the stronger disturbances.

4.2 Incorporating the relaxation assists in the piston-

die assembly

The signi�cance of relaxation assists in a piston-die assembly is investigated in PaperC. In the compaction model, the hammer mass is set to 1300 × particle mass whilethe piston mass and relaxation assist mass are one half and one quarter of the hammermass, respectively. The piston side in contact with the hammer has a radius of 18× particle radius. The hammer is given an initial velocity of 3 m/s. In all the casesone hundred aluminum particles of diameter 2R = 100 µm are used. All the loadingelements are made of steel with a high yielding stress, therefore, during loading, theseelements deform only elastically. The density, Young's modulus and Poissons's ratiofor piston and relaxation assists material are same as those for steel particles. Thecompaction process for various compaction systems (A to E in Figure 3.4) is simulated.The movement of the compaction ends, particle deformation and change in the energyof the loading elements are investigated. Furthermore, it is explained how relaxation

22 M. Shoaib

2 4 6 8 10

x 10−5

0

5

10

Time (s)

Par

ticle

vel

ocity

(m

/s)

Particle 8

Particle 20

Particle 32

Particle 44

Particle 56

Particle 68

Particle 80

Particle 92

(a) Change in particle velocity due to fluctuations

2 4 6 8 10

x 10−5

0

2

4

6

x 10−6

Time (s)

Par

ticle

def

orm

atio

n (m

)

P 8

P 20

P 32

P 44

P 56

P 68

P 80

P 92

(b)

Compaction dueto fluctuations

2nd cycle

Figure 4.3: Compaction of particles with voids of 0.1 radius between particles 55−56 andparticles 56−57. (a) Particle velocity, (b) particle deformation.


0 1 2 3

x 10−4

−1

0

1

2

3

4

Time (s)

Vel

ocity

(m

/s)

Hammer Piston

P0

P4P3P1

P2 P5 P6

(a)

0 1 2 3 4

x 10−4

0

1

2

x 10−4

Time (s)

Dis

pl. &

def

omat

ion

(m)

Hammer net displacementPiston net displacementSPD

P5

P6

(b)

P3

P2

P1P4

A

A = Room for separation

Figure 4.4: Compaction process for case B. (a) Loading elements velocity, (b) netdisplacement and compact deformation.

24 M. Shoaib

0 1 2 3 4

x 10−4

−2

0

2

4

6

Time (s)

Vel

ocity

(m

/s)

Hammer PistonRA1

(a)

P1

P2P3

P4

P5

P6

0 1 2 3 4

x 10−4

0

1

2

x 10−4

Time (s)

Dis

pl. &

def

orm

atio

n (m

)

Hammer net displacement

Piston net displacement

RA1 net displacement = SED

SPD

(b)

P1

P2

P4

P3 P5

P7

P6

A

A = Room for separation

Figure 4.5: Compaction process for case D. (a) Loading elements velocity, (b) netdisplacement and compact deformation.


assists o�er a smooth compaction − where the particles are compacted more evenlyat several compaction rounds − during the loading stage while reducing the particleseparation during the unloading stage.To explain the working of dynamic compaction system, two particular cases B and D(Figure 3.4) are discussed below. The loading elements in case B are the hammer anda piston. In the beginning of the compaction process, the hammer strikes the piston atthe velocity of 3 m/s. As a result, the piston velocity increases from zero to around 4m/s while the hammer velocity decreases to around 1 m/s, point P0 in Figure 4.4a. Thepiston compacts the particles until its velocity becomes zero at point P1. Then particlesexpand due to elastic unloading energy and push the piston back from point P1 to P2.Thereafter, the piston which moves freely until it is again struck by the hammer atP3. The piston velocity increases and particles are again compacted mostly plasticallyduring the second loading to point P4. The second elastic unloading stage ends at P5after which there is no change in SPD, see Figure 4.4b. However, the piston moves freelyuntil it is again struck by the hammer at P6. Beyond the point P6 the velocity of thepiston is negligible.

In case D, the loading elements are the hammer, the piston and relaxation assist #1 (RA1). In the beginning of the compaction process, the hammer strikes the pistonwhich hits the RA1 and, as a consequence, the hammer velocity decreases to 1 m/swhile the piston and RA1 gain velocity to around 1.5 m/s and 5 m/s, respectively(Figure 4.5a). RA1 compacts the particles until its velocity becomes zero, point P1 inthe Figure 4.5. Then the particles expand and push the RA1 back from P1 to P2. RA1moves freely from P2 to P3 where it is again struck by the piston. During this collisionpiston velocity decreases. However, it regains energy at the collision with the hammerat P5. This sequence of operations is repeated in the next two compaction rounds afterwhich the absolute velocity of the loading elements becomes small. The introductionof RA1 makes the compaction process smooth with the less particles expansion duringunloading stage as can be seen in Figure 4.5. Finally, the presence of RA1 reduces theroom for particles separation, and thus, is improving the compaction process.Figure 4.6 illustrates the improvements of incorporating di�erent loading elements;piston, RA1 and RA2 to the simple hammer die system (case A). Figure 4.6a showsthat the compaction takes place more evenly from case A to E. Similarly, SPD increasesin the same sequence, see Figure 4.6b. In addition, the room for separation reducesaccordingly. Finally the hammer velocity for di�erent cases are shown in the Figure4.6c. At the end of the compaction process, case E has minimum absolute velocitywhich indicates that most of the the hammer kinetic energy is transferred to the compactdeformation (including piston, RA1 and RA2).Adhesion between particles plays an important role during the unloading stage. Asunloading stage proceeds, particles expand due to elastic unloading energy and pushback the loading element. The contact force between the particles decreases and theseparticles start to move with di�erent velocities depending upon their position from theloading element. In case of without adhesion, most of the particles are separated as the

26 M. Shoaib

0 1 2 3 4

x 10−4

0

1

2

x 10−4

Time (s)

SE

D (

m)

Hammer

Hammer and piston

Hammer, piston and RA2


Hammer, piston, RA1 & 2

(a)

1 2 3 4 52.11

2.12

2.13

2.14

2.15

2.16

x 10−4

SP

D (

m)

Hammer

Hammer, piston

Hammer, piston, RA1

Hammer, piston, RA2

Hammer, piston, RA1&2

(b)

0 1 2 3

x 10−4

0

1

2

3

Time (s)

Vel

ocity

(m

/s)

Hammer

Hammer and piston



Hammer, piston, RA1& 2

(c)

Figure 4.6: Comparison between all the compaction models. (a) Displacement of compactionends, (b) compact deformation, (c) hammer velocity.


0

0.5

1x 10

−5

0

0.5

1x 10

−5

0

0.5

1x 10

−5

0 20 40 60 80 1000

0.51

x 10−5

Particle No

Sepa

ratio

n be

twee

n pa

rtic

les

(m)

0

0.5

1x 10

−5

(d)

(e)

(a)

(a)

(b)

(b)

1 2 3 4 50

0.2

0.4

0.6

0.8

1x 10

−4

Tot

al s

epar

atio

n (m

)

Case ACase BCase CCase DCase E

(b)

Figure 4.7: Particles separation when adhesion is taken into account. (a) Separation of aparticle from its predecessor, (b) total separation.

28 M. Shoaib

contact force between them becomes zero. However, in case of with adhesion, additionaltensile force is required to separate the particles. Therefore, only a few particles areseparated.

Figure 4.7a shows particle separation − the separation between two adjacent surfaces ofa particle and its predecessor − in case of adhesion between the particles. The resultsshown are at the end of �nal particle expansion. Obviously, the more loading elements(A to E), the less number particles are separated and the less are separation betweenthe particles. Figure 4.7b shows the total separation between all the particles for thesecases.

4.3 Compaction of a two-dimensional assembly of

particles

The dynamic loading of two-dimensional assembly of particles is investigated in PaperD. Three di�erent cases, as shown in the Figure 3.6, are investigated. In case A andC, 992 particles and in case B 100 particles are taken into account. Fewer particles areconsidered in case B since it is hard to visualize the position of more than, say, onehundred particles. The hammer impact velocity is 5 m/s which along with di�erentchoices of loading mass results in a compaction energy of 0.1 J/g to 0.6 J/g. In thepresent study, net force, particle velocity and displacement are taken positive alongpositive x and y-axes, otherwise negative. Similarly, compression particle deformationis positive. The sum of all the contact deformations of a particle is named particledeformation and the sum of all the particle deformation is named compact deformation.In simulation, aluminum particles with the radius of 50 µm are used. In the following,the dynamic loading of loosely packed random particle assembly, with the same initialposition as shown in the case B in Figure 3.6 is explained. The positions of the particlesduring compaction process are visualized in Figure 4.8 along with the hammer velocity atcorresponding times in Figure 4.9a. In the beginning of the compaction process, particlesmove to �ll the cavities in the container. Then particles rearrange their positions to �llthe gap between particles and �nally make almost the hexagonal shape with a fullypacked assembly, positions at P0, P1 and P2 in Figure 4.8. The hammer kinetic energydecrease at a slow rate during �lling of the cavities and rearrangement of the particles,as can be seen in Figure 4.9a. During the rearrangement period, particles deformationmainly depends upon their location in the assembly, see bar graphs in Figure 4.9. Uptill rearrangement point, P2 in Figure 4.8, particles which �ll the cavity are mainlyelastically compacted while mostly recover their original dimensions by unloading. Afterdensely packed, particles are heavily compacted. As particles are deformed at higherrate, therefore, hammer velocity decreases sharply, point P2 to P3 in Figure 4.9a.Similarly, there is less change in particles positions; compare positions P2 and P3 inFigure 4.8.The e�ects of friction and adhesion between particles and between particles and


0 0.2 0.4 0.6 0.8 1 1.2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2x 10

−3

x (m)

y (m

)

Position of particles at P0

0 0.2 0.4 0.6 0.8 1 1.2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2x 10

−3

x (m)

y (m

)


0 0.2 0.4 0.6 0.8 1 1.2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2x 10

−3

x (m)

y (m

)


0 0.2 0.4 0.6 0.8 1 1.2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2x 10

−3

x (m)

y (m

)


Figure 4.8: Particles positions at di�erent stages of hammer velocity, see Figure 4.9a.

30 M. Shoaib

0 0.2 0.4 0.6 0.8 1

x 10−4

−2

0

2

4

6

Time (s)

Ham

mer

vel

ocity

(m

/s)

P0

P3

P2P1 (a)

0 20 40 60 80 1000

0.5

1

1.5x 10

−6

Particle No

Par

ticle

def

orm

atio

n (m

)

Particle deformation at P1(b)

0 20 40 60 80 1000

0.5

1

1.5x 10

−6

Particle No

Par

ticle

def

orm

atio

n (m

)

Particle deformaiton at P2

(c)

0 20 40 60 80 1000

2

4

6x 10

−6

Par

ticle

def

orm

atio

n (m

)

Particle No

Particle deformation at P3

(d)

Figure 4.9: Dynamic compaction of random particles assembly. (a) Hammer velocity, whilebar graphs are particle deformation at corresponding points mentioned in (a).


0 0.5 1 1.5

x 10−4

−1

0

1

2

3

4

5

Time (s)

Ham

mer

vel

ocity

(m

/s)

Normal

With friction

With adhesion

S2

S1(a)

1 2 30

1

2

x 10−4

Com

pact

def

orm

atio

n (m

)

(b)

Normal

Withfriction

Withadhesion

Figure 4.10: E�ects of friction and adhesion on compaction process. (a) Hammer velocity,(b) compact deformation.

32 M. Shoaib

container walls are examined. The assembly consists of loosely packed random particlesof the same material and size, see case B in Figure 3.6. The e�ect of the the frictionalforce at a particular contact between two particles is to oppose the relative motion of thecontacting surfaces. During dynamic compaction, each particle makes several contactswith other particles. In the current scenario of loosely packed assembly, friction resiststhe particles rearrangement in the beginning of the compaction process, see stage S1 inFigure 4.10. Therefore, more kinetic energy of the hammer is used during rearrangementof particles which e�ects its velocity, Figure 4.10a. In the normal case without frictionand adhesion, particles rearrange their positions easier, earlier and velocity of hammerdecreases slower. However, at the end of compaction stage, S2 in Figure 4.10a, theparticles without friction move more readily into a dense, hexagonal like assembly, andare therefore resisting the hammer displacement strongly, resulting in a sharp hammervelocity reduction. On the other hand, the particles with friction move more slowerinto the dense, hexagonal like assembly, thus resulting in a later and slower hammervelocity decrease, see S2 in Figure 4.10a. In the present situation, there is more compactdeformation at the end of loading stage in friction case compared to the normal case,see Figure 4.10b. This compaction also includes small amount of recoverable elasticunloading deformation. Finally, adhesion between particles e�ects on the compactionprocess in a similar fashion as that of friction, Figure 4.10.

Chapter 5

Conclusions and outlook

In this study, discrete element method is used to investigate the dynamic loading ofa one and two-dimensional assembly of spherical particles. The elasto-plastic shockpropagation, particle velocity and deformation are simulated in a homogenous andnon-homogenous chains of particles. In the case of a two-dimensional hexagonal closepacking, simulation shows particles deformation when shock propagates through theparticles during di�erent cycles. Results show the force exerted by the particles onthe container walls and e�ects of the incident shock which travels in the direction ofdynamic load. The change in compact deformation with the hammer movement areplotted. In addition, the compaction of a loosely packed random particles assemblyis simulated. Results show how particles rearrange their positions to become closelypacked in the end the compaction process. Furthermore, changes in particles positionsand their deformation with the load movement are shown. In addition, the e�ects offriction and adhesion between particles are discussed. It is shown that both frictionand adhesion increase the overall particles deformation resulting in less hammer kineticenergy during unloading.Furthermore, the discrete element method is used in this study to investigate numericallythe signi�cance of relaxation assists in a one-dimensional model of a compactionassembly. It is shown that by using the relaxation assists, the locking of particles isimproved during the compaction process. Simulation results show the displacement ofthe compaction ends and particle deformation during the whole compaction process.It is found that particles are more compacted and the room for particle separationis reduced by using relaxation assists in the compaction assembly. The movement ofthe relaxation assists during the compaction process is also plotted which is di�cultto acquire experimentally. It is shown that the presence of relaxation assist o�ers;smooth compaction during loading stage, prevention of particles separation duringunloading stage and conversion of the higher kinetic energy of the hammer into particledeformation.

33

34 M. Shoaib

The present study is believed to enlighten several new aspects of elasto-plastic shockwaves in high-velocity compaction. Nevertheless, it would be interesting to expand thiswork in several directions including:

� Extension of the one and two -dimensional analysis to three-dimensions

� Examine the e�ects of each relaxation assistant separately

� Investigation of complex die geometries including the more realistic objects suchas gears, bearings and tools

� To investigate multi-stage high-velocity compaction including double actingcompaction

� To model more complex particles such as non-spherical particles and nano-particles

Chapter 6

Summary of Appended Papers

6.1 Paper A

Discrete element simulation of elasto-plastic shock wave propagation in

spherical particles

M. Shoaib and L. Kari

Elasto-plastic shock wave propagation in a one-dimensional assembly of spherical metalparticles is presented by extending well established quasi-static compaction models. Thecompaction process is modeled by a discrete element method while using elastic andplastic loading, elastic unloading and adhesion at contacts with typical dynamic loadingparameters. Of particular interest is to study the development of the elasto-plasticshock wave, its propagation and re�ection during entire loading process. Simulationresults yield information on contact behavior, velocity and deformation of particlesduring dynamic loading. E�ects of shock wave propagation on loading parameters arealso discussed. The elasto-plastic shock propagation in granular material has manypractical applications including the high-velocity compaction of particulate material.

6.2 Paper B

Simulating the dynamic loading of non-homogenous spherical particles using

discrete element method


Dynamic loading of a chain of non-homogenous spherical particles is presented by usingthe discrete element method. The dynamic response of particles is modeled by using

35

36 M. Shoaib

elastic and plastic loading, elastic unloading and adhesion at contacts. Of particularinterest is to study the transmission and re�ection of elasto-plastic shock wave througha chain having; particles of di�erent sizes and materials, voids between the particlesand particles with/without adhesion at contacts. Simulation yields information onshock propagation, particles velocity and their deformation during dynamic compaction.Furthermore, particles deformation during incident and re�ected shocks, particle velocity�uctuations due to voids between particles and e�ects of adhesion on particles separationduring unloading stage are simulated. Although the developed model is con�ned toone-dimensional studies, nevertheless useful for simulating high-velocity compactionprocesses.

6.3 Paper C

Simulation of high-velocity compaction process with relaxation assists using

the discrete element method

M. Shoaib, L. Kari and B. Azhdar

The discrete element method is used to investigate the high-velocity compaction processwith additional piston supports known as relaxation assists. It is shown that byincorporating the relaxation assists in the piston-die assembly, particles can be betterlocked during the compaction process. The simulation results reveal that relaxationassists o�er; smooth compaction during loading stage, prevention of the particleseparation during unloading stage and conversion of higher kinetic energy of hammer intoparticle deformation. Finally, the in�uences of various loading elements on compactionprocess and e�ects of presence of adhesion during unloading stage are investigated. Theresults support the �ndings of experimental work.

6.4 Paper D

High-velocity compaction simulation of a two-dimensional assembly of

spherical particles using the discrete element method


High-velocity compaction of a two-dimensional assembly of spherical particles isnumerically studied using the discrete element simulation technique. Three particularcases are investigated including closely packed hexagonal, loosely packed random and anon-homogenous assembly of particles of various sizes and materials. Of primary interestis the linking of particles deformation with the elasto-plastic shock wave propagation.


Simulation yields information on particle deformation during shock propagation andchange in overall particles compaction with the velocity of the hammer. Moreover,the force exerted by particles on the container walls and rearrangement of the looselypacked particles during dynamic loading are also investigated. Finally, e�ects of presenceof friction and adhesion on overall particles deformation and compaction process aresimulated.

Bibliography

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[2] B. Azhdar, B. Stenberg and L. Kari, �Determination of springback gradient inthe die on compacted polymer powders during high-velocity compaction,� PolymerTesting, vol. 25, pp. 114-123, 2006.

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[10] P.J. Digby, �The e�ective elastic moduli of porous granular rocks,� Journal ofApplied Mechanics, vol. 48, pp. 803-808, 1981.

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40 M. Shoaib

[12] C.S. chang and L. Ma, �Modeling of discrete granulates as microploar continua,�Journal of Engineering Mechanics, vol. 116, pp. 2703-2721, 1990.

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