Micromechanics of Powder Compaction andParticle Contact
Erik Olsson
Licentiate Thesis no. 115, 2013KTH School of Engineering Sciences
Department of Solid MechanicsRoyal Institute of TechnologySE-100 44 Stockholm Sweden
TRITA HFL-0534
ISSN 1104-6813
ISRN KTH/HFL/R-13/03-SE
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholmframlägges till o�entlig granskning för avläggande av teknologie licentiatexamen fredagenden 22 februari kl. 10:00 i seminarierummet Teknikringen 8D, Kungliga Tekniska Högskolan,Stockholm. Granskare är universitetslektor Pär Jonsén, Luleå tekniska universitet
Abstract
Cold compaction of powders followed by sintering is in the industry of hard materials a pop-
ular production route for cutting tools and machine parts of complex shapes. During the
compaction and during handling of the powder compact, defects can develop which a�ects the
strength of the �nal sintered product. In order to have a better understanding of the com-
paction process and predict the properties of the powder compact, the compaction is studied
numerically using the Discrete Element Method (DEM). In DEM, single powder particles are
modeled as an element and the most critical issue for obtaining accurate predictions is the
description of the contact force between two particles.
In Paper A is the e�ect of particle size distribution studied for spherical rigid plastic powder
particles. The in�uence from size distribution was found to be small and can be neglected for
narrow distributions. Comparisons with compaction experiments found in the literature were
also made and good agreement was found with the results from the simulations.
Paper B is focused on models for force-displacement relations for powder particles in con-
tact. Firstly, a model for describing loading of the contact is derived, taking the combined
elastic-plastic deformation into account leading to a complete theory for elastic, elastic-plastic
and �nally rigid plastic contact behavior. The rest of the paper is devoted to the adhesive
unloading of the contact. This problem is solved by �rst considering unloading in the absence
of adhesive forces and then add an adhesive pressure term to the solution. All derived results
are veri�ed using FE simulations, which for the adhesive case is made by introducing a cohe-
sive surface behavior between the particles.
In Paper C, compaction of industrially relevant spray dried cemented granules is studied
using DEM. The force-displacement relations for the granules are obtained by performing ex-
periments on the single granules. Firstly, a compaction test is performed on a single granule
giving information of the mechanical behavior at small indentation depths. At higher indenta-
tion depths, nanoindentation tests are made where the results are exported to a FE simulation
of the two granules in contact. The resulting force-displacement relations is then exported
to a DEM program where closed die compaction of the granules is simulated. The simulated
results is then compared with presently performed compaction experiments and an excellent
agreement is found.
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ii
Sammanfattning
Kallkompaktering av pulver med efterföljande sintring är en populär metod i industrin för
att tillverka hårda material såsom skärstål och maskinkomponenter med komplicerad form.
Under kompakteringen och under hanteringen av den pressade kroppen kan defekter uppstå
vilket kan påverka styrkan hos den färdiga sintrade produkten. För att få en bättre förståelse
för kompakteringsprocessen och för att kunna förutspå den pressade kroppens egenskaper har
kompakteringen studerats numeriskt med Diskreta Element Metoden (DEM). I DEM mod-
elleras de enskilda partiklarna där den viktigaste faktorn för att få noggranna förutsägelser
om materialbeteendet är modelleringen av kontaktkrafterna mellan pulverpartiklarna.
I Artikel A studeras inverkan av storleksfördelning hos sfäriska stelplastiska partiklar. In-
verkan av storleksfördelning visade sig vara liten och kan försummas om det är liten spridning
i partikelradien. Dessutom gjordes jämförelser med experiment i litteraturen vilka visade god
överensstämmelse med gjorda simuleringar.
Artikel B fokuserar på sambandet mellan kraft och förskjutning för pulverpartiklar i kon-
takt. Först härleds en modell för pålastning av kontakten med hänsyn tagen till kombinerad
elastisk och plastisk deformation. Den resterande delen av artikeln tillägnas adhesiv avlast-
ning av kontakten. Detta problem löses genom att först betrakta avlastning utan adhesiva
krafter och därefter lägga till en adhesiv tryckterm till lösningen. Alla härledda resultat är
veri�erade med FEM-simuleringar som i det adhesiva problemet löstes genom att introducera
ett kohesivt ytbeteende mellan partiklarna.
I Artikel C studeras, med hjälp av DEM, kompaktering av ett industriellt relevant mate-
rial; spraytorkade hårdmetallgranuler. Kraft-förskjutningssambanden för granulerna bestäms
genom att utföra mikromekaniska experiment på de enskilda granulerna. Först görs ett kom-
pressionsprov på de enskilda granulerna vilket ger information om materialbeteendet för små
intryckningsdjup. För större intryckningsdjup görs nanointryck där resultatet exporteras till
en FEM-modell av två granuler i kontakt. De resulterande kraft-förskjutningssambanden ex-
porteras därefter till ett DEM-program där enaxlig kompaktering av granulerna simuleras.
De simulerade resultaten jämförs med egna kompakteringsexperiment och jämförelsen visar
utmärkt övernsstämmelse mellan simuleringar och experiment.
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iv
Preface
The work in this licentiate thesis was carried out at the Department of Solid Mechanics at
KTH Royal Institute of Technology between August 2010 and January 2013. The work was
partly founded by the VINN Excellence Center Hero-M, �nanced by VINNOVA, the Swedish
Governmental Agency for Innovation Systems, Swedish industry, and KTH Royal Institute of
Technology which is gratefully acknowledged.
I would like to express my deepest gratitude to my supervisor Prof. Per-Lennart Larsson for
giving me the opportunity to start my PhD studies and for introducing me to the challenging
and interesting �eld of contact mechanics. I am really looking forward to the second half of
my PhD studies with him as a supervisor.
The department of Solid Mechanics is a very stimulating workplace because of my nice col-
leagues. Thank you all!
Many thanks to Per Lindskog, Carl-Johan Maderud, Daniel Petrini and Anders Stenberg at
Sandvik Coromant AB for letting me doing experiments at their lab. This has made the work
much more interesting.
Finally, I want to thank my family for always supporting me in my work and my girlfriend Lo
for always having patience with me, even in times of heavy programming.
Stockholm, January 2013
Erik Olsson
v
List of appended papers
Paper A: On the E�ect of Particle Size Distribution in Cold Powder CompactionErik Olsson and Per-Lennart LarssonJournal of Applied Mechanics 79, 2012, 051017
Paper B: On Force-Displacement Relations at Contact Between Adhesive Elastic-PlasticBodiesErik Olsson and Per-Lennart LarssonAccepted for publication in Journal of the Mechanics and Physics of Solids
Paper C: A Numerical Analysis of Cold Powder Compaction Based on MicromechanicalExperimentsErik Olsson and Per-Lennart LarssonReport 533, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute of
Technology, Stockholm, Sweden Submitted for international publication
In addition to the appended papers, the work has resulted in the following publications andpresentations1:
Simulering av pulverkompaktering med olika fördelning av partikelstorlekar
Erik Olsson and Per-Lennart LarssonPresented at Svenska Mekanikdagar, Göteborg 2011 (Ea,OP)
E�ect of particle Size Distribution at Powder Compaction
Erik Olsson and Per-Lennart LarssonPresented at Euro PM 2011, Barcelona 2011 (Pp,POP)
Elastic-Plastic Powder Compaction Simulations
Erik Olsson and Per-Lennart LarssonPresented at PM2012, Yokohama 2012 (Pp,OP)
On the Appropriate Use of Representative Stress Quantities at Correlation of
Indentation Experiments
Erik Olsson and Per-Lennart LarssonSubmitted for international publication (R)
1Ea = Extended abstract, OP = Oral presentation, POP = Poster presentation, Pp = Proceeding paper,
R = Report
vi
Contents
Abstract i
Sammanfattning iii
Preface v
List of appended papers vi
Introduction 1
Micromechanical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Discrete Element Method (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Contact beween powder particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Concludning remarks and suggestions for future work . . . . . . . . . . . . . . . . . . 4
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Paper A
Paper B
Paper C
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viii
Micromechanics of Powder Compaction and Particle Contact
Introduction
Powder compaction followed by sintering is a commonly used method in the industry for
producing hard materials and machine components with complex shape. However, the idea
of compacting metal powders is not new; in ancient Egypt, tools were produced by metal
powders and the Inca Indians produced jewelry by compacting gold powder. Today, one of
the biggest advantages with the powder compaction method is that the produced part requires
a minimum of machining after production. The typical production route is
• Filling a die with powder
• Uniaxially compacting the powder in the die. The compaction pressure must be high
enough to give the pressed compact, the green body, enough strength for safe handling
• Sintering i. e. heat treatment at a temperature close to but below the melting point of
the powder material. During this process the powder particles will bond together and
thus give the component a much higher strength.
During the sintering, the compact will change its dimensions. This dimensional change is
dependent on, among other things, the density of the pressed compact and varies through
the green body. Due to this, it is of great interest to be able to predict the (local) density
at a given compaction pressure. It is also important to be able to predict �aws during the
�lling and compaction stages, which can be voids or cracks, because they could remain as
defects or weak zones in the �nal sintered product. These predictions can be made by a costly
experimental characterization or by modeling supported by a smaller number of experiments.
Micromechanical modeling
Powder compaction can be modeled using a macroscopic model by treating the powder as a
continuum or using a micromechanical model by taking the properties of the individual powder
particles into account. One issue with a macroscopic model is that the constitutive description
is very complicated and the identi�cation of material parameters from experiments becomes
di�cult. Instead, a micromechanical model can be used where the known material parameters
of the powder particles is utilized. Pioneering work in this �eld of micromechanical modeling
1
Erik Olsson
was presented by Wilkinsson and Ashby (1975) followed by work by Fleck et al. (1992) and
Fleck (1995). Based on studies of contact between visco-plastic spheres, Biwa and Storåkers
(1995) and Storåkers et al. (1997), more detailed and analytical micromechanical studies was
performed by Larsson et al. (1996) and Storåkers et al. (1999) including creep, e�ect of size
ratio and determination of yield surfaces. However, these micromechanical models are based
on simplifying assumptions, maybe the most limiting of them all is the assumption of a�ne
motion i. e. that the movement of one particle is solely given by the macroscopic strain �eld.
Discrete Element Method (DEM)
In order to relax the assumption of a�ne motion, the compaction can be simulated using
the Discrete Element Method, abbreviated DEM. In DEM, developed by Cundall and Strack
(1979), each particle is modeled as a single object and the local contact forces determines the
motion of the particle. This is done by explicitly integrating the equations of motion for each
particle. In each time step during the integration, three di�erent issues must be solved:
• Find new collisions between the particles
• Compute the contact force given the position of the contacting particles
• Explicitly integrate the equations of motion, preferably using a Verlet type algorithm
Due to the small time steps needed in DEM simulations, and with available computer power,
it is impossible to simulate all particles in a pressed product. Instead, only small subvolumes
can be simulated, with an upper limit of a few ten thousand particles.
DEM has several advantages when modeling powder compaction. Firstly, no restriction is
made on the movement of the particles. It is also straightforward to introduce tangential con-
tact forces (friction) and allowing the particles to have rotational degrees of freedom. Studies
based DEM have been performed for investigating the accuracy of the analytical models,
Skrinjar and Larsson (2004b,a) and Martin et al. (2003), and good agreement was found for
isostatic compaction but not for closed die compaction. It is also possible to determine the
compact strength in DEM allowing for adhesive bonding between the particles, Martin (2004)
and Pizette et al. (2010).
2
Micromechanics of Powder Compaction and Particle Contact
Contact between powder particles
The most critical issue, for accuracy, in all micromechanical models of powder compaction is
the contact description of two powder particles in contact. In order to be able to derive closed
form expressions, the particles are assumed to be spherical which is a good approximation
for atomized powder and spray dried granules. The solution to the problem when two elastic
spheres are in contact is known since more then one century ago, by Hertzian contact theory,
Hertz (1881). However, the assumption of elastic contacts is only valid very early during
compaction and can often be neglected.
A much more suitable approximation is that the spheres behaves rigid plastically, i. e. all
elastic e�ects can be neglected. A common material model in this case is a power-law relation
between the stresses and strains. Under such circumstances, Biwa and Storåkers (1995) and
Storåkers et al. (1997) discovered that the problem is self-similar and they derived a closed
form solution for the contact force as function of the indentation depth if the particles are of
the same material but possibly having di�erent radii. This model was later generalized by
Skrinjar et al. (2007) for spheres of di�erent materials. In paper B, this model is generalized
one step further by incorporating combined elastic-plastic deformation which is needed when
modeling hard materials like ceramics.
An accurate model for the elastic unloading of two particles in contact is needed to predict
the springback of a pressed powder compact. Further, if the adhesive bonding between the
particles is included, it is also possible to predict the strength of the compact and simulate
crack initiation. Mesarovic and Johnson (2000) derived a model for describing rigid ideally
plastic adhesive contacts between spherical particles. In Paper B, this model is, among other
things, generalized to strain hardening materials with the aim of predicting the compact
strength of industry relevant powder materials in future studies.
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Erik Olsson
Concluding remarks and suggestions for future work
A numerical study of cold compaction of powders has been presented including suggestions
for experiments for parameter identi�cation. Still, work remains in order to make the method
useful for the industry. One type of problems where DEM could be appropriate is the predic-
tion of crack initiation when handling the green body. The necessary models on the contact
level are presented in Paper B, but more work is needed regarding experimental determination
of the fracture parameters and a discussion on how to de�ne a crack in DEM simulations.
The main issue when using FEM for simulating compaction problems is, as mentioned earlier,
the calibration of the constitutive model which requires a costly experimental procedure. This
could be solved by performing the experiments virtually using DEM and extract the needed
material parameters for the FEM simulations.
Another interesting issue is to couple FEM and DEM into one single analysis tool. This
would make it possible to study whole components where the majority of the volume is sim-
ulated using FEM but in some small region of interest, for instance around an inward corner
where cracks are expected, DEM can be used to get a more detailed description.
4
Bibliography
Biwa, S., Storåkers, B., 1995. Analysis of Fully Plastic Brinell Indentation. Journal of the
Mechanics and Physics of Solids 43 (8), 1303�1333.
Cundall, P. A., Strack, O. D. L., 1979. A Discrete Numerical Model for Granular Assemblies.
Geotechnique 29, 49�62.
Fleck, N. A., 1995. On the Cold Compaction of Powders. Journal of the Mechanics and Physics
of Solids 43 (9), 1409�1431.
Fleck, N. A., Kuhn, L. T., McMeeking, R. M., 1992. Yielding of Metal Powder Bonded by
Isolated Contacts. Journal of the Mechanics and Physics of Solids 43 (9), 1139�1162.
Hertz, H., 1881. Über die Berührung Fester Elastischer Körper. Journal für die Reine und
Angewandte Mathematik 92, 156�171.
Larsson, P.-L., Biwa, S., Storåkers, B., 1996. Analysis of Cold and Hot Isostatic Compaction.
Acta Materialia 44 (9), 3655�3666.
Martin, C. L., 2004. Elasticity, Fracture and Yielding of Cold Compacted Metal Powders.
Journal of the Mechanics and Physics of Solids 52 (8), 1691�1717.
Martin, C. L., Bouvard, D., Shima, S., 2003. Study of Particle Rearrangement During Powder
Compaction by the Discrete Element Method. Journal of the Mechanics and Physics of
Solids 51 (4), 667�693.
Mesarovic, S. D., Johnson, K. L., 2000. Adhesive Contact of Elastic-Plastic Spheres. Journal
of the Mechanics and Physics of Solids 48 (10), 2009�2033.
Pizette, P., Martin, C. L., Delette, G., Sornay, P., Sans, F., 2010. Compaction of Aggregated
Ceramic Powders: from Contact Law to Fracture and Yield Surfaces. Powder Technology
198 (2), 240�250.
Skrinjar, O., Larsson, P.-L., 2004a. Cold Compaction of Composite Powders with Size Ratio.
Acta Materialia 57 (7), 1871�1884.
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Erik Olsson
Skrinjar, O., Larsson, P.-L., 2004b. On Discrete Element Modelling of Compaction of Powders
with Size Ratio. Computional Materials Science 31 (1�2), 131�146.
Skrinjar, O., Larsson, P.-L., Storåkers, B., 2007. Local Contact Compliance Relations at
Compaction of Composite Powders. Journal of Applied Mechanics 74 (1), 164�168.
Storåkers, B., Biwa, S., Larsson, P.-L., 1997. Similarity Analysis of Inelastic Contact. Inter-
national Journal of Solids and Structures 34 (24), 3061�3083.
Storåkers, B., Fleck, N. A., McMeeking, R. M., 1999. The Visco-plastic Compaction of Com-
posite Powders. Journal of the Mechanics and Physics of Solids 47 (4), 785�815.
Wilkinsson, D., Ashby, M. F., 1975. Pressure Sintering by Power Law Creep. Acta Metallurgica
23 (11), 1277�1285.
6
Micromechanics of Powder Compaction and Particle Contact
Summary of appended papers
Paper A: On the E�ect of Particle Size Distribution in Cold Powder Compaction.
In this paper, the e�ect of particle size distribution in powder compaction is studied numer-
ically using the discrete element method. The particles are assumed to be spherical and are
constitutively described by rigid plastic material behavior. The radii of the particles are as-
sumed to follow a truncated normal distribution. Both isostatic and closed die compaction
are studied and the in�uence of particle size distribution is small in both cases and can be
neglected for particles that are close to monosized. The simulations are compared with two
sets of experiment from the literature and good agreement is found both for fundamental
properties, like the average number of contacts per particle, and properties of more practical
interest, like macroscopic compaction pressure
Paper B: On Force-Displacement Relation at Contact Between Elastic-Plastic Adhesive Bod-
ies.
This paper is devoted to the modeling of contact between two powder particles. The parti-
cles are assumed to be of an elastic-plastic material and the aim is to �nd force-displacement
relations suitable for DEM simulations. Firstly, a model of two elastic-plastic particles in
contact is derived with account taken of the elastic-plastic deformation. The model is partly
based on results from investigations of Brinell indentation and accounts for strain hardening
e�ects. The adhesive unloading of the particles, which can be used in simulations of powder
compact strength, is solved in two steps; �rst unloading in the absence of adhesion is studied
and thereafter an adhesive term is added to the contact pressure. The model for the adhesive
term is derived using fracture mechanics arguments and is based on one parameter, the frac-
ture energy. Finally the model of adhesive unloading is veri�ed by adding a cohesive surface
behavior between the two particles in contact and good agreement is found when comparing
with the derived analytical expressions.
7
Erik Olsson
Paper C: A Numerical Analysis of Cold Powder Compaction Based on Micromechanical
Experiments.
In this paper, the compaction behavior of cemented carbide granules is studied numerically and
experimentally. The material model of the powder granules is determined by micromechanical
experiments. Firstly, the material behavior at low strains is determined using a granule
compression test. For information at high strains, which are needed in powder compaction
simulations, nanoindentation tests are made. The material model is used in a FE simulation
of two powder granules in contact and the force-displacement relations so determined are
exported to a DEM program. The performed DEM simulations shows excellent agreement
with presently performed compaction experiments in the range where the DEM simulations
are expected to be valid.
8