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.

i

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

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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.

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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