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Micromechanics of Powder Compaction

Erik Olsson

Doctoral thesis no. 88, 2015KTH School of Engineering Sciences

Department of Solid MechanicsRoyal Institute of TechnologySE-100 44 Stockholm Sweden

TRITA HFL-0565

ISSN 1104-6813

ISRN KTH/HFL/R-14/12-SE

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan i Stockholmframlagges till offentlig granskning for avlaggande av teknisk doktorsexamen fredagen den13 februari kl. 10:00 i sal F3, Kungliga Tekniska Hogskolan, Lindstedtsvagen 26, Stockholm.

Abstract

Compaction of powders followed by sintering is a convenient manufacturing method for prod-

ucts of complex shape and components of materials that are difficult to produce using con-

ventional metallurgy. During the compaction and the handling of the unsintered compact,

defects can develop which could remain in the final sintered product. Modeling is an option

to predict these issues and in this thesis micromechanical modeling of the compaction and

the final components is discussed. Such models provide a more physical description than a

macroscopic model, and specifically, the Discrete Element Method (DEM) is utilized.

An initial study of the effect of particle size distribution, performed with DEM, was

presented in Paper A. The study showed that this effect is small and is thus neglected in

the other DEM studies in this thesis. The study also showed that good agreement with

experimental data can be obtained if friction effects is correctly accounted for.

The most critical issue for accurate results in the DEM simulations is the modeling of

normal contact between the powder particles. A unified treatment of this problem for particles

of a strain hardening elastic-plastic material is presented in Paper B. Results concerning

both the elastic-plastic loading, elastic unloading as well as the adhesive bonding between

the particles is included. All results are compared with finite element simulation with good

agreement with the proposed model.

The modeling of industry relevant powders, namely spray dried granules is presented

in Paper C. The mechanical behavior of the granules is determined using two types of

micromechanical experiments, granule compression tests and nanoindentation testing. The

determined material model is used in an FEM simulation of two granules in contact. The

resulting force-displacement relationships are exported to a DEM analysis of the compaction

of the granules which shows very good agreement with corresponding experimental data.

The modeling of the tangential forces between two contacting powder particles is studied

in Paper D by an extensive parametric study using the finite element method. The outcome

are correlated using normalized parameters and the resulting equations provide the tangential

contact force as function of the tangential displacement for different materials and friction

coefficients.

Finally, in Paper E, the unloading and fracture of powder compacts, made of the same

granules as in Paper C, are studied both experimentally and numerically. A microscopy

study showed that fracture of the powder granules might be of importance for the fracture

and thus a granule fracture model is presented and implemented in the numerical model.

The simulations show that incorporating the fracture of the granules is essential to obtain

agreement with the experimental data.

i

ii

Sammanfattning

Pressning av pulver foljt av sintring ar ett smidigt satt att tillverka komponenter av komplex

form eller av material dar konventionell metallurgi inte fungerar. Under pressningen och

under hanterandet av den pressade men annu inte sintrade komponenten kan defekter uppsta

som kan finnas kvar efter sintring. Modellering ett satt att forutsaga dessa defekter och i

denna avhandling diskuteras mikromekanisk modellering av pressningen och den pressade

komponenten. Denna typ av modeller ger en mer fysikalisk beskrivning av problemet an

makroskopiska modeller och specifikt har Diskreta Elementmetoden (DEM) anvants.

En inledande studie av inverkan av pulverpartiklarnas storleksfordelning gjordes med

DEM och ar presenterad i Artikel A. Studien visade att denna effekt ar liten och ar darfor

forsummad i de andra DEM studierna i denna avhandling. Studien visade ocksa att det

ar mojligt att fa god overensstammelse med experimentell data om friktionseffekter pa ett

korrekt satt ar inkluderat i analysen.

Den viktigaste komponenten for korrekta resultat i DEM simuleringar ar hur kontakten

mellan tva pulverpartiklar beskrivs. En modell for detta problem som pa ett enhetligt satt

behandlar tojningshardnande elastiskt-plastiska material ar presenterad i Artikel B. Resul-

tat for elastisk-plastisk palastning, elastisk avlastning samt adhesiv bindning av kontakten

diskuteras. Alla resultat jamfors med FEM-simuleringar som visar god overensstammelse

med den foreslagna modellen.

Modellering av pulver som anvands i industrin, sa kallade spraytorkade granuler, presen-

teras i Artikel C. Det mekaniska beteendet hos granulerna bestammes genom tva typer av

mikromekaniska experiment, kompressionsprov pa de enskilda granulerna samt nanointryckn-

ing. Den framtagna materialmodellen anvandes sedan i en FEM simulering av tva pulvergran-

uler i kontakt med varandra och de resulterande kraft-forskjutningssambanden exporterades

till en DEM analys var resultat visade mycket god overensstammelse med motsvarande ex-

perimentell data.

Hur tangentiella krafter mellan tva pulverpartiklar i kontakt ska modelleras studeras i

Artikel D med hjalp av en omfattande parameterstudie genomford med FEM. Utfallet av

denna studie korreleras med hjalp av normaliserade parametrar och de resulterande ekvation-

erna ger den tangentiella kontaktkraften som funktion av den tangentiella forskjutningen for

pulverpartiklar av olika material och olika friktionskoefficienter.

Slutligen, i Artikel E, studeras avlastning av och brott i pressade provkroppar bade

numeriskt och experimentellt. Provkropparna ar tillverkade av samma granuler som stud-

erades i Artikel C. En mikroskopstudie visade att brott i de enskilda granulerna kan vara

viktigt for brottegenskaperna och av denna anledning presenterades en modell for brott i

granulerna och implementerades i den numeriska analysen. Simuleringsresultaten visade att

det ar nodvandigt att ta hansyn till att granulerna spricker for att fa overensstammelse med

experimentell data.

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iv

Preface

The work in this doctoral thesis was carried out at the Department of Solid Mechanics at

KTH Royal Institute of Technology between August 2010 and December 2014. The work was

partly funded by the VINN Excellence Center Hero-M, financed by VINNOVA, the Swedish

Governmental Agency for Innovation Systems, Swedish industry, and KTH Royal Institute of

Technology which is gratefully acknowledged.

First and foremost, I would send my deepest gratitude to my supervisor Professor Per-

Lennart Larsson. His door is always open, both literally and figurative, for discussions about

the work but also just for a nice chat. Something that I also really appreciate is the very

short feedback times for paper drafts etc. Thank you very much!

Collaboration and discussions with people working in industry has been very rewarding

for me as a PhD-student and has been very important for the research presented in this thesis.

I would especially thank Svend Fjordvald, Per Lindskog, Carl-Johan Maderud, Daniel Petrini

and Anders Stenberg at Sandvik Coromant AB and Stefan G Larsson at SECO Tools AB for

excellent experimental assistance and for providing the material used in the experiments.

I also would like to thank all my colleagues at the department of Solid Mechanics KTH. I

have really enjoyed the discussions around the fika table, both when it comes to research as

well as discussions about events in Sweden and the rest of the world.

Last, but not least, I would like to thank my family and my girlfriend Sara for having

patience with me during the periods with, probably self-imposed, high workload.

Grasberg, Christmas 2014

v

List of appended papers

Paper A: On the Effect of Particle Size Distribution in Cold Powder CompactionErik Olsson and Per Lennart LarssonJournal of Applied Mechanics, Transactions of the ASME 79(5), 2012, Article number 051017.

Paper B: On Force-Displacement Relations at Contact between Elastic-Plastic AdhesiveBodiesErik Olsson and Per Lennart LarssonJournal of the Mechanics and Physics of Solids 61(5), 2013, 71-18.

Paper C: A Numerical Analysis of Cold Powder Compaction Based on MicromechanicalExperimentsErik Olsson and Per Lennart LarssonPowder Technology 243, 2013, 1185-1201.

Paper D: On the Tangential Contact Behavior at Elastic-Plastic Spherical Contact ProblemsErik Olsson and Per Lennart LarssonWear 319(1-2), 2014, 110-117.

Paper E: Micromechanical Investigation of the Fracture Behavior of Powder MaterialsErik Olsson and Per Lennart LarssonReport 564, Department of Solid Mechanics, KTH Engineering Sciences, Royal Institute ofTechnology, Stockholm, Sweden, 2014. Submitted for international publication.

vi

In addition to the appended papers, the work has resulted in the following publications andpresentations1:

Simulering av pulverkompaktering med olika fordelning av partikelstorlekarErik Olsson and Per-Lennart LarssonPresented at Svenska Mekanikdagar, Goteborg 2011 (Ea,OP).

Effect of particle Size Distribution at Powder CompactionErik Olsson and Per-Lennart LarssonPresented at Euro PM 2011, Barcelona 2011 (Pp,POP).

Elastic-Plastic Powder Compaction SimulationsErik Olsson and Per-Lennart LarssonPresented at PM2012, Yokohama 2012 (Pp,OP).

On the Appropriate Use of Representative Stress Quantities at Correlation ofIndentation ExperimentsErik Olsson and Per-Lennart LarssonTribology Letters 50(2), 2013, 221-232 (Jp).

Study of Springback of Green Bodies using Micromechanical Experiments andthe Discrete Element MethodErik Olsson and Per-Lennart LarssonPresented at 3rd International Conference on Particle-Based Methods Fundamentals andApplications, Particles 2013; Stuttgart (Pp,OP).

Mikromekanisk analys av pulverpressningErik Olsson and Per-Lennart LarssonPresented at Svenska Mekanikdagar, Lund 2013 (Ea,OP).

Micromechanics of Green Body FractureErik Olsson and Per-Lennart LarssonPresented at Euro PM 2014, Salzburg 2014 (Pp,POP).

Discrete Element Simulations of Powder CompactionErik Olsson and Per-Lennart LarssonPresented at EMPA PhD Students’ Symposium 2014, St. Gallen 2014 (POP).

A Numerical Study of the Mechanical Behavior at Contact between Particles ofDissimilar Elastic-Ideally Plastic MaterialsPer-Lennart Larsson and Erik OlssonJournal of Physics and Chemistry of Solids 77, 2015, 92-100 (Jp).

1Ea = Extended abstract, OP = Oral presentation, POP=Poster presentation, Pp = Proceeding paper,Jp=Journal paper

vii

Contribution to the papers

Paper A: Developed the DEM code, performed the numerical simulations and wrote largeparts of the paper under supervision of Per-Lennart Larsson.

Paper B: Developed large parts of the analytical models, performed FE-simulations andwrote the paper under supervision of Per-Lennart Larsson.

Paper C: Performed the experiments under supervision of personnel at Sandvik. Performedthe numerical simulations and wrote the paper under supervision of Per-Lennart Larsson.

Paper D: Developed the analytical models, performed FE-simulations and wrote the pa-per under supervision of Per-Lennart Larsson.

Paper E: Performed the experiments under supervision of personnel at Sandvik. Performedthe numerical simulations and wrote the paper under supervision of Per-Lennart Larsson.

viii

Contents

Abstract i

Sammanfattning iii

Preface v

List of appended papers vi

Contribution to the papers viii

Introduction 1

Modeling of powder compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Discrete Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Contact between particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Modeling of powder granules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Bibliography 15

Summary of appended papers 19

Paper A

Paper B

Paper C

Paper D

Paper E


Introduction

The use of powders for producing metal components has a long history. In the ancient Egypt,

tools were produced by compacting metal powder and the Incas produced jewelery from pow-

ders from precious metals. However, it was first in the middle of the 19th century when powder

metallurgy became a commonly used production route with products like copper coins and

tungsten wires for light bulbs. Today, the method has grown and, according to the European

Powder Metallurgy Association (EPMA), just the European powder metallurgy market has

a turnover of over 6 billion euros and worldwide over one million tonnes of powder products

are produced [1].

There are two main reasons for utilizing powder metallurgy in a production process. First,

the component can be manufactured to the final shape directly requiring very little or small

machining. This is utilized in the sintered steel technology where components with complex

geometry, like gears, are produced using steel powder. The second reason for using powder

metallurgy is that it enables the production of materials that are difficult to produce us-

ing conventional techniques due to high melting temperatures. phase segregations etc. This

applies to typical hard materials like ceramics, the above mentioned tungsten wires and ce-

mented carbides used for machining tools which are pictured in Figure 1. It is the latter

application that is the main focus of the applied parts of this thesis.

Figure 1: Some examples of tools for metal cutting manufactured of cemented carbide powders.

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Erik Olsson

Independent of the application, the production route for different powder products is very

similar and can be divided into three steps:

• Filling a die with powder. The die should be shaped so that the shape of the sintered

component is as close as possible to the required shape of the final product.

• Compression of the powder in the die which is mostly done uniaxially. Higher density

of the compact usually gives better mechanical properties of the final product. The

compact after pressing, the so called green body, has weak mechanical properties but

must be strong enough for handling.

• Sintering where the compact is heat treated at a temperature close but at least below

the melting point of one of the constituents. During this process, the powder particles

will bond together and give the product a much higher mechanical strength.

There are a few issues during this process that needs to be controlled in order to get a

final product with desired shape and properties. During the sintering step, the compact will

substantially change its dimensions. This dimensional change is dependent on, among other

things, the density distribution after pressing. Due to this, it is obvious that it is important

to predict the (local) density distribution in the compact after pressing in order to design

the pressing die for achieving the correct dimensions of the final product. Furthermore,

different types of defects can develop during the process like voids after filling and cracks in

the green body during handling. These defects will create weak zones in the compact after

sintering and are thus undesirable. All these issues can be controlled by a costly experimental

characterization or by modeling supported by a fewer number of experiments.

Modeling of powder compaction

From a modeling point of view, the compaction can be divided into three steps, [2]. The

first stage is the filling and is denoted stage 0. During the second step, Stage I, the powder

assembly is compressed using external pressure and lasts as long as the powder mechanics

can be characterized by individual contacts between the particles. In the last step, stage II,

the powder behaves more like a solid and the assumption that each particle contact can be

treated independently is invalid.

The standard engineering tool for numerically modeling a mechanical problem is the Fi-

nite Element Method (FEM) due to the fact that the method is easy to adapt to different

geometries, boundary conditions etc. Because of this, the obvious choice of method in the

industry for modeling the production of powder components is FEM. However, in order to get

reliable predictions, a constitutive model for the powder that can describe all features, like

non-linear material behavior and compaction induced anisotropy, must be used. Haggblad

2


[3] discusses three types of suitable material models with different complexity and thus with

different capabilities to capture the powder behavior. However, an increasing complexity of

the material model leads to an increase in the number of material parameters that must be

determined by experiments. One way of reducing the number of experiments to identify these

parameters is to use inverse modeling of a relatively simple experiment, for example uniaxial

compression of powder in a cylindrical die, [4]. However, the stress state in a compact of

simple geometry is different from the state in a compact of a complex shape, like a cutting

tool, and thus simple experiments do not activate all parameters that are needed to describe

the powder in more complex situation than a cylinder, for example a cutting insert [5].

Another route would be to derive the material model using micromechanical arguments,

which relies on a much simpler constitutive description of the powder particles. Under the as-

sumption of affine motion, i. e. that each particle moves according to the applied macroscopic

strain, and assuming how the number of contacts on each particle evolves, it is possible to

derive analytical expressions. Pioneering work in this field was performed by [6] followed by

studies by Fleck et. al [7] and Fleck [8]. Based on theoretical studies of visco-plastic spheres

in contact by Biwa and Storakers [9] and Storakers et. al [10], more detailed micomechanical

studies were performed by Larsson et. al [11] and Storakers et. al [12] including creep, parti-

cles of different sizes and investigation of yield surfaces. However, all these studies are based

on simplifying assumptions with affine motion probably being the most limiting one. This

is manifested as the accuracy of the predictions using the analytical models is much worse

at stress states that contain a deviatoric component, for instance, the industrially relevant

uniaxial compression case.

Discrete Element Method

In order to relax some of the assumptions in the analytical models, such as affine motion and

the evolution of the number of contacts per particle, the Discrete Element Method (DEM)

can be used. In DEM, originally developed by Cundall and Strack [13], each powder particle

is modeled as a single object and the contact forces acting on each particle determine its

motion by Newton´s second law. In each time step of the DEM simulations, the following

tasks need to be accomplished

• Determine new positions of the particles by integrating the equations of motions using

the forces from the previous time step.

• Find new contacts between the particles.

• Determine new contact forces between the particles using the updated positions of the

particles.

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Erik Olsson

The time integration for determining the position, xi, the velocity, vi, and the acceleration,

ai, of particle i is preferably performed using an explicit Verlet type algorithm by

xi(t+ ∆t) = xi(t) +

(vi(t) +

1

2ai(t)∆t

)∆t (1)

vi(t+ ∆t) = vi(t) +ai(t) + ai(t+ ∆t)

2∆t (2)

ai(t+ ∆t) =Fi(t)

mi(3)

with Fi(t) being the sum of forces, normal and tangential, acting on particle i and mi is the

mass of particle i. If rotational degrees of freedom also are considered, analogous expressions

are used for the rotation, angular velocity and angular acceleration. One problem with explicit

methods is that they are conditionally stable for too large time steps ∆t. It was shown in

[13] that the largest allowed time step is given by

∆t < 2

√mmin

k(4)

where mmin is the mass of the lightest particle and k = dF/dh is the contact stiffness. Using

relevant values for metallic powders in Eq. 4, it becomes obvious that the time step becomes

too small and the simulation times will be too long. However, under quasi-static conditions,

it was shown by Thornton and Antony [14] that the particle masses could be upscaled several

order of magnitude and thus increase the time step given by Eq. 4 without affecting the

response considerably. The validity of quasi-static conditions can be checked by monitoring

the kinetic energy of the system which must be relatively small in order to avoid the influence

of inertia effects. Still, even with mass scaling, only a few tens of thousand particles can

be simulated within a reasonable time (a few days) on a normal desktop computer and thus

DEM is only intended to be used to determine material properties using small sub-volumes

and not to simulate a whole component consisting of millions of particles.

DEM has more advantages, compared to the analytical models, than that the simplifying

assumptions can be avoided. It allows for a straightforward inclusion of friction forces and

also complex contact models can be included conveniently. Furthermore, effects of different

boundary conditions can be studied without considerable efforts, for instance, a rigid die wall.

Early investigations of powder pressing using DEM include a 2D study by Redanz and Fleck

[15] and a full 3D study by Heylinger and MecMeeking [16]. Using the contact model in [10],

comprehensive studies, including both isostatic and uniaxial compaction and involving par-

ticles of different materials and sizes, were made by Martin et. al [17], Martin and Bouvard

[18] and Skrinjar and Larsson [19, 20]. Furthermore, studies regarding the unloading and

4


strength of powder compacts were presented by Martin [21, 22] and later by Pizette et. al

[23], in order to predict dimensional changes during unloading and cracks in the green body.

Contact between particles

The most important issue in all micromechanical models of powder compaction is the mod-

eling of contact between the powder particles (and between a particle and a pressing tool

wall) and explicitly the contact force as function of the displacement of the particles. Several

different effects need to be accounted for like plastic compression, elastic unloading, contact

bonding as well as tangential forces due to friction. A sketch of a typical force-displacement

relationship in the normal direction is visualized in Figure 2. In the case of DEM simulations,

the evaluation of these models needs to be very computationally efficient as these relations

are evaluated billions of times in a simulation with a few thousand particles.

Figure 2: A sketch of a typical force-displacement relationship needed in an analysis of powder compactionwith plastic compression, elastic unloading as well as contact bonding.

Elastic plastic loading

If the particles behave elastically during the compression, the prominent solution by Hertz

[24] would be applicable. However, an elastic model would only be appropriate during the

filling stage; even under the very initial stage of compaction the contact deforms plastically

which is a much more challenging problem to analyze. More than one century after the so-

lution by Hertz [24] was presented, the loading of contact between spheres of material with

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Erik Olsson

negligible elastic deformation was fully understood. This applies to typically soft materials

like aluminum and copper. By utilizing the self-similarity at spherical contact the problem

was analyzed, assuming deformation plasticity, by Hill et. al. [25] and later, more accurately

assuming plastic flow theory, by Biwa and Storakers [9]. A further extension to viscoplastic

spheres of different sizes was presented in Storakers et. al [10] which was used in Paper A

in a study of the effect of particle size distribution and the simulation results showed good

agreement with experimental data for aluminum powder.

In the intermediate region, where neither elastic nor plastic deformation of the contact can be

neglected, a semi-analytical treatment is still possible which was developed in Paper B. The

materials that applies to this behavior is typically hard materials like ceramics. A summary

of this analysis is presented below. The treatment is based on the behavior of two normal-

ized quantities determined from a Brinell hardness test, namely the normalized hardness H

defined as

H =F

πa2σrep(5)

and the area parameter c2 defined as

c2 =a2

2Rh(6)

In Eqs. 5-6, F is the indentation force, a being the radius of the contact, R being the

radius of the (rigid) indenter and σrep a representative stress defined as the yield stress at a

strain of a/R. Johnson [26, 27] studied the variation of these two quantities as function of a

dimensionless parameter Λ given by

Λ =E

(1 − ν2)σY

a

R(7)

where E and ν are the elastic modulus and the Poisson’s ratio of the material and σY being

the initial yield stress. The behavior of H and c2 as function of log Λ is sketched in Figure 3

where three regimes can be identified.

Figure 3: Sketch of the behavior of H and c2 as function of the parameter Λ. The Figure is taken from [28].

6


The first regime, level I (Λ < 3), is the elastic regime where plastic deformations of the

contact can be neglected and the behavior is accurately described by the Hertzian contact

theory. The third regime, level III, is characterized with constant values for both H and

c2 which is the stage where the self-similarity solution [9, 10] is applicable. In stage II,

both elastic and plastic deformations of the contact are of importance and modeled by linear

functions describing H and c2 together with a limiting value for H. The parameters for the

linear functions are determined from high-accuracy finite element simulations by Mesarovic

and Fleck [29, 30] and together with the assumption that the contact surface between two

(unequal) spheres is spherical, the contact force as function of the penetration is derived. The

proposed formulas show good agreement with finite element simulations at small deformations.

At large deformations, H and c2 decrease which is not accounted for in the present analysis.

Furthermore, the finite element study justifies the assumption of a spherical contact area.

The Brinell indentation problem was then further analyzed in a separate paper by Olsson

and Larsson [31] leading to a unified theory for different materials.

Unloading and adhesive bonding

Accurate models for the elastic unloading and the bonding of the contacts are important

in order to predict the springback of the compacts during unloading and ejection as well as

cracks in the compact. Bonding is introduced in the model by assuming that the separation

of the contact requires an energy per separated contact area denoted G.

For elastic contacts, adhesive bonding is well understood by the JKR model [32] applica-

ble for contacts involving small and stiff spheres and the DMT model [33] for larger, more

compliant spheres. These models were later bridged by a cohesive zone solution by Maugis

[34]. Adhesion between spheres that have deformed plastically is less studied in the litera-

ture. A study of contacts involving materials with no strain hardening and negligible effects of

elastic deformation was presented by Mesarovic and Johnson [35] using the same assumption

on the adhesive traction as in the JKR model. In Paper B, an extension of this model is

presented accounting for strain hardening. Furthermore, the model accounts for large con-

tact deformation, contrary to the model in [35], as the corresponding linear elastic fracture

mechanics problem is studied taking the geometry explicitly into account.

To verify the proposed model, the derived solution was compared with finite element sim-

ulations using the cohesive surface model which showed very good agreement for different

materials and bonding energies. One example from Paper B, for strain hardening equal

spheres, is presented in Figure 4. The agreement between the linear elastic fracture mechan-

ics based analytical solution and the cohesive surface model, when a linear theory should be

valid, gives further confidence in the use of the cohesive surface model for more complicated

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Erik Olsson

Figure 4: Comparison of the analytical model in Paper B and the cohesive surface model using FE simula-tions. The results are pertinent to contact between two equal and strain hardening spheres.

cases. An example is when the adhesive traction causes large scale yielding of the contact

during adhesive pull-off. This was utilized in Paper E where also the material model was

more complicated than the power-law model used in Paper B.

Friction

Friction is important to account for when modeling powder compaction as friction between

the particles will prevent the rearrangement of the particles. Another important effect is that

friction forces between the powder and the die walls will cause density gradients in the pressed

compact. The behavior of the friction forces in the case of an elastic body in contact with

a plane is well understood since the studies by Cattaneo [36] and Mindlin [37]. Their model

relies on the assumption that the normal pressure and the tangential stresses are uncoupled

which is a good approximation for contacts involving elastic materials.

In DEM simulations, a stick-slip friction model is commonly used. In the stick-region, a

linear force displacement relation is used for the tangential force as function of the tangential

displacement up to the limiting force according to Coulomb friction FT = µFN . However, it

is difficult to find explicit values of the stiffness used for the linear model and even more rare,

how to relate the stiffness to material properties of the simulated particles. Moreover, the use

of a linear stick-slip model is questionable. For this reason, a large parametric study of the

problem of a deformable elastic-plastic sphere, with different material properties, compressed

and sheared against a rigid plane, is presented in Paper D. Focus is on parameters such

as the change in normal force during shearing, initial tangential stiffness and the tangential

displacement at full slip. The outcome of the different calculations are correlated using nor-

malized parameters like H, Eq. 5, and Λ, Eq. 7. This results in equations that provide

8


the tangential force as function of the tangential displacement for a general elastic-plastic

material behavior and with different coefficient of friction. The proposed model is verified

by studying two spheres in contact that are sheared against each other. The material of the

spheres and the friction coefficient are different from the materials used in the parametric

study for the derivation of the model. The outcome of this verification is shown in Figure 5,

where a very good agreement is seen between the FE simulations and the proposed model.

Figure 5: The model proposed in Paper D compared with FE simulations for different coefficients of friction.A sketch of the two-particle tangential contact problem is provided to the right.

Modeling of powder granules

The material studied in the two most applied papers in this thesis, Paper C and Paper E, is

spray dried cemented carbide granules. The granules are produced by mixing small tungsten

carbide and cobalt particles with a liquid polymeric binder. This mixture is then spray dried

to granules which are spherical and around 100 µm in size. Two different powders are studied

with slightly different properties, Powder A and Powder B, which are used in the Swedish

industry. Powder A granules consist of cemented carbide particles of 0.6 µm in size whereas

the corresponding size for Powder B granules is 2.2 µm. The amount of binder, which is

the same for both granules, polyethyleneglycol, is 2 wt.% for Powder A and 2.1 wt.% for

Powder B. The granule manufacturing process leads to a material with unknown mechanical

properties that must be determined experimentally.

Micromechanical experiments

To determine the mechanical properties of the granules two types of micromechanical exper-

iments was used in Paper C; granule compression experiments and nanoindentation of the

granule material. In the granule compression test, a single granule is compressed between

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Erik Olsson

two (rigid) plates until fracture. The compression force as function of the displacement of

the compressing plate is monitored continuously. The evaluation of the experiments, using

the self-similarity solution [10] together with FE simulations showed that the granules behave

rigid ideal-plastic in the deformation range obtained during the granule compression exper-

iment. However, when using a rigid ideal-plastic model with the yield stress determined in

the granule experiments in a DEM simulation, the predicted compaction pressure becomes

too low compared to compaction experiments. Thus, hardening is expected at larger strains

than obtained in the granule compression experiments.

An indentation experiment provides the yield stress at high plastic strains. The granules

were embedded in an epoxy binder and indented using a nanoindenter with a continuous

registration of the indentation force as function of the indentation depth. The benefit using

such an approach is that the Young’s modulus of the granule material can be determined if

the unloading of the indenter is analyzed. The strain hardening was evaluated by assuming

an irregular stress-strain curve for the material and using the results by Larsson [38]. The

results showed that the yield stress was about 30 times larger at 35 % plastic strain than what

was seen in the granule compression test. Based on the experimental results, the following

stress-strain relationship was proposed

σ =

εE εE ≤ σY

σY σY /E ≤ εpl ≤ εH

σY + σ0 (εpl − εH)1/m εpl ≥ εH

(8)

In order to fulfill the value of the yield stress at 35 % plastic strain σ0.35, determined from

the nanoidentation experiments, σ0 becomes

σ0 =σ0.35 − σY

(0.35 − εH)1/m(9)

where σY is the yield stress determined from the granule compression experiments and εH

and m are fitting parameters. The force-displacement relations, needed in DEM, is then

derived by simulating two contacting particles in FEM with the above described material

model. Using such an approach, large deformations is explicitly accounted for, in contrast to

the analytical self-similarity model which relies on small deformation theory.

Granule fracture model

In a microscopy study of fracture surfaces in crushed powder compacts, presented in Paper

E, it was seen that fracture of the individual powder granules could be important to describe

the strength and crack initiation in the compacts. Therefore a granule fracture criterion and

10


a model for the post-fracture behavior is needed in DEM. The proposed fracture criterion

consist of the requirements:

• The compression force must be larger than the fracture force in the granule compression

experiments

• At least one force must cause plastic deformation of the contact

• There must be a plane in the granule with no compressive perpendicular forces. This

plane, if all requirements are fulfilled, will be the fracture plane.

Several other approaches were tried and discussed in Paper E but numerical experiments

in combination with experimental observations showed that the above model worked best for

the studied materials. The post-fracture behavior is described by assigning different stiffness

reductions depending on the angle between the force of interest and the fracture plane. It

is assumed that perpendicular to the fracture plane, the granule has no tensile stiffness and

if no compressive forces act perpendicular to the fracture plane, the compressive stiffness in

that plane is zero.

Simulation results

Below, some results, based on the granule model, are presented and compared with exper-

iments. In the experiments, cylindrical compacts are pressed using a movable upper punch

and a fixed lower punch. The simulations use the same quotient between the radius, R0,

and the height, H0, of the cylindrical compact in order to obtain the correct density gradient

due to friction between the powder and the die wall. All simulations are made with 8 000

monosized particles as it was shown in Paper A that the effect of different particle sizes is

small. Bonding is introduced between the particles by the cohesive surface model described

in Paper B and the tangential forces are computed using a slight simplification of the model

presented in Paper D.

The compaction pressure to obtain a specific relative density, here defined as the density

of the pressed compact divided by the sintered density of the final product, is presented in

Figure 6 for Powder A. The friction coefficient between the powder and the die wall was

determined in a small parameter study in Paper C to be 0.2. The simulations show a very

good agreement with the experimental data up to a relative density of D = 0.45. At higher

density, the assumption that each particle contact can be treated individually is invalid and

thus a too low compaction pressure is predicted numerically.

Using Young’s modulus determined in the nanoidentation experiments, the springback of the

components can be investigated numerically. The results in Figure 7, presenting the spring-

back in the pressing direction defined as the height of the unloaded and ejected compact, Hu,

11

Erik Olsson

Figure 6: The numerically simulated compaction pressure as function of the relative density of the componentcompared with the outcome of two different compaction experiments. The pressure in the radial direction wasnot possible to determine in the experiments. The presented data is pertinent to Powder A.

divided with the in-die height, H0. The results shows that bonding between the particles is

essential to obtain values of the predicted springback that reasonably agree with experimental

data.

Figure 7: Comparison between the experimentally measured springback in the axial direction and the corre-sponding values determined numerically.

Figure 8 presents a comparison of the crushing force, Pc, as function of the displacement of the

crushing plate, ∆, when an ejected compact is crushed in the pressing direction as pictured in

Figure 8. It is obvious from the comparison that including fracture of the powder granules is

essential in order to get the right behavior of the crushing force at increasing displacements.

12


Figure 8: Comparison of the crushing simulations and the crushing experiments presented in Paper E.

Concluding remarks

The compaction of powders and the resulting compacts are studied from a micromechanical

point of view. The numerical study is made using the Discrete Element Method (DEM) which

allows for rearrangement of the powder particles as well as it is straightforward to implement

complex force-displacement relationships between the powder particles. A comparison with

previously published experiments in the literature, presented in Paper A, showed that DEM

could predict the behavior at both isostatic and uniaxial load cases. The study in Paper A

also showed that the particle size distribution is of minor importance and thus, the studies

in Paper C and Paper E were performed with monosized particles.

A unified treatment of normal contact between strain hardening elastic-plastic powder parti-

cles was presented in Paper B and extended to tangential contact in Paper D. The resulting

equations provide the contact forces as function of the displacement without doing any time

consuming FE simulations.

For powders of materials with an unknown stress-strain relationship, for instance, the studied

spray dried granules, a granule compression test together with nanoindentation testing gives

very useful data in order to derive the needed force-displacement relations. Furthermore, in

order to predict cracking of powder compacts made of these granules, fracture of the granules

themselves must be included in the simulations.

13

Erik Olsson

Suggestions for future work

There are several things that can be studied in order to make DEM simulations of powder

compaction useful in the industrial process. First of all, it would be interesting to use DEM

to identify parameters in the macroscopic constitutive models for powder compaction. In the

work by Andersson et. al [39], inverse modeling was used for this purpose but only a few of

the parameters in the model they used could be identified in this way. Here DEM simulations

could be a convenient way to investigate more complicated stress states than uniaxial com-

paction that are difficult to obtain experimentally and by doing so, activate more parameters

in the constitutive model. In the same context, it would be interesting to implement the

fracture model derived in Paper E in such an analysis to define a fracture criterion suitable

for FE simulations.

Regarding the contact force models, most of the efforts should be spent on tangential force

models. An obvious extension of the model in Paper D is to incorporate varying normal

and tangential forces handling the same load cases as the elastic model by Cattaneo [36] and

Mindlin [37]. More work needs certainly to be done in the case of tangential bonding and

especially in the case of mixed mode loading. The preliminary model presented in Paper E

was just an initial attempt to determine if tangential bonding would have an effect and it was

clearly demonstrated that it has.

One of the biggest limitations of a DEM analysis is the assumption that each contact can

be treated individually which gives problems at high packing densities (> 85%− 90%) corre-

sponding to about 45 % relative density for the cemented carbide powders. A contact model

for DEM simulations was presented by Harthong et. al. [40] which takes the interaction of

neighboring contacts into account. It would be interesting to extend the validity range of

the present DEM simulations by incorporating that model for the cemented carbide granules.

Also, the models for unloading, adhesion and particle fracture should then be treated accord-

ingly.

Finally, one interesting extension would be to couple DEM with the finite element method

by simulating the main part of the component with FEM but in a small area of interest, for

example an inward corner where cracking could occur, use DEM to get a refined description.

14


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Characterization of Powder Materials, J Test Eval 43 (5) (2015) .

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Erik Olsson

18


Summary of appended papers

Paper A: On the effect of particle size distribution in cold powder compaction.

In this paper, the effect of particle size distribution in powder compaction is studied using the

discrete element method. Both isostatic compaction and closed die compaction are studied

during the entire loading-unloading process. Particle rotation and frictional effects are ac-

counted for in the analysis. The particles are constitutively described by rigid plasticity and

assumed to be spherical with the size of the radii following a truncated normal distribution.

The results show that size distribution effects are small on global compaction properties like

compaction pressure if the size distribution is small. Furthermore, the size distribution had no

influence at all on the macroscopic behavior at unloading. To verify the model, comparisons

were made with two different sets of experiments found in the literature where the particles

were of varying sizes. Good agreement was found both as regards fundamental properties

like the average number of contacts per particle and for more important properties from a

practical point of view such as the 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 particles

are assumed to be of an elastic-plastic material and the aim is to find force-displacement re-

lations suitable for DEM simulations. First, a model of two elastic-plastic particles in contact

is derived accounting for the elastic-plastic deformation. The model is partly based on results

from investigations of the Brinell indentation problem and accounts for strain hardening ef-

fects. The adhesive unloading of the particles, which can be used in simulations of powder

compact strength, is solved in two steps; first 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 verified by adding a cohesive surface

behavior between the two particles in contact and good agreement is found when comparing

with the derived analytical expressions.

19

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 microme-

chanical experiments. First, the material behavior at low strains is determined using a granule

compression test. For information of the behavior 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 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.

Paper D: On the tangential contact behavior at elastic-plastic spherical contact problems.

In this paper, the problem of tangential contact between an elastic-plastic sphere and a rigid

plane is studied analytically and numerically with the specific aim to derive force-displacement

relations to be used in numerical simulations of granular materials. The simulations are

performed for both ideal-plastic and strain hardening materials with different yield stresses

and including large deformation effects in order to draw general conclusions. The results

are correlated using normalized quantities pertinent to the correlation of indentation testing

experiments leading to a general description of the tangential contact problem. Explicit

formulas for the normal and tangential forces are presented as a function of the tangential

displacement using data that are easily available from axi-symmetric analyses of spherical

contact. The proposed model shows very good agreement when compared with the FE-

simulations.

20


Paper E: Micromechanical Investigation of the Fracture Behavior of Powder Materials.

This paper presents an experimental and numerical study of fracture of compacted powder

assemblies using a micromechanical approach. In the experimental investigation, the compacts

are crushed in two different directions to account for general stress states and a microscopy

study shows that fracture of the powder granules plays a significant role in the fracture process.

The numerical analysis is based on the Discrete Element Method (DEM) and a novel approach

is presented to account for the fracture of the particles in the numerical model. The force-

displacement relations for two particles in contact, that are needed in DEM, are derived using

micromechanical experiments together with finite element analyses of the contact problem.

The contact model accounts for plastic compression, elastic unloading and adhesive bonding

together with friction and tangential bonding. The model shows a very good agreement with

the experimental data both for the elastic behavior during unloading of the contact and, if

failure of the particles is accounted for, the fracture of the compacts.

21

Erik Olsson

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