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
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.
iii
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
Micromechanics of Powder Compaction
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.
1
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
Micromechanics of Powder Compaction
[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.
3
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
Micromechanics of Powder Compaction
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
5
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
Micromechanics of Powder Compaction
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
7
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
Micromechanics of Powder Compaction
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
9
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
Micromechanics of Powder Compaction
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
Micromechanics of Powder Compaction
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
Micromechanics of Powder Compaction
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Micromechanics of Powder Compaction
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.
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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 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
Micromechanics of Powder Compaction
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