Metallic powders are used to create new materials for many
industrial manufacturing processes. Any number of metals can
be used in the mixture to change the characteristics of the final
product, in terms of strength, flexibility, durability, etc.
For a material with a large number of particles, particle densification
is very important because there is a correlation between density
and the material properties. For example, the electrodes used in
rechargeable batteries consist of densely packed particles (also
called as active materials). The electrons inside the individual
particles are charged. This means that the electrical properties,
such as battery capacity and charging efficiency, are closely
related to the particle density. However, the mechanical properties
also have an influence, so it is equally important to understand
the mechanical properties of the product in order to improve its
reliability and durability, and so on.
There are many conditions involved in creating a material full of
particles, such as the shape and size distribution of the particles,
and the type of material. Therefore, the material properties are not
easily optimized using only actual testing; rather, it should be
combined with simulation.
Many researchers have analyzed the process of particle filling. Most
of them used the discrete element method (DEM),. The DEM
was considered highly compatible with these analyses because it
can efficiently calculate the motion of individual particles, which
are assumed to be rigid. The inelastic behavior that occurs during
the compression process is then represented by the interaction
between the particles. However, since these parameters are not
directly related to real phenomena, it is necessary to calibrate
them using actual tests. The finite element method (FEM), on
the other hand, can directly express the deformation behavior of
particles, which allows us to analyze phenomena that are closer
to the real problem.
This article presents an example of analyzing the particle packing
problem using the finite element method. First, I will discuss
the issue of particle packing to establish the initial state before
compression. Multiscale.Sim™ , an add-on tool for Ansys®
Workbench™ for multiscale analysis, was used to create the
model. Next, an example of the analysis is shown whereby a rigid
wall is placed around the micromodel created and then densified
Here, we will observe the change in the microstructure during
compression and the relationship between the compression force
and the fill rate. Finally, the
resulting macroscopic material
behavior is adapted using the
law. With the equivalent material
properties obtained in this way,
you can perform the analysis of
the macrostructure using the
Before presenting an example of
the analysis of the compression process, we will consider the initial
composition of the particles in the powder.
The initial placement of the particles is very
important because, if there are multiple
bullet gaps or bites between particles in
the microstructure from the beginning,
it can compromise the accuracy of the
compression process analysis. The particles
should be packed with as few gaps and bites
as possible. By observing the initial packing
state of various distributions of particle
sizes before compression analysis, it is also
possible to predict trends in the maximum
particle packing rate after compression.
As a result, we adjusted the distribution of
the particle sizes considered here for three
different types (uniform, gaussian and
bias) and used Multiscale.Sim to fill the
particles to the limits of the instrument for
each condition. Each condition was defined
so that no overlap was allowed between the
particles. Fig. 1 summarizes the appearance
and particle size distribution of the model.
If the number of particles that make up the
microstructure model is too small, it is not
possible to create a model with a specified
particle size distribution.
Therefore, we adjusted the size of the unit cell so that about 3,000
particles could be created. Consequently, we confirmed that it is
possible to create a model similar to the input size distribution.
A solid model with a uniform particle size was found to have a
volume content of about 63%. Generally, we know the theoretical
maximum volume content is 74% when the particles are regularly
aligned. On the other hand, however, when the particles are
randomly scattered, the upper limit of the theoretical volume
content is considered to be 64%, which is almost identical to the
results obtained by Multiscale.Sim.
The results demonstrate that Multiscale.Sim is capable of packing
the particles to the limit. In the case of size distribution, the spaces
created between relatively large particles are filled with smaller
particles, so a larger volume content can be achieved than in the
case of particles of uniform size. It should be possible to create
an even higher volume content by increasing the probability of
the presence of statistically small particles. The results of this
analysis are consistent with those considerations.
The compression process analysis for densification is performed
using the particle-filled microstructure model created in the
previous section. The microstructures analyzed in this study are
the result of extracting only a small portion of the actual structure.
In order to evaluate the apparent behavior of the material based
on such a partial model, the homogenization method is generally
In this analysis, the extracted microstructure is used as a unit cell,
and conditions of periodic symmetry are defined in all directions
on its outer surface. However, if the particle-free region is not
filled with other materials, such as resin, it is inevitable that small
particles will escape from the unit cell during compression.
Therefore, an alternative is to create a rigid wall that touches
each of the six outer surfaces of the unit cell. One of these rigid
walls is then loaded by forced displacement and the unit cell is
compressed. The compression displacement was applied to about
45% of the macro strain and then returned to its pre-compression
position. The other five rigid walls were fully constrained by the
displacement degrees of freedom. The elasto-plastic properties of
the Mises yield function were defined for the individual particles.
The friction coefficient for all interfaces between each part was
set to 0.3.
These models were used to perform an implicit analysis using
Ansys Workbench. In order to ensure the stability of the analysis,
a transient analysis was conducted to take into account the effect
of inertia. For the loading history, the compression and unloading processes were configured to take 1
second each. These analysis conditions
are shown in Fig. 2.
Fig. 3 shows the deformation shapes of
the microstructure and the distribution
of the equivalent plastic strain at various
points in the compression process. The
analysis was performed on cases with
no size distribution and cases with a
Gaussian distribution. The displacement
constraints defined on the rigid walls
created conditions that were close to
hydrostatic pressure from a macroscopic
point of view. Large plastic strains can be observed inside the
microstructure despite the fact that the particles do not yield
due to hydrostatic stress. This means that the heterogeneity of
the microstructure generates a complex stress field within the
microstructure, even though the macroscopic conditions are
Fig. 4 shows the macroscopic stress-strain characteristics
calculated from the reaction force of the rigid wall under
compressive displacement and the history of porosity in the unit
cell. In the compression process of such porous materials, the
tangential gradient tends to increase rapidly with increasing strain.
This is due to the fact that the voids inside the microstructure are
reduced by compression as shown Fig. 5.
This reduces the inhomogeneity and brings the inside of the
microstructure closer to the same hydrostatic pressure conditions
as the macroscopic external force.
In this paper, the accuracy could not be verified because the actual
measurements were not made under the same conditions as the
analysis. However, since the stress-strain shape shows the same
tendency as the generally acknowledged facts, it is qualitatively
considered to have captured the actual behavior to some extent. In addition, as shown in fig. 4 and fig. 5, we believe that the changes
in the inelastic material behavior during press forming due to
differences in particle size distribution qualitatively capture the
Fig. 4 – Macroscopic stress strain curve as a result of the compression analysis
Fig. 5 – History of the volume fraction of the void in the microstructure during
Up to this point, we have analyzed the material behavior during the
compression process using a microstructure model composed of
However, this approach is computationally expensive and can only
focus on extremely limited areas for analysis.
Modeling for heterogeneity is not appropriate for actual productlevel
objects. Therefore, at the end of this article, we attempt to
replace the heterogeneous microstructure with a homogeneous
To do so, the behavior of the macroscopic material obtained in
the previous section must be adapted to the material composition
radius shown in Fig. 4. Powdered materials composed of particles
show remarkable elastoplastic properties even under hydrostatic
conditions. These properties cannot be expressed by the Mises
yield function, where deviation stress triggers the generation of
In this paper, Drucker Prager’s capped model  was used. This
is a typical constitutive law for materials to represent the yielding
behavior dependent on hydrostatic stress (accessible with the
command “TB,EDP,,,,CYFUN” in Ansys software). Fig. 6 shows the
curve fit analysis flow. The homogeneous blocks defined according
to the Drucker Prager constitutive laws were compressed by the
same boundary conditions as in the previous section. The material
constants were adjusted so that the stress-strain properties obtained
in this analysis correspond to the response
of the heterogeneous microstructure model
analysis evaluated in the previous section.
Ideally, this adjustment work should be done
by optimization analysis, but in this case, it
was performed manually. In future, we plan
to implement a fitting algorithm based on
the optimization analysis in Multiscale.Sim
The result of the fitting is shown as a solid
line in Fig. 4. There is a slight error in the
results. However, the tendency for the stress
to increase exponentially with increasing
plastic deformation, and the tendency for the unloading curve
to move downwards in a convex manner, are well captured. The
effect of adjusting the material properties to change the particle
size distribution is also well represented. Although the fitting
results may not yet be sufficient, we do not think that this is a
problem caused by the ability to express the constitutive law of the
material. We believe that more accurate fitting will be possible in
future by incorporating an optimization algorithm.
Fig. 6 – Analysis flow to identify the material constants for macro scale analysis
The results of the analysis showed that the material behavior
during compression, and the degree of densification that can
be achieved by compression, can vary greatly depending on the
size distribution of the particles. Because of the wide range of
parameters that determine the material behavior of powders, it is
not realistic to design materials based on actual measurements
alone. It is expected that the efficiency of material design will be
greatly improved by using the virtual material testing analysis
Newsletter EnginSoft Year 17 n°3
By Koji Yamamoto | CYBERNET SYSTEMS CO.
Koji Yamamoto is Technology Specialist and Developer of
Multiscale.Sim at Cybernet System Co.,Ltd.
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