Thesis

Modeling of Rigid Foams
¨
JOONAS KOLL
Doctoral Thesis
Stockholm, Sweden 2015
TRITA-AVE 2015-21
ISSN 1651-7660
ISBN 978-91-7595-562-9
Kungliga Tekniska H¨ogskolan (KTH)
Department of Aeronautical and Vehicle Engineering
SE-100 44 Stockholm, Sweden
Akademisk avhandling som med tillst˚
and av Kungl Tekniska H¨ogskolan framl¨agges till
offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen i L¨attkonstruktioner
fredagen den 22 maj kl 9.15 i D3, Lindstedtsv¨agen 5, Kungliga Tekniska H¨ogskolan,
Stockholm.
© Joonas K¨
oll, 2015
Tryck: E-Print AB
iii
Acknowledgments
The work presented in this thesis has been carried out at the Department
of Aeronautical and Vehicle Engineering at KTH Royal Institute of Technology.
Financial support was kindly granted by the Swedish Research Council.
I thank my supervisor Dr. Stefan Hallstr¨om for introducing me to the amazing
world of foams and for giving me the opportunity to write this thesis. Your guidance
and support is gratefully acknowledged. I want to express my gratitude to all of my
present and former colleagues at the department. In particular I want to thank my
two office mates during my time as a PhD student at KTH, Sohrab and Christof. I
appreciate your friendship, support, and all the interesting discussions we had. A
special thanks goes to Ylva for your endless care and support. Last but not least I
want to thank my family for all your love and support.
Joonas K¨
oll
Stockholm in 2015
v
Abstract
The morphology and elastic properties of foams are investigated using models accounting
for the cellular structure. Different modeling approaches are evaluated and compared to
the cellular structures of real foams. The aim is to find an approach that produces realistic
models that can be adjusted to match morphology measures taken form real foams. The
elastic properties are computed using finite element analysis and the results are compared
with both experimental results from mechanical testing and existing analytical models. A
generalization of the existing analytical models is suggested for better agreement with the
modeling results.
In Paper A Voronoi and equilibrium foam models are generated and investigated. The
Voronoi models are found to have a high content of short edges and small faces while
conversion into equilibrium foams eliminates such small geometrical features. It is also
seen that different seed point distribution algorithms generally result in different model
topologies.
In Paper B the relations between the sphere packing fraction and the resulting degree
of volumetric polydispersity are studied as well as the relations between polydispersity and
a number of morphology parameters. Both Voronoi and equilibrium foams are investigated
and compared with data from real foams. Such comparisons indicate that the used method
is somewhat limited in terms of polydispersity but provides a controlled way of varying
the foam morphology.
In Paper C equilibrium models are used to study the influence of polydispersity, relative
density and distribution of solid on the elastic properties of foams. It is found that the
elastic moduli are very weakly coupled to the polydispersity while the influence from
the relative density and distribution of solid is considerably stronger. Existing analytical
models from the literature are fitted to the results and generalizations are suggested for
better agreement over the investigated range of relative density and distribution of solid.
In Paper D equilibrium foam models with prescribed polydispersity are generated,
analyzed and compared to real foams. The polydispersity of real foams is measured in
micro-CT images and the results are applied to the models. The elastic properties are
computed and the results are compared with experimental results and previous models with
significantly lower polydispersity. No clear relation is found between the elastic properties
and the polydispersity of equilibrium foams.
vii
Sammanfattning
Morfologi och elastiska egenskaper f¨
or skum unders¨
oks h¨
ar med hj¨
alp av modeller
som tar h¨
ansyn till materialens cellstruktur. Olika modelleringsalternativ utv¨
arderas och
j¨
amf¨
ors med cellstrukturen hos riktiga skum. M˚
als¨
attningen ¨
ar att hitta en metod som
ger realistiska modeller vars geometri kan anpassas till olika typer av riktiga skum. De
elastiska egenskaperna ber¨
aknas med hj¨
alp av finita element analys och resultaten j¨
amf¨
ors
dels med experimentella resultat och dels med befintliga analytiska modeller. F¨
or en b¨
attre
overensst¨
ammelse med modelleringsresultaten f¨
oresl˚
as en generalisering av de befintliga
¨
analytiska modellerna.
I Papper A genereras och unders¨
oks Voronoi och j¨
amviktsmodeller av skum. Voronoimodellerna konstateras ha en onormalt h¨
og andel korta kanter och sm˚
a ytor som elimineras n¨
ar
modellerna konverteras till j¨
amviktsmodeller. Det konstateras ocks˚
a att olika algoritmer
f¨
or sf¨
arpackning resulterar i olika modelltopologier.
I Papper B unders¨
oks sambanden mellan packningsgraden i sf¨
ardistributioner och
den resulterande cellvolymspridningen i modellerna, samt dess relation till ett antal
morfologiparametrar. B˚
ade Voronoi och j¨
amviktsskum analyseras och j¨
amf¨
ors med data
fr˚
an riktiga skum. Resultaten visar att endast en berg¨
ansad cellvolymsspridning kan
uppn˚
as med den valda metodiken men den tillhandah˚
aller ¨
and˚
a ett s¨
att att kontrollerat
variera morfologin.
I Papper C unders¨
oks ett antal parametrars inflytande p˚
a dom elastiska egenskaperna.
Dessa parametrar utg¨
ors av cellvolymsspridning, relativa densitet och f¨
ordelning av bulkmaterial mellan cellv¨
aggar och kanter. Kopplingen mellan cellvolymsspridningen och de
elastiska egenskaperna visar sig vara mycket svag. Den relativa densiteten och f¨
ordelningen
av bulkmaterial har d¨
aremot ett mycket starkare inflytande. Analytiska modeller fr˚
an
litteraturen anpassas till resultaten och generaliseringar av dessa f¨
oresl˚
as f¨
or en b¨
attre
overensst¨
ammelse ¨
over hela spannet av relativ densitet och bulkmaterialf¨
ordelning.
¨
I Papper D genereras och analyseras j¨
amviktsmodeller med p˚
atvingad cellvolymsspridning och modeller j¨
amf¨
ors med riktiga skum. Cellvolymsspridningen i riktiga skum m¨
ats
i datortomografibilder och resultaten anv¨
ands i modellerna. De elastiska egenskaperna
ber¨
aknas och resultaten j¨
amf¨
ors dels med experimentella resultat och dels med tidigare
modeller med v¨
asentligt l¨
agre cellvolymsspridning. Inga tydliga relationer kan ses mellan
de elastiska egenskaperna och cellvolymsspridningen i modellerna av j¨
amviktsskum.
Dissertation
This doctoral thesis is based on an introduction to the area of research and the
following appended papers:
Paper A
J. K¨
oll and S. Hallstr¨
om. Generation of stochastic foam models for numerical
analysis. Journal of Cellular Plastics, 50(1) 37-54, 2014.
Paper B
J. K¨
oll and S. Hallstr¨
om. Influence of sphere packing fraction on polydispersity
and morphology of Voronoi-partitioned foam models. Manuscript submitted for
publication.
Paper C
J. K¨
oll and S. Hallstr¨
om. Elastic properties of equilibrium foams. Manuscript
submitted for publication.
Paper D
J. K¨oll and S. Hallstr¨om. Elastic properties from equilibrium foam models with statistically representative polydispersity. Manuscript to be submitted for publication.
ix
x
Parts of this thesis have also been presented as follows:
J. K¨oll and S Hallstr¨om. Morphology effects on constitutive properties of foams. In
Proceedings of the 18th International Conference on Composite Materials (ICCM-18),
Jeju Island (Korea), 2011.
Division of work between authors
Paper A
K¨
oll implemented the modeling principles through development of software tools,
performed the simulations and wrote the paper. Hallstr¨om initiated and guided the
work and contributed to the paper with valuable comments and revisions.
Paper B
K¨oll performed the simulations and wrote the paper. Hallstr¨om initiated and guided
the work and contributed to the paper with valuable comments and revisions.
Paper C
K¨
oll developed the preprocessing software, performed the simulations, suggested
the generalized scaling laws and wrote the paper. Hallstr¨
om initiated and guided
the work and contributed to the paper with valuable comments and revisions.
Paper D
K¨oll performed the image processing and the simulations, analyzed the results from
the mechanical testing and wrote the paper. Hallstr¨
om initiated and guided the
work and contributed to the paper with valuable comments and revisions.
xi
Contents
I Introduction
1
1 Background
1.1 Applications for rigid foam materials . . . . . . . . . . . . . . . . . .
3
5
2 Cellular geometry
2.1 Characterization of the cellular structure . . . . . . . . . . . . . . . .
2.2 Generation of model geometries . . . . . . . . . . . . . . . . . . . . .
7
7
10
3 Elastic properties
17
3.1 FE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Summary of appended papers
23
5 Future work
25
Bibliography
27
II Appended papers
29
xiii
Part I
Introduction
1
1 Background
A range of different models have been used to study the properties of foams,
from relatively simple single cell approximations to very advanced geometrical
representations consisting of large clusters of cells. Gibson & Ashby developed
analytical models describing the mechanics of foams [1]. These models were based
on the idealized single-cell geometries illustrated in Figure 1.1 and offer relations
that describe how the elastic properties and strength scale with relative density for
open- and closed-cell foams. They also introduced the concept of distribution of
solid between beams and plates to account for possible bulk material concentrations
along the cell edges.
Figure 1.1: Illustration of the single-cell geometries used by Gibson &
Ashby to develop analytical models describing the mechanics of open- and closed-cell foams.
Several single cell polyhedral models exist that pack to fill space, i.e. are
geometrically periodic. For a dry foam the edge-connectivity is four and the faceconnectivity is three [2] and that is also the connectivity usually found in other real
foams. Most of the periodic polyhedral cells on the other hand have other edgeand/or face-connectivities, but not the tetrakaidecahedron illustrated in Figure 1.2a,
also known as the Kelvin cell. It is the product of Kelvins search for the one ideal
cell that partitions space with the minimum use of surface area [3]. The mechanical
properties of Kelvin cells have been studied [4–9] but there is always the questions
3
4
J. K¨oll
(a)
(b)
Figure 1.2: Illustration of the Kelvin cell (a) and the Weaire-Phelan
model (b).
whether or not it is representative of real foam materials with their broad range of
cell shapes and sizes. The botanist Matzke undertook historical experimens where
he characterized and documented the shape of individual bubbles in monodisperse
liquid foams [10, 11]. He presented a range of different cell architectures from his
experiments but his findings did not include any Kelvin cells.
The Weaire-Phelan model illustrated in Figure 1.2b is the most surface-area
effective way known today of partitioning space into cells with equal volumes [12].
Having slightly less surface area per unit volume than the Kelvin cell it shows
that a combination of several cells can be more effective from a surface energy
perspective. The mechanical properties of the Weaire-Phelan model have also been
(a)
(b)
Figure 1.3: Illustration of Voronoi (a) and equilibrium foam (b).
Introduction
5
studied [4, 9] but the eight cells in the model are still limited to only two different
cell architectures.
Recent work has focused on stochastic models based on Voronoi [13] and Laguerre
(weighted Voronoi) partitioning as well as equilibrium foams [8, 9, 14–23]. The
aim with such approaches is to develop geometrically realistic models of rigid
polydisperse foam materials. The work presented in this thesis is focused on Voronoi
and equilibrium foams illustrated in Figure 1.3.
1.1
Applications for rigid foam materials
Today rigid foam materials can be found almost anywhere. They are widely used in
buildings as insulation underneath concrete floors and they are on of the dominating
types of chock-absorbing packing-material used for protecting electronic devices
and home appliances during storage and transportation. On a more advanced level
rigid foams are used as load carrying core members in sandwich structures typically
found in sports equipment as for instance skis and boards, in wind turbine blades,
boat hulls, various floor panels, and in the cargo area body panels of box trucks, to
name a few examples. The common objective for sandwich applications is the need
for high bending stiffness in combination with low weight.
Lately there has been an increasing demand for high performance polymer foam
core material from the wind turbine industry. The increase is related to the global
efforts of switching to renewable energy sources. The automotive industry is in
need of developing lighter vehicles due to increasing restrictions on emissions. There
is already an increase in the use of composite materials for replacement of metal,
especially in ancillary components. It is probable that emission regulations will
eventually force the automotive industry to a change of technology, even for main
structural components. Foam core sandwich construction can be an alternative
for any component where bending stiffness is critical. It is therefore possible that
there will be an increasing demand for foam core material also from the automotive
industry.
2 Cellular geometry
Foam materials are stochastic structures that consist of a space filling cluster of
cells. Such clusters are built up from an interconnected network of walls creating a
closed-cell foam or an interconnected network of struts creating an open-cell foam.
The cells vary in size, shape and orientation creating the irregular structure that is
typical for foams.
2.1
Characterization of the cellular structure
Microscopy
Traditional optical microscopy has been available for a long time and gives some
possibility to study the cellular structure of foams. It provides a 2D view of a
cross-section of the material as illustrated in Figure 2.1a. With a Scanning Electron
Microscope (SEM) hight resolution images can be produced that also show the
topography of the surface. That gives depth to the image as illustrated in Figure 2.1b.
SEM provides more information about the cellular structure than optical microscopy
but is still limited to what can be observed from the surface. It is however useful
for preliminary estimates of for instance cell sizes and wall thicknesses.
(a)
(b)
Figure 2.1: Optical microscope image (a) and SEM image (b) of Rohacell IG-F.
7
8
J. K¨oll
Computed tomography
The internal 3D structure of foams can be studied using X-ray computed tomography
(CT) [7, 15, 24–26]. The principle is illustrated in Figure 2.2. It is a technology
that uses multiple radiographic images (Figure 2.3a), taken around a single axis of
rotation, to reconstruct the internal 3D structure of the scanned object. A stack of
tomography images is created where each image is representing a one pixel thick
slice of the object (Figure 2.3b). The slices are finally combined to a 3D voxel image
representing the structure of the scanned object (Figure 2.3c).
Detector
X-ray source
Specimen
Figure 2.2: The principle of computed tomography.
Image processing
Scientific image processing and analysis tools are needed to extract morphology
parameters from reconstructed CT images [26, 27]. First the system of cell walls
and/or edges needs to be separated from the enclosed empty space (voids) through
thresholding. Then the 3D image needs to be segmented into individual cells
through what is called water shedding. After the segmentation is done the cells can
be analyzed individually.
Thresholding is a process that determines what is void and what is structure,
so that the two can be separated. Using simple thresholding all voxels with an
intensity value below a specified single threshold are considered to be background
and are thus removed while all voxels with an intensity value above the threshold
are retained. The problem with simple thresholding is that stray voxels with low
intensity values are removed even if they obviously belong to the structure. Inversely,
stray background voxels with high intensity values might be retained. In hysteresis
Introduction
(a)
9
(b)
(c)
Figure 2.3: Radiograph image (a), reconstructed slice (b) and reconstructed CT image (c) of Rohacell IG-F.
10
J. K¨oll
thresholding two threshold values are used, one lower and one upper. Voxels with
an intensity below the lower threshold are always considered to be background and
voxels with an intensity above the upper threshold are always considered to be
part of the structure. The voxels that fall between the two thresholds are retained
only if connected to an other voxel with intensity above the upper threshold. This
gives a smoother and more realistic separation between background and structure.
Figure 2.4a shows an example of a micro-CT image that has been thresholded using
hysteresis thresholding.
Water shedding is a process that segments the structure into individual cells.
Starting from seeds approximately centered inside the individual cells the algorithm
simultaneously builds particles by adding voxels to each seed until the it reaches
the cell walls. If walls or parts of walls are missing the algorithm restores walls
where the growing particles meet. In order to do that the algorithm needs additional
information i.e. the cell centers and in which directions to shed the water. The
first step is therefore to calculate an euclidian distance map that is exemplified
in Figure 2.4b. It is computed from the thresholded image so that every voxel is
assigned an intensity value corresponding to the distance to the nearest cell wall.
In the ideal case this would give one maxima inside every cell with the intensity
values strictly declining towards the cell walls that have zero intensity. In reality
many cells will contain several local maxima. The surplus maxima can be removed
using the H-maxima transform [28]. The transform reduces local maxima until they
reach the intensity level of the surrounding or the maximum suppression distance
h. By selecting the appropriate value for h, an image can be generated with one
approximately centered local maximum within every cell. This image can then
be used for the water shedding to segment the individual cells as illustrated in
Figure 2.4c. The segmented image can in turn be used for extracting morphology
parameters as for instance individual cell volumes.
2.2
Generation of model geometries
Voronoi structures
For 3D Voronoi structures the partitioning is based on a set of seed points distributed
in a model space where each cell is defined by all points that are closer to one
particular seed point than to any other. In order to have the seeds relatively evenly
distributed they can be obtained from centra of distributions of densely packed hard
spheres, as illustrated in Figure 2.5. Inherently all Voronoi cells are strictly convex
polyhedra with flat faces as can be seen in Figure 1.3a. Voronoi partitions can be
created with for instance the QHull software [29]. A mathematical definition of the
partitioning is given in Equation (1) in Paper A and more examples of such models
are shown in Figure 3 in the same paper.
Even though Voronoi foams have some resemblance with real foam materials
they are also associated with a geometrical problem since they inherently contain
small geometrical features in form of short cell edges and small cell faces. The actual
Introduction
(a)
(b)
(c)
Figure 2.4: Thresholded image (a), EDT image (b) and segmented
image (c) of Rohacell IG-F.
11
12
J. K¨oll
Figure 2.5: Illustration of randomly packed hard spheres.
size can be down to the limit formed by the numerical precision in the geometry
description. Such small features in the geometry description will create problems
later when the mechanical properties are to be computed. For the FE analysis the
geometry needs to be meshed with finite elements and for accurate and successful
analysis the mesh needs to be smooth and fulfill some requirements on internal
angles, aspect ratios and the variation of element size. Such smooth meshing is in
practice very hard to achieve for irregular Voronoi foams.
The small geometrical features origin from the seed point locations. A Voronoi
Figure 2.6: Illustration of the cause of small geometrical features in
Voronoi foams.
Introduction
13
vertex has an exactly equal distance d to four neigboring seed points in 3D and three
different neighbors in 2D. For practical reasons the source of the problem will here be
illustrated in 2D but the principles are the same in 3D. As illustrated in Figure 2.6, a
short edge emerges when a fourth seed point happens to be located at approximately
the same distance r + d to the Voronoi vertex as the true neighbors so that d << r.
The size depends on d and the corresponding event in 3D happens when a fifth
seed point is located at approximately the same distance. Proportionally increasing
triangular constellations of seed points in 2D and tetrahedral constellations in 3D
reduces the problem. It can be achieved by using densely packed disks or spheres as
source for the seeds. The share of tetrahedrally organized spheres increases with
densification when a sphere packing approaches the random close packing limit and
peaks at a fraction of about 0.3 [30]. That is the reason why random close packing
reduces the problem but does not eliminate it.
Equilibrium foams
Equilibrium foams resemble liquid foams with a low liquid content, i.e. dry foams.
They are at equilibrium from a cell wall surface energy point of view. Corresponding
models can be generated with the Surface Evolver software [31]. A combination of
surface area minimization and topology transitions is used to find the equilibrium
structure. An initial geometry needs to be provided and all equilibrium foams
presented in this thesis were generated from Voronoi partitions. When run to
convergence the process also entirely eliminates problem with small geometrical
features found in the Voronoi foams, which is confirmed by the results presented in
Figures 11 and 12 in Paper A. Figure 2.7 illustrates the geometry in a few different
stages during the conversion and more examples of equilibrium foams are shown in
Figure 7 in Paper A.
Converting Voronoi foams into equilibrium foams with Surface Evolver is a delicate process associated with a risk of having both some robustness and repeatability
issues. The nature and consequences of these issues is discussed in Paper A but one
consequence is that it is very difficult to automate the process. For every single
model the conversion into equilibrium state is a time consuming hands on process.
Geometry variation
The polydispersity of Voronoi foams can to a certain extent be controlled through
varying the packing fraction of the sphere distributions from which the seed points
are extracted. The polydispersity is in turn related to several key morphology
parameters as for instance the number of face per cell and the spread in edge
length and face area, while other parameters, as for instance average edge length
and face area, are unaffected. Such relations are shown and discussed in Paper B.
The achievable polydispersity is however limited. As shown in Paper B a relative
standard deviation of cell volume up to 0.5 can be achieved.
14
J. K¨oll
(a)
(b)
(c)
(d )
Figure 2.7: Illustration of different stages during the conversion from
Voronoi to equilibrium foam, chronologically ordered from
(a) to (d ). The initial Voronoi state is shown in (a). In (c)
an ongoing topology transition can be seen while (b) and
(d ) show the structure before and after that transition.
Converting the Voronoi foams to equilibrium foams strongly reduces the influence
of polydispersity and brings the morphology parameters to a virtually constant level.
The results presented in Paper B also indicate that the equilibrium models are in
better agreement with real foams while the conversion to equilibrium foams was
found to be difficult for models with a relative standard deviation in cell volume
above 0.25.
As shown in Paper D higher spread in cell volume can be achieved for equilibrium
foams by applying direct cell volume constraints during the conversion from Voronoi
foams. The benefit with this approach is that predefined cell volume distributions
can be used. As shown in the paper the individual cell volumes as well as their
Introduction
15
statistical distribution in real foams can be determined using micro-CT and image
processing. That way models with truly or statistically representative cell volumes
can be generated.
3 Elastic properties
The elastic properties of foams can be computed with FE-models using foamstructure geometries based on the principles explained in section 2. The mechanical
behavior of the models can be verified by comparing with results from mechanical
testing. From the FE results scaling laws can be derived that describe how the
elastic properties scale with relative density and distribution of solid.
3.1
FE analysis
The mechanical properties of closed-cell foams can be computed with FE using plate
models where shell elements are assigned to the cell walls. Material concentrations
along the cell edges can be modeled by relocating material from the shell elements
to additional beam elements placed along the edges. Placing all material in the
beam elements creates models of open-cell foam.
The geometry descriptions provided by the softwares QHull and Surface Evolver
cannot be directly imported by commercial FE software nor is there any other
software available for converting them to importable formats. The amount of data
in the geometry descriptions is too large for manual conversion to be an alternative
why a specialized preprocessing software package was developed to handle the
necessary file formats and geometry descriptions. The preprocessor also handles the
application of periodic boundary conditions and loads.
After preprocessing all FE-models were analyzed with the Abaqus standard
solver. The principles described here can however be applied with other FE solvers
as well.
Preprocessing
All Voronoi foams presented in this thesis were created with the QHull software
that can produce output in the off-file format. An algorithm that converts from
the off-file format to a Surface Evolver input file (fe-file) was developed as part
of the preprocessing package. In off-file format polygonal surfaces are defined by
vertices and bodies are defined by a set of polygonal surfaces while in the fe-file
lines are defined by vertices, triangular surfaces are defined by lines and bodies
are defined by a set of triangular surfaces. The output from QHull is defined in
17
18
J. K¨oll
Euclidian space while a 3-torus space is used for the surface Evolver input file to
handle the periodicity. In Euclidian space there is one way to connect two points
with a straight line while in a 3-torus space there are 27 different possibilities. When
placing a geometry from Euclidian space in 3-torus space parts of the geometry will
typically at first be located outside the 3-torus. All vertices need to be moved inside
the 3-torus and torus wrapping definitions need to be added to all line definitions.
The torus wrapping definitions state which torus borders are crossed by a line and
in which direction they are crossed.
After the conversion to equilibrium foam Surface Evolver produces an output
file (dmp-file) that in principle follows the same geometry definitions as the fe-file.
Another algorithm was developed that converts the geometry back to Euclidian
space. The same algorithm also changes geometry description from a cell-by-cell
definition to a single interconnected structure suitable for FE-analysis. It identifies
all periodic node pairs in the model and defines periodic equation constraints for
them. Six different sets of constraints are defined for six principal load-cases, axial
loading in three orthogonal directions as well as shear loading in three orthogonal
planes.
The periodic boundary conditions are in the following illustrated in 2D but the
principles are the same when expanded to 3D. For a periodic geometry with the
periodicity L, exemplified in Figure 3.1a, the nodes located on the boundaries can
be grouped depending on which side of the geometry they belong to as
a =
[1, 2, 3, 4]
(3.1)
b =
[10, 9, 8, 7]
(3.2)
c =
[1, 12, 11, 10]
(3.3)
d =
[4, 5, 6, 7] .
(3.4)
R1 and R2 are reference nodes used for homogenizing the global displacements
between different periodic node-pairs. For the case of axial loading the periodic
boundary conditions can be defined as
ua(i) − ub(i)
=
0
(3.5)
vc(i) − vd(i)
=
0
(3.6)
θa(i) − θb(i)
=
0
(3.7)
θc(i) − θd(i)
=
0
(3.8)
for the x- and y-displacements and the nodal rotations respectively. The homogenized
global displacements u and v are controlled by R1 and R2 through the relations
uc(i) − ud(i) + u =
0
(3.9)
vb(i) − va(i) + v
0.
(3.10)
=
When a homogenized strain u/L is applied in the x-direction, as illustrated in
Figure 3.1b, it results in a homogenized lateral strain v/L in the y-direction and
the Poisson’s ratio is given by
Introduction
19
2
4
1
3
L
12
5
L
11
6
9
7
10
8
y
R1
R2
x
(a)
2
4
1
3
L+u
12
5
L+v
11
6
9
7
10
8
u
y
R1
R2
v
x
(b)
u
2
4
1
u
3
5
12
L+v
11
6
L+v
9
7
10
8
u
y
(c)
R1
R2
v
x
Figure 3.1: Illustration of a periodic geometry (a) under axial loading
(b) and shear loading (c).
20
J. K¨oll
u
νxy = − .
v
(3.11)
For the case of shear loading the nodal rotations are controlled as in Equations 3.7
and 3.8 The shear loading is applied by the displacement u through the relations
ua(i) − ub(i) + u =
0
(3.12)
vc(i) − vd(i) + u =
0
(3.13)
and the contractions are homogenized by the displacement v through the relations
va(i) − vb(i) + v
=
0
(3.14)
uc(i) − ud(i) + v
=
0.
(3.15)
The preprocessor also computes the parameters needed for defining the plate
and beam cross-sections illustrated in Figure 3.2. Using the given distribution of
solid and the total edge length and wall area in the model, the cross-section areas
and the area moments of inertia are determined. Adding also material properties
the preprocessor finally creates a complete Abaqus input-file. An example of a
finished model under axial loading is illustrated in Figure 3.3. After the analysis,
homogenized forces and displacements can easily be extracted from the two reference
nodes R1 and R2.
Figure 3.2: Illustrations of the cross-sections perpendicular to a cell
edge.
3.2
Experimental verification
The models were compared with experimental results from mechanical testing. Both
block shear tests and tensile test were conducted and analyzed. The experimental
verification of the models is discussed in Paper D.
Introduction
21
Figure 3.3: Illustrations of a FE model showing displacements under
axial loading.
3.3
Analytical model
The elastic properties depend on the relative density and the distribution of solid.
Analytical expressions can be derived to describe such relations so that further
simulations don not have to be performed. In Paper C a generalization of Gibson &
Ashby’s analytical model is suggested based on the FE results,
E∗
= f1 (φ)
Es
G∗
= f2 (φ)
Es
ρ∗
ρs
2
ρ∗
ρs
2
+ g1 (φ)
ρ∗
ρs
(3.16)
+ g2 (φ)
ρ∗
,
ρs
(3.17)
where fi is a second order polynomial, gi is a first order polynomial, φ is the
distribution of solid, ρ is the density, E is the Young’s modulus, G is the shear
modulus and the indices s and ∗ denote solid material properties and foam properties
respectively. The determination of fi and gi is discussed in Papers C and D.
4 Summary of appended papers
Paper A
Voronoi and equilibrium foam models are generated and investigated for their
potential use in numerical analysis using finite element methods. A drawback with
the Voronoi models is found to be the high content of short edges and small faces
while conversion into equilibrium foams is found to eliminate such small geometrical
features. It is also seen that different seed point distribution algorithms generally
result in different model topologies.
Paper B
The relations between the sphere packing fraction and the resulting degree of
volumetric polydispersity is investigated for Voronoi foams. Further, the relations
between polydispersity and a number of morphology parameters are investigated
for both Voronoi and equilibrium foams. Comparisons with data form real foams
indicate that the used method is somewhat limited in terms of polydispersity but it
provides a controlled way of varying the foam morphology.
Paper C
Equilibrium finite element models are used to study the influence of polydispersity,
relative density and distribution of solid on the elastic properties of foam materials.
It is found that the elastic moduli are very weakly coupled to polydispersity. The
influence from relative density and distribution of solid is considerably stronger.
Analytical models from the literature are fitted to the finite element results. A
generalization of the analytical models is suggested for better agreement over the
investigated range of relative density and distribution of solid.
Paper D
Polydisperse finite element equilibrium foam models are generated, analyzed and
compared to real foams. The polydispersity of real foams is measured from micro-CT
23
24
J. K¨oll
images and applied to the models. Finite element analysis is used to study the elastic
properties and the results are compared to experimental results from mechanical
testing. The results are also compared to previous models with significantly lower
polydispersity. No pronounced relation is found between the elastic properties and
the polydispersity of equilibrium foams.
5 Future work
A reoccurring issue when comparing the models to polymer foams is that accurate
bulk material properties are often unknown, even in the linear elastic regime. The
properties can change during the foaming process and accurately measuring them in
the finished foam is challenging due to the small dimension of cell walls and struts.
Finding methods for determining the full non-linear bulk material properties is one
of the future challenges in order to reach reliable predictions for large deformations
and the strength of foams.
The FE-models can also be further improved by adding the contribution from
internal gas pressure. Another future subject is to determine the reasons why
Laguerre models predict a relation between the polydispersity and the stiffness of
the foam and equilibrium models do not.
One possible path when it comes to the robustness issues when converting
from Voronoi to equilibrium foams is to rearrange seed point distributions through
relaxation of Delauny tetrahedra in Surface Evolver. No topology transitions are
likely to take place and only a corse mesh is needed. It could dramatically reduce
the number of topology transitions needed for the subsequent conversion of the
associated Voronoi foam to an equilibrium foam and make the process much more
stable.
25
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Part II
Appended papers
29