NUMERICAL CHARACTERISATION OF HOLLOW SPHERE
COMPOSITES BASED ON PERFORATED INCLUSIONS
MOHD AYUB BIN SULONG
UNIVERSITI TEKNOLOGI MALAYSIA
NUMERICAL CHARACTERISATION OF HOLLOW SPHERE COMPOSITES
BASED ON PERFORATED INCLUSIONS
MOHD AYUB BIN SULONG
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Engineering (Mechanical)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
NOVEMBER 2011
iii
The credit of this work goes to my eminent supervisor, PROFESSOR DR.-ING.
ANDREAS ÖCHSNER D.SC. for his generosity in guiding me through the research,
to my beloved parents and wife for their motivational support.
iv
ACKNOWLEDGEMENT
I would like to thanks all of my friends who have supported me during
completing this work, especially to Meysam, Moones and Huda which helped me to
get everything started. I also would like to express my gratitude to UTM and FKM
members who have supported continuously throughout this research, Research
Management Center (RMC) for the facilities provided, Office of Postgraduate
Studies (SPS) for their great administration task, UTM library for striving to serve
the best service of resources. Last but not least many thanks to Malaysian Ministry of
Higher Education (MOHE) for supporting this research financially.
v
ABSTRACT
Metallic hollow sphere structures (MHSS) are a new type of reinforced
materials and can be classified as an advanced composite material. A modified
metallic hollow sphere MHS geometry which introduced the perforation becomes the
main model in this research. This structure is called a perforated hollow sphere
structures (PHSS) which is opened to be infiltrated by the matrix to fully embed it
and form a composite. PHSS composites offer a new field of mechanical properties
compared to cellular structures studied by other researchers. Emphasis will be given
to determine the influence of the modified perforation diameter of PHSS composite
in terms of macroscopic mechanical properties (e.g. Young’s modulus and Poisson’s
ratio). In addition, the mechanical properties of PHSS composites were also
compared to hollow sphere (HS) composites (with and without filled matrix). A
perforation introduced in the sphere shells obviously changes the mechanical
properties of the PHSS composite, e.g. Young’s modulus and Poisson’s ratio. The
result of the investigation revealed that these values decrease as the perforation
diameter increases. PHSS composite models were simulated based on the unit cell
approach by means of the Finite Element (FE) method. This method can reduce the
costs of experimental tests and provides more information on possible mechanical
properties of perforated hollow sphere structures (PHSS) composites. Nevertheless,
experimental tests are still necessary and should be conducted in the future for
validation purpose.
vi
ABSTRAK
Struktur Sfera Logam Berongga adalah jenis baru bahan pengukuh dan boleh
dikelaskan sebagai bahan komposit termaju. Geometri Sfera Logam Berongga yang
telah diubahsuai iaitu mempunyai lubang menjadi model utama bagi kajian ini.
Struktur ini dipanggil sfera berongga berlubang terbuka untuk dimasuki oleh matriks
untuk menerapkan sepenuhnya dan membentuk bahan rencam. Komposit sfera
berongga berlubang menawarkan satu sifat baru mekanikal berbanding struktur sel
yang dikaji oleh penyelidik lain. Tumpuan kajian ini adalah untuk menentukan
pengaruh diameter penembusan komposit sfera berongga berlubang yang telah
diubahsuai dari segi ciri-ciri makroskopik mekanikal (contohnya modulus Young dan
nisbah Poisson). Di samping itu, sifat-sifat mekanik komposit sfera berongga
berlubang juga dibandingkan dengan komposit sfera berongga (dengan atau tanpa
matriks isian). Penembusan yang diperkenalkan dalam cengkerang sfera merubah
sifat-sifat mekanik komposit sfera berongga berlubang dengan ketara, contohnya
Modulus Young dan nisbah Poisson berkurangan kerana kenaikan diameter
penembusan. Model komposit sfera berongga berlubang disimulasikan berdasarkan
pendekatan sel unit dengan menggunakan analisis kaedah unsur terhingga. Kaedah
ini boleh mengurangkan kos ujian ujikaji dan memberikan maklumat lanjut
mengenai sifat-sifat mekanikal yang mungkin bagi komposit sfera berongga
berlubang. Walau bagaimanapun, ujikaji sebenar masih diperlukan dan perlu
dijalankan pada masa hadapan bagi tujuan pengesahan.
vii
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF ABBREVIATIONS
xvii
LIST OF SYMBOLS
xviii
LIST OF APPENDICES
1
xxi
INTRODUTION
1
1.1 Introduction
1
1.2 Problem Identification
2
1.3 Objective
3
1.4 Scope of Study
3
viii
2
1.5 Gantt Chart
4
1.6 Summary
5
LITERATURE REVIEW
6
2.1 Introduction
6
2.2 Prospective for Lightweight Cellular Metals
6
2.2.1 High specific stiffness and strength
7
2.2.2 Good Energy Absorbers
9
2.2.3 Good Thermal Conductors
9
2.2.4 Structural Vibration
2.3 Perforated Hollow Sphere Structures
11
2.4 Literature Assessment
15
2.5 Finite Element Method
18
2.5.1 Representative Volume Element
3
10
19
2.6 Summary
20
RESEARCH METHODOLOGY
21
3.1 Introduction
21
3.2 Modelling of PHSS
21
3.3 Finite Element Approach: Geometry, Mesh,
27
Boundary Conditions and Materials
3.3.1 Geometry
28
3.3.2 Mesh
35
3.3.3 Boundary Conditions
38
3.3.4 Materials
42
3.4 Mechanical Properties
44
3.4.1 Young’s Modulus and Poisson’s Ratio
44
3.4.2 Deformation of the Model
46
3.5 Thermal Properties
47
3.6 The average Density for Composite materials
49
3.7 Summary
51
ix
4
5
RESULTS & DISCUSSION
52
4.1 Introduction
52
4.2 Young’s Modulus and Poisson’s Ratio
52
4.3 Thermal Conductivity
60
4.4 Summary
65
CONCLUSIONS AND FUTURE STUDY
66
5.1 Conclusions
66
5.2 Future Study
67
REFERENCES
71
APPENDICES A-B
78
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
3.1
Summary of the considered geometry................................. 30
3.2
Specific mechanical properties of the spheres and binder.. 42
3.3
Heat conductivity properties of base materials................... 42
xi
LIST OF FIGURES
FIGURE NO.
1.1
TITLE
PAGE
Cellular metals: a) M-Pore® (aluminium sponge);.............. 2
b) Alporas® (aluminium foam); c) Brass foam
1.2
Gantt chart.....................................................................
4
2.1
Properties of cellular metals.............................................
7
2.2
Evolution of engineering materials...................................
8
2.3
Comparison of different unit cell stacking and.............
8
their influence on the relative strength
2.4
Deformation of 316L MHS under crush test.................
9
2.5
Temperature distribution inside the MHSS........................ 10
2.6
Mode shapes gained by numerical simulation.................... 10
2.7
a) Connection of single spheres; ......................................
11
b) Multi-sphere network
2.8
Simplified models for sintered hollow sphere..................... 13
structures: a) Flattened contact region;
b) Point contact; c) Syntactic hollow sphere
structure (new development)
2.9
Simplified models of PHSS: a) Non-perforated shell; ......
b) Perforation allowing the matrix to fill the cavity;
c) PHSS fully embedded in the matrix
14
xii
2.10
Single hollow spheres: a) closed surface (common.............. 14
configuration); b) with perforated surface (new
development); (© by Glatt GmbH, Dresden, Germany)
2.11
Modeling of structures by gathering discrete elements......... 18
(a) One-dimensional elements. (b) Two-dimensional plane
stress elements
2.12
Representative volume element of a simple cubic cell......... 25
Structure
3.1
Flowchart of the research methodology................................ 22
3.2
The derivation the one-eighth of a unit cell used in the........ 23
Finite Element Analysis (FEA) package
3.3
Schematic diagram for the PHSS embedded in the matrix:.. 24
a) PHSS and matrix with distance of adhesively bonded
necking, d (i.e. 0.12 mm); b) PHSS and matrix with double
distance of adhesively bonded necking, 2d; c) Flattened
contact area of PHSS
3.4
PHSS composite exploded view: a) Inner matrix;................ 25
b) PHSS; c) Plate-shaped matrix; d) Outer matrix
3.5
Schematic illustration of primitive cubic sphere.................. 26
arrangement of perforated hollow spheres: a) bonded
syntactic perforated hollow sphere structure (PHSS);
b) sintered syntactic PHSS (on the right of each
schematic representation, the arrangement of the spheres
in 2D is shown to clarify the distance between adjacent
spheres)
3.6
Front view of PHSS............................................................... 28
3.7
Front view of inner matrix..................................................... 29
3.8
Front view of outer matrix..................................................... 29
3.9
Isometric view of plate-shaped matrix...................................30
xiii
3.10
Finite element mesh of a perforated hollow sphere:.............. 31
a) arrangement with completely immersed sphere in the
matrix; b) sintered arrangement (the matrix mesh is not
shown for clarity)
3.11
Configuration of different hole diameters for bond.............. 32
gap, a (refer to Figure3.17) starting with arrangement of the
PHSS with the smallest hole on the shell surface, hole
diameter is increased from; a) HSS (non-perforated
configuration) by b) PHSS with hole diameter equal to 25%
of dl, c) PHSS with hole diameter equal to 50% of dl,
d) PHSS with hole diameter equal to 75% of dl and e) PHSS
with hole diameter equal to dl = 1.36 mm
3.12
Configuration of different hole diameters for bond............... 33
gap, 2a starting with arrangement of the PHSS with the
smallest hole on the shell surface, hole diameter is increased
from; a) HSS (non-perforated configuration) by b) PHSS
with hole diameter equal to 25% of dl, c) PHSS with hole
diameter equal to 50% of dl, d) PHSS with hole diameter
equal to 75% of dl and e) PHSS with hole diameter equal to
dl = 1.36 mm
3.13
Configuration of different hole diameters for sintered......... 34
starting with arrangement of the PHSS with the smallest
hole on the shell surface, hole diameter is increased from;
a) HSS (non-perforated configuration) by b) PHSS
with hole diameter equal to 25% of dl, c) PHSS with hole
diameter equal to 50% of dl, d) PHSS with hole diameter
equal to 75% of dl and e) PHSS with hole diameter equal to
dl = 1.36 mm
3.12
PHSS shell transformed from solid to three-dimensional.... 35
Hex-meshing
xiv
3.15
Results of convergence study on mesh refinement; a) the... 36
graph shows that the value of the thermal conductivity
became stable when the number of nodes in the meshed
model exceeds 9000 nodes b) on left is the one-eighth of
a simple cubic arrangement model with less number of
nodes and on the right is the model with finer mesh
3.16
Results of convergence study on mesh refinement for the.... 37
sintered arrangement i.e. using automatically generated
tetrahedral mesh; a) the graph shows that the value of the
Young’s modulus became stable when the number of
nodes in the meshed model reaches 14000 nodes b) on left
is the one-eighth of a primitive cubic arrangement model
with less number of nodes and on the right is the model
with finer mesh
3.17
Finite element mesh and applied boundary conditions of a... 39
syntactic perforated hollow sphere cubic unit cell: a)
arrangement with completely immersed sphere in the
matrix; b) sintered arrangement (the darker grey elements
belong to the sphere while the lighter grey elements belong
to the matrix
3.18
Finite element mesh and applied boundary conditions........ 40
of a syntactic perforated hollow sphere cubic unit
cell: a) arrangement with completely immersed sphere in
the matrix; b) sintered arrangement (the darker grey
elements belong to the sphere while the lighter grey
elements belong to the matrix)
3.19
Schematic diagram to indicate the difference between......... 41
perforated and empty shell syntactic hollow sphere
structures: a) perforated sphere with bond gap, b) empty
sphere with bond gap, c) sintered perforated sphere, and
d) sintered empty sphere
xv
3.20
a) Micrograph of a completely single hollow sphere;......... 47
b) Micrograph with several hollow spheres; c) Micrograph
of the wall of a hollow sphere (the micrographs are
recorded and analysed with the Zeiss AxioVision 4.6.3.
Sp1 Software in Aalan Univ. Germany).
3.21
Schematics stress-strain diagram showing linear elastic...... 44
deformation for loading and unloading cycles
3.22
The graphical interpretation of the Poisson’s ratio.............. 46
3.23
Original and deformed model with bond gap arrangement.. 47
3.24
The heat transfer principles interpreted into electrical......... 48
circuits
4.1
Young’s modulus of syntactic PHSS versus fraction of ...... 53
hole diameter: a) Using steel (AISI 8000) as the material
of the shell (as a function of the hole diameter, a fraction
of 0 denotes to the shell without holes, with a hole diameter
of d = 1.36 mm) and embedded in epoxy matrix (a = 0.12
mm); b) Using aluminium as the material of the shell and
embedded in epoxy matrix.
4.2
Poisson’s ratio of syntactic PHSS versus fraction of............ 54
hole diameter: a) Using steel (AISI 8000) as the material
of the shell (as a function of the hole diameter, a fraction
of 0 denotes to the shell without holes, with a hole
diameter of d = 1.36 mm) and embedded in epoxy matrix
(a = 0.12 mm); b) Using aluminium as the material of the
shell and embedded in epoxy matrix
4.3
Young’s modulus of syntactic PHSS versus average........... 55
density: a) Using steel (AISI 8000) as the material of the
shell embedded in epoxy matrix (a = 0.12 mm); b) Using
aluminium as the material of the shell and embedded in
epoxy matrix
xvi
4.4
Poisson’s ratio of syntactic PHSS versus average................ 56
density: a) Using steel (AISI 8000) as the material of the
shell embedded in epoxy matrix (a = 0.12 mm); b) Using
aluminium as the material of the shell and embedded in
epoxy matrix
4.5
Elastic properties dependence on the loading direction: .... 57
(a) partially-bonded MHSS, (b) syntactic MHSS
4.6
Numerical compression computation results of .................. 58
the closed-cell Alporas® aluminium
4.7
Young’s modulus of PHSS as a function of the average....... 58
density: (a) primitive cubic arrangement with links; (b)
sintered primitive cubic arrangement
4.8
Thermal conductivity of syntactic PHSS versus fraction of.. 62
hole diameter: a) bonded configuration (a hole diameter
fraction of 0 corresponds to the shell without holes,
a = 0.12 mm); b) sintered configuration
4.9
Thermal conductivity of syntactic PHSS versus average..... 63
density: a) bonded configuration (a hole diameter fraction
of 0 corresponds to the shell without holes, a = 0.12 mm);
b) sintered configuration
4.10
Schematic thermal distribution inside the one-eighth........... 64
of the unit cell of bonded and sintered arrangement;
a) Unit cell with bond gap; b) Sintered unit cell
5.1
The common metallic crystal structures................................ 68
5.2
Table of a milling machine made from hollow..................
69
sphere composite, steel plate and carbon laminates
5.3
Finite element models and robot arms made from.............
(a) aluminium alloy and (b) hollow sphere composite
69
xvii
LIST OF ABBREVIATIONS
CFRP -
Carbon fiber reinforced plastic
MHSS -
Metallic hollow sphere structures
FCC
-
Face-centered cubic
BCC
-
Body-centered
PC
-
Primitive cubic
Hex
-
Hexagonal
SPHB -
Split Hopkinson Pressure Bar
HSS
Hollow sphere structures
-
PHSS -
Perforated hollow sphere structures
SSP
Spherical sphere structures
-
MHSC -
Metallic hollow sphere composites
LMC -
Lattice Monte Carlo
RVE
-
Representative Volume Element
UCs
-
Unit cells
ave
-
Average
ma
-
Matrix
eff
-
Effective
sp
-
Sphere
xviii
Ep
-
Epoxy resin
FEM -
Finite element method
Al
-
Aluminium
St
-
Steel
AISI
-
American Iron and Steel Institute
TPS
-
Transient Plane Source
CT
-
Computed tomography
xviii
LIST OF SYMBOLS
Latin minuscules
a
-
Bond gap (the thickness of epoxy matrix between the spheres)
bs
-
the radius of sintered contact area
dl
-
Perforation hole diameter
ε
-
Strain
εtrans
-
Transverse strain
εaxial
-
Axial strain
∆x
-
Distance between two surfaces
σ
-
Total reaction stress from the applied displacement (MPa)
l
-
Length of specimen
∆l
-
Applied displacement (boundary condition) or displacement
obtained from the applied load
∆lx
-
Displacement resulted from the applied load in x-direction
k
-
Thermal conductivity
ri
-
Shell inner radius
rs
-
Shell outer radius
ro
-
Shell outer radius
t
-
Shell thickness
υ
-
Poisson’s ratio
xix
x
-
Coordinate in the unit space
ρave
-
Average density
ρrel
-
Relative Density
ρso
-
Density of the sphere shell material
xx
Latin capitals
̇
-
Total heat flux
-
Convective heat flux
̇
-
Conductive heat flux
A
-
Area for heat flux calculation
E
-
Young’s modulus (MPa)
K
-
Kelvin
GB
-
Gigabyte
̇
RAM -
Random access memory
T1 , T2 -
Constant temperature boundary conditions
V
-
Volume
Vfree
-
Total volume of the void(s) inside the unit cell
Vma
-
Total volume of the matrix inside the unit cell
Vrel
-
Relative volume of the voids
Vso
-
Total volume of the solid material inside the unit cell
Vsp
-
Total volume of the spherical shell(s) inside the unit cell
VUC
-
Volume of the unit cell
xxi
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
Sample Calculation of the Volume Fraction of the Matrix.......
78
B
Sample Calculation of the average density for composite .......
81
Material
1
CHAPTER 1
INTRODUCTION
1.1
Introduction
The idea of artificial cellular and porous materials originated from nature which
creates structural optimization with respect to weight and load-carrying capacities. Bones,
cork, wood, honeycombs and foams are natural materials to name a few, structured to have
the wonderful properties according to their needs. Due to their unique cellular structure, for
years people have been working on the development of artificial cellular materials in order to
fulfill the potential materials demand in the near future. Starting in 1960s, the geometry of
honeycombs was identically converted into aluminium structures as cores of lightweight
sandwich panels in the aviation and space industries [1]. In 1970, the concept of porous and
cellular metals first emerged [2-4]. The combination of specific mechanical and physical
properties in the cellular materials makes the newfound composite varying from the ordinary
dense metal. Cellular metals are being thoroughly investigated since they have a wide range
of different possible arrangements and forms of cell structures. Open- and closed-type
classical metal foams were illustrated in Figure 1.1 taken from literature [5-6].
2
a)
b)
c)
Figure 1.1: Cellular metals: a) M-Pore® (aluminium sponge); b) Alporas® (aluminium foam);
c) Brass foam [5-6] .
The usage of composite materials in various industries including marine, aerospace
and chemical process plant shows that this alternative material is capable to replace
traditional ferrous materials. Composite materials comprise of the reinforced phase bounded
within a matrix or binder, e.g. Carbon Fiber Reinforced Plastic (CFRP) and Fibre glass. There
are various reinforcing materials in terms of shape such as fibers, whiskers, cloth, braids,
dispersed particles, and flakes [7-9]. For this research project, the characteristic of hollow
spheres immersed in a polymer matrix was investigated.
1.2
Problem Identification
Classical engineering materials utilized in many industrial fields reach their
limitations in properties thus, new developments are required. The increasing demands can be
satisfied in many fields with introducing advanced structured materials. For instance,
syntactic foams are of a promising candidate in this context. The prediction and optimization
of physical properties require the development of accurate and justified computational
models from which constitutive equations and material properties can be derived. By means
of an advanced commercial finite element analysis code, this research has comprehensively
investigated the trend and behavior of hollow sphere structure composites based on
perforated inclusions.
3
1.3
Objective
The primary objective of this thesis is to develop adequate computational models
based on different unit cell approaches. Optimized meshes should be determined based on
mesh refinement analysis. The following physical parameters should be predicted for
different geometrical properties and material sets;
1.4
i.
Average mechanical properties (i.e. elastic properties) and
ii.
Average heat transfer properties (i.e. heat conductivity).
Scope of Study
The scope of this research is as follows:
i.
Generate finite element models for the hollow sphere composites;
ii.
Run simulations for different parameters;
iii.
Evaluation and interpretation of the numerical results.
2010
Jan Feb Mar Apr May Jun
Jul Aug Nov Oct Nov Dec Jan
Literature review
and background study
Modeling & meshing
1st
2nd
3rd c
ration
Preliminary simulation
Full Analysis
Results and discussion
Writing and publishing
Figure 1.2: Gantt chart
2011
Feb Mar Apr May Jun
5
1.6
Summary
This chapter introduces the past and current development on hollow sphere structures.
Initiating with successfully transformed natural honeycombs geometry with aluminium core,
the investigation on the advanced materials continues rapidly with the novel PHSS. The shell
of the HSS with the perforated structure offers a variety of specific mechanical and physical
properties to be explored. The scopes and objective of this research were also highlighted in
this chapter. Last but not least, the Gantt chart for this thesis was also included.
6
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
This chapter covers the literature review on perforated hollow sphere structures which
being the main focus in this thesis. Latest development of porous and cellular materials by a
handful of researchers was also highlighted in this chapter. The knowledge of the literature
review of PHSS, which covers the basic of porous structures, will give the reader deeper
understanding of the following chapters.
2.2
Prospective for Lightweight Cellular Metals
Cellular metals demonstrate a variety of outstanding specific mechanical and physical
properties such as energy absorption [10] and high specific stiffness [11]. In conjunction with
their specific properties, these cellular materials have considerable potential for a vast usage
in the future. Taking metallic hollow sphere structures (MHSS) into practice in automotive
industry, filler of hollow profiles to absorb the impact energy in the car front bumper [12] for
instance, engineers and designers are also aiming for reduction in weight of the vehicle.
Reduction in weight will bring the reduction in fuel consumption and improvement of the
7
driving dynamic hence increasing in passive safety which derives from the energy absorption
capacity at low stress levels. Furthermore, the driving comfort can be increased by acoustic
and thermal insulation in conjunction with the damping of driving vibrations [13]. Cellular
metals are also popular due to their high strength-to-weight ratio and their high gas
permeability combined with a high conductivity [1]. The Venn’s diagram, (cf. Figure 2.1)
shows the summary of the benefits of cellular metals.
High specific stiffness
and strength
Structural
vibration,
Acoustic
damping
Cellular
Metals
Good
Energy
Absorbers
Good Thermal
Conductors
Figure 2.1: Properties of cellular metals (adopted from [14])
2.2.1 High specific stiffness and strength
Cellular metals are engineered to meet the demand of the high-end technology usage
i.e. aerospace, Formula-1 and aero plane. The main criterion of the above mentioned
industries is to have light yet strong material in order to optimize their need. The evolution of
engineering material from the beginning is summarized in Figure 2.2 [15]. The classical
metals experienced a decrease in importance while a rise of advanced composites started by
late 1950 and kept improved by various scientists and engineers up to date. Sanders and
8
Gibson (2003) addressed that face-centered cubic(FCC) stacking hollow sphere structures is
more superior than body-centered (BCC) and primitive cubic (PC) packing in terms of elastic
properties [16]. The following diagram (cf. Figure 2.3) summarises the results.
Figure 2.2: Evolution of engineering materials (adopted from [15])
Figure 2.3: Comparison of different unit cell stacking and their influence on the relative
strength (adopted from [16])
9
2.2.2 Good Energy Absorbers
A study by using the crush tests on 316 L metallic hollow sphere structures have been
conducted by Taşdemirci (2010). The following snapshot, Figure 2.4 from the test by using
Split Hopkinson Pressure Bar (SPHB) shows how cellular materials are being able to deform
up to three-fourth from their original size [17]. It is evident that cellular metals are good
energy absorbers, which can be used to enhance the car safety for both driver and passengers.
Figure 2.4: Deformation of 316L MHS under crush test (adopted from [17])
2.2.3 Good Thermal Conductors
Fiedler et al. (2008) conducted experimental and numerical analyses of thermal
conductivity behavior in adhesively bonded metallic hollow sphere and concluded that the
adhesive in MHSS acts as a good insulator [18]. The temperature distribution in the bonded
and syntactic configuration illustrated in Figure 2.5 indicates that the heat distribution in the
metallic shell is approximately constant.
10
Figure 2.5: Temperature distribution inside the MHSS (adopted from [18])
2.2.4 Structural Vibration
Low density property found in most cellular materials makes the composite suitable to
fit in the component where excessive accelerations occur. Merkel and colleagues (2009) [19]
addressed the vibroacoustic behavior of hollow sphere structure numerically and
experimentally and found a good correlation between both approaches. The mode shapes of
the finite element analysis models are shown in Figure 2.6. Baumeister and Molitor (2009)
[12] had demonstrated the potential usage of metallic hollow sphere composite in industrial
machine parts e.g.: industrial robot arm. In addition, cellular materials are also able to restrain
unnecessary sound and act as an acoustic insulator.
11
Figure 2.6: Mode shapes gained by numerical simulation (adopted from [19])
2.3
Perforated Hollow Sphere Structures
The main idea of introducing holes on the sphere shells is to make the inner sphere and
volume functional [20]. With the perforation in the shells, the matrix used for the composite
will not linger around the shell anymore but, the matter can now fill the inner part of the
sphere thus automatically change the physical and also the specific mechanical properties of
the compound. Instead of using HSS with air cavities as the reinforcing particles in the
matrix, now it is replaced by the perforated hollow sphere with a matrix as a filler. The
connection between the spheres can be simplified as shown in Figure 2.7(a). The formation
that occurred between the spheres and the matrix determines the properties on the
macroscopic level [21].
12
a)
b)
Figure 2.7: a) Connection of single spheres; b) Multi-sphere network (adopted from [21])
The schematic diagram shown in Figure 2.7(b) explains how the hollow spheres can be
joined together to produce composite materials. There are three main procedures of joining
the spheres without perforation, namely sintering, neck bonding and infiltrating the spheres.
Sintering of metallic precursor spheres by applying heat and pressure is the first method [21].
Examples of sintered metal hollow spheres for titanium and 316L [22-23] are stated in the
references. The contact area due to the sintering process is to a certain extent flattened or
even curved resulting in difficulties to generate the computer models. Secondly, the spheres
can be bonded by using a liquid phase which forms a neck region. However, the minimum
distance between the two spheres does not need to be reduced to completely to zero. In [24],
a minimum distance between 0.12 and 0.13 mm is reported for adhesively bonded spheres of
diameter 1.0 mm. The third method of joining the spheres is to infiltrate the interstitial spaces
in a three-dimensional network of hollow spheres with a liquid resin such as epoxy or a low
melting temperature metal [21].
The so-called ‘syntactic’ foams are produced since the spheres are ‘constructed’, as
opposed to the normal foaming process which results in a chaotic arrangement of cavities in a
single material [25]. In this research project, the third method of bonding the metal spheres
was taken into account since the main objective of the research is to investigate how the
inclusions of the matrix in the sphere cavity affect the properties of the cellular metals. It
must be noticed that the final arrangement of the spheres particles was not only periodically
arranged as illustrated schematically in Figure 2.8(c), since in real life the array of the spheres
will be randomly dispersed. Hence, the flattened region and point contact cases is also
13
possible to occur for the PHSS as shown in Figure 2.8(a, b). Figure 2.9 illustrates the
transformation of HSS based on perforated inclusions as the matrix ideally embedded inside
the spheres. The difference between the shell surface of HSS and PHSS is illustrated in
Figure 2.10.
a)
b)
c)
Figure 2.8: Simplified models for sintered hollow sphere structures: a) Flattened contact
region; b) Point contact; c) Syntactic hollow sphere structure (new development) (adopted
from [21])
14
b)
a)
c)
Figure 2.9: Simplified models of PHSS: a) Non-perforated shell; b) Perforation allowing the
matrix to fill the cavity; c) PHSS fully embedded in the matrix (adopted from [21])
a)
b)
Figure 2.10: Single hollow spheres: a) closed surface (common configuration); b) with
perforated surface (new development); (© by Glatt GmbH, Dresden, Germany) (adopted
from [20])
15
2.4
Literature Assessment
Cellular metals demonstrate a variety of outstanding specific mechanical and physical
properties. In conjunction with their special properties, these cellular materials have
considerable potential for the vast usage in the future. Furthermore, driving comfort can be
improved by acoustic and thermal insulation in conjunction with the damping of driving
vibrations [26]. In addition, the cellular metals are also desirable due to their high strength-toweight ratio [27].
Several techniques have been discovered since a century ago in order to manufacture
hollow sphere structures [27]. In recent years, many researchers and engineers carried out
experiments and simulations to come out with low-cost yet viable method to produce hollow
sphere structures in bulk. Palmer et al. (2007) [28] concentrated on the pressure infiltration
technique producing a low-density material with high energy absorption. Augustin and
colleague (2009) [29] addressed the so-called “green spheres” to produce hollow spheres
(HS) and hollow sphere structures (HSS) involving polystyrene core, coating and binder as
the key materials.
In the early times, the so-called spherical sphere structures (SSP) were used as main
reinforcements for polymers. Muliana and Kim (2007) [30] predicted nonlinear viscoelastic
responses of composites reinforced with solid spherical particles by using silicon carbide
embedded in aluminium matrix. Zhang and Ma (2009) [31] proposed the characterisation of
syntactic carbon foam containing hollow carbon microspheres in phenolic resin. While
Dudina and co-workers (2009) [32] have reported that their work on adding metallic glass
into Mg alloy shows in increased mechanical strength without critical loss of ductility instead
of Mg alloy alone.
The researchers continue to discover more opportunities within the attractive hollow
sphere structures, where now the spheres are connected together rather than mixed in the
matrix. Öchsner and Fiedler (2009) [21] discussed two configurations of the spheres
connection. The connection of the spheres can be either bonded with a neck region or in
syntactic formation (the spheres arrangement constructed inside the matrix). This work also
16
described three main procedures of joining the single hollow spheres; sintering, bond-neck
and infiltrating. Starting with investigations on the mechanical properties, Sanders and
Gibson (2003) [16] have reported thoroughly about the mechanics of body-centered cubic
(BCC) and face-centered cubic (FCC). Baumeister and Molitor (2009) [12] used two
different cold hardening epoxide resin systems in preparation of metallic hollow sphere
composite (MHSC) to inspect the Young’s modulus and the tensile strength. Gasser and
colleagues (2003 and 2004) [33-35] conducted both experimental and numerical analyses on
elastic and plastic criterion of regular and periodic stacking of hollow spheres. Speich et al.
(2010) [36] investigated the large plastic deformation behavior of metallic hollow sphere
structures (MHSS) based on the finite element method calibrated with the experimental
results. Later, Fiedler and Öchsner (2008) [37] described the mechanical properties of
adhesively bonded metallic hollow sphere structures (MHSS). Macroscopic properties and
damage mechanisms of stainless steel hollow sphere foam followed by X-ray tomography
have been carried out by Lhuissier and colleagues (2009) [38]. Taşdemirci et al. (2010) [17]
studied the compression behavior of a 316 L metallic hollow sphere (MHS) structure both
numerically and experimentally. The experiment conducted by using a Split Hopkinson
Pressure Bar (SHPB) test apparatus has shown very similar crushing characteristics with
numerical results. Veyhl et al. (2011) [39] addressed qualitatively similar characteristics with
quantitative differences of the macroscopic mechanical properties of the open-cell between
M-Pore® sponge and closed cell Alporas® foam. Hollow sphere structures undergone a rework, where Hosseini et al. (2011) [40] introduced perforation holes in order to open the
inner sphere volume and surface. Then, the numerical analyses of the initial yield surface
were conducted on the so-called perforated hollow sphere structures (PHSS) with primitive
cubic pattern.
The discovery of properties for modern engineering materials is not only done on
basic mechanical properties, but scientists also look into thermal properties where the product
is expected to be used. For this, Elomari and co-workers (1998) [41] used aluminum-matrix
composites containing thermally oxidized SiC particles and found out that the effect of
particle size is quite obvious in the pressure-infiltrated composites: the larger the particles,
the greater the thermal expansion of the composite. Farnsworth et al. (2010) [42] reported
that the Lattice Monte Carlo (LMC) can be used to analyze multiphase materials and the
study was on thermal diffusion in syntactic hollow-sphere structures. Fiedler et al. (2008) [18,
43] studied the thermal conductivity of random and metallic hollow sphere structures by
17
using the Lattice Monte Carlo (LMC) approach for the numerical study and the Transient
Plane Source method (TPS) is applied in order to perform thermal measurements on
experimental samples. They reported that a good agreement of the findings is observed
between numerical and experimental analyses. Solórzano et al. (2009) [44] conducted a
comparative study between experimental and numerical findings on metallic hollow sphere
structures (MHSS). Hosseini and colleagues (2009) [14] predicted the thermal conductivity of
perforated hollow sphere structures (PHSS). Fiedler et al. (2009) [45] extended the previous
investigations by giving special focus to the influence of the sintered contact area and the
relative density of PHSS.
Besides the common properties investigation that has been done on hollow sphere
structures, there are a few investigations which address another angle of exploring this ‘new
type’ composite on other fields. Merkel et al. (2009) [19] conducted a comprehensive
numerical and experimental investigation on vibration and acoustic behaviour of hollow
sphere structures. Vesenjak and colleagues (2009) [46] numerically investigated the dynamic
behaviour of metallic hollow sphere structures and reported that syntactic structures exhibit
distinctly higher value of stiffness and stress in comparison to partial MHSS. While Winkler
et al. (2010) [47] investigated the vibration behavior of adhesively bonded MHSS plates.
Winkler et al. (2010) then conducted an acoustic study regarding the capability of the
metallic hollow sphere structures to bear the sound using the time delay between two signals
and on eigenmode analysis method. Their experimental and numerical simulations gave 0.1%
difference for sound velocity.
In recent companion studies, mechanical and thermal studies have been carried out on
perforated hollow sphere structures without matrix. Here, special focus was given on the
influence of the matrix and the relative density on the thermal properties. In the scope of this
work, three-dimensional finite element analysis was used in order to investigate mechanical
properties and the effective thermal conductivity of primitive cubic (PC) unit cell models of
PHSS embedded in the matrix (i.e. epoxy) and the obtained values were compared to
structures without holes.
18
2.5
Finite Element Method
The term ‘finite element method’ was used for the very first time in 1960 [48],
however the origin of the approach is credited to the work of Castigliano, Maxwell, Navier,
Ritz and others in 19th century. The foundation of the modern finite element method is and
being simultaneously brought by two groups; Argyris and co-workers (1990) [48] in Europe
and Clough et al. (1965) [49] in USA. Figure 2.11 interpreted by Clough (1965) shows that
finite element method in the beginning can solve the structural beam and truss elements and
the method can be further extended by using two- and three-dimensional elements [50].
Figure 2.11: Modeling of structures by gathering discrete elements. (a) One-dimensional
elements. (b) Two-dimensional plane stress elements (adopted from [15])
Nowadays, the finite element method has been approved as a practical engineering
tool and with the help of fast-growing computer-aided technologies, huge numbers of finite
element codes have been generated to simulate the real structure response. Computer
simulation can open the new field of applications of PHSS based on matrix inclusions by
predicting the properties. Simulating the structures numerically is cost-effective and of course
can give us more informative knowledge about the possible abilities of PHSS and its
applications, even before the real specimens become available.
19
2.5.1 Representative Modeling of Cellular Materials
In the case of cellular materials, the influence of edge effects at the free surface of the
specimen to characterize macroscopic values for the huge numbers of cells in the real
arrangement can be avoided by using the “representative volume element” (RVE)
simplification [51]. A RVE of a unit cell of particular porous structures can represent the
whole specimen in order to determine the mechanical properties and their mathematical
characterization. In this approach, a RVE will consist of a minimum amount of unit cells
(UCs) as illustrated in Figure 2.12. By using a unit cell representation for the model, if the
normal strains in the same principal material system applied, the common relationships for
isotropic solids will remain applicable [52]. However, the precise values of the material
parameters cannot be gained by this particular approach since the dependency on the pore
arrangement (i.e. another possibility of lattice configuration such as FCC, BCC and Hex) and
on the direction of the loads applied (i.e. tilted to the x- and y-axes) exists. This modelling
technique has been successfully applied on several investigations for instance, the plastic
behaviour of porous metals explanation (Green, 1972)[53], for the numerical analysis of the
deformation delocalization in metallic foams (Meguid, 2002) [54] and on the elasto-plastic
behaviour of porous metals (Öchsner, 2009) [55].
Figure 2.12: Representative volume element of a simple cubic cell structure
(adopted from [55])
20
2.6
Summary
This chapter elaborates on the literature review of perforated hollow sphere structures
and its prospective properties lie within the interesting behavior. An overview of PHSS
manufacturing and other fields of investigation (i.e. acoustic and dynamic) out of mechanical
and thermal properties was addressed. The potential of the cellular metals to replace
traditional dense metal in the future in terms of their specific mechanical and outstanding
physical properties. The birth of the idea for PHSS was also elaborated and the possible
connection of the hollow spheres was discussed. In addition, this chapter also elaborates on
the fundamentals of mechanical and thermal properties concerning their respective governing
equations involved in this investigation. The deformation of the simulated specimen of PHSS
is also discussed. The last part of this chapter includes the elaboration on the usage of
representative volume elements (RVE) and its rationale in modeling cellular materials.
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
This chapter explains in details about the construction of the perforated hollow sphere
structures which were used in the numerical analysis software. The whole process was
compressed in a flowchart illustrated in Figure 3.1. The chapter covers the modeling concept
and approach, geometry, mesh, applied boundary conditions and the materials used for the
composite. The research project used the commercially available finite element software
MSC.Marc® (MSC Software Corporation, Santa Ana, CA, USA) and the respective
subroutine was used to extract the result of the post-processing models at the macroscopic
level.
3.2
Modeling of PHSS
The size of the system of the equation can be reduced when the so-called unit-cell
approach is chosen where the computational structure is reduced from a larger or infinite
amount of randomly arranged spheres to a single unit-cell which is commonly based on
22
typical space groups as in crystallography [56]. Thus, the analysis was done on a unit cell and
assumptions on the boundary conditions were made so that the nearest possible actual
situation of the cell can be replaced by the computational model.
START
 Geometry generation by
using CAD Software
 Variation of geometrical
parameters
NO
TRANSFER
YES
Finite element software:



Mesh,
Boundary Conditions,
Material
SUBROUTINE
Solution for the system of
equations
Evaluation of results
END
Figure 3.1: Flowchart of the research methodology
23
By referring to the three dimensional (3D) diagram of the syntactic perforated hollow sphere
structure in Figure 3.2, the simplest form of a unit cell can be drawn with symmetrical
condition. As a result, an even cut of one-eighth of the entire sphere is obtained to provide
simplicity in modelling and optimizing the computation time during the analysis.
Figure 3.2: The derivation the one-eighth of a unit cell used in the Finite Element Analysis
(FEA) package
24
Since the aim of this research was to focus on the effect of the inclusions of the matrix phase
in the PHSS cavities, there are three possible configuration of the composite which was
considered as shown in the schematic diagram in Figure 3.3.
a)
b)
2d
d
c)
Figure 3.3: Schematic diagram for the PHSS embedded in the matrix: a) PHSS and matrix
with distance of adhesively bonded neck, d (i.e. 0.12 mm); b) PHSS and matrix with double
distance of adhesively bonded neck, 2d; c) Flattened contact area of PHSS (adopted from
[24])
25
Figure 3.4 shows the exploded view of a unit cell used in this thesis which consists of four
parts namely; a) Inner matrix, b) Metallic sphere shell, c) Plate-shaped matrix, and d) Outer
matrix. Due to the symmetrical behavior of the unit cell of the model, all the geometry, mesh,
boundary conditions and the material applied were only applied on one-eighth of the entire
model. The computer models generated in this project were referred to the previous research
done by Hosseini et al. [20] in terms of size and parameter. The similarities in the scope of
the size of the model i.e. perforation holes, hollow sphere structures and necking distance
allow an easy comparison of the findings although the meshes were differently generated.
Figure 3.5 illustrates the arrangement of a stack of perforated hollow spheres embedded in a
matrix (i.e. epoxy).
d)
c)
b)
a)
Figure 3.4: PHSS composite exploded view: a) Inner matrix; b) PHSS; c) Plate-shaped
matrix; d) outer matrix
26
The model used in this work is a simplified approach which neglects detailed features (where
the miniscule pores in the metal sphere wall resulted from manufacturing process) of the real
specimens cf. 3.20 but allows a first quick prediction of physical properties.
Figure 3.5: Schematic illustration of primitive cubic sphere arrangement of perforated
hollow spheres: a) bonded syntactic perforated hollow sphere structure (PHSS); b) sintered
syntactic PHSS (on the right of each schematic representation, the arrangement of the spheres
in 2D is shown to clarify the distance between adjacent spheres)
27
3.3
Finite Element Approach: Geometry, Mesh, Boundary Conditions and Materials
The assumed cubic symmetry of the primitive cubic arrangement allows that only
one-eighth of a unit cell is modeled in the primary orientation, thus reducing the computation
time. For a three-dimensional meshing, principally tetrahedral or hexahedral elements can be
employed. Previous investigations [24, 57] have shown that hexahedral elements yielded
superior performance. However, due to the geometric complexity, this research used an
automatically generated tetrahedral mesh for sintered spheres, cf. Figure 3.8 (b) and for the
unconnected sphere models, typical hexahedral meshes were generated as in Figure 3.8 (a).
Preliminary investigations on the tetrahedral meshing were performed and it was found that
the used meshes are in the converged region. The entire models composed of matrix and
sphere consist of 29592 to 67080 elements for both configurations, with the largest hole
radius of 0.68 mm. Subsequent models were generated where the size of the perforation hole
was reduced to 75%, 50%, 25% and 0% (i.e. without perforation as limiting case for
comparison) of the maximum radius as shown in Figure 3.11 – 3.13. Figure 3.15 shows the
influence of mesh density on the calculated results for the heat conductivity. It can be seen
that the thermal conductivity is converging to a stable value as the mesh density increases.
Thus, as shown in Figure 3.15(a), a choice of 9000 nodes is reasonable to calculate the heat
conductivity. Higher number of nodes will yield more accurate results but the calculation
time will increase exponentially. In this research the highest number of nodes that is selected
according to the computer with 3.00 GB RAM. Figure 3.15(b) illustrates the significant effect
of mesh density of the model on the smoothness of the curves that contributes to the accuracy
of the calculation. The model on the left in Figure 3.15(b) with less number of nodes has
bigger corners on its curves compared to the smooth profile found on the model with denser
number of nodes. Figure 3.16 shows the convergence trend of the results for the tetrahedral
meshing generated in this paper for sintered configurations. The chosen number of nodes is
14000 where the error will be less than 1%.
28
3.3.1 Geometry
The basic idea of constructing the computer model of the PHSS embedded in the
matrix is to divide them into four major parts, illustrated in the figures will represent the
matrix phase which is assumed to ideally fit in the PHSS cavity. Thus, the shape of the inner
matrix, plate-shaped matrix and the outer matrix will follow the profile of the PHSS as it is.
Each of the major parts then will be merged together in the commercially available finite
element software MSC.Marc®. Thus it is very important to set one reference point of origin.
The point of origin of (0, 0, 0) is set and is taken as a reference point in this research project
to make sure the assembling process will be easy afterwards. The following diagrams (Figure
3.6-3.9) show the dismantled PHSS to clearly indicate the dimensions of the numerical
model.
t
dl
rl
Figure 3.6: Front view of PHSS
29
ri
Figure 3.7: Front view of inner matrix
ro
Figure 3.8: Front view of outer matrix
30
t
Figure 3.9: Isometric view of plate-shaped matrix
The physical parameters of the model generated in this research project are listed in
Table 3.1. Figure 3.10 shows the finite element mesh of the sphere shell and the key
dimensions to build the model.
Table 3.1: Summary of the considered geometry
Dimension
Value [mm]
ri
1.25
rl
1.35
ro
1.47
dl
1.36
bs
0.3
t
0.10
a
0.12
31
Figure 3.10: Finite element mesh of a perforated hollow sphere: a) arrangement with
completely immersed sphere in the matrix; b) sintered arrangement (the matrix mesh is not
shown for clarity)
32
a)
b)
c)
d)
e)
Figure 3.11: Configuration of different hole diameters for bond gap, a (refer to Figure3.17)
starting with arrangement of the PHSS with the smallest hole on the shell surface, hole
diameter is increased from a) HSS (non-perforated configuration) by b) PHSS with hole
diameter equal to 25% of dl, c) PHSS with hole diameter equal to 50% of dl, d) PHSS with
hole diameter equal to 75% of dl and e) PHSS with hole diameter equal to dl = 1.36 mm
33
a)
b)
c)
d)
e)
Figure 3.12: Configuration of different hole diameters for bond gap, 2a arrangement of the
PHSS with the smallest hole on the shell surface, hole diameter is increased from a) HSS
(non-perforated configuration) by b) PHSS with hole diameter equal to 25% of dl, c) PHSS
with hole diameter equal to 50% of dl, d) PHSS with hole diameter equal to 75% of dl and e)
PHSS with hole diameter equal to dl = 1.36 mm
34
a)
b)
c)
d)
e)
Figure 3.13: Configuration of different hole diameters for sintered arrangement with the
smallest hole on the shell surface, hole diameter is increased from a) HSS (non-perforated
configuration) by b) PHSS with hole diameter equal to 25% of dl, c) PHSS with hole
diameter equal to 50% of dl, d) PHSS with hole diameter equal to 75% of dl and e) PHSS
with hole diameter equal to dl = 1.36 mm
35
3.3.2 Mesh
The mesh of the model plays a vital role in gaining reliable results in a finite element
analysis. Every part in the model should have identical yet homogeneous meshing for the best
results. Otherwise, if the meshing is not consistent i.e. in terms of the location of the nodes
contact points, it can mess up the entire calculation done by the computer. For the threedimensional meshing, principally tetrahedral or hexahedral elements can be employed.
However, investigations [24, 57] have shown that hexahedral elements yield superior
performance. Thus, for this research project, three-dimensional mapped-meshing with
approach on the solid surface was used. Mapped-meshing approach may produce elements of
the mesh brick-shaped throughout the model.
Figure 3.14: PHSS shell transformed from solid to three-dimensional Hex-meshing
To have a good mesh, several steps must be followed until the model is divided so
that they have four corners on the surface. A finer mesh is desirable for perpendicular angle
instead of tilted ones as it is not stable. The configurations (cf. Figure 3.14) considered oneeighth of a PHSS divided further to 12 pieces. The division process was not done randomly
since it must be designed so that the symmetrical characteristic remains to reduce the time of
generation of the entire mesh. It is noted that each piece in Figure 3.14 has four corners so
that they can be easily mapped-meshed [58]. A convergence study has been done on the
mock model of the PHSS embedded in the matrix showing that the result will be stable until
36
the number of nodes reaches 9000 and above as shown in Figure 3.15 since the difference is
less than 1%. The regularly subdivided mesh can be observed in Figure 3.15 (b). Figure 3.16
illustrates the convergence study result for the mesh of tetrahedral type applied on the spheres
of sintered arrangement and the Young’s modulus converged at the point of 14000 nodes.
6.0
Conductivity, k [10-4W/(mm∙ K)]
a)
5.5
chosen model
5.0
4.5
4.0
3.5
3.0
2.5
Bonded, 2a
St-Ep
2.0
0
10000
20000
30000
40000
50000
Number of nodes
b)
Figure 3.15: Results of convergence study on mesh refinement; a) the graph shows that the
value of the thermal conductivity became stable when the number of nodes in the meshed
model exceeds 9000 nodes b) on left is the one-eighth of a primitive cubic arrangement
model with less number of nodes and on the right is the model with finer mesh
37
5500
Young's modulus (MPa)
a)
chosen model
5000
4500
4000
3500
Sintered
St-Ep
3000
0
10000
20000
30000
40000
50000
60000
Number of nodes
b)
Figure 3.16: Results of convergence study on mesh refinement for the sintered arrangement
i.e. using automatically generated tetrahedral mesh where the graph shows that the value of
the Young’s modulus became stable when the number of nodes in the meshed model reaches
14000 nodes b) on left is the one-eighth of a primitive cubic arrangement model with less
number of nodes and on the right is the model with finer mesh
38
3.3.3 Boundary Conditions
All considered boundary conditions in this research project were applied only on each
of the outer surface of the meshed model. The so-called periodic boundary condition was
introduced since the term ‘periodic’ represents a periodic continuation of two of the model
perpendicular to the xy- and xz- planes, respectively. Therefore, for these two surfaces all the
nodes on their corresponding surface were tied to one ‘master’ node. This means that all
nodes on this surface will have the same displacement (u = const.). Theoretically, when the
node is tied together, every node on those sides will behave in similar manner or have a
constant deformation with their master. The summary of the boundary conditions is
demonstrated in Figure 3.17. The displacement is applied in y-direction by considering a
value of 0.1 mm. The finite element models of the syntactic perforated hollow sphere
structure consist of two basic configurations where a perforated hollow sphere structure
(PHSS) is mixed with a matrix i.e. an epoxy matrix, in the manufacturing process. For the
first configuration, spheres without directly contacting each other are arranged in a primitive
cubic periodic pattern where a minimum distance (matrix gap) of length 2a between adjacent
spheres occurs (due to symmetry, the length shown in Figure 3.17 (a) is equal to 1a). In
addition, this first configuration was modified by the option that the minimum distance
between adjacent spheres can vary and in a second set of computational models this distance
was increased by a factor two. Thus, a similar arrangement as shown in Figure 3.15 (a) was
used where the distance was increased to 2a. The second possible configuration consists of
directly connected spheres as illustrated in Figure 3.17 (b) where flattened contact areas
between the spheres were created due to a joining sintering process. Due to the symmetry of
the applied load and the geometry, reflective boundary conditions (ux = uy = uz = 0) were
applied at three perpendicular surfaces, i.e. along the symmetry planes of the singles sphere.
The influence of neighboring cells in the primitive cubic pattern is considered by the
so-called repetitive boundary conditions. Reflective boundary condition means that every
node located on the described surface is constrained to not move in the direction of the
surfaces’ normal vector and repetitive boundary condition means that every node located on a
linked surface must experience the same displacement perpendicular to the surface [59]. The
thin lines on the right side in Figure 3.17 and Figure 3.18 represent the ‘periodic’
continuation of the spheres in PHSS packing. For the heat transfer analysis, zero heat flux or
39
( ̇ = 0) is imposed on the left and right wall of the one-eighth of a unit cell and T1 and T2 on
the bottom and top surfaces is set to 300K and 320K respectively.
Figure 3.17: Finite element mesh and applied boundary conditions of a syntactic perforated
hollow sphere cubic unit cell: a) arrangement with completely immersed sphere in the matrix;
b) sintered arrangement (the darker grey elements belong to the sphere while the lighter grey
elements belong to the matrix)
40
Figure 3.18: Finite element mesh and applied boundary conditions of a syntactic
perforated hollow sphere cubic unit cell: a) arrangement with completely immersed
sphere in the matrix; b) sintered arrangement (the darker grey elements belong to the
sphere while the lighter grey elements belong to the matrix)
41
A more realistic model for the spheres without perforation holes can be
obtained by assuming that the inner volume of the spheres remains empty, i.e.
without any reinforcing matrix. This is schematically shown in Figure 3.19 (b) and
(d).
Figure 3.19: Schematic diagram to indicate the difference between perforated and
empty shell syntactic hollow sphere structures: a) perforated sphere with bond gap, b)
empty sphere with bond gap, c) sintered perforated sphere, and d) sintered empty
sphere
42
3.3.4 Materials
The materials used in this project research are the same with the one used by
Hosseini et al. [60] in order to compare the findings. The summary of the materials is
tabulated in Table 3.2 and Table 3.3.
Table 3.2: Specific mechanical properties of the spheres and binder [13]
Material
Young’s
Poisson’s ratio
Density
modulus (MPa)
(-)
(kg/dm3)
Steel
110000
0.30
6.95
Aluminium
58820
0.34
2.67
Epoxy
2460
0.36
1.13
Table 3.3: Heat conductivity properties of base materials. [18, 20]
Material
Conductivity [W/(mm∙ K)] Density (kg/dm3)
Steel
0.05
6.95
Aluminium
0.232
2.67
0.000214
1.13
Epoxy
43
Figure 3.20: a) Micrograph of a completely single hollow sphere; b) Micrograph
with several hollow spheres; c) Micrograph of the wall of a hollow sphere (the
micrographs are recorded and analysed with the Zeiss AxioVision 4.6.3.Sp1
Software in Aalan Univ. Germany). (adopted from [28])
As shown in Figure 3.20, one can see that the wall of hollow spheres is
porous. This porosity is the result of vaporing of EPS (Expanded Poly Styrol)
granulates from inside of hollow sphere during manufacturing. Therefore, reduced
values taken from literature [40]have been used for the base materials properties. It
has been shown that the behaviour of the base material in wall thickness of PHSS is
isotropic [29].
44
3.4
Mechanical Properties
3.4.1 Young’s Modulus and Poisson’s Ratio
Young’s Modulus The ratio of the stress over strain is known as Young’s
modulus. This is known as Hooke’s law, introduced by Robert Hooke back in 1678
[61], and the constant of proportionality E (GPa or equivalent to psi) is the Young’s
modulus .
Stress
Unload
Slope = E
1
Load
Strain
Figure 3.21: Schematics stress-strain diagram showing linear elastic deformation for
loading and unloading cycles [30]
( ⁄ )
= = (∆ ⁄ )
Where,
σ
= the total reaction force from the applied displacement
divided by the cross-sectional area
l
= the initial length of the specimen
∆l
= applied displacement boundary condition
F
= summation of reaction forces
ε
= strain
[3.1]
45
Poisson’s ratio The negative ratio of the fraction of contraction (strain) and
the fraction of elongation (strain) is called Poisson’s ratio. The Poisson’s effect is
named after the contraction and elongation ratio phenomenon. Theoretically,
Poisson’s ratio for isotropic materials should be in the range of 0 to 0.5. Furthermore,
the maximum value for (or that value for which there is no net volume change) is
0.50 [62]. For many metals and metal alloys, the value range is between 0.25 and
0.35 as can be found in literature [63-65]. The three-dimensional representation of a
cube constructed in Figure 3.22 demonstrates the existence of transverse and axial
strains whenever a metallic block is elongated.
= −
d
d
=−
d
d
=−
d
d
[3.2]
Where,
= The resulting Poisson’s ratio
d
= The transverse strain [negative for axial tension
(stretching), positive for axial compression i.e. y and zdirection]
d
= The axial strain [positive for axial tension, negative
for axial compression i.e. x-direction]
∆l
= applied displacement
46
Figure 3.22: The graphical interpretation of the Poisson’s ratio
3.4.2 Deformation of the Model
The original and deformed model is shown in Figure 3.23. The colors show
the difference of the reaction force experienced by the nodes at a different location.
The lighter grey region represents the higher reaction force exerted onto the area. All
the models show a similar pattern of deformation thus proving the consistency of the
boundary conditions applied. It can be concluded that the surface contacted with x-,
y- and z-plane remains to their respective planes (symmetry boundary condition) and
just elongates to y-direction since the displacement of 0.1 mm was applied in this
analysis. In Figure 3.23, it is noted that there are two faces (i.e. xy- and yz- faces)
contracted constantly since periodic boundary conditions were applied on the
respective surfaces.
47
Figure 3.23: Original and deformed model with bond gap arrangement
3.5
Thermal Properties
Thermal Conductivity Dated back to 1807, Jean Baptiste Joseph Fourier has
given a contribution in formulating the transient process of heat conduction,
described by a differential equation [66]. As shown in Figure 3.24, the heat can be
transferred through three mechanism (i.e. thermal conduction, ̇
convection, ̇
; and thermal radiation, ̇
). The total heat flux, ̇
; thermal
(the rate of
heat energy transfer through a given surface) can be determined in terms of the outer
temperatures (i.e. T1 and T4), providing the effective thermal conductivity of the
48
system. A in equation 3.3 denotes the area of the control surface and ∆x is the
distance between these surfaces.
=
̇
∆
∙
[3.3]
Figure 3.24: The heat transfer principles interpreted into electrical circuits [13]
In this thesis, we only investigated the thermal conductivity of PHSS based on matrix
inclusions where the significant conduction in solids is based on lattice vibration and
movement of free electrons.
Thus, the equation to compute the conduction of
thermal energy based on Fourier’s law is as follows;
̇
=− ∙
∆
∆
where the parameter k is known as thermal conductivity.
[3.4]
49
3.6
The Average Density for Composite Materials
From the relative density equation proposed by Gibson and Ashby (1997)
[67], the average density of cellular materials can be computed through following
formulae. For the cellular materials, physical properties are usually described as a
function of their relative density.
=
Where,
(3.5)
: The density of the cellular material
: The density of the sphere shell material
In literature [11], the scaling relations or power-laws for different physical properties
were discussed. In this thesis, both the masses of the solid material and cellular
materials is identical; the RHS of equation 3.5 is divided by the mass to state the
relative density in terms of volume. The equation is also known as the relative solid
volume.
=
Where,
(3.6)
: The volume of the solid material (e.g. the spheres, the bond
material)
: The volume of a unit cell
50
The overall volume of a unit cell in HSS in this study including the solid and free
volume can be expressed as:
=
+
(3.7)
By referring to Figure 3.3, the solid volume consists of the perforated sphere shells
(index ‘sp’) and the bond material (index ‘ma’). Now the total volume of a unit cell
can be calculated by:
=
+
(3.8)
It must be noted on the assumption made that the densities of the sphere shell and the
matrix (bond) material are identical, (i.e. ρso = ρsp = ρma) hence, the relative density is:


=



=
+

(3.9)


In the case that the sphere shell material and the bond material are different
(e.g. steel and epoxy), instead of relative density which is no longer possible, an
average density, ρ can be used:
=
∙
+
∙
(3.10)
51
3.7
Summary
This chapter highlights the development of the PHSS model generated by
computer aided drawing (CAD) software; its simplification and assumptions. The
possible arrangement of the hollow spheres in the matrix is also explained. A set of
models was generated using the commercial available finite element software
MSC.Marc®. The overview of the basic criteria of FEA approach including the
considered geometry, mesh, applied boundary conditions and the base materials
chosen are also treated. The chapter ends up with the derivation of average density
for common cellular materials.
52
CHAPTER 4
RESULTS AND DISCUSSION
4.1
Introduction
The numerical model of the PHSS made of steel and aluminium spheres
embedded in the epoxy matrix has undergone the numerical analysis. A set of
numerical results is presented in this chapter. The results gained are compared to the
previous companion work including the discussions on the trend of the graphs. The
details of the findings of the simulation are discussed further in the sub-chapter.
4.2
Young’s Modulus and Poisson’s Ratio
Figures 4.1 and 4.2 summarize the macroscopic Young’s modulus and
Poisson’s ratio of both cases i.e. bonded and sintered arrangements of perforated
hollow sphere structures for steel-epoxy and Aluminium-epoxy composites. It can be
seen that Young’s modulus (cf. Figures 4.1a and 4.1b), increases as the hole diameter
53
decreases. This can be explained by the fact that the stiffer metal is more replaced by
the softer epoxy matrix.
6000
a)
Sintered
Young's modulus (MPa)
5500
5000
Bond gap, a
4500
4000
3500
Bond gap, 2a
3000
St-Ep
2500
2000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fraction of hole diameter
Young's modulus (MPa)
b)
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
Sintered
Bond gap, a
Bond gap, 2a
Al-Ep
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fraction of hole diameter
Figure 4.1: Young’s modulus of syntactic PHSS versus fraction of hole diameter: a)
Using steel (AISI 8000) as the material of the shell (as a function of the hole
diameter, a fraction of 0 denotes to the shell without holes, with a hole diameter of d
= 1.36 mm) and embedded in epoxy matrix (a = 0.12 mm); b) Using aluminium as
the material of the shell and embedded in epoxy matrix.
54
0.350
a)
Bond gap, 2a
Sintered
Poisson's ratio
0.345
0.340
Bond gap, a
0.335
0.330
0.325
St-Ep
0.320
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fraction of hole diameter
0.355
b)
Poisson's ratio
0.350
Sintered
0.345
Bond gap, 2a
0.340
Bond gap, a
0.335
0.330
0.325
Al-Ep
0.320
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fraction of hole diameter
Figure 4.2: Poisson’s ratio of syntactic PHSS versus fraction of hole diameter: a)
Using steel (AISI 8000) as the material of the shell (as a function of the hole
diameter, a fraction of 0 denotes to the shell without holes, with a hole diameter of d
= 1.36 mm) and embedded in epoxy matrix (a = 0.12 mm); b) Using aluminium as
the material of the shell and embedded in epoxy matrix
55
6000
a)
Sintered
Bond gap, a
Bond gap, 2a
Young's modulus (MPa)
5500
5000
4500
4000
3500
3000
St-Ep
2500
2000
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Average density (kg/dm3)
4500
Young's modulus (MPa)
b)
Sintered
Bond gap, a
Bond gap, 2a
4000
3500
3000
2500
Al-Ep
2000
1.16
1.18
1.20
1.22
1.24
1.26
1.28
1.30
1.32
Average density (kg/dm3)
Figure 4.3: Young’s modulus of syntactic PHSS versus average density: a) Using
steel (AISI 8000) as the material of the shell embedded in epoxy matrix (a = 0.12
mm); b) Using aluminium as the material of the shell and embedded in epoxy matrix
56
0.350
a)
Poisson's ratio
0.345
0.340
0.335
0.330
Sintered
Bond gap, a
Bond gap, 2a
0.325
St-Ep
0.320
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Average density (kg/dm3)
b)
0.355
Poisson's ratio
0.350
0.345
0.340
Sintered
Bond gap, a
Bond gap, 2a
0.335
Al-Ep
0.330
1.15
1.20
1.25
1.30
1.35
Average density (kg/dm3)
Figure 4.4: Poisson’s ratio of syntactic PHSS versus average density: a) Using steel
(AISI 8000) as the material of the shell embedded in epoxy matrix (a = 0.12 mm); b)
Using aluminium as the material of the shell and embedded in epoxy matrix
57
Comparing the both bond gap and sintered configurations, it is evident that the
directly connected sphere arrangement is clearly stiffer than the spheres with a bond
gap. Furthermore, the entire stiffness of the structure decreases as the bond gap
increases (bond gap a to 2a). The pattern of the graphs plotted (cf. Figure 4.1)
correlates well with the findings in [37, 39, 59] in terms of the trend as shown in the
following Figures 4.5, 4.6 and 4.7.
Figure 4.5: Elastic properties dependence on the loading direction: (a)
partially-bonded MHSS, (b) syntactic MHSS (adopted from [37])
58
Figure 4.6: Numerical compression computation results of the closed-cell Alporas®
aluminium foam (adopted from [39])
Figure 4.7: Young’s modulus of PHSS as a function of the average density: (a)
primitive cubic arrangement with links; (b) sintered primitive cubic arrangement
(adopted from [60])
59
The trend of Poisson’s ratio as shown in Figure 4.2 is opposed to the general
behavior of Young’s modulus. As such, an increase of the hole diameters increases
Poisson’s ratio. This comes from the fact that with increasing hole diameter more
matrix is accumulated in the unit cell. The trend of the results obtained follows the
study conducted by Hosseini et al. [37, 59] on ‘neck bonded’ PHSS with primitive
cubic arrangement. In both diagrams (cf. Figures 4.1 and 4.2), the behavior of
sintered and bonded structures exhibits a linear tendency. The filled markers on the
vertical axis are shown to indicate the results of the sphere shells with no matrix
inside in order to compare with these new type configurations. The schematic
diagram to show the difference between the filled and empty shell is illustrated in
Figure 4.1 and Figure 4.2.
The values gained from the simulation of the empty shell are relatively lower
in the case of elastic modulus compared to the shell filled with the matrix. In
contrary, in the case of Poisson’s ratio, the result of the empty shells is slightly higher
than that of the shells accumulated with the matrix. Thus, from this observation, it
can be concluded that the matrix elements clearly stiffen the perforated hollow
sphere structures. Figures 4.3 and 4.4 show the influence of the average density (hole
diameter) and different joining methods on Young’s modulus and Poisson’s ratio for
steel-epoxy and Aluminium-epoxy combinations. It can be seen from Figures 4.3(a)
and 4.3(b), the Young’s modulus increases as the average density increases (hole
diameter decreases) for all considered joining techniques. Moreover, the behavior of
these structures seems to be nearly continuous in terms of strength from sintered to
bond gap, a and bond gap 2a arrangement.
While for the Poisson’s ratio, the values decrease with increasing average
density (decreasing hole diameter) (cf. Figure 4.4). The structures with bond gap
arrangements behavior tend to follow a linear shape whereas the sintered structures
follow some kind of linear trend with a slight different of the values. In order to
avoid a discontinuity in the distribution, the spheres without a perforation hole are
still assumed to be filled with the epoxy matrix. This assumption can be questioned
from the point of manufacturing but results in a much smoother curve for the limiting
case. For the mechanical properties, the rule of mixtures [68] of PHSS composite is
applied to obtain the value of Young’s modulus, E in the beginning of this work. This
calculation also useful to check whether numerical approach is needed or not since,
60
the rule of mixtures itself can be used to calculate the Young’s modulus of large
particle composites. From this rule, we can calculate the value of E of the composite
(Ec) in terms of elastic moduli of the matrix phase (Em) and the particulate or metallic
sphere phase (Ep), by using the following formulation;
=
+
(1 −
)
[4.1]
where vm is the volume fraction of the matrix phase. However, this equation is not
applicable for this complex geometry composite since the Young’s modulus value
obtained seems to be too far from the numerical values.
4.2
Thermal Conductivity
In the following, the results of the finite element analyses are shown. Figure
4.8 summarises the macroscopic conductivity of syntactic perforated hollow sphere
structures for bonded and sintered arrangements. It can be seen in Figure 4.8(a)
(configurations with different bond gaps) that there is dominant influence of the
matrix and the macroscopic thermal conductivity of the composite is in the same
order of magnitude as the thermally isolating matrix. Different sphere materials, i.e.
aluminium or steel, or the size of the bond gap, i.e. a or 2a, have only a minor
influence on the macroscopic conductivity and the entire behavior is determined by
the matrix material. The situation is completely opposed in the case of sintered
configurations, cf. Figure 4.8(b), where relatively higher conductivity values are
obtained since the heat can easily flow through the interconnected metallic spheres.
For all the configurations, i.e. bonded or sintered, the same trend is obtained where
the thermal conductivity decreases with increasing perforation diameter. This
decrease in conductivity comes from the fact that with increasing hole diameter,
more matrix is accumulated in the unit cell, thus enforcing the insulating effect. The
filled markers in Figure 4.8 at a hole fraction of zero (i.e. spheres without
61
perforation) indicate the results for sphere shells with no matrix inside or in other
words for ‘empty’ spheres. These models were additionally created to investigate if a
filling matrix inside the hollow spheres significantly influences the heat conductivity
for unperforated spheres. As can be seen in Figure 4.8, the empty (represented by
solid markers) and filled unperforated spheres reveal practically the same
conductivity which is a direct result of the assumed matrix, i.e. a thermal isolator.
Figure 4.9 shows the same results but in a more common graphical representation in the
context of composite materials. Here, the thermal conductivity is represented as a
function of the average density (densities of the single components are weighted with
their volume fraction to calculate the average density value). It can be seen in both
diagrams of Fig 4.9 that the thermal conductivity increases now as the average density
increases (i.e. hole diameter decreases). The numerical result obtained in this thesis
shows a good agreement with the existing results in [20, 44]. Comparing the graph of
thermal conductivity versus the hole diameter fraction and the thermal conductivity
versus the average density, similar conclusions can be taken in regards to the
influence of the joining technique and sphere shell material. Let us highlight at the
end of this section that the two investigated configurations, i.e. where either the
spheres are not in direct contact or directly connected due to a sintering process, do
not give - from a practical point of view - a significant difference in the overall
weight but can change the macroscopic thermal conductivity by several orders of
magnitude. This offers a great opportunity to design and tailor an advanced
structured material according to different requirements, i.e. thermally isolating or
conducting.
62
Figure 4.8: Thermal conductivity of syntactic PHSS versus fraction of hole
diameter: a) bonded configuration (a hole diameter fraction of 0 corresponds to the
shell without holes, a = 0.12 mm); b) sintered configuration.
63
a)
b)
Figure 4.9: Thermal conductivity of syntactic PHSS versus average density: a)
bonded configuration (a hole diameter fraction of 0 corresponds to the shell without
holes, a = 0.12 mm); b) sintered configuration.
64
The schematic temperature distribution as shown in Figure 4.10 illustrates how the
steady state temperature gradient varies inside bonded and sintered PHSS. Due to the
high thermal conductivity in the metallic base material (i.e. sphere shell), the
temperature surrounding the metallic shell is nearly constant for the bond gap
arrangement. Meanwhile for the sintered arrangement, the temperature is also
distributed within the sphere shell since a small portion of the inter-connected shell is
Figure 4.10: Schematic thermal distribution inside the one-eighth of the unit cell of
bonded and sintered arrangement; a) Unit cell with bond gap; b) Sintered unit cell.
65
exposed to the heat. The diagram visualizes that for bond gap arrangement, the
isotherm (boundaries between the two contrast grey areas) occur with short space
indicating that the adhesive (i.e. epoxy) acts as a good insulator. A similar trend was
found in [18] by Fiedler et al. where the metallic hollow sphere structures used by
them consists of steel and epoxy as the base materials.
4.4
Summary
This chapter elaborates on the mechanical properties and the deformation
behavior of the models after the considered geometry, materials and boundary
conditions applied. Then, the numerical results of Young’s modulus and Poisson’s
ratio of the composites are comprehensively discussed. Both steel and aluminium
spheres combined with epoxy as the matrix phase give identical pattern of
mechanical and thermal properties.
66
CHAPTER 5
CONCLUSION AND FUTURE STUDY
5.1
Conclusions
Mechanical Properties Hollow sphere structures based on perforated
inclusions have been examined numerically to determine the Young’s modulus and
the Poisson’s ratio. The findings in this work were compared to the previous
companion study as the dimensions and the base materials used are the same as in
literature [59-60, 69]. It can be seen that from the comparison, the value of the
Young’s modulus obtained in this study for all cases (i.e. bond gap, a; bond gap, 2a;
and sintered arrangement) is slightly higher than what is reported by Öchsner et al.
(2010). In addition, the value of the Young’s modulus is proportional to the size of
the perforation hole diameter and the average density, whereas the bigger the hole the
weaker the composites will be. As a conclusion, the matrix component accumulated
inside the sphere via perforation holes helps to stiffen the structures. In contrast, the
value of the Poisson’s ratio is lower compared to the result of PHSS without matrix
inclusions (i.e. empty PHSS). Similar to the Young’s modulus, the result of the
Poisson’s ratio follows the linear pattern when it is plotted against the size of the
perforation hole and average density. In addition, the results gained by using different
67
combination of the base material (i.e. Steel-epoxy and aluminium-epoxy) give a
similar tendency for parameters namely Young’s modulus and Poisson’s ratio.
Thermal Properties The thermal conductivity of the PHSS with matrix
inclusions was studied and compared to the work done by Hosseini et al. [20, 69]. It
can be seen that a good agreement were shown between the both results. As a
consequence, increasing in the size of the perforation hole (average density
increases), the thermal conductivity of the composite decreases. It can be concluded
that, the matrix component does not only bind the spheres, but also strengthen the
insulating effect.
5.1
Future Study
Metallic Crystal Structures As discussed with Professor Dr. –Ing. Andreas
Öchsner, there is still a lot more research that can be done on the PHSS in various
fields. Investigations of HSS based on perforated inclusions can be further extended
to different periodic arrangements (cf. metallic crystal structures). The study of PHSS
can cover different model structures taken from common crystal lattices including
FCC, BCC, Hex (cf. Figure 5.1) and the real structure obtained from microscopic
Computed Tomography (CT) scans.
68
PC (done)
FCC
BCC
HEX
Figure 5.1: The common metallic crystal structures
PHSS Applications The lightweight and low strength to weight ratio
properties found in PHSS will open a vast field of its applications. As discussed
before in Chapter 2, cellular and porous materials can be used as crash absorbers,
thermal isolators, lightweight constructions and also vibration absorbers in machine
part with excessive accelerations. The studies conducted by Baumeister et al. (2004)
have shown that metal foams can be utilized as machine building parts (i.e. robot arm
and milling machine table). Many parts have been produced with good accuracy and
tested with good results [70-71]. Example of the parts built by Baumeister at al.
illustrated in Figures 5.2 and 5.3.
69
Figure 5.2: Table of a milling machine made from hollow sphere composite, steel
plate and carbon laminates (adopted from [70])
Figure 5.3: Finite element models and robot arms made from (a) aluminium alloy
and (b) hollow sphere composite (adopted from [71])
70
Experimental Investigations The novel perforated hollow sphere structures
are currently under development. So far, there is no available experimental data on
PHSS with matrix inclusions to compare with the simulated findings. However,
Fiedler et al. (2008) have carried out three different methodologies (i.e. experimental,
numerical and analytical) on the thermal conductivity of metallic hollow sphere
structures and showed a good agreement between the results gained [18]. In addition,
the cost of experimental studies can be reduced by simulating the new type by
numerical analysis. The numerical study could also open vast possible applications
and properties prediction of PHSS via simulations where extensive information can
be obtained. Nevertheless, experimental test is indispensably necessary and is still
under development on PHSS. There is an excellent prospective to explore PHSS
based on matrix inclusions in this particular field concerning the mechanical, thermal
and material and their applications.
71
References
[1]
A. Öchsner and C. Augustin, "Introduction," in Multifunctional Metallic
Hollow Sphere Structures: Manufacturing, Properties and Application, ed:
Springer-Verlag Berlin Heidelberg, 2009, pp. 1–3.
[2]
N. N., "Metal foam," Composites Science and Technology, vol. 2, p. 11.,
1971.
[3]
H. Bray, "Design opportunities with metal foam," Engineering Materials and
Design, vol. 16, pp. 16-19, 1972.
[4]
R. R. Barton, et al., "Use of reticulated metal foam as flash-back arrestor
elements" Chemical Engineer - London, vol. 291, pp. 708-708, 1974.
[5]
J. Banhart, "Manufacture, characterisation and application of cellular metals
and metal foams," Progress in Materials Science, vol. 46, pp. 559–632, 2001.
[6]
F. Topin, et al., "Contents: Adv. Eng. Mater. 9/2006," Advanced Engineering
Materials, vol. 8, pp. 767-772, 2006.
[7]
A. K. Kaw, Mechanics Composite Materials Second Edition. Boca Raton:
CRC Press, Taylor & Francis Group, 2006.
[8]
H. Altenbach, et al., Mechanics of Composites Structure Elements. Berlin:
Springer, 2004.
[9]
E. G. Wolff, Introduction to the Dimensional Stability of Composite
Materials. Lancaster: DEStech Publications, Inc., 2004.
[10]
A. G. Evans, et al., ""Multifunctionality of cellular metal systems", ,"
Progress in Mtls. Sci., vol. 43, pp. 171-221, 1998.
[11]
M. F. Ashby, et al., Metal foams: a design guide vol. 23. Oxford, UK:
Butterworth-Heinemann, 2000.
72
[12]
E. Baumeister and M. Molitor, "Hollow Spheres in Composite Materials and
Metallic Hollow Sphere Composites (MHSC)," in Multifunctional Metallic
Hollow Sphere Structures: Manufacturing, Properties and Application, ed:
Springer-Verlag Berlin Heidelberg, 2009, pp. 109–136.
[13]
T. Fiedler, "Numerical and Experimental Investigation of Hollow Sphere
Structures in Sandwich Panels," Ph.D Thesis, University of Aveiro, Portugal,
2007.
[14]
S. M. H. Hosseini, et al., "Finite element simulation of the thermal
conductivity of perforated hollow sphere structures (PHSS): Parametric
study," Materials Letters, vol. 63, pp. 1135–1137, 2009.
[15]
W. O. Soboyejo, "Introduction to Advanced Materials," in Advanced
Structural Materials: Properties, Design Optimization, and Applications, ed
New York: Taylor & Francis Group, LLC, 2006.
[16]
W. S. Sanders and L. J. Gibson, "Mechanics of BCC and FCC hollow-sphere
foams," Materials Science and Engineering A, vol. 352 pp. 150-161, 2003.
[17]
A. Taşdemirci, et al., "Split Hopkinson pressure bar multiple reloading and
modeling of a 316 L stainless steel metallic hollow sphere structure,"
International Journal of Impact Engineering, vol. 37, pp. 250–259, 2010.
[18]
T. Fiedler, et al., "Numerical and experimental analysis of the thermal
conductivity of metallic hollow sphere structures," Materials Letters, vol. 62,
pp. 1204–1207, 2008.
[19]
M. Merkel, et al., "On the Vibroacoustic Behaviour of HSS," in
Multifunctional
Metallic
Hollow
Sphere
Structures:
Manufacturing,
Properties and Application, ed: Springer-Verlag Berlin Heidelberg, 2009, pp.
71-88.
[20]
S. M. H. Hosseini, et al., "Finite element simulation of the thermal
conductivity of perforated hollow sphere structures (PHSS): Parametric
study," Materials Letters, vol. 63, pp. 1135–1137, 2009.
[21]
A. Öchsner and T. Fiedler, "Geometrical Properties of Hollow Sphere
Structures,"
in Multifunctional
Metallic
Hollow Sphere
Structures:
Manufacturing, Properties and Applicationn, ed: Springer-Verlag Berlin
Heidelberg, 2009, pp. 31–46.
73
[22]
O. Anderson, et al., Novel metallic hollow sphere structures: processing and
properties. In: Banhart, J., Ashby, M.F., Fleck, N.A. (eds.) Metal foams and
porous metal structures. Bremen: MIT Publishing, 1999.
[23]
O. Anderson, et al., "Novel Metallic Hollow Sphere Structures," Advanced
Engineering Materials, vol. 2, pp. 192-195, 2000.
[24]
T. Fiedler, et al., "Modelling the mechanical behaviour of adhesively bonded
and sintered hollow-sphere structures," Mechanics of Composite Materials,
vol. 42, pp. 559-570, 2006.
[25]
A. R. Luxmoore and D. R. J. Owen, Syntactic foams. New York: Macmillan
Publishing Co., 1982.
[26]
T. Fiedler, "Numerical and Experimental Investigation of Hollow Sphere
Structures in Sandwich Panels," Ph.D, University of Aveiro, Portugal, 2007.
[27]
C. Augustin and W. Hungerbach, "History and Production of Hollow
Spheres,"
in
Multifunctional
Metallic
Hollow
Sphere
Structures:
Manufacturing, Properties and Application, ed: Springer-Verlag Berlin
Heidelberg, 2009, pp. 5–30.
[28]
R. A. Palmer , et al., "Pressure infiltrated syntactic foams—Process
development and mechanical properties," Materials Science and Engineering
A, vol. 464, pp. 85–92, 2007.
[29]
C. Augustin and Wolfgang Hungerbach, "Production of hollow spheres (HS)
and hollow sphere structures (HSS)," Materials Letters, vol. 63, pp. 1109–
1112, 2009.
[30]
A. H. Muliana and J. S. Kim, "A concurrent micromechanical model for
predicting nonlinear viscoelastic responses of composites reinforced with
solid spherical particles," International Journal of Solids and Structures, vol.
44, pp. 6891–6913, 2007.
[31]
L. Zhang and J. Ma, "Processing and characterization of syntactic carbon
foams containing hollow carbon microspheres," CARBON, vol. 47, pp. 14511456., 2009.
[32]
D. V. Dudina, et al., "A magnesium alloy matrix composite reinforced with
metallic glass," Composites Science and Technology vol. 69, pp. 2734-2736,
2009.
[33]
S. Gasser, et al., "Uniaxial tensile elastic properties of a regular stacking of
brazed hollow spheres," Scripta Materialia, vol. 48, pp. 1617–1623, 2003.
74
[34]
S. Gasser, et al., "Finite elements computation for the elastic properties of a
regular stacking of hollow spheres," Materials Science and Engineering A,
vol. 379, pp. 240–244, 2004.
[35]
S. Gasser, et al., "Microplastic yield condition for a periodic stacking of
hollow spheres," Scripta Materialia, vol. 50, pp. 401–405, 2004.
[36]
M. Speich, et al., "Large deformation of core materials for sandwich panels.
Große
Verformung
von
Kernmaterialien
für
Sandwichelemente,"
Materialwissenschaft und Werkstofftechnik, vol. 41, pp. 270-275, 2010.
[37]
T. Fiedler and A. Öchsner, "On the anisotropy of adhesively bonded metallic
hollow sphere structures," Scripta Materialia, vol. 58, pp. 695–698, 2008.
[38]
P. Lhuissier, et al., "Quasistatic mechanical behaviour of stainless steel
hollow sphere foam: Macroscopic properties and damage mechanisms
followed by X-ray tomography," Materials Letters, vol. 63, pp. 1113–1116,
2009.
[39]
C. Veyhl, et al., "Finite element analysis of the mechanical properties of
cellular aluminium based on micro-computed tomography," Materials
Science and Engineering A, vol. 528, pp. 4550–4555, 2011 2011.
[40]
S. M. H. Hosseini, et al., "Numerical investigation of the initial yield surface
of perforated hollow
sphere structures (PHSS) in a primitive cubic pattern," Finite Elements in Analysis
and Design, vol. 47, pp. 804–811, 2011.
[41]
S. Elomari, et al., "Thermal Expansion Behavior of Particulate Metal-Matrix
Composites," Composites Science and Technology, vol. 58, pp. 369-376,
1998.
[42]
J. Farnsworth, et al., "A lattice Monte Carlo analysis on thermal diffusion in
syntactic
hollow-sphere
thermischer
Diffusion
structures.
in
‘Lattice
syntaktischen
Monte
Carlo’
Analyse
Hohlkugelstrukturen,"
Materialwissenschaft und Werkstofftechnik, vol. 41, pp. 283-286, 2010.
[43]
T. Fiedler, et al., "Numerical analyses of the thermal conductivity of random
hollow sphere structures," Materials Letters, vol. 63, pp. 1125–1127, 2008.
[44]
E. Solórzano, et al., "Thermal conductivity of metallic hollow sphere
structures: An experimental, analytical and comparative study," Materials
Letters, vol. 63, pp. 1128–1130, 2009.
75
[45]
T. Fiedler, et al., "A refined finite element analysis on the thermal
conductivity of perforated hollow sphere structures," Computational
Materials Science, vol. 47, pp. 314–319, 2009.
[46]
M. Vesenjak, et al., "Dynamic Behaviour of Metallic Hollow Sphere
Structures,"
in Multifunctional
Metallic
Hollow Sphere
Structures:
Manufacturing, Properties and Application, ed: Springer-Verlag Berlin
Heidelberg, 2009, pp. 137–158.
[47]
R. Winkler, et al., "Finite Element Vibration Analysis of MHSS Based on 3D
Tomography Image Processing," in Materials with Complex Behaviour,
Advanced Structured Materials, ed Heidelberg Springer-Verlag Berlin 2010,
pp. 243-256.
[48]
N. C. Knowles, "Finite element analysis," Computer-Aided Design, vol. 16,
pp. 134-140, 1984.
[49]
R. W. Clough, "Original formulation of the finite element method," Finite
Elements in Analysis and Design, vol. 7, pp. 89-101, 1990.
[50]
R. Clough, "The finite element method in structural mechanics," in Stress
Analysis, O. C. Zienkiewicz and G.S. Holister, Eds., ed: John Wiley and
Sons, Ltd., 1965, pp. 85-87.
[51]
J. Lemaitre, "A Course on Damage Mechanics," ed Berlin: Springer, 1996.
[52]
T. Fiedler, et al., "Structural Modeling of the Mechanical Behavior of
Periodic Cellular Solids: Open-Cell Structures," Mechanics of Composite
Materials, vol. 41, pp. 277-290, 2005.
[53]
R. J. Green, "A plasticity theory for porous solids," International Journal of
Mechanical Sciences, vol. 14, pp. 215-224, 1972.
[54]
S. A. Meguid, et al., "FE modelling of deformation localization in metallic
foams," Finite Elements in Analysis and Design, vol. 38, pp. 631-643, 2002.
[55]
A. Öchsner and G. Mishuris, "Modelling of the multiaxial elasto-plastic
behaviour of porous metals with internal gas pressure," Finite Elements in
Analysis and Design, vol. 45, pp. 104-112, 2009.
[56]
M. D. Graef and M. E. McHenry, Structure of Materials: An Introduction to
Crystallography, Diffraction, and Symmetry. New York: Cambridge
University Press, 2007.
76
[57]
S. E. Benzley, et al., "A Comparison of All-Hexagonal and All-Tetrahedral
Finite Element Meshes for Elastic and Elastic Plastic Analysis
" presented at the In Fourth International Meshing Roundtable, Al-Buquerque, New
Mexico, 1995.
[58]
FEMAP Version 7.0 Examples, Copyright © 1986-1999 by Enterprise
Software Products Inc.
[59]
A. Öchsner, et al., "Numerical Simulation of the Mechanical Properties of
Sintered and Bonded Perforated Hollow Sphere Structures (PHSS)," J. Mater.
Sci. Technol., vol. 26, pp. 730-736, 2010.
[60]
S. M. H. Hosseini, "Numerical Characterization of Perforated Metallic
Hollow Sphere Structures," Master, Universiti Teknologi Malaysia, Malaysia,
2009.
[61]
J. Rychlewski, "On Hooke's law," Journal of Applied Mathematics and
Mechanics, vol. 48, pp. 303-314, 1984.
[62]
A. Kumar, et al., "Correlation between ultrasonic shear wave velocity and
Poisson's ratio for isotropic solid materials," Acta Materialia, vol. 51, pp.
2417-2426, 2003.
[63]
D. C. William, Jr., "Mechanical Properties of Metals," in Materials Science
and Engineering An Introduction Sixth Edition, ed: John Wiley & Sons. Inc.,
2003, p. 121.
[64]
H. M. Lee and W.-J. Lee, "Poisson's ratio of pure metals and their nonmetallic compounds," Scripta Metallurgica et Materialia, vol. 25, pp. 965968, 1991.
[65]
J. P. Romain, et al., "Relation between the Grüneisen ratio and the pressure
dependence of Poisson's ratio for metals," Journal of Physics and Chemistry
of Solids, vol. 37, pp. 1159-1165, 1976.
[66]
T. Narasimhan, "Fourier’s heat conduction equation: History, influence, and
connections," Journal of Earth System Science, vol. 108, pp. 117-148, 1999.
[67]
L. J. Gibson and M. F. Ashby, Cellular solids: structures & properties.
Cambridge: Cambridge University Press, 1997.
[68]
E. Jacquet, et al., "Calculation of the transverse modulus of a unidirectional
composite material and of the modulus of an aggregate. Application of the
rule of mixtures," Composites Science and Technology, vol. 60, pp. 345-350,
2000.
77
[69]
S. M. H. Hosseini, et al., "Numerical prediction of the effective thermal
conductivity of perforated hollow sphere structures," Defect and Diffusion
Forum vol. 283-286, pp. 6-12, 2009.
[70]
E. Baumeister, "Hohlkugelkomposit - Charakterisierung thermischer und
mechanischer Eigenschaften eines neuen Leichtbauwerkstoffes. Hollow
sphere composite - characterisation of thermal and mechanical properties of a
new lightweight material " PhD thesis University of Magdeburg, Otto von
Guericke, 2004.
[71]
E. Baumeister, et al., "Lightweight, hollow-sphere-composite (HSC)
materials for mechanical engineering applications," Journal of Materials
Processing Technology vol. 155–156 pp. 1839–1846, 2004.
Appendix A - Sample Calculation of the Volume Fraction of the Matrix
78
The volume fraction of the matrix
The matrix volume fraction of the matrix in composite material can be calculated by using
the following equation [3] .
Matrix volume fraction,
Φm
=
(Φm) old
<
(Φm) new
(Φm) 1.47mm
<
(Φm) new
(3.1)
Figure A: The schematic diagram to illustrate one-eighth of the cubic unit cell model of
syntactic perforated hollow sphere structures arrangement.
Sample calculation:
For s = 1.47 mm
To calculate the volume of one-eighth of the sphere (cf. Figure 3.19),
vsp
=
=
(3.2)
( .
)
-
( .
)
Appendix A - Sample Calculation of the Volume Fraction of the Matrix
79
= 1.288 mm3 – 1.023 mm3
= 0.266 mm3
To calculate the volume of one-eighth of the matrix inside the unit cell,
=
−
= (1.47) −
vm
= va + vb
(3.2)
vm
= [vcube - (vsp)] + vb
(3.3)
+
( .
)
+
( .
)
= 2.911mm3
Thus,
Matrix volume fraction,
(Φm)1.47 mm
=
.
.
.
= 0.916
Now, we want to choose a reasonable value for new s (i.e. the new length of the unit cell). In
this study, we will use the new value of s = 1.59 mm. This new value of the length of the
cubic unit cell expected to be more than 10% in terms of the matrix volume fraction so that a
significant different of the results will be achieved.
For s = 1.59 mm the same equation is applied,
vsp
=
=
( .
)
-
( .
)
Appendix A - Sample Calculation of the Volume Fraction of the Matrix
80
= 1.288 mm3 – 1.023 mm3
= 0.266 mm3
vm
= va + vb
= [vcube - (vsp)] + vb
=
−
+
( .
= (1.59) −
)
+
( .
)
= 3.755 mm3
Thus,
The new matrix volume fraction,
(Φm) 1.59 mm
(Φm) old = 1.47 mm
0.916
=
.
.
.
=
1.182
<
(Φm) new = 1.59 mm
<
1.182 (which means is 18.2%
more matrix than the old model
with s = 1.47 mm)
Hence, the length of the new cubic unit cell is chosen as equal to 1.59 mm to build the second
model of the syntactic perforated hollow sphere structure configuration.
Appendix B - Sample Calculation of the average density for composite material
81
The average density for composite material
The average density for composite material can be calculated by using the following equation
ρ=
∙
∙
+
(3.6)
where, vma: Matrix Volume
vsp: Sphere Volume
vuc: Entire Box
Sample calculation:
For bond gap configuration,
= 1.59 mm × 1.59 mm × 1.59 mm
vuc
= 4.019679 mm3
vsp
=
= 0.2656 mm3
= vuc – vsp
vma
= 4.019679 - 0.2656
= 3.754079 mm3
Thus,
∙
ρ=
ρ=
.
.
+
∙
∙ 1.13 +
.
.
∙ 6.95
ρ = 1.055 + 0.459 = 1.514 kg/dm3