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Synthesis and mechanical evaluation of micro-scale truss structures formed from self-propagating polymer waveguides

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Synthesis and mechanical evaluation of micro-scale truss structures formed from self-propagating polymer waveguides

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SYNTHESIS AND MECHANICAL EVALUATION OF MICRO-SCALE TRUSS
STRUCTURES FORMED FROM SELF-PROPAGATING POLYMER
WAVEGUIDES

by

Alan J. Jacobsen

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)

December 2007

Copyright 2007 Alan J. Jacobsen
ii
ACKNOWLEDGEMENTS

I would first like to thank my lovely wife, Caitlin, for all her support and care
over the last few years. I would also like to thank my parents, brother, and sister, for
directing my course in life in their own unique way. I am who I am, and made it to
this point because of them.
At USC, I am very grateful for all the advice and input of my advisor, Prof.
Steve Nutt. His willingness to let me take a risk and guide me through new research
has made my graduate school experience positive and memorable. In Prof. Nutt’s
group, I would like to thank Byungmin Ahn and Shad Thomas for their assistance
with the equipment and Cameron Massey, who always stimulates good ideas. In the
USC community, I would also like to thank Prof. Kassner and Prof. Hogen-Esch for
participating on my committee.
This research was supported by and done in conjunction with HRL
Laboratories, so there are a number of people there I would like to acknowledge and
thank. First, I extend my gratitude to Bill Barvosa-Carter, who has contributed in
many ways to this body of research. I am also very thankful for the support and
encouragement of Leslie Momoda. She was instrumental in getting me to go back to
school and making sure I reached my goals along the way. Others I would like to
thank at HRL are Chaoyin Zhou and Bob Doty. Most consider Chaoyin a chemist,
but I tend to believe he is a magician. And Bob, the very skillful microscopist,
created all the great SEM images shown in this dissertation. I would also like to
iii
thank Peter Brewer, Adam Gross, Robert Cumberland, Sky Van Atta, Kevin Kirby,
David Kisailus (now at UC Riverside), and the president and CEO Matt Ganz for all
their support, assistance, and advice along the way.
Lastly, I would like to acknowledge and thank the National Physical Science
Consortium (NPSC) for fellowship support.
iv
TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

LIST OF TABLES vii

LIST OF FIGURES viii

ABSTRACT xiii

CHAPTER 1: INTRODUCTION 1
1.1 Cellular materials 1
1.1.2. Mechanics of cellular materials 2
1.1.2.1 Elastic modulus 3
1.1.2.2 Compression strength 4
1.2 Motivation for ordered open-cellular structures 6
1.2.1 Existing approaches 7
1.4 Motivation for research 9
1.5 Scope of dissertation 10

CHAPTER 2: MICRO-TRUSS SYNTHESIS 12
2.1 Introduction 12
2.2 Self-propagating polymer waveguides 13
2.2.1 Photopolymer selection and characterization 14
2.2.1.1 UV transmission/absorption spectrum 14
2.2.1.2 Index of refraction 15
2.2.1.3 Density 16
2.2.2 Single waveguide formation 16
2.3 Polymer micro-truss fabrication 18
2.3.1 Three-dimensional patterning of waveguides 18
2.3.2 Sample synthesis procedures 21
2.3.2.1 Uniformity through thickness 22
2.3.3 Micro-truss thickness limitations 24
2.3.4 Varying waveguide angle, thickness and spacing 24
2.4 Summary 26

CHAPTER 3: COMPRESSION BEHAVIOR 27
3.1 Introduction 27
3.2 Analytical predictions for modulus and peak strength 27
3.3 Experimental 29
v
3.3.1 Micro-truss samples for compression loading 29
3.3.2 Unit cell configuration 31
3.3.3 Relative density 32
3.3.4 Parent material properties 33
3.3.5 Compression experiments 35
3.4 Results and discussion 36
3.4.1 Compressive Modulus 37
3.4.2 Peak strength 39
3.4.3 Peak strength with high temperature post-cure in air 42
3.5 Summary 47

CHAPTER 4: SHEAR BEHAVIOR 50
4.1 Introduction 50
4.2 Analytical predictions for shear modulus and shear strength 52
4.3 Experimental 56
4.3.1 Micro-truss samples for shear loading 56
4.3.2 Parent material properties 58
4.3.3 Shear experiments 61
4.4 Results and discussion 62
4.4.1 Shear modulus 65
4.4.2 Shear strength 68
4.5 Summary 74

CHAPTER 5: HEXAGONAL-BASED UNIT CELL ARCHITECTURES 76
5.1 Introduction 76
5.2 Micro-truss fabrication with hexagonal mask pattern 77
5.2.1 Unit cell architecture 79
5.2.2 Relative density 81
5.3 Prediction of compressive modulus and peak strength 82
5.4 Experimental 85
5.4.1 Parent material properties 85
5.4.2 Polymer micro-truss samples for compression loading 85
5.4.3 Compression experiments 88
5.5 Results and discussion 88
5.5.1 Compressive modulus 90
5.5.2 Peak strength 93
5.5.3 Sample size effect 95
5.5.4 Comparison between unit cell structures 96
5.6 Summary 98

vi
CHAPTER 6: CONCLUSIONS AND FUTURE WORK 100
6.1 General conclusions 100
6.2 Future studies 101

REFERENCES 103

BIBLIOGRAPHY 107

vii
LIST OF TABLES

Table 3-1 Summary of fabrication parameters and resulting micro-truss
dimensions samples tested under compression 29

Table 3-2 Comparison of the predicted and measured compression
modulus (±SD) of the micro-truss samples. 38

Table 4-1 Summary of the density, truss angle and post-cure temperature
of the micro-truss samples fabricated for shear testing. 57

Table 4-2 Summary of the shear properties for the different micro-truss
samples. 65

Table 5-1 Summary of the measured parameters for the micro-truss
structures tested under compression. 87

Table 5-2 Summary of the measured compressive properties. 90
viii
LIST OF FIGURES

Figure 1-1 Example of open cellular materials: (a) man-made aluminum
foam [2] and (b) bone [3]. 2

Figure 1-2 Relative modulus of cellular materials as a function of relative
density. Stretch/compression-dominated lattice structures
theoretically provide a significant increase in modulus in
comparison to random cellular foams [4]. 4

Figure 1-3 (a) Model of random cellular material the fails by (a) buckling
of the cell struts or (b) plastic yielding [4]. 5

Figure 1-4 Relative strength of cellular materials as a function of relative
density. Stretch/compression-dominated lattice structures
theoretically provide an increase in strength in comparison to
random cellular foams [4]. 6

Figure 1-5 Examples of metallic lattice materials formed by (a) perforated
sheet metal forming [10] and (b) investment casting [20]. 7

Figure 1-6 Schematic representation of wire or hollow tube lay-up process
used to create metallic lattice structures [13]. 8

Figure 1-7 Nanoporous materials with an ordered cellular structure formed
by interference lithography. 9

Figure 2-1 The UV transmission spectrum of the thiol-ene photopolymer
and the absorption spectrum of the free-radical photoinitiator.
The photopolymer exhibits high transmission at the peak
absorption of the photoinitiator, which enables the formation
of long-aspect-ratio polymer waveguides. 15

Figure 2-2 A single self-propagating polymer waveguide. a) A light
micrograph of a single waveguide formed with collimated
UV light directed through a circular aperture into a volume of
monomer contained in a Teflon reservoir 25 mm deep. The
propagation of the polymer waveguide ceased prior to reaching
the back surface of the mold, and b) an SEM image showing
the narrowing tip of the waveguide with a droplet of excess
monomer. 17
ix

Figure 2-3 (a) Schematic of the set-up for creating a parallel array of
self-propagating polymer waveguides from a single collimated
light beam, and (b) the top surface of the mask with a square
pattern of circular apertures. 19

Figure 2-4 (a) Shown here is a schematic representation of four polymer
waveguides intersecting at a single node that were formed from
large-area collimated beams narrowed through four separate
circular apertures. (b) Illustration of a micro-truss structure that
is four unit cells thick (in the z-axis). 20

Figure 2-5 A light micrograph of a polymer micro-truss structure 7.8 mm
thick formed by this technique. 21

Figure 2-6 SEM images of different polymer micro-truss structures with
repeating octahedral unit cells. The microstructures vary in
relative density (ρ/ρ
s
), diameter (d), angle (θ ) of the waveguide
with respect to the mask surface, and the thickness (t) of the
resulting cellular material. (a) Micro-truss with (ρ/ρ
s
) = 0.21,
d ≅ 180 μm, θ ≅ 60°, t = 7.8 mm, and (b) a single unit cell.
(c) Similar unit cell to (a) but at approximately 1/3 scale:
(ρ/ρ
s
) = 0.26, d ≅ 80 μm, θ ≅ 60°, t = 3.2 mm. (d) Top view
of micro-truss with (ρ/ρ
s
) = 0.22, d ≅ 150 μm, θ ≅ 60°, t = 7.3 mm,
and (e) increased magnification of the top surface. (f) Micro-truss
with a buckled vertical waveguide: (ρ/ρ
s
) = 0.11, d ≅ 150 μm,
θ ≅ 50° + 90°, t = 4.6 mm. 23

Figure 3-1 Image of polymer micro-truss compression sample sandwiched
between two quartz plates. 31

Figure 3-2 The octahedral-type unit cell configuration for all the samples
tested under compression. The darker truss members are shared
with the respective adjacent unit cell 32

Figure 3-3 Nominal compressive stress-strain response of three solid
samples with different curing conditions. A plot of the tangent
modulus as a function of nominal strain (left axis) shows the
nonlinear behavior of the polymer. 34

x
Figure 3-4 Compression behavior of five polymer micro-truss samples
with different relative density and/or waveguide angle.
Each sample initially exhibits linear elastic behavior prior to
reaching their respective peak strength. With subsequent
loading, the cellular structures reach a plateau stress and
eventually densification. 36

Figure 3-5 (a) Close-up of the linear elastic region, and (b) the slopes for
micro- trusses with the same waveguide angle are equivalent
when normalized by the respective density. 37

Figure 3-6 Calculated truss-member failure stress in the compressed
samples plotted as a function of slenderness ratio. The predictions
for failure by material yielding, inelastic buckling (based on the
tangent modulus), and Euler buckling are also included and the
data indicate these structures fall in the inelastic buckling regime.
The peak stress for all the samples tested is well below that
predicted by inelastic buckling theory. 40

Figure 3-7 SEM images showing the buckling behavior of the polymer
micro-truss structures. 41

Figure 3-8 (a) The nominal compressive stress-strain curve and tangent
modulus, and (b) the storage modulus as a function of
temperature for solid samples post-cured at 130°C and 250°C. 44

Figure 3-9 Surface oxidation layer on the solid polymer sample. 45

Figure 3-10 Comparison of the compressive response for the micro-truss
samples post-cured at 130°C (vacuum) and 250°C (air). 46

Figure 4-1 Top view of the unit cell defining the shear force direction ψ
in the x-y plane. 52

Figure 4-2 A schematic of a partial unit-cell displaced Δ at node A from
the shear force V. 53

Figure 4-3 Comparison of the tensile and compressive response of the
solid photo-polymer with different post-cure conditions. 59

Figure 4-4 Single-lap shear fixture used to test the micro-truss samples. 62

xi
Figure 4-5 The nominal shear stress-strain curves for samples
(a) post-cured at 130°C (vacuum) with θ = 59° and a
shear load direction ψ = 0°, (b) post-cured at 130°C
(vacuum) with θ = 51° and a shear load direction ψ = 0°,
and (c) with θ = 59° but different post-cure temperature
and/or shear load direction. 64

Figure 4-6 SEM images displaying different shear failure modes in the
micro-truss structures. (a) Sample 4, (b) Sample 2, (c) close-up
of a crack on a buckled truss member in Sample 2, and
(d) Sample 7. 69

Figure 4-7 Truss member fracture surfaces of Sample 8 which had a
shear force direction ψ = 0°. 72

Figure 4-8 Truss member fracture surfaces of thermally oxidized
Sample 9, in which the shear force direction was ψ = 45°. 73

Figure 5-1 (a) Schematic of the set-up for creating micro-truss structures
with an interconnected array of self-propagating waveguides
and (b) the top view of the mask with a hexagonal pattern of
circular apertures. 79

Figure 5-2 The two unit cell architectures that were formed using the mask
with apertures in a hexagonal pattern. (a) The unit cell formed
with three incident UV exposure beams and (b) the unit cell
formed with six exposure beams. 80

Figure 5-3 (a) Top view, (b) perspective view, and (c-d) side views of the
polymer micro-truss structures formed with three UV exposure
beams. (e) Top view and (f) perspective view of the architecture
formed with six intersecting waveguides. 81

Figure 5-4 The predicted truss-member failure stress as a function of
slenderness ratio for the photopolymer used to form the
micro-truss structures. 83

Figure 5-5 Image of a polymer micro-truss sample sandwiched between
two quartz plates. 88

Figure 5-6 A comparison of the compressive response for the two
different unit cell architectures. 89
xii

Figure 5-7 A representative unit cell (a) adjacent to the initiating substrate
surface and (b) adjacent to the waveguide terminating surface
for the structures with six-fold symmetry. These SEM images
show the change in waveguide angle through the thickness of
the micro-truss structure due to relaxation during processing. 92

Figure 5-8 (a) A structure with three-fold symmetry and (b structure with
six-fold symmetry compressed just beyond the point of initial
truss member buckling. (c) A close-up image of a buckled truss
member in (b). 93

Figure 5-9 The repeatability of the compressive strain-strain curve for
samples with six-fold symmetry. 95

Figure 5-10 A comparison of the elastic and post-buckling response
between the architectures in this study and an octahedral unit cell
with a similar truss member angle (θ ≅ 51°). Note the octahedral
unit cell is approximately twice the relative density of Samples 2
and 4. 98

xiii
ABSTRACT

Materials with significant porosity, generally termed cellular materials, have
considerably lower bulk density than their solid counterparts. However, at the cost
of reducing the mass of a material by introducing porosity, mechanical properties
such as the strength and elastic modulus are significantly diminished. Ordered
cellular structures generally exhibit an increase in modulus and peak strength relative
to random cellular configurations by changing the mode of deformation from
bending-dominated to stretch/compression-dominated within the microstructure
during elastic loading. Nevertheless, techniques to fabricate three-dimensional
ordered open-cellular materials, particularly with feature sizes ranging from tens to
hundred of microns, are limited. Presented in this dissertation, is a new technique to
create cellular materials with a truss architecture from a three-dimensional
interconnected pattern of self-propagating polymer waveguides. The self-
propagating effect enables the rapid formation (< 1 min) of thick (> 5 mm) three-
dimensional open-cellular micro-truss structures from a single two-dimensional
exposure surface. The process also affords significant flexibility and control of the
resulting truss microstructure.
The structure-property relationships in these new polymer micro-trusses have
been investigated, correlating compressive and shear behavior with structural
features, such as density, cell size, truss angle, and unit cell architecture. The
compression and shear modulus compare well with analytical predictions; however
xiv
the measured peak strength was significantly lower than predicted. The deviation of
the measured peak strength from the idealized predictions was attributed to
imperfections in the structure and the nonlinear behavior of the solid polymer. The
affect of imperfections can be reduced by developing unit cell architectures with
increased waveguide connectivity that ultimately increase nodal stability and
decrease the waveguide truss member slenderness ratio.

1

CHAPTER 1: INTRODUCTION
1.1 Cellular materials
Cellular materials are materials with significant porosity, generally having an
open volume greater than 70%. By introducing porosity into a material, unique
properties arise, such as ultra-low density, high surface area per unit volume, and
improved impact absorption. In nature, materials that exploit these unique properties
include bone, wood, and natural sponges. Such natural occurring materials have also
provided inspiration for man-made cellular materials, which exist in virtually all
material forms, including polymers, metals, and ceramics [1].
The majority of cellular materials have a random cellular architecture, as in
foams, and depending on the fabrication process, the cellular material can have either
an open or closed cellular configuration. The difference in these two configurations
is the accessibility of the open volume. A closed-cellular material contains isolated
pores closed off from neighboring pores by thin solid membranes and an open-
cellular material has a continuous open volume [1]. Examples of a man-made and
naturally occurring open-cellular material are shown in Figure 1-1.
2
(a) (b)

Figure 1-1 Example of open cellular materials: (a) man-made aluminum foam [2] and (b)
bone [3].

1.1.2. Mechanics of cellular materials
Open-cellular materials are desired for weight-critical and multifunctional
structural applications because of their low density and accessible open volume.
However, the mechanical properties of an open-cellular material, such as the strength
and elastic modulus, are significantly diminished in comparison to their solid
counterparts. The two most important factors that influence the mechanical
properties of a cellular material are its relative density, which is the ratio of the
cellular material density (ρ) to the density of the parent solid (ρ
s
), and the cellular
structure. In earlier work, Gibson and Ashby [1] showed that randomly oriented
cellular materials, such as foams, exhibit bending-dominated behavior of the cell
struts under applied mechanical load. Simple mechanics dictate that this “soft” mode
of deformation is less efficient in load carrying capacity than stretch/compress-
dominated behavior, as exemplified in conventional, large-scale truss structures,
such as building frames and bridges. By ordering the microstructure of a cellular
3
material to resemble these truss structures, it is possible to change the mode of
deformation of the cell struts (or truss members) from bending-dominated to stretch
or compress-dominated [4]. The effect of the strut deformation mode on the elastic
modulus and strength of a cellular material is significant and is discussed in the
following sections.
1.1.2.1 Elastic modulus
The elastic modulus E of a bulk cellular material can be approximated by,
E = C
E
E
s
(ρ /ρ
s
)
r
(1.1)
where E
s
is the elastic modulus of the solid material from which the cellular material
is comprised, and C
E
and r are scaling parameters related to the cellular structure [1,
5]. This generalized equation is valid for both closed and open cellular materials.
The scaling parameter r is a function of how the material deforms during elastic
loading. Based on a simple model for random open-cellular structures that exhibit
bending-dominated behavior, the elastic modulus is proportional to the square of the
relative density, i.e. r = 2 [1]. However, the elastic modulus of a cellular material
that exhibits stretch/compression-dominated behavior is directly proportional to the
relative density (r = 1) [6]. The scaling parameter C
E
is a function of the geometry
of the material and how that geometry is oriented in relation to the loading direction
[1, 5, 7, 8]. In Figure 1-2, the elastic modulus as a function of density for
stretch/compression-dominated microstructures are shown in comparison to existing
materials.
4
Relative Density (ρ /ρ
s
)
Relative Modulus (E /E
s
)
Relative Density (ρ /ρ
s
)
Relative Modulus (E /E
s
)
Relative Density (ρ /ρ
s
)
Relative Modulus (E /E
s
)

Figure 1-2 Relative modulus of cellular materials as a function of relative density.
Stretch/compression-dominated lattice structures theoretically provide a significant increase
in modulus in comparison to random cellular foams [4].

1.1.2.2 Compression strength
Estimating the strength of a cellular material is much more complex than the
modulus due to the existence of various failure mechanisms and unpredictable
defects. For elastomeric foams, or rigid polymer foams that have struts with
sufficiently long slenderness ratios, buckling of the cell struts is the dominate failure
mechanism when loaded under compression. The model in Figure 1-3a can be used
to develop the following relationship to predict strength σ
el
,
σ
el
= C
el
E
s
(ρ /ρ
s
)
2
(1.2)
where C
el
is a constant of proportionality determined empirically.

5

(a) (b)
Figure 1-3 (a) Model of random cellular material the fails by (a) buckling of the cell struts or
(b) plastic yielding [4].

For cellular materials that have bending-dominated behavior and fail through
plastic deformation, as shown in Figure 1-3b, the failure strength (σ
p
) can be
estimated by the following,
σ
p
= C
1
σ
ys
(ρ /ρ
s
)
3/2
(1.3)
where again, C
1
is a constant of proportionality, and σ
ys
is the yield strength of the
solid material. The compression strength of cellular materials with ordered
microstructures that exhibits stretch or compress-dominated behavior is given by,
σ = C
2
σ
ys
(ρ /ρ
s
) (1.4)
if the material fails by yielding. Here, the constant C
2
is a function of the geometry
of the material and how that geometry is oriented in relation to the loading direction
[6, 8]. As with the elastic modulus, the strength of a cellular material that can
suppress bending-dominated behavior (and fails by material yielding) is linearly
proportional to relative density. Figure 1-4 is a comparison of the strength of open-
6
cellular materials that exhibit bending-dominated and stretch/compression-
dominated behavior of the cell struts and fail by material yielding.
Relative Density (ρ /ρ
s
)
Relative Strength (σ /σ
s
)
Relative Density (ρ /ρ
s
)
Relative Strength (σ /σ
s
)

Figure 1-4 Relative strength of cellular materials as a function of relative density.
Stretch/compression-dominated lattice structures theoretically provide an increase in strength
in comparison to random cellular foams [4].

1.2 Motivation for ordered open-cellular structures
The theory described above provides motivation to develop cellular materials
with architectures that can suppress bending-dominated behavior. By designing the
microstructure of a cellular material to enable stretch or compress-dominated
behavior, a significant increase in modulus and strength can be realized without
increasing the density. Also, the benefit of suppressing bending-dominated behavior
becomes more significant as the relative density of the cellular materials is
decreased.
7
1.2.1 Existing approaches
Different techniques have been developed to create metallic lattice materials
comprised of truss architectures [8-11], and these have been reviewed by Wadley
[12, 13]. These techniques include investment casting, perforated metal sheet
forming, and wire or hollow tube lay-up, resulting in open-cellular materials with
millimeter-sized features. The architecture of these ordered metallic structures
enables the truss members, or struts, to stretch or compress during elastic loading,
and the resulting mechanical properties of these cellular structures indicate that they
may have utility as ultra-low relative density sandwich cores, while retaining open
porosity for multifunctional applications [6, 9, 14-19]. Examples of metallic lattice
materials fabricated by perforated sheet metal forming and investment casting are
shown in Figure 1-5, and a schematic of the wire or hollow tube lay-up process is
shown in Figure 1-6.
(a) (b)

Figure 1-5 Examples of metallic lattice materials formed by (a) perforated sheet metal
forming [10] and (b) investment casting [20].

8

Figure 1-6 Schematic representation of wire or hollow tube lay-up process used to create
metallic lattice structures [13].

While metallic lattice open-cellular structures typically have feature sizes on
the order of millimeters, a relatively new technique in lithography enables
production of three-dimensionally ordered open-cellular polymer materials with truss
features on the nanometer scale. This technique, commonly termed holographic
interference lithography, involves exposure of a photoresist to an optical interference
pattern created from multiple non-coplanar laser beams [21]. A similar technique,
which can easily be scaled to large areas, utilizes conformable phase masks rather
than multibeam holography [22]. Recent efforts have focused on exploring the
mechanical behavior of such cellular materials with nanometer length scale features.
For example, Jang et al. [23] showed that the ordered open-cellular architecture,
coupled with the nanometer size-scale, can greatly enhance the ductility of a
generally brittle polymer photoresist. The major limitation of this technique is
9
thickness of the resulting material, which is typically less than 10 μm. Examples of
open-cellular structures formed by this process are shown in Figure 1-7.

Figure 1-7 Nanoporous materials with an ordered cellular structure formed by interference
lithography.

Fabrication techniques to create micro-scale open-cellular materials are
limited. One possible approach utilizes soft lithography in combination with
electrodeposition to create a two-dimensional grid that is subsequently folded to
form a three-dimensional truss structure [24]. However, like similar techniques used
to make larger scale metallic lattice structures [10] this approach is limited to
forming structures with a single unit cell thickness. For thicker, multilayered
structures, a lamination process is required.
1.4 Motivation for research
Of the research devoted towards ordered, open-cellular materials, there is no
existing technique that can rapidly create polymer cellular materials with a truss
architecture that has feature sizes in the micrometer-size regime. Polymer foams
10
have utility in numerous load-bearing applications; therefore, creating similar
materials with improved mechanical properties is beneficial. In addition, reducing
the size-scale of the truss features (for example truss members < 10 μm) will
undoubtedly give rise to interesting property changes, including a higher modulus
due to molecular alignment of the polymer chains and suppression of fracture due to
an overall increase in surface energy (and thus increased strength).
From a materials standpoint, polymer foams are commonly used as
precursors or templates to create metallic, ceramic, or carbon foams [25, 26]. Thus,
developing a polymer precursor with a truss architecture, will enable ordered cellular
architectures in various material forms.
As with any new material, mechanical characterization is necessary if the
material is to be considered for load bearing applications. Also, in addition to simple
characterization, understanding how the material behaves under various loading
conditions provides insight into further improving the mechanical properties.
1.5 Scope of dissertation
This research was sponsored by HRL Laboratories, LLC (Malibu, CA), so
consequently the motivation for discovery and understanding was driven by
industrial needs. In the broadest sense, the need was defined simply as “lighter,
stronger materials”. The implied need was lighter, stronger materials that are low-
cost and processible on a useful scale. This need, coupled with the motivation for
new research described in Section 1.4, defined the scope of this dissertation.
11
Chapter 2 describes a new process to create micro-scale truss structures (or
micro-trusses) from a three-dimensional interconnected pattern of self-propagating
polymer waveguides. Chapter 3 and 4 are devoted towards understanding the
compression and shear properties, respectively, of these new materials. In Chapter 5,
new unit cell architectures are explored and their compression properties are
compared to the unit cell architecture described and studied in the previous chapters.
Chapter 6 includes the general conclusions than can be drawn from this research and
possibilities for future research.
12

CHAPTER 2: MICRO-TRUSS SYNTHESIS
⊕

2.1 Introduction
Innovative scalable techniques have been developed to fabricate open-
cellular materials with truss features ranging from hundreds of nanometers to a few
microns [22, 27, 28] – or on the order of millimeters [12]. However, these
techniques are not well suited to fabricate micro-scale structures with feature sizes
ranging from tens to hundred of microns. Presented in this chapter, is a new class of
cellular structures formed from a three-dimensional interconnected pattern of self-
propagating polymer waveguides. In contrast to existing lithographic techniques,
[21-23, 27-30] this self-propagating effect enables the rapid formation (< 1 min) of
thick (> 5 mm) three-dimensional open-cellular structures from a single two-
dimensional exposure surface. The process also affords significant flexibility and

⊕
This chapter is based on work published in the following paper: Jacobsen et al., “Micro-scale truss
structures formed from self-propagating photopolymer waveguides.” Advanced Materials (2007)
DOI: 10.1002/adma.200700797. Note the approaches and techniques described here may be
protected by the US Patent Pending.
13
control of the geometry and configuration of the resulting cellular structure, which in
turn, provides control of the bulk physical and mechanical properties.
2.2 Self-propagating polymer waveguides
A self-propagating polymer waveguide can be formed from a single point
exposure of light in a suitable photomonomer and can yield a high-aspect-ratio
polymer fiber (length / diameter > 100) in seconds with approximately constant
cross-section over its entire length [31, 32]. This self-propagating phenomenon is a
result of a self-focusing effect caused by a change in the index of refraction between
the liquid photomonomer and solid polymer during the polymerization reaction [31-
33]. Upon exposure of light in the appropriate wavelength range – typically UV for
most photosensitive monomers – polymerization begins at the point of exposure and
the subsequent incident light is essentially trapped in the polymer because of internal
reflection, as in optical fibers. This self-trapping effect tunnels the light towards the
far end of the already-formed polymer, further propagating the polymerization front
within the liquid monomer [34]. The diameter of the waveguide is dependent on the
exposed area, and the length is primarily dependent on the incident energy of the
light and the photo-absorption properties of the polymer [35]. Eventually, the
polymer itself will absorb enough light in the critical wavelength range to terminate
waveguide propagation.
14
2.2.1 Photopolymer selection and characterization
The key attributes of a photopolymer used to form high-aspect-ratio self-
propagating waveguides include a refractive index change between the liquid
monomer and solid polymer and maximum possible transparency at the wavelength
of light used to initiate polymerization. Here we use a thiol-ene monomer system
⊕

polymerized with 0.05 wt % 2,2-dimethoxy-2-phenylacetophenone (Aldrich) as the
free-radical photoinitiator.
2.2.1.1 UV transmission/absorption spectrum
A thin layer of the resin (~ 50 μm) was applied to a 380 μm-thick quartz
substrate and polymerized under UV light (~ 7.5 mW/cm
2
) generated from a 500W
mercury arc lamp (Oriel Corporation). The transmission spectrum of the
photopolymer film was measured with a UV-Vis-NIR spectrophotometer (Perkin-
Elmer). Separately, the absorption spectrum of 2,2-dimethoxy-2-
phenylacetophenone (dissolved in acetonitrile) was measured. As shown in Figure
2-1, the photopolymer exhibits excellent transparency at the wavelength of
maximum absorption for the photoinitiator.

⊕
The formulation of the thiol-ene monomer system is considered proprietary by HRL Laboratories,
LLC.
15
100
80
60
40
20
0
% Transmission (Photopolymer)
450 400 350 300 250 200
Wavelength (nm)
2.5
2.0
1.5
1.0
0.5
0.0
Absorbance (Photoinitiator)
100
80
60
40
20
0
% Transmission (Photopolymer)
450 400 350 300 250 200
Wavelength (nm)
2.5
2.0
1.5
1.0
0.5
0.0
Absorbance (Photoinitiator)

Figure 2-1 The UV transmission spectrum of the thiol-ene photopolymer and the absorption
spectrum of the free-radical photoinitiator. The photopolymer exhibits high transmission at
the peak absorption of the photoinitiator, which enables the formation of long-aspect-ratio
polymer waveguides.

2.2.1.2 Index of refraction
The refractive indices of the thoil-ene monomer system and polymer were
measured with a Leitz-Jelley refractometer. The light source used with the
refractometer was a collimated Halogen lamp. The refractive index of the monomer
was measured as 1.510 and the refractive index of polymer was 1.556. This change
in index of refraction is a result of the density change between the liquid monomer
and solid polymer, and is considered sufficient to induce the described self-
propagation effect [32].
16
2.2.1.3 Density
The density of the polymer was determined using a gas pycnometer
(Micromeritics). To ensure accuracy, the density was measured ten times and the
results yielded ρ
s
= 1.34 ± 0.01 g/cm
3
.
2.2.2 Single waveguide formation
Previous studies on waveguide formation utilized a fiber optic, lens
apparatus, or focusing mask to create a point source of light which initiated the self-
propagating formation of the polymer fiber through the monomer [31-35]. However,
as shown here, this effect can be achieved using a broad spectrum collimated light
source (generated from a 500 mercury arc lamp) directed through a mask with a
simple circular aperture. If such a mask is placed over a volume of photomonomer
and the area of the circular aperture is exposed to collimated light, a single
waveguide can be formed, as shown in Figure 2-2. Although the monomer was
exposed to the entire emission spectrum of the mercury arc lamp, the photoinitiator
generating the free radicals was sensitive only to the UV range shown by the
absorption spectrum in Figure 2-1.
17
1 mm
(a)
1 mm
(a)

500 μm
(b)
500 μm 500 μm
(b)

Figure 2-2 A single self-propagating polymer waveguide. a) A light micrograph of a single
waveguide formed with collimated UV light directed through a circular aperture into a
volume of monomer contained in a Teflon reservoir 25 mm deep. The propagation of the
polymer waveguide ceased prior to reaching the back surface of the mold, and b) an SEM
image showing the narrowing tip of the waveguide with a droplet of excess monomer.

18
The waveguide in Figure 2-2 is approximately 13.5 mm long and 260 μm in
diameter. It was formed on glass (which separated the monomer and mask) with an
exposure time of 100 seconds, long enough to ensure the maximum length was
achieved. The fluence measured at the mask surface was ~ 7.5 mW/cm
2
. The
diameter of the resulting waveguide was slightly larger than the diameter of the
single aperature (200 μm) because the prolonged exposure caused the waveguide to
“thicken”. The waveguide was then rinsed in toluene to remove any excess
monomer. As shown in Figure 2-2b, the waveguide tapers towards the terminus, a
consequence of the tunneling effect. However, after rinsing, a droplet of excess
monomer remained at the tip of the waveguide. Also, excess monomer remained at
the base of the waveguide, leading to a broadened contact area with the glass.
2.3 Polymer micro-truss fabrication
2.3.1 Three-dimensional patterning of waveguides
If a single mask with an array of apertures is covering a volume of an
adequate photomonomer, and the mask is exposed to an angled collimated beam that
will polymerize the monomer, an array of waveguides can be formed, as shown in
Figure 2-3. For this set-up, the angle of the resulting waveguides (θ ) is always
greater than the angle of the incident beam with respect to the surface (α), because
the index of refraction of the mask material (quartz) is greater than that of air.

19
α
θ
L
a
L
a
Mask (top)
Collimated
UV Light
Polymer
Waveguide
Liquid
Monomer
Quartz
(a)
(b)
α
θ
L
a
L
a
Mask (top)
Collimated
UV Light
Polymer
Waveguide
Liquid
Monomer
Quartz
(a)
(b)

Figure 2-3 (a) Schematic of the set-up for creating a parallel array of self-propagating
polymer waveguides from a single collimated light beam, and (b) the top surface of the mask
with a square pattern of circular apertures.

If a mask with a two-dimensional pattern of apertures is simultaneously
exposed to multiple collimated beams originating from different directions, an array
of waveguides will form from each of the collimated beams. Waveguides will
initiate at each aperture in the approximate direction of each collimated beam and
polymerize together at points of intersection, or nodes, as shown in Figure 2-4a.
Shoji and Kawata [36] showed that two self-propagating waveguides intersecting at
an angle greater than 9° will propagate through one another without merging into a
single waveguide.
20
Quartz
Collimated UV light
Mask
Polymer waveguide
Photomonomer
Node
Quartz
Collimated UV light
Mask
Polymer waveguide
Photomonomer
Node

(a)

x
y
z
x
y
z

(b)

Figure 2-4 (a) Shown here is a schematic representation of four polymer waveguides
intersecting at a single node that were formed from large-area collimated beams narrowed
through four separate circular apertures. (b) Illustration of a micro-truss structure that is four
unit cells thick (in the z-axis).
21

If the collimated beams expose the entire mask surface, a micro-truss can be
formed. Subsequent removal of the uncured liquid monomer will result in a self-
supporting polymer micro-truss structure, such as the one schematically shown in
Figure 2-4b.

Figure 2-5 A light micrograph of a polymer micro-truss structure 7.8 mm thick formed by
this technique.

2.3.2 Sample synthesis procedures
The micro-truss sample shown in Figure 2-5 was fabricated following the
procedures described below. The thiol-ene resin was contained in a Teflon reservoir
8.5 mm deep and the top surface of the monomer was covered with a 1.5 mm thick
quartz plate and separate mask (Photo Sciences, Inc.). The cellular structure was
formed by exposing the mask, which consisted of a square pattern of circular
apertures, with four separate collimated UV beams. The circular apertures were
patterned over a 50 mm x 50 mm area. Each aperture was 150 μm in diameter with a
22
spacing L
a
= 1350 μm (see Figure 2-3). The collimated UV light was generated from
a 500W mercury arc lamp (Oriel Corporation) with a fluence of ~ 7.5 mW/cm
2
at the
mask surface. The collimated beams each had an equal incident angle off the mask
surface but were rotated 90° about the mask normal. The exposure time for this
particular structure was 36 s. After exposure, the uncured monomer was rinsed in
toluene, and the polymer structure was post-cured for 24h at 130°C under vacuum to
remove any remaining solvent.
The resulting micro-truss structure in Figure 2-5 is 7.8 mm thick and features
a repeating octahedral-type unit cell. The measured relative density of this structure
is 21%, which was calculated by dividing the density of the micro-truss by the
density of the solid polymer. The height of this structure was a function of the
monomer reservoir depth (8.5 mm), which halted waveguide propagation. The slight
difference between the measured micro-truss height and reservoir depth is attributed
to polymer shrinkage and structural relaxation during the post-cure process.
2.3.2.1 Uniformity through thickness
Figure 2-6a is an SEM image of the terminating surface of the micro-truss
structure shown in Figure 2-5, which shows reasonable waveguide uniformity
through the thickness. The precise exposure time of 36 s was determined
experimentally. Shorter exposure times led to waveguides that did not propagate the
full depth of the monomer reservoir or decreased in diameter over the propagating
length. Longer exposure times first caused over-curing at the terminating surface,
23
and as the time was increased, the diameter of the polymer waveguides would
increase until the entire monomer polymerized.

1500 μm
(a)
1500 μm 1500 μm
(a)

860 μm
(b)
860 μm 860 μm
(b)

270 μm
(c)
270 μm 270 μm
(c)

750 μm
(d)
750 μm 750 μm
(d)

380 μm
(e)
380 μm 380 μm
(e)

380 μm
(f)
380 μm 380 μm
(f)

Figure 2-6 SEM images of different polymer micro-truss structures with repeating
octahedral unit cells. The microstructures vary in relative density (ρ/ρ
s
), diameter (d), angle
(θ ) of the waveguide with respect to the mask surface, and the thickness (t) of the resulting
cellular material. (a) Micro-truss with (ρ/ρ
s
) = 0.21, d ≅ 180 μm, θ ≅ 60°, t = 7.8 mm, and
(b) a single unit cell. (c) Similar unit cell to (a) but at approximately 1/3 scale: (ρ/ρ
s
) = 0.26,
d ≅ 80 μm, θ ≅ 60°, t = 3.2 mm. (d) Top view of micro-truss with (ρ/ρ
s
) = 0.22, d ≅ 150 μm,
θ ≅ 60°, t = 7.3 mm, and (e) increased magnification of the top surface. (f) Micro-truss with
a buckled vertical waveguide: (ρ/ρ
s
) = 0.11, d ≅ 150 μm, θ ≅ 50° + 90°, t = 4.6 mm.

24
2.3.3 Micro-truss thickness limitations
Whereas the microstructure is shaped from a pattern of interconnected
waveguides, the maximum thickness of these cellular materials is determined from
the ultimate length of the waveguides formed. As described earlier, the self-
propagating effect by which long-aspect-ratio polymer fibers are fabricated enables
the formation of thick, three-dimensional periodic open-cellular structures from a
single, two-dimensional exposure surface. There is a fundamental limit on the
maximum thickness of the microstructures formed using this technique; however,
with further optimization of the process, cellular materials approaching 15-20 mm
thick should be possible from a single two-dimensional exposure surface.
2.3.4 Varying waveguide angle, thickness and spacing
A wide variety of similar octahedral-type truss structures can be fabricated
with this technique, and the cell geometry can be easily altered by changing the
angle, diameter, and/or spacing of the individual waveguides, as shown by the
structures in Figures 2-6 b-f. The angle of the waveguides is controlled through the
incident angle of the collimated light with respect to the mask surface; however, the
minimum angle is limited by the refractive index difference between air and the
mask substrate material due to Snell’s Law. The diameter of the aperture, along with
any diffraction of light through the distance between the mask and monomer surface,
determines the diameter of the waveguides. Thus, the open volume fraction of the
25
resulting cellular material is a function of both the diameter of the waveguides and
the aperture spacing on the mask.
The microstructures shown in Figure 2-6 were fabricated with intersecting
waveguides at angles of 50-60° from the mask substrate surface and diameters of 80-
180 μm. The spacing between the apertures on the different masks varied between 5
and 10 times the aperture diameter, resulting in cellular structures with solid volume
fractions between 11% and 26%. Microstructures with increased or reduced open
volume fractions are possible. The upper bound on open volume fraction is
determined by the self-supporting ability of the structure, and the lower bound is
limited by the ability to remove the uncured monomer after the polymerization
process. The cellular structure in Figure 2-6f had an additional exposure from a
collimated beam (~ 2.2 mW/cm
2
) directed perpendicular to the mask surface. The
beam transmission loss associated with reflection at the quartz mask surface was
greater for the angled collimated beams, and thus the intensity of the vertical
collimated beam was reduced to form waveguides of approximately equal diameter
for a given exposure time. If the vertical and angled collimated beams had equal
fluence at the mask surface, the vertical exposure would cause the monomer to
completely polymerize before the angled waveguides could propagate the full depth
of the reservoir. The exposure time (40 s for this micro-truss structure), and the
intensity ratio between the vertical and angled collimated beams were determined
experimentally. The monomer reservoir depth for this micro-truss structure was 5.5
26
mm, but the thickness measured after post-cure was 4.6 mm. The buckled vertical
waveguide shown in Figure 2-6f was attributed to dimensional changes that arose
during the post-cure process.
2.4 Summary
The approach developed here can be used to rapidly create and control a wide
variety of polymeric micro-truss structures. These micro-truss structures can be used
as-fabricated, or as templates to form other cellular materials. For example, the
micro-truss structures can be converted to ceramic, metallic or graphitic cellular
materials using processes commonly applied to convert polymer foams [25, 26]. The
ability to fabricate micro-truss structures of different materials, coupled with the
geometric flexibility intrinsic to the process, would afford materials engineers
enormous flexibility for designing cellular structures with a wide range of
mechanical and physical properties.

27

CHAPTER 3: COMPRESSION BEHA VIOR
⊕

3.1 Introduction
In this chapter, the compressive properties of the polymer micro-truss
structures were investigated. Compression experiments were conducted on
structures of different density, cell size, and truss angle to determine the dependence
of strength and modulus on these parameters. In addition, the experimental results
were compared with relevant theory predicting compressive behavior of such
microstructures. Based on these results, conclusions were drawn on the failure
mechanism of these materials under compression.
3.2 Analytical predictions for modulus and peak strength
Assuming the initial strain of the bulk micro-truss structure is transferred as
pure axial strain to the truss members (with pinned end constraints), a work (energy)
balance leads to the following relationship [6, 11]:

⊕
This chapter is based on work published in the following paper: Jacobsen et al., “Compression
behavior of micro-scale truss structures formed from self-propagating polymer waveguides.” Acta
Materialia, (2007) DOI: 10.1016/j.actamat.2007.08.036.
28
E = E
s
sin
4
θ (ρ /ρ
s
) (3.1)
where E is the compressive modulus of the micro-truss material. This relationship
implies that for a given solid polymer and waveguide angle (θ ), the ratio E/ρ for
different micro-truss structures should be constant.
A force balance in the z-direction provides an equation relating the stress in
the micro-truss material to the stress in each solid truss member (σ
s
) [6].
σ = σ
s
sin
2
θ (ρ /ρ
s
) (3.2)
This relationship can be used to predict the peak strength of the micro-truss material
(σ
p
) knowing the stress in the solid truss members that will cause buckling, yielding,
or fracture. For extremely short truss members that fail in axial compression, σ
s
=
σ
y
, where σ
y
is the yield strength of the solid polymer. For long, slender truss
members that fail by elastic (Euler) buckling:

2 2
2
) / (
,
r l k
E
s
Euler b s
π
σ σ = = (3.3)
where k is a constant that depends on the end constraint of the truss members [37].
For truss members with slenderness ratios (l/r) between those governed by axial
compression and elastic buckling, the tangent modulus theory can be used to predict
inelastic buckling behavior. In this case, the stress that causes inelastic buckling
(σ
b,inelastic
) for a given slenderness ratio can be predicted by replacing E
s
in Equation
(3.3) with the tangent modulus (E
t
) of the solid material [10, 37].
29
3.3 Experimental
3.3.1 Micro-truss samples for compression loading
Five polymer micro-truss samples were fabricated using the procedures
described in the previous chapter. The mask aperture dimensions and the incident
collimated light angle (α) for each sample are summarized in Table 3-1. Identical
fabrication parameters were used for Samples 1 and 2. Samples 3 and 4 were
fabricated using the same mask aperture radius and spacing as in Sample 1 and 2, but
with a reduced incident light angle. Sample 5 was fabricated using the same incident
light angle as Sample 1 and 2, but with a larger aperture radius and spacing.
Although the unit cell dimensions were larger for this sample, the ratio of aperture
spacing to radius remained constant, which in theory should yield a micro-truss with
a density equivalent to the other samples.

Table 3-1 Summary of fabrication parameters and resulting micro-truss dimensions samples tested
under compression
UV Light Calculated
Radius
(μm)
Spacing,
L
a
(μm)
Incident
Angle,
α (deg)
Density,
ρ (g/cm
3
)
H
(μm)
L
(μm)
Waveguide
Radius,
r (μm)
Waveguide
Angle,
θ (deg)
Waveguide
Angle,
θ (deg)
1 50 900 40 0.179 ± 0.004 1450 ± 50 880 ± 30 65 ± 10 60 ± 2 59 ± 1.2
2 50 900 40 0.216 ± 0.004 1450 ± 50 880 ± 30 65 ± 10 60 ± 2 59 ± 1.2
3 50 900 20 0.169 ± 0.003 1120 ± 30 910 ± 10 65 ± 10 50 ± 2 51 ± 0.8
4 50 900 20 0.184 ± 0.004 1120 ± 30 910 ± 10 65 ± 10 50 ± 2 51 ± 0.8
5 75 1350 40 0.160 ± 0.003 2029 ± 30 1370 ± 20 95 ± 5 60 ± 2 59 ± 0.6
Sample
Polymer Waveguide Micro-truss
Measured
Fabrication Parameters
Aperture

30
After exposure and removal of the uncured monomer in toluene, the micro-
truss structures were heated to 130°C for 24hrs under vacuum to ensure full cure of
the polymer and removal of the solvent. The bulk samples were then cut to size
using a razor blade. The dimensions of Samples 1-4 were approximately 20 mm x
20 mm x 5 mm thick. All sample dimensions were measured using a digital caliper
with 0.01 mm precision and the variation in each measured dimension was ±0.1 mm.
Sample 5 had the same compression area, but was fabricated in a deeper mold,
yielding a micro-truss with a thickness of 7.75 mm. The densities reported in Table
3-1 were determined from the mass and outer volume of these samples.
After fabrication of the micro-truss structures, they remained attached to the
quartz plate that separated the mask from the monomer. Prior to the compression
experiments, the opposite surface was attached to a similar quartz plate by applying a
thin layer of photomonomer to the quartz and exposing the entire structure to UV
light. This rigidly constrained the nodes at these surfaces during compression.
Figure 3-1 is an image of Sample 5 sandwiched between two quartz plates.
31
20 mm
Quartz
Polymer micro-truss
20 mm
Quartz
Polymer micro-truss

Figure 3-1 Image of polymer micro-truss compression sample sandwiched between two
quartz plates.

3.3.2 Unit cell configuration
All the micro-truss samples had a repeating octahedral-type unit cell as
shown in Figure 3-2 As discussed in the previous chapter, this unit cell structure was
formed by exposing a mask with a square pattern of equally spaced, identical circular
apertures to four collimated beams. The collimated beams had the same incident
angle (α) but were rotated 90° apart with respect to the mask normal. The dimension
L represents the unit cell length and width, while H is the unit cell height. For this
configuration, the length of the waveguides between intersecting nodes (l) is equal,
as is the angle of the waveguides (θ ).

32
θ
L
H
l
L
x
y
z
Intersection
Node
Truss Member
θ
L
H
l
L
x
y
z
θ
L
H
l
L
x
y
z
Intersection
Node
Truss Member

Figure 3-2 The octahedral-type unit cell configuration for all the samples tested under
compression. The darker truss members are shared with the respective adjacent unit cell.

The unit cell dimensions (L, H), waveguide radius (r), and waveguide angle
(θ ) of the samples were determined from SEM micrographs. Because it was
difficult to capture the exact waveguide angle, θ was calculated from the relation θ =
tan
-1
(H/L). These values are included in Table 3-1, and were used in future
calculations.
3.3.3 Relative density
The properties of cellular materials are strongly dependent on the relative
density, which is the ratio of the density of the cellular material (ρ) to the density of
the parent solid (ρ
s
). The relative density is also a measure of the solid volume
fraction of a cellular material, and hence can be calculated from the geometry of a
periodic unit cell.
33

θ θ
π π
ρ
ρ
sin cos
2 2
2
2
2 2 2
2 8
l
r
H L
H L r
s
=
+
= (3.4)
By simply changing the mask aperture radius and/or spacing of the polymer
truss members, one can alter the relative density of the resulting microstructure.
However, even with an identical set-up and exposure time, the process is sensitive to
variables that cannot be precisely controlled, resulting in cellular structures with
substantially different densities. These unexpected variations in the density can be
caused from fluctuations in the light intensity of the collimated light source and the
age or handling conditions of the photomonomer. An additional processing
condition which can also vary the density of these structures is incomplete removal
of the uncured monomer after exposure, a task that becomes progressively more
difficult as the unit cell size is reduced. As shown later, these variations in density
affect the compressive properties of the resulting micro-truss structures.
3.3.4 Parent material properties
As shown in the previous section, to predict the mechanical behavior of a
cellular material, one must first determine the properties of the solid material of
which it is comprised. For most engineering metals, a tensile test can be used to
accurately determine the compressive elastic modulus and yield strength; however,
for polymers, the compressive properties can vary greatly from the tensile properties
[38]. Because all the polymer micro-truss structures were tested under compression
34
in this work, compression tests were conducted in accordance with standard methods
[39] to characterize the solid polymer.
A thiol-ene monomer system was used for the fabrication of all micro-truss
structures. After UV exposure and removal of the uncured monomer, the polymer
micro-truss structures were post-cured at 130°C for 24hrs under vacuum. The exact
UV exposure used to form the micro-truss structures could not be duplicated for the
solid polymer samples, and since the mechanical properties of a polymer can be
affected by changing the curing parameters, three solid samples were prepared using
different curing conditions. One sample was exposed to UV light generated from a
mercury arc lamp (~ 7.5 mW/cm
2
) for 30 s and then heated to 130°C for 24 hrs under
vacuum. The second sample was cured by UV light only (~ 200 s at the same
power) and the third sample was cured by heat only (130°C for 24 hrs).
3.0
2.5
2.0
1.5
1.0
0.5
0.0 Tangent Modulus, dσ/dε (GPa)
60x10
-3
50 40 30 20 10 0
Nominal Strain, ε
70
60
50
40
30
20
10
Nominal Stress, σ (MPa)
UV and Heat
Heat only
UV only
3.0
2.5
2.0
1.5
1.0
0.5
0.0 Tangent Modulus, dσ/dε (GPa)
60x10
-3
50 40 30 20 10 0
Nominal Strain, ε
70
60
50
40
30
20
10
Nominal Stress, σ (MPa)
UV and Heat
Heat only
UV only

Figure 3-3 Nominal compressive stress-strain response of three solid samples with different
curing conditions. A plot of the tangent modulus as a function of nominal strain (left axis)
shows the nonlinear behavior of the polymer.
35
As shown in Figure 3-3, the compressive response of the polymer is
nonlinear, so the compressive modulus E
s
is taken as the maximum tangent of the
stress-strain curve [39]. The difference in the compressive modulus for the three
samples is less than 5%, indicating that the effect of the different curing parameters
for this polymer is negligible. The average modulus from the three samples is E
s
=
2.4 GPa. The effect on the ultimate yield strength is slightly greater, but the
difference is still within 11% (average: σ
y
= 65 MPa).
The density of the polymer was determined from the mass and volume of the
solid compression samples, and from the polymer micro-truss structures using a gas
pycnometer. Both techniques yielded ρ
s
= 1.34 g/cm
3
.
3.3.5 Compression experiments
Quasi-static compression tests were conducted on the polymer micro-truss
samples described in the previous section. The experiments were performed on a
hydraulic load frame at a strain rate of 2 x 10
-3
s
-1
. The load was measured from a
15kN built-in load cell with an accuracy of ±1%, and the displacement was recorded
using a laser extensometer with 0.001 mm precision. The accuracy of all reported
compression stress data was ±2%.

36
3.4 Results and discussion
The nominal stress-strain data for Samples 1-5 is shown in Figure 3-4. All
samples exhibited axial compression prior to initial truss buckling, and subsequent
deformation was bending-dominated because of misalignments generated from the
buckled truss members. Bending-dominated behavior in a cellular material will
generally lead to a constant stress plateau [1, 4]; however, for these micro-truss
structures, the remaining periodicity from the unbuckled truss members caused slight
undulations in the stress plateau. Eventually the cellular micro-truss structures
reached a densification stage. As expected, the strain at the onset of densification
was dependent on the relative density of the cellular material.

5
4
3
2
1
0
Compressive Stress, σ (MPa)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Nominal Strain, ε
Peak Strength
Plateau Stress
Densification
1: ρ /ρ
s
= 13.5%
2: ρ /ρ
s
= 16.2%
3: ρ /ρ
s
= 12.7%
4: ρ /ρ
s
= 13.8%
5: ρ /ρ
s
= 12.0%
5
4
3
2
1
0
Compressive Stress, σ (MPa)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Nominal Strain, ε
Peak Strength
Plateau Stress
Densification
1: ρ /ρ
s
= 13.5%
2: ρ /ρ
s
= 16.2%
3: ρ /ρ
s
= 12.7%
4: ρ /ρ
s
= 13.8%
5: ρ /ρ
s
= 12.0%

Figure 3-4 Compression behavior of five polymer micro-truss samples with different
relative density and/or waveguide angle. Each sample initially exhibits linear elastic
behavior prior to reaching their respective peak strength. With subsequent loading, the
cellular structures reach a plateau stress and eventually densification.

37
3.4.1 Compressive Modulus
As shown in Figure 3-5a, the micro-truss samples exhibited linear elastic
behavior prior to the onset of buckling. The compressive modulus for each sample
was determined from the slope of the linear regression curve generated from the
nominal stress-strain data in the range 0.01 ≤ ε ≤ 0.02. Table 3-2 summarizes these
data, along with the predicted moduli determined from Equation (3.1).

4
3
2
1
0
Compressive Stress, σ (MPa)
50x10
-3
40 30 20 10 0
Nominal Strain, ε
2: E = 182 MPa, σ
p
= 3.5 MPa
1: E = 152 MPa, σ
p
= 2.9 MPa
5: E = 123 MPa,
σ
p
= 2.3 MPa
4: E = 103 MPa, σ
p
= 2.2 MPa
3: E = 96 MPa, σ
p
= 1.5 MPa
(a)
4
3
2
1
0
Compressive Stress, σ (MPa)
50x10
-3
40 30 20 10 0
Nominal Strain, ε
2: E = 182 MPa, σ
p
= 3.5 MPa
1: E = 152 MPa, σ
p
= 2.9 MPa
5: E = 123 MPa,
σ
p
= 2.3 MPa
4: E = 103 MPa, σ
p
= 2.2 MPa
3: E = 96 MPa, σ
p
= 1.5 MPa
4
3
2
1
0
Compressive Stress, σ (MPa)
50x10
-3
40 30 20 10 0
Nominal Strain, ε
2: E = 182 MPa, σ
p
= 3.5 MPa
1: E = 152 MPa, σ
p
= 2.9 MPa
5: E = 123 MPa,
σ
p
= 2.3 MPa
4: E = 103 MPa, σ
p
= 2.2 MPa
3: E = 96 MPa, σ
p
= 1.5 MPa
(a)

16
14
12
10
8
6
4
2
0
Specific Stress, σ/ρ [MPa/(g/cm
3
)]
50x10
-3
40 30 20 10 0
Nominal Strain, ε
θ ≅ 60°
θ ≅ 50°
(b)
16
14
12
10
8
6
4
2
0
Specific Stress, σ/ρ [MPa/(g/cm
3
)]
50x10
-3
40 30 20 10 0
Nominal Strain, ε
θ ≅ 60°
θ ≅ 50°
16
14
12
10
8
6
4
2
0
Specific Stress, σ/ρ [MPa/(g/cm
3
)]
50x10
-3
40 30 20 10 0
Nominal Strain, ε
θ ≅ 60°
θ ≅ 50°
(b)

Figure 3-5 (a) Close-up of the linear elastic region, and (b) the slopes for micro-trusses with
the same waveguide angle are equivalent when normalized by the respective density.
38
Table 3-2 Comparison of the predicted and measured compression modulus (±SD) of the micro-truss
samples.

Prediction Measured % error
1173 152 ± 0.7 + 13.9%
2 209 182 ± 1.1 + 14.7%
3 110 96 ± 1.0 + 15.0%
4 120 103 ± 0.6 + 16.7%
5 155 123 ± 0.5 + 25.8%
Compression Modulus, E (MPa)
Sample

The predicted modulus consistently exceeds the measured modulus by
approximately 15% for Samples 1-4 and 25% for Sample 5. However, the analytical
prediction is based on an idealized geometry and includes the assumption that all the
strain energy during deformation of the bulk sample is equal to the combined strain
energy in the truss members due only to axial deformation. Not accounted for in this
prediction is any reduction in predicted modulus due to edge effects and additional
minor modes of deformation, such as bending, caused by imperfections in the
structure. The increased error in the predicted modulus for Sample 5 can be
explained by its microstructure. This sample has a lower relative density, which
increases the truss member’s susceptibility to non-ideal deformation.
From Equation (3.1), the ratio E/ρ should be constant for polymer micro-truss
samples constructed from waveguides of equal angle. In Figure 3-5b, the nominal
compressive stress is divided by the density of each respective sample. As shown in
39
the plot, the slope of the specific stress-nominal strain curve (i.e., E/ρ) for samples of
equal θ are collinear prior to reaching peak stress.
3.4.2 Peak strength
The compression data for the solid polymer and the buckling theory
discussed in previous sections were used to predict the truss-member failure stress
for different slenderness ratios (l/r). The predictions were based upon material
yielding, inelastic buckling (calculated from the tangent modulus of the solid
polymer), and Euler buckling. Pinned end constraints were assumed (k = 1) for the
buckling predictions. The actual stress in the truss member at initial failure was
calculated from Equation (3.2) using the measured peak stress and relative density of
each sample. The predicted truss member failure stress is plotted in Figure 3-6 as a
function of slenderness ratio, along with the measured values. As shown, the
slenderness ratios of the samples tested fall within the inelastic bucking regime;
however, as shown in Figure 3-6, initial failure occurs at a truss-member stress well
below the values predicted from inelastic buckling theory.

40

Figure 3-6 Calculated truss-member failure stress in the compressed samples plotted as a
function of slenderness ratio. The predictions for failure by material yielding, inelastic
buckling (based on the tangent modulus), and Euler buckling are also included and the data
indicate these structures fall in the inelastic buckling regime. The peak stress for all the
samples tested is well below that predicted by inelastic buckling theory.

The theoretical predictions assume an ideal column (truss member) loaded at
the centroid of its cross section [37]. For an ideal, symmetric micro-truss structure
with ideal loading conditions, truss members connected at a node should buckle
uniformly. Also, during such uniform buckling, each node should remain parallel to
the z-axis, even without truss members connecting adjacent nodes in the x-y plane.
However, direct observations of samples during compression revealed that non-
symmetric buckling and off-axis displacement of the interior nodes occurred. This is
evident in post-compression micrographs of Sample 5, shown in Figures 3-7a and 3-
7b. The sample was compressed to the densification stage, which eventually caused
fracture in a layer of interior nodes parallel to the compression surface (these are the
41
top layer of nodes in the micrographs). Upon unloading, the polymer micro-truss
partially recovered to the state shown in the images.

750 μm
(a)
750 μm 750 μm
(a)

540 μm
(b)
540 μm 540 μm
(b)

Figure 3-7 SEM images showing the buckling behavior of the polymer micro-truss
structures.

The non-uniform buckling behavior leads to a reduced peak stress and is
attributed to microstructural imperfections and the nonlinear (viscoelastic) behavior
of the polymer. The imperfections cause non-ideal load transfer between the truss
members and generate bending moments at the nodes. This also accounts for the
difference in the measured micro-truss moduli from idealized predictions. However,
the moduli are determined at low strain levels (ε < 0.02) before any instability is
reached. As the strain level is increased, the nonlinear behavior of the polymer
becomes a factor and causes a reduction in the solid polymer modulus. This
42
decrease in modulus, coupled with the imperfections, will generate greater bending
and/or twisting of the truss members and increase the non-ideal load transfer at the
nodes. Unlike a true octahedral unit cell [4], the octahedral-type structure shown in
Figure 3-2 does not have truss members in the x-y plane that would resist off-axis
nodal movement. This leads to a structurally instability that is reached at a stress
level well below the predicted stress required to buckle a single truss member under
ideal conditions.
3.4.3 Peak strength with high temperature post-cure in air
In the previous section, the peak strength for the micro-truss structures was
dictated by the non-ideal buckling behavior of the truss members. To suppress
buckling and thus increase the peak strength of these micro-trusses, one must reduce
imperfections in the microstructure and/or increase the stiffness of the truss-members
by increasing the associated moment of inertia (I) and/or compressive modulus (E).
Currently, the cross-sectional geometry of the truss members and slight
imperfections in the microstructure are considered inherent to the fabrication
process, so increasing the polymer modulus is the only viable option to increase the
truss member peak strength.
A higher temperature or longer post-cure cycle can increase the crosslink
density of a thermoset polymer, but this generally will have little effect on the elastic
modulus, which is primarily dependent on the molecular backbone [38, 40, 41].
However, thermo-oxidation reactions of polymers, which commonly occur at high
43
temperatures in the presence of oxygen, generally alter the molecular backbone and
can cause an increase in modulus [42, 43].
To quantify the effect of a higher temperature post-cure on the compressive
modulus of the photopolymer, a solid polymer sample was first cured under UV light
(30 s at ~7.5 mW/cm
2
) and then post-cured at 250°C for 24 hrs in air. This
temperature was selected based on thermogravimetric analysis of the photopolymer,
which indicated rapid mass loss (indicating polymer degradation) occurred above
300°C. After post-cure, the solid polymer was dark amber, representative of a
surface oxidation reaction.
Compression tests were conducted on the solid polymer sample following the
same procedures used previously for solid samples, and the data is shown in Figure
3-8a. The high temperature post-cure cycle in air had little effect on the peak tangent
modulus (an increase of < 5%), but the yield strength of the polymer increased by
over 17%. This phenomenon was attributed to a higher crosslink density. To
corroborate these results, a dynamic mechanical analysis (DMA) was conducted on
additional solid polymer samples using a 3-point bending fixture. As shown in
Figure 3-8b, the room temperature storage modulus is approximately the same for
the polymer post-cured under different conditions. However, the solid sample post-
cured at 250°C in air exhibits an increase in both the glass transition temperature (T
g
)
and the storage modulus above T
g
, which is characteristic of a greater crosslink
density [44].
44

Figure 3-8 (a) The nominal compressive stress-strain curve and tangent modulus, and (b) the
storage modulus as a function of temperature for solid samples post-cured at 130°C and
250°C.

The color change caused by post-curing the polymer indicated that a surface
oxidation reaction had occurred, and the reaction depth reflected the extent of
oxygen diffusion [45-48]. The surface oxidation layer was approximately 50μm
45
thick (Figure 3-9) – occupying only a small fraction of the overall volume of the
solid polymer samples. Thus, the measured modulus of the oxidized solid polymer is
not representative of the surface oxidation layer.

25 μm
Surface
oxidation
layer
Oxidation layer
transition
Un-oxidized
polymer
25 μm 25 μm
Surface
oxidation
layer
Oxidation layer
transition
Un-oxidized
polymer

Figure 3-9 Surface oxidation layer on the solid polymer sample.

Given the size scale of the micro-truss members, the oxidized layer should
penetrate through virtually the entire truss cross-section during the alternative post-
cure cycle. Therefore, any change in the modulus of the oxidized polymer should
affect the modulus of the micro-truss structures.
An additional micro-truss sample (Sample 6) was fabricated under the same
conditions as Sample 1 and 2, but post-cured at 250°C in air rather than 130°C under
vacuum. During post-cure, the mass of the micro-truss structure decreased by 18%
and the thickness decreased by 9% (the area did not change because it was
46
constrained by the quartz plates). The density of the solid polymer comprising the
micro-truss structure after post-cure increased to 1.42±0.01 g/cm
3
(measured by gas
pycnometry). The increase in density of a polymer due to thermo-oxidation is
commonly attributed to the incorporation of “heavy” oxygen atoms, which replace
carbon and hydrogen in the molecular structure [49]. Taking into account this
change in density of the polymer, the resulting micro-truss sample had a relative
density of 14.6%.

Figure 3-10 Comparison of the compressive response for the micro-truss samples post-
cured at 130°C (vacuum) and 250°C (air).

47
After the high temperature post-cure, Sample 6 was compressed to
densification, and the nominal stress-strain data is presented in Figure 3-10. Both the
compressive modulus and peak strength (E = 230 MPa, σ
p
= 5.94 MPa) increased
substantially after the high temperature post-cure (compared to Samples 1 and 2),
indicating an increase in the modulus of the oxidized polymer. We assumed
Equation (3.1) overestimated the modulus of the micro-truss structure by 15%
(similar to Samples 1 and 2), then used this equation, along with the measured
compressive modulus of Sample 6, to estimate the modulus of the oxidized solid
polymer. The calculation yielded E
s, oxid
= 3.2 GPa, which is substantially different
from the non-oxidized polymer modulus (E
s
= 2.4 GPa). Also, according to ideal
buckling theory, the stress at initial buckling of the truss members should scale
linearly with the modulus. However, Sample 6 exhibited a greater proportional
increase in peak strength than in modulus, when compared to Samples 1 and 2. The
thermo-oxidation of the polymer apparently suppressed the non-linear elastic
behavior. Alternative factors such as residual (internal) stresses arising during the
heat treatment may also be involved.
3.5 Summary
Compression tests were conducted on polymer micro-truss structures formed
from a three-dimensional interconnected array of self-propagating polymer
waveguides. Because of the highly ordered cellular architecture of these materials, a
simple analytical model can be used to predict the compressive modulus. The model
48
yielded relatively good agreement between predicted modulus values and those
measured experimentally. However, the peak strength at initial truss member
buckling was 1.5 - 3.5 MPa, well below idealized predictions. The lower-than-
expected peak strength values were attributed to imperfections in the as-cured
structure and the nonlinear compressive behavior of the solid polymer. To increase
the solid polymer modulus and thus decrease the propensity for the truss members to
buckle, a higher temperature post-cure cycle (in air) was conducted on an additional
micro-truss sample. The surface oxidation reaction of the polymer during the post-
cure cycle increased the modulus of the polymer by approximately 40%, which
almost doubled the peak strength of the micro-truss structure. The extremely small
truss member diameter permitted near-complete oxidation of the polymer during the
post-cure, which enabled the observed increase in micro-truss strength.
The results highlight potentially important effects that size scale can have on
both polymer and non-polymer cellular structures. In this case, we observe a
significant change in structural strength resulting from a material change caused by a
surface oxidation reaction. Similar changes in structural properties can be effected
by other approaches. For example, thin coatings could be applied to the surface of
the micro-trusses to alter mechanical properties. Alternatively, the polymer truss
structure could be used as a sacrificial template and replaced with other types of
materials, such as metals, ceramics, or graphite. With such additions and/or
substitutions, new cellular materials with unique properties can be designed and
49
manufactured. Further reduction in the size scale of the truss members will
undoubtedly give rise to interesting property changes as well, including a higher
modulus due to molecular alignment of the polymer chains and suppression of
fracture due to an overall increase in surface energy (and thus increased strength). In
addition, the new fabrication technology utilized in the present study constitutes a
new paradigm for designing and synthesizing lightweight micro-structures with
enhanced properties and the capacity to introduce multiple functions.
50

CHAPTER 4: SHEAR BEHA VIOR
⊕

4.1 Introduction
Shear loading conditions are commonly encountered in engineering
applications, and thus understanding the shear response of new materials and
structures is essential. The shear behavior of cellular materials, i.e. materials with
significant porosity, is governed primarily by the cellular architecture and the
uniaxial properties of the solid material of which they are comprised [1]. For a
cellular material with a randomly oriented architecture, Gibson and Ashby [1] have
shown that the cell struts tend to exhibit bending under mechanical load. However,
for ordered, truss-like cellular architectures, bending of the cell struts can be
suppressed, improving the shear modulus and strength of the overall bulk material
[4, 13, 50].

⊕
This chapter is based on work included in the following paper: Jacobsen et al., “Shear behavior of
micro-scale truss structures formed from self-propagating polymer waveguides.” Acta Materialia. In
review.

51
Investigations on the shear behavior of micro-truss-type cellular materials
that exhibit stretch-dominated deformation modes are fairly limited, partly because
of the shortage of suitable synthesis methods. One technique for creating cellular
materials with micrometer-scale truss features involves folding a two-dimensional
grid (formed using soft-lithography and subsequent electrodeposition) into a three-
dimensional truss-structure. Brittan et al. [24] tested a beam formed through this
technique under four-point bending, which exerted a constant shear force on a
portion of the truss structure. Although, the bending stiffness and strength were less
than a simple box beam of equivalent weight, the truss structure afforded opportunity
for optimization, which they reported could greatly enhance the mechanical
performance.
The metallic lattice structures described in Chapter 1 have also been studied
under shear loading conditions. Different cellular architectures have been
investigated experimentally, and the results indicate a strong correlation to analytical
predictions based on stretch-dominated behavior [6, 9, 15, 17, 20, 51].
In this chapter, we investigate the shear behavior of micro-truss structures
formed from an interconnected pattern of self-propagating polymer waveguides. The
unit cell configuration of the micro-truss structures investigated in this work was
equivalent to the structures discussed in the previous chapter (see Figure 3-2). Shear
loading was applied to each sample using a single-lap shear fixture, and the
measured moduli were compared to analytical predictions. In addition, the failure
52
mechanisms for each structure were closely examined to enable an accurate
prediction of the peak shear strength.

Figure 4-1 Top view of the unit cell defining the shear force direction ψ in the x-y plane.

4.2 Analytical predictions for shear modulus and shear strength
A top view of the repeating unit cell is shown in Figure 4-1. The shear force
(V) is defined in the x-y plane and the direction of the shear force is identified by the
angle ψ. Based on the symmetry of the octahedral unit cell, the shear response will
be periodic in ψ = π/2. A force-displacement analysis can be conducted on a 1/4
unit cell, as shown schematically in Figure 4-2.
53

Figure 4-2 A schematic of a partial unit-cell displaced Δ at node A from the shear force V.

A shear force V that generates a displacement Δ at node A (in the direction ψ
= 0°) will create a tensile reactive force (F
t
) in truss member AB and a compressive
reactive force (F
c
) in truss member AD. Truss members AC and AE will exhibit a
reactive moment (M) under this loading condition if nodes C and E are considered
fixed. To predict the effective shear modulus (G) of the structure, Equation (4.11)
was derived, assuming the external work (V Δ) on the structure was equal to the
work required to axially deform truss members AB and AD and to bend truss
members AC and AE. The steps to reach Equation (4.11) are shown below.
Assuming the magnitude of F
t
and F
c
are equal and the change in length (dl)
of the truss members in tension and compression is equal, a work balance can be
written as:
φ Md d F V
tm
2 2 + = Δ l ) )( ( (4.1)
where F
tm
= F
t
= F
c
. Assuming
54
l
b
F M = , (4.2)
EI
F
d
b
2
2
l
= φ , and (4.3)
3
3
l
Δ
=
EI
F
b
, (4.4)
and substituting Equations (4.2-4.4) into Equation (4.1) gives,

3
2
9
2
l
Δ
+ Δ = Δ
EI
F V
tm
) cos ( ) )( ( θ . (4.5)
Knowing σ = F/A and applying Hooke’s Law leads to the following,

3
2
9
2
l
Δ
+ =
EI
r E V
tm s
) )(cos ( θ π ε , (4.6)
where E
s
is the modulus of the solid polymer and ε
tm
is the absolute strain in the truss
members in tension and compression. Knowing ε
tm
= (dl/l) and dl = Δ cos θ,

3
2 2
9 2
l l
Δ
+
Δ
=
EI r E
V
s
θ π cos ) (
. (4.7)
Now, assuming the shear force V is applied of the area of the quarter unit cell (L
2
)
which means the shear stress is τ = V/L
2
, and assuming the shear modulus G = τ /γ
where the shear strain γ = Δ / (H/2), then

⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
=
2 3 2
2 2
2
9
L
H EI
L
H r E
G
s
l l
θ π cos ) (
. (4.8)
The relative density of a quarter unit cell is,

H L
r
s
2
2
2 l π
ρ
ρ
= . (4.9)
55
Substituting Equation (4.9) into Equation (4.8) gives,

⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
2
2
2
2
8
9
l l 8
r H E
G
s
s
θ
ρ
ρ
cos . (4.10)
Simplifying Equation (4.10) gives the equation,

⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
2
2
2
9
2
l 8
θ
θ
ρ
ρ sin
sin
r E
G
s
s
. (4.11)
If applied to the entire unit cell, the relative density of the structure defined as,

θ θ
π
ρ
ρ
sin cos
2 2
2
2
l
r
s
= . (4.12)
The term (9/2)(r sin θ / l)
2
in Equation (4.11) is the contribution to G from
the reactive moments generated from truss members AC and AE. For the unit cell
geometries under consideration here, this contribution is negligible. If a similar
analysis is conducted for a shear force in the direction ψ = 45° and only axial
deformation of the truss members is considered, an equivalent expression for the
effective shear modulus can derived. This indicates that because of the symmetry of
the microstructure, the elastic response due to a shear load applied in the x-y plane is
effectively independent of ψ [6].
While the effective shear modulus is independent of the shear force direction,
the shear strength of the octahedral microstructure depends on ψ. Assuming pinned
nodes, the shear strength (τ
p
) is given by,
56
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
s
p
p
ρ
ρ
ψ
θ
σ
τ
cos
sin 2
4
(4.13)
where τ
p
is periodic in ψ by symmetry of the unit cell [6]. The minimum shear
strength occurs at ψ = nπ/2 (for n = 0, 1, 2,…) and the maximum occurs at ψ =
(2n+1) π/4. In Equation (4.13), the peak axial failure stress in the truss members
(σ
p
) depends on the initial failure mode. Competing mechanisms that will ultimately
determine σ
p
include buckling of the truss members and tensile yielding or fracture
of the polymer.
4.3 Experimental
4.3.1 Micro-truss samples for shear loading
A total of nine micro-truss samples were fabricated for shear testing. All
samples were produced in using the fabrication set-up and procedures discussed in
Chapter 2. The mask consisted of a square pattern of circular apertures spaced 900
μm apart, each with a radius of 50 μm. The thickness of each micro-truss structure
was approximately 5 mm (3 unit cells), which was ultimately determined from the
depth of the mold containing the monomer. All samples were cut to approximately
20 mm x 20 mm square with a razor blade, and the sample dimensions were
measured using a digital caliper with 0.01 mm precision. The tolerance of each
measured dimension was ±0.1 mm.
57
After micro-truss fabrication, the structure remained attached to the quartz
plate that separated the mask and monomer. An additional quartz plate was fixed to
the opposite free surface with a thin layer (~50 μm) of the same photo-polymer. The
samples were then post-cured for 24 hrs at either 130°C under vacuum or 250°C in
air, as indicated in Table 4-1. After post-cure, the quartz plates were cut to the size
of the sample with a diamond wafering saw.

Table 4-1 Summary of the density, truss angle and post-cure temperature of the micro-truss samples
fabricated for shear testing.
Sample
Density,
ρ (g)
Measured
Truss Angle,
θ (deg)
Calculated
Truss Angle,
θ (deg)
Post-Cure
Temp*
(°C)
10.165 60 ± 2 59 ± 1.2 130
20.198 60 ± 2 59 ± 1.2 130
30.207 60 ± 2 59 ± 1.2 130
40.168 50 ± 2 51 ± 0.8 130
50.201 50 ± 2 51 ± 0.8 130
60.217 50 ± 2 51 ± 0.8 130
70.207 60 ± 2 59 ± 0.8 130
80.206 60 ± 2 59 ± 0.8 250
90.199 60 ± 2 59 ± 0.8 250
* Samples post-cured at 130°C were under vacuum
* Samples post-cured at 250°C were in air

The micro-truss structures were fabricated with a truss angle θ ≅ 50° or θ ≅
60°. Because it was difficult to measure the exact angle from the micrographs, the
58
reported truss angles in Table 4-1 were calculated from θ = tan
-1
(H/L), where H and
L represent the average unit cell dimensions measured from the micrographs. The
densities provided in Table 1 were calculated from the measured outer volume and
mass of each respective structure and the known density of the solid polymer.
Although theoretically the density of each structure should be equal, the process
itself led to variations in density of up to 30% as discussed in Chapter 2.
4.3.2 Parent material properties
The photo-monomer used for polymer micro-truss fabrication was a thiol-ene
monomer system, which included 0.05 wt% 2,2-dimethoxy-2-phenylacetophenone
(Aldrich) as the photoinitiator. As described in the previous section, the shear
response of micro-truss structures with octahedral unit cells will depend on the
tensile and compressive behavior of the solid polymer. Previously, we examined
the compressive behavior of this photopolymer. However, because the modulus and
strength of a polymer can vary greatly for tension and compression [38] we
conducted tensile tests following standard methods [16] to characterize the solid
polymer.
Different post-curing treatments were used for the micro-truss samples
subjected to shear testing, so multiple tensile samples of the photopolymer were
fabricated under these different conditions. Four solid polymer tensile samples were
first cured under collimated UV light generated from a mercury arc lamp. An
exposure time of 30 s at a power of approximately 7.5 mW/cm
2
was used and
59
measured from a UV photometer sensitive to wavelengths between 300-400 nm
(International Light, Inc). Two of the samples were then post-cured at 130°C for 24
hrs under vacuum, while two other samples were post-cured at 250°C in air. The
higher-temperature post-cure in air was intended to oxidize the polymer and thus
increase the modulus [42, 43]. In Figure 4-3, the tensile stress-strain curves for these
samples are compared to compression data obtained from the polymer processed
under the same conditions.

Figure 4-3 Comparison of the tensile and compressive response of the solid photo-polymer
with different post-cure conditions.

The tensile moduli for the four samples were determined from the initial
slope of their respective stress-strain curves, and the accuracy of all reported stress
60
data is ±2%. The average tensile modulus (E
s, t
) of the samples post-cured at 130°C
and 250°C was 2.2 GPa and 2.3 GPa, respectively – consistent with the measured
compressive modulus (E
s, c
= 2.4 GPa). As with the compression samples, the slight
increase in the modulus measured from the samples post-cured at 250°C in air is
attributed to an increase in cross-link density and the thin surface oxidation layer (~
50μm). Because the surface oxidation layer is only a small fraction of the overall
volume of the solid polymer, the tensile modulus measured from these samples does
not adequately represent the modulus of this oxidized region. We assert that the
tensile modulus of the oxidized polymer should be 3.2 GPa, based on the
compression experiments on oxidized micro-truss samples discussed in the previous
chapter.
The tensile yield strength of the polymer was 30-40% less than the respective
compressive yield strength, a finding that is consistent with observations for similar
polymers [38]. The tensile samples post-cured at 130°C under vacuum deformed
plastically prior to fracture. The onset of plastic deformation in the samples post-
cured at 250°C in air was delayed, and the fracture in these samples occurred at a
much lower strain. Both the increased cross-link density and an oxidized surface are
known to decrease the fracture toughness of a polymer, promoting brittle failure
[43].
The density of the non-oxidized polymer was measured using a gas
pycnometer and yielded a value of ρ
s
= 1.34 ± 0.01 g/cm
3
. The density of the
61
oxidized region of the polymer was measured from a micro-truss sample that had
been post-cured at 250°C in air. The high surface area per unit volume and the small
length scale of the micro-truss features resulted in nearly complete oxidation of the
polymer during the post-cure cycle. Thus the density measurement from the solid
polymer comprising the micro-truss sample post cured at 250°C, which yielded 1.42
± 0.01 g/cm
3
, was taken as the density of the oxidized polymer.
4.3.3 Shear experiments
A single-lap shear fixture was used to test the polymer micro-truss samples
under quasi-static shear loading conditions. The samples were bonded to steel shear
plates with an acrylic adhesive (3M Scotch-Weld DP810NS) and the shear plates
were attached to a hydraulic load frame as shown in Figure 4-4. This configuration
provided a load line approximately through the diagonal of the samples. The cross-
head displacement rate was 0.5 mm/min and the load was measured with a 15kN
built-in load cell. The relative displacement of the two shear plates was measured at
the sample with a laser extensometer.
62

Figure 4-4 Single-lap shear fixture used to test the micro-truss samples.

4.4 Results and discussion
The shear stress τ in the micro-truss samples was calculated as the measured
shear load V divided by the area of the sample attached to the shear plates (A). The
average shear strain γ was equal to Δ / t, where Δ is relative displacement measured
between the two shear plates and t was the thickness of the micro-truss structure
[52].
63
The nominal shear stress-stain curves for all samples described in Table 4-1
are shown in Figure 4-5 a-c. Samples 1-6 and Sample 8 were tested with a shear
load direction ψ = 0°. Samples 7 and 9 were tested with a shear load direction ψ =
45°. Figure 4-5a and 4-5c compare the shear response of micro-truss samples of
different density but equal truss member angle, shear loading direction, and post-cure
cycle. Figure 4-5c is a comparison of the oxidized (250°C post-cure in air) and non-
oxidized (130°C post-cure under vacuum) samples with different shear load
directions. The noticeable difference in the ultimate shear strain for Samples 1-7 is
related to the distribution of deformation through the thickness of each structure
during shear loading. The two oxidized samples (Samples 8 and 9) fractured prior to
any significant plastic deformation.

Figure 4-5 The nominal shear stress-strain curves for samples (a) post-cured at 130°C
(vacuum) with θ = 59° and a shear load direction ψ = 0°.
64

Figure 4-5, Continued (b) post-cured at 130°C (vacuum) with θ = 51° and a shear load
direction ψ = 0°, and (c) with θ = 59° but different post-cure temperature and/or shear load
direction.
65
4.4.1 Shear modulus
The shear modulus G for each sample was determined from the average slope
of the respective nominal shear stress-strain curve. Linear regression was used to
determine the slope between 25% and 75% of the respective peak shear stress values.
At a stress below 25% of the maximum for each sample, effects from the fixture on
the measured load were apparent. At a stress level above 75% of the maximum, the
shear stress-strain response for most samples began to deviate noticeably from linear
behavior. The measured shear moduli, along with the predicted shear moduli
calculated from Equation (4.11), are shown in Table 4-2.

Table 4-2 Summary of the shear properties for the different micro-truss samples.
Prediction Measured % error
1 12.3 59 0 28 22 ± 0.3 +28% 1.0 37
2 14.8 59 0 33 29 ± 0.1 +14% 1.2 38
3 15.4 59 0 35 31 ± 0.2 +10% 1.3 38
4 12.5 51 0 34 19 ± 0.1 +81% 0.9 30
5 15.0 51 0 41 29 ± 0.2 +43% 1.4 38
6 16.2 51 0 45 40 ± 0.3 +11% 1.6 35
7 15.4 59 45 35 34 ± 0.3 +1% 1.5 30
8 14.5 59 0 45 35 ± 0.1 +29% 2.7 83
9 14.0 59 45 44 47 ± 0.2 -6% 2.8 65
Shear Modulus, E (MPa)
Sample
Shear Force
Direction,
ψ (deg)
Measured
Shear Strength,
τ
p
(MPa)
Calculated Truss
Member Stress,
σ
p
(MPa)
Calculated
Truss Angle,
θ (deg)
Relative
Density
ρ/ρ
s

Samples 1-3, which had the same truss member angle θ and were tested
under the same shear load direction ψ, showed an increase in the deviation between
the predicted and measured shear modulus with a decrease in relative density.
66
Samples 4-6 exhibited a similar trend. These data indicate that as the relative density
of micro-truss structures with equal unit-cell geometry decreases, the structures
become more susceptible to non-ideal deformation, such as truss member bending
and twisting.
The data in Table 4-2 also implies that for micro-truss structures of
approximately equal relative density, there is a greater deviation between the
measured and predicted modulus when the truss member angle θ is reduced
(Samples 4-6). For a fixed micro-truss thickness, the polymer waveguides that form
the truss members have to propagate farther if there is a reduction in truss member
angle. Although theoretically, the shear modulus should increase as θ approaches
45°, the necessary increase in waveguide propagation distance leads to a greater
likelihood of initial structural defects, such as truss member curvature and/or
misalignment. This causes a greater reduction in the measured modulus, as reflected
in the data.
Sample 8 was post-cured in an oxidizing environment (250°C in air for 24
hrs), and had approximately the same relative density and truss member angle as
Sample 2. Thermo-oxidation of the polymer leads to an increase in solid polymer
modulus, which is directly proportional to the shear modulus of these structures, as
shown in Equation (4.11). As with Samples 1-6, the difference between predicted
and measured modulus values was attributed to non-ideal deformation, such as
bending of the truss members, prior to reaching the peak strength.
67
Two samples were tested with a shear force direction ψ = 45° (Samples 7 and
9). The measured modulus values for both samples were much closer to the
predicted values compared with Samples 1-6 and 8. While the shear modulus in
principle should be independent of ψ, these observations can be understood in terms
of the load distribution between the truss members for the different shear load
directions. When the shear force is applied in the direction ψ = 0°, nearly all the
load is carried by only half of the truss members (those parallel to the shear load, at
ψ = 0°). The truss members perpendicular to the shear load (at ψ = ± 90°),
contribute little to the shear resistance, as determined from Equation (4.11). If the
shear force is applied in the direction corresponding to ψ = 45°, the load is
distributed approximately equally between all truss members (half are in tension and
half are in compression). Thus, for a given applied shear load, the actual maximum
stress in the truss members is greater when ψ = 0° than when ψ = 45°. Under ideal
linear elastic conditions, this difference in load distribution would only affect the
shear strength of the micro-truss. However, we conclude that for polymer micro-
trusses, a more uniform distribution of load between the truss members reduces the
degree of non-ideal deformation in the truss members, thus increasing the measured
modulus.
68
4.4.2 Shear strength
The shear strength of the polymer micro-truss structures depends on the
initial failure mode of the truss members. As shown in Figure 4-3, the tensile yield
strength of the solid polymer is significantly less than the compressive yield strength,
and thus initial failure is expected in those truss members under tension. However,
in studies on the compression properties of these structures, buckling of the truss
members occurred at a stress level well below predicted values. These competing
mechanisms – compression buckling and tensile failure of the truss members – will
ultimately determine how the structures fail.
Table 4-2 shows the measured shear strength τ
p
for each sample, as well as
the estimated peak stress in the truss members at initial shear failure. The average
peak stress in the truss members was calculated by solving for σ
p
in Equation (4.13).
Samples 1-6 with a shear load direction ψ = 0° had an estimated average truss
member stress between 30-40 MPa at initial failure, which is approaching the tensile
strength of the polymer. This indicates that shear failure was likely to initiate from
the onset of tensile yielding, which subsequently caused the compressive members to
carry additional load and ultimately buckle.
After reaching the peak shear strength, continued loading produced a stress
plateau, followed ultimately by truss rupture. The net strain over which the stress
plateau occurred was markedly different between samples. Micrographs of the
structures after complete shear failure showed the total plateau strain was dependent
69
on how uniformly the deformation was distributed through the thickness of the
structure. For example, Figure 4-6a shows Sample 4, which had the greatest plateau
strain, after shear testing. The micrograph shows that truss members under
compression buckled uniformly through the thickness of the structure. In contrast,
Sample 2 produced roughly half the total plateau strain compared to Sample 4, and
the failure was localized in a single cell layer, as shown in Figure 4-6b. Figure 4-6c
is a close up of a fracture surface on Sample 2 that did not propagate through the
entire truss member.
860 μm
(a)
860 μm 860 μm
(a)

(b)
750 μm
(c)
(b)
750 μm 750 μm
(c)

50 μm
(c)
50 μm 50 μm
(c)

750 μm
(d)
750 μm 750 μm
(d)

Figure 4-6 SEM images displaying different shear failure modes in the micro-truss
structures. (a) Sample 4, (b) Sample 2, (c) close-up of a crack on a buckled truss member in
Sample 2, and (d) Sample 7.

70
As shown in Equation (4.13), the shear strength for a micro-truss sample with
a shear load direction ψ = 45° should be greater (by 41%) than an equivalent sample
with a shear load direction ψ = 0°. Samples 3 and 7 have equivalent unit cell
structures and relative density, yet the shear strength of Sample 7 (with ψ = 45°) is
only 15% greater than that of Sample 3 (with ψ = 0°). Examination of Sample 7
after shear testing (Figure 4-6d) reveals that failure occurred by polymer yielding at
the nodes along a single unit cell layer. The different mode of failure for this
structure accounts for the discrepancy between measured and predicted strength
values.
The two oxidized samples (8 and 9) showed a significant increase in shear
strength compared to the other samples, and this was consistent with expectations
based on previous compression experiments. Again, the analytical expressions
predicted different shear strengths for these two samples based on different shear
load directions. However, as with Samples 3 and 7, the shear strengths were
approximately equal. Also, as shown in Figure 4-5c, the brittle nature of these
samples led to abrupt failure at the peak load without the subsequent stress plateau
that was characteristic of the unoxidized samples.
Inspection of the fracture surfaces of Samples 8 and 9 provides insight into
the process by which these structures failed. Fractography defines three distinct
regions associated with the dynamics of a fracture surface: mirror, mist, and hackle
[53]. The mirror region is the smooth surface that forms during initial slow crack
71
growth in a material. As the stress in the material increases and the critical stress
intensity factor (K
Ic
) is reached, the crack becomes unstable and rapid crack growth
ensues. This causes the hackle region of a fracture surface. The mist region is the
transition between mirror and hackle and is often difficult to clearly identify [53].
Figure 4-7a is a micrograph of a fractured region in Sample 8, and Figure 4-7
b-d are close-up images of the fracture surfaces for adjacent individual truss
members. The direction of the shear load V (ψ = 0°) indicates the fracture surface
shown in Figure 4-7b corresponds to a truss member under compression. Based on
comparisons with fractography of optical fibers [54],we conclude that the fracture
surface shown in Figure 4-7b was caused by slight bending of the truss member
under compression. The mirror regions in Figure 4-7 c-e represent the degree of
crack propagation prior to failure. As cracks formed under tension and
bending/twisting in these truss members, the load carried in the compression member
increased. Upon failure of the compression truss members, the sudden increase in
load on the remaining truss members led to rapid failure, causing the distinct hackle
regions in Figure 4-7 c-e.
72
(d)
(b)
(e)
(c)
600 μm
V
(a)
(d)
(b)
(e)
(c)
600 μm
V
(a)

60 μm
(b)
60 μm 60 μm
(b)

43 μm
(c)
43 μm 43 μm
(c)

50 μm
(d)
50 μm 50 μm
(d)

38 μm
(e)
38 μm 38 μm
(e)

Figure 4-7 Truss member fracture surfaces of Sample 8 which had a shear force direction ψ
= 0°.

The fracture surfaces of a single unit cell in the thermally oxidized Sample 9
is shown in Figure 4-8a. Figure 4-8 b-e are enlargments of the individual fractured
truss members. The location of the mirror and hackle regions in these truss members
with respect to the shear load direction indicates that bending and twisting of all truss
members occurred prior to failure. In addition, the similarity in all four fracture
surfaces suggests a more uniform distribution of load between the truss members in
73
comparison to Sample 8. Theory predicts that failure in Sample 9 should occur at a
greater stress than Sample 8. However, the complex state of stress in each truss
member generated from the constraint at the nodes was not taken into account in the
analytical prediction. We believe this complex state of stress contributed to the
reduced shear strength observed for the micro-truss structure.
(b)
(c)
(d)
(e)
230 μm
V
(a)
(b)
(c)
(d)
(e)
230 μm
V
(a)

43 μm
(b)
43 μm 43 μm
(b)

43 μm
(c)
43 μm 43 μm
(c)

43 μm
(d)
43 μm 43 μm
(d)

43 μm
(e)
43 μm 43 μm
(e)

Figure 4-8 Truss member fracture surfaces of thermally oxidized Sample 9, in which the
shear force direction was ψ = 45°.
74
4.5 Summary
Shear experiments were conducted on polymer micro-truss structures. The
results showed that for structures of equivalent unit cell geometry, the measured
shear modulus deviated from predicted values in inverse proportion to the relative
density. When the shear force was applied in a direction that distributed the load
uniformly between all truss members (ψ = 45°), the sensitivity to non-ideal
deformation was reduced, and the measured shear modulus values were consistent
with analytical predictions.
The shear strength of the micro-truss structures depended on the tensile and
compressive strength of the polymer, as well as on the failure mode of the structure.
For the unoxidized polymer micro-truss structures, tensile yielding promoted
bucking of the truss members under compression. The ultimate plateau strain after
initial truss member failure was determined by non-localized distribution of buckling
members through the thickness of the structure. When buckling was distributed
evenly through the thickness, uniform plateau shear strains of up to 60% are
possible. However, when buckling was localized primarily in a single layer, this
plateau strain was reduced two-fold or more. Thus, to maximize energy absorption
from shear failure – which is directly proportional to the area under the shear stress-
strain curve – designing and fabricating a structure in which failure is uniformly
distributed through the thickness is essential.
75
To increase the shear strength and modulus of these structures, micro-truss
samples were post-cured in an oxidizing environment. The shear strength and
modulus increased as expected, but the trusses were embrittled and complete fracture
occurred in the structure at the peak load. While this behavior is undesirable for
energy absorption, similar post-cure cycles could be utilized to produce ultra-
lightweight open-celled structures with high shear and bending strength and stiffness.
Additional improvements in the overall shear properties of the polymer micro-truss
structures should be possible through further reductions in feature size and new
processing methods that promote molecular alignment along truss members,
analogous to polymer fiber processing, thereby enhancing the tensile properties of
the trusses.

76

CHAPTER 5: HEXAGONAL-BASED UNIT CELL
ARCHITECTURES
⊕

5.1 Introduction
The mechanical properties of cellular materials depend strongly on the
architecture and physical distribution of the solid material from which the cellular
structure is comprised. Open-cellular materials with a random architecture, such as
polymer foams, have been shown to exhibit bending-dominated behavior in the cell
struts during elastic loading [4]. To increase the modulus and strength of cellular
solids, open-cellular truss topologies have been proposed [50], with the majority of
research focusing on metallic, three-dimensional truss structures [8, 12, 13, 15, 16,
20, 50].
The metallic truss topologies studied to date stem from the available
fabrication techniques, which include investment casting, perforated metal sheet

⊕
This chapter is based on work included in the following paper: Jacobsen et al., “Micro-scale truss
structures with three-fold and six-fold symmetry formed from self-propagating polymer waveguides,”
In preparation.
77
forming, and wire or hollow tube lay-up [12, 13]. Some of the unit cell
configurations that can be produced by these fabrication techniques include
tetrahedral [10], pyramidal [18], and three-dimensional Kagome structures. Hyun et
al. [51] used finite element simulations to show that the Kagome architecture was
superior to the tetrahedral-based structures under compression and shear loading.
Wang et al. [55] verified these results experimentally, concluding the three-
dimensional Kagome architecture outperformed both the tetrahedral and pyramidal-
based truss structures.
Although the general three-dimensional Kagome structure has been shown to
be mechanically efficient, fabrication techniques for such structures have been
limited. In this chapter, a mask with a hexagonal pattern of apertures is used to
create two distinct unit cell architectures with three-fold and six-fold symmetry,
which can be reduced to Kagome sub-cells. These cellular structures were tested
under compression to investigate the effect of waveguide connectivity on the
mechanical performance of these materials.
5.2 Micro-truss fabrication with hexagonal mask pattern
A polymer waveguide can be formed within a photosensitive monomer from
a single point exposure of light [31-33, 36]. As discussed in Chapter 2, exposing a
volume of photomonomer to multiple, angled collimated UV (ultraviolet) light
beams through a mask with a two-dimensional pattern of apertures, multiple self-
propagating waveguides originate from each aperture. The number of waveguides
78
formed at each aperture and the direction and angle of these waveguides depends on
the number of collimated exposure beams and the angle and direction of the beams at
the mask surface. The waveguides naturally intersect during the formation process,
so if a two-dimensional exposure surface is patterned appropriately, it produces an
ordered three-dimensional open-cellular architecture. A schematic representation of
this process is shown in Figure 5-1a. In previous chapters, the micro-truss cellular
materials formed from this process utilized a mask with a square pattern of circular
apertures to create a repeating octahedral-type unit cell. By substituting a mask with
a hexagonal pattern of apertures (Figure 5-1b), new architectures with three-fold and
six-fold symmetry can be created. The cell highlighted in Figure 5-1a is two unit
cells thick and displays the Kagome sub-cell that can be formed.

79
a
Collimated UV light
Liquid
monomer
Polymer
waveguide
3D Kagome
architecture
Quartz
a
Collimated UV light
Liquid
monomer
Polymer
waveguide
3D Kagome
architecture
Quartz

Top View
Collimated
UV light
Hexagonal
mask pattern
Major axis
Minor axis
L
h
b
Top View
Collimated
UV light
Hexagonal
mask pattern
Major axis
Minor axis
L
h
Top View
Collimated
UV light
Hexagonal
mask pattern
Major axis
Minor axis
L
h
b

Figure 5-1 (a) Schematic of the set-up for creating micro-truss structures with an
interconnected array of self-propagating waveguides and (b) the top view of the mask with a
hexagonal pattern of circular apertures.

5.2.1 Unit cell architecture
Two of the unit cell structures that can be created using a hexagonal mask
pattern are shown in Figure 5-2. Figure 5-2a is a unit cell formed from three equally
angled exposure beams, where each beam is aligned along the major axis of the hex
pattern (see Fig. 1b) but rotated 120° apart with respect to the mask normal. The
tetragonal architecture of the waveguides in this unit cell will form a Kagome sub-
cell if the structure is two unit cells thick, as shown in Figure 5-1a. Figure 5-2b is
80
the unit cell that is formed if six angled exposure beams are used. Each of these
beams is also aligned with a major axis of the hex pattern but rotated 60° with
respect to the mask normal.
L
H
θ
l
3n
(a)
L
H
θ
l
3n
(a)

(b)
H
L
θ
Primary nodes
Secondary
nodes
l
6n
(b)
H
L
θ
Primary nodes
Secondary
nodes
l
6n

Figure 5-2 The two unit cell architectures that were formed using the mask with apertures in
a hexagonal pattern. (a) The unit cell formed with three incident UV exposure beams and (b)
the unit cell formed with six exposure beams.

Notice that the formation of six self-propagating waveguides from each
aperture on the mask surface leads to a three-dimensional structure with increased
waveguide connectivity within the described unit cell volume. This unit cell
structure is comprised of two types of intersecting nodes: primary and secondary
nodes. The primary nodes are formed from six intersecting waveguides and the
secondary nodes are created from the intersection of two waveguides. The
secondary nodes are located at the midpoint of each waveguide connecting adjacent
primary nodes. Micrographs of cellular structures formed with these unit cell
architectures are shown in Figure 5-3.

81
(a) (b) (c)

(d) (e) (f)

Figure 5-3 (a) Top view, (b) perspective view, and (c-d) side views of the polymer micro-
truss structures formed with three UV exposure beams. (e) Top view and (f) perspective
view of the architecture formed with six intersecting waveguides.

5.2.2 Relative density
Relative density is defined as the density of a cellular material (ρ) divided by
the density of the solid material (ρ
s
) from which it is comprised. By definition, it is
also a measure of the solid volume fraction of a cellular material, which can be
determined from the geometric parameters of the repeating unit cell, such as
82
waveguide radius (r), length (l), and angle (θ). An equation used to estimate the
relative density for a structure with three-fold symmetry (Figure 5-2a) is given
below.

2
3
2
2
2 2 2
3
3 2
2
3 3
9
n
n
s
H L
H L
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
l
r r
θ θ
π π
ρ
ρ
sin cos
(5.1)
The structures with six-fold symmetry have twice the number of waveguides
per unit cell and increased waveguide connectivity. Taking into account the reduced
truss member length (l
6n
= 0.5 l
3n
) between nodes, the relative density for this unit
cell architecture is calculated from the following equation.

2
6
2
6
3
n
n
s
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
l
r
θ θ
π
ρ
ρ
sin cos
(5.2)

5.3 Prediction of compressive modulus and peak strength
The compressive modulus of an open-cellular material (E) that exhibits
compression-dominated behavior in the struts during elastic loading can be predicted
from the following expression [6, 11],
E = E
s
sin
4
θ (ρ /ρ
s
) (5.3)
where E
s
is the modulus of the solid material. The stress in each solid truss member
(σ
s
) can be calculated from the compressive stress applied to the cellular material
(σ), the waveguide angle, and the relative density [6].
83
) / ( sin
s
s
ρ ρ θ
σ
σ
2
= (5.4)
The peak stress of the bulk cellular material can be predicted from Equation (5.4),
assuming the maximum compressive stress that can be sustained in the truss
members before buckling, yielding, or fracture is known. The truss member failure
stress under ideal conditions can be predicted as a function of slenderness ratio (l/r),
as shown in Figure 5-4 (see Chapter 3 for discussion on material yield, inelastic
buckling, and Euler buckling).

60
50
40
30
20
10
0
Truss-Member Failure Stress (MPa)
50 40 30 20 10 0
Slenderness Ratio, l/r
Inelastic buckling
Euler buckling
Material
yielding
60
50
40
30
20
10
0
Truss-Member Failure Stress (MPa)
50 40 30 20 10 0
Slenderness Ratio, l/r
Inelastic buckling
Euler buckling
Material
yielding

Figure 5-4 The predicted truss-member failure stress as a function of slenderness ratio for
the photopolymer used to form the micro-truss structures.

Truss members with small slenderness ratios (< 5) will fail by material
yielding. As the slenderness ratio is increased, failure will transition to inelastic
buckling and eventually Euler buckling [37].
84
If the number of load bearing truss members in a unit cell volume is doubled
and the angle of the truss members θ remains equal (as is the case from Figure 5-2a
to Figure 5-2b), the radius of each truss member must be reduced by a factor of √2/2
to maintain a constant relative density. Thus, increasing the number of truss
members without changing the relative density will not change the stress in each
truss member, only the slenderness ratio. If secondary nodes are neglected, the
slenderness ratio of the truss members will increase by a factor of √2, which in turn
will decrease the maximum theoretical failure stress of the cellular structure.
However, the secondary nodes formed at l/2 in the structures with six-fold symmetry
will decrease the slenderness ratio of each truss member by √2/2, thereby increasing
the theoretical failure stress in comparison to the structures with three-fold
symmetry.
If the waveguide overlap at the nodes is considered when the number of truss
members in a unit cell volume is doubled, the actual reduction in waveguide radius
necessary to maintain a constant relative density is less than a factor of √2/2. The
mass that would occupy this overlapping region can be redistributed to increase the
truss member radius, and thus decrease the stress in each truss member. This effect
becomes more significant if the number of waveguides intersecting at each nodal
region is further increased. However, increasing the number of waveguides that
intersect at a node creates a stress concentration at that nodal region, which may
85
cause material yielding at the node prior to reaching the expected truss member
failure stress.
In reality, these micro-truss structures have imperfections, such as waveguide
misalignment, that will lower the actual truss member failure stress. Also, as a result
of the formation process, the nodal regions can overcure, reducing the stress
concentration effect. Compression experiments were conducted on multiple micro-
truss structures with the repeating unit cells shown in Figure 5-2 to better understand
these competing mechanisms.
5.4 Experimental
5.4.1 Parent material properties
All micro-truss samples were formed with a thiol-ene polymer system
previously characterized (see Chapter 2 and 3). The modulus of the solid polymer E
s
is 2.4 GPa and the density ρ
s
is 1.34 ± 0.01 g/cm
3
.
5.4.2 Polymer micro-truss samples for compression loading
Six micro-truss samples were fabricated for compression testing using a
hexagonal mask pattern, as described in Section 5.2. The aperture radius was 75 μm
and the spacing L
h
between adjacent apertures was 1731 μm. Three samples
(Samples 1-3) were produced using three incident UV beams, resulting in a
microstructure with the repeating unit cell shown in Figure 5-2a. The remaining
three samples (Samples 4-6) were produced with six incident UV beams to form
86
micro-truss structures with the unit cell shown in Figure 5-2b. Each collimated beam
was generated from a mercury arc lamp with a fluence of ~ 7.5 mW/cm
2
at the mask
surface. The angle of each incident light beam off the mask substrate surface
remained constant for all samples (~ 20°). Due to light refraction in the quartz, the
resulting waveguide angle θ was approximately 51°.
The mold containing the liquid monomer during micro-truss fabrication was
approximately 5 mm deep. During the formation process the waveguides first
propagate to the bottom surface of the mold and increased exposure time causes each
waveguide to thicken. Eventually, if exposure is continued, the entire volume of
monomer will cure.
The exposure time for Samples 1-3 was 70 s. Significantly shorter exposure
times did not yield a structure that was sufficiently self-supporting when removing
the uncured monomer in toluene. This relatively long exposure time formed
waveguides with a radius of approximately 120 μm. The exposure time for Samples
4-6 was 35 s, which was determined experimentally to yield micro-truss structures
with a density approximately equal to Samples 1-3.
All samples were post-cured for 24 hours at 130°C under vacuum while still
attached to the 1.5 mm thick quartz plate that separated the mask and monomer.
After post-cure, each sample was cut into the shape of a hexagon with a razor blade.
The surface area dimensions and thickness were measured within ± 0.1 mm using
digital calipers. The mass of each sample attached to the quartz plate was measured
87
on a scale with 0.001 g accuracy. The mass of each quartz plate was measured prior
to sample fabrication, and subtracted to determine the mass of each sample.
Samples 1-3 had a hexagonal “footprint” with 12 unit cells per edge (~ 20
mm). Samples 4, 5, and 6 were cut in hexagonal shapes with 9, 10, and 11 unit cells
per edge, respectively, to determine if edge effects played a significant role on the
compression properties of samples this size. The measured micro-truss parameters
for all samples tested under compression are summarized in Table 5-1. As discussed
in Chapter 2, the small variation in density between similar samples was attributed to
sensitivities in the process.
To constrain the nodes at the waveguide terminating surface of each sample
during compression, an additional quartz plate was attached to the open surface of
the structures with a thin layer of the same photopolymer used to fabricate the
structures. A compression sample with six-fold symmetry sandwiched between two
quartz plates is shown in Figure 5-5.

Table 5-1 Summary of the measured parameters for the micro-truss structures tested under
compression.
No. of
incident UV
light beams
Density
ρ (g/cm
3
)
Relative
density
ρ / ρ
s
(%)
Waveguide
length
l (mm)
Waveguide
radius
r (μm)
Waveguide
angle*
θ (deg)
No. of Unit
Cells / Edge
1 3 0.081 6.1 2.8 120 ± 5 51 ± 1 12
2 3 0.087 6.5 2.8 120 ± 5 51 ± 1 12
3 3 0.090 6.7 2.8 120 ± 5 51 ± 1 12
4 6 0.087 6.5 2.8 90 ± 5 51 ± 2 9
5 6 0.085 6.3 2.8 90 ± 5 51 ± 2 10
6 6 0.086 6.4 2.8 90 ± 5 51 ± 2 11
Sample
Measured micro-truss parameters
* Waveguide angle measured relative to initiating substrate surface

88

Figure 5-5 Image of a polymer micro-truss sample sandwiched between two quartz plates.

5.4.3 Compression experiments
The six samples described in the previous section were subjected to
compressive loading using a hydraulic load frame and a constant strain rate of 2 x 10
-
3
s
-1
. The displacement was measured using a laser extensometer with 0.001 mm
precision. The applied force was measured using a load cell with an accuracy of
±1%. The accuracy of all reported compression stress data is ±2%.
5.5 Results and discussion
All six samples exhibited a linear elastic region during the initial stage of
compression. As the compressive strain increased, the stress increased until the truss
members buckled, after which there was a marked decrease in stress. The stress
reached a minimum at approximately 20% strain for each sample and then gradually
89
increased until the strain reached approximately 60%, at which point densification
led to a rapid increase in stress. Figure 5-6 is the nominal compressive stress-strain
data for two samples (Sample 2 and 4) of equal measured relative density, but
different unit cell architectures. The increase in peak strength and modulus of
Sample 4 is attributed to the reduced slenderness ratio in the six-fold symmetric
structure, as discussed in the following sections.

2.0
1.5
1.0
0.5
0.0
Compressive Stress, σ (MPa)
0.8 0.6 0.4 0.2 0.0
Nominal Strain, ε
Sample 4: E = 43 MPa, σ
p
= 0.71 MPa, ρ /ρ
s
= 6.5%
Sample 2: E = 25 MPa, σ
p
= 0.52 MPa, ρ /ρ
s
= 6.5%
Peak Strength
Plateau Stress
Densification
2.0
1.5
1.0
0.5
0.0
Compressive Stress, σ (MPa)
0.8 0.6 0.4 0.2 0.0
Nominal Strain, ε
Sample 4: E = 43 MPa, σ
p
= 0.71 MPa, ρ /ρ
s
= 6.5%
Sample 2: E = 25 MPa, σ
p
= 0.52 MPa, ρ /ρ
s
= 6.5%
Peak Strength
Plateau Stress
Densification

Figure 5-6 A comparison of the compressive response for the two different unit cell
architectures.

90
5.5.1 Compressive modulus
The compressive modulus E for the micro-truss samples was determined
from the average slope of the nominal stress-strain curve. The average slope was
measured between 25% and 75% of the respective peak stress values to avoid the
nonlinear behavior arising from preloading affects at low stresses and the onset of
buckling at higher stress levels. The measured compressive moduli are shown in
Table 5-2.

Table 5-2 Summary of the measured compressive properties.
1 35 ± 0.2 0.48 13.0
2 25 ± 0.2 0.52 13.2
3 31 ± 0.2 0.54 13.3
4 43 ± 0.3 0.71 18.1
5 42 ± 0.2 0.76 20.0
6 43 ± 0.4 0.74 19.1
Sample
Compressive
modulus
(MPa)
Peak
strength
(MPa)
Average peak truss
member stress
(MPa)

The modulus values for the samples with three-fold symmetry (Samples 1-3)
varied significantly (up to 40%). From Equation (5.3), the predicted modulus for a
micro-truss sample with waveguides at θ = 51° and a relative density of 6.5% is 57
91
MPa, while the measured values range between 25-35 MPa. The discrepancy in
measured and predicted values, along with the inconsistency of the measured
modulus values, is attributed to the long slenderness ratio of the truss members (l/r >
20) coupled with inherent imperfections. As l/r is increased, the truss members are
more susceptible to bending during elastic loading. The degree to which the truss
members bend prior to buckling will significantly affect the modulus of the micro-
truss material. This phenomenon relates to the original principal of developing open-
cellular structures that suppress bending during elastic loading, introduced earlier.
The samples with six-fold symmetry exhibit consistent measured modulus
values. However, these values are approximately 33% less than the modulus values
predicted from Equation (5.3). Closer examination of the microstructure revealed
the waveguide angle θ decreased through the sample thickness in structures
produced with this unit cell architecture. Although the incident angle of UV light
was constant for all samples in this study, the thinner truss members in Samples 4-6
allowed the structure to relax during subsequent processing steps. Figure 5-7a and
Figure 5-7b clearly show the difference in waveguide angle between the unit cell
adjacent to the initiating substrate (5-7a) and the unit cell at the terminating surface
of the waveguides (5-7b). An additional indication of relaxation appeared in the
measured thickness of Samples 4-6, which was an average of 4% less than the
thickness of Samples 1-3.

92

(a) (b)

Figure 5-7 A representative unit cell (a) adjacent to the initiating substrate surface and (b)
adjacent to the waveguide terminating surface for the structures with six-fold symmetry.
These SEM images show the change in waveguide angle through the thickness of the micro-
truss structure due to relaxation during processing.

From Equation (5.3), the modulus scales with sin
4
θ and thus is sensitive to
variations in waveguide angle. This was verified in Chapter 2 where micro-truss
structures with different waveguide angles were tested under compressive loading.
The change in truss member angle through the thickness accounts for the
disagreement between the measured and predicted moduli. For example, if an
average angle of θ = 46° is used in Equation (5.3), the predicted modulus for a
micro-truss with a relative density of 6.5% is 42 MPa (compared with 57 MPa when
θ = 51°).
Comparing the measured modulus values for samples with equal relative
density but differing architectures illustrates the effect of suppressing bending during
93
elastic
d

5.5.2 Peak strength
The peak strength for the polymer micro-truss materials is defined as the
to initial buckling failure. The measured peak strength for
ed
oth
(b) (c)

Figure 5-8 (a) A structure with three-fold symmetry and (b structure with six-fold symmetry
compressed just beyond the point of initial truss member buckling. (c) A close-up image of a
buckled truss member in ).
loading. Increasing the connectivity of the waveguides (as in the unit cell
shown in Figure 5-2b) reduced the effective slenderness ratio of the truss members,
thus decreasing the susceptibility to bending. Although the structures with six-fol
symmetry had a significantly reduced waveguide angle due to relaxation, the
measured modulus was still 20%-70% greater than the alternative unit cell structure.
maximum stress prior
each sample is listed in Table 5-2. SEM images of representative samples deform
to approximately 20% strain are shown in Figure 5-8. The images show that for b
unit cell architectures, the waveguides buckle at or near the midpoint between
adjacent nodes.

(a)
(b
94

ample e

the measured peak stress was repeatable.
, the
tress
described above is that the measured
the
The estimated stress in a single truss member at the peak strength of each
s was calculated using Equation (5.4), and these values are tabulated in Tabl
5-2. When the slenderness ratio of the truss members in the unit cell was decreased,
the average peak strength increased by over 40%. However, the maximum stress in
each truss member was still less than the theoretical buckling stress for ideal loading
conditions, as indicated in Figure 5-4.
For both unit cell architectures,
This was expected for Samples 4-6, which exhibited similar elastic response in
compression. However, based on the variation in modulus between Samples 1-3
repeatability in the peak strength of these samples was surprising. Based on the
measured cross-sectional area of the nodal regions, the maximum compressive s
at each node was calculated to determine if the yield stress was exceeded. The nodes
in Samples 1-3 and Samples 4-6 experienced a maximum stress of approximately 28
MPa and 47 MPa, respectively, which are below 65 MPa, the measured yield
strength of the polymer (see Figure 5-4).
One possible explanation for results
modulus of structures with long-slenderness-ratio truss members is dominated by
initial curvature in the waveguides and the peak strength is dominated by waveguide
misalignment at the nodes. With fewer intersecting waveguides at each node, nodal
stability presumably decreases, leading to rotations that initiate buckling at lower
95
stress levels. These results require further investigation of the role of imperfections
in these micro-truss structures.
5.5.3 Sample size effect
For all samples, the unit cell size was relatively large in comparison to the
overall bulk sample dimensions. Therefore, Samples 4-6 were cut to different sizes
to determine if edge effects played a significant role in the measured compressive
response. Sample 4 had a total of 9 unit cells per edge, while Sample 5 had one
additional unit cell per edge, and Sample 6 had two additional unit cells per edge.
The stress-strain response for these three samples, shown in Figure 5-9, was
repeatable and thus the sample dimensions did not affect the results.

2.0
1.5
1.0
0.5
0.0
Compressive Stress, σ (MPa)
0.8 0.6 0.4 0.2 0.0
Nominal Strain, ε
Samples 4 - 6
2.0
1.5
1.0
0.5
0.0
Compressive Stress, σ (MPa)
0.8 0.6 0.4 0.2 0.0
Nominal Strain, ε
Samples 4 - 6

Figure 5-9 The repeatability of the compressive strain-strain curve for samples with six-fold
symmetry.

96
5.5.4 Comparison between unit cell structures
In this study, a single hexagonal mask pattern was used to fabricate micro-
truss structures with the two unit cell structures shown in Figure 5-2. The
compression data demonstrated that the six-fold symmetric structures have higher
measured strength and modulus values than the structures with three-fold symmetry.
This increase in mechanical performance is attributed to the increased waveguide
connectivity in the unit cell, thereby reducing the effective slenderness ratio.
Although theoretically the slenderness ratio should only affect the peak strength and
not the modulus, we have shown that long-slenderness-ratio waveguides are more
susceptible to imperfections. Thus, unit cell architectures with the shortest
slenderness ratio for a given relative density result in higher strength and stiffness
values.
In the previous chapters, a square mask pattern was used to fabricate micro-
truss structures with an octahedral-type unit cell. An equation similar to Equation
(5.1) and (5.2) was derived to determine the relative density based on the geometric
parameters of this unit cell (see Chapter 3). Assuming an equal relative density and
truss member angle θ, a relationship between the slenderness ratios for the
octahedral-type unit cell (l/r)
4n
and the hex-based unit cell with six-intersecting-
waveguides can be derived.

2
4
2
2
6
2
2 3
n n
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
l
r
l
r
θ θ
π
θ θ
π
sin cos sin cos
(5.5)
97

n n 4 6
93 0
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
r
l
r
l
. (5.6)
This relationship indicates that for an equivalent relative density and truss
member angle, the slenderness ratio of the truss members in the unit cell shown in
Figure 5-2b is slightly less that the octahedral-type unit cell. In addition, the
hexagonal-based unit cells considered in this chapter should exhibit greater isotropy
in shear [55].
In Figure 5-10, the compressive behavior of Samples 2 and 4 is compared to
the behavior of a sample with a repeating octahedral-type unit cell. Although the
density of Samples 2 and 4 are approximately half the density of the octahedral
sample, the post-buckling response of the samples is interesting to compare. The
octahedral-based structure exhibits a sharp decrease in stress after initial buckling.
In contrast, the unit cells with three-fold or six-fold symmetry, which are a product
of multiple interconnected Kagome sub-cells, display a softer transition to the
plateau stress. This finding confirms previous studies on Kagome architectures [51,
55].
98
2.0
1.5
1.0
0.5
0.0
Compressive Stress, σ (MPa)
60x10
-3
50 40 30 20 10 0
Nominal Strain, ε
Sample 4
Sample 2
E = 96 MPa, σ
p
= 1.5 MPa, ρ /ρ
s
= 12.6%
2.0
1.5
1.0
0.5
0.0
Compressive Stress, σ (MPa)
60x10
-3
50 40 30 20 10 0
Nominal Strain, ε
Sample 4
Sample 2
E = 96 MPa, σ
p
= 1.5 MPa, ρ /ρ
s
= 12.6%

Figure 5-10 A comparison of the elastic and post-buckling response between the
architectures in this study and an octahedral unit cell with a similar truss member angle (θ ≅
51°). Note the octahedral unit cell is approximately twice the relative density of Samples 2
and 4.

5.6 Summary
The process of interconnecting self-propagating polymer waveguides has
been used to create micro-truss structures with two unit cell architectures. The unit
cell architectures were formed using the same two-dimensional mask pattern, but
differed in the number of incident UV exposure beams. One structure was formed
using three exposure beams, which resulted in a unit cell with three-fold symmetry.
The second structure, formed from six exposure beams, had a unit cell with six-fold
symmetry and featured primary nodes with six intersecting waveguides and
secondary nodes with two intersecting waveguides. Both sample structures were
comprised of multiple three-dimensional Kagome structures.
99
Compression experiments were conducted on polymer micro-truss samples
with the described unit cell architectures. All samples had approximately equal
relative density (ρ /ρ
s
= 6.5%) and truss member angle (θ ≅ 51°). The measured
compression modulus of the structures with six-fold symmetry was approximately
20-70% greater than the modulus values for the structures produced with three-fold
symmetry. The discrepancy in the measured and predicted moduli, as well as the
inconsistency in the modulus of the three-fold-symmetric structures, was attributed
to the difference in the truss member slenderness ratios between the competing
architectures. The secondary nodes in the six-intersecting-waveguide structures
reduced the slenderness ratio of the truss members without increasing the relative
density. This reduced the structure’s susceptibility to imperfections, such as initial
waveguide curvature. The smaller slenderness ratio also increased the peak strength
of the structures by an average of 42%, although these values were still below
idealized predictions.

100

CHAPTER 6: CONCLUSIONS AND FUTURE WORK
6.1 General conclusions
This research has focused on the development of a new process to form
polymer micro-truss structures and the mechanical characterization of the resulting
cellular materials. The process is based on interconnecting self-propagating polymer
waveguides in a three-dimensional pattern. By taking advantage of the self-
propagating phenomenon and the inherent ability of polymer waveguides to intersect
during formation, three-dimensional open-cellular structures were formed from a
single two-dimensional exposure surface on the order of seconds.
Compression and shear experiments were conducted on polymer micro-truss
samples formed from the described process. Various physical parameters were
altered in the samples, such as density, unit cell size, and truss member (waveguide)
angle, to better understand how these materials behave under different loading
conditions. The experimental results were compared to simple analytical predictions
of the strength and elastic modulus for each respective sample. The predictions were
101
based on pure stretch- or compression-dominated behavior of the waveguide truss
members prior to initial failure. The measured elastic response of the micro-truss
structures indicates these materials are capable of suppressing the bending-
dominated behavior that is characteristic of random cellular materials. However, the
elastic response, and to a greater extent, the peak strength, are highly sensitive to
structural imperfections, such as misalignment and the nodal regions and initial
waveguide curvature. Thus improving the process to minimize these imperfections
should improve the strength and modulus of these materials.
The majority of micro-truss samples fabricated and tested in this work had a
repeating octahedral-type unit cell. However, the flexibility of this synthesis process
to create micro-truss structures with different unit cell architectures was also
demonstrated. Two addition architectures, one with three-fold symmetry and one
with six-fold symmetry, were tested under compression and compared to the
octahedral structures. Although these samples had a lower compression modulus
and peak strength due to a lower relative density, the six-fold symmetric structures
are comprised of truss members with the shortest slenderness ratios (for a given
relative density), and thus have the greatest strength potential. Also, the micro-truss
structures with six-fold symmetric unit cells should exhibit greater isotropy in shear.
6.2 Future studies
The process presented in this dissertation has led to new materials that show
promise for increasing the modulus and strength of lightweight cellular solids.
102
However, many questions have been left unanswered. Can the polymer micro-truss
materials be used as templates to form a metal, ceramic, or carbon cellular material
with a truss architecture, and if so, do the mechanical properties of these materials
better align with idealized predictions? What other unit cell architectures are
possible with this technique, and how can manipulating the unit cell architecture lead
to structures that, for example, preferentially deform or fail in a specific manner?
These are just a few of the unanswered questions that surround this new process and
material.
In my opinion, however, the single most important unanswered question is
the effect of size-scale? If the process can be improved to reduce the feature sizes –
such as truss member diameter – below say, 10 μm, does this size-scale significantly
impact the bulk physical properties? Can the mechanisms that improve the
mechanical properties of fibrous materials over their bulk counterparts be realized in
the truss members of these structures?

103
REFERENCES
[1] L. J. Gibson and M. F. Ashby, Cellular solids: structure & properties. 2nd ed.
1997, Cambridge, UK: Cambridge University Press.

[2] ERG Materials and Aerospace Corporation. Website:
http://www.ergaerospace.com/

[3] P. Hansma. Website: http://hansmalab.physics.ucsb.edu/

[4] M. F. Ashby, Phil. Trans. R. Soc. A. 2006, 364, 15.

[5] R. Lakes, Nature. 1993, 361, 511.

[6] V. S. Deshpande and N. A. Fleck, Int. J. Solids Struct. 2001, 38, 6275.

[7] J. L. Grenestedt, International Journal of Solids and Structures. 1999, 36, 1471.

[8] D. T. Queheillalt and H. N. G. Wadley, Acta Mater. 2005, 53, 303.

[9] S. Chiras, D. R. Mumm, A. G. Evans, N. Wicks, J. W. Hutchinson, K.
Dharmasena, H. N. G. Wadley, and S. Fichter, Int. J. Solids and Struct. 2002,
39, 4093.

[10] G. W. Kooistra, V. S. Deshpande, and H. N. G. Wadley, Acta Mater. 2004, 52,
4229.

[11] M. Zupan, V. S. Deshpande, and N. A. Fleck, Eur. J.Mech. A. 2004, 23, 411.

[12] H. N. G. Wadley, Adv. Eng. Mater. 2002, 4, 726.

[13] H. N. G. Wadley, Phil. Trans. R. Soc. A. 2006, 364, 31.

[14] S. Hyun and S. Torquato, Journal of Materials Research. 2002, 17, 137.

[15] J. C. Wallach and L. J. Gibson, Int. J. Solids Struct. 2001, 38, 7181.

[16] N. Wicks and J. W. Hutchinson, Int. J. Solids Struct. 2001, 38, 5165.

[17] N. Wicks and J. W. Hutchinson, Mech. Mater. 2004, 36, 739.

104
[18] F. W. Zok, H. J. Rathbun, Z. Wei, and A. G. Evans, Int. J. Solids Struct. 2003,
40, 5707.

[19] F. W. Zok, S. A. Waltner, Z. Wei, H. J. Rathbun, R. M. McMeeking, and A. G.
Evans, Int. J. Solids Struct. 2004, 41, 6249.

[20] H. N. G. Wadley, N. A. Fleck, and A. G. Evans, Compos. Sci. Technol. 2003,
63, 2331.

[21] M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J.
Turberfield, Nature. 2000, 404, 53.

[22] S. Jeon, Y. S. Nam, D. J. L. Shir, A. Hamza, and J. A. Rogers, Appl. Phys. Lett.
2006, 89, 253101 1.

[23] J. H. Jang, C. K. Ullal, T. Choi, M. C. Lemieux, V. V. Tsukruk, and E. L.
Thomas, Adv. Mater. 2006, 18, 2123.

[24] S. T. Brittain, Y. Sugimura, O. J. A. Schueller, A. G. Evans, and G. M.
Whitesides, J. Microelectromech. Syst. 2001, 10, 113.

[25] A. R. Studart, U. T. Gonzenbach, E. Tervoort, and L. J. Gauckler, J. Am.
Ceram. Soc. 2006, 89, 1771.

[26] O. B. Olurin, D. S. Wilkinson, G. C. Weatherly, V. Paserin, and J. Shu,
Compos. Sci. Technol. 2003, 63, 2317.

[27] F. Romanato, R. Kumar, and E. D. Fabrizio, Nanotechnology. 2005, 16, 40.

[28] Y. K. Yoon, J. H. Park, and M. G. Allen, J. Microelectromech. Syst. 2006, 15,
1121.

[29] M. Deubel, G. V. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M.
Soukoulis, Nat. Mater. 2004, 3, 444.

[30] M. Geissler and Y. Xia, Adv. Mater. 2004, 16, 1249.

[31] A. S. Kewitsch and A. Yariv, Appl. Phys. Lett. 1996, 68, 455.

[32] A. S. Kewitsch and A. Yariv, Opt. Lett. 1996, 21, 24.

[33] M. Kagami, T. Yamashita, and H. Ito, Appl. Phys. Lett. 2001, 79, 1079.
105
[34] T. M. Monro, C. M. D. Sterke, and L. Poladian, J. Mod. Opt. 2001, 48, 191.

[35] T. Yamashita, M. Kagami, and H. Ito, J. Lightw. Technol. 2002, 20, 1556.

[36] S. Shoji and S. Kawata, Appl. Phys. Lett. 1999, 75, 737.

[37] J. M. Gere and S. P. Timoshenko, Mechanics of Materials. 1984, Monterey,
CA: Wadsworth, Inc.

[38] L. E. Nielsen, Mechanical Properties of Polymers and Composites. Vol. 2.
1974, New York: Marcel Dekker, Inc.

[39] ASTM International, Standard No. 695-02a, Standard test method for
compressive properties of rigid plastics. 2002.

[40] V. V. Kozey and S. Kumar, J. Mater. Res. 1994, 9, 2717.

[41] V. V. Kozey, H. Jiang, V. R. Mehta, and S. Kumar, Journal of Materials
Research. 1995, 10, 1044.

[42] M. Celina, A. C. Graham, K. T. Gillen, R. A. Assink, and L. M. Minier, Rubber
Chem. Technol. 2000, 73, 678.

[43] J. R. White, C. R. Chimie. 2006, 9, 1396.

[44] A. Rudin, The elements of polymer science and engineering. 2nd Ed. ed. 1999,
San Diego: Academic Press.

[45] N. Grassie and R. McGuchan, Eur. Polym. J. 1971, 7, 1357.

[46] M. Rodriguez-Vazquez, C. M. Liauw, N. S. Allen, M. Edge, and E. Fontan,
Polym. Degrad. Stab. 2006, 91, 154.

[47] Y. T. Shieh, H. T. Chen, K. H. Liu, and Y. K. Twu, J. Polym. Sci., Part A:
Polym. Chem. 1999, 37, 4126.

[48] B. E. Tiganis, L. S. Burn, P. Davis, and A. J. Hill, Polym. Degrad. Stab. 2002,
76, 425.

[49] V. Langlois, M. Meyer, L. Audouin, and J. Verdu, Polym. Degrad. Stab. 1992,
36, 207.

106
[50] V. S. Deshpande, M. F. Ashby, and N. A. Fleck, Acta Mater. 2001, 49, 1035.

[51] S. Hyun, A. M. Karlsson, S. Torquato, and A. G. Evans, Int. J. Solids Struct.
2003, 40, 6989.

[52] ASTM International, Standard No. 273-00, Standard test method for shear
properties of sandwich core materials.

[53] D. Hull, Fractography: observing, measuring, and interpreting fracture surface
topography. 1999, Cambridge U.K.: Cambridge University Press.

[54] C. P. Chen and T. H. Chang, Mat. Chem. Phy. 2002, 77, 110.

[55] J. Wang, A. G. Evans, K. Dharmasena, and H. N. G. Wadley, International
Journal of Solids and Structures. 2003, 40, 6981.

107
BIBLIOGRAPHY
Ashby, M. F. (2006). "The properties of foams and lattices." Phil. Trans. R. Soc. A
364: 15-30.

ASTM International, Standard No. 273-00, (2000). Standard test method for shear
properties of sandwich core materials.

ASTM International, Standard No. 695-02a, (2002). Standard test method for
compressive properties of rigid plastics.

Brittain, S. T., Y. Sugimura, et al. (2001). "Fabrication and mechanical performance
of a mesoscale space-filling truss system." J. Microelectromech. Syst. 10(1):
113-120.

Campbell, M., D. N. Sharp, et al. (2000). "Fabrication of photonic crystals for the
visible spectrum by holographic lithography." Nature 404: 53-56.

Celina, M., A. C. Graham, et al. (2000). "Thermal degradation studies of a
polyurethane propellant binder." Rubber Chem. Technol. 73(4): 678-693.

Chen, C. P. and T. H. Chang (2002). "Fracture mechanics evaluation of optical
fibers." Mat. Chem. Phy. 77: 110-116.

Chiras, S., D. R. Mumm, et al. (2002). "The structural performance of near-
optimized truss core panels." Int. J. Solids and Struct. 39: 4093-4115.

Deshpande, V. S., M. F. Ashby, et al. (2001). "Foam topology bending versus
stretching dominated architectures." Acta Mater. 49: 1035-1040.

Deshpande, V. S. and N. A. Fleck (2001). "Collapse of truss core sandwich beams in
3-point bending." Int. J. Solids Struct. 38: 6275-6305.

Deubel, M., G. V. Freymann, et al. (2004). "Direct laser writing of three-dimensional
photonic-crystal templates for telecommunications." Nat. Mater. 3: 444-447.

ERG Materials and Aerospace Corporation, (2007). Website:
http://www.ergaerospace.com.

Geissler, M. and Y. Xia (2004). "Patterning: principles and some new
developments." Adv. Mater. 16(15): 1249-1269.
108

Gere, J. M. and S. P. Timoshenko (1984). Mechanics of Materials. Monterey, CA,
Wadsworth, Inc.

Gibson, L. J. and M. F. Ashby (1997). Cellular solids: structure & properties.
Cambridge, UK, Cambridge University Press.

Grassie, N. and R. McGuchan (1971). "Pyrolysis of polyacrylonitrile and related
polymers - III." Eur. Polym. J. 7: 1357-1371.

Grenestedt, J. L. (1999). "Effective elastic behavior of some models for 'perfect'
cellular solids." International Journal of Solids and Structures 36: 1471-1501.

Hansma, P. (2007). Website: http://hansmalab.physics.ucsb.edu

Hull, D. (1999). Fractography: observing, measuring, and interpreting fracture
surface topography. Cambridge U.K., Cambridge University Press.

Hyun, S., A. M. Karlsson, et al. (2003). "Simulated properties of Kagome and
tetragonal truss core panels." Int. J. Solids Struct. 40: 6989-6998.

Hyun, S. and S. Torquato (2002). "Optimal and manufacturable two-dimensional,
Kagome-like cellular solids." Journal of Materials Research 17(1): 137-144.

Jang, J. H., C. K. Ullal, et al. (2006). "3D polymer microframes that exploit length-
scale-dependent mechanical behavior." Adv. Mater. 18: 2123-2127.

Jeon, S., Y. S. Nam, et al. (2006). "Three dimensional nanoporous density graded
materials formed by optical exposures through conformable phase masks."
Appl. Phys. Lett. 89: 253101 1-3.

Kagami, M., T. Yamashita, et al. (2001). "Light-induced self-written three-
dimensional optical waveguide." Appl. Phys. Lett. 79(8): 1079-1081.

Kewitsch, A. S. and A. Yariv (1996). "Nonlinear optical properties of photoresists
for projection lithography." Appl. Phys. Lett. 68(4): 455-457.

Kewitsch, A. S. and A. Yariv (1996). "Self-focusing and self-trapping of optical
beams upon photopolymerization." Opt. Lett. 21(24): 24-26.

109
Kooistra, G. W., V. S. Deshpande, et al. (2004). "Compressive behavior of age
hardenable tetrahedral lattice truss structures made from aluminium." Acta
Mater. 52: 4229-4237.

Kozey, V. V., H. Jiang, et al. (1995). "Compressive behavior of materials: Part II.
High performance fibers." Journal of Materials Research 10(4): 1044-1061.

Kozey, V. V. and S. Kumar (1994). "Compression behavior of materials: Part I.
Glassy polymers." J. Mater. Res. 9(19): 2717-2726.

Lakes, R. (1993). "Materials with structural hierarchy." Nature 361: 511-515.

Langlois, V., M. Meyer, et al. (1992). "Physical aspects of the thermal oxidation of
crosslinked polyethylene." Polym. Degrad. Stab. 36: 207-216.

Monro, T. M., C. M. D. Sterke, et al. (2001). "Topical review: catching light in its
own trap." J. Mod. Opt. 48(2): 191-238.

Nielsen, L. E. (1974). Mechanical Properties of Polymers and Composites. New
York, Marcel Dekker, Inc.

Olurin, O. B., D. S. Wilkinson, et al. (2003). "Strength and ductility of as-plated and
sintered CVD nickel foams." Compos. Sci. Technol. 63: 2317-2329.

Queheillalt, D. T. and H. N. G. Wadley (2005). "Cellular metal lattices with hollow
trusses." Acta Mater. 53: 303-313.

Rodriguez-Vazquez, M., C. M. Liauw, et al. (2006). "Degradation and stabilisation
of poly(ethylene-stat-vinyl acetate): 1 - Spectroscopic and rheological
examination of thermal and thermo-oxidative degradation mechanisms."
Polym. Degrad. Stab. 91: 154-164.

Romanato, F., R. Kumar, et al. (2005). "Interface lithography: a hybrid lithographic
approach for the fabrication of patterns embedded in three-dimensional
structures." Nanotechnology 16: 40-46.

Rudin, A. (1999). The elements of polymer science and engineering. San Diego,
Academic Press.

Shieh, Y. T., H. T. Chen, et al. (1999). "Thermal degradation of MDI-based
segmented polyurethanes." J. Polym. Sci., Part A: Polym. Chem. 37: 4126-
4134.
110
Shoji, S. and S. Kawata (1999). "Optically-induced growth of fiber patterns into a
photopolymerizable resin." Appl. Phys. Lett. 75(5): 737-739.

Studart, A. R., U. T. Gonzenbach, et al. (2006). "Processing routes to macroporous
ceramics: a review." J. Am. Ceram. Soc. 89(6): 1771-1789.

Tiganis, B. E., L. S. Burn, et al. (2002). "Thermal degradation of acrylonitrile-
butadiene-styrene (ABS) blends." Polym. Degrad. Stab. 76: 425-434.

Wadley, H. N. G. (2002). "Cellular metals manufacturing." Adv. Eng. Mater. 4(10):
726-733.

Wadley, H. N. G. (2006). "Multifunctional periodic cellular metals." Phil. Trans. R.
Soc. A 364: 31-68.

Wadley, H. N. G., N. A. Fleck, et al. (2003). "Fabrication and structural performance
of periodic cellular metal sandwich structures." Compos. Sci. Technol. 63:
2331-2343.

Wallach, J. C. and L. J. Gibson (2001). "Mechanical behavior of a three-dimensional
truss material." Int. J. Solids Struct. 38: 7181-7196.

Wang, J., A. G. Evans, et al. (2003). "On the performance of truss panels with
Kagome cores." International Journal of Solids and Structures 40: 6981-6988.

White, J. R. (2006). "Polymer ageing: physics, chemistry or engineering? Time to
reflect." C. R. Chimie 9: 1396-1408.

Wicks, N. and J. W. Hutchinson (2001). "Optimal truss plates." Int. J. Solids Struct.
38: 5165-5183.

Wicks, N. and J. W. Hutchinson (2004). "Performance of sandwich plates with truss
cores." Mech. Mater. 36: 739-751.

Yamashita, T., M. Kagami, et al. (2002). "Waveguide shape control and loss
properties of light-induced self-written (LISW) optical waveguides." J.
Lightw. Technol. 20(8): 1556-1562.

Yoon, Y. K., J. H. Park, et al. (2006). "Multidirectional UV lithography for complex
3-D MEMS structures." J. Microelectromech. Syst. 15(5): 1121-1130.

111
Zok, F. W., H. J. Rathbun, et al. (2003). "Design of metallic textile core sandwich
panels." Int. J. Solids Struct. 40: 5707-5722.

Zok, F. W., S. A. Waltner, et al. (2004). "A protocol for characterizing the structural
performance of metallic sandwich panels: application to pyramidal truss
cores." Int. J. Solids Struct. 41: 6249-6271.

Zupan, M., V. S. Deshpande, et al. (2004). "The out-of-plane compressive behaviour
of woven-core sandwich plates." Eur. J.Mech. A 23: 411-421.

Abstract (if available)

Abstract Materials with significant porosity, generally termed cellular materials, have considerably lower bulk density than their solid counterparts. However, at the cost of reducing the mass of a material by introducing porosity, mechanical properties such as the strength and elastic modulus are significantly diminished. Ordered cellular structures generally exhibit an increase in modulus and peak strength relative to random cellular configurations by changing the mode of deformation from bending-dominated to stretch/compression-dominated within the microstructure during elastic loading. Nevertheless, techniques to fabricate three-dimensional ordered open-cellular materials, particularly with feature sizes ranging from tens to hundred of microns, are limited. Presented in this dissertation, is a new technique to create cellular materials with a truss architecture from a three-dimensional interconnected pattern of self-propagating polymer waveguides. The self-propagating effect enables the rapid formation (< 1 min) of thick (> 5 mm) three-dimensional open-cellular micro-truss structures from a single two-dimensional exposure surface. The process also affords significant flexibility and control of the resulting truss microstructure.

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Asset Metadata

Creator Jacobsen, Alan J. (author)

Core Title Synthesis and mechanical evaluation of micro-scale truss structures formed from self-propagating polymer waveguides

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 10/31/2009

Defense Date 10/22/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cellular materials,lithography,mechanical properties,OAI-PMH Harvest,polymer waveguides,rapid prototyping

Language English

Advisor Nutt, Steven R. (committee chair), Hogen-Esch, Thieo E. (committee member), Kassner, Michael E. (committee member)

Creator Email ajacobse@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m896

Unique identifier UC1135734

Identifier etd-Jacobsen-20071031 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-559761 (legacy record id),usctheses-m896 (legacy record id)

Legacy Identifier etd-Jacobsen-20071031.pdf

Dmrecord 559761

Document Type Dissertation

Rights Jacobsen, Alan J.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu