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Optimization of Highly Architectured
Stereolithographic Microtrusses
by
Adam Gregory Bird
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Department of Materials Science and Engineering
University of Toronto
© Copyright by Adam Gregory Bird 2015
ii
Optimization of Highly Architectured
Stereolithographic Microtrusses
Adam Gregory Bird
Master of Applied Science
Department of Materials Science and Engineering
University of Toronto
2015
Abstract
Stereolithography allows architectural freedom which can be used to fabricate optimal architectures with
the potential for significant enhancements in structural efficiency. In this study, buckling behaviour of
compressive struts is explored. Experimental failure stress values for stereolithographic polymer tubes are
found to agree with existing predictive models for Euler and local shell buckling, validating the
methodology. Experimental testing on a space frame compressive strut design proposed in literature
reveals end constraints between fixed and free, and stereolithographic design freedom is considered to
reduce over-engineered features, improving performance. Finally, a novel sandwich wall tubular strut
design is introduced: experimental results show successful inhibition of local shell buckling while also
enhancing Euler buckling performance (improvements of 45% and 30% in failure strength when
compared to equal-mass simple tubes, respectively). A new failure mode termed “wall splitting” is
identified, and a preliminary model is developed to predict the increases in failure stress.
iii
Acknowledgements
The author would like to express his gratitude to Professor Glenn D. Hibbard for his support and guidance
throughout this thesis. The assistance of Ante Lausic, Khaled Abu Samk, and Craig Steeves is also
gratefully acknowledged. Their experience was most valuable.
Funding from NSERC, ORF, the Queen Elizabeth II scholarship, and the Materials Science and
Engineering Department at the University of Toronto is gratefully acknowledged.
The technical assistance of Dan Grozea and Sal Boccia is also very much appreciated.
The author especially appreciates the excellent discussions with Matthew Daly, Jean Hsu, and Martin
Magill. Their advice, insight, and alternative viewpoints spurred many ideas. Arno Glasser was a great
help with final editing.
Finally, a special thank you to family and friends – and critically, the U of T rowing and Nordic ski
teams, and the road cycling community of southern Ontario. They provided the perfect physical outlet to
allow fresh new thoughts.
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Table of Contents
Abstract ......................................................................................................................................................... ii
Acknowledgements ...................................................................................................................................... iii
Table of Contents ......................................................................................................................................... iv
List of Symbols ........................................................................................................................................... vii
List of Tables ............................................................................................................................................. viii
List of Figures .............................................................................................................................................. ix
1. Introduction ........................................................................................................................................... 1
1.1 Stereolithography .......................................................................................................................... 1
1.2 Microtruss Architectures ............................................................................................................... 2
1.3 Thesis Objective ............................................................................................................................ 7
2. Background Information ....................................................................................................................... 8
2.1 Failure Mechanisms of Tubes in Compression ............................................................................. 8
2.1.1 Euler Buckling .................................................................................................................... 11
2.1.2 Local Shell Buckling ........................................................................................................... 12
2.1.3 Competing Failure Mechanisms ......................................................................................... 13
2.2 Failure Mechanisms of Space Frame Compressive Struts .......................................................... 15
2.3 Optimality ................................................................................................................................... 21
3. Methods and Materials ........................................................................................................................ 25
3.1 Mechanical Testing ..................................................................................................................... 25
3.1.1 Load Frame ......................................................................................................................... 25
3.1.2 Digital Image Correlation ................................................................................................... 25
3.2 Stereolithographic Polymer Material Properties ......................................................................... 27
3.2.1 Finishing Process ................................................................................................................ 27
3.2.2 Printer Resolution ............................................................................................................... 28
3.2.3 Compressive Behaviour and Repeatability within a Print .................................................. 28
3.2.4 Polymer Property Variation Between Prints ....................................................................... 30
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3.2.5 Effects of Finishing Protocol on Polymer Properties .......................................................... 33
3.2.6 Effects of Printed Height on Polymer Stiffness .................................................................. 38
3.2.7 Material Property Summary ................................................................................................ 41
4. Polymer Tube Compression ................................................................................................................ 42
4.1 Characterization of Failure Modes .............................................................................................. 42
4.2 Equal-Mass Polymer Tubes in Compression .............................................................................. 47
4.3 Tubes of Greater Mass ................................................................................................................ 56
5. Novel Compressive Structures ............................................................................................................ 65
5.1 Mechanical Testing and Modeling of Space Frame Compressive Struts .................................... 65
5.1.1 Experimental Analysis ........................................................................................................ 65
5.1.2 Theoretical Improvement on the Basic Structure ................................................................ 72
5.2 Mechanical Testing and Modeling of Sandwich Wall Tubes ..................................................... 77
5.2.1 Proof of Concept ................................................................................................................. 78
5.2.2 Exploration of Gap and Number of Webs ........................................................................... 83
5.2.3 Sandwich Wall Tubes in Euler Buckling ............................................................................ 88
6. Conclusion .......................................................................................................................................... 90
6.1 Simple Polymer Tubes ................................................................................................................ 90
6.2 Space Frame Compressive Struts ................................................................................................ 92
6.3 Sandwich Wall Tubes ................................................................................................................. 92
7. Future Work ........................................................................................................................................ 93
7.1 Simple Polymer Tubes ................................................................................................................ 93
7.2 Space Frame Compressive Struts ................................................................................................ 93
7.3 Sandwich Wall Tubes ................................................................................................................. 93
References ................................................................................................................................................... 95
Appendix A: Compression Machine Compliance ...................................................................................... 97
Appendix B: Repeats of Tube Compression Tests ..................................................................................... 98
Appendix C: Repeats of Space Frame Compressive Strut Tests .............................................................. 100
vi
Appendix D: Mechanical Testing of Hollow Polymer Microtrusses ........................................................ 101
vii
List of Symbols
A Cross-sectional area
B Numerical constant for overall Euler buckling modification for space frame
Et Tangent modulus
FEuler Euler critical buckling force
I Second moment of area
K Effective length constant
L Overall structure length
L0 Length of constituent struts in space frames
M Maxwell’s number
b Number of struts
f Dimensionless force
f0 Dimensionless force on single constituent strut in a space frame
j Number of joints
k Spring constant of constituent struts in a space frame
n Number of octahedral in a space frame
n Wall thickness of a tube
Ratio of wall thickness to inner radius of a tube
r Strut radius in a space frame
r Inner radius of a tube
Ratio of inner radius to length of a tube
v Dimensionless volume
γ Local shell buckling correction factor
ν Poisson’s ratio
σEuler Euler critical buckling stress
σLSB Local shell critical buckling stress
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List of Tables
Table 4.1 Specifications of ten tubes of equal mass and length, but varying radius and wall thickness.
Note that tube 1j is actually a solid rod. ...................................................................................................... 47
Table 4.2 Specifications of two sets of ten tubes. Within each set, tubes are of equal mass and length, but
varying radius and wall thickness. .............................................................................................................. 56
Table 5.1 Specifications of the nine space frame compressive struts. Note that as the number of octahedra
is increased, the length of each constituent strut decreases accordingly to maintain a constant overall
length of 70 mm. ......................................................................................................................................... 67
Table 5.2 Specifications of the first set of sandwich wall tubes, as well as the simple tube base case (1f).
The length, inner radius, and mass are equal for all structures. .................................................................. 79
Table 5.3 Specifications of two sets of 11 tubes. Within each set, there is a comparison involving the
value of gap and the number of webs. As well, each set contains a simple tube base case to provide a
baseline for comparison. ............................................................................................................................. 84
Table 5.4 Specifications of the set of sandwich wall tubes designed to fail by Euler buckling, as well as
the simple tube base case (4f). The length, inner radius, and mass are equal for all structures. ................. 88
Table D.1 Specifications of constituent struts used in various microtruss compression tests. ................. 101
ix
List of Figures
Figure 1.1 Example of a microtruss sheet, showing a network of struts and nodes. This particular
microtruss employs a combination of tetrahedral and octahedral cells. Note that the image shown was
generated by the author using SolidWorks software..................................................................................... 3
Figure 1.2 Two-dimensional representation of bending-dominated (left, for which M=-1) and stretch-
dominated (right, for which M=0) structures [2]. ......................................................................................... 3
Figure 1.3 A range of three-dimensional shapes, along with an indication as to whether or not they are
stretch-dominated [3]. ................................................................................................................................... 5
Figure 1.4 Material property chart showing the drop in relative strength for a corresponding drop in
relative density. At low densities, stretch-dominated materials show an advantage over bending-
dominated materials [3]. ............................................................................................................................... 6
Figure 1.5 Material property chart showing the drop in relative modulus for a corresponding drop in
relative density. At low densities, stretch-dominated materials show an advantage over bending-
dominated materials [3]. ............................................................................................................................... 6
Figure 2.1 Left, an image of an aluminum rod which has failed compressively by Euler buckling. Note the
characteristic arc shape. Right, an aluminum tube which has failed compressively by local shell buckling.
One local shell buckling ring has completed at the top of the tube, while a second ring has started to bulge
partway down the tube. Note that the samples shown in this image were tested as part of undergraduate
laboratories at the University of Toronto. ..................................................................................................... 8
Figure 2.2 Parameters used to describe a tube: the wall thickness n, the inner radius r, and the length L.
Note that the image shown was generated by the author using SolidWorks software. ................................. 9
Figure 2.3 Schematic representation of various tube geometries corresponding to the dimensionless
parameters and . Note that the image shown was generated by the author using SolidWorks software.
.................................................................................................................................................................... 10
Figure 2.4 The process by which predictions for Euler buckling stress are made. The cyan line shows the
stress-strain curve of the polymer material. The three red lines depict the Euler critical buckling stress
x
equation for an value of 0.05 and a range of values. Note that the image shown was generated by the
author using MATLAB software. ............................................................................................................... 12
Figure 2.5 The process by which predictions for local shell buckling stress are made. The cyan line shows
the stress-strain curve of the polymer material. The three blue lines depict the local shell critical buckling
stress equation for an value of 0.1 and a range of values. Note that the image shown was generated by
the author using MATLAB software. ......................................................................................................... 13
Figure 2.6 The predicted failure stress for both Euler buckling and local shell buckling plotted over a
range of potential tube geometries. Note that the image shown was generated by the author using
MATLAB and OriginPro software. ............................................................................................................ 14
Figure 2.7 Failure map showing the active failure mode for a range of tube geometries. Note that the
image shown was generated by the author using MATLAB software. ...................................................... 15
Figure 2.8 A space frame compressive strut consists of a tetrahedra on each end with a number of
octahedra in between. It can be defined by the overall length L, strut radius r, strut length L0, and number
of octahedra used to span the length (in this figure, n=4). Note that the image shown was generated by the
author using SolidWorks software. ............................................................................................................. 16
Figure 2.9 Left, the end of a space frame compressive strut. Right, Euler buckling has occurred on an end
tetrahedral strut of this structure. Note that the image shown is a photograph taken by the author. .......... 19
Figure 2.10 Left, a space frame compressive strut. Right, Euler buckling has occurred on the overall
structure. Note that the image shown is a photograph taken by the author. ................................................ 19
Figure 2.11 A continuous ring constructed from the octahedral space frame design. By calculating the
elastic energy stored in this structure, the equivalent Euler buckling prediction equation can be inferred
[9]. ............................................................................................................................................................... 20
Figure 2.12 Comparison of the optimal designs of four different compressive structures. For each value of
dimensionless force, the lowest value of dimensionless volume required to resist that force is shown for
each structure. It is interesting to note that the design which results in the most efficient structure switches
as indicated by the dotted black line: to the right, a tube is the most efficient design; to the left, a hollow
space frame compressive strut is the most efficient design. Note that the image shown was generated by
the author using MATLAB software. ......................................................................................................... 23
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Figure 3.1 Polymer tube which has been prepared for DIC analysis by applying a spray-paint speckle
pattern to its surface. ................................................................................................................................... 26
Figure 3.2 Stress-strain data from five compression coupons prepared together. Note that the material
behaviour is similar so the lines are difficult to distinguish. ....................................................................... 29
Figure 3.3 Summarized compressive material properties from five compression coupons prepared
together. Note that the data points are clustered, showing good repeatability. ........................................... 30
Figure 3.4 Stress-strain data from 13 compression coupons, each from a separate print cycle. Note that
there is considerably more variation in material properties when compared to the repeatability observed in
Figure 3.2. ................................................................................................................................................... 32
Figure 3.5 Summarized compressive material properties from 13 compression coupons, each from a
separate print cycle. Note that the material properties are not as repeatable as they were for samples
produced in the same print cycle (Figure 3.3)............................................................................................. 33
Figure 3.6 Stress-strain curves for 10 compression coupons. Each coupon was allowed to sit for a
different amount of time between removal from the finisher and mechanical testing. ............................... 35
Figure 3.7 Progression of Young's modulus with sitting time between finishing and testing. The stiffness
of the material increases significantly until a plateau is reached. ............................................................... 36
Figure 3.8 Progression of 0.2% offset yield stress with time allowed to sit between finishing and testing.
.................................................................................................................................................................... 37
Figure 3.9 Stress-strain curves for 10 compression coupons of varying height: the standard 25 mm tall
compression coupon, and vertically scaled versions down to 2.5 mm in height. Note in particular the
reduced stiffness and increased yield stress of the 0.1 and 0.2 scaled coupons (green and blue,
respectively). ............................................................................................................................................... 39
Figure 3.10 Young's modulus increases with height for a series of vertically scaled compression coupons.
The height fraction range from 0.1 to 1.0 corresponds to a height change from 2.5 mm to 25 mm. .......... 40
Figure 3.11 0.2% offset yield stress decreases with height for a series of vertically scaled compression
coupons. A height fraction range from 0.1 to 1.0 corresponds to a height change from 2.5 mm to 25 mm.
.................................................................................................................................................................... 41
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Figure 4.1 Two tubes designed to verify failure modes: the left tube was predicted to fail by Euler
buckling, while the right tube was predicted to fail by local shell buckling. .............................................. 42
Figure 4.2 Force displacement curves for a tube which failed by (a) Euler buckling and (b) local shell
buckling. Insets show fractured tubes after testing. .................................................................................... 44
Figure 4.3 Digital image correlation analysis performed on a stereolithographic polymer tube expected to
fail by Euler buckling. The top images show the tube at the various stages of compression, with colours
corresponding to local strain values on the surface of the tube. The bottom graph shows the strain values
along the length of the tube (denoted by the thick black line on each tube image). The tube experiences
uniform strain until bifurcation occurs where the central outer face of the tube enters a state of tension.
Note that the high values of compressive strain at the ends of the red curve represent the corners of the
tube which have been crushed against the platen........................................................................................ 45
Figure 4.4 Digital image correlation analysis performed on a stereolithographic polymer tube expected to
fail by local shell buckling. The top images show the tube at the various stages of compression, with
colours corresponding to local strain values on the surface of the tube. The bottom graph shows the strain
values along the length of the tube (denoted by the thick black line on each tube image). The tube
experiences uniform strain until alternating bands of high and low strain appear near each end of the tube.
.................................................................................................................................................................... 46
Figure 4.5 The ten tubes in the first set, ranging from wide tubes with thin walls to narrow tubes with
thick walls. .................................................................................................................................................. 47
Figure 4.6 (a) Stress-strain curves of the four tubes which failed by local shell buckling, as well as the
tube which had the greatest failure stress. (b) Stress-strain curves of the five tubes which failed by Euler
buckling, as well as the tube which had the greatest failure stress. Note that the modulus is approximately
equal between all tubes until buckling occurs. Also note that tube 1a is has the greatest inner radius and
the lowest wall thickness, while tube 1j has the smallest inner radius and the greatest wall thickness. ..... 49
Figure 4.7 Experimental failure stress of the ten tubes. All tubes are of equal mass and equal length. It is
important to note that the displayed change in is accompanied by a hidden change in to maintain a
constant mass. ............................................................................................................................................. 51
Figure 4.8 Euler buckling failure stress predictions using effective length constants of 0.5 (fixed) and 1.0
(free). It is clear that for tubes expected to fail by Euler buckling, the experimental failure stress matches
more closely to the predictions using fixed end conditions. ....................................................................... 52
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Figure 4.9 Euler and local shell buckling failure stress predictions compared to the experimental failure
stress values. ............................................................................................................................................... 53
Figure 4.10 Comparison of Euler and local shell buckling predictions to experimental data for two print
cycles. Separate tubes and material compression data were used for each print cycle. .............................. 54
Figure 4.11 The ten tubes in the second set, ranging from wide tubes with thin walls to narrow tubes with
thick walls. .................................................................................................................................................. 57
Figure 4.12 The ten tubes in the third set, ranging from wide tubes with thin walls to narrow tubes with
thick walls. .................................................................................................................................................. 57
Figure 4.13 (a) Stress-strain curves of the five tubes in the second set which failed by local shell buckling,
as well as the tube which had the greatest failure stress. (b) Stress-strain curves of the four tubes in the
second set which failed by Euler buckling, as well as the tube which had the greatest failure stress. ....... 58
Figure 4.14 (a) Stress-strain curves of the five tubes in the third set which failed by local shell buckling,
as well as the tube which had the greatest failure stress. (b) Stress-strain curves of the four tubes in the
third set which failed by Euler buckling, as well as the tube which had the greatest failure stress. ........... 59
Figure 4.15 Predicted Euler and local shell buckling failure stresses compared to experimental failure
stresses for the second set of tubes.............................................................................................................. 60
Figure 4.16 Predicted Euler and local shell buckling failure stresses compared to experimental failure
stresses for the third set of tubes. ................................................................................................................ 61
Figure 4.17 Predictive model for local shell buckling. Note that all four tubes shown have an value of
0.15. Two comparisons are shown: the lower two lines show that for many tubes, the doubling of wall
thickness results in near-doubling of failure stress; the upper two lines show that for very robust tubes, the
doubling of wall thickness results in a relatively small change to failure stress. ........................................ 64
Figure 5.1 Example of a space frame compression strut. Note that to enable the strut to stand vertically for
compressive testing, polymer discs were built in to the ends of the structure during fabrication. ............. 66
Figure 5.2 The nine space frame compressive struts, ranging from five octahedra with long constituent
struts to 13 octahedra with shorter constituent struts. All structures are 70 mm in overall length. ............ 67
xiv
Figure 5.3 (a) Force-displacement curves of the three space frame compressive struts which failed by
Euler buckling of the end tetrahedral struts, as well as the structure which had the greatest failure force.
(b) Force-displacement curves of the four space frame compressive struts which failed by Euler buckling
of the overall structure, as well as the structure which had the greatest failure force. ................................ 68
Figure 5.4 Experimental failure force of the eight space frame compressive struts. Note that the structure
with six octahedra was damaged during preparation and removed from results. All structures are of equal
mass and equal length. It is important to note that the displayed change in number of octahedra is
accompanied by a hidden change in the radius and length of the constituent struts to maintain a constant
mass and overall length. The black dotted line indicates the transition in active failure mode between
Euler buckling of the end tetrahedral struts and Euler buckling of the overall structure. ........................... 69
Figure 5.5 Euler buckling failure stress predictions for both the end tetrahedral struts and the overall
structure using effective length constants of 0.6 for the struts and 0.75 for the overall structure. ............. 71
Figure 5.6 Comparison of the optimal designs of three different compressive structures. For each value of
dimensionless force, the lowest value of dimensionless volume required to resist that force is shown for
each structure. The solid space frame compressive strut is made lighter by allowing for different values of
radius for the constituent struts during the optimization process. Note that this improvement does not
make the solid space frame into a desirable structure: it is still significantly less structurally efficient than
a simple tube. .............................................................................................................................................. 74
Figure 5.7 Comparison of the optimal designs of three different compressive structures. For each value of
dimensionless force, the lowest value of dimensionless volume required to resist that force is shown for
each structure. The hollow space frame compressive strut is made lighter by allowing for variable values
of radius for the constituent struts during the optimization process. It is important to note that this
optimization was not actually performed – an estimated hypothetical line is shown based on performance
enhancements calculated for the solid structure. By making this change, the loading scenarios for which
the space frame shows greater structural efficiency than a tube is expanded into heavier loadings. .......... 76
Figure 5.8 Basic design of a tube whose wall is constructed similar to a sandwich panel: an inner and
outer wall connected by a series of webs. ................................................................................................... 77
Figure 5.9 Parameters used to describe a sandwich wall tube: the left schematic shows the cross-section of
the structure, while the right schematic shows a magnified section of the tube wall. The structure shown
here has 72 webs. ........................................................................................................................................ 78
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Figure 5.10 Magnified portions of the cross-sections of the six structures. The five images on the left
show the walls of the sandwich wall tubes with varying gap, while the image on the right show the wall of
the simple tube. Note that the simple tube has the same mass as each of the five sandwich wall tubes. ... 80
Figure 5.11 Stress-strain curves for the six tubes tested in this set. Tubes 1a-1e are sandwich wall tubes
with varying values for the gap. Tube 1f is a simple tube. ......................................................................... 80
Figure 5.12 Single I-beam slice, 72 of which comprise a sandwich wall tube. The second moment of area
of this I-beam is calculated about its neutral bending axis. ........................................................................ 81
Figure 5.13 Local shell buckling failure stress predictions compared to the experimental failure stress
values. Note that the right five experimental points correspond to sandwich wall tubes while the point on
the left corresponds to the equal-mass simple tube. .................................................................................... 82
Figure 5.14 Comparison of experimental failure stress to predicted local shell buckling stress. Two sets of
sandwich wall tubes are shown: one with 72 webs and one with 36 webs. Note that the experimental data
point on the left corresponds to the simple tube base case. Also note that local shell buckling predictions
are made for both 72 and 36 webs, but they are very similar due to the similar second moments of area for
the walls of each structure. The dotted lines connecting experimental data points serve only to guide the
eye. .............................................................................................................................................................. 85
Figure 5.15 Comparison of experimental failure stress to predicted local shell buckling stress. Two sets of
sandwich wall tubes are shown: one with 72 webs and one with 108 webs. Note that the experimental data
point on the left corresponds to the simple tube base case. Also note that local shell buckling predictions
are made for both 72 and 108 webs, but they are very similar due to the similar second moments of area
for the walls of each structure. The dotted lines connecting experimental data points serve only to guide
the eye. ........................................................................................................................................................ 86
Figure 5.16 Compressive testing of sandwich wall tubes with high values of the gap resulted in splitting
of the inner and outer wall away from the webs. ........................................................................................ 87
Figure 5.17 Comparison of experimental failure stress to predicted Euler buckling failure stress for
sandwich wall tubes designed to fail by Euler buckling. Note that the experimental point on the left
corresponds to the simple tube base case. ................................................................................................... 89
Figure 6.1 Euler and local shell buckling failure stress predictions are shown over a range of tube
geometries as well as a range of material properties. The upper, more pale surfaces show predictions
xvi
made for the stiffest observed stereolithographic polymer (Young’s modulus 1780 MPa) while the lower,
darker surfaces show predictions made for the least stiff observed stereolithographic polymer (Young’s
modulus 1470 MPa). ................................................................................................................................... 91
Figure 7.1 Comparison of stress-strain curves for a sandwich wall tube which failed by wall splitting
(blue) and a simple tube with the same length and mass (red). Note that the structure shows progressive
fracture, with a significant stress value maintained to strain values in excess of 15%. The shaded areas
show energy absorption. The sandwich wall tube shows greatly enhanced energy absorption when
compared to the simple tube which fractured. ............................................................................................ 94
Figure B.0.1 Comparison of experimental failure stress to Euler and local shell buckling predictions for
repeat sets of the second set of tubes. ......................................................................................................... 98
Figure B.0.2 Comparison of experimental failure stress to Euler and local shell buckling predictions for
repeat sets of the third set of tubes. ............................................................................................................. 99
Figure C.0.1 Comparison of experiment to prediction for a repeat set of the structures analyzed in Figure
5.7. Note that two structures here were broken during preparation and are excluded. ............................. 100
Figure D.0.1 Confinement shim placed around truss blocks during compression testing. ....................... 102
Figure D.0.2 First microtruss construction technique: top and bottom faces are constructed from the same
network of struts used in the core. ............................................................................................................ 103
Figure D.0.3 Microtruss block with upper and lower faces replaced with solid sheets. Note that holes were
added to these sheets to allow support wax to be removed. ...................................................................... 104
Figure D.0.4 Microtruss block with filleting to thicken the tube walls at the interface with the plates.... 105
Figure D.0.5 Microtruss block with no tube intersection at the nodes...................................................... 106
Figure D.0.6 Comparison of experimental failure force to predictions for Euler and local shell buckling
for microtrusses in compression. Note that predictions were made using a combination of the standard
models and basic trigonometry to sum the contributions from all struts. ................................................. 107
1
1. Introduction
1.1 Stereolithography
Stereolithographic additive manufacturing is an emerging technology with exciting potential for
producing complex, structural components. However, the properties generated by this new manufacturing
technique, along with their repeatability, require systematic exploration. This presents an opportunity to
study stereolithography as a manufacturing method for structural components, while simultaneously
revealing novel architectures previously off limits due to manufacturing constraints.
For example, previous work done by Schaedler et al. [1] has used a self-propagating photopolymer
waveguide technique to fabricate ultra-lightweight microtruss materials with impressive structural
properties. This technique uses collimated ultraviolet light and a patterned mask to solidify a liquid
photomonomer to form a network of struts. However, there are limitations to the architectural complexity
that can be achieved using this waveguide technique: nodal strengthening, complex strut designs, and
other enhanced features are simply not possible. This thesis explores the use of an alternative method to
fabricate polymeric microtrusses allowing complete architectural freedom: stereolithographic additive
manufacturing.
Stereolithographic additive manufacturing deposits material layer-by-layer to produce polymer parts.
While there are practical limitations such as layer thickness, in principle this technique is capable of
fabricating parts of any desired architecture with no limits on complexity. All experiments in this thesis
were performed on parts made by the 3D Systems ProJet HD3500 rapid prototyper. This machine
employs a multi-jet system to lay down liquid-state thermoset polymer (trade name VisiJet M3 Crystal)
part material and liquid wax support material (VisiJet S300). These two components are then cured using
an ultraviolet lamp to form a solid part.
The process begins by preparing a three-dimensional CAD file for the desired part using SolidWorks.
This part file is imported to the printer and converted into a file describing the series of discrete layers
which will form the final desired part. An aluminum base plate is used as a substrate onto which the part
is printed. The multi-jet printhead lays down material onto a moving stage to allow material deposition
anywhere in the two-dimensional plane. Repetitive printing of successive layers generates the third
(vertical) dimension. The printing process can be summarized in the following series of five steps:
1. A thin and uniform layer of support wax is printed to smooth imperfections in the aluminum
substrate plate and enhance adhesion of the part to the substrate.
2
2. The printer jets deposit liquid polymer to form a single layer of the desired part.
3. The printer jets deposit liquid wax support material to form a layer of support in regions which
are empty in the current layer but will contain polymer in higher layers.
4. An ultraviolet lamp is used to expose the liquid polymer and wax. This cures the liquid material,
forming a solid part.
5. The multi-jet printhead assembly is raised by a single layer thickness in order to print the
subsequent layer of the part.
Steps two through five are repeated to build successive layers of the part until completion. When the
printing operation is complete, an oven is used to melt the support wax material and reveal the desired
polymer part.
There are a variety of uncertainties and challenges surrounding the use of stereolithography for structural
components. It is expected that batch to batch variation in the series of steps could lead to variation in
material performance. The effect of surface roughness is another issue, as is the variation in material
properties between parts of different sizes. Despite these challenges, the architectural freedom afforded by
the use of stereolithography, and the possibility of coating these components with ultra high strength
nanomaterials, opens a new realm of possibility to the efficient design of complex structures.
1.2 Microtruss Architectures
Microtrusses (Figure 1.1) are an attractive option when low density, structural materials are desired.
When compared to other low density materials such as foams, microtrusses offer the potential for
improved strength and stiffness for a particular mass [2]. This improvement comes from the fact that
whereas foams are bending-dominated structures, microtrusses are stretch-dominated structures. The
network of struts and nodes comprising a microtruss forms a periodic cellular structure with high nodal
connectivity.
3
Figure 1.1 Example of a microtruss sheet, showing a network of struts and nodes. This particular microtruss
employs a combination of tetrahedral and octahedral cells. Note that the image shown was generated by the author
using SolidWorks software.
Figure 1.2 Two-dimensional representation of bending-dominated (left, for which M=-1) and stretch-dominated
(right, for which M=0) structures [2].
4
A given structure can be classified as stretch- or bending-dominated based on its relative number of struts
(b) and joints (j). For the two-dimensional case [3], a structure is stretch-dominated if:
M = b – 2j + 3 = 0
and for the three-dimensional case [3], if:
M = b – 3j + 6 = 0
This is most easily demonstrated with the two-dimensional case. Examining Figure 1.2 [2], it is clear that
a force applied to the bending-dominated structure (M<0) will result in rotation (“bending”) of the struts
about the joints. Conversely, the application of a force to the stretch-dominated structure (M=0) will result
in pure compression or pure tension in all struts. This correlates with the predictions from the equation:
while both structures have four joints, the stretch-dominated structure has five struts while the bending-
dominated structure has just four.
While this concept is simpler to visualize in two dimensions, the same ideas hold in three dimensions.
Figure 1.3 shows a variety of three-dimensional structures and whether or not they are stretch-dominated.
A microtruss composed of a periodic repetition of the stretch-dominated structures (e.g. tetrahedral and
octahedral) will itself be a stretch-dominated structure and yield the benefits attached to that
categorization.
5
Figure 1.3 A range of three-dimensional shapes, along with an indication as to whether or not they are stretch-
dominated [3].
As material is removed from a three-dimensional structure to lower the relative density, the relative
strength and relative stiffness will decrease (Figures 1.4-1.5). The rate of loss in stiffness or strength with
a corresponding decrease in density will vary depending on the way in which the material is removed. If
the density is lowered by forming a bending-dominated structure such as a foam, the strength will
decrease at a slope of 1.5 while the stiffness will decrease at a slope of 2.0. If, however, the density is
decreased by forming a stretch-dominated structure such as a microtruss, the strength and stiffness will
both decrease at a slope of 1.0 [3]. The result of this is that at low relative densities, stretch-dominated
structures will exhibit strength and stiffness characteristics which are significantly (potentially an order of
magnitude) higher than bending-dominated structures of similar material and relative density.
The structural efficiency of microtrusses makes them desirable in a variety of industries including
aerospace and automotive [4]. The potential to dramatically reduce the mass of structural components
while meeting load-related requirements is an exciting possibility.
6
Figure 1.4 Material property chart showing the drop in relative strength for a corresponding drop in relative density.
At low densities, stretch-dominated materials show an advantage over bending-dominated materials [3].
Figure 1.5 Material property chart showing the drop in relative modulus for a corresponding drop in relative density.
At low densities, stretch-dominated materials show an advantage over bending-dominated materials [3].
7
1.3 Thesis Objective
Microtrusses have a complex geometry and as such can be difficult or impossible to fabricate using
traditional methods and incorporating high performing materials. The application of stereolithographic
additive manufacturing to this challenge has the potential to be highly beneficial. This technique yields
complete freedom over the microtruss architecture, but also over the geometry of the individual struts
comprising the microtruss. The shape generated through additive manufacturing can go on to be used as a
template for the electrodeposition of structural metals such as nanocrystalline nickel [5].
An exploration of the ideal architecture for microtruss materials is critical. As stretch-dominated
structures, all struts in a microtruss experience either pure tension or pure compression. While the
strength of the tensile members is limited simply by material properties and cross-sectional area, the
compressive members present the added complexity of structural instability. As such, in order to optimize
a microtruss to find the lightest possible structure to resist a given loading scenario, the initial step must
be to optimize the structure of the compressive members.
Microstrusses and their constituent struts offer an excellent platform on which to explore the interplay
between stereolithographic additive manufacturing and structural components.
The goal of this thesis is to explore a variety of novel compressive strut designs while enhancing the
understanding of the stereolithographic additive manufacturing technique. The thesis is organized in the
following manner: in Chapter 2, background information is presented, particularly on the failure
mechanisms of structures; in Chapter 3, the polymer material and the methods used to study it are
described; in Chapter 4, polymer tubes are studied in compression; Chapter 5 examines a space frame
compressive strut design from literature and presents a novel sandwich wall tube design; Chapters 6 and 7
provide conclusions and discuss future work, respectively.
8
2. Background Information
This background section is a combination of a literature review and preliminary analysis performed by the
author to compare competing strut designs. Some figures were generated by the author to enhance the
reader’s understanding of the relevant theories.
2.1 Failure Mechanisms of Tubes in Compression
The potential failure mechanisms of simple tubes loaded in compression have been studied thoroughly
[6][7][8], and can be divided into two distinct types: Euler buckling and local shell buckling. Euler
buckling is a general arcing of a tube as a whole, and occurs, for example, when a pipe cleaner is loaded
compressively. It is a symptom of a tube being too narrow. Local shell buckling, on the other hand, is a
phenomenon in which the walls of the tube wrinkle and bend in on themselves, and occurs, for example,
when a can is crushed. It is a symptom of the walls of a tube being too thin. Figure 2.1 shows these two
failure mechanisms for aluminum rods and tubes.
Figure 2.1 Left, an image of an aluminum rod which has failed compressively by Euler buckling. Note the
characteristic arc shape. Right, an aluminum tube which has failed compressively by local shell buckling. One
local shell buckling ring has completed at the top of the tube, while a second ring has started to bulge partway
down the tube. Note that the samples shown in this image were tested as part of undergraduate laboratories at the
University of Toronto.
9
Both types of buckling can be modeled based on material properties and tube geometry. These models
are based around a tube as described in Figure 2.2: the tube length L, the tube inner radius r, and the tube
wall thickness n. Two dimensionless parameters will be used to simplify subsequent equations: is the
ratio of inner radius to length, while is the ratio of wall thickness to inner radius. These dimensionless
parameters will prove useful as they will allow the failure stress of a tube to be predicted over all design
space, regardless of scale. Figure 2.3 is a conceptual representation of the relation between the
dimensionless parameters and the tube geometry.
Figure 2.2 Parameters used to describe a tube: the wall thickness n, the inner radius r, and the length L. Note that the
image shown was generated by the author using SolidWorks software.
10
Figure 2.3 Schematic representation of various tube geometries corresponding to the dimensionless parameters and
. Note that the image shown was generated by the author using SolidWorks software.
11
2.1.1 Euler Buckling
The force at which a column in compression will fail by the Euler buckling mode is calculated using the
equation:
where FEuler is the Euler critical buckling force, Et is the tangent modulus of the constituent material, I is
the second moment of area of the column, and KL is the effective length of the column [6]. This last term
deserves some explanation: while L is the simple length of the column, K is the effective length constant.
If the ends of the column are free to rotate at both ends, K has a value of 1.0. Conversely, if the ends of
the column are both fixed, the effective length of the column is reduced and K has a value of just 0.5.
The equation can be altered slightly to calculate the Euler critical buckling stress of the column:
For the case of a tube:
The original equation for Euler critical buckling stress can now be re-stated in terms of the dimensionless
parameters and :
With this equation, the failure stress by Euler buckling of a tube of any given geometry and material
properties can be calculated. The use of a computational software package such as MATLAB is required
to incorporate the tangent modulus of the material. This is done by finding the intersection point of two
equations: for each strain value, the tangent modulus of the material is calculated and the Euler critical
buckling stress of the tube is found. For low values of strain, this value is greater than the actual stress
experienced by the material, and the tube does not buckle. Once a sufficient strain value is reached, the
Euler critical buckling stress reaches a value equal to the material stress, and it is at this point that the tube
buckles. This process is visualized in Figure 2.4.
12
Figure 2.4 The process by which predictions for Euler buckling stress are made. The cyan line shows the stress-
strain curve of the polymer material. The three red lines depict the Euler critical buckling stress equation for an
value of 0.05 and a range of values. Note that the image shown was generated by the author using MATLAB
software.
2.1.2 Local Shell Buckling
The process for calculating the local shell buckling stress is analogous to the process followed to calculate
the Euler buckling stress in the previous section. The major difference is, of course, the initial equation
[8]:
where ν is the Poisson’s ratio of the material (assumed 0.35 for the polymer used in this study) and γ is a
correction factor [7]:
13
This equation can easily be re-stated in terms of the dimensionless parameter :
A computational software package can now be used in a similar way as was employed for Euler buckling.
An example of the prediction process for local shell buckling is depicted in Figure 2.5.
Figure 2.5 The process by which predictions for local shell buckling stress are made. The cyan line shows the stress-
strain curve of the polymer material. The three blue lines depict the local shell critical buckling stress equation for
an value of 0.1 and a range of values. Note that the image shown was generated by the author using MATLAB
software.
2.1.3 Competing Failure Mechanisms
While both Euler buckling and local shell buckling are potential failure modes for a tube in compression,
the failure mode which actually occurs for a particular tube will depend on the geometry of that tube. For
given dimensions (and corresponding values of and ) the failure mode which occurs first will be that
14
which has the lowest predicted failure stress. For example, a tube which is weaker in the Euler buckling
mode will fail by Euler buckling.
It is useful to plot the predicted failure stress for Euler buckling and local shell buckling over a range of
potential values of and . The result of this is shown in Figure 2.6.
Figure 2.6 The predicted failure stress for both Euler buckling and local shell buckling plotted over a range of
potential tube geometries. Note that the image shown was generated by the author using MATLAB and OriginPro
software.
By taking the lower of the two stress values plotted on Figure 2.6, the predicted failure mode is found for
each geometry. The resulting failure map is shown in Figure 2.7. As expected, Euler buckling occurs for
lower values of , corresponding to more slender tubes.
15
Figure 2.7 Failure map showing the active failure mode for a range of tube geometries. Note that the image shown
was generated by the author using MATLAB software.
2.2 Failure Mechanisms of Space Frame Compressive Struts
One novel way to hold a compressive load is a space frame, as examined by Farr et al. [9]. These space
frame compressive struts are essentially a series of octahedra with a tetrahedra on each end (Figure 2.8).
Local Shell
Euler
16
Figure 2.8 A space frame compressive strut consists of a tetrahedra on each end with a number of octahedra in
between. It can be defined by the overall length L, strut radius r, strut length L0, and number of octahedra used to
span the length (in this figure, n=4). Note that the image shown was generated by the author using SolidWorks
software.
The structure can be defined by its overall length L, the number of octahedral used to span the length n,
and the radius of each strut r. When such a structure is loaded in compression, some struts will be in a
state of compression (end tetrahedral struts and the outer six struts of each octahedra) while others will be
in a state of tension (in-plane struts between each octahedra).
17
Before the design of these structures is explored further, it is important to introduce two parameters which
can, in principle, be used to evaluate the structural efficiency of any compressive structure: dimensionless
force f, and dimensionless volume v. These are defined as:
where F is the compressive force applied to the structure, Y is the Young’s modulus of the constituent
material, and V is the volume of material comprising the structure. Conceptually, f can be thought of as
the dimensionless compressive force on the structure while v is the dimensionless volume of material
comprising the structure. These terms remove the effects of material properties and focus solely on the
structural aspect. They also remove the effect of scale. The value f denotes a particular loading scenario:
gentle loading scenarios involve small forces over great distances while heavy loading scenarios involve
greater forces and shorter distances. For example, a lamppost may hold a load of 200 N over 6 m, while a
couch leg may hold a load of 1500 N over just 0.2 m. The optimal design, which is very different for each
case, can be characterized by the dimensionless volume of material required for it. This term, v, can
effectively be thought of as the mass of the necessary structure. In short, for a given loading scenario (f),
the optimal design is that which is capable of resisting the loading scenario while using the smallest mass
of material (v).
Using simple trigonometry, some features of the structure can be calculated:
where L0 is the length of each strut and Fcom is the compressive force on the tetrahedral compressive struts
at each end of the structure. It is important to note that while the compressive force on the end tetrahedral
struts is twice that on the octahedral compressive struts, all struts are set to have the same radius. This is a
simplification in the literature which is addressed in more detail later in this thesis.
The compressive stiffness of the overall structure is found to be [9]:
18
where k is the compressive stiffness of an individual strut, calculated by:
In order to optimize this structure, the potential failure modes must be identified. Assuming a structure
which is lightly loaded (i.e. struts are relatively slender) the two potential failure modes are Euler
buckling of the individual struts (specifically, the end tetrahedral compressive struts, which carry the
greatest load) (Figure 2.9) and Euler buckling of the overall structure (Figure 2.10).
19
Figure 2.9 Left, the end of a space frame compressive strut. Right, Euler buckling has occurred on an end tetrahedral
strut of this structure. Note that the image shown is a photograph taken by the author.
Figure 2.10 Left, a space frame compressive strut. Right, Euler buckling has occurred on the overall structure. Note
that the image shown is a photograph taken by the author.
20
Euler buckling of simple cylindrical struts is simple to analyze, as discussed previously:
Note that for the purposes of this optimization, the original authors have assumed free end conditions
(K=1.0), allowing the effective length constant to be removed from the equation.
The analysis of Euler buckling on the overall space frame structure is somewhat more complex. The
original authors have proposed a modification to the Euler buckling equation, whereby [9]:
where B is a numerical constant approximated to have a value of 0.245 through simulations involving the
energy contained in a relaxed ring of the space frame (Figure 2.11) [9].
Figure 2.11 A continuous ring constructed from the octahedral space frame design. By calculating the elastic energy
stored in this structure, the equivalent Euler buckling prediction equation can be inferred [9].
21
The optimal design is that which results in simultaneous failure by both modes. A new dimensionless
parameter is useful here. The dimensionless force on a single tetrahedral compressive strut is:
Now, the optimal design is found to have [9]:
where the floor function appears due to the integer nature of the number of octahedra, n.
Finally, the dimensionless parameters f and v can be determined [9]:
These two parameters would form the basis of an analysis on the optimized structure.
The original authors have also performed an optimization on a structure identical to that described above
but with hollow tubular struts [10].
2.3 Optimality
The term “optimality” will be used frequently throughout this thesis. For a given loading scenario, the
optimal design is that which, within certain design constraints, adequately resists the applied load while
having the lowest mass.
An equivalent and more straight-forward interpretation is that the optimal design is that which, amidst a
series of equal-mass alternatives, is capable of resisting the greatest load. This can be explored by
considering a series of designs within a certain design space – for example, variations on a simple tube.
While all designs are of equal mass and length, the fixed amount of material can be placed differently in
the cross section. A comparison between competing designs can be visualized by noting the failure load
22
of the design as a particular parameter is varied. It is important to note that as that parameter is varied,
other parameters shift accordingly to maintain a constant mass for the overall design. This technique will
be employed later in this thesis.
For a structure which has two competing failure mechanisms, the optimal will occur at the geometry for
which the expected failure stress for each mechanism is equal. This concept is valid when the failure
stresses of the two mechanisms vary inversely: by re-arranging material to strengthen one failure
mechanism, the other failure mechanism is weakened, and vice versa. This concept simplifies the process
of finding an optimal design, and is in contrast to more complex optimization processes such as that
performed on filled metal tubes in compression: due to the “dead weight” of the tube filling, Kuhn-Tucker
techniques are required to find the optimal design [11].
It is very interesting to compare the structural efficiency of a variety of structures: rods, tubes, and the
space frame structures introduced in Section 2.2. The dimensionless parameters f and v provide an elegant
way of doing just that. Recall that f can be thought of as how gentle or heavy the loading scenario is,
while v can be thought of as the mass of material required to resist that load. For a given value of f, the
smallest value of v is preferred.
Here, four designs will be compared: a simple solid column; a hollow tubular column; the solid space
frame compressive strut; and the hollow version of this space frame. The relations for f and v for these
structures can be obtained in an analogous way to that described in Section 2.2. Figure 2.12 shows a
comparison of these four designs. It is critical to understand that this graph shows only the optimal design
of each structural: for a given loading scenario f, the minimum amount of material v required to resist that
load within each set of design constraints is shown.
This final point deserves some explanation: of course it is possible to design a structure with a wide range
of mass values to resist a given loading scenario. For example, suppose that a steel tube must support a
load of 1 kN over a length of 10 cm. One possible geometry which would satisfy this loading requirement
is a very wide tube with thin walls. Another is a very thin tube with thick walls. On Figure 2.12, these two
designs would lie on the graph at a much greater value of v than the optimal line for a tube. This is
because these two geometries are not optimal. The data shown in Figure 2.12 shows only the minimum
values of v possible for each architecture – these minimum values are the optimal designs.
23
Figure 2.12 Comparison of the optimal designs of four different compressive structures. For each value of
dimensionless force, the lowest value of dimensionless volume required to resist that force is shown for each
structure. It is interesting to note that the design which results in the most efficient structure switches as indicated by
the dotted black line: to the right, a tube is the most efficient design; to the left, a hollow space frame compressive
strut is the most efficient design. Note that the image shown was generated by the author using MATLAB software.
Examining this figure, several interesting observations can be made.
First, it is clear that a simple solid rod is a relatively inefficient structural design: for any loading scenario,
a structure employing a rod design would be much heavier than alternative designs. However, this
performance penalty decreases for heavier loading scenarios. This makes sense intuitively: more complex
designs thrive at gentle loadings when structures become more slender, because the prevention of
structural instabilities becomes more critical. For an extremely heavy loading scenario such as a 2 cm tall
block on which an elephant steps, a solid puck-shaped rod will do the job as efficiently as is possible:
24
using a tube to hold the elephant’s weight will not result in a lighter structure, because the puck is not
susceptible to Euler buckling. For heavy loading scenarios, material constraints, not shape constraints,
become the limiting factor.
It is also quite clear that the solid space frame structure yields disappointing results: for any loading
scenario, it performs significantly worse than a simple tube.
The critical finding from this graph is the cross-over point between the tube and the hollow space frame
structure, occurring at an f value of approximately 10-8
. For relatively heavy loading scenarios, a tube is
the most efficient structure. However, for more gentle loading scenarios, the hollow space frame structure
gives an advantage: potentially up to an order of magnitude savings in mass. The lesson here is that there
is no single optimal structure: different architectures will yield the best performance for different loading
scenarios. To lend some meaning to the cross-over value: for a steel chair leg of length 0.4 m, this
transition would occur at a load of 300 N or approximate 30 kg. While the complexity of these structures
would likely prevent their use for such mundane objects as chair legs, their advantages would prove
hugely beneficial for very gently loaded structure such as solar sails or other spacecraft.
While this theoretical analysis is very useful in assessing the potential of a variety of structural designs,
there is a critical need for experiments on the structures. Variable material properties, flaws, and
simplifications in the models all require experimentation to be fully understood. In this thesis, the
architectural freedom afforded by stereolithographic additive manufacturing will be employed to better
understand the experimental behaviour of the structural designs.
25
3. Methods and Materials
3.1 Mechanical Testing
Compressive mechanical testing was used extensively in this thesis to study a variety of structures.
3.1.1 Load Frame
A Shimadzu AG-I 50 kN machine was used for all compressive testing performed for this thesis. This
machine is capable of performing compressive tests at a range of stroke rates and load cell ranges.
Most structures tested were 70 mm in length and were tested with a stroke rate of 0.8 mm/min,
corresponding to a strain rate of 0.011 min-1
. For structures shorter than 70 mm, the stroke rate was
adjusted to maintain a comparable strain rate.
The load cell range from 0.5 kN up to 50 kN was possible. As smaller load cell ranges yielded greater
accuracy in results, the smallest load cell still sufficiently high to test the strongest structure in a series
was used.
Through experiment, the value to correct for machine compliance was obtained. The stiffness of the
machine was found to be 65700 N/mm (see Appendix A).
3.1.2 Digital Image Correlation
In order to visually study the failure mechanisms of certain structures, digital image correlation (DIC) was
used. This technique uses high-definition cameras to optically analyze the surface of the structure and
extract local strain values throughout the compressive test.
A GOM 5M digital image correlation system using Aramis analysis software was used [12]. This system
uses two cameras to achieve depth perception and thus three-dimensional image capture. The local strain
values are calculated by tracking the relative movement of tightly-spaced points on the surface of the
sample. These points are applied in the form of a speckle pattern using spray paint: first, the sample is
painted white to form a uniform background; next, a gentle mist of black spray paint is applied to the
sample, forming a speckled pattern of black dots on a white background. An example of a tube with
speckle pattern applied is shown in Figure 3.1.
26
Figure 3.1 Polymer tube which has been prepared for DIC analysis by applying a spray-paint speckle pattern to its
surface.
27
3.2 Stereolithographic Polymer Material Properties
Details of the printing process are described to begin this section. Next, it was important to explore
several aspects of the material properties of the stereolithographic polymer which were not previously
well understood. These include: compressive behaviour and its repeatability within a single print; polymer
property variation between prints; the effects of the post-printing procedure on material properties; and
the effects of printed height on material properties. Understanding these factors was critical to be able to
analyze further results on more complex parts.
3.2.1 Finishing Process
The printing process described in Section 1.1 yields the desired polymer parts with wax attached to any
overhanging areas, all affixed to the aluminum substrate plate.
To expose the polymer parts and release their desired shape, a “finishing” process is employed. The first
step in this process is to remove the parts from the aluminum plate. This is done by placing the entire
piece in a refrigerator (approximately 2 °C) for approximately 10 minutes. The differential thermal
expansion of the aluminum and the support wax is sufficient to separate the parts from the aluminum
substrate. The next step is to remove the support wax. A variety of methods were tested for wax removal.
Techniques employing liquids (water, with or without heat and soap) to wash the wax away from the parts
were somewhat effective for simple parts. However, removing wax from the interior of hollow parts such
as tubes proved difficult using liquids due to surface tension: liquids were prevented from entering small
pores. The simple yet effective technique used instead employed prolonged exposure to an elevated
temperature. Since the wax melted at a temperature of 60 °C [13], a ProJet Finisher oven set to 65 °C was
used. It is important to note that the polymer material itself has a heat distortion temperature at 0.45 MPa
of just 56 °C. As such, care had to be taken to avoid deformation of the polymer parts. In addition, it was
found that hollow structures with a wall thickness less than approximately 200 µm could not be prepared
without an unacceptable amount of deformation. This temperature-related effect defined the lower bound
on wall thickness.
The parts were placed on absorbent tissues on a metal grating in the oven. The bulk of the wax was
removed in liquid form within the first hour. The remaining residue of liquid wax clinging to the polymer
surface was removed using a combination manual wiping and evaporation. Depending on the complexity
of the part, this process could require between one day and one week to remove all support wax. The gas-
phase nature of the evaporative wax removal allowed waxy residue to be removed from the interior of
hollow components – a feat not possible using absorption and challenging using liquid rinses.
28
3.2.2 Printer Resolution
The 3D Systems ProJet HD3500 was capable of printing in two distinct modes: High Definition (HD) and
Ultra High Definition (UHD), with the difference being the resolution of the parts and the maximum size
of the print. For this thesis, all parts were produced using the UHD mode. This mode yielded a resolution
of 750 x 750 x 890 dots per inch (29 µm layers) with a maximum print size of 127 x 178 x 52 mm [13].
3.2.3 Compressive Behaviour and Repeatability within a Print
Compressive coupons were used to extract the compressive stress-strain curve of the stereolithographic
polymer used in this thesis. These data were crucial as they yielded values for Young’s modulus and
tangent modulus for use in the variety of predictive models used in this thesis.
In accordance with ASTM standard D695-15 [14], cylindrical compression coupons with a height of 25
mm and a diameter of 13 mm were used with a strain rate of 0.011 min-1
.
Five compression coupons printed and prepared together were tested, and the resulting stress-strain curves
and summarized Young’s modulus and 0.2% offset yield stress are shown in Figures 3.2-3.3. It is clear
that compressive material properties show excellent repeatability for samples prepared together: Young’s
modulus has an average value of 1560 MPa with a standard deviation of 30 MPa, while the 0.2% offset
yield stress has an average value of 37.3 MPa with a standard deviation of 0.3 MPa.
This analysis suggests that samples prepared together consist of material with similar properties, and thus
can be directly compared to each other. This will prove crucial in the comparative analysis of more
complex structures.
29
Figure 3.2 Stress-strain data from five compression coupons prepared together. Note that the material behaviour is
similar so the lines are difficult to distinguish.
30
Figure 3.3 Summarized compressive material properties from five compression coupons prepared together. Note that
the data points are clustered, showing good repeatability.
3.2.4 Polymer Property Variation Between Prints
It has been established in the previous section that the compressive properties of the stereolithographic
polymer show good repeatability within a single print cycle. However, throughout this thesis a large
number of parts were tested from many different print cycles. While the printer settings, nominal source
material, and wax removal procedures were held constant, it is still possible that material properties could
vary between print cycles. To explore this, a compression coupon was prepared and tested as a part of
every print cycle performed. This had two benefits: first, the material properties could be verified for
modeling the structures in each print; second, a large amount of data was generated to provide insight into
the question of repeatability between prints.
31
Figures 3.4-3.5 show stress-strain curves and summarized material properties from 13 compression
coupons corresponding to 13 separate print cycles. As described above, the physical properties of the
polymer have the potential to vary between prints. This could be due to minor variation in the source
material, time that the parts are left in the finisher, or cooling rate when removed from the finisher.
These compression coupons from different print cycles yield a greater spread in material properties than
do coupons from the same print cycle examined in the previous section: the Young’s modulus has an
average value of 1570 MPa with a standard deviation of 120 MPa and a range from 1470 MPa up to 1830
MPa, while the 0.2% offset yield stress has an average value of 35.8 MPa with a standard deviation of 3.9
MPa and a range from 31.4 MPa up to 44.6 MPa.
This analysis suggests that material properties vary significantly between print cycles. As such, it is not
necessarily possible to directly compare structures from different print cycles, as the material from which
they are made may have different properties. This makes it complicated to repeat experimental results. It
also means that if a direct comparison between two structures is desired, those structures should be
prepared in the same print cycle.
32
Figure 3.4 Stress-strain data from 13 compression coupons, each from a separate print cycle. Note that there is
considerably more variation in material properties when compared to the repeatability observed in Figure 3.2.
33
Figure 3.5 Summarized compressive material properties from 13 compression coupons, each from a separate print
cycle. Note that the material properties are not as repeatable as they were for samples produced in the same print
cycle (Figure 3.3).
3.2.5 Effects of Finishing Protocol on Polymer Properties
The previous section has shown that material properties vary between print cycles. To further explore the
cause of this variation, the finishing (i.e. wax-removal) protocol was examined. Ten compression coupons
were printed in the same print cycle. These ten coupons were removed from the printer and placed in the
finisher oven at the same time. They were all removed from the finisher oven 10 hours later, with all wax
removed. The 10 samples were then allowed to sit for varying amounts of time before testing. One sample
was tested after just 4 minutes; one after 14 minutes; one after 27 minutes; one after 37 minutes; one after
47 minutes; and five after 7 days. Figure 3.6 shows the progression in stress-strain curves for these
samples as the amount of time sitting between finisher oven and testing is increased. Figures 3.7-3.8 show
the change in Young’s modulus and 0.2% offset yield stress with sitting time, respectively.
34
It is clear that the polymer material becomes stiffer as it is left to sit for more time: from a value of 640
MPa up to a value of 1590 MPa. On a short time scale, this effect is likely a result of increased
temperature: a compression coupon at 65 °C in the finisher oven likely does not cool to room temperature
before testing just 4 minutes later. However, the temperature would likely drop quite quickly and is
unlikely to explain the changes between coupons left to sit for longer times. It appears that some type of
polymer aging takes place as the coupons sit. The trend of this effect appears to be somewhat logarithmic,
with the material properties reaching a plateau after some time.
This analysis suggests that all printed parts should be left to sit for sufficient time before undergoing
mechanical testing. As most parts tested have a significantly greater surface area to volume ratio than the
compression coupons, cooling will occur at a much higher rate. In order to ensure that the plateau
material properties have been reached, all parts should be allowed to sit for at least several hours between
finishing and testing. This procedure was already in place for the data collected in Section 3.2.4. Variation
in sitting time does not explain the variation in material properties between print cycles.
This experiment was repeated in an identical manner except for the amount of time that the coupons were
left in the finisher oven: parts were left for one week at high temperature instead of just 10 hours. Results
were similar between these two experiments: material properties did not change appreciably with the
amount of time the parts spent in the finisher, provided that wax removal was complete. As such, the
amount of time that the parts are left in the finisher should depend only on completeness of wax removal.
35
Figure 3.6 Stress-strain curves for 10 compression coupons. Each coupon was allowed to sit for a different amount
of time between removal from the finisher and mechanical testing.
36
Figure 3.7 Progression of Young's modulus with sitting time between finishing and testing. The stiffness of the
material increases significantly until a plateau is reached.
37
Figure 3.8 Progression of 0.2% offset yield stress with time allowed to sit between finishing and testing.
38
3.2.6 Effects of Printed Height on Polymer Stiffness
Preliminary testing led to a suspicion that the physical properties of the polymer may vary depending on
the height of the part. A possible reason for this is that if two parts of different heights are printed
together, the shorter part will continue to receive UV exposure while the taller part is being printed. This
is best described with a simple example. Suppose that two columns are printed together: one with a height
of 10 mm and one with a height of 50 mm. Recall that after each layer of liquid polymer is deposited, it is
cured by ultraviolet radiation. This radiation is applied uniformly across the entire print area. As such,
during the deposition of the upper 40 mm of the taller column, the shorter column will receive an extra
“dose” of ultraviolet radiation despite the fact that this part is already complete. While this phenomenon is
not well understood, it is conceivable that this ultraviolet-curable material is sensitive enough to
ultraviolet radiation that some modification is incurred upon it by this extra radiation.
To explore this idea further, a series of vertically scaled compression coupons were prepared and tested in
compression. These cylindrical samples all had a diameter of 13 mm, while the height of the samples
ranged from 2.5 mm to 25 mm. A strain rate of 0.011 min-1
was used for all samples. The stress-strain
curves for these samples are shown in Figure 3.9. The Young’s modulus extracted from each sample is
shown in Figure 3.10. This graph suggests that the stereolithographic polymer contained in taller
structures is stiffer than that contained in shorter structures: Young’s modulus ranges from 1500 MPa for
the full-height compression coupon down the just 1190 MPa for the 2.5 mm tall vertically scaled version.
Figure 3.11 shows the trend of decreasing 0.2% offset yield stress with increasing coupon height, ranging
from 46.7 MPa down to 35.3 MPa.
This has potential consequences for the analysis of certain structures: predictive models of tall structures
(e.g. 70 mm in height) which use material properties derived from compression coupons just 25 mm in
height may be inaccurate. This issue is discussed further in future sections of this thesis.
It is important to note that the aspect ratio of the compression coupons changes as they are vertically
scaled. It is possible that this affects the material property results obtained. Due to the end constraint for
the coupon against the platens, barreling occurs. As aspect ratio changes, this barreling will have different
effects on non-uniformity of strain within the sample. The Saint-Venant principle states that samples must
be sufficiently long in order to achieve uniform strain throughout the cross-section of the sample [15].
39
Figure 3.9 Stress-strain curves for 10 compression coupons of varying height: the standard 25 mm tall compression
coupon, and vertically scaled versions down to 2.5 mm in height. Note in particular the reduced stiffness and
increased yield stress of the 0.1 and 0.2 scaled coupons (green and blue, respectively).
40
Figure 3.10 Young's modulus increases with height for a series of vertically scaled compression coupons. The height
fraction range from 0.1 to 1.0 corresponds to a height change from 2.5 mm to 25 mm.
41
Figure 3.11 0.2% offset yield stress decreases with height for a series of vertically scaled compression coupons. A
height fraction range from 0.1 to 1.0 corresponds to a height change from 2.5 mm to 25 mm.
3.2.7 Material Property Summary
In the section above, several observations have been made about the mechanical properties of the
stereolithographic polymer. The first key conclusion is that material properties show good repeatability
within a print cycle but poor repeatability between print cycles. As such, direct comparisons between
structures should only be made for structures prepared in the same print cycle, and a compression coupon
should be prepared with each print cycle to extract material properties for that cycle. The next conclusion
is that while the material properties do not change depending on time spent in the finisher (assuming
complete wax removal) they do vary with sitting time between finishing and testing. All samples should
be left to sit for at least six hours to ensure that plateau material properties have been reached. Finally,
material properties depend on the height of the printed part, which must be considered when comparing
predictions to experiments in future chapters.
42
4. Polymer Tube Compression
The theoretical predictions of Euler and local shell buckling stresses based on the stereolithographic
polymer material properties can be compared to experimental data. This is a useful experiment as tubes
provide an excellent platform on which to study the interaction between stereolithographic polymer
material properties and geometric effects.
4.1 Characterization of Failure Modes
While the expected failure modes for a tube include Euler buckling and local shell buckling,
stereolithographic polymer is a novel material and it is possible that other failure modes will dominate. In
order to explore the active failure modes in stereolithographic polymer tubes, a pair of tubes were printed
and prepared: one tube had an value of 0.015 and an value of 1.9 – a very slender tube which was
expected to fail by Euler buckling; the second tube had an value of 0.18 and an value of 0.049 – a
wide tube with very thin walls that was expected to fail by local shell buckling. Both tubes had a length of
70 mm. These two tubes are shown in Figure 4.1.
Figure 4.1 Two tubes designed to verify failure modes: the left tube was predicted to fail by Euler buckling, while
the right tube was predicted to fail by local shell buckling.
43
These two tubes were mechanically tested in compression at a strain rate of 0.011 min-1
. For reference,
the force-displacement data for each tube is shown in Figure 4.2. Throughout each test, the local strain
values across the surface of each tube were captured using digital image correlation (DIC). Figure 4.3
shows the DIC analysis performed on the tube expected to fail by Euler buckling. At the beginning of the
test, the strain is zero along the length of the tube, as expected. As the tube is compressed, the tube
experiences uniform compression up to a strain value of approximately 1.5%. Beyond this value,
bifurcation is observed where the central outer face of the tube enters a state of tension, reaching tensile
strain values in excess of 8%. Other areas of the tube remain in compression. The tube forms an arced
shape. This pattern is characteristic of Euler buckling, and confirms that stereolithographic polymer tub es
can potentially experience this failure mode. Figure 4.4 shows the DIC analysis performed on the tube
expected to fail by local shell buckling. At the beginning, the tube has a uniform strain of zero along the
length of the tube. As the tube is compressed, the tube experiences relatively uniform compression. At a
certain point (past approximately 1% of uniform compressive strain), alternating bands of high and low
strain appear near the ends of the tube. These local shell buckling bands show strain values alternating
between -1.2% and -4.5%, while the remainder of the tube continues to experience uniform compression.
This pattern is characteristic of local shell buckling, and confirms that stereolithographic polymer tubes
can potentially experience this failure mode. It should be noted that this tube fractured after these images
were captured: the polymer tube was not able to develop complete local shell buckling bands as are
observed, for example, in certain metal tubes (Figure 2.1).
44
Figure 4.2 Force displacement curves for a tube which failed by (a) Euler buckling and (b) local shell buckling.
Insets show fractured tubes after testing.
45
Figure 4.3 Digital image correlation analysis performed on a stereolithographic polymer tube expected to fail by
Euler buckling. The top images show the tube at the various stages of compression, with colours corresponding to
local strain values on the surface of the tube. The bottom graph shows the strain values along the length of the tube
(denoted by the thick black line on each tube image). The tube experiences uniform strain until bifurcation occurs
where the central outer face of the tube enters a state of tension. Note that the high values of compressive strain at
the ends of the red curve represent the corners of the tube which have been crushed against the platen.
46
Figure 4.4 Digital image correlation analysis performed on a stereolithographic polymer tube expected to fail by
local shell buckling. The top images show the tube at the various stages of compression, with colours corresponding
to local strain values on the surface of the tube. The bottom graph shows the strain values along the length of the
tube (denoted by the thick black line on each tube image). The tube experiences uniform strain until alternating
bands of high and low strain appear near each end of the tube.
47
4.2 Equal-Mass Polymer Tubes in Compression
Using the predictive models described in Section 2.1, the failure stress of tubes can be predicted. In order
to compare these predictive values to experimental failure stress, ten tubes were printed. These ten tubes
were all of the same length (70 mm) and the same mass (average 0.78 g, standard deviation 0.04 g). The
difference between the tubes was where in the cross-section the material was located: the given mass
could be used to form a narrow tube with thick walls, or the tube could be made wider with a
corresponding decrease in wall thickness. The specifications of the ten tubes are shown in Table 4.1, and
they are visualized in Figure 4.5.
Table 4.1 Specifications of ten tubes of equal mass and length, but varying radius and wall thickness. Note that tube
1j is actually a solid rod.
Figure 4.5 The ten tubes in the first set, ranging from wide tubes with thin walls to narrow tubes with thick walls.
48
Due to the finite resolution of the printer, the requested dimensions were not guaranteed in reality.
Measurement using calipers was difficult due to the soft nature of the polymer. However, post-printing
measurements on more robust samples confirmed accuracy within a range of approximately 50 µm. The
two largest independent dimensions (length and outer radius) along with the measured mass were used to
find the mass-calculated wall thickness. It should be noted that while the ten tubes were designed to all
have equal mass, in reality the mass shows an increasing trend as wall thickness is decreased. This is
likely due to slight over-printing by the 3D printer. Comparing requested to mass-calculated wall
thicknesses, over-printing by approximately 0.02 mm appears common. This equates to an increase in
wall thickness of 15% for the thinnest-walled tube. This effect is mitigated by calculating wall thickness
using tube mass. Since these real dimensions are used in all predictive models, comparison between
prediction and experiment is still valid.
The tubes were tested in compression at a stroke rate of 0.8 mm/min, equating to a strain rate of 0.011
min-1
. Measured mass and tube length were used to calculate cross-sectional area in order to convert force
to stress. Stress and strain values were used in place of force and displacement values to simplify
comparison between tubes of different lengths and masses: all tubes with the same values of and will
fail at the same stress, but not at the same force.
The stress-strain curves from these compression tests are shown in Figure 4.6. Note that for clarity, tubes
which failed by local shell buckling are shown on Figure 4.6(a) while tubes which failed by Euler
buckling are shown on Figure 4.6(b). The tube which had the greatest failure stress is shown on both
graphs. It is interesting to note that the modulus of all tubes is approximately equal until one of the failure
modes is activated. This is expected: for these tubes, modulus is a material effect, and is not influenced by
any shape effect until buckling occurs.
49
Figure 4.6 (a) Stress-strain curves of the four tubes which failed by local shell buckling, as well as the tube which
had the greatest failure stress. (b) Stress-strain curves of the five tubes which failed by Euler buckling, as well as the
tube which had the greatest failure stress. Note that the modulus is approximately equal between all tubes until
buckling occurs. Also note that tube 1a is has the greatest inner radius and the lowest wall thickness, while tube 1j
has the smallest inner radius and the greatest wall thickness.
50
The first observation is in relation to the shape of the stress-strain curve for different tube geometries.
Referring to Figure 4.6(a), the widest tubes with the thinnest walls typically fail in a brittle manner,
characterized by a sharp drop in the stress-strain curve past the maximum. Note that in literature, tubes
which fail by local shell buckling generally display an oscillating trend in their stress-strain curves as
successive bands of material are buckled [7]. However, the relatively brittle nature of the
stereolithographic polymer used in this study seems to result in fracture of the tube once a significant
local shell buckling bulge is formed.
Referring to Figure 4.6(b), the most slender tubes with the thickest walls display a drop in strain past the
maximum followed by somewhat of a plateau. This shape is characteristic of Euler buckling.
The next observation is that there is a significant variation in stress-strain curve shape and magnitude
between tubes within a set, despite all tubes having identical mass, length, and cross-sectional area. Using
polymer compression data from a compression coupon printed and treated alongside the ten tubes,
predictions for the Euler and local shell buckling failure stresses were calculated. Note that these
predictions are purely related to the failure stress of each tube for each failure mechanism. Specifically,
the maximum stress is predicted. As such, the maximum stress of each experimental stress-strain curve
was extracted.
In order to easily compare peak stress values from the ten tubes in the set, a single parameter must be
used to define all tubes. In this case, is used. Recall that is defined as the inner radius of the tube
divided by the length of the tube, and that the length and mass are equal for all tubes. The only difference
between the tubes is where in the cross-section the material is placed. It is very important to note that
while the followed data is graphed with respect to , the value of also varies between tubes. Recall that
is defined as the wall thickness of the tube divided by the inner radius of the tube. In general, given a
constant mass and length, an increase in corresponds to a decrease in . Figure 4.7 shows the
experimental failure stress of the ten tubes as a function of their .
51
Figure 4.7 Experimental failure stress of the ten tubes. All tubes are of equal mass and equal length. It is important
to note that the displayed change in is accompanied by a hidden change in to maintain a constant mass.
52
The predictions of failure stress for local shell buckling and Euler buckling can now be compared to this
data. Note that predictions were made using material compression data from a coupon printed and
finished alongside the tubes to match material properties as closely as possible. The reader is referred to
Section 2.1 for the equations used in these predictive models. It is important to note that one value used in
the Euler buckling model has yet to be specified: the effective length constant, K. This value describes the
nature of the end constraints on the tube. A value of 1.0 for K indicates that the ends are pinned: they are
free to rotate. A value of 0.5 for K indicates that the ends are fixed and are unable to rotate [16].
Figure 4.8 shows the Euler buckling predictions, comparing effective length constants of 1.0 and 0.5. It is
clear that the experimental failure stress for those tubes expected to fail by Euler buckling (i.e. tubes with
low values of ) match more closely the prediction based on fixed end conditions (K=0.5). This is
reasonable given the fact that compressive tests on the tubes are conducted with simple flat steel plattons
pressing on either end of each tube. This wide contact patch between tube end and platen prevents the
tube end from rotating, effectively giving the tube fixed end conditions.
Figure 4.8 Euler buckling failure stress predictions using effective length constants of 0.5 (fixed) and 1.0 (free). It is clear that
for tubes expected to fail by Euler buckling, the experimental failure stress matches more closely to the predictions using fixed
end conditions.
53
Figure 4.9 Euler and local shell buckling failure stress predictions compared to the experimental failure stress
values.
54
Figure 4.10 Comparison of Euler and local shell buckling predictions to experimental data for two print cycles.
Separate tubes and material compression data were used for each print cycle.
Figure 4.9 shows the experimental data along with predictions for Euler buckling (using K=0.5) and
predictions for local shell buckling.
It is desirable to repeat this experiment to confirm results. Unfortunately, as discussed in Section 3.2.4,
the stereolithographic polymer material properties show slight yet significant variation between print
cycles. As such, repeated results cannot be directly compared to initial results. However, the experiment
can be repeated and analyzed on its own to confirm trends. This repetition was performed: the ten tubes
were re-printed along with a compression coupon to extract material properties. The data from this new
print cycle is shown in Figure 4.10. The trends within each print show good repeatability.
Recall that mass and length are held constant across the ten tubes shown: higher values of indicate wide
tubes with thin walls, while lower values of indicate slender tubes with thicker walls. As the cross-
section of the tube is altered, the failure stress changes. In the case of Euler buckling, wider tubes have
more material far from the neutral bending axis and are predicted to fail at a greater stress. Conversely,
thinner walls are more susceptible to local shell buckling, and so tubes with a high value of are
predicted to fail at a lower stress for this mechanism.
55
The experimental results generally agree well with the analytical predictions. As expected, tubes fail at
the failure stress predicted by the weaker of the two competing failure mechanisms. The optimal design –
that which exhibits the greatest failure stress under the constraint of a constant mass – occurs at or near
the architecture for which the analytical predictions give equal values for failure stress between the two
failure mechanisms.
For many tubes, the failure stress is under-predicted. The hypothesized reason for this relates to the
properties of the stereolithographic polymer:
The modulus from the compression coupons used in the predictive model may in reality be lower than the
modulus of the polymer comprising the tubes. The stiffness extracted from the elastic region of the ten
tubes has an average value of 1920 MPa and a standard deviation of 70 MPa. This is high when compared
to the range of compression coupons tested in Section 3.2.4: an average value of 1570 MPa with a
standard deviation of 120 MPa and a range from 1470 MPa up to 1830 MPa. This could be explained by
the height effects described in Section 3.2.6. The ten tubes are 70 mm in height, which is significantly
greater than the 25 mm height of the compression coupons. This could mean that the tubes are comprised
of a polymer with a greater stiffness than the polymer in the compression coupons. This would have the
consequence that the predictions for the tubes are made using material properties from a material which is
noticeably less stiff than the material tested experimentally, resulting in under-prediction.
It is interesting to note that despite under-prediction, the optimal tube geometry (i.e. the value of which
yields the greatest failure stress) is well predicted. This is possible due to the fact that the input from the
material property data (i.e. tangent modulus) appears in the same order in both the Euler buckling
equation and the local shell buckling equation (see Section 2.1). As such, for the geometry at which the
two failure stresses are equal, the material properties effectively cancel out.
56
4.3 Tubes of Greater Mass
The analysis performed in the previous section was repeated on two more sets of ten tubes. While the
same concept was used (all tubes of equal mass and equal length), these two new sets of tubes had greater
mass: one set contained tubes with an average mass of 2.11 g and a standard deviation of 0.04 g; the other
set contained tubes with an average mass of 5.98 g and a standard deviation of 0.05 g. The specifications
of these tubes are shown in Table 4.2, and they are visualized in Figures 4.11-4.12.
Table 4.2 Specifications of two sets of ten tubes. Within each set, tubes are of equal mass and length, but varying
radius and wall thickness.
57
Figure 4.11 The ten tubes in the second set, ranging from wide tubes with thin walls to narrow tubes with thick
walls.
Figure 4.12 The ten tubes in the third set, ranging from wide tubes with thin walls to narrow tubes with thick walls.
58
The stress-strain curves for these two sets of tubes are shown in Figures 4.13-4.14.
Figure 4.13 (a) Stress-strain curves of the five tubes in the second set which failed by local shell buckling, as well as
the tube which had the greatest failure stress. (b) Stress-strain curves of the four tubes in the second set which failed
by Euler buckling, as well as the tube which had the greatest failure stress.
59
Figure 4.14 (a) Stress-strain curves of the five tubes in the third set which failed by local shell buckling, as well as
the tube which had the greatest failure stress. (b) Stress-strain curves of the four tubes in the third set which failed by
Euler buckling, as well as the tube which had the greatest failure stress.
60
The predictions for Euler buckling and local shell buckling are shown alongside the experimental failure
stress for the second and third sets of tubes in Figures 4.15-4.16. Repeats of these tests can be found in
Appendix B.
Figure 4.15 Predicted Euler and local shell buckling failure stresses compared to experimental failure stresses for the
second set of tubes.
61
Figure 4.16 Predicted Euler and local shell buckling failure stresses compared to experimental failure stresses for the
third set of tubes.
62
In general, the conclusions from the analysis of these tubes are similar to those for the initial set: failure
stresses are under-predicted, but the trend and optimal are predicted accurately.
One interesting note is that the heavier tubes which failed by local shell buckling did not fracture as the
lighter local shell buckling tubes from the first set of tubes did. This is in contrast to similar experiments
from literature: research by Bele et al. on 20nm grain size Ni tubes in compression showed that thicker
local shell buckling tubes experienced fracture while thinner local shell buckling tubes experienced
progressive buckling involving multiple folds [17]. This contrasting behaviour between nano-nickel and
stereolithographic polymer tubes suggests that material effects may play a role in this phenomenon.
The main point to note from the data from sets of heavier tubes is that the failure stresses are universally
higher, but with an upper limit. This is most clear for the third set of tubes, for which nine of the ten tubes
fail at a stress greater than 50 MPa, yet none exceed a failure stress of 60 MPa. This peculiarity is easily
explained: heavier, more “robust” tubes are more resistant to shape related failures, but begin to encounter
issues of material limitation. An extreme example would be a pancake- or puck-shaped structure loaded in
compression. This structure will not encounter any shape-related buckling or instability. It will simply
follow the stress-strain curve of the constituent material. For the first (and lightest) set of tubes studied
here, stress-strain curves follow that of the constituent material until a shape-related instability (i.e. Euler
or local shell buckling) arises. For the heavier sets of tubes (especially the third set), material limitations
are reached before shape-related effects are activated. This is even clear from the predictive model:
Suppose that there is a tube which fails by local shell buckling and has an value of 0.15 and an value
of 0.02. This tube is predicted to fail at a stress of 13.2 MPa (Figure 4.17). Now suppose that there is a
second tube which still has an value of 0.15 but has an value of 0.04: the wall thickness has been
doubled. This tube is predicted to have a failure stress of 28.2 MPa: by doubling the wall thickness, the
failure stress has increased by 110%.
Now suppose that there is a tube which still fails by local shell buckling, but is much more robust: it still
has an value of 0.15 but has an value of 0.2. This tube is predicted to fail at a stress of 54.9 MPa
(Figure 4.17). Finally, suppose that there is a tube which still has an value of 0.15 but has an value of
0.4: the wall thickness has been doubled from the already quite robust tube. This tube is predicted to have
a failure stress of 57.7 MPa: by doubling the wall thickness of the robust tube, the failure stress has
increased by just 5%.
The key point here is that robust tubes are limited not only by shape effects, but also by material effects.
Tubes which fail in the plastic region of the material’s stress-strain curve are already exploiting the
63
material to its fullest extent. The result of this is that for robust tubes, a wide range of geometries will all
display relatively similar failure stresses.
While this plateau stress phenomenon is well understood, it does pose challenges for future studies. It is
clear that the optimal tube geometry is more obvious from the first (lightest) set of tubes than from the
third (heaviest) set of tubes: the heavier set shows little difference in failure stress between tubes of
widely varying geometry. The ideal solution, then, would be to use a lighter tube set as constituent struts
to explore more complex geometries (such as microtrusses or space frames) where it is desirable to locate
an optimal architecture. Unfortunately, limitations in the current stereolithographic manufacturing
technique preclude this endeavor. For the lightest tube set, many tubes have wall thicknesses less than 500
µm. Recall that this thickness is for tubes 70 mm in length. In order to incorporate tubes into more
complex architectures, tube lengths of approximately 15 mm are required (this is simply a function of the
maximum part size possible with existing equipment). In order to maintain the geometry while scaling
down the tubes, wall thicknesses under 100 µm would be required. However, the equipment used in this
study is incapable of reliably manufacturing structures with a wall thickness under 200 µm. Due to this
constraint, a heavier set of tubes (similar to the third set) was considered for use in more complex
architectures, despite the shortcoming of an unclear optimal for these structures.
64
Figure 4.17 Predictive model for local shell buckling. Note that all four tubes shown have an value of 0.15. Two
comparisons are shown: the lower two lines show that for many tubes, the doubling of wall thickness results in near-
doubling of failure stress; the upper two lines show that for very robust tubes, the doubling of wall thickness results
in a relatively small change to failure stress.
65
5. Novel Compressive Structures
Compared to a simple solid rod, a tube represents a significant increase in structural efficiency (i.e. the
compressive force that can be resisted by the structure for a given mass). However, it is possible to further
enhance structural efficiency by adding new levels of freedom to the design of compressive structures.
The architectural freedom allowed by stereolithographic additive manufacturing presents the opportunity
to design, manufacture, and test some of the structures. This section introduces two such structures: first,
a design from literature is examined; second, a new design concept is developed.
5.1 Mechanical Testing and Modeling of Space Frame Compressive
Struts
5.1.1 Experimental Analysis
Space frame compressive strut designs envisioned by Farr et al. [9] have been introduced in Section 2.2.
In the theoretical analysis of these structures, it was discovered that solid space frame compressive struts
offered disappointing performance: they were significantly less structurally efficient than simple tubes.
However, hollow versions of these space frame struts offered considerable reduction in weight when
compared to tubes for situations of gentle loading. Unfortunately, the hollow version of these structures
cannot be investigated using available printing techniques: in order to fabricate these hollow structures on
a scale possible with the 3D printer used in this study, the wall thicknesses required for optimal designs
would be well below the 200 µm minimum thickness possible with the current technique. Despite this, the
understanding of these structures can be enhanced by investigating the experimental compressive
behaviour of solid space frame compressive struts.
Recall that two potential failure mechanisms exist for these structures: first, Euler buckling of an
individual tetrahedral compressive strut; second, Euler buckling of the structure as a whole. Analogous to
the investigation of tubes in the previous section for which a range of tube geometries of equal mass were
compared, a range of space frame compressive struts of equal mass can be studied. It is important to note
that these space frame structures will not stand vertically on their own due to the pointed shape of the
ends. To allow the structures to stand vertically for compressive testing, small discs of thickness 2 mm
and radius 7.6 mm were printed on the ends of each strut (Figure 5.1). The overall strut length for the
current study was chosen to be 70 mm to simplify comparison with previous results on simple tubes. The
mass of the strut (including end discs) had an average value of 1.34 g with a standard deviation of 0.03 g.
These values were chosen as they result in struts which are expected to fail in the elastic region of the
constituent material. As discussed in Section 4.3, this allows for a significant amount of improvement in
66
shape-related failure load without being constrained by material failure (i.e. the ~60 MPa failure strength
material “ceiling” will not mask architectural improvements).
Figure 5.1 Example of a space frame compression strut. Note that to enable the strut to stand vertically for
compressive testing, polymer discs were built in to the ends of the structure during fabrication.
In the case of tubes, there were two free parameters to describe the geometry: inner radius and wall
thickness. For space frame compressive struts, the two free parameters are the number of octahedra, n,
and the strut radius, r. As the number of octahedra must be a discrete, integer value, this was set to
various values and the strut radius was varied accordingly to maintain a constant mass across designs. The
specifications of the nine space frame structures tested are shown in Table 5.1, and they are visualized in
Figure 5.2. Two identical sets of these structures were printed and tested separately to confirm results.
67
Table 5.1 Specifications of the nine space frame compressive struts. Note that as the number of octahedra is
increased, the length of each constituent strut decreases accordingly to maintain a constant overall length of 70 mm.
Figure 5.2 The nine space frame compressive struts, ranging from five octahedra with long constituent struts to 13
octahedra with shorter constituent struts. All structures are 70 mm in overall length.
The nine space frame compressive struts were tested in compression at a stroke rate of 0.8 mm/min,
equating to a strain rate of 0.011 min-1
. The force-displacement curves for the nine structures are shown in
Figure 5.3. Note that due to the complexity of the structures, a stress value was not easily defined. The
expected failure mechanisms of individual strut Euler buckling and overall structure Euler buckling were
confirmed visually during experimentation. Examples of these two failure modes are shown in Figures
2.9-2.10.
68
Figure 5.3 (a) Force-displacement curves of the three space frame compressive struts which failed by Euler buckling
of the end tetrahedral struts, as well as the structure which had the greatest failure force. (b) Force-displacement
curves of the four space frame compressive struts which failed by Euler buckling of the overall structure, as well as
the structure which had the greatest failure force.
69
As with the tube analysis in the previous chapter, the model for the space frame compressive struts
predicts only the failure force. As such, the maximum force was extracted for each space frame and is
shown in Figure 5.4. Just as the analysis on the tubes in the previous chapter was plotted in terms of , the
structures here are plotted in terms of the number of octahedra. It is important to note that this change is
accompanied by a hidden change in the length and radius of the constituent struts to maintain a constant
overall length and mass.
Figure 5.4 Experimental failure force of the eight space frame compressive struts. Note that the structure with six
octahedra was damaged during preparation and removed from results. All structures are of equal mass and equal
length. It is important to note that the displayed change in number of octahedra is accompanied by a hidden change
in the radius and length of the constituent struts to maintain a constant mass and overall length. The black dotted line
indicates the transition in active failure mode between Euler buckling of the end tetrahedral struts and Euler
buckling of the overall structure.
70
The failure force for each mechanism (Euler buckling of a strut and Euler buckling of the overall
structure) can be calculated using the models described in Section 2.2. Young’s modulus from a
compression coupon printed, finished, and tested alongside the space frames was used in the models.
As expected, the predicted failure force for Euler buckling of the end tetrahedral struts is lower for
structures with a lower number of octahedra – the struts in these structures are longer and more slender. In
addition, the predicted failure force for Euler buckling of the overall structure is lower for structures with
a greater number of octahedra – these structures are more slender overall. It was expected that the
experimental failure force would match the lower of the two predicted competing failure mechanisms.
However, there is an unknown value required by the models: the effective length constant, K, for the
struts and for the overall structure. The upper and lower limits to this value are free (K=1.0) and fixed
(K=0.5), respectively.
Recall that the effective length constant, K, was found to be 0.5 for simple tubes in compression, with
ends against flat platens. It would initially seem that the struts in these space frame structures, being
bonded to surrounding polymer, would have ends more fixed than simple platen ends. However, the
polymer nodes can flex and rotate, so in reality the effective length constant may be between 0.5 and 1.0.
The experimental data matches this hypothesis. While a precise fit is difficult given the amount of data
available, an effective length constant of ~0.60 for the struts and ~0.75 for the overall structure seems to
yield the closest fit to experiment (Figure 5.5). This seems reasonable given that the contact area on the
ends of an individual strut is more significant than that between the structure and the end discs. The struts
are more constrained at their ends than the overall structure is at the end discs.
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Figure 5.5 Euler buckling failure stress predictions for both the end tetrahedral struts and the overall structure using
effective length constants of 0.6 for the struts and 0.75 for the overall structure.
A similar analysis for a repeat set of structures is shown in Appendix C. These results confirmed the
general trend from the initial test.
In conclusion, experimental results reveal a simplification in the model relating to end conditions. While
solid space frame compression struts are not particularly structurally efficient, the understanding gained
from testing them can be transferred to hollow space frame compressive struts, which show great
potential, in the future.
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5.1.2 Theoretical Improvement on the Basic Structure
The basic structure tested experimentally in the preceding sections is defined in a straightforward but sub-
optimal way: while the forces on the end tetrahedral struts are twice as high as the forces on the
octahedral compressive struts, all struts are given the same radius. This simplification means that the
octahedral struts are heavier than necessary. This recalls a fundamental principle of optimization: in order
to avoid some features of a structure being heavier than required, a structure in which all failure
mechanisms occur simultaneously is desired.
One way to improve on the basic structure is to alter the initial optimization process to allow for an extra
degree of freedom: the number of octahedra (n), the radius of the tetrahedral compressive struts (rt), and
the radius of all remaining struts (ro) are all free variables in the optimization procedure. We now have
three equations for failure force in the structure (Euler buckling of end tetrahedral struts, Euler buckling
of octahedral compressive struts, and Euler buckling of the overall structure):
For Euler buckling of the individual struts, the strut radius can be solved in terms of the strut length:
where:
Now the number of octahedra (n) must be solved by considered Euler buckling of the overall structure.
Combining several basic equations from above and Section 2.2:
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which eventually gives:
Now the expression for f and v for use in comparison of optimal structures can be found.
where:
Note that these equations are parameterized in terms of ft. Using these expressions, the performance
enhancement can be displayed using a graph similar to Figure 2.12. First, the new variable radius solid
space frame is compared to the standard solid space frame (Figure 5.6). A simple rod is also shown for
reference. Recall that this graph shows only the optimal design for each architecture.
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Figure 5.6 Comparison of the optimal designs of three different compressive structures. For each value of
dimensionless force, the lowest value of dimensionless volume required to resist that force is shown for each
structure. The solid space frame compressive strut is made lighter by allowing for different values of radius for the
constituent struts during the optimization process. Note that this improvement does not make the solid space frame
into a desirable structure: it is still significantly less structurally efficient than a simple tube.
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While the allowance for variable strut radius in the optimization process results in an improvement over
the standard solid space frame compressive strut, it is important to note that this structure is still
significantly less structurally efficient than a simple tube. However, the hollow version of a space frame
compressive strut is already an attractive option, having an efficiency advantage over a simple tube for
gentle loading. By employing the techniques described in this section in an analogous way to the hollow
space frame (i.e. allowing for variable strut radii and wall thickness in the optimization), this already
attractive architecture could be even further improved. While the mathematics behind this adjustment was
not performed, Figure 5.7 shows the hypothetical improvement, compared to a simple tube. First, such an
adjustment would improve the structural efficiency of hollow space frame compressive struts. More
critically, it would increase the range of loading scenarios for which hollow space frames show benefits
when compared to simple tubes, increasing the attractiveness of such structures.
This alteration would pose an interesting challenge due to the nature of certain manufacturing techniques:
for example, using a combination of stereolithography and electrodeposition would allow for control over
the radius of individual struts but less control over specific wall thicknesses.
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Figure 5.7 Comparison of the optimal designs of three different compressive structures. For each value of
dimensionless force, the lowest value of dimensionless volume required to resist that force is shown for each
structure. The hollow space frame compressive strut is made lighter by allowing for variable values of radius for the
constituent struts during the optimization process. It is important to note that this optimization was not actually
performed – an estimated hypothetical line is shown based on performance enhancements calculated for the solid
structure. By making this change, the loading scenarios for which the space frame shows greater structural
efficiency than a tube is expanded into heavier loadings.
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5.2 Mechanical Testing and Modeling of Sandwich Wall Tubes
Considering the simple case of tubes in compression, the two major failure mechanisms are Euler
buckling and local shell buckling. In Chapter 4, it has been shown that the optimal design of a simple tube
is that which yields equal failure stresses for the two competing failure modes. Given this upper bound on
the compressive strength of a tube, the only way to obtain structures with greater compressive strength is
to alter the fundamental design.
Compared to simple rods, tubes are efficient structures due to the placement of material far from the
neutral bending axis. This serves to inhibit Euler buckling. Unfortunately, the creation of thin tube walls
introduces the new failure mode of local shell buckling. Following this same logical thinking, local shell
buckling may be able to be inhibited by increasing the effective thickness of the walls of a tube at
constant mass. To do this, a new structure is proposed which uses a sandwich panel-like structure for a
tube wall (Figure 5.8).
Figure 5.8 Basic design of a tube whose wall is constructed similar to a sandwich panel: an inner and outer wall
connected by a series of webs.
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Figure 5.9 Parameters used to describe a sandwich wall tube: the left schematic shows the cross-section of the
structure, while the right schematic shows a magnified section of the tube wall. The structure shown here has 72
webs.
In essence, this structure is creating be taking a simple tube, splitting the tube wall in half, spreading these
two thinner walls apart, and filling the gap between the two walls with a webbed structure. This new
structure can be defined in a similar manner to a simple tube: the length of the tube L, the inner radius of
the tube r1, the thickness of the inner and outer walls t, the number of webs #, the thickness of each web
tweb, and the gap between the two walls G (Figure 5.9).
5.2.1 Proof of Concept
This new sandwich wall tube has the potential to inhibit local shell buckling due to the movement of
material away from the bending axis active in local shell buckling (this is the bending axis that runs
around the centre of the tube wall). In order to understand the behaviour of these structures, their
resistance to the two existing failure modes (Euler buckling and local shell buckling) must be explored. In
addition, a new failure mode is expected.
To study the resistance of these structures to local shell buckling, a base geometry very susceptible to
local shell buckling was used. By choosing a base geometry which fails at a stress well in the elastic
region of the material, a significant amount of improvement is possible, which will make potential
improvement easier to observe. (Conversely, choosing a base geometry which fails at 50 MPa would
leave little room for improvement. A structure like this becomes limited by material effects instead of
shape effects. This concept is explained in detail in Section 4.3.) A simple tube with a relatively high of
0.667 and a relatively low of 0.040 is expected to fail by local shell buckling at a stress of
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approximately 25 MPa and satisfied the requirements for the base geometry. This tube had a length of 30
mm, an inner radius of 20 mm, and a wall thickness of 0.8 mm.
First, a series of structures was printed with the same mass, length, and inner radius as the base case. It
was assumed that the optimal design for the webs would be that which attached the inner and outer walls
the most uniformly (i.e. a greater number of thin webs was preferred to a lesser number of thicker webs).
As such, the web thickness was set to 200 µm – the minimum thickness which the 3D printer was able to
reliably produce. The number of webs was set to 72 using approximation guided by preliminary testing
with far fewer webs.
Five sandwich wall tube geometries were fabricated for this first series. Structures were varied by
choosing different gaps between the inner and outer walls. It should be noted that as the gap increases, the
amount of material devoted to the webs also increases, and thus the thickness of the inner and outer walls
decreases. The dimensions of the structures examined in this first test are shown in Table 5.2, and they are
visualized in Figure 5.10.
The structures were tested in compression with a stroke rate of 0.4 mm/min, equating to a strain rate of
0.013 min-1
. The resulting stress-strain curves are shown in Figure 5.11.
Table 5.2 Specifications of the first set of sandwich wall tubes, as well as the simple tube base case (1f). The length,
inner radius, and mass are equal for all structures.
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Figure 5.10 Magnified portions of the cross-sections of the six structures. The five images on the left show the walls
of the sandwich wall tubes with varying gap, while the image on the right show the wall of the simple tube. Note
that the simple tube has the same mass as each of the five sandwich wall tubes.
Figure 5.11 Stress-strain curves for the six tubes tested in this set. Tubes 1a-1e are sandwich wall tubes with varying
values for the gap. Tube 1f is a simple tube.
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In order to model these structures, a modification to the standard local shell buckling prediction must be
made. Due to the type of failure experienced with local shell buckling (i.e. structural instability), it is
reasonable to consider the second moment of area of the tube wall about the centre line of the tube wall.
In order to do this, the sandwich-wall tube is divided into a number of slices equal to the number of webs
(in this case, 72) so that each slice is similar to an I-beam. The second moment of area of this I-beam is
then calculated about its neutral bending axis (Figure 5.12). An analogous calculation is performed on a
slice of the wall of a simple tube to find the second moment of area of that slice. By comparing these two
calculations, the wall thickness of the simple tube which would yield the same second moment of area as
that for the sandwich wall tube is calculated. This wall thickness is used to calculate an “effective ” for
the sandwich-wall tube. This effective value of can now be used in the standard local shell buckling
equation described in Section 2.1.2. However, the cross-sectional area of the sandwich wall tube will be
lower than that of its equivalent simple tube. To correct for the corresponding discrepancy in stress, a
term is added to the standard local shell buckling equation. Specifically, the local shell critical buckling
stress equation is multiplied by the ratio of the cross-sectional area of the equivalent simple tube to the
cross-sectional area of the sandwich wall tube. Finally, this modified version of the original local shell
buckling model is used to calculate a predicted local shell buckling failure stress for the effective
dimensions.
Figure 5.12 Single I-beam slice, 72 of which comprise a sandwich wall tube. The second moment of area of this I-
beam is calculated about its neutral bending axis.
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The experimental failure stresses are shown along with the predictions for local shell buckling in Figure
5.13. Note that the data is shown in terms of the effective value of .
Figure 5.13 Local shell buckling failure stress predictions compared to the experimental failure stress values. Note
that the right five experimental points correspond to sandwich wall tubes while the point on the left corresponds to
the equal-mass simple tube.
Several observations can be made from this analysis. First, the local shell buckling model under-predicts
the experimental failure stress value for the simple tube. This is expected considering the similar results in
Sections 4.2-4.3. Given that under-prediction is expected, there appears to be a penalty applied to the
sandwich wall tubes – their experimental values are not under-predicted by experiment. Despite this
penalty, the sandwich wall tubes do show an improvement in structural efficiency for higher values of
(corresponding to higher values of the gap): the sandwich wall tubes are the same mass as one of the
simple tubes, yet they demonstrate an increase in failure stress from 36.0 MPa up to 43.3 MPa (an
increase in performance of approximately 20%).
The penalty to the sandwich wall tubes is likely due to the introduction of a “flaw” – the relatively
pristine simple tube structure has been upset by splitting the wall in half. Due to this penalty, tubes with
low values of the gap perform slightly worse than the simple tube. The gains afforded by inhibiting local
shell buckling are less significant when the inner and outer tube walls are only slightly pulled apart. The
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modified local shell buckling predictive model does seem to accurately reflect this increasing trend in
failure stress for greater values of the gap. While this behaviour is encouraging, it leads to the question of
how far this phenomenon can continue: it is reasonable to assume that at some point, spreading the inner
and outer tube walls further and further apart must result in some new failure mechanism and a
corresponding decrease in stress. Specifically, there must be a peak in the failure stress corresponding to
an optimal design.
5.2.2 Exploration of Gap and Number of Webs
The trend of increasing failure stress with increasing gap was observed in the previous section. In order to
explore the continuation of this trend, tubes with greater values of gap were tested. As it was expected
that a new failure mechanism would be encountered for high values of the gap, the number of webs used
in the structure also warranted exploration: given the nature of these structures, the new failure
mechanism was likely to involve the web structure connecting the inner and outer walls. While the
thickness of the webs had already been set as previously discussed, the number of webs would impact the
strength of the connection between the two walls.
Two sets of tubes were fabricated and tested in compression. Both sets used the same simple tube base
case and corresponding length and mass as the previous test. In both tests, gap values ranging from 0.5
mm to 2.5 mm were tested – significantly greater values than tested in the previous test. One set
compared tubes with 72 webs to tubes with 36 webs, while the second set compared tubes with 72 webs
to tubes with 108 webs. 108 webs proved to be an upper limit: sandwich walls tubes with the standard
length and mass with greater than 108 webs resulted in wall thickness below the minimum value of 200
µm (this was due to a greater proportion of the available material being used for the webs). The
specifications of the second and third sets of sandwich wall tubes are shown in Table 5.3.
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Table 5.3 Specifications of two sets of 11 tubes. Within each set, there is a comparison involving the value of gap
and the number of webs. As well, each set contains a simple tube base case to provide a baseline for comparison.
The experimental failure stresses are shown along with the predictions for local shell buckling in Figures
5.14-5.15. One key insight from this analysis is that as expected, a peak value of failure stress is
experimentally observed. This must be due to the activation of a new failure mode. Examining the
polymer sandwich wall tubes with high values of the gap just after buckling, clear splitting of the tube
walls is present (Figure 5.16). However, this failure mechanism is not well understood: it is not obvious
which precise geometric regimes (i.e. combination of gap and number of webs) will activate this failure
mode. In addition, it would seem reasonable that splitting of the walls would be inhibited more by a
greater number of webs, but the enhancements made by increasing the number of webs are not limited to
the region in which the tubes failed by wall splitting.
The second key insight from this analysis is that increasing the number of webs generally increases failure
stress: the sandwich wall tubes with 72 webs were significantly stronger than those with 36 webs, while
the sandwich wall tubes with 108 webs were marginally stronger than those with 72 webs. This is
intriguing given the fact that the predicted values are actually slightly lower for a greater number of webs:
more webs results in less material in the inner and outer walls and a corresponding decrease in second
moment of area of the tube wall. However, this observation can be rationalized by considering an extreme
example: if the sandwich wall tubes were constructed using just one or two webs, the structure would
85
behave approximately like two separate thin-walled tubes and fail at a considerably lower stress. The
webs are critical to unite the inner and outer walls and enable the structure to take advantage of its
theoretical inhibition of local shell buckling. Judging by the poor performance of the sandwich wall tubes
with 36 webs, it is clear that a sufficient number of webs is critical to exploit the strength gains possible
with this architecture.
In some cases, the strength increase of the strongest sandwich wall tube over the simple tube base case is
26.2 MPa up to 38.0 MPa – an increase of approximately 45%.
Figure 5.14 Comparison of experimental failure stress to predicted local shell buckling stress. Two sets of sandwich
wall tubes are shown: one with 72 webs and one with 36 webs. Note that the experimental data point on the left
corresponds to the simple tube base case. Also note that local shell buckling predictions are made for both 72 and 36
webs, but they are very similar due to the similar second moments of area for the walls of each structure. The dotted
lines connecting experimental data points serve only to guide the eye.
86
Figure 5.15 Comparison of experimental failure stress to predicted local shell buckling stress. Two sets of sandwich
wall tubes are shown: one with 72 webs and one with 108 webs. Note that the experimental data point on the left
corresponds to the simple tube base case. Also note that local shell buckling predictions are made for both 72 and
108 webs, but they are very similar due to the similar second moments of area for the walls of each structure. The
dotted lines connecting experimental data points serve only to guide the eye.
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Figure 5.16 Compressive testing of sandwich wall tubes with high values of the gap resulted in splitting of the inner
and outer wall away from the webs.
88
5.2.3 Sandwich Wall Tubes in Euler Buckling
The main impetus behind the design of these sandwich wall tubes was the possibility of inhibiting local
shell buckling. However, in order to yield gains in the strength of the optimal design in relation to optimal
simple tubes, Euler buckling must also be considered. In order to examine this, sandwich wall tubes
highly susceptible to Euler buckling were fabricated. A simple tube base case was used which had a
relatively low of 0.017 and a relatively high of 1.17. This tube was expected to fail by Euler buckling
at a stress of approximately 24 MPa and satisfied the requirements for the base geometry: there was lots
of room for improvement without encountering material limitations. This tube had a length of 70 mm, an
inner radius of 1.2 mm, and a wall thickness of 1.4 mm.
Five sandwich wall tubes with varying values of the gap were fabricated, along with the simple tube base
case. The specifications for these tubes are shown in Table 5.4. The number of webs for these tubes was
set to 18: this value was deemed sufficient yet not so great that the entire tube wall was filled with webs.
Table 5.4 Specifications of the set of sandwich wall tubes designed to fail by Euler buckling, as well as the simple
tube base case (4f). The length, inner radius, and mass are equal for all structures.
The tubes were tested in compression at a stroke rate of 0.8 mm/min, equating to a strain rate of 0.011
min-1
. Predictions for the Euler buckling stress of these tubes were made using the original Euler buckling
equation described in Section 2.1.1. The second moment of area of the sandwich wall tube about its
neutral bending axis was calculated for use in the original equation. The experimental data and Euler
buckling predictions are shown in Figure 5.17. The analysis is plotted with respect to second moment of
area in order to compare the sandwich wall tubes to the simple tube.
From this analysis, it appears that the effects of converting tubes to sandwich wall tubes are similar for
Euler buckling as they were local shell buckling. As predicted by the model, increasing the value of the
gap increases the second moment of area of the tube and increases the tube’s failure stress by Euler
buckling. However, there is also a penalty to converting the tubes to sandwich wall tubes, likely due to
the introduced “flaw”. The sandwich wall tube with the lowest value of the gap has a lower failure stress
than the equivalent simple tube. This reveals the complexity of the structure of these tubes.
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These results suggest that in addition to the enhancements already observed by sandwich wall tubes in
local shell buckling, this new design also yields enhancements in terms of Euler buckling. The simple
tube failure stress of 26.4 MPa was increased to 34.2 for the strongest sandwich wall tube – an increase of
almost 30%. Further, this increase has the potential to expand for higher values of the gap – the optimal
has not yet been reached.
While a full optimization has yet to be performed, the structural efficiency of the sandwich wall tube
design is very promising.
Figure 5.17 Comparison of experimental failure stress to predicted Euler buckling failure stress for sandwich wall
tubes designed to fail by Euler buckling. Note that the experimental point on the left corresponds to the simple tube
base case.
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6. Conclusion
6.1 Simple Polymer Tubes
Experimental testing of polymer tubes in compression shows good agreement with predictions from
analytical models for Euler and local shell buckling. As expected, the optimal architecture was that which
resulted in equal failure stresses for the two failure modes.
The predictive models for Euler and local shell buckling can be considered along with the new
understanding of the range of mechanical properties displayed through various print cycles. Predictions
can be made throughout the range of possible material properties to give new real-world failure stress
predictions for potential 3D printed polymer tubes (Figure 6.1). It is interesting to note that since the
material property appears on the same order in the predictive models for both failure modes, the optimal
geometry does not change with material properties.
Mechanical testing on heavier sets of tubes reinforced the concept that optimal architectures are less clear
when material limitations are encountered. In order for experiments to show a clear optimal, it is ideal to
test structures which fail in the material’s elastic regime. This ensures that shape effects rather than
material effects are the crucial factor in structural failure.
The under-prediction of experimental failure stress draws attention to the complexity of using
stereolithography for structural designs: variation in material modulus with printed part height makes
accurate prediction difficult.
91
Figure 6.1 Euler and local shell buckling failure stress predictions are shown over a range of tube geometries as well
as a range of material properties. The upper, more pale surfaces show predictions made for the stiffest observed
stereolithographic polymer (Young’s modulus 1780 MPa) while the lower, darker surfaces show predictions made
for the least stiff observed stereolithographic polymer (Young’s modulus 1470 MPa).
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6.2 Space Frame Compressive Struts
Experimental testing of space frame compressive struts validated the basic concepts underpinning the
analytical model but revealed simplifications in relation to Euler buckling end conditions. In reality, the
effective length constants for both the overall structure and the constituent struts appear to be somewhere
between fixed (K=0.5) and free (K=1.0).
A modification was made to the theoretical optimization for solid space frame compressive struts,
increasing structural efficiency by allowing different radii for different struts. This enhancement suggests
the potential for analogous enhancements to the already very structurally efficient hollow space frame
compressive strut design.
6.3 Sandwich Wall Tubes
A novel sandwich wall tube design has been shown to effectively inhibit local shell buckling while also
resulting in strengthening of the Euler buckling failure mode. The gap between the inner and outer tube
walls was found to have an optimal value, above which a new failure mechanism termed wall splitting
became active. Despite the introduction of this new failure mode and a general penalty in strength due to
the introduction of flaws, the sandwich wall design still offered improvements in local shell buckling
strength exceeding 45% when compared to simple tubes of equal mass and length. Finally, increasing the
number of webs used to connect the inner and outer tube walls resulted in enhanced strength regardless of
the value of the gap.
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7. Future Work
7.1 Simple Polymer Tubes
While simple polymer tubes in compression have been shown to behave in a manner similar to that
predicted by models, the behaviour of tubes integrated into microtrusses is not clear. Added complexity of
strut end conditions and collapse of the tube walls are the main points of uncertainty. While some
preliminary work has been done to explore these issues (see Appendix D) further experimental and
analytical work is required.
7.2 Space Frame Compressive Struts
The experimental behaviour of solid space frame compressive struts has enhanced the understanding of
this type of structure. However, the real structural performance gains come by using hollow versions of
these structures. While existing stereolithographic additive manufacturing techniques are not able to
fabricate optimal hollow space frame structures due to practical lower limits on wall thickness, other
methods of fabrication exist. Using stereolithographic parts as pre-forms onto which metal is deposited
through electrodeposition would result in effectively hollow metal structures, and is an exciting
possibility in the future.
7.3 Sandwich Wall Tubes
A new failure mode of wall splitting was identified, but no model exists to predict for which geometries
and values of stress this failure will occur. Development of an analytic model for this failure mechanism
would allow more complete optimization of these novel structures, potentially leading to further
improvements in structural efficiency and weight savings. Finite element analysis is one tool which may
be helpful in developing this model. In addition, filleting of the web-wall interface may reduce stress
concentration and strengthen this fracture failure mode.
Experimental testing on sandwich wall tubes which failed by wall splitting led to an interesting
observation: whereas many stereolithographic polymer structures failed by fracturing soon after reaching
their peak stress value, the wall splitting failure resulted in a slow, progressive failure (Figure 7.1),
leading to exciting possibilities for energy absorption. Much of the structure remained pristine while
fracture was confined to the end of the tube in a moving band of failure. Additional work could be done to
raise the stress value of the plateau, and explore the energy absorption potential of these structures.
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Figure 7.1 Comparison of stress-strain curves for a sandwich wall tube which failed by wall splitting (blue) and a
simple tube with the same length and mass (red). Note that the structure shows progressive fracture, with a
significant stress value maintained to strain values in excess of 15%. The shaded areas show energy absorption. The
sandwich wall tube shows greatly enhanced energy absorption when compared to the simple tube which fractured.
95
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Appendix A: Compression Machine Compliance
As with any mechanical testing apparatus, the Shimadzu AG-I 50 kN has its own stiffness that must be
considered when analyzing the force-displacement data from compression tests. Specifically, the
measured values of displacement must be reduced by the amount of that displacement that is due to the
compression of the machine itself.
In order to find the stiffness of the machine, a compression test was performed with the upper and lower
platens touching: no sample was placed in between. The stiffness of the machine extracted from this test
was found to be 65700 N/mm.
In order to correct the force-displacement data for each compression test (e.g. on compression coupons to
extract material properties or on struts to test various architectures), the measured displacement values
measured by the machine were reduced using the following equation:
where dS is the true displacement of the sample, dT is the displacement measured in the test, and F is the
force measured in the test.
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Appendix B: Repeats of Tube Compression Tests
Figures B.0.1-B.0.2 show predicted Euler buckling and local shell buckling failure stresses along with
experimental values for the second and third sets of tubes. Note that each graph shows two separate,
repeated experiments using separate tubes and compression coupon data.
Figure B.0.1 Comparison of experimental failure stress to Euler and local shell buckling predictions for repeat sets
of the second set of tubes.
99
Figure B.0.2 Comparison of experimental failure stress to Euler and local shell buckling predictions for repeat sets
of the third set of tubes.
100
Appendix C: Repeats of Space Frame Compressive Strut
Tests
Figure C.0.1 Comparison of experiment to prediction for a repeat set of the structures analyzed in Figure 5.7. Note that two
structures here were broken during preparation and are excluded.
101
Appendix D: Mechanical Testing of Hollow Polymer
Microtrusses
Preliminary work was done on the incorporation of simple tubes into microtrusses to begin to understand
the complexities in modeling that arise when studying compressive struts in a network instead of on their
own.
The basic geometry studied was a single sheet of tetrahedral/octahedral microtruss in a hexagonal block
containing three complete tetrahedral unit cells loaded in compression. The exact construction of this
block is the subject of study and is discussed below. This general geometry was chosen for economy of
printing (larger structures would be significantly more expensive) while still capturing truss behaviour. In
order to show improvements made by using tubular struts instead of solid struts, one solid strut and five
tubular struts were tested. All struts designs had equal mass and length. The five tubular struts
theoretically contained several geometries susceptible to Euler buckling and several geometries
susceptible to local shell buckling, as well as an optimal structure. The tube geometries are shown in
Table D.1.
Table D.1 Specifications of constituent struts used in various microtruss compression tests.
To reduce edge effects (i.e. approximate conditions of an infinite sheet) a solid polymer confinement shim
was placed around the truss during compression (Figure D.0.1).
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Figure D.0.1 Confinement shim placed around truss blocks during compression testing.
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First, a series six of structures (one solid and five hollow) were constructed as shown in Figure D.0.2 and
tested in compression. It was found that all these structures containing hollow struts failed by fracture of
the top and bottom struts loaded on their sides. The desired and predicted failure mechanisms (Euler and
local shell buckling of the compressively loaded struts) did not occur. The side-loaded struts in the upper
and lower faces were not able to transmit the force. As such, the potential enhancements in structural
efficiency of tubes compared to rods in compression were not realized.
Figure D.0.2 First microtruss construction technique: top and bottom faces are constructed from the same network of
struts used in the core.
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To avoid the pitfall of side-loaded crushing of the upper and lower faces, these faces were replaced with
flat, solid sheets as shown in Figure D.0.3. Once again, the desired failure modes were not active. These
structures failed by fracture at the interface between the ends of the tubular struts and the upper/lower
sheets. Tubular struts were weaker than the solid version.
This interface fracture is likely encouraged by stress concentration at the interface, and potential surface
roughness resulting from the printing process.
Figure D.0.3 Microtruss block with upper and lower faces replaced with solid sheets. Note that holes were added to
these sheets to allow support wax to be removed.
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To inhibit fracture at the strut/plate interface, various techniques were employed to reduce the stress in
this area. First, the ends of the hollow struts were thickened slightly by filleting (Figure D.0.4). This still
resulted in fracture at the tube/plate interface.
Figure D.0.4 Microtruss block with filleting to thicken the tube walls at the interface with the plates.
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Finally, to reduce the stress at the tube/plate interface further, the nodal geometry was changed (Figure
D.0.5). The struts were placed such that there was no intersection of the tubes – each tube interfaced with
the plate on its own. This effectively increased the interface area (and reduced the stress) between the core
and the upper/lower plates. This prevented fracture at the tube/plate interface, and resulted in performance
enhancements of the hollow trusses when compared to the solid truss (Figure D.0.6). Euler buckling and
local shell buckling were observed. It is important to note that the tube geometries used in these
microtrusses are quite “heavy” – they fail in the material’s plastic regime. As discussed previously, this
diminishes the clarity of the optimal design.
It is clear that more work must be done to effectively integrate strut designs into microtrusses. The
preliminary exploration discussed here shows that changes to node geometry and points of load contact
can help to unleash the potential of structural efficiencies in the constituent struts. Adjustments to the
original models to account for these factors as well as others (such as a change to strut end constraints)
are also required.
Figure D.0.5 Microtruss block with no tube intersection at the nodes.
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Figure D.0.6 Comparison of experimental failure force to predictions for Euler and local shell buckling for
microtrusses in compression. Note that predictions were made using a combination of the standard models and basic
trigonometry to sum the contributions from all struts.
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