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.

iv

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

v

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

viii

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

xi

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

xii

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

xv

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.

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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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97

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.

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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.