DEVELOPING UNIT CELL DESIGN GUIDELINES ...

DEVELOPING UNIT CELL DESIGN GUIDELINES FOR MESO-SCALE PERIODIC CELLULAR MATERIALS

A Dissertation Presented to the Graduate School of Clemson University

In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Mechanical Engineering

by Mohammad Fazelpour December 2016

Accepted by: Dr. Joshua D. Summers, Committee Chair Dr. Prabhu Shankar, Committee Co-Chair Dr. Georges M. Fadel Dr. Gang Li Dr. Vincent Y. Blouin

ABSTRACT

Periodic cellular materials can have superior mechanical effective properties such as higher stiffness, strength, and/or flexibility compared to other materials with similar density. There is a significant body of research focusing on designing meso scale periodic cellular material structures with superior effective elastic properties; however, there is only limited research on how to guide designers on developing improved unit cell topologies and shapes, for a given set of loading requirements and conditions. To address this challenge, this dissertation introduces unit cell design guidelines. The guidelines codify design knowledge to provide direction and recommendations to engineers who are trying to change the topology and shape of unit cells to improve a targeted measure of performance. The guidelines investigate the effects of changes to topology and shape characteristics of unit cells subjected to in-plane shear loading. The investigated topology and shape characteristics include side connections, transverse connections, curved beams, and vertical legs. In order to validate that the unit cell design guidelines apply to a variety of unit cells with similar topology characteristics, each guideline is applied to a variety of unit cells and numerical simulations are used to validate that the performance of the unit cell is improved by the change. More importantly, this dissertation introduces a formalized and systematic method to develop unit cell design guidelines to achieve desired mechanical effective properties of a meso scale periodic cellular materials by changing the topology and shape of a unit cell. In order to support unambiguous discussion of cellular topology, a vocabulary is developed to describe two-dimensional periodic cellular materials.

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The vocabulary represents the 2D

periodic cellular materials through descriptions of the unit cell and of the tiling of the unit cell arrangements. A designer study is conducted to evaluate the usability of the vocabulary by engineers. The results of designer study support the idea that the vocabulary is usable by engineers who generally do not have a background in periodic cellular materials. Two user studies are conducted to evaluate the effect of subjectivity and sequencing of the guidelines. The subjectivity results show that the modified unit cells developed by different engineers using the same guideline and initial topology have similarities. However, the degree of similarity is affected by the content of the guidelines, how the guidelines are written, and the initial unit cells. The sequencing of guidelines is also investigated to evaluate if changing the order of guidelines affects modified unit cells. Ultimately, the guidelines are validated through numerical simulations and controlled studies with human subjects while the guideline development method is demonstrated through one case study with novice engineer developing new guidelines under tension loading.

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DEDICATION

Dedicated to my wife because without her love and encouragement, I would not able to finish my studies and to my patents for supporting me to reach this far in my education.

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ACKNOWLEDGMENTS I sincerely thank my adviser Dr. Joshua Summers. Without his support, advice, patience, and encouragement, I would not able to finish this dissertation. He mentored me not only on research, but also to think critically, learn how to do research, take the ownership of my work, and grow personally and professionally. I would like also thank my co-adviser Dr. Prabhu Shankar with countless hours he have spent to discuss different aspects of my research with me. He mentored me on learning ABAQUS, Python scripting, developing my research questions, and hypotheses. More importantly, our fruitful discussions strengthened my confidence and arguments to defend my research. I would also like to thank Dr. Georges Fadel, Dr. Gang Li, and Dr. Vincent Blouin for serving on my committee, providing me feedback, and having in-depth discussions to improve my research quality. I need sincerely thank the Department of Mechanical Engineering at Clemson University to provide me financial assistance for the continuation of my PhD studies. I thank all CEDAR members, specifically Marcus Yoder and Apurva Patel for discussions and help on my research. The last but not least, I would like to thank my siblings, Maryam and Reza, for helping me to grow and supporting me spiritually throughout the most difficult moments of my life.

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TABLE OF CONTENTS

Page ABSTRACT .................................................................................................................................... ii DEDICATION ............................................................................................................................... iv ACKNOWLEDGMENTS .............................................................................................................. v TABLE OF CONTENTS ............................................................................................................... vi LIST OF TABLES .......................................................................................................................... x LIST OF FIGURES ..................................................................................................................... xiv Chapter 1 INTRODUCTION TO PERIODIC CELLULAR MATERIALS................................. 1 1.1

Periodic Cellular Materials and their Engineering Applications ..................................... 1

1.2

Topology, Shape, Size Design - Definitions .................................................................... 3

1.3

Research Gaps: Motivation of Developing Unit Cell Design (UCD) Guidelines............ 4

1.4

Research Questions and Hypothesis ................................................................................ 6

1.5

Intellectual Merit .............................................................................................................. 8

1.6

Dissertation Outline.......................................................................................................... 9

Chapter 2 A VOCABULARY FOR DESCRIBING PERIODIC CELLULAR MATERIALS .. 11 2.1

Overview ........................................................................................................................ 11

2.2

The Need for a Classification to Represent Periodic Cellular Materials ....................... 11

2.3

Representation in Engineering Design ........................................................................... 14

2.4

Overview of the Vocabulary to Describe Periodic Cellular Materials........................... 18

2.5

Periodic Cellular Material Representation -- Level 1 .................................................... 19

2.5.1

Two-Dimensional (2D) Unit cell ............................................................................ 20

2.5.2

Tiling of Unit Cells ................................................................................................. 20

2.6

Periodic Cellular Material Representation – Level 2 ..................................................... 21

2.6.1

Boundary ................................................................................................................. 21

2.6.2

Interior..................................................................................................................... 22

2.6.3

Connection Type ..................................................................................................... 23

2.7

Validation of the Periodic Cellular Material Vocabulary .............................................. 24

2.7.1

Perceptual Orthogonality ........................................................................................ 25

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2.7.2

Completeness .......................................................................................................... 28

2.7.3

Depth ....................................................................................................................... 32

2.8 Exploring the Usability of the Vocabulary to Represent Periodic Cellular Materials – A Designer Study .......................................................................................................................... 35 2.9

Summary and Future Work ............................................................................................ 37

Chapter 3 REVIEW ON DESIGN METHODS FOR PERIODIC CELLULAR MATERIALS 40 3.1

Analytical Method .......................................................................................................... 40

3.2

Simulation Method ......................................................................................................... 45

3.3

Parametric Optimization Method ................................................................................... 46

3.4

Topology Optimization Method ..................................................................................... 50

3.5

Selection Method............................................................................................................ 54

3.6

Intuitive Design Method................................................................................................. 55

3.7

Conclusion...................................................................................................................... 56

Chapter 4 A UNIT CELL DESIGN GUIDELINE DEVELOPMENT METHOD FOR MESOSCALE PERIODIC CELLULAR MATERIALS UNDER SHEAR LOADING ........................ 59 4.1

Overview ........................................................................................................................ 59

4.2

The need to Develop Unit Cell Design Guidelines ........................................................ 59

4.2.1

Design methods for periodic cellular material structures ....................................... 62

4.2.2

Topology and shape characteristics of unit cells .................................................... 64

4.3 Evolution of Meso-Scaled Periodic Cellular Material Structures for Non-Pneumatic Tires - A Case Study ................................................................................................................. 64 4.4

Two Unit Cell Design Guidelines from the Case Study ................................................ 67

4.4.1

Guideline #1 – Connection between Unit Cells ...................................................... 67

4.4.2

Guideline #2 – Transverse Connections ................................................................. 69

4.5

Validation of Generalization of guidelines using numerical simulations ...................... 72

4.5.1

General Validation Procedure ................................................................................. 72

4.5.2

Validation of Guideline #1...................................................................................... 74

4.5.3

Validation of Guideline #2...................................................................................... 75

4.6 Generalization of a Unit Cell Design Guideline Development Method (UCDGD Method) ..................................................................................................................................... 76 4.7

Demonstration of the Unit Cell Design Guideline Development Method ..................... 83

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4.7.1

Guideline#3 Development ...................................................................................... 83

4.7.2

Guideline#4 Development ...................................................................................... 86

4.8

Conclusion and Future Work ......................................................................................... 89

Chapter 5 A DEMONSTRATION ON HOW RESEARCHERS USE THE UNIT CELL DESIGN GUIDELINE DEVELOPMENT METHOD – A case study ........................................ 92 Chapter 6 EXPLORING THE SUBJECTIVITY AND SEQUENCING OF UNIT CELL DESIGN GUIDELINES – A USER STUDY............................................................................... 98 6.1

Overview ........................................................................................................................ 98

6.2

Designing the User Study............................................................................................... 98

6.3

Hypothesis ...................................................................................................................... 99

6.4

Variables......................................................................................................................... 99

6.4.1

User study I – Subjectivity .................................................................................... 102

6.4.2

User study II – sequencing .................................................................................... 103

6.5

Study Participants ......................................................................................................... 103

6.6

Study Procedure ........................................................................................................... 104

6.7

Data Analysis ............................................................................................................... 108

6.7.1

Metrics to evaluate similarities among unit cells after change ............................. 109

6.7.2

Inter-rater reliability analysis ................................................................................ 109

6.7.3

Results for subjectivity of unit cell design guidelines .......................................... 111

6.7.3.1

Guideline#1 Results ....................................................................................... 111

6.7.3.2


6.7.3.3


6.7.3.4


6.7.4

Results for sequencing effect of unit cell design guidelines ................................. 125

6.7.4.1

Results for Guideline#1 Guideline#2 ....................................................... 125

5.7.4.2

Results for Guideline#1 Guideline#3 ........................................................ 127

5.7.4.3

Results for Guideline#1  Guideline#4 ....................................................... 129

5.7.4.4


5.7.4.5


5.7.4.6


6.7.5

Overall Sequencing Results .................................................................................. 138

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6.8

Conclusion and Future Work ....................................................................................... 138

Chapter 7 CONCLUSION AND FUTURE WORK ................................................................. 141 7.1

Answering the Research Questions .............................................................................. 141

7.2

Concluding Reflections ................................................................................................ 143

7.3

Limitations and Future Work ....................................................................................... 145

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LIST OF TABLES

Table

Page

Table 2.1: Pictorial expression of PCM representation with the classification scheme by [51] .. 16 Table 2.2: Textual expression of PCM representation with the classification scheme by [51] .... 17 Table 2.3: Orthogonality matrix of periodic cellular material vocabulary ................................... 26 Table 2.4: Hexagonal honeycomb, unit cell illustration, abstraction and representation ............. 29 Table 2.5: Hexagonal chiral honeycomb, unit cell illustration, abstraction and representation [23] ....................................................................................................................................................... 30 Table 2.6: S-type, unit cell illustrations, abstraction, and representation [62] ............................. 31 Table 2.7: Honeycomb with diamond cell, unit cell illustrations, abstraction, and representation[42] ......................................................................................................................... 32 Table 2.8: Representation of a parallelogram and its illustration ................................................. 34 Table 2.9: Representation of a rhombus unit cell and its illustration ........................................... 35 Table 2.10: The percentage of descriptors as expected based on their definitions for both honeycomb and S-type .................................................................................................................. 37 Table 3.1: Comparison of design methods of periodic cellular materials .................................... 57 Table 4.1: Material properties of 7075-T6 Aluminum alloy ........................................................ 73 Table 4.2: Shear flexure and corresponding thickness of UC for the validation of Guideline #1 at effective shear modulus of 10 MPa .............................................................................................. 75 Table 4.3: Shear Flexure and corresponding thickness of UC for the validation of Guideline #2 at effective shear modulus of 10 MPa .............................................................................................. 76 Table 4.4: Shear flexure and corresponding thickness of UC for the validation of Guideline #3 at effective shear modulus of 10 MPa .............................................................................................. 85 Table 4.5: Shear flexure and corresponding thickness of UC for the validation of Guideline #4 at effective shear modulus of 10 MPa .............................................................................................. 89

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LIST OF TABLES (CONTINUED) Table 5.1: Effective yield strain and corresponding thickness of UC for the validation of Guideline #5 at effective young’s modulus of 10 MPa ................................................................ 95 Table 5.2: Effective yield strain and corresponding thickness of UC for the validation of Guideline #6 at effective Young’s modulus of 10 MPa ............................................................... 97 Table 6.1: Number of responses collected for each guideline and shape ................................... 107 Table 6.2: Number of responses collected for each case in sequencing study ........................... 108 Table 6.3: Initial inter-rater reliability using Cohen's kappa ...................................................... 110 Table 6.4: Kappa values for guidelines applied to the hexagon ................................................. 110 Table 6.5: Kappa values for guidelines applied to the rhombus ................................................. 110 Table 6.6: Results for Guideline#1 – Distribution of the number of shapes which are followed, not followed, and blank spaces for Guildeline#1 - The percent of shapes followed Guildeline#1 for honeycomb and rhombus ...................................................................................................... 112 Table 6.7: Distribution of similar shapes which are followed Guildeline#1 with the number of their occurrences for honeycomb and rhombus .......................................................................... 113 Table 6.8: Results for Guideline#2 – Distribution of the number of shapes which are followed, not followed, and blank spaces for Guildeline#2 - The percentage of shapes followed Guildeline#2 for honeycomb and rhombus................................................................................. 114 Table 6.9: Distribution of similar shapes which are followed Guildeline#2 with the number of their occurrences for honeycomb and rhombus .......................................................................... 116 Table 6.10: Distribution of similar shapes which are followed Guildeline#2 with the number of their occurrences for honeycomb and rhombus – continued ...................................................... 117 Table 6.11: Results for Guideline#3 – Distribution of the Number of shapes which are followed, not followed, and blank spaces for Guildeline#3 - The Percentage of Shapes Followed Guildeline#3 for Honeycomb and Rhombus .............................................................................. 118 Table 6.12: Distribution of similar shapes which are followed Guildeline#3 with the number of their occurrences for honeycomb and rhombus .......................................................................... 119

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LIST OF TABLES (CONTINUED) Table 6.13: Distribution of similar shapes which are followed Guildeline#3 with the number of their occurrences for honeycomb and rhombus .......................................................................... 120 Table 6.14: Results for Guideline#4 – Distribution of the number of shapes which are followed, not followed, and blank spaces for Guildeline#4 - The percentage of shapes followed Guildeline#4 for honeycomb and rhombus................................................................................. 123 Table 6.15: Distribution of similar shapes which are followed Guildeline#4 with the number of their occurrences for honeycomb and rhombus .......................................................................... 124 Table 6.16: Sequencing results for Guidelines#1=>2 and 2=>1 – Distribution of the Number of ..................................................................................................................................................... 125 Table 6.17: Sequencing results for distribution of similar shapes which are followed Guildeline#1=>2 and 2=>1 with the number of their occurrences for honeycomb and rhombus ..................................................................................................................................................... 126 Table 6.18: Sequencing results for Guidelines#1=>3 and 3=>1 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>3 and 3=>1 for honeycomb and rhombus ........................................................... 127 Table 6.19: Sequencing results for distribution of similar shapes which are followed Guildeline#1=>3 and 3=>1 with the number of their occurrences for honeycomb and rhombus ..................................................................................................................................................... 128 Table 6.20: Sequencing results for Guidelines#1=>4 and 4=>1 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>4 and 4=>1 for honeycomb and rhombus ........................................................... 130 Table 6.21: Sequencing results for distribution of similar shapes which are followed Guildeline#1=>4 and 4=>1 with the number of their occurrences for honeycomb and rhombus ..................................................................................................................................................... 131

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LIST OF TABLES (CONTINUED) Table 6.22: Sequencing results for Guidelines#2=>3 and 3=>2 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>4 and 4=>1 for honeycomb and rhombus ........................................................... 132 Table 6.23: Sequencing results for distribution of similar shapes which are followed Guildeline#2=>3 and 3=>2 with the number of their occurrences for honeycomb and rhombus ..................................................................................................................................................... 133 Table 6.24: Sequencing results for Guidelines#2=>4 and 4=>2 – Distribution of the number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#2=>4 and 4=>2 for honeycomb and rhombus ........................................................... 134 Table 6.25: Sequencing results for distribution of similar shapes which are followed Guildeline#2=>4 and 4=>2 with the number of their occurrences for honeycomb and rhombus ..................................................................................................................................................... 135 Table 6.26: Sequencing results for Guidelines#3=>4 and 4=>3 – Distribution of the number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#3=>4 and 4=>3 for honeycomb and rhombus ........................................................... 136 Table 6.27: Sequencing results for distribution of similar shapes which are followed Guildeline#3=>4 and 4=>3 with the number of their occurrences for honeycomb and rhombus ..................................................................................................................................................... 137 Table 6.28: Percent of modified unit cells accounted for in the top three clusters ..................... 138

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LIST OF FIGURES

Figure

Page

Figure 1.1: (A) A Hexagonal honeycomb (periodic cellular material), (B) A Foam (stochastic cellular material) ............................................................................................................................. 2 Figure 1.2: Applications of meso-scaled periodic cellular materials in one-dimensional wing morphing applications (left) and non-pneumatic tire (right) – Adopted from [12] and [14] .......... 3 Figure 1.3: Shape, dimension, and topology design of a cantilever beam (adapted from [19]) ..... 4 Figure 1.4: Dissertation roadmap .................................................................................................. 10 Figure 2.1: Examples of misinterpretation through current status of explaining the PCM. (A) Square super cell constructed from mix of squares and triangles [42]; (B) An alternative cell with the combination of squares and triangles; (C) diamond cell [42]; (D) An alternative diamond cell without transverse connections. .................................................................................................... 13 Figure 2.2: Honeycomb with (A) Square Cells and (B) Triangle Cells; (C) A Triangle UC; (D) The Edge Connection between a UC and its Adjacent Structural Member; (E) The Edge Connection between two Vertexes; (F) all edge connections that intersect with a vertex. .......... 14 Figure 2.3: Hierarchal periodic cellular material vocabulary ....................................................... 19 Figure 2.4: Elements of unit cell vocabulary ................................................................................ 20 Figure 2.5: The Simply (A) and non-simply (B) connected unit cells [58] .................................. 21 Figure 2.6: The size change of the boundary around the circle unit cell ...................................... 22 Figure 2.7: Examples of different interior changes: topology, shape, and size changes. ............. 23 Figure 2.8: Example of vertex and edge connections for hexagonal honeycomb. The vertex and edge connections are indicated by slide circles and highlighted lines, respectively. The dash lines are UC boundaries......................................................................................................................... 24 Figure 2.9: Example of how topology changes affect the vertex and edge connections for hexagonal honeycomb. The vertex and edge connections are indicated by slide circles and highlighted lines, respectively. The dash lines are UC boundaries. ............................................ 24

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LIST OF FIGURES (CONTINUED) Figure 2.10: Simple (A) and Non-simple (B) connectivity optimization results – Adopted from [58] ................................................................................................................................................ 27 Figure 2.11: Relation of the size of boundary and the size of interior.......................................... 28 Figure 2.12: Representation of two UC design guidelines through removing (A) Side connection between two unit cells and (B) transverse connections in the interior ......................................... 39 Figure 3.1: Two-dimensional regular hexagonal honeycomb [1] ................................................. 41 Figure 3.2: Wire-woven bulk Kagome cellular materials [66] ..................................................... 42 Figure 3.3: Analytical design algorithm [3] .................................................................................. 44 Figure 3.4: Effective shear modulus vs. yield strain for honeycomb [73] .................................... 45 Figure 3.5: Algorithm for parametric optimization of MPCM [75] ............................................. 48 Figure 3.6: Topology optimization algorithm [85] ....................................................................... 52 Figure 3.7: MPCM configurations designed with topology optimization [86] ............................ 53 Figure 3.8: MPCM Prototypes for the Lunar Tweel Project a) Bristle, b) Segmented cylinder, c) Helical coil [21] ............................................................................................................................ 56 Figure 4.1: Different MPCMs and their UCs. UCs are highlighted with dash line. ..................... 60 Figure 4.2: Non-pneumatic tire with solid shear beam [108] ....................................................... 65 Figure 4.3: The evolution of meso-scale periodic cellular materials for Tweel project [15] ....... 66 Figure 4.4: (A) Side connection between two adjacent circles; (B) two disconnected circles ..... 68 Figure 4.5: Abstraction illustration for disconnecting the unit cells from the side connection – Guideline#1 ................................................................................................................................... 68 Figure 4.6: Guideline #1 is applied to hexagonal honeycomb, chiral, and octagon through disconnecting the UCs from side connection. The type of connections can be either vertex or edge. .............................................................................................................................................. 69 Figure 4.7: Examples of transverse connections for hexagonal honeycomb and square cells with dash line. ....................................................................................................................................... 70

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LIST OF FIGURES (CONTINUED) Figure 4.8: Removing the transverse connections (highlighted in red) leads to a structure that cannot carry the shear load. .......................................................................................................... 70 Figure 4.9: Abstraction illustration for removing the transverse connections – Guideline #2. .... 71 Figure 4.10: Guideline #2 applied to hexagonal honeycomb, square cells with mix of triangle, rhumbas, and octagon through removing the transverse connections of the UCs. ....................... 71 Figure 4.11: Top nodes on hexagonal honeycomb and circle UCs; the direction of displacement boundary condition. ...................................................................................................................... 73 Figure 4.12: Unit Cell Design Guideline Development Method (GL: Guideline; UC: Unit Cell)78 Figure 4.13: An illustration of the fully documented Guideline #1 .............................................. 82 Figure 4.14: Hexagonal honeycomb with vertical legs as initial UC on the left and without legs on the right. ................................................................................................................................... 84 Figure 4.15: Guideline #3 is applied to circle, rhombus, and octagon through removing the vertical legs of UCs ....................................................................................................................... 85 Figure 4.16: Abstraction illustration for removing the top and bottom legs – Guideline #3 ........ 86 Figure 4.17: Guideline #4 applied to hexagonal, octagon, rhombus, and octagon with legs through replacing the UCs with different curved beams. ............................................................. 88 Figure 4.18: Abstraction illustration for replacing the UC with curved beams – Guideline #4 ... 89 Figure 5.1: Guideline#5 Abstraction............................................................................................. 93 Figure 5.2: Guideline #5 applied to bristles that led to circles, rhombus, and hexagon. .............. 94 Figure 5.3: Guideline#6 abstraction .............................................................................................. 96 Figure 5.4: Guideline #6 applies to circles, rhombus, hexagon, and octagon. ............................. 96 Figure 6.1: Problem — unit cell re-design for shear beam [14] ................................................. 100 Figure 6.2: Four unit cell design guidelines to improve shear flexure ....................................... 101 Figure 6.3: Hexagonal honeycomb and rhombus as the initial unit cells ................................... 102 Figure 6.4: A sheet sample of applied Guidline#1 to hexagonal honeycomb for subjectivity experiment................................................................................................................................... 105

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LIST OF FIGURES (CONTINUED) Figure 6.5: A sheet sample of applied Guidline#2-4 to hexagonal honeycomb for sequencing experiment................................................................................................................................... 107

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CHAPTER 1 INTRODUCTION TO PERIODIC CELLULAR MATERIALS This chapter defines cellular materials and describes their importance and applications in engineering. Furthermore, in order to pose research questions, the research opportunities are found and discussed. The dissertation outline and road map is presented in the last section of this chapter. 1.1 Periodic Cellular Materials and their Engineering Applications A cellular material is made of interconnected links of solid edges and faces of cells which have voids or empty spaces within each unit cell [1]. The motivation of using cellular materials came from the examples of these structures in nature including cork, balsa, sponge, cancellous bone, and hexagonal wax cells created by bees [1]. Cellular materials increase the range of material properties by higher stiffness, higher flexibility, and lower heat conductivity at lower relative density than average fully dense solids [1–5]. From a geometric standpoint, there are two types of cellular materials: periodic and stochastic [6]. A periodic cellular material is made of repeatable unit cells (UCs) across a structure; however, a stochastic cellular material (or foam) is typically built of non-repeatable three-dimensional UCs as shown in Figure 1.1 [1]. Periodic cellular materials are classified into two types: (i) prismatic elements or honeycombs which are made through extruding two-dimensional unit cells, (ii) truss structures that are made of threedimensional assembly of struts [6]. This dissertation focuses on designing periodic cellular materials at the meso-scale unit cells which are structures with thickness in the range of 0.1-10 mm and beam length in the order of centimeters [5,7–9].

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A B Figure 1.1: (A) A Hexagonal honeycomb (periodic cellular material), (B) A Foam (stochastic cellular material)

With regards to applications, there are two types of meso-scaled periodic cellular materials (MPCM) that have been typically explored in the literature: lightweight structure and compliant structure [10]. Lightweight structures are rigid structures which use voids to achieve low relative density with high stiffness and sometimes even multifunctional capabilities such as heat exchanger [11], energy [5], and acoustic absorptions [1]. Compliant structures are flexible structures which transfer forces, displacement, or motion from one point to another in the elastic region. A compliant structure can be designed as a monolithic or single-piece structure such as contact-aided cellular mechanism for morphing wing application [12,13] and hexagonal honeycomb for the shear beam of non-pneumatic tire [9,14,15] as shown in Figure 1.2.

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Figure 1.2: Applications of meso-scaled periodic cellular materials in one-dimensional wing morphing applications (left) and non-pneumatic tire (right) – Adopted from [12] and [14]

Monolithic compliant structures are made by one single part and therefore have less cost of assembly, less friction, and wear [10]. This dissertation focuses on the compliant application of MPCM. 1.2 Topology, Shape, Size Design - Definitions A unit cell is defined as the basic structural element of the periodic cellular materials. The UC is represented by topology, shape, and size [16–18].

Topology defines the material

arrangements and voids between the elements of a structure [19]. The difference between shape, dimension, and topology design of a simple cantilever beam under a given loading is shown in Figure 1.3. By changing the orientation of the sides and/or replacing them with different curves, a new shape can be designed (Figure 1.3B). By adjusting the dimension of the original beam, the beam can be designed parametrically while maintaining its original topology and shape (Figure 1.3C). In some cases, the shape and dimension optimization cannot meet the requirements, it is necessary to alter the inside of the boundaries of structures by creating voids, and adjusting the elements, which is topology design. The topology design of the cantilever beam shown in Figure

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1.3D presents how voids could be introduced into the material in order to achieve the design goals based on the load and boundary conditions. To meet structural requirements, shape and dimension of a structure with constant topologies could be adjusted. However, topology design involves changing the connectivity and continuity of material inside the boundaries of structure.

A. Original Beam

B. Shape Design

C. Dimension Design

D. Topology Design

Figure 1.3: Shape, dimension, and topology design of a cantilever beam (adapted from [19])

1.3 Research Gaps: Motivation of Developing Unit Cell Design (UCD) Guidelines Two research gaps identified for this research are: Research Gap 1: To develop unit cell design guidelines for periodic cellular materials. The motivation of this research stems from a previous effort to replace the hyper-elastic material

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of conventional tires with linear elastic material to reduce the tire’s rolling resistance [20–27]. A case study was conducted on different MPCM that were developed for the shear beam of a nonpneumatic tire (Tweel) designed at Clemson University between 2007 and 2010 [15]. The engineers developing the MPCM did not follow a clear and systematic process, but rather relied heavily on size optimization of pre-existing topologies [5,28–31], topologies design through topology optimization [19,32–35], and intuitive methods. Each of these methods has limitations. The size optimization cannot lead to a new topology. Topology optimization has modelling challenges: i) modelling periodic boundary conditions, ii) choosing the basic UC shape, and iii) tiling of UCs [34]. The challenges associated with intuition includes trial and error, random solutions, and more importantly no guidance during the design process [36,37]. These issues make the intuition method tedious and time-consuming. Guidance can help designers to make an informed decision during any stage of design process. There is also limited research found in the literature for design guidelines to address how to design a unit cell topology for targeted effective properties [3]. A detailed review on design methods for periodic cellular materials will be presented in Chapter Chapter 3. Furthermore, guidelines have been developed and widely used in other areas of design: manufacturing [38], assembly [38], product evolution [39], adaptive manufacturing [40], environment [41], and aesthetic [37]. These guidelines have been developed for different objectives, including: simplify design process, standardize product, standardize test procedures, automate assembly, improve product quality, and reduce time and cost of assembly [37]. Research gap 1 poses research questions, which are going to discuss in the section1.4.

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Research Gap 2: To develop vocabulary for describing periodic cellular materials. The current status of representing MPCM is mostly limited to graphical illustrations and limited textual descriptions. For example, the most cited work on cellular materials by Ashby and Gibson defines “the honeycomb of the bees” as the “regular array of prismatic hexagonal cells” [1]. Based on their definition, the MPCMs with different UC are called honeycombs, even though they have totally diverse effective properties [1,42].

The lack of a formalized language leads to

misinterpretations when describing a MPCM. To address this gap, a vocabulary is developed for describing periodic cellular materials and discussed in details in chapter Chapter 2. 1.4 Research Questions and Hypothesis In this section, research questions and hypothesis are put forward. After discussing the motivations and research gaps, the first research question arises about how unit cell design guidelines can be developed. Research Question 1. How can unit cell design guidelines be developed? Research Hypothesis 1. Unit cell design guidelines can be developed through the unit cell design guideline development method. Null Hypothesis 1. Unit cell design guidelines cannot be developed through the unit cell design guideline development method. The method is proposed based on the preliminary UCD guidelines built on a case study on the evolution of cellular materials for non-pneumatic tire [15,43]. The hypothesis is tested through developing guidelines from the method. The research hypothesis (RH) 1 is not provable regardless

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of how many UCD guidelines are developed from the method. Therefore, the null hypothesis 1 is tested. Null hypothesis is rooted in logic. The idea is no hypothesis in empirical science is completely correct; but if not falsified, it can support the hypothesis. The idea was named falsifiability and proposed by Karl Popper, a famous philosopher of science in the 20th century [44]. Therefore, Null hypothesis 1 is tested in this dissertation. By falsifying the Null hypothesis 1, which states that design for unit cell guidelines cannot be developed through the proposed method, the RH 1 is supported. While research question (RQ) 1 is about developing the guidelines, research question 2 and 3 deals with using the guidelines. Research Question 2. Are unit cell design guidelines subjective? Research Hypothesis 2. There are similarities between developed topologies. Low subjectivity is expected. Null Hypothesis 2. There is no similarities between developed topologies. High subjectivity is expected. RQ 2 explores the subjectivity of the unit cell design guidelines. Subjectivity means how much the user’s personal opinion can affect his/her judgments on a particular issue such the use of the guidelines. It is hypothesized that there will be similarities between developed topologies, which imply that low subjectivity is expected. RQ 2.1 explores the effect of initial UC on the subjectivity of guidelines. It is hypothesized that the initial unit cell topology has no effect on the subjectivity of the UCD guidelines.

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Research Question 2.1. Does the initial unit cell topology affect the subjectivity of the UCD guidelines? Research Hypothesis 2.1. The initial unit cell topology has effect on the subjectivity of the UCD guidelines. Null Hypothesis 2.1. The initial unit cell topology has no effect on the subjectivity of the UCD guidelines. RQ 3 explores the effect of sequencing of the unit cell design guidelines on the outcome UC. It is hypothesized that sequencing will not result in similar topology outcomes. Research Question 3. Does sequencing of the unit cell design guidelines matters with regards to the outcome? Research Hypothesis 3. Sequencing will result in similar topology outcomes. Null Hypothesis 3. Sequencing will not result in similar topology outcomes. 1.5 Intellectual Merit The first intellectual merit of this research lies in developing UCD guidelines to support systematic re-design of unit cell shapes and topologies for MPCM. Engineers in the area of designing cellular materials can use the UCD guidelines to develop new UCs for targeted structural behaviors. More importantly, the second intellectual merit of this research is to provide a UCD guideline development method which will be contributed to future guideline development. The

8

method can be used by engineers to develop new guidelines for applications with different loading scenarios, effective properties, and boundary conditions. The third intellectual merit of this research lies in MPCM taxonomy with a set of vocabularies that can be used to represent MPCM. This taxonomy helps designers to represent a MPCM and develop more guidelines through changing these topological parameters at a higher level of abstraction, similar to near field analogical reasoning recognized as an important design mechanism [45,46]. 1.6 Dissertation Outline The dissertation roadmap and overview is shown in Figure 1.4. In chapter Chapter 1, the foundation of cellular materials, research gaps, research questions, and hypothesis are presented. To address Research gap 2, a vocabulary for describing periodic cellular materials is presented in Chapter Chapter 2. A review on design methods for periodic cellular material development is presented in Chapter Chapter 2. The Unit Cell Design Guideline Development Method is presented and demonstrated through developing new guideline in Chapter Chapter 4. The effect of subjectivity and sequencing of unit cell design guidelines are explored in Chapter Chapter 6. Finally, conclusion and future work is discussed in Chapter Chapter 7.

9

Motivation and Problem Definition Developing unit cell design guidelines

  

What are MPCMs? Research motivation Research questions and hypothesis

Chapter Two Periodic cellular material vocabulary



A vocabulary for describing 2D periodic cellular materials

Chapter Three Review on design methods for periodic cellular materials



Review the existing design methods for PCMs



Method to Develop guidelines Demonstrate the method to develop guidelines

Chapter One Introduction to periodic cellular materials

Chapter Four Unit Cell Design Guideline Development Method

Closure

Using the guidelines

Chapter Five External Demonstration of Unit Cell Design guideline development method

Chapter Six Using Guidelines – user studies on subjectivity and sequencing

Chapter Seven Concluding remarks

 

Demonstrate how novice engineer use the method



Explore the effect of subjectivity and sequencing of the guidelines



Answer research questions Future work



Figure 1.4: Dissertation roadmap

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CHAPTER 2 A VOCABULARY FOR DESCRIBING PERIODIC CELLULAR MATERIALS 2.1 Overview A vocabulary is developed to explicitly represent two-dimensional periodic cellular materials (PCMs) to allow for unambiguous discussion of cellular topology. The vocabulary represents a PCM through descriptions of the unit cell and of the tiling of the unit cell arrangements. The unit cell is described through three characteristics: boundary, interior, and connections of the unit cell with its adjacent structural members. Each of these are further refined into sub-divisions. The vocabulary is evaluated against three metrics: completeness, perceptual orthogonality, and depth. Using this descriptive classification, researchers can more uniformly describe the development efforts of PCM that they are pursuing. 2.2 The Need for a Classification to Represent Periodic Cellular Materials A cellular material consists of interconnected links of solids, edges, and faces of cells which have voids or empty spaces within each cell [1]. More specifically, a periodic cellular material (PCM) is made of repeatable two-dimensional unit cells across structure, such as hexagonal honeycombs [1]. Finally, a unit cell (UC) is defined as the basic element in a definite volume that is used to create a periodic solid structure. The work presented in this chapter focuses on how to represent the unit cells of periodic cellular materials in an explicit, consistent, and unambiguous manner. This is achieved through the introduction of a new unit cell representation language.

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The current approach of describing periodic cellular materials (PCM) is mostly limited to graphical illustrations or simplistic textual descriptions. For example, a highly cited work on cellular materials defines “the honeycomb of the bees” as the “regular array of prismatic hexagonal cells” [1]. Based on this definition, the PCMs with different UC topologies and shapes are called honeycombs, however they have diverse effective properties resulting from many different topologies and shapes [42]. The lack of a formalized classification can lead to misinterpretations when representing or communicating a PCM. For instance, a periodic honeycomb is described as a square supercell constructed from mix of squares and triangles as shown in Figure 2.1(A) [42]. However, it is not clear how the squares and triangles are arranged. An alternative representation of square and triangle cells is shown in Figure 2.1 (B). Another example is periodic honeycombs with diamond cells (Figure 2.1 (C)) which is not clear how the UC topology seems and how the UCs are arranged [42]. An alternative is a diamond cell without lateral joint as shown in Figure 2.1 (D). Further, each of these four PC examples are not hexagonal as would be the honeycombs of bees. The current practice of generally describing with text and offering an illustration is not sufficient for clearly and explicitly communicating the unit cells between researchers. Another misinterpretation of representing the PCM lies in the following example. It is stated that square and triangular honeycombs can be made with edge connectivity of four and of six, respectively as shown in Figure 2.2 (A) and (B) [1]. However, the edge connectivity is not defined clearly whether it refers to the shared edge connections between a UC (Figure 2.2 (C)) and its adjacent structural members (Figure 2.2 (D)), any edge in the UC which connects two vertexes (Figure 2.2 (E)), or all edge connections that intersect with a vertex (Figure 2.2 (F)). This

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chapter aims to address this challenge in explicitly representing unit cells through a formalized vocabulary with a set of descriptors. This formalism will also enable other approaches to computational design synthesis in a more robust and neutral manner.

A

B

C

D

Figure 2.1: Examples of misinterpretation through current status of explaining the PCM. (A) Square super cell constructed from mix of squares and triangles [42]; (B) An alternative cell with the combination of squares and triangles; (C) diamond cell [42]; (D) An alternative diamond cell without transverse connections.

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A

B

C

D

E

F

Figure 2.2: Honeycomb with (A) Square Cells and (B) Triangle Cells; (C) A Triangle UC; (D) The Edge Connection between a UC and its Adjacent Structural Member; (E) The Edge Connection between two Vertexes; (F) all edge connections that intersect with a vertex.

2.3 Representation in Engineering Design Representations in engineering design have been developed to organize information, support communication, support reasoning, and facilitate archival and retrieval [47–53]. A representation is defined as a replacement of a reality with a symbol [51], the instantiation of a specific element of reality by a specific symbol is a model using the representational language. The language of a representation includes both the vocabulary and the grammar that includes the rules to relate the vocabulary elements. Representations found in engineering can be categorized considering how they are expressed, such as pictorial, symbolic, linguistic, virtual, or algorithmic [54]. Representations are valuable from different perspectives, such as in design automation

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development by reducing subjectivity, design education by providing students with easier understanding and detailed information within the defined domains, and design research by clearly articulating potential gaps and opportunities for further development [47–56]. Thus, the representations used to define engineering design models can be described through a set of attributes: vocabulary, structure, expression, purpose, and abstraction [51]. The vocabulary of a design representation is defined by three attributes: the types of elements, the vocabulary size, and flexibility of definition of the vocabulary. The types of the vocabulary elements defines whether the vocabulary is object-relation or object-relation-modifier. The size of vocabulary refers to the number of available element types. The flexibility of representation refers to whether the vocabulary is defined by the user or pre-defined. The representation is categorized by local and global grammars. The local grammar refers to possible options between entities and relations. The global model is defined by complicated rules for assembling the model. The representation can either be self-validating or rely on external validation for correctness. Textual, mathematical, graphical, and computational are four types of expression of representation. The purpose of using representation by designers is whether analysis, synthesis, or communication. The abstraction level of the representation is continuum form low to high [51]. Table 2.1 and Table 2.2 show the current status of representing PCM, pictorial and textual expressions, respectively, with the classification scheme by [51].

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Table 2.1: Pictorial expression of PCM representation with the classification scheme by [51]

Vocabulary:

Structure: Expression: Purpose:

Abstraction:

Type: entity-relation (material connected through vertices) Size: low (few instances of few types) Flexibility: pre-defined (the user has no flexibility to define structural elements and their connections) Local structure: attributed (links are connected between joints by picture) Global structure: (user defines the links, joints, and length values) Graphical: pictorial representation of the honeycomb Analysis: graphical representation is used to define variables and perform analysis. Communication: graphical representation is used for communication. High: the graphic has some hidden information (thickness and length of beams, how the unit cell is defined)

Table 2.2 shows the type of vocabulary is not clear and few instances of few types of vocabulary is found in the literature. Furthermore, no local structure or clear relationship between the words is found. The high abstraction of current textual expressions provides a vague textual representation of PCMs. The textual expression does not have enough vocabulary to communicate PCMs. Furthermore, the definition of available vocabulary is not clear. This provides an opportunity to develop a vocabulary to describe periodic cellular materials.

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Table 2.2: Textual expression of PCM representation with the classification scheme by [51]

The available vocabulary in the literature to describe the periodic cellular materials [1] “Two-dimensional, prismatic cellular materials with a regular and periodic microstructure are called honeycombs.” [42] “A stretched dominated lattice and its unit cell.” [57] “Honeycomb with square and triangle cells, with edge connectivity of four and six, can be made,” [1] “Our aim is to characterize their size, shape and topology: that is, the connectivity of the cell walls”[1] “The cell edge thickness is tc and the face thickness is tf” [1] Type: Not clear Vocabulary: Size: low (few instances of few types) Flexibility: pre-defined Local structure: no local structure Structure: Global structure: free (user defines the relation between words) Textual: limited words are available (honeycomb, edge connectivity, topology, Expression: size). Communication: textual representation is used for communication (not enough Purpose: words are available). High: the text provides a vague representation Abstraction: In addition to the representation of engineering design models, a hierarchical classification was developed to classify visual design representations (VDRs) [56]. The VDRs are divided into two general groups: two-dimensional and three-dimensional VDRs.

A two-dimensional

representation included either sketches or drawings. A three-dimensional representation can be either models or prototypes. Each of these classes are sub-categorized into groups to ensure the classification has enough depth [56]. The validation of the taxonomy was evaluated through fortythree interviews and printed checklists to see if there is an agreement on the definition of each taxon among engineers and academics. However, the usability of the taxonomy by engineers was not fully tested.

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In this chapter, a vocabulary is developed to represent periodic cellular material structures explicitly. In section 2.4 an overview of representation of periodic cellular materials is presented. In sections 2.52.6 the classification of PCM representation are discussed in two levels. The validation of the classification is discussed in section 2.7. In section 2.8, the usability of the vocabulary is evaluated by a designer study. Finally, the summary and future work are discussed in section 2.9. 2.4 Overview of the Vocabulary to Describe Periodic Cellular Materials The vocabulary of periodic cellular materials is organized in a hierarchy of descriptors with the unit cell and tiling approaches as being the highest level in the hierarchy. This vocabulary classes are shown in Figure 2.3.

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Figure 2.3: Hierarchal periodic cellular material vocabulary

A two-dimensional (2D) unit cell is characterized through three aspects as shown in: i) boundary, ii) interior, and iii) connections. The tiling of UCs which represents how the UCs are arranged is an important parameter that can affect the PCM effective properties [58]. Each of these terms, inside, boundary, connections, and tiling, are further refined into sub-divisions and defined in the following section. 2.5 Periodic Cellular Material Representation -- Level 1

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2.5.1 Two-Dimensional (2D) Unit cell The 2D UC is represented by the boundary, the interior, and the connections as shown in. The boundary encircles the UC, the interior defines the UC material distribution, and the connections joint the UC with its adjacent structural members. For example, the boundary encapsulates the rotated square (interior) in Figure 2.4. The UC is connected to adjacent structural UCs with four connections. Each of these can be modified independent of the other. For instance, the straight interior legs can be replaced with curved interior legs (Figure 2.4). Alternatively, the UC connections can be changed from vertex based to edge based. Finally, the boundary itself can be changed in size and orientation.

Interior

Boundary

Connections

Figure 2.4: Elements of unit cell vocabulary

2.5.2 Tiling of Unit Cells To assemble a PCM, the UC serves as the tessellations. The positioning that the unit cells are arranged to make a PCM refers to the tiling of unit cells [59]. The tiling of the unit cells is divided into simply-connected and non-simply connected UCs which are illustrated in Figure 2.5. As shown in Figure 2.5(A), in the simply-connected structure, corner nodes share the same boundary edge while in the non-simply connected arrangements, the top and bottom layers are connected in which each edge in the middle layer is shared between three UCs as shown in Figure

20

2.5(B). Figure 2.5 shows how a hexagonal honeycomb can be illustrated by the simple-connected and the non-simple connected arrangements defining different unit cells and the tiling. Figure 2.5 shows that change of tiling affects the interior, the boundary, and the connections of the unit cell.

(A)

(B)

Figure 2.5: The Simply (A) and non-simply (B) connected unit cells [58]

2.6 Periodic Cellular Material Representation – Level 2 2.6.1 Boundary The boundary that surrounds the interior can be defined by shape and size. The shape defines as the curvature and orientation of boundary sides; and the size defines by the dimensions of boundary sides. Examples of different boundary shapes include square, parallelogram, and rhombus. An example of boundary size is shown in Figure 2.6 through increasing the size or width of square boundary around the cylinder cellular material.

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Figure 2.6: The size change of the boundary around the circle unit cell

2.6.2 Interior The interior, material distribution inside the UC boundary, is described by topology, shape, and size [16,17]. The topology represents the connectivity of material within the interior. The shape deals with curvature and orientation of the structure elements. The size states the dimensions of the structure elements such as length, width, height, and diameter [16,17]. Figure 2.7 illustrates the examples of topology, shape, and size changes of a hexagonal unit cell.

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Figure 2.7: Examples of different interior changes: topology, shape, and size changes.

2.6.3 Connection Type The connection type represents how the unit cell is connected to its adjacent structural members or unit cells and is represented by vertexes or edges. The edge connection is defined with location, number, and length. The location represents the position of connections along the boundary. The number quantifies the count of connections which connect the unit cell with its adjacent structural members. The length refers to the size of the edge connection. The vertex connection is described by number and location. Figure 2.8 illustrates the example of edge and vertex connections for hexagonal honeycomb unit cell. The hexagonal honeycomb has two vertexes and two edge connections. The location of vertex connections are located on the top and bottom of unit cell boundary. The location of edge connections are located on the right and left of

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the unit cell boundary. These connections are different from the connection in the interior to represent the unit cell topology. In fact, the connections and the topology of interior are not fully decoupled. Figure 2.9 shows how topology changes of hexagonal honeycomb affect the number of vertex and edge connections. Figure 2.9 (A) shows the unit cell has two vertexes and two edges. However, removing transverse connections led to half hexagon (Figure 2.9 (B)) with only two vertex connections and no edge connection which connected the interior with adjacent structural members [43].

Unit cell

Figure 2.8: Example of vertex and edge connections for hexagonal honeycomb. The vertex and edge connections are indicated by slide circles and highlighted lines, respectively. The dash lines are UC boundaries.

Figure 2.9: Example of how topology changes affect the vertex and edge connections for hexagonal honeycomb. The vertex and edge connections are indicated by slide circles and highlighted lines, respectively. The dash lines are UC boundaries.

2.7 Validation of the Periodic Cellular Material Vocabulary

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In the modeling literature, validation is defined as “justification of knowledge claims” [60,61]. Essentially, this means that the justification and rationalization should be provided to demonstrate that declarations made about the knowledge are true.

Internal validation of

classifications and taxonomies can be evaluated through three points of view:

perceptual

orthogonality, completeness, and depth [50–52]. Perceptual orthogonality requires that each individual descriptor should be mutually exclusive of all other descriptors in the classification. Completeness refers to the vocabulary being exhaustive, or at least adequate, over the domain [50]. Here, the vocabulary should be able to represent PCM’s as found in the literature. Depth refers to the usability of the vocabulary. A deep classification with a lot of details is not convenient to use. However, there should be enough descriptors in the vocabulary to describe a PCM in a meaningful level. 2.7.1 Perceptual Orthogonality The classification proposed for periodic cellular material structures is evaluated through perceptual orthogonality. Each descriptor in the classification is chosen in a way that represents any unit cell entity uniquely. In other words, the information in each descriptor should not be found in any other descriptors. In order to evaluate the orthogonality, a matrix is developed and lists all descriptors that need to represent a PCM. The orthogonality matrix is represented in Table 2.3. The matrix represents the relationship between any descriptors in the rows with the ones in columns as in strong effect (solid circles), weak effect (hollow circles), and no effect (empty cell) [51]. The matrix ideally is nearly diagonal suggesting that each element is independent of the other elements.

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Tiling Non-simply connected

Simply connected

Edge

Vertex

Boundary Size

Shape

Size

Interior Shape

Topology Interior

Connection

Table 2.3: Orthogonality matrix of periodic cellular material vocabulary

Topology Shape

Boundary

Vertex

Tiling

Shape

Connection

Size

Size

Edge Simply connected Non-simply connected

The results show among all possible cases, around a fifth have weak relationships between the descriptors. The reason comes from the fact that these descriptors are used to represent different PCM and unit cell entities therefore some weak relationship between them are expected. Three weak relations are observed in Table 2.3.

The first one is the weak relation between the topology and the connection types of vertex and edge. This weak relationship is expected since the topology defines as the connectivity of structural elements which is somewhat related with the vertex and the edge connections on the

26

boundary. Figure 2.9 shows how topology changes affect the number of connections in hexagonal honeycomb unit cell. It reduced the number of connection from four, two vertex and two edge connections, to solely two vertex connections.

The second weak relationship can be observed between the tiling, the interior, and the connections. Previous research on topology optimization of the metamaterial shear beam of a nonpneumatic wheel has shown that the simply and non-simply connected arrangements of unit cells have an effect on the generated interior structure and connections of unit cells as shown in Figure 2.10 [58]. Different results, (A) and (B) in Figure 2.10, were obtained under the same shear beam thickness, the same effective shear modulus, and the same analysis but different tilings, simple connectivity (A) and non-simple connectivity (B).

(A)

(B)

Figure 2.10: Simple (A) and Non-simple (B) connectivity optimization results – Adopted from [58]

The last weak relationship is between the size of interior and the size of boundary; this weak relation arises from the fact that the size of interior elements is indirectly related to the size of

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boundary. This is shown in Figure 2.11. As the size of boundary increases, Figure 2.11 from (A) to (B), the size of interior structural elements can be larger. Therefore, the matrix is orthogonal.

Figure 2.11: Relation of the size of boundary and the size of interior

2.7.2 Completeness The completeness metric of a classification refers to its exhaustiveness over a domain. The literature discussed that the completeness of a classification is subjective and impossible to attain [50–52]. In other words, there is no proof that the vocabulary is able to describe all future PCMs. However it is feasible to evaluate a classification with the current available knowledge in the field. Therefore, the objective is to make sure the proposed classification is able to represent examples of PCMs available in the literature to the best current knowledge of the authors.

Initially, a vocabulary was provided to describe PCMs based on the current knowledge of authors. Then, the vocabulary was evaluated to explicitly represent PCMs and UCs found in the literature. The initial vocabulary was modified through adding or removing descriptors in case they were not able to represent a PCM or UC explicitly. Each time the vocabulary was modified, the validity of the classification was re-evaluated over the known PCMs and UCs. This process

28

continued until the hierarchal PCM classification as shown in Figure 2.3 can represent the PCMs. The PCMs that were used to evaluate the completeness of the classification are found in [1,2,7,15,42,58]. Four samples of these PCMs including hexagonal honeycomb (Table 2.4), chiral (Table 2.5), S-type (Table 2.6), and diamond (Table 2.7) periodic cellular materials are described through the vocabulary.

Table 2.4: Hexagonal honeycomb, unit cell illustration, abstraction and representation

PCM illustration

Boundary:

Connection:

Interior: Tiling:

Unit cell – Illustration

UC Abstraction – Illustration

Shape: rectangle Size: the length and width of rectangle Vertex: number: two; location: one at the middle of top (A) and one at the middle of bottom (B) of UC boundary Edge: number: two; location: one at the middle of left (D) and one the middle of right (C) of UC boundary; Topology: connection between A-C, C-B, B-D, A-D Shape: straight connections between A-E, E-C, C-F, F-B, B-D, D-E, E-A Size: the length and thickness of each structural elements Simply connected

29

Table 2.5: Hexagonal chiral honeycomb, unit cell illustration, abstraction and representation [23]

PCM illustration


UC Abstraction – Illustration A B

F Center

E

C D

Boundary: Connection:

Interior:

Tiling:

Shape: hexagon Size: length of sides Vertex: number: six; location: one at each hexagon vertices (A through F) Topology: connection between A-Center, B-Center, C-, D-, E-, and FCenter of hexagon Shape: A circle at the center of hexagon; straight connections between A-, B-, C-, D-, E-, F-, and Center; all straight connects are tangent to the circle. Size: the diameter of circle, length and thickness of each structural elements Simply connected

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Table 2.6: S-type, unit cell illustrations, abstraction, and representation [62]

PCM illustration

Boundary: Connection:

Interior:

Tiling:


UC Abstraction – Illustration

Shape: rectangle Size: the length and width of rectangle Vertex: number: two; location: one at the middle of top (A) and one at the middle of bottom (B) of UC boundary Topology: connection between A-B Shape: straight connections between A-C and B-D; C-D connection through a sine wave. (C and D are two vertexes between A and B). Size: the length of A-C and B-D; the amplitude of sine wave; and thickness of structural elements Simply connected

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Table 2.7: Honeycomb with diamond cell, unit cell illustrations, abstraction, and representation[42]

PCM illustration


UC Abstraction – Illustration A

D

B

C Boundary: Connection: Interior: Tiling:

Shape: rectangle Size: dimension of length and width of rectangle Vertex: number: four; location: one at the middle of each rectangle (UC boundary) side; A, B, C, and D at the top, right, bottom, and left, respectively. Topology: connection between A-B, B-C, C-D, D-A, B-D. Shape: straight line for each connections. Size: the length and thickness of each structural elements Simply connected

2.7.3 Depth The depth of a classification refers to how much detailed descriptors are required to represent PCMs. A deeper detailed classification needs more expensive search to represent the domain. For this research, two levels of representation is found enough to cover the PCM available found in the literature. The depth of the classification is evaluated into two steps: (1) represent different PCMs (illustration) with the vocabulary, and (2) illustrate PCMs with the provided vocabulary representations. For the first step, four PCMs and their unit cells are depicted in Table 2.4 -Table 2.7. Four examples selected from the literature include hexagonal honeycomb, chiral, S-type, and honeycomb with diamond cells. In each case, PCM, UC, UC abstraction, and the representation using the classification are represented in Table 2.4 - Table 2.7. The four main

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descriptors to represent include: interior, boundary, connection, and tiling. Each descriptor is represented with the higher level of representations. For example, interior is represented by topology, shape, and size. Boundary is represented by shape and size. The second step of depth validation evaluates the ability of the vocabulary to illustrate a PCM if a complete representation is given. To do this step, four complete representations provided in Table 2.4 - Table 2.7 are used for illustration. The observations show the representations lead to the PCM illustrations. Two more examples are provided to show how a PCM can be illustrated if a complete representation is given as shown in Table 2.8 and Table 2.9. Parallelogram and rhombus PCMs are illustrated with the given vocabulary. The above-mentioned examples in Table 2.8-Table 2.9 give credibility to the depth and the completeness of the classification.

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Table 2.8: Representation of a parallelogram and its illustration

Shape: parallelogram; angle between AD and AB is 60 degrees (A, B, C, and D are top left, top right, bottom right, and bottom left vertexes, respectively). Size: the length of base and height of parallelogram

Boundary:

Vertex: number: four; location: one at the each corner side of parallelogram (A, B, C, and D vertexes) Connection: Edge: number: four; location: one at the each side of parallelogram; length: equal to the length of each side of parallelogram Topology: connection between B and D Shape: straight connection between B and D Interior: Size: the length and thickness of each structural elements Non-simply connected Tiling: UC Abstraction – Unit cell – Illustration PCM illustration Illustration A

B

D

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Table 2.9: Representation of a rhombus unit cell and its illustration

Shape: rectangle Size: length of width and height Vertex: number: two; location: one at the middle of top (A) and one at the middle of bottom (B) of UC boundary Edge: number: two; location: one at top and one at the bottom side of Connection: rectangle at the UC boundary; length: the width of the rectangle (UC boundary). Topology: two connections between A and B Shape: rhombus Interior: Size: the side length of rhombus, thickness of each structural elements Simply connected Tiling: UC Abstraction – Unit cell – Illustration PCM illustration Illustration Boundary:

2.8 Exploring the Usability of the Vocabulary to Represent Periodic Cellular Materials – A Designer Study The objective of this study is to evaluate if engineers can use the vocabulary predictably to represent the PCMs. It is accepted that there is not a unique representation even using the standardized vocabulary for a PCM since various unit cells and tiling combinations are possible. However, regardless of the representation, it should be able to recreate the PCM.

There are four main descriptors used to represent the PCMs: boundary, connections, interior, and tiling. These four descriptors are sub-categorized into more detailed descriptors (Figure 2.3)

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which are defined for participants. These descriptors are listed in Table 2.10. Participants received a packet which includes the vocabulary, descriptors with definitions, as well as two examples. Two examples, diamond cells and parallelograms (Table 2.8 and Table 2.9), are provided for the participants to demonstrate how the vocabulary can be used to describe PCMs. Seven participants were asked to use the vocabulary and its descriptors as needed to represent two new PCMs: hexagonal honeycomb and S-type.

An inter-rater reliability test was conducted by independent raters to ensure the raters have consistent understanding of the protocol to judge the representations. The raters evaluated 112 descriptors based on the standard definitions provided in this study on a 0-1 scale. If a descriptor was used as expected based on its definition, the rater assigned 1, otherwise a 0 was given. In order to compare the consistency between the raters, a Cohen’s kappa analysis was done between two raters. Cohen’s kappa analysis is used to evaluate statistically the agreement between two raters in qualitative research [63]. The Cohen’s kappa results show 73.2% and 91.3% agreement between raters for hexagonal honeycomb and S-type, respectively. The overall average agreement between raters is 84.1% which shows a high agreement [63]. The minimum average agreement is 73.2% which is a substantial agreement between raters [63].

The percentage of descriptors that were used as expected based on their definitions in this study for both honeycomb and S-type are shown in Table 2.10. The boundary shape and size were each used as expected for every case. Connection edge and vertex were used as expected for 93% and 86% of the cases. The interior topology and size each were used as expected only 79% of all cases,

36

due to these descriptors not being involved by the participants. In 93% of all cases, tiling was used as expected.

Table 2.10: The percentage of descriptors as expected based on their definitions for both honeycomb and S-type

Descriptors

The percentage as expected based on the definitions 100% 100% 93% 86% 79% 86% 79% 93%

Boundary – Shape Boundary – Size Connection – Edge Connection – Vertex Interior – Topology Interior – Shape Interior – Size Tiling

In summary, eight descriptors are applied by seven participants to two PCMs. The average percentage of expected descriptor use was 89%. As a result, it is believed that the designer study gives credence to the usability of the vocabulary by engineers who generally would not have a background in PCMs.

2.9 Summary and Future Work In this chapter, the classification has developed for periodic cellular material representation. The classification is made of a vocabulary with set of descriptors and classified in two levels to represent a PCM. The first level represents the unit cell and the tiling. The unit cells are represented by the boundary, the connection, and the interior; and tiling is sub-categorized by the simply connected and non-simply connected. The second level has more descriptors to represent a unit cell explicitly. The internal validation of the classification was evaluated through

37

three metrics: completeness, perceptual orthogonality, and depth. An external validation of the vocabulary was conducted through a designer study to assess how engineers use the vocabulary to represent PCMs. Much research have been conducted on cellular materials to develop and evaluate methods to design topology, shape, and size of these periodic structures [3,9,27,32,58,62].

Recently,

researchers found that the boundary shape of unit cells and the tilling affect the topology design using the topology optimization [34,58]. The aim of this classification is to formalize the vocabulary needed to represent PCM and provide geometrical parameters that potentially can be changed to improve the targeted effective properties of PCMs. Examples of these geometrical parameters include the connections such as vertex and edge, the boundary shape, and the tiling. For instance, removing the side connection between two unit cells and removing transverse connections of a unit cell led to development of two UC design guidelines on improving shear flexure of unit cells while maintaining the effective shear modulus [43].

The transverse

connections refer to any diagonal or horizontal structural members of the interior of UC. The pictorial representations of two guidelines are presented in Figure 2.12.

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A

B Figure 2.12: Representation of two UC design guidelines through removing (A) Side connection between two unit cells and (B) transverse connections in the interior

Formalizing the representation of UCs facilitates UC design guideline development. For example, changing the type of connections from vertex to edge, the shape of UC boundary, and tiling of UCs can potentially lead to new UC design guidelines, which is identified as a research opportunity [43]. The given geometrical parameters and vocabulary in the classification can be used to explore new unit cells and more importantly help in developing design guidelines through changing or adjusting each vocabulary. Two UC design guidelines with their abstractions are developed and represented in [43]. The design guidelines, which is identified as a gap for future research, are aimed to help engineers in developing new PCMs for targeted effective properties.

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CHAPTER 3 REVIEW ON DESIGN METHODS FOR PERIODIC CELLULAR MATERIALS The design of MPCM to satisfy required effective mechanical properties is a challenging area in material science usually addressed by modifying the topology, shape, and size of the unit cells. In this chapter, the methods currently being used to design MPCM are introduced and discussed. Ultimately, it is found that there is a gap in design methods for developing new mesoscale unit cell topologies, as the current methods are limited to topology optimization and intuitive design method [3]. Addressing this gap is the focus of this dissertation. With new manufacturing technologies offering opportunities to create unique materials for product design, these design methods are anticipated to become ubiquitous. However, it is not clear which of the methods should be used for a specific design scenario. As a first step to support design method selection, this chapter classifies and organizes these methods, evaluating them against the steps involved, the inputs provided, their outcomes, and the technologies used. The methods are generally classified as closed form analytical, simulation (finite element), parametric optimization, topology optimization, selection method, and intuitive method. 3.1 Analytical Method The analytical method is developed for MPCM based on the seminal work on cellular material theory (CMT) [1]. This method provides the analytical method that uses the pre-defined topology and the geometric parameters of a two-dimensional, regular honeycomb seen in Figure

40

3.1 to determine effective mechanical properties including the modulus of elasticity, Poisson’s ratio, the shear modulus, the elastic buckling, and the plastic deformation [1].

Figure 3.1: Two-dimensional regular hexagonal honeycomb [1]

The CMT is premised on the assumption that when the bending moment in the cell walls reaches the plastic yield point of the material, the honeycomb will begin to fail. Thus, the overall performance of the MPCM depends on the mechanics of the individual unit cells comprising it. A significant limitation of this method is that the topology is defined a priori. While honeycombs are popular based on their known efficiencies in strength to density properties, new analytical models are needed if different topologies are explored. Moreover, there are geometric limitations parameter values in the CMT that should be addressed. The mechanical properties of three-dimensional cellular materials (foams) including bending, elastic buckling, and plastic collapsing of the cell walls has also been developed into models [64]. These models are based on the dependency of foam properties at low densities; however for higher densities, the correction coefficients were proposed. These foam-based models are empirically based, informed by experimentation to include the correlation and correction coefficients. A challenge with this method is that as new configurations (metal-foams defined by

41

new metals, different porosity, different density, and different manufacturing process) are introduced, new analytical models need to be developed. The results of analytical models for twoand three-dimensional analysis were verified through experiments [64,65]. More recent research determined the shear modulus of a wire worn cellular material and the results being verified through simulation and experimentation [66,67]. Wire-woven Bulk Kagome (WBK) is an innovative MPCM made of woven wires in six directions (Figure 3.2). Similar models have been developed for this column-based structure as opposed to the beam/plate based model of CMT. These analytical approaches in MPCM usually have focused on the balance of the forces and the moments of the unit cell to describe the mechanical properties as a function of parametric geometry.

Figure 3.2: Wire-woven bulk Kagome cellular materials [66]

Two critical assumptions are made for the analytical method: (1) the topology, and material and (2) the mechanical properties sought are pre-determined. For instance, the CMT model must be modified to support a new topology such as the chiral structure [23] or the model must be evolved to support hyper-elastic materials to create the MPCM [68]. Should a new mechanical property be of interest, such as vibro-acoustic damping [69], these models cannot be used. The

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algorithm of interest with respect to this method is presented in Figure 3.3. First, the algorithm for developing the model is important to expose to future engineers what steps are needed to develop models for new materials, structures, and target properties (Figure 3.3). An important step included in this flow chart includes the validation. This is typically done either through simulation using finite element analysis (FEA) or experimental testing [70]. A challenge with the validation is the extent to which the model can be validated (parametric ranges). Computational cost limits a complete detailed simulation for all possible parametric combinations and the expense of physically testing all combinations is often prohibitive of full validation through experimentation. However, some validation is needed to support this model. This flow chart exposes to the engineer what is needed to be considered in developing the analytical model. The robustness of this method is limited by the initial UC topology.

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Define Topology

Define Parameters

Define Target Property Define Constitutive Mechanical Model Define Material Limits

Set Model Limits

Validate Experimentally

Figure 3.3: Analytical design algorithm [3]

This algorithm can be automated to support parametric exploration of the configurations [71,72]. The material and topology are limited to the specific analytical model. Thus, the designer has limited flexibility in choosing different materials or topologies. These explorations can be done to create databases of available configurations to meet property ranges, similar to a proposed material selection chart in Figure 3.4. Alternatively, the algorithm can be used to support optimization to converge to specific targeted properties by refining the input parameters, which is discussed in Section 3.3.

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Figure 3.4: Effective shear modulus vs. yield strain for honeycomb [73]

3.2 Simulation Method Numerical simulation has increasingly been used to verify the analytical results for the MPCM. For instance, the CMT model might be extended to accommodate both regular and auxetic honeycombs and then finite element analysis (FEA) can be used to verify the results [9]. In this work, multiple parametric geometries are modeled and the shear yield strain and stress were determined. The goal of the study was to minimize the relative density of the MPCM by maximizing the shear yield strain at a specific effective shear modulus. As a result, the honeycomb dimensions and material were selected based on design requirements of shear stress, strain, and low relative density. The simulations were simplified approximations of the MPCM, with Timoshenko’s beam elements (B22) rather than detailed models of 3D elements. This was done to support faster simulation.

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Moving beyond the validation of analytical models, an advantage of the simulation is with the increased flexibility of exploring different topological structures and materials without redefining an analytical model. In similar research, a strain energy method was applied to develop a hyper-elastic relation for presenting the nonlinear behavior of macroscopic strain [68]. The objective was to explore different materials, both linear elastic and hyper-elastic, as constitutive materials for the honeycombs. The analytical model is not capable of supporting this type of exploration, as the model is limited to specific types of materials. However, in a simulation model, the underlying mesh, boundary conditions, and loading can be kept the same with the material properties changed. A further advantage of this method is that the simulation model of the MPCM can be integrated into other simulation models. For instance, the MPCM for the non-pneumatic tire can be integrated into a complete non-pneumatic tire (Tweel) model and further integrated into a tiresoil interaction model [14,74]. The contact pressure of the non-pneumatic tire is directly related to the shear flexure of the MPCM and can be explored for different geometries to meet the design requirements. The algorithms for the simulation method are not significantly different from the analytical approach (section 3.1). The major distinction lies in how the analysis is done, either with a simplified behavior model or in a detailed elemental simulation model. 3.3 Parametric Optimization Method

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To improve the design outcome of the analytical and simulation methods, optimization algorithms have been considered extensively [20,62,75,76]. Much of the efforts have been focused on iterating the analytical and simulation methods to specify parameters, determine the result mechanical property performance, compare the results to the target, and then to change the parameters. Different optimization algorithms are used including particle swarm, LevenbergMarquardt, simplex, and genetic algorithms [5,62,76,77]. The optimal design of a wire-woven truss panel under three failure modes was achieved using two different design optimization algorithms with regard to the slenderness ratio and the weight of the wire-woven unit cell [76]. In this work, parametric design optimization was done using spreadsheet software rather than conventional optimization tools. This was possible with a simple analytical model and limited parameters to explore. The preliminary results showed that the performance of the optimized wire-woven truss structure in term of load over weight is as good as the conventional honeycomb sandwich panels. In this manner, analytical methods with optimization can be used for fast virtual prototyping when the behaviors are well understood. Two design methods were proposed for compliant MPCM with graded stiffness given varying magnitudes of loads and locations of irregularities in the MPCM [75]. They proposed a parametric optimization method for different cell wall thicknesses and distance between cells to obtain targeted stiffness and absorbed energy through elastic displacements under static and quasistatic loads. The nonlinear behavior of materials, large-scale deformations, and cell wall contact were considered in the analysis. The flowchart is the design method depicted in Figure 3.5.

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Specify MPCM Type

Determine Number of Layers

Determine Design Parameters in Unit Cell

Formulate Optimization Problem

Create FE Model

Use Optimization Iteration (Changing Parameters)

Figure 3.5: Algorithm for parametric optimization of MPCM [75]

Rather than using optimization as a tool for finding optimal configurations, others have used optimization as a tool to find what target effective properties could be sought in the MPCM design [78]. This inversion illustrates that there are different design activities in MPCM design, including understanding what target values could be sought. For instance, to reduce the effects of hysteresis loss of the low rolling resistance non-pneumatic shear beam, six configurations of different materials are explored [78]. Rather than defining a specific material, such as replacing a known material of polyurethane with a MPCM, a “fictitious” material is optimized for the given geometric parameters with respect to material properties, such as the effective shear modulus.

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A comparative study is conducted on the size optimization of four MPCMs to meet the design requirements for a non-pneumatic tire’s (NPT) shear beam with 10% shear flexure at 10 MPa effective shear modulus[62]. Four MPCMs, honeycomb, auxetic, sinusoidal auxetic, and Stype are parametrically optimized using the following optimization algorithms: (i) particle swarm; (ii) genetic algorithm (GA); and (iii) FAST–SIMPLEX (using response surface method). The design variables are thickness, angle, and the amplitude of the sinusoidal wave. The results show the S-type unit cell has higher shear flexure than the three other UCs and can be configured to meet the design requirements [62]. MPCMs are typically used to achieve dual targeted performance, such as strength and low density. These targets can be used to formulate multi-objective optimization problems for multifunctional MPCM [79,80]. Two analytical models, the corrugated wall approach and volume averaging techniques, were used to develop the thermal and structural characteristics of a twodimensional MPCM exhibition force convection heat transfer [75]. The convection heat transfer coefficient was calculated in addition to the shear modulus based on the cell geometry and flow speed for various cell geometries including square, triangle, and conventional honeycombs. Comparing non-dimensional thermal, structural, and thermo-mechanical characteristics, it was concluded that hexagonal honeycombs and six-connected triangle cells provide optimal performance for small wall thickness and high heat flux, respectively. In this way, parametric optimization on several topologies and UC arrangements can be used to evaluate the overall potential of a topology. Similarly, different MPCMs were used to explore mechanical performance (stiffness and buckling) and thermal properties [11,80]. In this latter method, sequential quadratic

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programming was used to solve the compromise decision support problem defined as the multiobjective optimization problem. Simulation and analytical methods were used in the optimization framework. 3.4 Topology Optimization Method Topology optimization is a design method that seeks to change the underlying architecture or topology of the MPCM. This can be done by reformulating the optimization problem to exhaustively define the space in which the material can exist in the MPCM, typically through discretization of the space. Then, material is added or removed from each cell as the algorithm evolves to a steady state of targeted performance. Homogenization and topology optimization methods have been developed by numerous researchers during the past couple of decades [32,58,81–85]. Often, the topological optimization method employs an analytical or mathematical model, such as in defining a composite bar with periodic MPCM [32].

The solutions for the

homogenization equations are used to drive the topology optimization often augmented with simulations and FEA [82]. This method is illustrated with beam and plate examples [83]. Thus, the optimized structure is based on analytically calculated mechanical properties, such as shear modulus and strain energy for a specific volume portion, while the finite element analysis frequently is being used to verify the results [82]. As an alternative example, volume averaging analysis and homogenization have been used to optimize the topology of MPCM used in the nonpneumatic shear beam [25,58].

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Two typical goals of topology optimization method are to minimize the strain energy in the MPCM and the second is to enhance the strength of the structure under certain loading and boundary conditions [85]. Strain energy can be used to assess the relation between strain and stress. The integration of the strain energy and the dual optimization methods maximize the stiffness of the two-dimensional and three-dimensional unit cells. This integrated approach is illustrated in Figure 3.6.

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FE Discretization of the overall space

FE Modeling of the unit cell including BC’s

Determine the elastic tensor of the unit cell

Calculate the overall structure compliance

Sensitivity analysis of density

Topology optimization of the element density to minimize the structural compliance

NO

Update density of the unit cells

Convergence

YES

Plot density contour

Figure 3.6: Topology optimization algorithm [85]

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In exploring MPCM for the non-pneumatic tire, the targeted effective material properties previously determined [26] are used to explore different possible topologies of MPCMs [25,58]. Examples of a few MPCM are shown in Figure 3.7. These were all generated from the same seed concept, but with varying thicknesses and associated shear moduli. While configurations are found, it is important to note that most of the MPCM developed are topologically similar to each other and that no new topologies were found when compared to the intuitive method that led to identifying the bristle, the auxetic, the honeycomb, the chiral, or the cylindrical concepts [3].

Figure 3.7: MPCM configurations designed with topology optimization [86]

Another popular application in topology optimization is the determination of the displacement as an output and as a specific point of a given structure [84]. Both linear and nonlinear formulations for the topology of the compliant structures can be used, but applying

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nonlinear formulation generates improved stress distribution, making the structure stiffer over a wide range of deflections than the linear model does. The topology optimization method yields a coarse solution for the MPCM. These results then need to be smoothed and a regular periodic structure created. Moreover, defining the loading scenario has proven to be challenging for problems where there are different loading conditions and where stiffness and compliance are both sought. An advantage is that the explored design space is broader than in the other methods. A challenge with this approach is the loading scenarios that defines the problem. While the first example of plates and beams is statically loaded, the second example for the shear beam is continuously changing (the MPCM of shear beam is under shear in each direction). Formulating the problem is challenging due to modelling the periodic boundary conditions, basic shapes of the unit cells [34] and tiling the unit cells [59]. 3.5 Selection Method Due to the computationally expensive and sometimes impractical use of topology and multivariable optimization methods for designing large scale structures, one method has been proposed for size matching and scaling elemental MPCMs to populate a material region [8]. In this method, finite element analysis is applied to the solid part of interest, replacing the material elements with unit cells from a pre-defined library. To demonstrate the method, three sample problems of varying levels of difficulty with the goal of reducing both volume and compliance are studied.

The results found that this method can be used successfully for two- and three-

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dimensional lattice MPCMs. This general method has been evaluated with different synthesis algorithms [87] and has been extended with inclusion of a particle swarm and Levenberg– Marquardt optimization algorithms [88]. This method has been used for optimizing meso-scaled lattice structures filled beams and morphing airfoils for weight, compliance distribution, and motion [89].

Furthermore, the method replaces the finite element cells having specific

deformations with meso-scaled lattice structures that can achieve these deformations.

The

limitations of this method lie in the heuristic selection process of the lattice structure and the ability to model curved and non-rectangular surfaces and unit cells [8]. 3.6 Intuitive Design Method The intuitive design method for developing new MPCMs, intuitively offers new configurations and tests them through physical testing, such as in the early work on the lunar nonpneumatic tire development [90,91]. The inspiration for these topologies might be drawn from biological sources, such as the lightweight yet strong hexagonal honeycomb of bees and chiral honeycomb, which was inspired from a 2D lattice model of hexagonal molecules [92,93]. Alternatively, the inspiration might come from idea generation tools such as C-Sketch [94], the Gallery Method [95], or brainstorming [96]. After a configuration is conceptualized, it is modeled and then tested experimentally to see if the performance target is achieved. This method is useful when initially exploring a new application area where the behavior of the system is not well understood. Thus, physical prototyping is more useful in understanding how the system works than in the development parametrically tuned solutions. While limited, this method can lead to

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innovative solutions quickly, as evidenced by the lunar Tweel project for undergraduates where different concepts were intuitively explored [90]. These included (a) the bristle, (b) the segmented cylinder, and (c) the helical coil as shown in Figure 3.8.

(a)

(b)

(c)

Figure 3.8: MPCM Prototypes for the Lunar Tweel Project a) Bristle, b) Segmented cylinder, c) Helical coil [21]

3.7 Conclusion Designing MPCMs has been investigated by numerous researches for various applications over the last couple of decades. Table 3.1 shows a summary of the design methods of MPCMs. Different design methods have been proposed, applied and verified by different researches to choose geometry, material distribution, structure, and material to design effectively MPCMs for different goals and functions. The design methods have included numerical methods such as topology optimization, parametric optimization, and selection methods.

Experimental and

analytical methods were used for verifying numerical methods. Most of the research in the area of MPCMs has been done on optimization and finite elements. The applications have focused on

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lightweight structures, compliant structures, with few exploring combined mechanical, thermal, or other properties. Table 3.1: Comparison of design methods of periodic cellular materials Method

Requirement

Limitation

Generate Unit Cell Topology

Intuitive [21]

Overall design domain

Unable to explore the performance

Yes

Analytical [1]

Pre-defined geometry

Unable to create new topologies

No

Simulation [9]

Pre-defined geometry, loadings, BCs


No

Parametric Optimization [20]

Pre-defined geometry, loadings, BCs


No

Topology Optimization [34]

Design domain, loadings, BCs

Modeling repeatable boundary conditions, shape of basic elements,

Yes

Selection [8]

Design domain, Unit cell library, loading, BCs

Unable to create new topology

No

Only the topology related methods, which include topology optimization and intuitive methods, provide support for developing new topologies. The intuitive method is not able to evaluate the performance during the design process and topology optimization has limitations of numerical instabilities, modeling periodic boundary conditions, the basic shape of cells, and tiling the cells. Only the selection method is focused on populating unit cells and performing size optimization between different MPCMs. A recent approach has been developed which seeks to develop strain-energy based design guidelines for developing new MPCMs [97].

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There are some overlaps between the design methods. One reason is the nature of the model. For example, simulation (FEA) and topology optimization are based on numerical algorithms but their design processes are different; in some cases, FEA has been used to validate the topology optimization results. Parametric optimization and multifunctional design have an overlap on multifunctional design process. Different design parameters are defined and parametric optimization could be the next step to reach an optimum solution towards the problem. Ultimately, there were no efforts found regarding the development of design guidelines for creating new unit cells systematically. The systematic design of new topologies is identified as a new, critical research gap in the literature which is the focus of this dissertation. Furthermore, the mesostructures can be used to design human control systems and biosensors [98–101].

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CHAPTER 4 A UNIT CELL DESIGN GUIDELINE DEVELOPMENT METHOD FOR MESO-SCALE PERIODIC CELLULAR MATERIALS UNDER SHEAR LOADING

4.1 Overview Much research has been conducted on effective elastic properties of meso-scaled periodic cellular material (MPCM) structures; however, there is only limited research providing guidelines on how to develop improved unit cell (UC) topologies and shapes, for a given set of loading requirements and conditions. This dissertation develops guidelines to improve the shear flexibility of a MPCM while maintaining the effective shear modules by changing the topology and shape of a unit cell. The guidelines are intended to use design knowledge for helping engineers by providing recommendations at any stage of the design process. In this dissertation, the guidelines are developed by changing topology characteristics to achieve a desired effective property of a MPCM structure. The effect of individual members such as side connection and transverse connection of MPCM structure when subjected to in-plane shear loading are investigated through conducting a set of numerical simulation on UCs with similar topology and shape characteristics. Based on the simulation results, the unit cell design guidelines are developed to provide recommendations to engineers on improving shear flexure of MPCM during the design process. Ultimately, a unit cell design guideline development method is offered and demonstrated by developing two new design guidelines. 4.2 The need to Develop Unit Cell Design Guidelines

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A periodic cellular material is made of interconnected links of solid edges and faces of unit cells which have empty spaces within each unit cell in a periodic pattern [1]. A unit cell (UC) is defined as the smallest cell in a periodic structure [102]. Therefore, a MPCM is made of repeatable UCs. Figure 4.1 illustrates different types of MPCMs and their corresponding UCs.

Figure 4.1: Different MPCMs and their UCs. UCs are highlighted with dash line.

Meso-length scale UCs are structures with beam thickness in the range of 0.1-10 mm and beam length in the order of centimeters [8]. Due to the superior effective properties such as high strength and/or flexibility at low relative density, the use of cellular materials has applications in crushing energy absorption [5,103], light-weight structures [8,104], heat exchanger [1,2], insulation [1], and compliant structures [9,62]. A compliant structure (a typical class of MPCM) is a flexible structure which transfers forces, displacement, or motion from one point to another one [13,105]. A compliant structure can be designed as a monolithic or single-piece structure such as contact-aided cellular mechanism for morphing wing application [13] and hexagonal honeycomb for the shear beam of nonpneumatic tire [9]. Monolithic compliant structures are made in one piece and have potentially

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lower production costs, less friction, and wear [10]. Because this research has a background in non-pneumatic tire development, it focuses on monolithic compliant applications of MPCMs [15]. The motivation for this research stems from a previous effort to replace the hyper-elastic material of a conventional tire with a linear elastic material to reduce the tire’s rolling resistance [9,15,21,27,62,106]. A case study was conducted on different MPCMs that are developed for the shear beam of a non-pneumatic tire developed at Clemson University between 2007 and 2010 [15]. The Clemson researchers relied heavily on intuition to generate new topologies and then evolved them through a build (model) and test (simulate) loop. The main challenges associated with the intuitive method are that it is tedious and time-consuming. Moreover, the developed experience of one designer cannot be used by another designer unless it is formalized and then communicated. This dissertation aims to communicate formal guidelines for intentional redesign of UC topology and shape. The term ‘redesign’ is used in this chapter because the guidelines are intended to provide recommendations to improve the effective properties of an existing UC topology. Guidelines have been developed and widely used in other areas of design: design for manufacturing [38], design for assembly [38], design for product evolution [39], design for adaptive manufacturing [40], design for environment [41], design for aesthetic [107]. These guidelines have been developed for different objectives including: simplifying design processes, standardizing products, automating assembly, improving product quality, and reducing time and cost of assembly [107]. Generally, the guidelines are formulated as a conditionally based rule: IF situation X exists, THEN change to condition Y.

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The unit cell design (UCD) guidelines presented in this dissertation guide engineers to intentionally change the shape and topology of a unit cell to improve a desired structural behavior. In this chapter, the structural behavior refers to the effective properties of MPCMs under in-plane shear loading. This chapter presents four unit cell design guidelines for increasing shear flexure, which is the quantification of maximum shear strain a structure can withstand before yielding [62]. More importantly, the unit cell design guideline development (UCDGD) method is explicitly defined and the applicability of the method is demonstrated through developing two UCs design guidelines. Both academic researchers and engineers in industry can benefit from this research. First, researchers in the area of material design can use the UCDGD method to develop UCD guidelines and new UCs for targeted applications. Engineers in the industry can use the UCD guidelines to develop new UCs. Outstanding effective properties of UCs can lead to products which could be stiffer, or more flexible, or have lower relative density than conventional materials. Products with lower relative density create lightweight structures. For example, lightweight products in the transportation industry result in lower fuel consumption and consequently produce less CO2 emissions. Furthermore, this research lies in educating engineers with important parameters which are critical in MPCM design. 4.2.1 Design methods for periodic cellular material structures The unit cell design affects the effective properties of cellular material structure. Currently, the available methods for structure design focus on size, shape, and topology optimization [16,17]. Size refers to dimension of structural elements such as diameter, thickness, or length. Shape

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represents the curvature and orientation of structural elements.

Topology represents the

connectivity of material in the design domain [4,16,17]. Size optimization parametrically changes variables defining the thickness, diameter, length, and other dimensions of structural elements to meet design objective(s) under given constraint(s). The shape and topology of a structure are kept unchanged during size optimization. The example of design requirements could be maximum compliance [62], minimum displacement, or minimum mass[8]. Shape optimization uses parametric curves through optimal vertex locations [108,109]. If the size and shape optimizations of a pre-existing topology cannot meet the design objectives, topology optimization can be used to design a UC topology which incorporates optimum size and shape as well. Topology optimization methods have been used to design a MPCM by changing material distribution through removing and adding materials from each UC based on certain boundary and load conditions [58,110]. The associated challenges include: (1) modelling of periodic boundary conditions of UCs, (2) choosing the basic shape of the UCs, and (3) tiling of UCs [34]. Due to high required computational time to conduct topology and multi-variable optimizations, the size matching and scaling method for lattice structures was developed based on a predefined unit cell library and finite element analysis [8,77]. The limitations of this method derive from the heuristic selection process of the lattice structure and the ability to model curved and non-rectangular surfaces and unit cells [8]. Another method to develop new topologies for MPCM structures is the intuitive method [21]. The intuitive method uses idea generation tools to develop new topologies and then test them

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through finite element analysis or physical prototyping [3]. The challenges associated with this method includes several time consuming trial and error solution search, random solutions, and more importantly no guidance to engineers during the design process [36,107]. 4.2.2 Topology and shape characteristics of unit cells The guidelines in this chapter provide guidance for changing the topology characteristics of UCs to improve shear flexure, which is the performance metric. The topology characteristics refer to the structural elements that represent the connectivity of materials in the UC. Only one topology characteristic is changed for each guideline. The connectivity between two adjacent UCs and transverse connections inside a single UC are two topology characteristics that can be changed to improve shear flexure. A recent work introduces a vocabulary to represent the MPCM explicitly [18]. In the next section, a case study is discussed which inspires the motivation of two UCD guidelines presented in section 4.4. 4.3 Evolution of Meso-Scaled Periodic Cellular Material Structures for Non-Pneumatic Tires - A Case Study A case study is conducted on the evolution of meso-scaled periodic cellular materials for non-pneumatic tire development [15]. The two objectives of this case study are: (1) identifying factors that affect the performance of MPCMs, and (2) developing design guidelines for creating the new UC topologies. Through a collaborative research investigation exploring the low-rolling resistance nonpneumatic tire (NPT) known as “Tweel” for lunar and terrestrial applications, different design

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methods have been applied to replace the polyurethane shear beam of NPT with a linear elastic material [20–23,25,26,58,111]. The project goal was to develop a MPCM as the replacements of the Tweel polyurethane shear beam such that the new design can withstand cyclic loadings and extreme temperatures (40 K to 400 K). An additional goal was to reduce the NPT rolling resistance by 50% while maintaining the structural compliance in the elastic range under certain pressure contact between the wheel and the road. The Tweel consists of a shear beam between two inextensible membranes connected to a central hub through deformable spokes as shown in Figure 4.2. Because of the extreme temperature requirement, metals were explored to replace the existing polyurethane.

Figure 4.2: Non-pneumatic tire with solid shear beam [112]

The next challenge was to find a suitable size, shape, and topology for a unit cell of MPCM. Thus, different topologies were explored, parametric optimization was used to improve individual topologies, and eventually the unit cell was populated to create the shear beam. The topology evolution of MPCM for NPT shear beam between 2006 and 2010 is shown in Figure 4.3 [15].

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Specifically, twelve MPCMs were developed and explored at varying levels of detail. The evolution between these concepts was driven primarily by lessons learned, elicitation of new requirements, performance limits of the concepts, and the manufacturability of the concepts [15]. Helical Coil Fall 2006

Bristle Fall 2006

Straw Segmented Cylinder Fall 2007 Fall 2006

Foam Spring 2008

Honeycomb Summer 2008

Auxetic Fall 2008

Bristle Summer 2009

Chiral 2009

S-type Summer 2010

Sinusoidal Auxetic Summer 2009

Figure 4.3: The evolution of meso-scale periodic cellular materials for Tweel project [15]

The segmented cylinders failed under testing due to stress concentrations from assembly and from inextensible membrane buckling [21]. It was hypothesized that reducing the distance between the unit cells would increase the segmented cylinder resistance towards failure. This led to the straws concept which consisted of a stack of cylinders. The performance of the straws concept was not evaluated due to challenges with the connections between the cylinder stack and inextensible membranes and the cylinders themselves. The straws were placed in contact with each other to reduce the loads at individual cylinders in contact with the inner inextensible membrane. Next, the straws were envisioned as monolithic or made of a continuous material, reducing the impact of the joints. Finally, this concept led to an important takeaway which was the introduction of the layering of UCs. The straw concept directly inspired the use of hexagonal

66

honeycombs as an alternative structure for the Tweel shear beam. From this evolution of UC concepts, is seems that the effect of side joint connection should be explored when considering shear flexure of MPCMs. More details on the case study and how the evolution proceeds are discussed in [15]. The final UC developed and explored was the ‘S-type’ [97]. A stain energy-based analysis was conducted to compare available strain energy in the three layers of each unit cells of regular honeycomb, auxetic honeycomb, and sinusoidal auxetic. It was found that the strain energy distribution on the unit cells affects shear flexure and the S-type was proposed to reduce the moment impact of bending. The results showed that the S-type has higher shear flexure than hexagonal, auxetic, and sinusoidal auxetic UCs [62,97]. It was hypothesized that the lack of transverse connections is the main reason that the S-type UC has higher shear flexure than three other UCs. From this UC evolution, the traverse connections between UC and their role in shear flexure may be explored or formalized as a guideline to design MPCMs. 4.4 Two Unit Cell Design Guidelines from the Case Study Recognizing the evolution and change of the UCs from the Tweel project, this section attempts to formalize two design guidelines. 4.4.1 Guideline #1 – Connection between Unit Cells The first topology characteristic formalized into a guideline is based on changing the side connections which connect two adjacent UCs as shown in Figure 4.4. In Figure 4.4 (A), circles are considered as the UCs. Each circle has four vertex connections. It is expected that removing

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the side connection increases the shear flexure. The disconnected circles are shown in Figure 4.4 (B). Throughout this chapter, the arrows on top of the UCs represent the direction of applied displacement boundary condition. After disconnecting the circles from the side joint, each circle has two vertex connections as shown in Figure 4.4 (B).

Figure 4.4: (A) Side connection between two adjacent circles; (B) two disconnected circles

The abstraction of this change is illustrated in Figure 4.5. Therefore, Guideline #1 is written as: ‘If more flexibility in the direction of shear loading is desirable and the unit cells are connected from the side joint, disconnect the unit cells from the side joint.’ This guideline is applicable when unit cells are connected from the side joint and the objective is to increase in-plane shear flexure.

Figure 4.5: Abstraction illustration for disconnecting the unit cells from the side connection – Guideline#1

To generalize the guideline, it is applied to three other UCs: hexagonal honeycomb, chiral, and octagon as shown in Figure 4.6. Applying the Guideline #1 to chiral shapes leaves straight ligaments disconnected when the side connections are removed; since these ligaments are not

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connected to the left and right UCs, they are not able to carry the load. For this reason, they are removed after applying the guideline. Initial UC Hexagonal Honeycomb

UC After Change

Chiral

Octagon

Figure 4.6: Guideline #1 is applied to hexagonal honeycomb, chiral, and octagon through disconnecting the UCs from side connection. The type of connections can be either vertex or edge.

4.4.2 Guideline #2 – Transverse Connections Transverse connections are the second topology characteristics explored for guideline development. They are defined as horizontal, diagonal, or oblique structural elements in the interior of UC. Examples of transverse connections are illustrated for hexagonal honeycomb and square cells made from mix of squares and triangles in Figure 4.7. Hexagonal honeycomb and square cells have four and five transverse connections, respectively.

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Figure 4.7: Examples of transverse connections for hexagonal honeycomb and square cells with dash line.

It is expected that removing the transverse connections of a UC can increase the in-plane shear flexure. Then, the guideline # 2 is written as: ‘If more flexibility in the direction of shear loading is desirable and the unit cell has internal transverse connections, remove the transverse connections in the interior of the unit cell in case the unit cell keeps its integrity.’ If too many transverse connections are removed, it is possible for a unit cell to lose all integrity, and become incapable of carrying any load. An example of this issue is shown in Figure 4.8 under Guideline # 2. Therefore, care should be taken to “heal” the internal shape after applying the guideline to ensure that UC integrity is maintained under loading and boundary conditions.

Figure 4.8: Removing the transverse connections (highlighted in red) leads to a structure that cannot carry the shear load.

The graphical abstraction of Guideline #2 through removal of the transverse connections, a type of topology change, is generically illustrated in Figure 4.9.

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Figure 4.9: Abstraction illustration for removing the transverse connections – Guideline #2.

Figure 4.10 generalizes the guideline by showing how it can be applied to four UCs with transverse connections, hexagonal honeycomb, square cells with the mix of squares and triangles, rhombus, and octagon. Guideline #2 is applicable when the UC has transverse connection(s) and the objective is to improve the shear flexure. Initial UC Hexagonal Honeycomb

UC After Change

Square cells with mix of squares and triangles

Rhombus

Octagon

Figure 4.10: Guideline #2 applied to hexagonal honeycomb, square cells with mix of triangle, rhumbas,

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and octagon through removing the transverse connections of the UCs.

Figure 4.6 and Figure 4.10 provide examples for Guidelines #1 and #2. These demonstrate that the guidelines are applicable to different UCs with similar topology characteristics. In the next section, the generalization of guidelines are validated through numerical simulations to determine whether the application of the guidelines improves shear flexure. 4.5 Validation of Generalization of guidelines using numerical simulations 4.5.1 General Validation Procedure Numerical simulation (finite element analysis) is used to validate the generalization of guidelines. This section describes a general procedure used to validate the guidelines. The next section describes specific applications of this general procedure. The objective of the simulations is to compare the UCs and demonstrate how topology changes of UCs affect the performance metric in order to validate the generalization of guidelines. Simulations are conducted using a commercial finite element software, ABAQUS 6.14. Because shear flexure is the performance metric for these guidelines, a displacement boundary condition of 10% of the height of the UC is applied to the top nodes at the x-direction as shown in Figure 4.11. Zero displacement at the top nodes in the y-direction is considered. The top nodes of the UCs are defined as all of the nodes at the maximum height of the UCs. In Figure 4.11, examples of the top nodes are illustrated with the dash line. The bottom nodes of the UCs

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are fixed in the x- and y-directions as shown at the bottom of honeycomb and circle UCs in Figure 4.11.

y x

Figure 4.11: Top nodes on hexagonal honeycomb and circle UCs; the direction of displacement boundary condition.

In this chapter, all UCs have a consistent height of 12.70 mm and a depth of 19.05 mm [62]. The width is considered consistent between each two initial UCs and UCs after change. It is important to note that all simulation and geometric assumptions came from the previous research on non-pneumatic tire [9,26]. All UCs are modeled of 7075-T6 Aluminum alloy with material properties presented in Table 4.1 [9]. Table 4.1: Material properties of 7075-T6 Aluminum alloy Base Material

7075-T6 Aluminum alloy

Density Young’s Shear Modulus Poisson’s Yield Strength σys [kg/m3] Modulus [GPa] [GPa] ratio νs [MPa] 2800

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27

0.33

503

A flexible beam element (B22) in ABAQUS 6.14 is chosen for simulation to consider shear deformation [9,113]. B22 is a three-node quadratic beam element and each node has three degrees of freedom: x-displacement, y-displacement, and rotation. Geometric nonlinearity is considered due to the large-displacement effect. A post-processing code is developed to determine the shear flexure (yield shear strain) and effective shear modulus of the UCs. The detailed description on how shear flexure and effective shear modulus are calculated can be found in [62].

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The simulations were run on the UCs before and after the guidelines were applied. Both sets of simulations have the same boundary conditions and post-processing. Disconnecting the side joints of the UCs and removing the transverse connections decreased the effective shear modulus (or stiffness) and increased the shear flexure of the UCs. Different effective shear moduli would provide an inconsistent comparison. Therefore, the wall thickness of the UCs resulting from the guidelines was increased. A thickness was chosen such that the effective shear modulus was held constant. In order to address the issue, the beam thickness of unit cell after change is increased to reach the same effective shear modulus as well as the shear reaction forces. The thickness modification of the UC after change modifies the mass of the UC. However, the objective of simulation is to compare the flexibility of UCs based on changing the topology. As such, mass variation or relative density changes is out of scope this research. Conducting simulations at the same effective shear modulus provides an opportunity to compare the UC’s shear flexure only due to their shape/topology variations. Shear flexure target of 10% at effective shear modulus of 10MPa came from the original design requirements of the shear beam of non-pneumatic tire [62]. 4.5.2 Validation of Guideline #1 For guideline #1, a set of simulations are conducted to calculate shear flexure for the initial UCs and the UCs after change. Guideline #1 is applied to circles, hexagonal honeycombs, chiral, and octagon as shown in Figure 4.6. The shear flexure results for side-connected and disconnected UCs under in-plane shear loading are presented in Table 4.2. For Guideline #1, the results show shear flexure improvement of 34% - 131%. The fact that higher shear flexure is obtained with

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higher thickness for the UCs after applying the guideline is viewed as a positive benefit from a manufacturing stand point. Table 4.2: Shear flexure and corresponding thickness of UC for the validation of Guideline #1 at effective shear modulus of 10 MPa Initial UC Circle Hexagonal Honeycomb Chiral Octagon

Shear flexure Thickness (mm) Shear flexure Thickness (mm) Shear flexure Thickness (mm) Shear flexure Thickness (mm)

0.038 0.438 0.044 0.130 0.036 0.182 0.035 0.184

UC After change 0.088 0.810 0.059 0.675 0.080 0.435 0.061 0.655

Result of Change Improved Increased Improved Increased Improved Increased Improved Increased

4.5.3 Validation of Guideline #2 For Guideline #2, a set of simulations are conducted to calculate shear flexure for the initial UCs and the UCs after change. Guideline #2 applies to hexagonal honeycombs, square cells with mix of squares and triangles, rhombus, and octagon as shown in Figure 4.10. The shear flexure results for the initial UCs and UCs after change are presented in Table 4.3. For Guideline #2, the results show shear flexure improvement for all UCs of 46% - 109%. A similar observation is made in terms of thickness values as what is made for results from Guideline #1. Specifically, for square cells, a thickness value of 0.005mm is practically infeasible with any existing manufacturing process whereas with modification, such UCs are manufacturable with additive manufacturing techniques considering a typical manufacturing constraint on limiting the wall thickness above 0.1 mm [9].

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Table 4.3: Shear Flexure and corresponding thickness of UC for the validation of Guideline #2 at effective shear modulus of 10 MPa 0.032

UC After change 0.067

0.200

0.550

0.032

0.064

0.005

0.520

0.046

0.067

0.392

0.465

0.033

0.061

0.270

0.655

Initial UC Hexagonal Honeycomb Square cells - mix triangles Rhombus

Octagon


Result of Change Improved Increased Improved Increased Improved Increased Improved Increased

The results in both Guidelines #1 and #2 show that removing side or transverse connections improves the shear flexure of structures with the fact that the UCs before and after change have the same effective shear modulus. This observation can be justified by the fact that transverse or side connections add additional restrictions to deformation. Without them the UCs can deform further reaching the yield point. Validating the generalization of guidelines shows that they are applicable to different UCs with similar topology characteristics; however, there is no guarantee that the guidelines are applicable and/or effective in other cases. The next section provides a synthesis of a method to develop the guidelines. This method is extracted based on how these two guidelines were developed and tested. 4.6 Generalization of a Unit Cell Design Guideline Development Method (UCDGD Method) Although this chapter presents two guidelines, more guidelines are possible. Guidelines could be written for any combination of loading scenario, effective property, and topology change.

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The possible loading scenarios include shear, compression, tension, and bending. The possible effective properties of cellular materials include effective Young’s modulus for tension or compression, effective shear modulus, and effective Poisson’s ratio [1]. A method for developing unit cell design guidelines is presented in Figure 4.12. This method formalizes the process used in Section 4.4 to facilitate the development of future unit cell design guidelines.

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1 .

Define Requirements: target loads, BC’s, properties

2

Choose an initial unit cell (existing)

3

Determine a topology characteristic of the UC to modify

4

Apply the modification to the UC

5

Run simulation on initial UC (A) and after change (B)

6

No

B>A

A>B

Yes 7

Yes

Write initial GL; If A, then B

Write initial GL; If B, then A

8

Apply the initial guideline on three other unit cells

9

Run simulation on other initial UCs and UCs after change

10

No

No Similar Trend? Yes

11

Formally Document Guideline

Figure 4.12: Unit Cell Design Guideline Development Method (GL: Guideline; UC: Unit Cell)

In the first step of the method, the guideline developer defines the requirements for the change of the UC. The requirement should be defined as an increase (or decrease) of an effective property, while another property is held constant, given loads and boundary conditions. It is recommended that single requirements be identified, though the impact of combined requirements

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defined guidelines is reserved for future study. The requirement for the guidelines presented in Section 4.4, is to increase the shear flexure of the structure while the shear modulus is maintained. The shear load is applied at the top of the UC and the base of the UC is held fixed. These requirements and conditions again were derived from the design requirements of the shear beam of the non-pneumatic tire [9]. The second step is to choose an initial UC from the literature or intuition. Examples of initial unit cells can be hexagonal honeycomb, chiral, or segmented cylinder. It is important to note that the guidelines developed here and through this proposed method are evolved from existing UCs. Other methods are still needed to define the initial UC. Alternatively, the result of one application of a guideline can be the initial UC for another guideline. The third step is to choose a topology characteristic of the initial UC. In the guidelines presented in Section 4.4, the topology characteristics include the UC connections with its adjacent UCs and interior elements as depicted with examples in Figure 4.4 and Figure 4.7, respectively. There are two types of UC connections with its adjacent UCs: vertexes, and edges [18]. The fourth step is to modify the initial UC’s topology characteristic into a new UC. For any single guideline and initial topology, there may be many ways to apply the guideline to modify the UC. The degree of subjectivity of the guideline and the size of the modification space is not explicitly addressed in this method, though it should be explored in future studies. The fifth step is to conduct numerical simulations (finite element analysis) to determine the values of effective properties for the initial UC and the UC after change. It is recommended that the simulations be done with similar initial assumptions and choices as the simulations

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described above. The rationale for this is to provide a standard comparison basis when developing many guidelines. The sixth step is to examine the results of the fifth step to check if the change makes the improvement that the guideline was intended to make. In other words, if the result of B has improvement over the result of A (B > A); then, the initial guideline is documented as an “if A, then B” statement (the seventh step). If the topology change has an adverse effect on the performance criteria, if B is worse than A (B < A), then the initial guideline is documented as an “if B, then A” statement (the seventh step). If changing the topology characteristic has no effect on the performance criteria or in other words result of A is similar to B, no guideline can be written. In this case, either the topology characteristic or the initial UC need to be changed and repeat the simulation and comparison between A and B (go to step two or three). The eighth step is the generalization of the initial guideline. The initial guideline is applied to three other initial UCs which have the same topology characteristic. The initial UCs can be found in the literature or developed by the user through trying different topologies. It is assumed that three additional topologies can provide a suitable level of confidence that this guideline could be applied in the future. However, this assumption has not been tested. Through deployment of the design guideline development method, perhaps this assumption can be validated through use. The ninth step is to perform simulations to compare the initial UCs and the modified UCs. If a similar trend is observed (the tenth step), the guideline can be documented (the eleventh step). The guideline is documented with a guideline name, explanation, an ‘if, then’ statement, a graphical abstraction, an illustrational example, a justification, and the list of references. The ‘if,

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then’ statement can be written as If {conditions}, then {consequence}. The condition is the requirement(s) and the topology characteristic that needs to be changes. The consequence describes the topology change recommendation should be made to the initial unit cell to obtain the UC after change. A sample of fully documented guideline is illustrated in Figure 4.13. This format is similar to that defined in [114,115].

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Guideline Number: 1 Date: 04/22/2015 Guideline Name: Effect of removing side connection between unit cells under shear loading Guideline Explanation In tessellating unit cells, the connection between the unit cells is critical. It is recommended to remove side connections between the unit cells to improve the flexibility under shear loading. Guideline condition (If statement) If more flexibility in the direction of shear loading is desirable, Guideline consequence (Then statement) then, disconnect the unit cells from the side connection. Illustration

Abstract

Guideline Justification Through removing the side connections, the UCs which carry the load path face fewer restrictions to deformation under shear loading. Thus, the UCs can deform further before reaching the yielding point. Reference [1] Simulation results in Table 4.2. [2] Fazelpour, M., and Summers, J. D., 2014, “Evolution of Meso-structures for Nonpneumatic Tire Development: A Case Study,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME, Buffalo, NY, p. V02BT03A002. [3] Stowe, D., Conger, K., Summers, J. D., Joseph, P., Thompson, B., and Matthews, J., 2008, “Designing a Lunar Wheel,” Proceedings of the ASME 2008 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Brooklyn, New York, pp. DETC2008–49981. Figure 4.13: An illustration of the fully documented Guideline #1

The justification behind the guidelines is an important aspect that can be used to explain the guidelines for future users. The rationale behind the guidelines explain why making a change

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to the initial UC leads to meet the desired performance criteria. This can help the developer of the guideline by answering this question: why changing the topology characteristic can meet the desired performance criteria? In the next section, the unit cell design guideline development method is evaluated through developing two new guidelines. 4.7 Demonstration of the Unit Cell Design Guideline Development Method To demonstrate the UCDGD method, it is used to develop two new guidelines. While the first two guidelines were developed before the method was formalized, these two guidelines presented in this section are defined by strictly following the process. The first guideline is defined step by step following the method. 4.7.1 Guideline#3 Development For the first step, increasing the shear flexure is used as the desired performance criteria for Guidelines 3 and 4.

For the second step, hexagonal honeycombs with detached side

connections are used as the initial UCs as shown in Figure 4.14. The top and bottom legs are chosen as topology characteristics to modify (step 3). The UC after change is obtained by removing the top and bottom legs and using the core of the cell to connect the top and bottom boundaries as shown in Figure 4.14 (step 4). The next step is to conduct simulations and compare the results. The simulation details are discussed earlier in the validation of generalization section (step 5). For an effective shear modulus of 10MPa, the shear flexure of honeycomb with legs is 0.042 (at thickness of 0.710 mm) and for honeycomb without legs, 0.059 (at thickness of 0.675

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mm). This change increases the shear flexure by 40%. This comparison suggests that the guideline has potential (step 6). Top Leg UC Core Bottom Leg

Figure 4.14: Hexagonal honeycomb with vertical legs as initial UC on the left and without legs on the right.

Since removing the top and bottom legs increases the shear flexure as expected, the initial guideline #3 is written (step 7): ‘If more flexibility in the direction of shear loading is desirable and the unit cell has top and/or bottom legs, remove the top and/or bottom legs and use the core of the cell to connect the top and bottom of boundary.’ To generalize, the guideline is applied to three initial UCs (step 8). The topology changes are presented in Figure 4.15. The simulation results are presented in Table 4.4 (step 9).

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Initial UC

UC After Change

Circle

Rhombus

Octagon

Figure 4.15: Guideline #3 is applied to circle, rhombus, and octagon through removing the vertical legs of UCs

Table 4.4: Shear flexure and corresponding thickness of UC for the validation of Guideline #3 at effective shear modulus of 10 MPa Hexagonal Honeycomb Circle Rhombus Octagon


Initial UC 0.042 0.710 0.045 0.765 0.058 0.480 0.047 0.755

UC After change 0.059 0.675 0.084 0.810 0.067 0.465 0.061 0.655

Result of Change Improved Decreased Improved Increased Improved Decreased Improved Decreased

The results in Table 4.4 show shear flexure increase for the UCs after change between 1687%. The results for four UCs show a similar trend (step 10). Therefore, Guideline #3 is

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documented (step 11). The abstraction of Guideline #3 through removing top and bottom legs is presented in Figure 4.16.

Figure 4.16: Abstraction illustration for removing the top and bottom legs – Guideline #3

The reason behind Guideline#3 can be explained as follows. In the core of the unit cell, there are two parallel load paths that share the load. Because the top and bottom legs have no parallel load path, they must carry the entire load. Removing them, allows the UC to deform further before the yield point. 4.7.2 Guideline#4 Development The idea for Guideline#4 stems from developing the S-type UC that was introduced and optimized in [62,97]. It is hypothesized that using curved beams can decrease stress due to bending moments and therefore improve the shear flexure of UCs. Increasing the shear flexure is used as the requirement or desired performance criteria for the guideline #4 (step 1). For the second step, two hexagons without side connections are chosen as the initial UCs as shown in Figure 4.17. For the third step, straight edges of the UC interior are considered to be replaced. For the fourth step, the UC after change is obtained by replacing the straight edges with circles as shown in Figure 4.17. The next step is to conduct simulations and compare the results. The simulation details are discussed earlier in the validation of generalization section (step 5). For the effective shear modulus of 10MPa, the shear flexure of honeycomb

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without legs is 0.059 (at thickness of 0.675 mm) and for circles, 0.084 (at thickness of 0.810 mm). This change increases the shear flexure by 42%. This comparison suggests that the guideline has potential (step 6). Since replacing the UCs with circles increases the shear flexure as expected, Guideline#4 is written as (step 7): ‘If more flexibility in the direction of shear loading is desirable, replace the straight edges of the unit cell interior with curve beams to connect from the top to the bottom of the unit cell in a direction normal to the applied shear loading. Examples of curve beams include a circle, a semi-circle, and an S-shape.’ To generalize, the guideline is applied to three initial UCs (step 8). Figure 4.17 presents the topology changes. Simulation results (step 9) are presented in Table 4.5. For the fourth guideline, different curved beams including circles, semi-circles, S-type, and S-type without legs are evaluated versus non-curved UCs as shown in Figure 4.17.

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Initial UC Hexagonal Honeycomb

Circle

UC After Change

Octagon

Semi-circle

Zigzag

S-type without legs

Octagon

S-type

Figure 4.17: Guideline #4 applied to hexagonal, octagon, rhombus, and octagon with legs through replacing the UCs with different curved beams.

The results in Table 4.5 show the curved beams increase the shear flexure by 34-57%. The results for four UCs show a similar trend (step 10). Therefore, Guideline #4 is documented (step 11).

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Table 4.5: Shear flexure and corresponding thickness of UC for the validation of Guideline #4 at effective shear modulus of 10 MPa


Initial UC Hexagonal Honeycomb 0.059 0.675 Octagon 0.061 0.655 Zigzag 0.049 0.440 Octagon 0.061 0.655

UC After change Circle 0.084 0.810 Semi-circle 0.082 0.800 S-type without leg 0.077 0.620 S-type 0.083 0.495

Result of change Improved Increased Improved Increased Improved Increased Improved Decreased

The abstraction of Guideline #4 through replacing the interior of initial unit cell (A) with the curved beams (B) is illustrated in Figure 4.18. The initial unit cell may include only interior with straight edges while unit cell after change (B) includes interior with curved beams.

Interior with Straight Edges

Interior with Curved Beam

(B)

(A)

Figure 4.18: Abstraction illustration for replacing the UC with curved beams – Guideline #4

4.8 Conclusion and Future Work Four UCD guidelines are developed to improve the shear flexure of MPCM without changing effective shear modulus. The first two guidelines are inspired through the case study [15] which suggested that disconnecting the UC from its side joints and removing transverse

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connections would improve the shear flexure of UCs. Based on the first two guidelines, the UCDGD method is proposed and demonstrated by developing two new guidelines. The second two guidelines recommend eliminating the vertical legs and using curved beams to increase the shear flexure. As a result of applying the guidelines, the shear flexure is improved in a range between 15-131% with the average of 62%. One limitation of the current work is that selecting an initial unit cell and a topology characteristic to change is still a heuristic step. Another limitation is that the guidelines are only guaranteed to be effective for the four cases that were actually simulated. In choosing different test cases, the developer seeks to extend the applicability of the guidelines. Further, it is recognized that simulating two UCs is not the same as simulating the complete MPCMs, but the results of general trends are indicative and suggestive. In modifying the UC, then the mesostructured designer can populate the material region of interest and optimize the sizes. More research is needed to determine the limitations of modeling UC’s to obtain MPCM results. Different loading scenarios such as compression and tension, and other effective properties of MPCMs may be explored in future work. The combination of different loading scenarios, for example, shear and compression, can also be considered in future work. Moreover, the subjectivity of the guidelines has not yet been tested. It is not clear whether two engineers would generate different topologies by applying the same guidelines to the same initial UCs. This remains as future work and is out of scope for this dissertation. The four guidelines presented in this dissertation are a demonstration of the UCDGD method; however, further validation of the UCDGD method can be considered in future work. The validation square for design methods

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consists of four phases and six steps is considered to validate the UCDGD method [61]. Four phases include: theoretical structural validity, empirical structural validity, empirical performance validity, and theoretical performance validity. The structural validation consists of three steps: (1) obtaining methods’ constructs validity, individually and integrated; (2) obtaining method’s consistency; (3) obtaining the example problems. The performance validation consists of three steps: (4) obtaining suitability of method for some example problems; (5) obtaining that suitability is related to applying the method; (6) obtaining suitability of method beyond example problems [61]. Among these six steps, obtaining method’s consistency, Step#2, is conducted through several guideline development demonstrations. The rest of the steps are left for future work.

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CHAPTER 5 A DEMONSTRATION ON HOW RESEARCHERS USE THE UNIT CELL DESIGN GUIDELINE DEVELOPMENT METHOD – A CASE STUDY This chapter provides an external demonstration of the Unit Cell Design Guideline Development (UCDGD) Method through a case study. A senior mechanical engineering student who has taken only one introductory Mechanics of Materials course was asked to follow the UCDGD method to develop unit cell design guidelines. The student was a visiting student from Stanford university and spent two months of summer as a guest researcher to work on developing guidelines. Lectures on mechanics of periodic cellular materials, finite element analysis, and ABAQUS tutorials were provided to the student to have necessary background knowledge on using the method and modeling unit cells. The lectures and learning fundamentals took around two weeks. The rest of two months (six weeks), the student worked to using the method to develop guidelines that led to two new guidelines. The loading scenario for the both guideline was tension. Displacement boundary conditions was applied to the top nodes as they Both guidelines consider a tension loading condition with an applied displacement boundary condition of 10% strain on the top nodes in the y-direction. The bottom nodes had no displacement in x- and y-directions. Except the boundary conditions, the material, element, and the rest of simulation assumptions are the same as discussed in Section 4.5.1. The first guideline, Guideline#5, is written as: ‘If more flexibility or higher yield strain in the direction of tension loading is desirable, replace bristles or straight beams parallel to the direction of the tension loading with non-straight beams such as rhombus, circle, and hexagon.’

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If higher yield strain under tension loading is desirable, Guideline#5 recommends replacing bristle or straight beam unit cells with non-straight beams such as rhombus, circle, and hexagon to connect from the top to the bottom of unit cell boundary in the parallel direction of the applied tension loading. The abstraction of guideline#5 is illustrated in Figure 5.1.

Figure 5.1: Guideline#5 Abstraction

In order to show the generalization of Guideline#5, it applies to different unit cells and the bristle unit cells are replaced with non-straight beams. The examples of non-straight beams include circle, rhombus, and hexagon. The generalization of Guideline#5 was shown in Figure 5.2.

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Initial UC

UC After Change

Figure 5.2: Guideline #5 applied to bristles that led to circles, rhombus, and hexagon.

The simulation results are presented in Table 5.1. The results show in all cases, the unit cells after change have higher effective yield strain. The justification of this guideline is due to bending-dominated behavior of unit cells after change while the bristle has a strong stretchingdominated behavior [116]. Bending-dominated cellular materials are appropriate for flexible or mechanism applications while stretching-dominated have applications in the light-weight structures which are typically stiff and strong [116].

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Table 5.1: Effective yield strain and corresponding thickness of UC for the validation of Guideline #5 at effective young’s modulus of 10 MPa

Yield strain Thickness (mm) Yield strain Thickness (mm) Yield strain Thickness (mm) Yield strain Thickness (mm)

Initial UC

UC After change

Bristle 0.0069 0.0027 Bristle 0.0069 0.0027 Bristle 0.0069 0.0027 Bristle 0.0069 0.0027

Circle 0.046 0.423 Rhombus 0.057 0.660 Hexagon 0.044 0.292

Increase in effective yield strain, %

0.010 0.0035

566%

726%

537%

45%

The second developed guideline, Guideline#6, is written as: ‘If higher yield strain (or flexibility parallel to tension loading) under the tension loading is desirable, remove horizontal connections between unit cells.’ Guideline#6 abstraction is illustrated in Figure 5.3 that shows how the horizontal connection is removed. In order to show the generalization of Guideline#5, it applies to different unit cells as shown in Figure 5.4. the horizontal connection, which is a vertex connection [18], is removed between two unit cells which are circles, rhombus, hexagon, and octagon as shown in Figure 5.4.

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Figure 5.3: Guideline#6 abstraction

Initial UC

UC After Change

Figure 5.4: Guideline #6 applies to circles, rhombus, hexagon, and octagon.

To verify the generalization of Guideline#6, a set of simulations were conducted and the results of yield strain under tension loading for the both initial and after change unit cells are in Table 5.2 for a consistent effective Young’s modulus of 10 MPa. The results show in all four cases, the effective yield strain increases 38%-171% with an average of 87%. The justification of

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Guideline#6 is due to removing the horizontal connections between unit cells allow to pull the vertical beams parallel to applied tension loading. Table 5.2: Effective yield strain and corresponding thickness of UC for the validation of Guideline #6 at effective Young’s modulus of 10 MPa

Effective yield strain Thickness (mm) Effective yield strain Thickness (mm) Effective yield strain Thickness (mm) Effective yield strain Thickness (mm)

Initial UC

UC After change

Connected Circles 0.024 0.298 Connected Rhombus 0.021 0.040 Connected Hexagon 0.032 0.0027 Connected Octagon 0.018 0.056

Dis-connected circles 0.039 0.514 Dis-connected Rhombus 0.057 0.660 Dis-connected Hexagon 0.044 0.292 Dis-connected Octagon 0.032 0.278

Increase in effective yield strain, % 63%

171%

38%

77%

This case study provides an external demonstration of the UCDGDM through two guidelines developed by a novice engineer who did not have background in mechanical of cellular materials and ABAQUS.

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CHAPTER 6 EXPLORING THE SUBJECTIVITY AND SEQUENCING OF UNIT CELL DESIGN GUIDELINES – A USER STUDY

6.1 Overview The objective of this user study is to evaluate the subjectivity and sequencing of unit cell design (UCD) guidelines. The unit cell design guidelines outline a set of instructions on changing important parameters to guide engineers in intentionally redesigning shapes and topologies for unit cells when considering a desired structural behavior. Guidelines are inherently subjective, therefore, the subjectivity of the guidelines is evaluated. Subjectivity refers to how much the user’s personal opinion can affect their judgment on using the guidelines. Additionally, the effect of sequencing of guidelines is evaluated. In the sequencing study, the effect of the order of guidelines on the outcome shape when they are applied consecutively is evaluated. This chapter presents two user studies designed to address the subjectivity and sequencing of the (UCD) guidelines. 6.2 Designing the User Study A user study is defined as a controlled empirical investigation on human subjects and their resulting assessments of a problem or phenomena to test a hypothesis. Defining the objective of a user study is a critical step in designing a user study. This study has two objectives: (1) evaluate the similarities between the outcome unit cells developed by applying unit cell design guideline to a pre-defined unit cell, (2) evaluate the effect of sequencing of guidelines. For designing a valid user study, four factors must be considered carefully: hypothesis, variables, participants, and

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statistical analysis [117]. In the following section, each of these terms are defined and choices are justified. 6.3 Hypothesis A hypothesis is an informed, testable guess or explanation of a given phenomenon [118]. The first step to design a user study is hypothesis formulation, which relies on the objective of user study. A clear and explicit hypothesis helps define the variables that need to be tested, and relationships between the variables. The evaluation of subjectivity and sequencing of the UCD guidelines triggered the following research questions, research hypotheses, and null hypotheses: Research Question 2. Are the unit cell design guidelines subjective? Research Hypothesis 2. There will be similarities between developed shapes. Low subjectivity is expected. Null Hypothesis (NH0) 2. There will be no similarities between developed shapes. High subjectivity is expected. Research Question 3. Does sequencing of the unit cell design guidelines affect modified shape? Research Hypothesis 3. Sequencing will result in similar modified shapes. Null Hypothesis (NH0) 3. Sequencing will not result in similar modified shapes. It is hypothesized that there will be similarities between resulting topologies, which implies that low subjectivity is expected. 6.4 Variables

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The controlled variables for this study include one problem, four UCD guidelines, and two initial unit cell shapes (UC). The problem is chosen to motivate the participants by stating that there is a need to design new unit cells for non-pneumatic tire developed by tire company Michelin [15]. NASA is also interested in non-pneumatic tire development for lunar rover application [15]. Almost all participants have similar level of knowledge and expertise regarding tires. Also, the non-pneumatic tire is the primary motivation for developing the guidelines to increase shear flexibility of periodic cellular material. The design problem is presented in Figure 6.1. While the objective of this study is not to solve the design problem, the design problem describes the need for guidelines and provides the students with motivation to participate by explaining the need for the non-pneumatic tire.

Figure 6.1: Problem — unit cell re-design for shear beam [14]

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Four UCD guidelines are considered for this study. These guidelines provide recommendations for topology or shape changes to increase shear flexibility of the unit cells. To evaluate the effect of sequencing, two guidelines are chosen to be applied consecutively by each participant. The initial unit cells are provided to apply the first guideline while the unit cells created as an outcome of the first guideline are used as initial unit cells for the second guideline. The guidelines for this user study are presented in Figure 6.2. Guideline #1: ‘If more flexibility in the direction of shear loading is desirable and the unit cells are connected from the side joint, disconnect the unit cells from the side joint.’ Guideline #2: ‘If more flexibility in the direction of shear loading is desirable and the unit cell has internal transverse connections, remove the transverse connections in the interior of the unit cell while ensuring that the unit cell keeps its integrity.’ Guideline #3: ‘If more flexibility in the direction of shear loading is desirable, replace the straight edges of the unit cell interior with curve beams to connect from the top to the bottom of the unit cell in a direction normal to the applied shear loading. Examples of curve beams include a circle, a semi-circle, and an S-shape.’ Guideline #4: ‘If more flexibility in the direction of shear loading is desirable and the unit cell has top and/or bottom legs, remove the top and/or bottom legs and use the core of the cell to connect the top and bottom of boundary.’ Figure 6.2: Four unit cell design guidelines to improve shear flexure

These guidelines were applied to two shapes: hexagonal honeycomb, and rhombus. These shapes, chosen as the initial unit cells for this study, are shown in Figure 6.3 [1].

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Rhombus

Hexagonal Honeycomb

Figure 6.3: Hexagonal honeycomb and rhombus as the initial unit cells

The design problem and the guidelines should be clear to the participants. This was evaluated by conducting a pilot study where the participants were asked to apply the guidelines to the unit cells. At the end, a short survey was conducted among participants to investigate any lack of clarity on the problem and the guidelines. After reviewing the feedback from the pilot study, Guideline#2 was modified by adding a sentence to describe what is meant by transverse connections. 6.4.1 User study I – Subjectivity The primary objective of this user study was to evaluate the subjectivity of the guidelines as they are applied to a unit cell. Four guidelines were tested in this study on two shapes, resulting in eight topologies. In order to have a statistically significant sample size of 30, a total of 240 topologies were needed. There were 60 participants available for this study so each participant was assigned five cases of applying one of the four guidelines to one of the two shapes. In order to prevent any bias incurred from the order and frequency of guidelines and shapes assigned, the packets were created by listing all possible instances of a guideline (1 of 4) applied to a shape (1

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of 2), and randomly selecting five of those instances to create a packet. The packets were then tallied to ensure that each of the 8 instances were fairly represented. 6.4.2 User study II – sequencing In addition to reviewing the subjectivity of individual guidelines, the effect of the order in which the guidelines are applied in a sequence was also a topic of interest. The resulting shape after a sequential application of two guidelines was investigated. Similar to the subjectivity experiment, four guidelines, and two shapes were used, resulting in 12 distinct cases that need to be examined. In order to have a statistically significant data set for each case, a total of 360 will need to be produced. With the assumption of an available participant pool of 60, the experiment packets were designed such that each participant will complete 6 of the 12 cases. The assignment of the cases was done in a similar manner to the subjectivity experiment. 6.5 Study Participants The user studies are designed for implementation among senior undergraduate students of the Department of Mechanical Engineering at Clemson University. At the time of experiment, they were enrolled in ME4010 – senior mechanical engineering design course, fall 2016. All students are approximately between 20-24 years of age, enrolled in the fourth year of mechanical engineering; therefore, they have a similar background in mechanical engineering design and mechanics of materials. They generally need one more year to complete their studies and graduate. A total of 129 students are enrolled in senior mechanical engineering design will participate in the study. Since the objective is to evaluate the personal opinion of students with regards to the

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guidelines, therefore, each the students carry out the study individually. In this study, the differences between gender, race, and culture are not controlled, however, careful attention is given to the problem, guidelines, and initial unit cells to be independent of gender, race, and nationality biases. A short lecture is delivered to provide a consistent level of knowledge to all participants before beginning the study. There is no concern regarding the previous experienced participants on using UCD guidelines because limited research on UCD guidelines was found in the literature to the current knowledge of the authors. The environment of the experiment is the normal scaleup classroom that the students attend regularly for lectures and design reviews. This helps the students being comfortable with the environment. 6.6 Study Procedure The experimental procedure consists of three main phases: the pre-study lecture, the activity, and packet collection. The environment of the experiment is the normal up-scale classroom that the students attend for lectures and design reviews. The class is equipped with a projector to present the lecture for students; all cares will be considered to eliminate the distraction sources (e.g. the students will be asked to turn off their cell phones during the user study). When all students have arrived and are seated in the classroom, they will receive a short lecture, lasting roughly ten minutes, on the design problem, instructions on how UCD guidelines work, and four examples of applying two UCD guidelines consecutively. The examples in the lecture are

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excluded from the user study to avoid affecting students’ judgments. After the mini-lecture, participants are allowed to ask any questions they may have. Once all questions are addressed, the experiment packets will be distributed among the participants. For the subjectivity experiment, each packet includes stapled sheets of paper including the problem description in one sheet (Figure 6.1) and five guideline sheets. Each sheet has one guideline in the form of text (an “if-then” statement) and graphical abstraction, one initial unit cell, and one empty spaces to draw the unit cell after applying the guideline. The students are expected to draw the unit cell after modifying them based on the provided guideline. A sample of one of the guideline sheets collected from the students is shown in Figure 6.4.

Figure 6.4: A sheet sample of applied Guidline#1 to hexagonal honeycomb for subjectivity experiment

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For sequencing experiment, which is conducted in a different session from the subjectivity experiment, each packet includes stapled sheets of paper including the problem description in one sheet (Figure 6.1) and six guideline sheets. Each guideline sheet has two guidelines in the form of text (an “if-then” statement) and graphical abstraction, one initial unit cell, and two empty spaces to draw the unit cells after applying guidelines. The sample of one of collected sheets for the sequencing experiment is shown in Figure 6.5. The students are expected to draw the unit cells after modifying them based on the provided guidelines. The students have 25 and 50 minutes to carry out the subjectivity and sequencing experiments, respectively. These time limits were selected based on the number of guidelines being applied in each experiment. In the subjectivity experiments, each participant applies five guidelines. Two minutes are considered for applying each guidelines plus two minutes reading the problem, therefore the students are expected to take approximately 12 minutes; however, a time limit of 25 minutes is considered to make sure all students can finish the experiment. Similarly, for the sequencing experiment, each student applies 12 guidelines, therefore a time limit of 50 minutes is considered. It is critical to ensure that the students are able to finish the study in the allotted time, so two pilot studies were conducted among CEDAR (Clemson Engineering Design Application Research) members to evaluate the feasibility of finishing the study in the given time. The results of pilot studies showed 30 minutes was a reasonable amount of time to finish both experiments; however, the students were given the entirety of class time (75 minutes). Once the student participants have completed the activity, the packets are collected, and the resulting unit

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cells generated are used for data analysis. Table 6.1 shows the number of responses collected for each guideline and shape. Table 6.1: Number of responses collected for each guideline and shape

Hexagon

Rhombus

Guideline 1

40

39

Guideline 2

38

39

Guideline 3

38

41

Guideline 4

41

36

Total

157

155

Figure 6.5: A sheet sample of applied Guidline#2-4 to hexagonal honeycomb for sequencing experiment

The number of the responses received for sequencing study for each case are listed in Table 6.2. As shown in Table 6.2, the intended number of 360 responses was not achieved because some

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participants were absent during the experiment. However, the experiment still yielded enough responses to statistically compare the guideline outcomes in the cases where they were applied first to those where they were applied second. Table 6.2: Number of responses collected for each case in sequencing study

Hexagon

Rhombus

Guideline 1 => Guideline 2

9

14


15

12


13

14


13

12


13

10


13

12


13

15


11

10


15

12


12

12


15

17


7

11

149

151

Total 6.7 Data Analysis

The results are evaluated based on the similarities of the unit cells after applying the guidelines. In order to evaluate the similarity, certain metrics need to be developed, upon which the similarity of the unit cells can be considered.

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6.7.1 Metrics to evaluate similarities among unit cells after change The interior of unit cell is defined by three entities: topology, shape, and size [18]. The topology of the unit cell represents the connectivity of material within the interior, the shape deals with curvature and orientation of the structure elements within the unit cell, and the size states the dimensions of the structure elements such as length, width, height, thickness, and diameter [16,17]. The interior is defined as the material surrounded by the unit cell boundary. The UCD guidelines provide recommendations on topology and/or shape changes to improve the shear flexure of unit cells [43]. Therefore, the connectivity of material within the interior (topology) and the curvature and orientation of the structure elements (shape) are considered as similarity metrics. If two unit cells after change have the same topology and shape, they are considered similar. Size or parametric change of unit cells is out of scope for the UCD guidelines; therefore the size of unit cells is not considered in the metric. 6.7.2 Inter-rater reliability analysis An inter-rater reliability test was done in order to ensure that the subjective comparison of unit cell shapes was consistent across different raters. Initially, two different raters were used to evaluate the participant responses based on a given rubric. One rater was a PhD candidate who designed and conducted the user studies. The other rater was an MSc graduate student in the Department of Mechanical Engineering with research background in design and function structures. Each rater reviewed ten student responses for each of the four guidelines, for two different shapes. According to the rubric, the raters identified whether the guidelines were followed or not, and categorized the shapes generated by students into different clusters.

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In order to compare the consistency between the raters, a Cohen’s kappa analysis was done between the two raters, and the results can be seen in Table 6.3. It should be noted that for guideline 3 on the rhombus, the actual percent agreement between raters was 70%, however, due to the observed agreement and the change agreement being equal, the kappa value for this instance was seen as zero. Table 6.3: Initial inter-rater reliability using Cohen's kappa

Guideline 1 Guideline 2 Guideline 3 Guideline 4 Average

Hexagon 100.0% 61.5% 61.5% 100.0% 80.8%

Rhombus 100.0% 11.8% 0.0% 100.0% 52.9%

As shown in the Table 6.3, the raters showed almost perfect agreement for guidelines applied to the hexagon, and moderate agreement for guidelines applied to the rhombus [119]. Following a discussion between the raters, the rubric was updated to make some clarification changes to the instructions as well as provide additional definitions for concepts that were more subjective. The inter-rater reliability test was redone with the updated rubric using three raters. The results can be seen in Table 6.4 and Table 6.5. Table 6.4: Kappa values for guidelines applied to the hexagon

Raters R1-R2 R1-R3 R2-R3

Guideline 1 100.0% 100.0% 100.0%

Guideline 2 100.0% 61.5% 100.0%

Guideline 3 100.0% 61.5% 100.0%

Guideline 4 100.0% 100.0% 100.0%

Average 100.0% 80.8% 100.0%

Table 6.5: Kappa values for guidelines applied to the rhombus

Raters R1-R2 R1-R3 R2-R3

Guideline 1 100.0% 100.0% 100.0%

Guideline 2 100.0% 61.5% 61.5%

Guideline 3 100.0% 100.0% 100.0%

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Guideline 4 100.0% 100.0% 100.0%

Average 100.0% 90.4% 90.4%

As shown in Table 6.4 and Table 6.5 the minimum average agreement was 80.8%. This is almost perfect agreement between raters and suggests that the rating rubric is robust, and will yield consistent results when used by different raters [120,121]. 6.7.3 Results for subjectivity of unit cell design guidelines The similarity evaluation is conducted into two steps. In the first step, the rater evaluates the modified unit cells and divides them into three categories: (1) guideline followed, (2) guideline not followed, and (3) blank sheet. (1)

The unit cells in which provided changes follow the guidelines. For example,

Guideline#1 asks to disconnect the unit cells from the side joint. The rater judges if the two unit cells are disconnected or not regardless of other topology and shape changes. In a few cases, a single unit cell is drawn; if that unit is disconnected from the side joint, it is considered as followed Guideline #1. (2)

The unit cells in which provided changes do not follow the guidelines. Guideline#1

asks to disconnect the unit cells from the side joint. If the two unit cells after change are still connected, it means it does not follow Guideline#1 no matter if other topology/shape changes occur. (3)

Sheets with blank spaces. Nothing is drawn in the given spaces.

6.7.3.1 Guideline#1 Results The results of Guideline#1, for honeycomb and rhombus, based on the numbers of unit cells in each of the three aforementioned categories are summarized in Table 6.6.

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Table 6.6: Results for Guideline#1 – Distribution of the number of shapes which are followed, not followed, and blank spaces for Guildeline#1 - The percent of shapes followed Guildeline#1 for honeycomb and rhombus

Honeycomb 40 39 1 0 97.50 3

Number of collected sheets Number of shapes followed GL#1 Number of shapes did NOT follow GL#1 Number of blank spaces Percentage of shapes followed GL#1 Number of clusters which have unit cells

Rhombus 39 38 1 0 97.44 4

For Guideline#1, there are 40 sheets collected for the honeycomb, and 39 sheets collected for the rhombus. As shown in Table 6.6, Guideline#1 was followed 97.50% of the time on the honeycomb and 97.44% of the time on the rhombus. In the second step, the unit cells sorted into the first category of the previous step, meaning that unit cells that followed the guideline, are evaluated based on the topology and shape similarities. The size or dimension of the sketches is not considered in the similarity metric evaluation because the guidelines only aim to change topology and shape of the unit cells. For this step, the unit cells are grouped into clusters, along with a short description of the topology and shape of the unit cells in the cluster. The result of this section is a number of clusters with a number of UCs after change in each cluster. As the number of cluster increases, the guideline is more subjective. The number of clusters for Guidleine#1 for honeycomb and rhombus is 3 and 4, respectively (Table 6.6). The different categories of unit cells identified for Guideline#1, for the honeycomb and the rhombus, are shown in Table 6.7.

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Table 6.7: Distribution of similar shapes which are followed Guildeline#1 with the number of their occurrences for honeycomb and rhombus

Honeycomb Illustration

Rhombus Illustration

A)

# 36

A)

# 33

B)

2

B)

3

C)

1

C)

1

D)

1

D)

Among the responses that followed Guideline #1, majority of the resulting unit cells were similar to shape A in Table 6.7; 89.74% for honeycomb and 86.84% for rhombus. This suggests that the application of Guideline#1 generated similar responses for the most part, and therefore the subjective interpretation of the guideline was similar for most of the participants. There are two shapes in Cluster B of honeycomb which are disconnected from the side joint as well as the vertical

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legs are also removed. There are three shapes in Cluster B of rhombus which has two disconnected rhombuses without middle transverse connections. In Table 6.7, there is also one shape in Cluster C, and one in Cluster D (without middle transverse connection) of rhombus which do not have vertical legs. These extra changes, removing middle transverse connection and/or vertical legs improve the shear flexure of unit cells as well. Numerical simulation results from previous research have shown removing transverse connections and/or vertical legs improve the shear flexibility of unit cells while maintaining the same effective shear modules [43,122]. 6.7.3.2 Guideline#2 Results Next, the responses for Guideline#2 were evaluated. A summary of the responses based on whether the guideline was followed or not is shown in Table 6.8. Table 6.8: Results for Guideline#2 – Distribution of the number of shapes which are followed, not followed, and blank spaces for Guildeline#2 - The percentage of shapes followed Guildeline#2 for honeycomb and rhombus

Honeycomb 38 29 6 3 76.32 9


Rhombus 39 35 4 0 89.74 8

For Guideline#2, 38 sheets were collected for the honeycomb, and 39 sheets for the rhombus. As shown in Table 6.8, Guideline#2 was followed 76.32% of the time on the honeycomb and 89.74% of the time on the rhombus. The number of clusters generated for Guidleine#2 for honeycomb and rhombus is 14 and 8, respectively (Table 6.6). The different shapes present in these clusters of unit cells are shown in Table 6.9 and Table 6.10. Among all the modified unit

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cells which followed Guideline #2, 72.41% of modified honeycomb unit cell, and 45.71% of modified rhombus unit cell have a shape and topology similar to that of shape A in Table 6.9, for honeycomb and rhombus, respectively. For, honeycomb, the most common modified unit cell was the result of removing the three transverse connections on the left side of the honeycomb. Similarly, for the rhombus, the most common modified unit cell was the result of removing the two transverse connection on the left, and the inside transverse connection. These modifications are expected as a result of following the guideline, and therefore suggest that a majority of the student participants correctly interpreted the subjective guideline.

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A)

# 21

A)

# 16

B)

1

B)

8

C)

1

C)

4

D)

1

D)

2

E)

1

E)

2

116

Table 6.10: Distribution of similar shapes which are followed Guildeline#2 with the number of their occurrences for honeycomb and rhombus – continued



F)

# 1

F)

# 1

G)

1

G)

1

H)

1

H)

1

I)

1

6.7.3.3 Guideline#3 Results Similar to Guideline#1 and Guideline#2, Guideline#3 was also applied on both honeycomb and rhombus unit cells. A summary of the responses based on whether the guideline was followed or not is shown in Table 6.11.

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Table 6.11: Results for Guideline#3 – Distribution of the Number of shapes which are followed, not followed, and blank spaces for Guildeline#3 - The Percentage of Shapes Followed Guildeline#3 for Honeycomb and Rhombus

Honeycomb 38 36 2 0 94.74 13


Rhombus 41 37 1 3 90.24 18

As shown in Table 6.11, a total of 38 sheets were collected for honeycomb unit cells, among which 94.97% of the unit cells followed the guideline. Similarly, 41 sheets were collected for the rhombus unit cells, and 90.24% of those were modified unit cells that followed the guideline. The modified unit cells that followed Guideline#3 can be classified based on shape and topology similarity, into 13 and 18 clusters, for honeycomb and rhombus, respectively. This high number of clusters suggests that although a majority of the student participants were able to follow the guidelines, there was a high level of subjectivity in the application of the guideline. The different clusters for both honeycomb and rhombus are shown in Table 6.12 and Table 6.13.

118




A)

# 16

A)

# 6

B)

9

B)

5

C)

1

C)

5

D)

1

D)

3

E)

1

E)

2

119




F)

# 1

F)

# 1

G)

1

G)

1

H)

1

H)

2

I)

1

I)

1

J)

1

J)

1

120

K)

1

K)

2

L)

1

L)

2

M)

1

M)

1

N)

1

O)

1

P)

1

121

Q)

1

R)

1

Among these clusters, two main clusters can be identified for the modified honeycomb unit cells: cluster A containing 44.44% of the total modified unit cells, and cluster B containing 25% of the total modified unit cells. However, for the rhombus unit cells, the top 3 clusters accounted for 43.24% of the total modified unit cells, and the remaining 15 clusters accounted for the rest. Similar to the honeycomb, this suggest that Guideline#3, albeit being followed more than 90% of the time, the modified unit cell generated have a high variety of shape and topology. 6.7.3.4 Guideline#4 Results Finally, the student responses for Guideline#4 were evaluated. A distribution of the responses is presented in Table 6.14. A total of 41 responses were collected for the honeycomb, 95.12% of which followed the given guideline. For the rhombus, 36 sheets were collected, out of which, 97.22% of the responses followed the guideline. As shown in Table 6.14, among the modified unit cells that followed the guideline, the responses were grouped into 3 clusters for the modified honeycomb unit cells, and 4 clusters for the modified rhombus unit cells. This clustering

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was done based on shape and topology similarity. Table 6.15 shows the different clusters identified from the modified unit cells collected for Guideline#4. Table 6.14: Results for Guideline#4 – Distribution of the number of shapes which are followed, not followed, and blank spaces for Guildeline#4 - The percentage of shapes followed Guildeline#4 for honeycomb and rhombus

Honeycomb 41 39 0 2 95.12 3


123

Rhombus 36 35 0 1 97.22 4




A)

# 37

A)

# 27

B)

1

B)

5

C)

1

C)

2

D)

1

D)

Among all shapes which are followed Guideline #4, 94.87% and 77.14% of the modified unit cells were similar to shape A (Table 6.15), for honeycomb and rhombus, respectively. This small number of clusters suggests that the subjective interpretation of the guideline does not vary largely between student participants. In comparison to Guideline#2 and Guideline#3, Guideline#1

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and Guideline#4 have much lower subjectivity, and produce similar resulting unit cell more consistently. 6.7.4 Results for sequencing effect of unit cell design guidelines In this section, the sequencing results are presented to evaluate the effect of applying the guidelines in a sequence. A sequence of two guidelines was used for this experiment, and compared to the results of the guidelines applied in the reverse order. The pairs tested were (12), (13), (14), (23), (24), and (34). 6.7.4.1 Results for Guideline#1 Guideline#2 The sequencing results for Guidelines# 1 => 2 and 2 => 1 for honeycomb and rhombus are presented in Table 6.16. Table 6.16: Sequencing results for Guidelines#1=>2 and 2=>1 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>2 and 2=>1 for honeycomb and rhombus

Number of collected sheets Number of shapes followed Guidelines Number of shapes did NOT follow Guidelines Percentage of shapes followed Guidelines Number of clusters which have unit cells

Honeycomb 1=>2 2=>1 9 15 5 10 4 5 55.56 66.67 4 3

Rhombus 1=>2 2=>1 14 12 11 11 3 1 78.57 91.67 3 4

The results show that for the honeycomb, 55.56% of the modified unit cells followed the guidelines when applied as 12, and 66.67% when applied as 21. Similarly, for the rhombus, the percentage of the modified unit cells that followed the guidelines were 78.57% and 91.67% for 12, and 21. The number of clusters identified for the modified unit cells based on shape and

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topology similarity are 3 or 4 for each set of modified unit cells as shown in Table 6.16. These different modified unit cells are shown in Table 6.17. Table 6.17: Sequencing results for distribution of similar shapes which are followed Guildeline#1=>2 and 2=>1 with the number of their occurrences for honeycomb and rhombus Honeycomb 1=>2 Illustration A )

# 1

Rhombus 2=>1 Illustration

# 5

1=>2 Illustration

2=>1 # 8

# 1

B )

2

4

1

1

C )

2

1

2

6

D )

1

3

For both initial unit cells, honeycomb and rhombus, three out of four clusters showed that the application of the two guidelines generated similar unit cells, regardless of the order of application. The three modified unit cells that were similar between guidelines 12 and guidelines 21 accounted for at least 83.33% and 72.27% of the total unit cells that followed the guidelines, for the honeycomb and rhombus respectively. This shows that the subjective interpretation of these guidelines has little variation with the sequence of applying the guidelines.

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5.7.4.2 Results for Guideline#1 Guideline#3 The results from sequential application of Guidelines#1 and Guideline#3 for honeycomb and rhombus are presented in Table 6.18. Table 6.18: Sequencing results for Guidelines#1=>3 and 3=>1 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>3 and 3=>1 for honeycomb and rhombus


Honeycomb 1=>3 3=>1 13 13 10 13 3 0 76.92 100 6 6

Rhombus 1=>3 3=>1 14 12 12 12 2 0 85.71 100 10 8

The results show that for the honeycomb 76.92% of the modified unit cells followed the guidelines when applied as 13, and 100% when applied as 31. Similarity, for the rhombus, 85.71% of the modified unit cells followed the guidelines as 13, and 100% as 31. The number of clusters identified for the modified unit cells based on shape and topology similarity are 6 for both cases of the honeycomb. However, for the rhombus, 13 yielded 10 different clusters, and 31 yielded 8 clusters (Table 6.18). The number of clusters identified for this sequence (13 and 31) is higher than that for 12 and 21, presumably, because of higher subjectivity of Guideline#3 when compared to Guideline#1 and Guideline#2. The different clusters generated for the modified unit cells are shown in Table 6.19.

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Table 6.19: Sequencing results for distribution of similar shapes which are followed Guildeline#1=>3 and 3=>1 with the number of their occurrences for honeycomb and rhombus Honeycomb 1=>3 Illustration


# 4

1=>3 Illustration

# 3

3=>1 Illustration

A)

# 2

B)

1

3

1

2

C)

1

2

1

2

D)

1

1

1

2

E)

1

1

1

1

F)

1

1

1

1

G)

1

1

1

1

H

1

1

128

# 2

I)

1

1

J)

1

As shown in Table 6.19, for honeycomb, 40% and 69.23% of the modified unit cells were in the top 3 clusters for 13 and 31 respectively. Similarly, for rhombus, 41.66% and 54.54% of the modified unit cells were in the top 3 clusters for 13 and 31 respectively. For honeycomb as an the initial unit cell, there are some similarities when changing the sequencing between Guidelines#1 and 3 as shown in clusters A, B, C, D, and E. However, for rhombus there are similarities in clusters A, C, D, and E. 5.7.4.3 Results for Guideline#1  Guideline#4 The results from sequential application of Guidelines#1 and Guideline#4 for honeycomb and rhombus are presented in Table 6.20.

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Table 6.20: Sequencing results for Guidelines#1=>4 and 4=>1 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>4 and 4=>1 for honeycomb and rhombus


Honeycomb 1=>4 4=>1 13 13 12 10 1 3 92.31 76.92 2 1

Rhombus 1=>4 4=>1 10 12 10 12 0 0 100 100 3 2

The results show that for the honeycomb 92.31% of the modified unit cells followed the guidelines when applied as 14, and 76.92% when applied as 41. Similarity, for the rhombus, 100% of the modified unit cells followed the guidelines as 14, and 100% as 41. The number of clusters identified for honeycomb are 2 for (14) and 1 for (41). Similarly, the number of clusters identified for rhombus are 3 for (14) and 2 for (41). The number of clusters identified in this case are lower number of clusters for (12) and (23) because both Guideline#1 and Guideline#4 are less subjective than Guideline#2 and Guideline#3, which is concluded in the subjectivity experiment. The different clusters of modified unit cells identified based on shape and topology similarity are shown in Table 6.21.

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Table 6.21: Sequencing results for distribution of similar shapes which are followed Guildeline#1=>4 and 4=>1 with the number of their occurrences for honeycomb and rhombus Honeycomb 1=>4 Illustration A)

# 11

B)

1


# 10

1=>4 Illustration

# 6

4=>1 Illustration

3

C)

# 10

2

1

As shown in Table 6.21, 100% of the modified unit cells are accounted for by the top 3 clusters in each case, for both, the honeycomb and the rhombus. The top cluster accounts for 91.67% and 100% of total modified honeycomb unit cells, for 14 and 41 respectively. 5.7.4.4 Results for Guideline#2  Guideline#3 The results from sequential application of Guidelines#2 and Guideline#3 for honeycomb and rhombus are presented in Table 6.22. .

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Table 6.22: Sequencing results for Guidelines#2=>3 and 3=>2 – Distribution of the Number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#1=>4 and 4=>1 for honeycomb and rhombus


Honeycomb 2=>3 3=>2 13 11 10 8 3 3 76.92 72.73 4 6

Rhombus 2=>3 3=>2 15 10 14 10 1 0 93.33 100 7 4

The results show that for the honeycomb 76.92% of the modified unit cells followed the guidelines when applied as 23, and 72.73% when applied as 32. Similarity, for the rhombus, 93.33% of the modified unit cells followed the guidelines as 23, and 100% as 32. The number of clusters identified for honeycomb are 4 for (23) and 6 for (32). Similarly, the number of clusters identified for rhombus are 7 for (23) and 4 for (32). The number of clusters identified in this case are higher than the number of clusters for (12) and (14) because both Guideline#1 and Guideline#4 are less subjective than Guideline#2 and Guideline#3, which is concluded in the subjectivity experiment. The different clusters identified for this case are shown in Table 6.23.

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# 1

2=>3 Illustration

A)

B)

3

3

2

3

C)

1

1

1

3

D)

1

1

3

1

E)

1

1

F)

1

1

G)

# 5

3=>2 Illustration

# 5

# 3

1

In the case of honeycomb unit cells, the top 3 clusters account for 90% and 62.50% of the total modified unit cells that followed the guidelines, for (23) and (32) respectively. Similarly,

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for the rhombus unit cells, the top 3 clusters account for 57.14% and 90%, for (23) and (32) respectively. For honeycomb, there are similarities in clusters A and B; and for rhombus, similarities are found in clusters A, B, and C. However, for both honeycomb and rhombus, there are some dissimilarities in D, E, F, and G clusters. 5.7.4.5 Results for Guideline#2  Guideline#4 The results from sequential application of Guidelines#2 Guideline#4 for honeycomb and rhombus are presented in Table 6.24. Table 6.24: Sequencing results for Guidelines#2=>4 and 4=>2 – Distribution of the number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#2=>4 and 4=>2 for honeycomb and rhombus


Honeycomb 2=>4 4=>2 15 12 7 7 8 5 46.67 58.33 2 2

Rhombus 2=>4 4=>2 12 12 11 12 1 0 91.67 100 5 4

The results show that for the honeycomb, 46.67% of the modified unit cells followed the guidelines when applied as (24), and 58.83% when applied as (42). Similarly, for the rhombus, 91.67% and 100% followed the guidelines for (24) and (42) respectively. Two clusters, based on shape and topology similarity, were identified for the modified honeycomb unit cells in both cases of (24) and (42). For the modified rhombus unit cells, 5 clusters were identified for (24) and 4 clusters for (42). The low number of clusters suggests that the subjectivity interpretation of the guidelines has little variation between participants, whereas the number of clusters being similar between the (24) and (42) suggests that the subjectivity of

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the guidelines does not change with the order in which these two guidelines are applied. The different clusters of modified unit cells identified for this case are shown in Table 6.25. Table 6.25: Sequencing results for distribution of similar shapes which are followed Guildeline#2=>4 and 4=>2 with the number of their occurrences for honeycomb and rhombus Honeycomb 2=>4 Illustration A)

# 6

B)

1


# 5

2

2=>4 Illustration

# 5

4=>2 Illustration

# 7

3

3

C)

1

1

D)

1

1

E)

1

In the case of honeycomb unit cells, the top 3 clusters account for 100% of the modified unit cells that followed the guidelines, in both cases. Similarly, for the rhombus unit cells, the top 3 clusters account for 81.81% and 91.66%, for (24) and (42) respectively. For honeycomb, there are similarities in clusters A; and for rhombus, similarities are found in clusters A and B. However, there are some dissimilarities in B for honeycomb and C, D, and E clusters for rhombus.

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5.7.4.6 Results for Guideline#3  Guideline#4 The results from sequential application of Guideline#3 and Guideline#4 for honeycomb and rhombus are presented in Table 6.26. Table 6.26: Sequencing results for Guidelines#3=>4 and 4=>3 – Distribution of the number of shapes which are followed, not followed, and blank spaces - The percentage of shapes followed guildeline#3=>4 and 4=>3 for honeycomb and rhombus


Honeycomb 3=>4 4=>3 15 7 14 6 1 1 93.33 85.71 7 3

Rhombus 3=>4 4=>3 17 11 17 11 0 0 100 100 7 3

The results show that for the honeycomb, 93.33% and 58.71% of the modified unit cells followed the guidelines when applied as (34) and (43), respectively. For the rhombus, 100% of the modified unit cells followed the guidelines in both case. Based on shape and topology similarity, 7 clusters were identified for the modified unit cells when the guidelines were applied as (34), and 3 when applied as (43), for both honeycomb and rhombus unit cells. This distribution of clusters suggests that using Guideline#4 before Guideline#3 results in a higher level of similarity between the modified unit cells. This is consistent with the subjectivity results that show that Guideline#4 is less subjective than Guideline#3 The different clusters identified for each case of applying Guideline#3 and Guideline#4 are shown in Table 6.27.

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# 4

3=>4 Illustration

# 8

4=>3 Illustration

A)

# 8

# 8

B)

1

1

3

2

C)

1

1

1

1

D)

1

2

E)

1

1

F)

1

1

G

1

1

2

In the case of honeycomb unit cells, the top 3 clusters account for 71.43% and 100% of the total modified unit cells that followed the guidelines, for (34) and (43) respectively. Similarly, for the rhombus unit cells, the top 3 clusters account for 70.59% and 100%, for (34) and (43)

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respectively. For honeycomb, there are similarities in clusters A and B; and for rhombus, similarities are found in clusters A, B, and C. However, for both honeycomb and rhombus, there are some dissimilarities in D, E, F, and G clusters. Advanced manufacturing technologies can be used to manufacture these novel unit cells developed in two previous user experiments [123–132]. 6.7.5 Overall Sequencing Results Different sequences of guideline application were tested to identify any cases where the order in which the guidelines were applied had an impact on the subjective interpretation of the guidelines. Additionally, cases where the application of certain guidelines before other guidelines were also of interest. Table 6.28 shows the percentage of student participant that followed the guidelines in each case, and the percentage of the modified unit cells present in the top 3 clusters in each case. Table 6.28: Percent of modified unit cells accounted for in the top three clusters

Honeycomb

Rhombus

Case

Original

Reversed

Original

Reversed

12

83.33%

100%

100%

72.77%

13

40%

69.23%

41.66%

54.54%

14

100%

100%

100%

100%

23

90%

62.50%

57.14%

90%

24

100%

100%

81.81%

91.66%

34

71.43%

100%

70.59%

100%

6.8 Conclusion and Future Work

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Two user studies are conducted to explore the effect of subjectivity and sequencing of four unit cells design guidelines to improve shear flexibility of unit cells. The results of subjectivity study show Guidelines#2 (removing the transverse connections) and #3 (using the curved beams) are more subjective than Guidelines#1 (disconnecting the unit cells from the side joint) and #4 (removing the vertical legs).

Additionally, more clusters or more dissimilar shapes for

Guidelines#2 and #3 than #1 and #4 are due to how the guidelines are written and each individual guideline by itself. Guideline#2 recommends to remove transverse connections. However, it does not tell users which one(s) of transverse connections should be removed. The honeycomb and rhombus unit cells have four and five transverse connections, respectively.

Guideline#3

recommends to replace the straight edges of the unit cells with curved beams. It also added examples of curved beams: a circle, a semi-circle, and an S-shape. However, it does not tell which ones of straight edges should be replaced with which curved beams. This is the main reason to increase subjectivity and dissimilarity among developed shapes for Guidelines#2 and 3. This subjectivity and dissimilarity can lead to create more diverse developed shapes which some of them have not been existed before. The results of sequencing show there are some similarities among developed shapes. However, some dissimilarities are also found due to the sequencing. The sequencing matters in cases one of two guidelines or the both are the identified subjective Guidelines (#2 and/or #3). In these cases, more clusters are identified from the collected shapes. However, in the sequence of more objective guidelines (#1 and #4) a few clusters are identified.

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The effect of subjectivity with respect to the number of guidelines applied in sequence is considered as a continuation of this research. These guidelines are human-driven design tools [133] versus computational design tools that are generally intended to automate problem solving in an algorithmic manner [58,134].

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CHAPTER 7 CONCLUSION AND FUTURE WORK

This chapter reviews answering the research questions in section 7.1, the concluding reflections in section 7.2 and limitations and future work in section 7.3. 7.1 Answering the Research Questions In this section, the research questions discussed in Chapter Chapter 1 are answered as follows: Research Question 1 (RQ 1). How can unit cell design guidelines be developed? Research Hypothesis 1 (RH 1). Unit cell design guidelines can be developed through the Unit Cell Design Guideline Development Method. Null Hypothesis 1 (RH0 1). Unit cell design guidelines cannot be developed through the Unit Cell Design Guideline Development Method. In Chapter Chapter 4, four unit cell design guidelines are developed to increase the shear flexibility of unit cells while keeping the effective shear modulus constant. The improvements are achieved by modifying the topology and shape of the unit cells. By rejecting the RH0 1, the main hypothesis (RH 1) was supported. In this case, the main hypothesis is supported two times (Guidelines#3 and#4) and the other two unit cell design guidelines (Guidelines#1 and#2) were also followed the development guideline process which are discussed in Chapter Chapter 4. Furthermore, the results of case study on how other engineers are developing unit cell design guidelines in Chapter 5 rejected the RH0 1 and supports RH 1.

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Research Question 2. Are unit cell design guidelines subjective? Research Hypothesis 2. There are similarities between developed topologies. Low subjectivity is expected. Null Hypothesis 2. There is no similarities between developed topologies. High subjectivity is expected. The subjectivity of guidelines is evaluated through a user study as discussed in Chapter Chapter 6. The results showed all four guidelines generated some (partially) similar modified unit cells. By showing this, the null hypothesis, RH0 2, was rejected. Rejecting RH0 2 supports Research Hypothesis 2 (RH 2). However, the number of clusters in Section 6.7.3 show Guidelines #1 and #4 are less subjective and Guidelines #2 and #3 are more subjective. Research Question 3. Does sequencing of the unit cell design guidelines matters with regards to the outcome? Research Hypothesis 3. Sequencing will result in similar topology outcomes. Null Hypothesis 3. Sequencing will not result in similar topology outcomes. The sequencing of guidelines is evaluated through a user study as discussed in Chapter Chapter 6. The results in Section 6.7.4 showed all twelve combinations of guidelines generated some (partially) similar modified unit cells. However, the level of similarities represented with the number of clusters is different among twelve cases. The results reject RH0 3 and support the Research Hypothesis 3 (RH 3).

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7.2 Concluding Reflections Three aspects of this dissertation are original: (1) a vocabulary to describe 2-D periodic cellular materials, (2) unit cell design guidelines that are seen for the first time in the area of material design, and (3) more importantly, the method to develop unit cell design guidelines. The classification has developed for periodic cellular material representation.

The

classification is made of a vocabulary with set of descriptors and classified in two levels to represent a PCM. The first level represents the unit cell and the tiling. The unit cells are represented by the boundary, the connection, and the interior; and tiling is sub-categorized by the simply connected and non-simply connected. The second level has more descriptors to represent a unit cell explicitly. The internal validation of the classification was evaluated through three metrics: completeness, perceptual orthogonality, and usability. An external validation of the vocabulary was conducted through a designer study to assess how engineers use the vocabulary to represent PCMs. The vocabulary helps engineers and researchers to unambiguously describe twodimensional periodic cellular materials. Four UCD guidelines are developed to improve the shear flexure of MPCM without changing effective shear modulus. The first two guidelines are inspired through the case study [15] which suggested that disconnecting the UC from its side joints and removing transverse connections would improve the shear flexure of UCs. Based on the first two guidelines, the UCDGD method is proposed and demonstrated by developing two new guidelines. The second two guidelines recommend eliminating the vertical legs and using curved beams to increase the

143

shear flexure. As a result of applying the guidelines, the shear flexure is improved in a range between 15-131% with the average of 62%. The results of subjectivity study show Guidelines#2 (removing the transverse connections) and 3 (using the curved beams) are more subjective than Guidelines#1 (disconnecting the unit cells from the side joint) and 4 (removing the vertical legs). Additionally, more clusters or more dissimilar shapes for Guidelines#2 and 3 than #1 and 4 are due to how the guidelines are written and each individual guideline by itself.

Guideline#2 recommends to remove transverse

connections; however it does not tell users which one(s) of transverse connections should be removed. The honeycomb and rhombus unit cells have four and five transverse connections, respectively. Guideline#3 recommends to replace the straight edges of the unit cells with curved beams. It also added examples of curved beams: a circle, a semi-circle, and an S-shape. However, it does not tell which ones of straight edges should be replaced with which curved beams. This is the main reason to increase subjectivity and dissimilarity among developed shapes for Guidelines#2 and 3. This subjectivity and dissimilarity can lead to create more diverse developed shapes which some of them have not been existed before. The results of sequencing show there are some similarities among developed shapes. However, some dissimilarities are also found due to the sequencing. The sequencing matters in cases one of two guidelines or the both are the identified subjective Guidelines (#2 and/or 3). In these cases more clusters are identified from the collected shapes. However, in the sequence of more objective guidelines (#1 and 4) a few clusters are identified.

144

7.3 Limitations and Future Work Formalizing the representation of UCs facilitates UC design guideline development. For example, changing the type of connections from vertex to edge, the shape of UC boundary, and tiling of UCs can potentially lead to new UC design guidelines, which is identified as a research opportunity [43]. The given geometrical parameters and vocabulary in the classification can be used to explore new unit cells and more importantly help in developing design guidelines through changing or adjusting each vocabulary. Two UC design guidelines with their abstractions are developed and represented in [43]. The design guidelines which is identified as a gap for future research are aimed to help engineers in developing new PCMs for targeted effective properties. One limitation of the Unit Cell Design Guideline Development method is that selecting an initial unit cell and a topology characteristic to change is still a heuristic step. Another limitation is that the guidelines are only guaranteed to be effective for the four cases that were actually simulated. In choosing different test cases, the developer seeks to extend the applicability of the guidelines. Further, it is recognized that simulating two UCs is not the same as simulating the complete MPCMs, but the results of general trends are indicative and suggestive. In modifying the UC, a designer can populate the material region of interest and optimize the sizes. More research is needed to determine the limitations of modeling UC’s to obtain MPCM results. Different loading scenarios such as compression and tension, and other effective properties of MPCMs may be explored to develop unit cell design guidelines in future work.

The

combination of different loading scenarios, for example, shear and compression, can also be

145

considered in future work. The four guidelines presented in this dissertation are a demonstration of the UCDGD method; however, further validation of the UCDGD method can be considered in future work. The validation square for design methods consists of four phases and six steps is considered to validate the UCDGD method [61]. Four phases include theoretical structural validity, empirical structural validity, empirical performance validity, and theoretical performance validity. The structural validation consists of three steps: (1) obtaining methods’ constructs validity, individually and integrated; (2) obtaining method’s consistency; (3) obtaining the example problems. The performance validation consists of three steps: (4) obtaining suitability of method for some example problems; (5) obtaining that suitability is related to applying the method; (6) obtaining suitability of method beyond example problems [61]. Among these six steps, obtaining method’s consistency, Step#2, is conducted through several guideline development demonstrations. The rest of the steps are left for future work. In the sequencing user study only two guidelines were applied each time. However, it is not clear what would be the effect of sequencing if this number increases to three or four. Therefore, the effect of subjectivity with respect to the number of guidelines applied in sequence is considered as a continuation of this research.

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