Bio Inspired Lightweight Composite Material Design for 3D Printing Kaushik Thiyagarajan South Dakota State University

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Theses and Dissertations

2017 Bio Inspired Lightweight Composite Material Design for 3D Printing Kaushik Thiyagarajan South Dakota State University

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Recommended Citation Thiyagarajan, Kaushik, "Bio Inspired Lightweight Composite Material Design for 3D Printing" (2017). Theses and Dissertations. 1703. http://openprairie.sdstate.edu/etd/1703

This Thesis - Open Access is brought to you for free and open access by Open PRAIRIE: Open Public Research Access Institutional Repository and Information Exchange. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Open PRAIRIE: Open Public Research Access Institutional Repository and Information Exchange. For more information, please contact [email protected]. BIO INSPIRED LIGHTWEIGHT COMPOSITE MATERIAL DESIGN FOR 3D PRINTING

Kaushik Thiyagarajan

A thesis submitted in partial fulfillment of the requirements for the

Master of Science

Major in Mechanical Engineering

South Dakota State University

2017

iii

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to my advisor Dr. Zhong Hu, for his guidance, knowledge, enthusiasm, support and most importantly his kindness and patience with me. Right from very first semester till my last semester, I have no words to express my gratitude for encouragement and kindness that he has given me throughout my graduate studies. I am grateful to have the opportunity to work with Dr. Letcher and being able to carry out my research work with an excellent mentor like him. He has been encouraging me from the very beginning and always tried to clear my misperceptions with an intellectual explanation. I would also like to thank South Dakota School of Mines and Technology CNAM center for supporting the research with necessary funds. Finally, my deepest thanks goes to State of South Dakota for creating such an innovation platform like CNAM center for aspiring scholars, professors and industries in accelerating technology.

I am so much grateful to the Mechanical Engineering Department at South

Dakota State University which has given me continuous financial support during my graduate study.

I am thankful to my parents for their faith in me and they have always supported me to pursue my dreams and desire. Without their consent and help, I wouldn’t be able to be here. My special thanks go to Dr. Michna supporting and believing me and honoring me as Teaching assistant for continued semesters between (Fall 2015- Spring 2016)

I strongly believe only because of all the above-mentioned people and departments; I was able to finish my study in best possible way.

CONTENTS

ABSTRACT ...... xii

Introduction ...... 1

Motivation ...... 3

Literature Review ...... 8

Hypothesis ...... 17

Material and Methods ...... 18

Fusion Deposition Molding ...... 19

Homogenization Theory ...... 23

3D Tessellation ...... 23

Representative Volume element...... 25

Characterization of bulk composites ...... 29

Visualization of manufacturing process ...... 29

Modeling methodologies ...... 30

Computer model generation ...... 32

Fiber definition ...... 34

Random Number generation ...... 35

Model simulation...... 37

A model for PLA reinforced Carbon Fiber ...... 43

Characterization of Cellular composites ...... 51

Identification of RVEs ...... 51 v

Characterization of RVEs by modeling ...... 55

RESULTS ...... 56

Properties from bulk composites ...... 56

Properties from Cellular Structures ...... 60

Verification ...... 68

Conclusion ...... 77

Literature cited ...... 78

ABBREVIATIONS

AM Additive Manufacturing mm millimeter

N Newton

GPa Giga Pascal

MPa Mega Pascal

ν Poisson’s ratio

σ Stress

ε Strain

E Young’s Modulus

PLA Polylactic Acid

DiFs Discontinuous Fibers vii

LIST OF FIGURES

Figure 1. Percentage distribution of material used in Boeing 787…..……...…..…………1

Figure 2. Local Motors 3D printed car with CFRP material…..……...…..………………3

Figure 3. Honeycomb Light Weight Seating Structure…..……...…..……………………4

Figure 4. Hierarchical structure of bone (a) and bamboo (b) …..……...…..……………..9

Figure 5. 3D anatomy for loblolly pine (a) …..……...…..………………………………10

Figure 6. SEM images displaying morphological variations for two different samples of:

(a) Cuttlebone 1 and (b) Cuttlebone 2) …..……...…..…………………………………..10

Figure 7. Skin–core, fibril bundle structure of spider silk. ( a) Schematic of the multi-fibril core of dragline silk that constitutes a spider web, on the order of micrometers………..11

Figure 8. Bee’s honeycomb as grown state …..……………………………...... 11

Figure 9. Bee’s honeycomb section view…..……………………………...... 12

Figure 10. Italian honeybee (Apis mellifera Ligustica) comb cell at (a) ‘birth’, and at (b)

2-days old, scale bar is 2 mm…..……...…..…………………………………...... 12

Figure 11. Section of bone structure from a vulture wing…..…………………………...13

Figure 12. Two-step mean-field homogenization procedure: The RVE is decomposed into a set of pseudo-grains, each of which is then individually homogenized (first-step).

Finally, the second homogenization is performed over all the pseudo-grains (second- step…………………………………………….…..……………………………...... 15

Figure 13. Polished cross section in the (y, z) plane……………………………………..17

Figure 14. Flow chart of development of a cellular structural composites………………18

Figure 15. Infill pattern for varying Fill density…..……………………………...... 19

Figure 16. A picture from local motors showing the fabrication of the 3D pinted car .....21

Figure 17. Infill pattern for varying Fill density…………………………………………22 viii

Figure 18. Infill pattern inspired from bone……………………………………………...22

Figure 19. 3D Tessellation…..……...…..…………………………………...... 24

Figure 20. 3D Tessellation……………………………………………………………….24

Figure 21. Array of truncated octahedron………………………………………………..25

Figure 22. A composite stucture with semi-circular fibers and matrix…………………..26 Figure 23. REV of above larger structure …..……………………………...... 26

Figure 24. REV of solid block 100% occupied…..……………………………...... 27

Figure 25. REV of truncated octahedron 73% occupied………………………………...27

Figure 26. REV of truncated octahedron 65% occupied………………………………...28

Figure 27. REV of truncated octahedron 55% occupied………………………………...28

Figure 28. REV of truncated octahedron 45% occupied………………………………...29

Figure 29. Two-step approach for bulk composite characterization: (a) step one: unidirectional fiber distribution, and (b) step two, elements with 3D random fiber distribution by Monte Carlo method……………..……………………………...... 31

Figure 30. One-step approach for bulk composite characterization: (a) 3D random fiber distribution, and (b) bulk composite 3D model……………..…………………………...31

Figure 31. Flow chart of development of random fiber placement pattern inside a bulk material……………..……………………………...... 33

Figure 32. Fiber Definition……………..……………………………...... 34

Figure 33. X coordinate described by random numbers………………………………....36

Figure 34. Y coordinate described by random numbers………………………………....36

Figure 35. Z coordinate described by random numbers………………………………....37

Figure 36. Line view of the REV with random distribution of SF in matrix – Front View……………………………………………………………………………………...38

Figure 37. Line view of the REV with random distribution of SF in matrix – Isometric View……………………………………………………………………………………...39

Figure 38 Volume plot of the REV with matrix material only ………………………….40 ix

Figure 39. Volume plot of the fibers only………………………………...... 41

Figure 40. Volume plot of the fibers only – Front View………………………………...42

Figure 41. View showing the meshing of the fiber materials……………………………43

Figure 42. Volume plot of the REV with fibers and matrix……………………………..45

Figure 43. Line plot of REV showing the randomly distributed fibers……………….....46

Figure 44. Volume plot of the random fibers…………………………………………....47

Figure 45. Volume plot of the matrix…………………………………………...... 48

Figure 46. Volume Mesh of REV with elements………………………………………..49

Figure 47. Material plot of REV clearly distinguish fiber material from the matrix material…………………………………………………………………………………..50

Figure 48. (a) Honeycomb, (b) 2D hexagon model and (c) REV for Honeycomb……...51

Figure 49. (d) Image based finite element modelling for homogenization……………...52

Figure 50. (a) Cuttlefish bone, (b) 2D cuttlefish bone model and (c) REV for cuttlefish bone……………………………………………………………………………………...53

Figure 51. (a) 3D truncated octahedron, (b) 3D truncated octahedron model and (c) REV of truncated octahedron model…………………………………………………………..54

Figure 52. Volume plot of the 2D hexagonal array……………………………………...55

Figure 53. Volume Mesh of 2D hexagonal array………………………………………..56

Figure 54. Tensile tests on bulk composites in printing direction and transverse to the printing direction: (a) PLA polymer, and (b) CFRPLA composites……………………..58

Figure 55. Compression tests on bulk composites in printing direction and transverse to the printing direction: (a) PLA polymer, and (b) CFRPLA composites…………………58

Figure 56. Digital microscopy images of carbon fiber orientations in 3D printed specimens: (a) along printing direction, and (b) transverse to the printing direction……59

Figure 57. Design of a two-dimensional hexagonal array topology: (a) without fillet, and

(b) with fillet……………………………………………………………………………..61 x

Figure 58. RVEs of the 2D hexagonal topology: (a) without fillet, and (b) with fillet...61

Figure 59. RVEs of the 2D Cuttlefish topology: (a) without fillet, and (b) with fillet…62

Figure 60. RVEs of the 3D Truncated Octahedron topology: (a) without fillet, and (b) with fillet………………………………………………………………………………..62

Figure 61. Young’s modulus for hexagon 2D model: (a) Ez (thickness), (b) Ex (planar), and (c) Ey (planar)………………………………………………………………………64

Figure 62. Young’s modulus for cuttlefish bone 2D model: (a) Ez (thickness), (b) Ex

(planar), and (c) Ey (planar)…………………………………………………………...... 65

Figure 63. Young’s modulus of 3D octahedron model: (a) Ex, (b) Ey, and (c) Ez……..66

Figure 64. Von Mises stress of 3D octahedron model: (a) by X-axis tensile test, (b) by Y- axis tensile test, and (c) by Z-axis tensile test…………………………………………..67

Figure 65. 3D printed 2D hexagon samples: (a) PLA, and (b) CFRPLA……………….68

Figure 66. Compression tests for hexagon model in Y-axis (vertical): (a) PLA, and (b) CFRPLA…………………………………………………………………………………69

Figure 67. Stress and strain curve from compression tests for hexagon model of 50% volume occupied: (a) PLA, and (b) CFRPLA…………………………………………...70

LIST OF TABLES

Table 1. Mechanical Properties of Structural Materials and Fibers .. …………..…………5

Table 2. Materials used for fabricating FRC in Stereo-lithography (SL)………………..20

Table 3. Summary of fiber configurations that are used in study………………………..32

Table 4. Properties of the individual Carbon Fiber and PLA Polymer resin used to construct bulk material model……………………………………………………………44

Table 5. Young's moduli of bulk composites reinforced with 15 vol% DiCF (z-axis: fiber direction for unidirectional fiber composites)……………………………………………57

Table 6. Bulk material properties by testing (15% DiCF in volume)……………………59

Table 7. Compressive Young’s modulus (GPa) of the hexagon structures by testing and modeling…………………………………………………………………………………70

Table 8. Summary of the specific stiffness for considered designs in thickness direction

(Ez – Young’s Modulus in Z direction for the 2D and 3D Topology) ………………….71

Table 9. Summary of the specific stiffness for considered designs in planar direction

(Ey – Young’s Modulus in Z direction for the 2D and 3D Topology) ………………….72

Table 10. Summary of the specific stiffness for considered designs in planar direction

(Ex – Young’s Modulus in Z direction for the 2D and 3D Topology) ………………….73

Table 11. Summary of the specific Young's modulus for some materials….…………...74

xii

ABSTRACT

BIO INSPIRED LIGHTWEIGHT COMPOSITE MATERIAL DESIGN FOR 3D

PRINTING

KAUSHIK THIYGARAJAN

2017

Lightweight material design is an indispensable subject in product design. The lightweight material design has high strength to weight ratio which becomes a huge attraction and an area of exploration for the researchers as its application is wide and increasing even in every day-to-day product.

Lightweight composite material design is achieved by selection of the cellular structure and its optimization. Cellular structure is used as it has wide multifunctional properties in addition to the lightweight characteristics. Applications of light weight cellular structures are wide and is witnessed in all industries from aerospace to automotive, construction to product design.

In this thesis, the one-step and two-step approaches for design and prediction of cellular structure's performance are presented for developing lightweight cellular composites reinforced by discontinuous fibers. The topology designs of a 2D honeycomb hexagon model, a 2D cuttlefish model, and a 3D octahedron model, inspired by bio material, are presented. Computer modeling based on finite element analysis was conducted on the periodic representative volume elements identified from the cellular structural models to characterize the designed cellular composites performance and properties. Additive manufacturing technique (3D printing) was used for prototyping the design, and experimental tests were carried out for validating the design methodology. 1

INTRODUCTION

The twenty first century is witnessing the most powerful technologies ever thought of. One such technology is Carbon Fiber Reinforced Plastic (CFRP) composite materials. CFRP materials are unique and is widely recognized for its g strength to weight ratio, very low coefficient of thermal expansion and few other properties like high resistance to chemical, biological changes [1]. Light weight quality of CFRP resulted in wide variety of applications in Transport, Construction, and Military to Sports Industry.

In Boeing 787 50% use of CFRP has resulted in 20% weight saving in comparison to conventional aluminum design [2]. The below picture show the percentage distribution of materials used in Boeing aircraft.

Figure 1. Percentage distribution of material used in Boeing 787.

Other technology is Additive Manufacturing process (AM) under Rapid

Prototyping (RP). AM process harvests the full-fledged Computer and Mechanical

Technology. 3D printing has now become a sustainable source for the efficient and flexible manufacturing for both prototype and mass manufacturing. 3D process can materials ranging from polymer, metals, ceramics, and composites. Composites are used in RP not only to make desired product but also to facilitate the process [3]. The peek application is it witnessed in the automotive industry where the CFRP material is used with AM process. Complex geometry components and assemblies can be manufactured with cellular meso-structure as single component or as an assembly which is impossible with traditional subtractive machining. Processing technologies are rapidly advancing and manufacturers now have the ability to control material architecture, or topology, at unprecedented length scales. This expands the design space and provides exciting opportunities for tailoring material properties through design of the material's topology.

Using a 3D additive manufacturing (AM) technology to fabricate test samples to verify computational model is a new idea, especially for printing composite materials with stiff fillers (e.g. continuous or discontinuous carbon fibers, tiny silicon carbide “whiskers”, carbon nanotubes, or graphene) within a soft matrix (polymer) [4-7]. Of particular significance is the way that fibers can be aligned, through control of the fiber aspect ratio (the length relative to the diameter) and the nozzle diameter. This marks an important step forward in designing engineering materials. On such example is local motors has already exhibited the prototype model of the 3D printed CFRP car as seen in below figure [8]. 3

Figure 2. Local Motors 3D printed car with CFRP material.

Motivation

Structures that we see in nature has evolved over several years such that it becomes strong and adaptive to given environment. This practical approach of referring from nature results us with a good start platform where we have a durable and reliable design which we can use an exoskeleton and tailor better properties and functionalities for our application. Nature inspired architecture is becoming more famous and excellent way to sort the sustainable structures. Designers can design the products like chair with all new creative possibilities as shown in below figure, 4

Figure 3. Honeycomb Light Weight Seating Structure [9]. Scientists are fascinated to observe and study the cellular structures of bio materials for past few decades. Cellular materials offer high strength-to-weight ratio, high stiffness, high permeability, excellent impact-absorption, and thermal and acoustic insulation [10]. Lightweight cellular composites, composed of an interconnected network of solid struts that form the edges or face of cells [10], are an emerging class of high performance structural materials that may find potential application in high stiffness sandwich panels, energy absorbers, catalyst support, vibration damping, and insulation

[11-17]. Cellular composites provide advantage of having a porous structure design and ability to alter our own property as a composite. Cellular composites are of significant interest due to their wide applications in lightweight structural components and thermal structural materials, and have the potential to revolutionize aerospace systems and capability [18]. The conventional way of CFRP manufacturing is tedious in comparison to the 3D printing. Also, the 3D printing allows the composite product to have a new 5 degree of freedom that conventional way of composite manufacturing lack. Today the need for having a light weight high strength material is increasing in exponential manner.

The carbon fiber reinforcement will reduce the weight up to 50 percent lighter that steel

[19]. The light weight property is helpful for the dynamic components like motors, shafts, turbine blades, automobiles, etc. In these key industries, a lot of fuel can be saved as lighter weight is needed to propel. Also, lightweight materials are gaining a lot of attention in sports, construction industry since fixing and moving these components are easier to conventional metal alloy structures.

Nature has a lot of light weight structural designs like human bone, cuttlefish bone, bamboo, and wood, silk and honeycomb, etc. The below table list few of the material properties of natural fibers, synthetic fibers, polymers that are widely used in composite material industry with conventional metals.

Table 2. Mechanical Properties of Structural Materials and Fibers [1].

Maximum Maximum Ultimate Young's Density Specific Specific Materials Tensile Modulus ρ Strength Modulus Stress σ (MPa) E (GPa) (g/cm3) 3 3 Kσ x 10 (m) KE x 10 (m)

Natural Fibers

Wood 160 23 1.5 24.5 1564.6

Bamboo 550 36 0.8 44.9 4591.8 0.40 -0.52 Cuttlefish 0.67 (Compressive) 4.4-8.0 0.036 1218.4 Bone [21] [20] 1.5 -9.3 Cancellous (Compressive) 0.01-1.57 1.0-1.4 0.9 160.2 Bone [9] & 1.5-28 (Tensile) Transverse Cortical 35 (Tensile) & 5 to 23 1.8-2.0 51.9 4.2 Bone [22] 160 (Compressive) 6

Longitudinal 240 (Compressive) & 283 (Tensile) Jute 580 22 1.5 88.7 1496.6 Cotton 540 28 1.5 82.6 1904.8 Wool 170 5.9 1.32 22.9 456.1 Natural silk 400 13 1.35 55.0 982.6 Spider silk 1750 12.7 1.097 195.7 1181.3

Fibers for advanced composites (diameter, micro-meter) Glass (3-9) 3100-5000 72-95 2.4-2.6 1325.2 3728.4 Carbon (5-11) High 7000 300 1.75 1248.7 17492.7 Strength High 2700 850 1.78 489.9 48727.4 Modulus Boron (100 - 2500-3700 390-420 2.5-2.6 980.6 16483.5 200)

Thermoplastic Polymers Polyester 60 2.5 1.32 4.6 193.3 (PC) Polysulfone 70 2.7 1.24 5.8 222.2 (PSU) Polyamide - 90-190 2.8-4.4 1.42 13.7 316.2 imide (PAI) Polyetheret herketone 90-100 3.1-3.8 1.3 7.8 298.3 (PEEK) Polylactic Acid (PLA) 53 3.5 1.25 8.2 285.7 [23]

Thermoset Polymers Epoxy 60-90 2.4-4.2 1.2-1.3 7.1 329.7 Polyester 30-70 2.8-3.8 1.2-1.35 5.3 287.2

Structural Materials - Metal Alloys Steel 400 -2200 180-210 7.8-7.85 28.8 2747.3 Aluminum 140-700 69-72 2.7-2.85 26.5 2721.1 Titanium 420-1200 110 4.5 27.2 2494.3

Structural Materials - Meatal wires (diameter, micro-meter) Steel (20 - 1500-4400 180-200 7.8-7.85 57.6 2616.4 1500) Aluminum 290 69 2.7-2.85 11.0 2607.7 (150) Titanium 1400-1500 120 4.5 34.0 2721.1 (100-800) From the Table 1 we are introduced to the new property terms Maximum specific strength and Maximum specific modulus. Maximum specific strength defines the length up-to which the material can carry its own weight or in simple terms strength to weight ratio. Similarly, Maximum specific modulus defines a material’s stiffness with respect to its density.

, = , =

σ E ′ ,E = , = ᵋ ρ

From the above table, we could see that the carbon fiber has very high maximum specific strength than the conventional structural materials. Hence a combination of carbon fiber in cellular structures will lead us to improve functionalities to our products.

Such innovative designs become complex and unconventional and it is real challenge for traditional subtractive manufacturing process to realize these designs. And AM techniques due to their unique additive nature becomes the best method to realize these designs with highest level of details. These key subjects are taken as the motivation for this research and to achieve an optimized cellular structure design for a lightweight high strength material.

Literature Review

To meet the lightweight deign application requirements, various design methods are adopted to generate topology of the highly porous composites, such as continuing improved material bounds approach for multiphase, multi-dimensional, isotropic/anisotropic and periodic/nonperiodic composites with different physical properties, topology optimization approach to material design which is frequently performed within a finite element framework and typically involves large number s of design variables, homogenization or inverse homogenization approach in the design of microstructural materials which has permitted an increased level of design capability and understanding of underlying material mechanisms, just name a few [24].

Cellular composites reinforced by discontinuous fibers (DiFs) are of particular interest as they offer great flexibility in tailoring specific physical properties by controlling the compositions, and/or microstructures of the constituent phases, and the fiber orientations. Periodic cellular composites consist of many identical base cells or representative volume elements (RVEs); manipulating the phase distributions and fiber orientation in these base cells provide an effective approach to the design of multifunctional composites. However, conventional design methods based on trial-and- error have been found cumbersome and time consuming. An effective approach to design cellular periodic composites reinforced by DiFs is to adopt the ideas behind biomimetics, which can encompass the essential aspects in materials design, system engineering, and even business models [24-44].

In this work, an approach involving homogenization, optimization, and validation of an efficient design of macro-/microcellular structural composites for additive 9 manufacturing (3D printing) will be presented for developing ultra-lightweight and high- strength cellular composites reinforced by DiFs. The topologies of the cellular composites will be designed based on the periodic RVEs and inspired by biomimetics.

Computer modeling will be conducted to characterize the performance and properties of the designed cellular structural composites, taking into account the materials' density/porosity. 3D printing techniques will be adopted

As the composite material design is inspired from nature, a closer observation of the microstructure will be helpful in understanding the material distribution. From nature, we should be learning how architecture, interfaces, and other key features, can be adapted to the engineering world for structural design and mechanical aspect of view [45]. The nature perspective is important as these structures have survived and evolved over the years to be a successful specimen. Thereby we come with a design which can have a long durability and good mechanical tolerance. The microstructure of human bone, bamboo, wood, cuttlefish bone, silk, honeycomb and vulture wing are listed below figures.

Figure 4. Hierarchical structure of bone (a) and bamboo (b) [46].

Figure 5. 3D anatomy for loblolly pine (a) [47]

Figure 6. SEM images displaying morphological variations for two different samples of: (a) Cuttlebone 1 and (b) Cuttlebone 2 [48]

Figure 7. Skin–core, fibril bundle structure of spider silk. (a) Schematic of the multi-fibril core of dragline silk that constitutes a spider web, on the order of micrometers [49]

Figure 8. Bee’s honeycomb [50]

Figure 9. Bee’s honeycomb [51]

Figure 10. Italian honeybee (Apis mellifera Ligustica) comb cell at (a) ‘birth’, and at (b) 2-days old, scale bar is 2 mm [50]

Figure 11. S ection of bone structure from a vulture wing [45]

The above figure shows a bone structure of a vulture wing. It is a combination of two strengthening and lightening mechanisms: that of the lightweight cellular bone structure (which, in fact, may be hollow); and this is combined with the geometrical truss sections that are frequently utilized in the design of bridges. In the vulture's case, the wing trusses also protrude into three dimensions, rather than the two-dimensional structures that are commonly designed for bridges [45]. From the above list of figure 4 to 11, listed are the nature’s own cellular structures. The above cellular structures are very strong in either the tensile or compression and very light in weight. Few of the above structures have fiber reinforcement to enhance strength properties. A closer observation to these cellular structures reveal that they are symmetrical in two dimension. Say if x and y are symmetrical the z direction will be just an extruded section of the cross section. The simple reason why bio materials are two-dimensional symmetry is because of the fact that these structures have living mater in and around them has to be facilitated with proper channels of nutrition and supplies that sustains the life in it and the third non- symmetrical direction servers this purpose where the nerves, blood vessels, roots, etc. can grow, feed and communicate with the cells and living matter. 14

On the other hand, the understanding the behavior of DiFs are extremely vital as our design foundation starts with DiFs material characterization. DiFs are delicate and their performance is enormously based on the below attributes,

• Orientation &.

• Aspect Ratio of DiFs

• Interaction between DiFs

These attributes lead to the complex situation and the experimental measurement will give us a clear property measurement but it is extremely ineffective as it expensive. So, finite element approach is used do the property prediction. method to characterize the effective mechanical properties of composites. Owing to the time-consuming and costly testing of experiments, however, the analytical or numerical prediction on the mechanical properties of composites is considered as an alternative, or at least a complement to the experimental measurement. Therefore, we firstly would like to use the approach: the direct finite element (FE) computation of the boundary-value problem (BVP) at a representative volume element (RVE) of composites (entitled RVE based FE method).

However, the RVE based FE method becomes extremely expensive if one or more of the following conditions are met: (1) material or geometric non-linearity; (2) complex microstructures (e.g., RVEs containing a large amount of fibers with the different orientations)[52]. Our DiFs falls in these categories and this complexity causes the increased computational power requirement and tedious time taking process to solve the problem. Theoretically, under a simple shear force most of the DiFs try to arrange themselves in the following direction of the polymer [53]. But this is highly influenced by the aspect ratio and percentage of volume by which the DiFs are used for the 15 reinforcement of the polymer and the velocity at which the polymer flows. Due to this many simplifying process are created by researchers such that the orientation of the fibers are accounted . One such simplification is by creating a two-step homogenization procedure shown in below figure 12.

Figure 12. Two-step mean-field homogenization procedure: The RVE is decomposed into a set of pseudo-grains, each of which is then individually homogenized (first-step). Finally, the second homogenization is performed over all the pseudo-grains (second-step)[52]

In first step homogenization RVE is broken into bunch of pseudo-grain and each of them is considered individually , and in second step homogenization of all pseudo- grains (‘aggregate’) is considered over the RVE. In first step the M–T (Mori–Tanaka) and

D-I (interpolative double inclusion) models for each two-phase pseudo-grain are considered and the best resulting from that the M–T and D-I models provide the best mean-field predictions on the effective stiffness of each pseudograin, and the Voigt and

Reuss models for the ‘aggregate’ in the second-step homogenization procedure resulting 16 from that the Voigt and Reuss models ensure physically acceptable results and the symmetric effective stiffness tensors.

Few other researchers try to find the fiber angle using image processing, from non- destructive testing using digital microscopy from a sample specimen [54] shown in figure

13 like, a) Optical reflection microscope:

Here short fiber reinforced plate is sliced in both longitudinal and transverse axis and measurement of semi major and minor axes of the ellipse are formed in cutting plane of the short fibers are measured. There by a quasi 3D reconstruction for calculating fiber cross-sectional area b) Confocal laser scanning microscopy:

Optical sectioning is done to follow a selected semi-ellipse for a typical distance of 20µm into the material. The displacement from the center of the ellipse leads to the fiber orientation. c) X-ray microtomography:

This technique allows to reconstruct the 3D space up to a volume of 20 mm3. As entire small volume is reconstructed we could have fiber orientation, length, curvature and fiber to fiber spatial statistics. 17

Figure 13. Polished cross section in the (y, z) plane [54]

So, from the literature review we can clearly see that the DiFs reinforced composite property estimation and prediction is complex and needs to simplified at the same time addresses the exact behavior of DiFs.

Hypothesis

Nature inspired bio materials two-dimensional topology cellular structures will have high specific stiffness and will lead to light weight material. The two-dimensional topology is enhanced to a three-dimensional symmetry cellular structure, such that the three-dimensional cellular structure will exhibit an overall better isotropic property even with random distribution of the DiFs. Both the two-dimensional and three-dimensional topology models are optimized such that optimization should result in lower stress for failure and increase the stiffness and strength. From the outcome of our hypothesis, we should have two kind of designs: two-dimensional topology with high strength to weight ratio, high specific stiffness, etc. and three-dimensional topology we have design that can 18 be used in application where near to isotropic properties are needed with improved stiffness.

MATERIAL AND METHODS

The below flowchart gives an overview idea about the process that is taken in this research work,

Figure 14. Flow chart of development of a cellular structural composites.

Once observations from nature is perceived array of cellular structures are referenced and CAD models of the structures are recreated. From the array, CAD model, 19 specific building block i.e. RVE is chosen and its CAD model is prepared for the simulation. Now the chosen REV will be characterized with material properties from

DiFs. Finite element simulations are run on characterized cellular structures and is further optimized for better stiffness. And the finalized design will be realized through AM process. The real-world design model is tested and result will be compared against the simulation and suitable changes and future work are established.

Fusion Deposition Molding

For a three-dimensional symmetrical cellular structure the conventional composite manufacturing process like Resin Transfer Molding (RTM), Vacuum Assisted Resin

Transfer Molding (VARTM), Resin Film Infusion (RFI), Injection and Compression

Molding, Filament Winding methods are tedious and is quite tough to manufacture these designs. The easiest and simplest way to manufacture this is through AM process. And

Filament Deposition Molding in AM process is chosen as the SDSU additive manufacturing lab facilities this technology. The below picture show the image of carbon fiber reinforce 3D printed specimen.

Figure 15. Infill pattern for varying Fill density [55]

Material options within the 3D printing space have expanded quite significantly over the past a few years, as companies have begun to realize that the potential that 3D printing provides and the practicality behind the technology are limited by the materials currently available. Recent reports identified several new carbon fiber-based materials for use in fused filament fabrication and fused deposition modeling (FFF/FDM) 3D printers.

These include Ultem and PEEK FDM filaments with chopped carbon fibers [56], carbon fiber-reinforced PLA, ABS, and Nylon, in addition to other innovative 3D printing products, such as high-performance 3D printing filament made using multi-walled carbon nanotubes for electronics applications [57]. A few materials rang is listed below,

Table 2. Materials used for fabricating FRC in Stereo-lithography (SL). [3].

Sl No Material Used Carbon fiber, bisphenol A-epoxy, 1 hydroxycyclohexyl phenyl ketone, lauroyl peroxide 2 Glass fiber, acrylic-based polymer 3 E-glas fiber, epoxy-based resin 4 E-glas fiber, acrylic-based resin 5 Carbon fiber, acrylic-based resin 6 Aramid, acrylic-based resin

Figure 16. A picture from local motors showing the fabrication of the 3D pinted car [58]

The Fill density in the 3D printing allows us to have the component fabrication with desired level of material and vacancy. The infill pattern selection is crucial as it describes the bonding structure of the component. The above figure 16 shows the fabrication of a 3D printed car and we could notice some two-different infill pattern.

Most of infill patterns available in current use are 2D symmetrical. This obviously results in the high strength in the third perpendicular direction. The cellular structure that’s developed in this study can be used as an infill such that we can have more customizations and with a three-dimensional infill pattern for having a closer isotropy and light weight for the 3D printed component is a step moving further. Some common infill pattern with varying fill density are shown below. 22

Figure 17. Infill pattern for varying Fill density [59]

Scientist has also discovered that the change in infill pattern inspired form bone has lightweight and has high strength to weight ratio compared to traditional infill. The different infill pattern that are used are shown below.

Figure 18. Infill pattern inspired from bone [60]

Homogenization Theory

A cellular structure is designed and various property like Young’s Modulus,

Poisson’s ratio, and Shear modulus is estimated for the cellular structure. A said earlier cellular structure is the basic skeletal structure over which the desired model is built in

3D printing. Homogenization theory allows us to have a small repeatable representative volume of the chosen cellular structure and various simulations are done for that volume element.

The advantage of having the homogenization theory saves a lot of computation time, cost of having higher version software. Since the discretization of the whole model in-to Finite Element model (FE) will cost a lot of time and money [61]. Thereby the homogenization method allows us to evaluate the effective properties of the periodic unit cell [62].

Using this homogenization theory, first we calculate the properties for the bulk composite material. And the calculated bulk composite materials are verified with properties from the 3DxTech CFRPLA filament [63]. The 3Dxtech filament has been chosen since it has the closest length to diameter ratio. More about the this will be discussed in the next section Characterization of bulk composites.

3D Tessellation

A tessellation is structure that is created when a shape is repeated in a plane thereby having no gaps. Few three-dimensional tessellation are shown in below figure. 24

Figure 19. 3D Tessellation [64]

Here for trial the truncated octahedron is chosen for initial simulation.

Figure 20. 3D Tessellation [65]

The truncated octahedron is modelled in ProE and the array is show below. 25

Figure 21. Array of truncated octahedron

Representative Volume element

Representative Volume Element (REV) is the smallest volume element that when repeated in all the directions in a periodic manner results in the larger structure designed.

REV is useful in identifying and estimating the properties of the larger structure. In our case RVE is cellular structure which is under a study. An example of the larger structure and its REV is shown below, 26

Figure 22. A composite stucture with semi-circular fibers and matrix

Figure 23. REV of above larger structure

Here our representative element for the truncated octahedron structure shown in figure ranging from 25-28 for different volume occupant percentage.

Figure 24. REV of solid block 100% occupied

Figure 25. REV of truncated octahedron 73% occupied

Figure 26. REV of truncated octahedron 65% occupied

Figure 27. REV of truncated octahedron 55% occupied

Figure 28. REV of truncated octahedron 45% occupied

CHARACTERIZATION OF BULK COMPOSITES

Visualization of manufacturing process

The key for having a proper computer model is to understand the manufacturing process thoroughly and recreate the details in the simulated model. The DiFs are feed to the flowing polymer from a hopper/funnel inside the single or twin scroll extruder. The slurry of resin and fiber is heated in different stages and passes through rollers. The molten liquid slurry gets its shape of a filament as it comes out from the single or twin scroll extruder. As discussed in the literature review, under a simple shear force the fiber tend to align itself in following direction. And the orientation of the DiFs depend on the aspect ratio and volume percentage by which it is occupied. 30

Modeling methodologies

For characterizing discontinuous carbon fiber reinforced bulk composites, two different approaches have been developed. The approach that we postulate here is so called two-step approach. In the first step, unidirectional (1D) random fibers were generated in a composite solid model, as shown in below figure 27-a, and the tests of tension/compression and shearing by computer modeling were conducted to extract the properties of the unidirectional fiber reinforced composites. In the second step, a composite solid model was generated and meshed in which each element represented an element of composites reinforced with unidirectional randomly oriented fibers generated by Monte Carlo method referred in figure 29-b and with the composite properties characterized from step one, as shown in below figure. The same tests of tension/compression and shearing by modeling were conducted to extract the properties of a bulk composites reinforced with 3D random discontinuous carbon fibers. The other approach is so called one-step approach, in which a 3D random fibers were directly generated in a bulk composite solid model represented in figure 30 a & b. The results from one-step approach is used to validate the two-step approach. as shown in below figure and material properties were extracted from the tests of tension/compression and 31 shearing by modeling.

Figure 29. Two-step approach for bulk composite characterization: (a) step one: unidirectional fiber distribution, and (b) step two, elements with 3D random fiber distribution by Monte Carlo method.

Figure 30. One-step approach for bulk composite characterization: (a) 3D random fiber distribution, and (b) bulk composite 3D model

Computer model generation

As the short fibers are feed into the extruder, the fibers will be dispersed and randomly aligned. A unit cell dimensions is chosen to accommodate appropriate fiber weight percentage. For ease in modelling and meshing the short fibers are designed to have squared cross section rather than a circular cross section. The key configuration used for various approaches are listed below,

Table 3. Summary of fiber configurations that are used in study

Approach L/d Ratio

1D Modeling 8.86 (Uni-directional fibers) 17.7

8.86 Two Step 17.7

9.45 One Step 18.9 3D Experiment 21.4

The specifics, like length to diameter ratio of short fiber is taken to be 8.86 for two step approach, and 9.45 for one step approach. The circular diameter of 11.284µm and length 100µm. Whereas the equivalent squared area is 100µm2, with a single side of

10µm. All parameters are defined as variables such that various sizes could be experimented. The flow chart of the modelling is shown below,

Figure 31. Flow chart of development of random fiber placement pattern inside a bulk material 34

Fiber definition

The fibers are generated in such a way that the generated random numbers are in x, y, z direction will be center of the cuboid. The range of the x, y, z directions are shown below,

In x: x - r < x < x + r

Similarly, in y: y - r < y < y + r

Similarly, in z: z – l/2 < z < z + l/2, where r, l are the radius and length of fiber respectively.

Figure 32. Fiber Definition

This unique definition of fiber aids us to have computations for various arrangements whether two fibers intersect, chop the fiber when it goes out of the unit cell, etc.

Random Number generation

Random number generation is a key step in our simulation. The random number in three directions define each of the fibers center as given by above definition. The random numbers are generated by the linear congruential generator, which is given by,

= +

Then the generated number of are normalized for the unit cell dimensions in respective axis. The other way of getting the random numbers is using rand() function in

ANSYS APDL. This function will let us to have a random number for an interval between two points.

The generated random numbers for each direction is given below in form of scattered plot. Here plot shows the distribution of random numbers with x axis being the count of how many random numbers and Y axis being the dimension in specified axis. Here in below plots the x axis will be 200 which indicates that we have 200 random numbers and

Y axis will hold the dimension for an axis for each of 200 numbers.

For example, the generated random numbers for each direction is given below, 36

X Random Numbers 0.11

0.09

0.07

0.05

0.03

0.01 X X directrion Width of unit cell

-25-0.01 25 75 125 175 225 N Numbers of SF

Figure 33. X coordinate described by random numbers

Y Random Numbers

0.11

0.09

0.07

0.05

0.03

0.01 Y directrionY Height of unit cell

-25-0.01 25 75 125 175 225 N Numbers of SF

Figure 34. Y coordinate described by random numbers

z Random Numbers

0.35

0.3

0.25

0.2

0.15

0.1

0.05

z directrion Height of unitcell 0 -25 25 75 125 175 225 -0.05 N Numbers of SF

Figure 35. Z coordinate described by random numbers

Model simulation

The ANSYS workbench is used to for modelling the 3D volume, mesh and test the boundary conditions.

Some of work progress in generation of fiber and matrix volume is shown below, 38

Figure 36. Line view of the REV with random distribution of SF in matrix – Front View

Figure 36, 37 show the line plot the fiber and matrix material. From the above figure, it very clear that the fiber generation is randomly spread and the fiber at the end of the REV boundary are chopped. The chopped fibers are encircled for identification purpose.

Figure 37. Line view of the REV with random distribution of SF in matrix – Isometric View

Figure 38 Volume plot of the REV with matrix material only

From the above figure 38, it very clear that the matrix material serves as the pocket holes to accommodate the short fibers. The holes are encircled for the identification purposes. 41

Figure 39. Volume plot of the fibers only

The above figure shows the random distribution of the short fiber in all the directions 42

Figure 40. Volume plot of the fibers only – Front View

Figure 41. View showing the meshing of the fiber materials

A brick mesh using the element SOLID 185 is shown for the generated fibers.

A model for PLA reinforced Carbon Fiber

A model with carbon fiber and PLA as matrix has been chosen and we have calculated material properties like Young’s modulus and Poisson’s ratio for all three directions. A tetrahedron element was chosen with 10 nodes using SOLID187 in

ANSYS. The aspect ratio and the fiber definition has been defined in the above sections.

The individual properties of the chosen polymer and carbon fiber is shown below,

Table 4. Properties of the individual Carbon Fiber and PLA Polymer resin used to construct bulk material model

Property PLA Carbon Type Polymer Fiber Ex 22.4e3 GPA 2.865e3 Ey 22.4e3 GPA Ez GPA 250e3 GPA Gxy 8.3e3 GPA 1.053e3 Gyz 22.1e3 GPA GPA Gxz 22.1e3 GPA νxy 0.35 νyz 0.36 0.027 νxz 0.027

All the fibers are oriented towards the z direction therefore x, y being transverse direction.

The below series of pictures show various stages involved in the model simulation, 45

Figure 42. Volume plot of the REV with fibers and matrix

Above figure shows the volume plot of REV containing both the fibers and matrix materials. From this figure, we could clearly see that many of the fibers are chopped at the edges of REV. This adaptation has enabled us to have a model close to the reality.

Figure 43. Line plot of REV showing the randomly distributed fibers

Figure 44. Volume plot of the random fibers

From above figure we could see that each fiber is distinct and randomly placed inside the matrix material. The randomness is more important as it is an influential parameter. If the randomness is not good, then we will have our properties absurd and weird. 48

Figure 45. Volume plot of the matrix

Figure 46. Volume Mesh of REV with elements

Figure 47. Material plot of REV clearly distinguish fiber material from the matrix material

CHARACTERIZATION OF CELLULAR COMPOSITES

Identification of RVEs

RVEs are useful in identifying and estimating the properties of the larger cellular structural composites. A brief explanation is given in the materials and methods section above. The RVEs which are considered for the study are modeled in CAD and then the

RVEs are selected. The following pictures shows the selected natural material with the corresponding CAD model and followed with the RVEs of those sections.

(a) (b)

Figure 48. (a) Honeycomb, (b) 2D hexagon model, (c) REV for Honeycomb and (d) REV Honeycomb with fillet

The cuttlefish bone pattern is known for its high strength light weight cellular structure. Hence cuttlefish bone pattern is modeled with the aid of the journal article,

“Cuttlebone: Characterization, Application and Development of Biomimetic Materials ”

[23] as shown,

Figure 49. (d) Image based finite element modelling for homogenization

(a) (b)

Figure 50. (a) Cuttlefish bone, (b) 2D cuttlefish bone model (c) REV for cuttlefish bone and (d) REV for cuttlefish bone with fillet

(a) (b)

Figure 51. (a) 3D truncated octahedron, (b) 3D truncated octahedron model, (c) REV of truncated octahedron model and (d) ) REV of truncated octahedron model with fillet

For evaluating the structural performance and material properties of the RVEs of the cellular structural composites, the material properties of the bulk composites must be first characterized and input into the characterization procedures of the RVEs of the cellular structural composites by modeling.

Characterization of RVEs by modeling

Based on the characterization results of the bulk composites, the material properties of the RVEs of the cellular composites can be predicted in the similar way, considering the fiber volume fraction, fiber aspect ratio, and fiber orientation formed in the 3D printing fabrication of the product. The following pictures demonstrate the volume plot and mesh for the corresponding volume plot the one of the 2D hexagonal topology model under study,

Figure 52. Volume plot of the 2D hexagonal array

Figure 53. Volume Mesh of 2D hexagonal array

RESULTS

Properties from bulk composites

Here we will discuss two major results which compromises of Simulation and experimental process. The first set of results summarizes the property estimation from one step and two step approach. These results will give us the very good insight of agile two step approach could be used in place of a stiff one step approach without sacrificing much of an overall understanding about the bulk material. The key results are summarized below are for bulk material properties of PLA base resin reinforced with

15% by volume of discontinuous carbon fibers (DiCFs) are listed in below table,

Table 5. Young's moduli of bulk composites reinforced with 15 vol% DiCF (z-axis: fiber direction for unidirectional fiber composites)

Fiber L/d Ex Ey Ez Approach Avg. E ratio (GPa) (GPa) (GPa) 1D Modeling 8.86 4.081 4.115 13.21 (Unidirectional random fibers) 17.7 4.158 4.172 23.22 3D Modeling 8.86 5.231 5.238 4.673 5.047 (Two step: Unidirectional random fibers +3D Monte 17.7 6.026 6.028 5.132 5.728 Carlo) 3D Modeling 9.45 5.391 5.187 5.098 5.226 (One step: 3D random fibers) 18.9 5.931 3D Experimental by 3D printing 21.4 4.711 (printing in +/-45 degree)

The above predicted properties are under the assumption that carbon fibers having transversely isotropic properties. The modeling results are compared with the experimental testing results using the 3D printing technique, test specimens were printed in ± 45° printing direction alternatively. For 3D random fiber models, due to the frictional fiber interaction induced in the one-step model, the bulk composite properties by the one- step approach is slightly higher than the results by the two-step approach. Due to the imperfection of the printing specimens, Young's modulus of the specimens by 3D printing is lower than the results by modeling on the idealized condition. Therefore, the results clearly state that the two step approach can be used in place of one step approach to estimate the properties of the composite bulk material.

For further investigation of the properties of the bulk composite specimens prepared by 3D printing, compression and tension tests in printing direction and transverse to the printing direction have been conducted for PLA polymer and carbon 58 fiber reinforced PLA (CFRPLA) composites based on the relevant ASTM standards [66], as shown in below Figs. 52 and 53, and results are listed in Table 6.

Figure 54. Tensile tests on bulk composites in printing direction and transverse to the

printing direction: (a) PLA polymer, and (b) CFRPLA composites

Figure 55. Compression tests on bulk composites in printing direction and transverse to the

printing direction: (a) PLA polymer, and (b) CFRPLA composites

Table 6. Bulk material properties by testing (15% DiCF in volume)

Ex (GPa) Fiber/printing direction Tensile test Compression test No PLA CFRPLA No PLA CFRPLA 1 3.219 7.424 1 2.476 4.296 2 3.099 7.756 2 2.543 3.902 3 3.193 Avg. 3.17 7.59 Avg. 2.509 4.099 Ey (GPa) Transverse to fiber/printing direction Tensile test Compression test No PLA CFRPLA No PLA CFRPLA 1 3.244 3.716 1 2.029 1.932 2 3.228 3.743 2 1.773 2.038 Avg. 3.236 3.729 Avg. 1.901 1.985

As the DiF composite filament used in 3D printing, the DiFs were aligned primarily along the printing path, as shown in below figure 56. From the microscopic images, it is very clear that the printing is essentially the fiber direction for the composites reinforced by DiFs.

Figure 56. Digital microscopy images of carbon fiber orientations in 3D printed specimens:

(a) along printing direction, and (b) transverse to the printing direction

It was found that the printing directions also refer to the fiber direction for DiF reinforced composites, and load types (tensile or compressive load) have great effects on the bulk composites stiffness. Tensile stiffness (Young's modulus), either for the case of

PLA or CFRPLA, are much higher than that of the compressive stiffness, and material stiffness in printing direction has higher stiffness than that in transverse to the printing direction. However, there is no significant difference in tensile stiffness between printing direction

Properties from Cellular Structures

The modeling results of 2D hexagon model, 2D cuttlefish bone model, and 3D octahedron model with CFRPLA composites and under the assumption of random fiber distribution in the matrix are shown considering the sharp edge and fillets in the corners of the designed topologies. Each REV is taken and a fillet is used to smooth all the edges and influence of the fillet is studied. Theoretical an introduction of fillet reduces the maximum stress value and thereby reducing the ability to fail and in turn increases the strength of REV. The below figures show REVs with and without fillet which are taken into study. 61

(a) (b)

Figure 57. Design of a two-dimensional hexagonal array topology: (a) without fillet, and (b) with fillet.

(a) (b)

Figure 58. RVEs of the 2D hexagonal topology: (a) without fillet, and (b) with fillet.

(b) (b)

Figure 59. RVEs of the 2D Cuttlefish topology: (a) without fillet, and (b) with fillet.

Figure 60. RVEs of the 3D Truncated Octahedron topology: (a) without fillet, and (b) with

fillet.

The Young’s modulus in the thickness direction, Ez, for the 2D model increases linearly as volume occupied increases, as shown in Figs. 61(a) and 62(a), since the cross- sectional area perpendicular to the load/thickness direction, i.e., the area of the plane, is 63 proportional to the volume occupied due to the 2D nature. The planar Young’s moduli,

Ex and Ey, for the 2D model are slightly nonlinearly increase as the volume occupied increases, as shown in Figs. 61(b) and (c), 62(b) and (c). This is due to the dimensional or configurational non-similarity and the influence from the fiber orientation within these geometries. Since the total height, width, and thickness are the same for all the RVEs, and the higher the volume occupied, the wider the beams of the RVEs are, i.e., the geometries are not similar as the volume occupied changes, resulting in a non-similarity between the changes of the RVEs’ beams cross-sectional areas (i.e., the changes of the volume occupied) and the changes of the RVEs’ beams configurations/dimensions. The slight nonlinearity relations of Young’s moduli vs. volume occupied are also presented in the 3D octahedron model due to the same reason of the non-similarity, as shown in Fig.

63. The optimized designs with fillets improved the planar stiffnesses for the 2D models

(as shown in Figs. 61(b) and (c) and 62(a) and (b)) and all of the three directional stiffnesses for the 3D model (as shown in Fig. 63), and reduced the maximum von Mises stress in the structures (as shown in Fig. 64). It also showed that the cuttlefish bone 2D model has higher planar stiffnesses than that of the hexagonal 2D model due to its structural uniqueness, especially in y-axis (vertical direction) which is the result of evolution for cuttlefishes to take the high pressure in the deep sea. The honeycomb hexagon model is the result of evolution for bees to maximize the space usage while maintaining certain stiffness.

(a)

(b)

(c)

Figure 61. Young’s modulus for hexagon 2D model: (a) Ez (thickness), (b) Ex (planar), and

(a)

(b)

(planar), and (c) Ey (planar). 66

(a)

(b)

(c)

Figure 63. Young’s modulus of 3D octahedron model: (a) Ex, (b) Ey, and (c) Ez. 67

(a)

(b)

(c) Figure 64. Von Mises stress of 3D octahedron model: (a) by X-axis tensile test, (b) by Y-axis tensile test, and (c) by Z-axis tensile test. 68

VERIFICATION

For validating the modeling approach adopted for cellular structure design, 3D printing was used to fabricate 2D hexagon specimens with PLA and CFRPLA materials, and microscope images of the specimens are shown in below figure,

Figure 65. 3D printed 2D hexagon samples: (a) PLA, and (b) CFRPLA

In-plane compression tests were conducted for the 2D hexagon specimens with

PLA and CFRPLA materials, as shown in Fig. 66, and failure sliding bands were found along the diagonal of the specimens. The typical cellular structural composites' stress- strain curves from compression tests were shown in Fig. 67. It shows that PLA and

CFRPLA have similar compressive stress-strain pattern. However, PLA specimens have higher ultimate strength, lower Young's modulus, and better ductility (plasticity) than that of the CFRPLA specimens. The testing and modeling data of Young's modulus, considering the fiber orientation in the specimens fabricated by 3D printing are listed in

Table 7. The in-plane stiff nesses in X and Y directions for the same material and same volume occupied fraction are different due to the asymmetric topology in nature, and the 69 in-plane stiffness of the CFRPLA is higher than that of the PLA in the same direction and same volume occupied fraction due to the reinforcement of the carbon fibers in CFRPLA.

The higher the volume occupied the higher the stiffness is. The Young's moduli of

CFRPLA by modeling are higher than that by experiment and the error percentage for

CFRPLA between modeling and experiment are higher than that for PLA. This is due to the facts that in modeling, the solid model is perfectly built layer by layer without bonding issue and void, and carbon fiber oriented perfectly as defined by the 3D printing path, while the specimens fabricated by 3D printing have some bonding and contact surface issues (voids) during layer by layer printing process (see Fig. 65(b), comparing with Fig.65(a)), and the fiber orientations could not exactly follow the printing path due to the mesh limitation as well as the fiber’s certain randomness in the printing process, as shown in Fig. 56(a). In addition, the volume fraction occupied for modeling are 0.7% to

2.8% higher than those of the specimens for the experiments, see Table 7.

Figure 66. Compression tests for hexagon model in Y-axis (vertical): (a) PLA, and (b) CFRPLA

(a) (b)

Figure 67. Stress and strain curve from compression tests for hexagon model of 50% volume occupied: (a) PLA, and (b) CFRPLA

Table 7. Compressive Young’s modulus (GPa) of the hexagon structures by testing and modeling

PLA CFRPLA Experiment, Simulation, Experiment, Simulation, Difference Difference GPa GPa GPa GPa Volume Direction (Deviation, (Deviation, (vol%) % (vol%) % GPa /vol%) GPa /vol%)

0.455 0.461 0.588 0.7245 Ex 1.32% 23.20% 50% (0.0083/52.6%) -51.50% (0.0523/50.0%) -51.50% Volume Occupied 0.505 0.55 0.622 0.86 Ey 8.91% 38.30% (0.0783/51.2%) -51.50% (0.0262/50.8%) -51.50%

1.291 1.6 1.855 1.299 Ex 0.62% 15.90% 80% (0.0455/81.0%) (80.9%) (0.0688/78.9%) -80.90% Volume Occupied 1.493 1.436 2.046 2.018 Ey 3.80% 1.37% (0.0205/80.0%) -80.90% (0.0014/78.1%) -80.90%

To verify our hypothesis that the, bio materials inspired two-dimensional topology cellular structures will have high specific stiffness and will lead to light weight material design and three-dimensional topology will have a more distributed isotropic behavior, considered topology models are evaluated with respect to its density. This will provide us a normalized platform where we can compare and infer the topology outcomes. Below tables 8, 10, 11 lists estimated properties considering the material density.

Table 8. Summary of the specific stiffness for considered designs in thickness direction

(Ez – Young’s Modulus in Z direction for the 2D and 3D Topology)

Occupied Density Specific E √E/ r Design Material Volume r Stiffness (GPa) (10 6m2s-2) % (g/cm 3) E/ r (10 6m2s-2)

PLA bulk 100.000 1.190 2.865 2.408 1.422 Bulk CFRPLA* Material 100.000 1.073 4.711 4.390 2.023 bulk 74.900 0.804 3.401 4.232 2.295 3D Tuncated 66.890 0.718 2.797 3.897 2.330 Octahedron CFRPLA Topology 57.400 0.616 2.185 3.548 2.400 46.300 0.497 1.574 3.168 2.525 74.070 0.795 4.243 5.339 2.592 2D Cuttlefish 66.090 0.709 3.786 5.339 2.744 Bone CFRPLA Topology 56.510 0.606 3.238 5.339 2.967 46.090 0.495 2.641 5.340 3.286 80.000 0.858 4.661 5.430 2.515 2D 70.000 0.751 4.075 5.425 2.688 Hexagonal CFRPLA Topology 60.300 0.647 3.506 5.419 2.894 50.200 0.539 2.912 5.406 3.168 Aluminum 100.000 2.700 69.000 25.556 3.077 Steel 100.000 7.9±0.15 200.000 25.316 1.790 Bulk Titanium 100.000 4.500 112.5±7.5 25.000 2.357 Material alloys Diamond 100.000 3.530 1220.000 345.609 9.895 (C) Note: * Carbon fiber average diameter of 7 µm, average length of 150 µm, and aspect ratio of 21.4. 15 vol% CF for CFRPLA;

Table 9. Summary of the specific stiffness for considered designs in planar direction

(Ey – Young’s Modulus in Z direction for the 2D and 3D Topology)

Occupied Density Specific E √E/ r Design Material Volume r Stiffness (GPa) (10 6m2s-2) % (g/cm 3) E/ r (10 6m2s-2)

Table 10. Summary of the specific stiffness for considered designs in planar direction

(Ex – Young’s Modulus in Z direction for the 2D and 3D Topology)

Occupied Density Specific E √E/ r Design Material Volume r Stiffness (GPa) (10 6m2s-2) % (g/cm 3) E/ r (10 6m2s-2)

PLA bulk 100.000 1.190 2.865 2.408 1.422 Bulk Material CFRPLA* 100.000 1.073 4.711 4.390 2.023 bulk 74.900 0.804 3.360 4.180 2.281 3D Tuncated 66.890 0.718 2.771 3.861 2.319 Octahedron CFRPLA Topology 57.400 0.616 2.158 3.504 2.385 46.300 0.497 1.549 3.117 2.505 74.070 0.795 3.355 4.221 2.305 2D Cuttlefish 66.090 0.709 2.779 3.919 2.351 Bone CFRPLA Topology 56.510 0.606 2.205 3.637 2.449 46.090 0.495 1.646 3.329 2.594 80.000 0.858 3.585 4.176 2.206 2D 70.000 0.751 2.728 3.632 2.199 Hexagonal CFRPLA Topology 60.300 0.647 1.997 3.087 2.184 50.200 0.539 1.325 2.460 2.137 Aluminum 100.000 2.700 69.000 25.556 3.077 Steel 100.000 7.9±0.15 200.000 25.316 1.790 Titanium Bulk Material 100.000 4.500 112.5±7.5 25.000 2.357 alloys Diamond 100.000 3.530 1220.000 345.609 9.895 (C) Note: * Carbon fiber average diameter of 7 µm, average length of 150 µm, and aspect ratio of 21.4. 15 vol% CF for CFRPLA;

From the above tables, comparing the specific stiffness and √E/ r values of the two- dimensional hexagonal topology and cuttlefish topology has higher performance than of its corresponding bulk material in Ez and Ey direction. Similarly, the three-dimensional topology of truncated octahedron has more distributed isotropic behavior in all three directions and is exceeding the performance of bulk material for √E/ r, this is important parameter in aerospace industry as weight saving are important. To note that the three- dimensional topology of truncated octahedron is still under performing than its corresponding bulk material with respect to specific stiffness. Design iterations can be done such that we could end up in having the properties higher than its bulk material and this is future work for this thesis.

Table 11. Summary of the specific Young's modulus for some materials

Density E/ r E/ r 2 E/ r 3 √E/ r E Material r (10 6m2s- (103m5kg- (m8kg- (10 6m2s- (GPa) (g/cm 3) 2) 1s-2) 2s-2) 2)

PLA bulk 1.19 2.865 2.408 2.023 1.7 1.422 CFRPLA* 1.073 4.711 4.39 4.092 3.813 2.023 bulk 2D hexagon PLA (51.2 0.609 0.505 0.829 1.36 2.233 1.167 vol%) in- plane X- axis** 2D hexagon PLA (52.6 0.626 0.455 0.727 1.161 1.856 1.078 vol%) in- plane Y- axis**

2D hexagon CFRPLA (50.8 0.545 0.622 1.141 2.093 3.84 1.447 vol%) in- plane X- axis**

2D hexagon CFRPLA (50.0 0.537 0.588 1.096 2.043 3.808 1.428 vol%) in- plane - axis**

2D hexagon CFRPLA (50.8 0.545 0.622 1.141 2.093 3.84 1.447 vol%) in- plane Y- axis** 75

2D cuttlefish bone PLA (51.3 0.61 0.829 1.358 2.224 3.643 1.493 vol%) in- plane X- axis ***

2D cuttlefish bone PLA (51.3 0.61 1.47 2.408 3.944 6.461 1.988 vol%) in- plane Y- axis ***

2D cuttlefish bone CFRPLA 0.61 1.926 3.155 5.168 8.466 2.275 (51.3 vol%) in- plane X- axis ***

2D cuttlefish bone CFRPLA 0.61 2.939 4.814 7.886 12.92 2.810 (51.3 vol%) in- plane Y- axis ***

3D Octahedron PLA (51.9 0.617 0.923 1.496 2.424 3.929 1.557 vol%) X- axis ***

3D Octahedron PLA (51.9 0.617 0.853 1.383 2.241 3.633 1.497 vol%) Y- axis ***

3D Octahedron PLA (51.9 0.617 0.943 1.529 2.477 4.015 1.574 vol%) Z- axis *** 76

3D Octahedron CFRPLA 0.556 1.853 3.331 5.988 10.76 2.448 (51.9 vol%) X- axis *** 3D Octahedron CFRPLA 0.556 1.682 3.023 5.433 9.766 2.333 (51.9 vol%) Y- axis *** 3D Octahedron CFRPLA 0.556 1.88 3.379 6.073 10.92 2.466 (51.9 vol%) Z- axis *** Aluminum 2.7 69 26 9.5 3.5 3.077 Steel 7.9±0.15 200 25±0.5 3.2±0.1 0.41±0.02 1.790 Titanium 4.5 112.5±7.5 25±2 5.55±0.35 1.23±0.08 25.000 alloys Diamond 3.53 1,220 346 98 28 9.895 (C) Note: * Carbon fiber average diameter of 7 µm, average length of 150 µm, and aspect ratio of 21.4. 15 vol% CF for CFRPLA; ** Experimental testing data;

The experimental and modeling data, and the data of some popular materials are summarized in Table 4, and considering the contribution of the material density ( r ), the specific stiffness (Young’s modulus over density, E/ r ), square root of young’s modulus to density ( √/ρ are also listed in the table. The 2D cuttlefish bone model has higher values of planar values of E, E/ r , E/ r 2, E/ r 3 and √/ρ than that of the 2D hexagon model, respectively. The 3D octahedron model has higher values of E, E/ r , E/ r 2, E/ r 3 and √/ρ in all three directions than that of the 2D hexagon model, respectively. These density related Young’s moduli of the 2D cuttlefish bone model in Y-axis direction is higher than that of the CFRPLA bulk composites, which is the result of evolution for 77 cuttlefishes to take high pressure in the deep sea. The 2D cuttlefish bone model and the

3D octahedron model have the potential to be optimized for better performance.

CONCLUSION

Cellular structural composite product design can be inspired by biomimetics. Bulk composites can be evaluated based on homogeneity combined with Monte Carlo method.

Cellular structural composites can be characterized based on representative volume elements (RVEs). The material properties can be predicted by conducting tensile/compression and shearing tests on the RVE of the cellular composites based on the bulk composite material properties predicted from the above steps. The cuttlefish bone 2D model has higher planar stiff nesses than that of the hexagon 2D model, especially in the vertical direction which is the result of evolution for cuttlefishes to take high pressure in deep sea. 3D topological models have overall better behavioral properties than the 2D topological model. 3D cellular composite product can be realized and fabricated by 3D printing. Optimization of topological model through fillet is simple and very effective in improving the performance like stiffness, Young’s modulus and drastically lowers the failures stress, Von Mises stress. Fillets in the structure design can improve the stiffness and lower the Von Mises stress of the cellular structures. From specific stiffness and √/ρ values it is very clear that the light weight material saving design is possible to model and realize such that they can offer a better property than traditional bulk composites at much lower weight.

Further work could be done by developing algorithm for 2D and 3D cellular structural composites; optimizing the design by sensitivity analysis of the topologies; 78 refining the modeling techniques by considering tension-compression asymmetry material model, material nonlinearity, large strain and large deformation, and buckling and failures, extending the approach for different fibers like glass fiber, Kevlar fiber. And much should be attributed to realization of topological models through advance fabrication process, improving the 3D printing quality.

LITERATURE CITED

1. Vasiliev, Valery V., and Evgeny V. Morozov. "Introduction." In Advanced

Mechanics of Composite Materials and Structural Elements. Third ed. Burlington:

Elsevier Science, 2013.

2. Justin Hale. Boeing 787 from the Ground Up. QTR_04 06 a quarterly publication.

BOEING.COM/COMERCIAL/AEROMAGAZIE.

www.boeing.com/commercial/aeromagazine/articles/qtr_4_06/article_04_2.html

3. Kumar, S., and J.-P. Kruth. "Composites by Rapid Prototyping Technology."

Materials & Design 31, no. 2 (2010): 850-56. doi:10.1016/j.matdes.2009.07.045.

4. Compton BG, Lewis JA. 3D-Printing of lightweight cellular composites. Adv Eng

Mater 2014;26:5930-5.

5. Hegab HA. Design for additive manufacturing of composite materials and potential

alloys: a review. Manuf Rev 2016;3:1-17.

6. Meisel NA, Williams CB, Druschitz A. Lightweight metal cellular structures via

indirect 3D printing and casting. In: Proceedings of the international solid freeform

fabrication symposium. TX; 2012. p. 162-76. 79

7. 3D printer and 3D printing news. Harvard engineers use new resin inks and 3D

printing to construct lightweight cellular composites. June 27, 2014 at

http://www.3ders.org/articles/20140627-harvard-en gineers-use-new-resininks-and-

3d-printing-to-construct-lightweight-cellular -composites.html . access online in

January 11, 2016

8. "Local Motors to 3D Print Cars in 12 Hours, Recycle Old Cars, & Research Printing

with Graphene & Metals." 3DPrint.com | The Voice of 3D Printing / Additive

Manufacturing. March 26, 2015. Accessed July 28, 2016.

https://3dprint.com/51866/local-motors-plans-3d-print/.

9. GROZDANIC, LIDIJA. "Watching You Chair- Honeycomb Lightweight Seating

Structure - EVolo | Architecture Magazine." EVolo Architecture Magazine. April 3,

2012. Accessed July 6, 2015. http://www.evolo.us/design/watching-you-chair-

honeycomb-lightweight-seating-structure/.

10. Gibson LJ, Ashby MF. Cellular solids: structure and properties. second ed.

Cambridge: Cambridge University Press; 1997.

11. Zok FW, Rathbun H, He M, Ferri E, Mercer C, McMeeking RM, et al. Structural

performance of metallic sandwich panels with square honeycomb cores. Phil Mag

2005;85:3207-34.

12. Hammetter CI. Pyramidal lattice structures for high strength and energy absorption.

J Appl Mech 2012;1:369-84.

13. Carty WM, Lednor PW. Monolithic ceramics and heterogeneous catalysts:

honeycombs and foams. Curr Opin Solid St M 1996;1:88-95. 80

14. Banhart J, Baumeister J, Weber M. Damping properties of aluminium foams. Mat

Sci Eng A 1996;205:221-8.

15. Kuhn J, Ebert HP, Arduini-Schuster MC, Büttner D, Fricke J. Thermal transport in

polystyrene and polyurethane foam insulations. Int J Heat Mass Tran 1992;35:1795-

801 .

16. EL-Dessouky HM, Lawrence CA. Ultra-lightweight carbon fibre/thermoplastic

composite material using spread tow technology. Compos Part B 2013;50:91-7.

17. Chiachio M, Chiachio J, Rus G. Reliability in composites e a selective re view and

survey of current development. Compos Part B 2012;43(3):902-13.

18. Council NR. NASA space technology roadmaps and priorities: restoring NASA's

technological edge and paving the way for a new era in space. 2012.

19. “Lightweight, heavy impact”. Mckinsey & Company. May 2013. Accessed June 21,

2015.

http://www.mckinsey.com/~/media/mckinsey/dotcom/client_service/automotive and

assembly/pdfs/lightweight_heavy_impact.ashx .

20. Milovac, Dajana, Tatiana C. Gamboa-Martínez, G. Gallego Ferrer, M. Ivankovic,

and H. Ivankovic. "HIGHLY POROUS

HYDROXYAPATITE/POLYCAPROLACTONE COMPOSITE SCAFFOLD:

PHYSICOCHEMICAL AND IN VITRO BIOLOGICAL PROPERTIES." In 16th

European Conference on Composite Materials, ECCM16 . 2014.

21. Denton, E. J., and J. B. Gilpin-Brown. "The Buoyancy of the Cuttlefish, Sepia

Officinalis (L.)." J. Mar. Biol. Ass. Journal of the Marine Biological Association of

the United Kingdom 41, no. 2 (1961): 319-42. doi:10.1017/S0025315400023948 81

22. Gérrard Eddy Jai Poinern, Ravi Krishna Brundavanam, Derek Fawcett, Nanometre

Scale Hydroxyapatite Ceramics for Bone Tissue Engineering, American Journal of

Biomedical Engineering, Vol. 3 No. 6, 2013, pp. 148-168. doi:

10.5923/j.ajbe.20130306.04.

23. Drumright, R. E., Gruber, P. R. and Henton, D. E. (2000), Polylactic Acid

Technology. Adv. Mater., 12: 1841-1846. doi:10.1002/1521-

4095(200012)12:23<1841::AID-ADMA1841>3.0.CO;2-E

24. Cadman J. The design of cellular materials inspired by nature-characterisation. In:

Design and fabrication. University of Sydney; 2012.

25. Wilk D. Biomimicry: innovation inspired by nature. Cah Du Cine 2002;6:52-5.

26. Sherrard KM. Cuttlebone morphology limits habitat depth in eleven species of Sepia

(Cephalopoda: sepiidae). Biol Bull-Us 2000;198:404-14.

27. Mayer G. Rigid biological systems as models for synthetic composites. Sci

2005;310:1144-7

28. Barthelat F, Espinosa HD. Elastic properties of nacre aragonite tablets. In: The 2003

SEM annual conference and exposition on experimental and applied mechanics;

2003. Charlotte, North Carolina.

29. Kolednik O, Predan J, Fischer FD, Fratzl P. Bioinspired design criteria for damage-

resistant materials with periodically varying microstructure. Adv Funct Mater

2011;21:3634-41.

30. Zhou S, Li Q. The relation of constant mean curvature surfaces to multiphase

composites with extremal thermal conductivity. J Phys D Appl Phys 2007;40: 6083-

93. 82

31. Kruijf ND, Zhou S, Li Q, Mai YW. Topological design of structures and composite

materials with multiobjectives. Int J Solids Struct 2007;44:7092-109.

32. Gower D, Vincent J. The mechanical design of the cuttlebone and its bathymetric

implications. Biomimetics 1996;4:37-58.

33. Cheung H-A, Lau K-T, Lu T-P, Hui D. A critical review on polymer-based

bioengineered materials for scaffold development. Compos part B 2007;38(3): 291-

300.

34. Huda S, Reddy N, Yang Y. Ultra-light-weight composites from bamboo strips and

polypropylene web with exceptional flexural properties. Compos Part B

2012;43(3):1658-64.

35. Lepore E, Faraldi P, Bongini D, Boarino L, Pugno N. Plasma and thermoforming

treatments to tune the bio-inspired wettability of polystyrene. Compos Part B

2012;43(2):681-90.

36. Sebastian W, Gegeshidze G, Luke S. Positive and negative moment behaviours of

hybrid members comprising cellular GFRP bridge decking epoxy-bonded to

reinforced concrete beams. Compos Part B 2013;45(1):486-96.

37. Koronis G, Silva A, Fontul M. Green composites: a review of adequate materials for

automotive applications. Compos Part B 2013;44(1):120-7.

38. Okereke MI, Akpoyomare AI, Bingley MS. Virtual testing of advanced composites,

cellular materials and biomaterials: a review. Compos Part B 2014;60: 637-62.

39. Dhand V, Mittal G, Rhee KY, Park S-J, Hui D. A short review on basalt fiber

reinforced polymer composites. Compos part B 2015;73:166-80. 83

40. Affatato S, Ruggiero A, Merola M. Advanced biomaterials in hip joint arthroplasty:

a review on polymer and ceramics composites as alternative bearings. Compos Part

B 2015;83:276-83.

41. Burns L, Mouritz AP, Pook D, Feih S. Bio-inspired hierarchical design of composite

T-joints with improved structural properties. Compos Part B 2015;69:222-31.

42. Fauziyah S, Soesilohadi RCH, Retnoaji B, Alam P. Dragonfly wing venous

crossjoints inspire the design of higher-performance bolted timber truss joints.

Compos Part B 2016;87:274-80.

43. Mittal V, Saini R, Sinha S. Natural fiber-mediated epoxy composites: a review.

Compos Part B 2016;99:425-35.

44. Sun X, Liang W. Cellular structure control and sound absorption of polyolefin

microlayer sheets. Compos Part B 2016;87:21-6

45. Mayer, G., and M. Sarikaya. "Rigid Biological Composite Materials: Structural

Examples for Biomimetic Design." Experimental Mechanics 42, no. 4 (2002): 395-

403. doi:10.1007/BF02412144.

46. Wegst UGK, Bai H, Saiz E, Tomsia AP, Ritchie RO. Bioinspired structural

materials. Nat Mater 2015;14:23e36.

47. Mayo SC, Chen F, Evans R. Micron-scale 3D imaging of wood and plant

microstructure using high-resolution X-ray phase-contrast microtomography. J

Struct Biol 2010;171:182e8.

48. Cadman J, Zhou S, Chen Y, Li Q. Cuttlebone: characterisation, application and

development of biomimetic materials. J Bionic Eng 2012;09:367-76. 84

49. Cranford, S. W. "Increasing Silk Fibre Strength through Heterogeneity of Bundled

Fibrils." Journal of The Royal Society Interface 10, no. 2 (2013): 20130148.

50. Karihaloo, B. L., K. Zhang, and J. Wang. "Honeybee Combs: How the Circular Cells

Transform into Rounded Hexagons." Journal of The Royal Society Interface 10, no.

86 (2013): 20130299.

51. "How to Raise Honeybees: A Beginner's Guide." Mother Earth News. Accessed

June 22, 2015. http://www.motherearthnews.com/homesteading-and-livestock/how-

to-raise-honeybees-zmaz85zsie.aspx

52. Tian, Wenlong, Lehua Qi, Changqing Su, Junhao Liang, and Jiming Zhou.

"Numerical Evaluation on Mechanical Properties of Short-fiber-reinforced Metal

Matrix Composites: Two-step Mean-field Homogenization Procedure." Composite

Structures 139 (April 1, 2016): 96-103. Accessed April 14, 2016.

doi:10.1016/j.compstruct.2015.11.072.

53. Jeffery, G. B. "The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid."

Proceedings of the Royal Society A: Mathematical, Physical and Engineering

Sciences 102, no. 715 (November 01, 1922): 161-79. Accessed April 14, 2016.

doi:10.1098/rspa.1922.0078.

54. Vincent, Michel, T. Giroud, A. Clarke, and C. Eberhardt. "Description and Modeling

of Fiber Orientation in Injection Molding of Fiber Reinforced Thermoplastics."

Polymer 46, no. 17 (2005): 6719-725. Accessed April 23, 2016.

doi:10.1016/j.polymer.2005.05.026.

55. "Mark One, the World's First Carbon Fiber 3D Printer Now Available for Pre-order."

3ders.org. February 18, 2014. Accessed June 23, 2015. 85

http://www.3ders.org/articles/20140218-mark-one-the-world-first-carbon-fiber-3d-

printer-now-available-for-pre-order.html .

56. Arevo, Perfecting the art of 3D printing. at http://www.arevolabs.com/, access online

in January 12, 2016.

57. T. Edwards, 3DXTech offers new carbon fiber-reinforced PLA and ABS filaments.

3DXMax CFR, 3D Printing News. December 15, 2014 at http://3dprint.

com/31019/3dxtech-carbon-fiber-filaments/, access online in January 12, 2016

58. "Behind the Wheel of World's First 3D-printed Car: Futuristic Vehicle Is Made in 44

Hours Using Just 40 Parts ... and It Costs £11,000." Daily Mail. September 15, 2014.

Accessed June 23, 2015. http://www.dailymail.co.uk/sciencetech/article-

2756181/History-making-The-futuristic-3D-printed-car-just-40-parts-costs-11-000-

takes-44-hours-build.html .

59. "Infill." Paul Sieradzki. January 19, 2015. Accessed June 21, 2015.

https://paulsieradzki.wordpress.com/graphic_design/r3-printing/infill/ .

60. Bauer, J., S. Hengsbach, I. Tesari, R. Schwaiger, and O. Kraft. "High-strength

Cellular Ceramic Composites with 3D Microarchitecture." Proceedings of the

National Academy of Sciences 111, no. 7 (2014): 2453-458.

doi:10.1073/pnas.1315147111.

61. Li, Eric, Zhongpu Zhang, C.c. Chang, G.r. Liu, and Q. Li. "Numerical

Homogenization for Incompressible Materials Using Selective Smoothed Finite

Element Method." Composite Structures : 216-32.

62. Dai, Gaoming, and Weihong Zhang. "Size Effects of Effective Young’s Modulus for

Periodic Cellular Materials." Science in China Series G: Physics, Mechanics and 86

Astronomy Sci. China Ser. G-Phys. Mech. Astron. 52, no. 8 (2009): 1262-270.

doi:10.1007/s11433-009-0151-9.

63. 3DXTECH, 3DXMAXTM CFR Carbon Fiber Reinforced PLA Filament Product

Data Sheet, Rev.12.2014v3, at www.3DXTech.com, access online in March 15,

2016

64. "Tessellation." - Wikimedia Commons. Accessed June 22, 2015.

https://commons.wikimedia.org/wiki/Tessellation#3D_tessellation .

65. "Tessellation." - Wikimedia Commons. Accessed June 21, 2015.

https://commons.wikimedia.org/wiki/Tessellation#/media/File:HC-A4.png .

66. ASTM D 3039. Standard test method for tensile properties of polymer matrix

composite materials. United States of America. In: Annual book of ASTM

standards; 2008