Biomimetic Design Based on BambooMT 11.07

D.V.W.M. de Vries

December 2010

Biomimetic Design Based on Bamboo

StudentD.V.W.M. de Vries

SupervisorsDr. Stefanie Feih

December 2010

Internship Project atRoyal Melbourne Institute of TechnologySchool for Aerospace, Mechanical andManufacturing Engineering

Education ProgramMaster Mechanical Engineering(Mechanics of Materials) atUniversity of Technology Eindhoven

Contents

Preface 9

Introduction 11

1 Biomimicry 131.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Hierarchical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Examples of Biomimicking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Complex Analysis and a Systematic Approach . . . . . . . . . . . . . . . . . . 16

2 Analysis of the Microstructure and Mechanical Properties of Bamboo 192.1 Microstructure of Bamboo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Introduction to Bamboo . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Hierarchical Structure of Bamboo . . . . . . . . . . . . . . . . . . . . . 192.1.3 Bamboo as a Functionally Graded Material (FGM) . . . . . . . . . . . 22

2.2 Mechanical Properties of Bamboo . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Failure Behaviour of Bamboo . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Objectives 27

4 Biomimetic Design and Manufacturing 294.1 Set-up for Biomimetic Designs of Bamboo . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Boundary Conditions for Biomimetic Design . . . . . . . . . . . . . . 294.1.2 Design Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.3 Biomimetic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Rapid Prototyping and Material Testing . . . . . . . . . . . . . . . . . . . . . 364.2.1 Rapid Prototyping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Materials Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.3 Sample Orientation in Rapid Prototyping . . . . . . . . . . . . . . . . 444.2.4 FEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.5 Test Set-up Configurations . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Test Plan 555.1 Test Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Test plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Bending Analysis 63

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6.1 4-Point-Bending Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Finite Element Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2.1 Meshing Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2.2 Opportunities and Improvements . . . . . . . . . . . . . . . . . . . . . 68

6.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Conclusions & Recommendations 71

Discussion 75

Bibliography 77

A M-files for Design of Models 81

B M-files for Testing of Materials 83

C Additional SolidWorks Drawings 85C.1 Testing Supply: Fixture for 3-Point-Bending . . . . . . . . . . . . . . . . . . . 85C.2 Tensile Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

D Costs of Rapid Prototyping 89

E Buckling Analysis 91E.1 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91E.2 FEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91E.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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List of Figures

1.1 Ashby maps for biological materials with the Young’s modulus E (top) andstrength σ (bottom) as a function of density ρ [5] . . . . . . . . . . . . . . . . 15

1.2 Overview of design strategies in nature and engineering [17] . . . . . . . . . . 161.3 Examples of biomimetic designs . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Systematic approach in biomimetic design of materials [19] . . . . . . . . . . 18

2.1 Split bamboo with internodal culm walls and partitioning nodes [1] . . . . . . 202.2 Hierarchical structure of bamboo [15] . . . . . . . . . . . . . . . . . . . . . . . 212.3 Transversal cross-section of bamboo with parenchyma cells [1, 19] . . . . . . . 212.4 Distribution of cell types as function of culm wall thickness [1] . . . . . . . . 222.5 Distribution of fibres (dark regions) in a bamboo culm [1] . . . . . . . . . . . 23

4.1 The two coordinate systems used for the model design . . . . . . . . . . . . . 324.2 General figure of fibre model III and IV with parameters needed to calculate

α and φf,layer i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Schematic distribution of volume fraction of fibres φf as function of dimension-

less thickness t and fibre radius Rf as function of layer number i for model IIIand IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Volume fraction of fibres φf as function of radial position R for model IV with4 layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5 Cross-sections of model IV (a) and bamboo (b) . . . . . . . . . . . . . . . . . 354.6 Example of slicing of a sample geometry in rapid prototyping . . . . . . . . . 374.7 Schematic representation of test set-up for tensile testing with extensometer . 394.8 Schematic representation of the normal material model (top) and simple

material model (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.9 Material model normal for material DM8410 . . . . . . . . . . . . . . . . . . 414.10 Test results from tensile tests on different materials . . . . . . . . . . . . . . . 434.11 Force-displacement response for 3-point-bending tests . . . . . . . . . . . . . 454.12 Boundary conditions and coordinate system for 3-point-bending, as used in

Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.13 Force-displacement response for 3-point-bending in both FEA and testing,

where numerical results are now utilising the material data from tensile tests 494.14 Force-displacement response for 3-point-bending in both FEA and testing, with

mesh refinement in axial direction . . . . . . . . . . . . . . . . . . . . . . . . 514.15 Force-displacement response for 3-point-bending in both FEA and testing, with

mesh refinement in radial direction . . . . . . . . . . . . . . . . . . . . . . . . 514.16 Indentation of fixture roller into structure in 3-point-bending . . . . . . . . . 52

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4.17 Contour plots showing the equivalent plastic strain for 3-point-bending (left)and 4-point-bending (right) tests with a support span of 150 mm in FEA . . 54

5.1 Material models (σ− ε -behaviour) for fibre-like (a) and matrix-like (b) materials 565.2 Image of the 4-point-bending test on one of the samples . . . . . . . . . . . . 585.3 Cross-section of Model IIb in SolidWorks . . . . . . . . . . . . . . . . . . . . 615.4 Cross-section of Model III in SolidWorks, with from left to right respectively

2, 3 and 4 fibre layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.5 Cross-section of Model IV in SolidWorks, with from left to right respectively

2, 3 and 4 fibre layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.6 Cross-section of Model IV in SolidWorks, with 3 fibre layers (left) and

photograph of the cross-section of the same model, manufactured with rapidprototyping (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1 Test results for 4-point-bending; with results for model IIb (top), model III(middle) and model IV (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Cross-section of model III with 2 fibre layers in Abaqus . . . . . . . . . . . . 676.3 Top view of a part of the created mesh of model III with 2 fibre layers in Abaqus 686.4 Schematic representation of the macroscopic constitutive behaviour of a

material, by J.A.W. van Dommelen and J.G.F. Wismans, Faculty MechanicalEngineering, University of Technology Eindhoven . . . . . . . . . . . . . . . . 70

6.5 Biomimetic design sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.1 Block scheme representing the matlab routine used to create the biomimeticdesigns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.1 Block scheme representing the matlab routine used to create the material models 84

C.1 Drawing for the fixture rollers used as support in bending tests . . . . . . . . 86C.2 Drawing for the tensile bar [30] . . . . . . . . . . . . . . . . . . . . . . . . . . 87

E.1 The load on the central node, applied on the structure by using a MPC . . . 92E.2 The ratio between the 1st mode eigenvalues from FEA and the analytical

solution of the eigenvalue versus the number of axial elements for columnswith a different slenderness ratio . . . . . . . . . . . . . . . . . . . . . . . . . 93

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List of Tables

2.1 Properties of Bamboo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Properties of Woods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Measured fracture toughness of natural and engineering materials [14] . . . . 26

4.1 Material properties of FullCure and Digital Materials used for rapid prototyp-ing with the Connex350TM printer, as specified by manufacturer . . . . . . . 38

4.2 Material properties obtained from tensile tests . . . . . . . . . . . . . . . . . 404.3 Comparison between horizontal and vertical sample orientation in the rapid

prototyping machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Properties of different test set-up configurations (obtained with FEA) . . . . 53

5.1 Radius of gyration and difference in effective stiffness for different models . . 59

6.1 Stiffness and maximum force for each model, obtained from 4-point-bendingtests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

D.1 Material costs for rapid prototyping . . . . . . . . . . . . . . . . . . . . . . . 89

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Preface

This internship project is performed as part of the master track Mechanics of Materialsin the master Mechanical Engineering, that I am following at the University of TechnologyEindhoven (TU/e). This internship project was undertaken from 1 September 2010 to 23December 2010 at the Royal Melbourne Institute of Technology, at the School for Aerospace,Mechanical and Manufacturing Engineering, located at Bundoora East, Melbourne.

First, I would like to thank my supervisor at RMIT, Dr. Stefanie Feih for her supportand advice in this project, and Prof. Dr. Ir. Marc Geers and Dr. Ir. Ron Peerlings fromthe TU/e, who helped me in organising this internship in Australia. Moreover, I want tothank Dr. Mike Burton, who supported me with Rapid Prototyping and Peter Tkatchyk,who guided me during several experiments in the Materials Testing Lab. Finally, I wouldlike to say thanks to all staff members and students of RMIT Bundoora and anyone else whosupported me whilst I have been working in this interesting field.

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Introduction

In 1859 Charles Darwin brought us new insights in the world of evolution. Nature has acontinuous thrive to improve its properties and adapt to its changing environment. This ismade possible by natural selection. Darwin’s work ‘On the Origin of Species, 1859’ explainsthis theory by the principle of natural selection, and his work is still one of the fundamentalprinciples on which the evolution theory is based. Organisms have developed during millionsof years, creating their own specific, most favourable properties for specific circumstances.Materials science and technology has also always had a thrive for improvement - where theapplication is defining the specific goal. In aerospace and aviation technology - one of themost sophisticated research areas - the focus is mostly on a high load bearing efficiency, i.e.a superb strength to weight ratio. Moreover, environmental issues nowadays put even morepressure on the development of lightweight structures. A relatively new research area inMaterials Science is biomimicking, which links technology to nature. Shortly, biomimickingis the design of materials based on biological structures.

Biological structures mostly have optimal features, depending on their environment andloading conditions. Several examples from nature can be given that offer high load bearingefficiency, see also Chapter 2. This is the reason why biomimicking offers a lot of potentialfor lightweight design. In this internship project, focus will be given to the biomimetic designbased on bamboo. The superb mechanical properties of bamboo result in a diverse set ofquestions. Special interest is given to the microstructure of bamboo. What are the reasonsbamboo is structured the way it is? Are there certain structural features that highly contributeto the mechanical behaviour, and can we learn from that? Can we create a biomimetic designthat is able to capture these structural features, mimicking the properties of bamboo indifferent loading modes and also creating the failure behaviour? And more importantly: arewe able to manufacture that?

In order to create realistic goals for this project, first more insight must be gottenabout biomimicking and the structure of bamboo. Therefore, in Chapter 1, the conceptsof biomimicking will shortly be explained. Some examples of biomimetic designs will begiven to state the opportunities of biomimetic design in Materials Science and Engineering.A systematic approach will give design guide lines for the design procedure in this project.Following that, the structure of bamboo will be analysed in more detail in Chapter 2. Thehierarchical structure will be presented and the microstructure is analysed. Together with themechanical properties and a review on the failure behaviour, this will generate a good startingpoint for the biomimetic design. The information of these two chapters is needed to state theobjectives of the internship project in more detail. This is done in Chapter 3. In section 4.1the design procedure is explained for this project. The different models that will be usedin an attempt to capture the structural features in the microstructure of bamboo that bringalong a certain mechanical behaviour are clarified. Boundary conditions that will put some

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restrictions on the design are given. Finally, the biomimetic designs are presented. Afterthe explanation of the design procedure, one of the most important aspects in this projectis the ability to manufacture these designs. The rapid prototyping process is explained insection 4.2. A couple of initial designs will be processed and tested in bending tests. Thesebenchmark experiments will give more insight in the properties of rapid prototyping. Inthis chapter, a couple of aspects are analysed to optimise test conditions and finite elementanalyses (FEA). As will be shown, both rapid prototyping, material choice and test set-upparameters will influence the test plan. This test plan is presented in Chapter 5. Selectedbiomimetic designs will be created, test set-up parameters will be given and a hypothesis isgiven regarding the test results. It will be interesting to see whether the objectives, stated inChapter 3, can be fulfilled. The latter can be found in Chapter 6, where test results are given.Besides that, numerical analyses are undertaken, to get more insight in test results. Are weable to predict the mechanical behaviour of this (anisotropic) structure precisely? Are theresults from testing as expected? And are we now able to answer the main objective questionsstated in Chapter 3? This will finally lead to the conclusions, which are given together withsome recommendations for future work on the biomimetic design of bamboo.

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Chapter 1

Biomimicry

As stated in the introduction, biological elements have progressed, improved and refinedthemselves by natural selection over millions of years. To create novel improvements intechnology, people have been inspired by nature for a long time. Inspiration by natureis the basic principle of biomimicking, a research field in Material Science that is growingrapidly [18, 21]. Other terms that refer to biomimicking are bionic, biomimetic and bio-inspired design. These terms are used throughout literature disorderly. As stated in theintroduction, biomimicking of materials involves approaches to synthesise materials inspiredby biological systems [18]. Biology creates many opportunities for improvements in materialdesigns: on growth and functional adaption, on damage repair and self-healing and onhierarchical structuring [17]. The latter opportunity is addressed in this project, with focuson the complex microstructure. The other aspects are definitely interesting but less relevantin relation to this survey. In this chapter, a short overview will be given of biomimicking ingeneral and some principles and examples will be shown, based on research work done in thepast.

“Biomimicking in Material Science is not a research field that copies features from biology;it is a fast developing research field in which one should learn from biological design principlesto create new, previously not possible aspects in material behaviour.” [21]

1.1 Background

There are plenty of examples of inventions that are inspired on nature, of which the earlyattempts at flying, inspired by birds, is one of the most obvious. In the same way, the earlywork of Leonardo da Vinci (1452-1519) and Galileo Galilei (1564-1642) is illustrative for thedesign procedure from observation from nature to technical applications. But also in MaterialScience, inspiration can be gotten from biological structures. Nowadays, there is a continuousthrive to make materials ‘lighter and stronger’, especially in Aerospace Engineering. This ismore and more driven by environmental issues, e.g. set by the EU initiative Clean Sky [29]. Aslimits in material research are being reached, improvements are possible by interaction withstructural design. Optimised structural design can save materials and will lead to diminishedemissions, e.g. when they are utilised for means of transportation. One of the essentialobjectives of biomimicking is to improve the load bearing efficiency [19].

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1.2 Hierarchical Structure

The fundamental principle that creates diverse opportunities for biomimicking in the scopeof Material Science is related to the statement in section 1.1: ‘improvements are possibleby interactions with structural design’. This principle can be understood by an analysisof the basic elements of which the biological materials are composed, in combination withtheir mechanical properties. The latter can certainly be clarified by application of Ashbydiagrams [5]. The Ashby maps are shown in figure 1.1. In here, the red coloured areasindicate the domain in which the properties of bamboo can be found, the biological materialon which material design is inspired in this study. As can be found from this diagram,the density ρ of biological materials is low compared to synthetic structural materials. Forexample, dry wood (southern pine) has an average density of 600 kg/m3, whereas copperhas a density of 8900 kg/m3 [2]. The density of natural systems rarely exceeds 103 kg/m3,while engineering materials have a density in the range from 104 to 1010 kg/m3. Besidesthat, the range of Young’s moduli is broad, covering 5 orders of magnitude from 10−3 to102 GPa. This is also the case for the strength, covering 4 orders of magnitude between 10−1

to 103 MPa. As can be obtained from this, there is a large variation in properties betweenbiological materials. By natural selection, they are all in one way or another optimised totheir specific environment. On top of the wide range of properties of biological materials,some have excellent specific strength (σ/ρ) (e.g. pure fibre of bamboo, feather shaft andwood), comparable with engineering alloys and metals [16]. Moreover, pure bamboo fibreeven has a higher specific modulus (E/ρ) than steel.

The large variation in properties of natural structures and good specific strength andmodulus might be quite surprising on first sight, especially when the elements, of which thebiological materials are composed, are considered. Natural materials consist of a relativelysmall scope of constituent elements by comparison to the scope of elements that are usedfor engineering materials. Furthermore, the production of inorganic materials in natureoccurs generally at ambient temperature and under isothermal and isobaric conditions [18].From the ages of copper, bronze and iron, to the industrial revolution based on steel andthe information age based on silicon semiconductors, all engineering materials require hightemperature processing. Nevertheless, biological materials - made of comparatively poorbasic elements - still have remarkable functional (specific) properties [17]. These functionalproperties, like the strength of some natural materials (e.g. shells, bones, wood), are derivedfrom its hierarchical structure rather than from the material itself. The hierarchical structureis the highly organised structure of a material, found at all different dimensional scales andoften smartly interacted between structures at the individual length scales. In biologicalmaterials, the hierarchical organisation is inherent to the design. Examples can be found inthe bone, abalone shell and crab structure [18], but bamboo has a highly organised structureon different dimensional scales as well. P. Fratzl [17] gives a clear overview of the strategiesused in nature and engineering to achieve a certain functionality, see figure 1.2.

From figure 1.2, hierarchical structuring in nature originates from the principles of self-assembly. In nature both the material and the whole organism grows, providing a control ofthe structure of materials at all dimensional scales. This hierarchical structuring from nano-to macro-level provides is able to create interesting specific material properties. It certainlyis an enormous challenge to manufacture such a highly organised hierarchical structure foran engineering material, and it will require more interaction between material engineers andmechanical engineers to create such a structural material [18]. However, it is clear that ‘we

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can learn from nature’. In section 1.4 a systematic approach will be presented to create abiomimetic design, however it must be stated that this methodology is not straightforward[20]. First, some examples will be given of biomimetic designs, which will emphasise theopportunities of biomimicking in Material Science.

Figure 1.1: Ashby maps for biological materials with the Young’s modulus E (top) andstrength σ (bottom) as a function of density ρ [5]

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Figure 1.2: Overview of design strategies in nature and engineering [17]

1.3 Examples of Biomimicking

In here, a few examples are presented of biomimetic designs to give an idea of the possibilitiesof biomimicking in Material Science. The first example is an aerospace application.Lightweight structures are inspired by the sandwich structures consisting of solid shells filledwith compliant cellular cores, as found in the cross-section of grass stems. The strength-to-weight ratio is optimised and the buckling resistance is increased. Besides that, also the bonestructure in birds wings provides a stiff lightweight design, on which sandwich structures aredesigned [18]. The latter can be seen in figure 1.3 (a). These kind of structures are widelyfound in avian materials.

Moreover, in literature biomimetic designs can be found of tubular structures based onbamboo, such as in [19], creating a better load bearing efficiency than a conventional tubularstructure and changing the failure mode in compression loading from local to global buckling.An example of a biomimetic design based on bamboo is given in figure 1.3 (b). The structureof bamboo is of special interest in this project, because its specific mechanical properties andfunctionally graded structure. In the next chapter, the microstructure of this natural materialis analysed in more detail. Besides the mentioned applications in Material Science (in whichbiomimicking is a relatively new approach (since 1980)), biomimicking has also been used asinspiration for different kind of applications, such as Velcro [18] and Kevlar [23].

1.4 Complex Analysis and a Systematic Approach

“Biomimicking in Material Science is not just a consequence of an observation of naturallyoccurring structures” [17].

This statement might need some more explanation. From the analysis of a biological structureat different length scales, one should notice that in nature plenty of boundary conditions can

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(a) Design of a sandwich structure [18]

(b) Biomimetic design based on bamboo [19]

Figure 1.3: Examples of biomimetic designs

be involved to create a particular hierarchical structure. Questions like ’What mechanicalproperty has been optimised?’ and ’Which structural design features have biological purposesonly, e.g. nutrition?’ may come up and answers to these kind of questions are notstraightforward. This makes biomimicking a complex design process, as basically the designanswer is known (the created biological structure), but the initial optimisation question(s)are complicated and might be interlaced too. Both physical and biological constraints of thebiological system have to be studied in biomimetic materials research. The multiple functionsof a single biological system make it such a complex system.

Although the optimisation question is initially unknown and may be quite complex asbiological systems can be optimised for different purposes, biomimetic materials researchstarts with the study of structure-function relationships. In this work, both the stressdistribution and failure modes of the internodal structure of bamboo in bending andcompression are of particular interest, as will be stated later in Chapter 3. Approachesfor biomimetic design still seem to be quite serendipitous. Currently, no general approachhas been developed for biomimetics, although a number of people are currently developing’standardised’ methods to search in literature for biological functional analogies to implement[20]. Nevertheless, some papers on bionic design use a systematic approach. In [19], thefollowing scheme is followed, which captures most basic steps for biomimetic design, seefigure 1.4. This scheme will roughly be followed in this thesis, starting with the analysisof the microstructure of bamboo in the next chapter. A biomimetic design will be createdthat is based on structural features, as found in bamboo. This biomimetic design will bemanufactured, tested, and finite element analyses (FEA) will be performed in order to validatethe similarity in mechanical behaviour between the design and the bamboo. One of theobjectives, stated later also in Chapter 3 is to capture some structural features in bamboo(e.g. the influence of fibres) that provide such a mechanical behaviour. In combinationwith manufacturing techniques, numerical analyses should give more insight in the structuraldesign of bamboo. The design sequence of figure 1.4 will be reviewed later. Conclusively,recommendations for further design optimisation will be given.

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Figure 1.4: Systematic approach in biomimetic design of materials [19]

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Chapter 2

Analysis of the Microstructure andMechanical Properties of Bamboo

Prior to starting a biomimetic design procedure, and discussing the objectives of this surveyin more detail, it is necessary to obtain information about the microstructure of bamboo - thenatural material on which biomimicking is based in this project. The hierarchical structureof bamboo will be analysed, and the role of fibre distribution will be explained in more detail.The microstructure can indicate the basic features in the design that are used to withstandnatural forces, like bending forces due to wind loads and compression forces due to the massof the bamboo stem. What does this microstructure look like? And what are the typicalmechanical properties and failure mechanisms of bamboo?

2.1 Microstructure of Bamboo

2.1.1 Introduction to Bamboo

Bamboo (Lat. Bambusa) is a group of perennial evergreens that belongs to the true grassfamily Poaceae, subfamily Bambusoideae, tribe Bambuseae. There exist 75 genera and about1250 species [1]. These tall grasses only produce a primary shoot without secondary growth,contrarily to most woods. The plant has a nodal structure (see figure 2.1); the internodalregions of the stem are hollow and the vascular bundles in the cross section are scatteredthroughout the stem instead of in a cylindrical arrangement. The absence of secondarygrowth causes the stems of bamboo to be columnar rather than tapering. Bamboos are someof the fastest growing plants in the world. Bamboos are of notable economic and culturalsignificance, especially in (South) East Asia, being used for building materials, as a foodsource, and as a versatile raw product [4].

2.1.2 Hierarchical Structure of Bamboo

The anatomical structure of bamboo culm determines its properties. At first sight, the culmshows a rather simple anatomical construction. Since growth and cellular differentiationoccurs fast, the tissue structure must be simple [1]. As stated in the introduction to bambooabove, there exist a lot of different bamboo species. However, variations in microstructureappear to be small and are considered to be less significant by comparison to the largestructural heterogeneity of the species.

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Figure 2.1: Split bamboo with internodal culmwalls and partitioning nodes [1]

Certain differences however, exist within theculm, and between species and genera. Thefocus here is to obtain the general trends ofthe microstructural features in bamboo.

The bamboo culm is characterised bynodes and internodes. The internodes ofalmost all bamboo species have a culm wallsurrounding a large cavity, called a lacuna[1]. In figure 2.1, a splitted bamboo culm isgiven, showing the typical nodal structure ofbamboo and the cavities. Because bambooproduces only a primary shoot, growthconditions apparently have little effect onthe culm structure, but ageing inducessome structural modifications. However,as stated before, in here only the generalmicrostructural trends are of interest. Inliterature a wide range of information can befound on the structure of bamboo, therefore

a reference is given to the papers [7] up to [16]. From this review, an analysis is made of thegeneral structure of bamboo. As other biological systems, bamboo has a hierarchical structure.This hierarchical structure of the internodal part of a bamboo culm is depicted schematicallyin figure 2.2 [15]. Bamboo is a highly organised multi-scale structured composite. In figure 2.2(a), the bamboo culm can be seen schematically. In the internodes, cells are strongly orientedaxially, which can be seen in figure 2.2 (b). The fibres are built up of many microfibril layers,which is shown in figure 2.2 (c). These microfibrils themselves have a cross-section that isapproximately pentagonal and are arranged in a honeycomb structure. A fibril contains manycontinuously elongated cellulose grains, staggered in twisted nature. The grains, which arethe basic building blocks of bamboo, are illustrated in figure 2.2 (d). All structures at differentdimensional scales will interact with each other and play a certain role in strengthening andtoughening.

Because the cells are strongly oriented axially in the internodal part, lateral movementof nutrients or liquids is greatly hindered. The nodes provide the transversal interconnectionwith their solid cross wall, called a diaphragm. In this thesis, the biomimetic design is basedon the microstructure of the internodal part of the bamboo culm. The nodal structure willtherefore not be considered in more detail.

A microscopic image of the transversal cross-section of the internode is shown in the leftside of figure 2.3. The ground tissue of a bamboo culm basically consists of parenchymacells, with embedded vascular bundles composed of metaxylem vessels, sieve tubes withcompanion cells, and fibres [1]. The region with a high density in the vascular bundle iscalled sclerenchyma and is composed of the cellulose microfibrils. These are groups of fibresand these are responsible for the bamboo strength. The veins are responsible for the transportof nutritious substances from the soil to all parts of the plant. The cellulose microfibrils aroundthe veins keep them straight along the whole culm. The lignin that surrounds the vascularbundles is called parenchyma, which acts as matrix and represents the weaker part of thebamboo composite [8]. This cell structure is also given in figure 2.3. Natural cellular materialsare often mechanically efficient: the honeycomb-like microstructure of wood, for instance,

20

Figure 2.2: Hierarchical structure of bamboo [15]

gives it an exceptionally high performance index for resisting bending and buckling [22].On average, a culm consists of about 52% parenchyma, 40% fibres and 8% conducting

tissue (vessels, sieve tubes, companion cells). These values may vary with species and withinternodal number. Nevertheless, some general microstructural features can be found whichare assumed to determine the superb mechanical properties of bamboo. One of the mostinteresting aspects is that bamboo can be seen as a natural functionally graded structure whichis both macroscopically and microscopically graded. The microscopically graded structure canbe seen in figure 2.3, where cell types vary considerably transversally across the culm wall [1].Also in axial direction of the bamboo culm bamboo is found to be a graded material, referredto as macroscopically graded. However, the focus will lie on the microscopically gradedstructure, thus on the cross-sectional area of the internodal part of the bamboo stem.

Figure 2.3: Transversal cross-section of bamboo with parenchyma cells [1, 19]

21

Figure 2.4: Distribution of cell types as function of culm wall thickness [1]

2.1.3 Bamboo as a Functionally Graded Material (FGM)

A FGM is characterised by a gradually varying material composition and structure overthe volume, changing the overall properties of the material. FGMs are widely found inthermo-mechanical systems, to reduce thermo-mechanical stresses. But they can also help toreduce stress concentrations upon certain mechanical loading conditions. As is indicated infigure 2.4, the cell type amount in bamboo varies over the cross-section of the bamboo culm,called the microscopically graded structure. Besides that, bamboo’s diameter, thickness andinternodal length have a graded structure too [13], referred to as the macroscopically gradedstructure. Thus bamboo can be seen as a natural FGM. Note that in this thesis, mainly themicroscopically graded structure is considered. Bamboo essentially consists of two materials,a fibre material and a surrounding matrix material, as stated before. The fibre material isindicated as dark regions in figure 2.5. From this figure and figure 2.4, it can be concluded thatthe bamboo fibres change both in volume concentration, shape and size over the culm wallthickness. The volume fraction of fibres φf is about 50% in the outer third of the culm wall [1].The fibre volume fraction at the most inner periphery is about 20% and the fibre volume atthe outer most periphery about 60% [1,14]. The overall fibre volume concentration, as statedbefore, is 40%. The reason that more fibres are found on the outer surface of bamboo, is thata higher strength is required in order to withstand wind loads, which is the most frequentloading condition of bamboo in nature [8]. Liese [1] obtained 4 different types of fibre bundleshapes. Because they are - for simplicity - not considered here for the biomimetic design (seeparagraph 4.1), these types are not set here into more detail. However, the fibre size changesover the cross-section. At the outer periphery, the size of the fibres is small compared withfibres at the inner periphery. This aspect will be taken into account for the biomimetic design.Possibilities of Digital Images Analysis (DIA) techniques can give more detailed insight inthe fibre distribution of specific bamboo species [8]. Another interesting study was done byNogata and Takahashi [16]. They show that the ingenious construction of bamboo is createdby self-optimising. Bamboo has a cell-based sensing system for sensing external mechanical

22

stimuli in order to create higher strength at high-stress positions in the stem. The latter willcreate a uniform strength distribution in bamboo, see also section 2.2.

The graded structure of bamboo leads to specific mechanical properties. An interestingaspect will be how the material behaviour of bamboo depends on this microstructural gradientin fibre volume concentration and fibre size. Is it possible to assign individual microstructuralfeatures in bamboo that are responsible for certain mechanical properties? And is it possibleto investigate that with a biomimetic design? Or can other aspects like the hierarchicalstructure, the fibre shape or even the structure at smaller length scales (cellular structure)not be dropped with respect to the overall behaviour? In the next paragraph, the (overall)mechanical properties of bamboo will be discussed.

Figure 2.5: Distribution of fibres (dark regions) in a bamboo culm [1]

2.2 Mechanical Properties of Bamboo

Since the bamboo culm shows a gradient in both fibre volume and shape over its transversalcross-section and the cells in the bamboo culm are strongly axially aligned too, the mechanicalproperties turn out to be highly anisotropic. Stiffness in axial direction is significantly higherthan in radial direction [1]. In the cross-sectional area of the bamboo culm, the mechanicalproperties show a gradient in radial direction. The denser area of fibres towards the outerperiphery increases the resistance against bending, by increasing the factor EI, where E is theYoung’s modulus and I the second moment of area. Furthermore, former studies [13] showthat the strength distribution in the cross-section of bamboo is proportional to the volumefraction of fibres across the culm thickness. This distribution accommodates a more equalstress distribution in the cross-section upon bending loading. For an indication of the stiffnessby the placement of fibres, also the radius of gyration is used, defined as

Rg =

√I

A(2.1)

in which I represents the second moment of area and A is the cross-sectional area. This radiuswill be used to predict the stiffness of different biomimetic designs.

For composite materials the rule of mixture is a common used principle to obtain averageproperties of the composite [13]. Bamboo can be seen as a natural composite. It can be arguedthat the rule of mixture is fully applicable, since the distribution of fibres is not uniform and

23

bonding between fibre and matrix material is not perfect by definition [8]. The rule of mixturestates that

ψc = ψfφf + ψm(1− φf ) (2.2)where ψ represents a specific (overall) mechanical property of the material and φf representsthe (overall) volume fraction of fibres. Since the volume fraction of fibres φf is not constant,this equation can be transformed to the following equation to obtain the properties at acertain radial position, such that the rule of mixture is applicable for bamboo [8]

ψc(t) = ψfφf (t) + ψm(1− φf (t)) with 0 ≤ t ≤ 1 (2.3)where t represents the non-dimensional thickness of the bamboo stem (t = 0 at the most innerradius and t = 1 at the most outer radius).

The Young’s modulus for the fibre material is 46 GPa and for the matrix material2 GPa [12, 14]. The distribution of the Young’s modulus and the tensile strength areinvestigated by [14], and show (approximately) the same distribution along the radius as thefibre volume concentration. This indicates that both Young’s modulus and tensile strengthare proportional to φf [14]. However, it was already noticed that the mechanical propertiesare highly anisotropic: the overall stiffness in axial direction of bamboo is much higher thanthe overall stiffness in radial direction, and also the stiffness in the radial direction showsa gradient. On top of that, it must be remarked that the anisotropic tubular structure ofbamboo greatly increases the propensity to fail due to non-linear effects. The latter is due toovalisation upon bending [25]. The most important mechanical properties of the bamboo fibreand matrix material (for this survey) are given in table 2.1. To give an idea of the superbmechanical properties of bamboo, these properties are related to properties of other woodstems in table 2.2 [14]. Moreover, as stated before in 1.2, the specific stiffness of bamboo fibreis better than that of steel. As can be obtained from table 2.2, the absolute value for theYoung’s modulus is significantly higher for bamboo than for other woods, whilst the specificmodulus E/ρ is similar. For the tensile strength, not only the absolute value is higher forbamboo, but also the specific tensile strength σ/ρ is more advantageous.

Table 2.1: Properties of Bamboo

Material Density ρ[kg/m−3] Young’s modulus E[GPa] Tensile strength σ[MPa]Fibre 1160 46 610Matrix 670 2 50

2.3 Failure Behaviour of Bamboo

The failure behaviour of bamboo depends obviously on the applied loading condition, and thefailure behaviour of this plant in bending is very ductile. The bamboo matrix is highly porousand the pores have a hexagonal structure, similar to the structure of the microfibrils. Poresexist between individual fibrils and in between fibres, and have different sizes and shapes [12].It is assumed that the porosity in a bamboo stem increases the level of stress that can beabsorbed, see also studies on the mechanical behaviour of cellular structures [6]. Note howeverthat for numerical analyses and manufacturing, a porous structure will most probably lead tocomplications. Assumed is that the stress distribution is quite homogeneous, both in radialand axial direction, due to the fibre distribution [16]. More research has to be done on thefailure of bamboo. The influence of the highly anisotropic structure on the different failure

24

Table 2.2: Properties of Woods

WoodsDensityρ[kg/m−3]

Young’smodulusE[GPa]

Emeanρmean

Tensilestrengthσ[MPa]

σmeanρmean

Cedar 290 - 460 4.4 - 9.8 0.0189 29.3 - 48.5 0.1037Fir 310 - 340 5.9 - 6.7 0.0194 30.7 - 33.8 0.0992Pine 350 - 420 6.5 - 8.8 0.0199 34.0 - 41.6 0.0982Spruce 380 7.3 - 8.5 0.0208 31.0 - 40.0 0.0934Hickory 560 - 670 8.9 - 11.4 0.0165 62.5 - 81.0 0.1167Oak 530 - 610 7.9 - 12.4 0.0178 47.7 - 74.9 0.1075Bamboo (fiber) 1160 46 0.0397 610 0.5259Bamboo (matrix) 670 2 0.0030 50 0.0746Bamboo (composite) 600 - 1100 11 - 17 0.0165 140 - 230 0.2176

mechanisms is not-well understood yet.In compression, the latter influence is investigated [25]. Note that in here the

macroscopically graded structure is analysed, i.e. the influence of the different stiffness inaxial and circumferential stiffness of the bamboo stem. The low circumferential stiffness andstrength (compared to the longitudinal stiffness) strongly affects failure due to local buckling.In here, also the effect of ovalisation of the cross-sectional area, which reduces the momentof inertia, is stated. For this thesis, the standardd buckling mode shapes are only brieflyanalysed with FEA, see appendix E.

The fracture behaviour (when loaded in tension) of a culm is different from that of wood;no spontaneous fracture occurs through the whole culm, the cracks becoming deflected in thedirection of fibres. This reduces the disadvantageous effect at the sites of strength loss. Thetoughness of bamboo is high compared to other woods. The highly anisotropic structure ofbamboo implies that the fracture behaviour depends on the location of the crack. The placewhere a fracture initiates is called the fracture origin. If fracture initiates at the matrix region,matrix-cracking is observed, and if the fracture origin lies in the fibre region, fibre-crackingoccurs. The first cracking type occurs when the following condition is satisfied [14]

σm =EmEf

σf (2.4)

in which the subscripts m and f refer to the matrix and fibre material respectively. Withuse of the properties in table 2.1 and 2.2, fracture characteristics of bamboo tend to be fibre-cracking rather than matrix-cracking, showing fibre pull-out on its fracture surface. From [14],it is found that the fracture toughness KIC is proportional to the volume fraction of fibresand forms a microscopically graded structure (i.e. in the culm cross-section). The measuredfracture toughness from this study is stated in table 2.3 together with the toughness of someother materials. Furthermore, it is showed that the fracture toughness forms a functionallygraded structure as well as the volume fraction of fibres, Young’s modulus and tensile strengthover the cross-section in radial direction.

25

Table 2.3: Measured fracture toughness of natural and engineering materials [14]

Material Fracture toughness KIC [MPa m1/2]Bamboo culm (maximum) 116.2Bamboo culm (average) 56.8Bamboo node 18.4Steel 217Al-Alloy 33Douglass fir 1.64Spruce 7.0

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Chapter 3

Objectives

From the analysis of the microstructure of bamboo, see Chapter 2, it was found that bamboois a biological composite that can be regarded as a Functionally Graded Material (FGM).Bamboo mainly consists of two materials, a fibre material surrounded by a matrix material.The volume fraction of fibres shows a gradient over the thickness of the bamboo stem, whichcan be characterised mathematically. Besides that, also the shape and size of the individualfibres change in the radial direction. This leads to a highly anisotropic structure, in whichthe fibre volume concentration, Young’s modulus, tensile strength and fracture toughness arefunctionally graded. With biomimicking, designs of FGM with uniform strength, optimalplacement of fibres, various microstructures, porous and/or cellular structures can be made,creating superb (specific) mechanical properties [16]. In Chapter 1, a systematic approachwas given for the biomimicking process. In here, this process will be explained in more detailfor this project, and the most interesting research questions will be stated.

As found in the literature review on bamboo (see Chapter 2), the microstructure ofbamboo contains biological features (e.g. nutrition sieves, vascular bundles making use ofcapillary forces). On top of that, the structure of bamboo is optimised for mechanical purposes(e.g. to sustain wind loads and carry own mass), resulting in good mechanical properties,certainly when related to their mass. The microscopically (and macroscopically) functionallygraded structure influences the mechanical behaviour, since the structure is highly anisotropic.The individual role of fibres, fibre size and fibre shape is unknown thereby. Mechanicalproperties will basically be investigated in here with bending tests, as this the most naturalloading condition applied to bamboo. For an analysis of compression loading (buckling),the reader is referred to Appendix E. In this project, biomimetic designs of bamboo has tobe created and manufactured, in order to learn more about the microstructural features ofbamboo. In here, the most important research question is therefore

‘Are we able to set up a biomimetic design procedure - using the availabletechniques - that creates a biomimetic design which can be used to capture themost important structural features that exist in bamboo?’

To achieve this, the biomimetic design procedure (see also figure 1.4) starts with the fibrevolume fraction distribution in bamboo stems as stated in the literature review on bambooin Chapter 2.

Different models will be created, in which the influence of the different microstructuralfeatures on the mechanical behaviour can be investigated separately, taking certain boundary

27

conditions (BC’s) into account. These BC’s come from both the microstructure of bamboo, aswell as the manufacturing method used. The manufacturing method that is going to be usedis rapid prototyping. With the new machine Connex350TM (see section 4.2) it is possibleto use two materials simultaneously. Another opportunity is to mix these two materials, bycreating so-called Digital Materials (DM’s). This creates even more options to create a gradedstructure, as will be seen in section 4.1. The use of different materials in one design creates alot of possibilities to investigate the microstructural behaviour of bamboo. The opportunitiesof this machine have to be explored and questions have come up that deal with:

• Built-up orientation? Does this influence the material behaviour in bending and/orcompression tests? Does the failure mode change?

• Accuracy, machine precision: is the prototype manufactured conform the design? Whatare the critical feature dimensions?

• Which materials should be used? What are the exact material parameters and whatare the possibilities with mixing of two materials?

The main objective related to the main research question is the ability of manufacturingthe biomimetic designs. The design procedure will further be explained in the next chapter.Besides manufacturing, finite element modelling will be used together with experimentaltesting, to validate test results and make reliable predictions of material behaviour ofbiomimicked structures possible. The numerical analysis can in the first instance also be seenas a similarity analysis. Besides that, it is interesting to see whether the mechanical behaviourof anisotropic structures can be predicted accurately by use of finite element methods (FEM).For the numerical analyses, optimum settings have to be found, like meshing techniques andboundary conditions. Furthermore, not only the mechanical behaviour in bending will beanalysed, also the failure behaviour of bamboo is not well-understood yet. Can the latterissue also be captured with this biomimetic design process?

Summarising, this work can be seen as an exploratory work into the opportunities ofbiomimetic design for structures with high load bearing efficiency (that are not designed foroptimal stiffness, like traditional materials), combined with the use of rapid prototyping.This survey will take place in different stages: starting with the information obtained fromthe literature review (see Chapter 2), to the creation of different models (see section 4.1)and manufacturing (see section 4.2), this finally leads to a similarity analysis in Chapter 6.Altogether, this study will give further insight in the biomimicking design procedure usingrapid prototyping.

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Chapter 4

Biomimetic Design andManufacturing

4.1 Set-up for Biomimetic Designs of Bamboo

From the analysis of the microstructure of bamboo stems (see Chapter 2), it is found thatthe distribution of fibres in the bamboo stem shows a gradient in radial direction (over thethickness of the stem). On average, the distribution of the volume concentration of fibres isfound to be 60% at the outer side of the stem and 20% at the most inner side of the wall. Theoverall volume concentration is 40%. In this biomimetic design, the fibre volume concentrationdistribution forms the basis of the different models that will be created. Assumed is that thedistribution of fibres along the thickness can be approximated by a quadratic function [8].

Besides that, not only the volume concentration of fibres is changed over the stem wallthickness, but also the diameter of the individual fibres. This was seen previously in figure2.5. The questions to be addressed are: what influence will this characteristic have on themechanical behaviour of the bamboo-like structure? Does the failure mode change?

4.1.1 Boundary Conditions for Biomimetic Design

The fibre distribution clearly results in a gradient in the stiffness, which has been investigatedin several studies [13, 14]. In this study, one of the objectives is to investigate why thegraded structure is organised in the way it is found in bamboo. Therefore, several cross-sectional geometries for tubular structures are designed, which will be manufactured withrapid prototyping and tested in bending tests. For future work, it might be interesting totest the mechanical behaviour of biomimetic designs under other loading conditions as well.For the design of the different models, some boundary conditions have to be fulfilled for alldifferent designs - such that the microstructure of bamboo can be mimicked and compared ina consistent manner, i.e. to make a decent comparison different biomimetic designs possible.These boundary conditions are stated below:

1. The volume concentration of fibres is given as a function of the (non-dimensional)thickness of the bamboo plant stem. This function has to fulfill some boundaryconditions; (1) The fibre volume concentration at the most inner side is 20%, (2) thefibre volume concentration at the most outer side is 60%, and (3) the overall fibrevolume concentration is 40%. If these volume concentrations can not be reached dueto modelling restrictions, the total volume concentration and the two boundary fibre

29

volume concentrations have to be scaled with a scaling factor; this concept will beexplained mathematically later in this section, and the design procedure can also befound in the MATLAB files stated in Appendix A.

2. The inner radius Rin of the tubular model is 5 mm and outer radius Rout is equal to10 mm, such that a consistent use of both total volume and fibre volume fraction isguaranteed.

4.1.2 Design Objectives

Prior to giving more details about the specific designs, the design objectives have to beexplained in more detail. There are different features that can influence the material behaviourof a bamboo-like structure. To investigate these, the ability to distinguish these features inthe designs has to be created, in order to get proper indications of the influences of thedifferent microstructural features on the mechanical behaviour. The different features thatwill be investigated are stated below, with the different models indicated with the Romannumbers.

A. Influence of gradient in fibre volume fraction over stem thickness:

(I) Model with homogeneous (mixed) material (no gradient in fibre volume concen-tration)

(II) Model with discrete layers with gradient in fibre concentration (conform to thedistribution of the fibre volume concentration distribution in bamboo), i.e. thisrepresents a functionally graded material

B. Influence of fibres

(III) Model with fibres of equal radius Rf , distributed along the main radius R with thesame gradient in fibre volume concentration over the thickness as used in modelII;

C. Influence of fibre diameter

(IV) Model with fibres of non-equal size, distributed along main radius with samegradient in fibre volume concentration as used in model II; The fibre sizedistribution follows a linear function of the number of layers, with the smallestfibres at the most outer side and the largest fibres at the most inner side, analogousto what is found in bamboo.

In models of type (A), no real fibres are present. In model (I) just one material is usedand no gradient in fibre volume fraction is present. This material is a mixture of the fibrematerial and the matrix material. In model (II), the gradient in the volume fraction can befound, conform to the distribution of the fibre volume concentration distribution in bamboo.Similarly to model (I), no actual fibres are modeled, but now the cross-section is built-up ofseveral concentric rings, which are made of different materials. These materials are mixturesof the fibre and the matrix material, in the proportions as found in bamboo at that particulardimensionless thickness. With use of more layers, the graded structure can be approximatedmore smoothly. This model most closely represents a functionally graded material. However,with the current manufacturing method (rapid prototyping, see paragraph 4.2) it is not

30

possible to create a continuously distribution as function of the thickness, since materials canonly be printed in discrete layers. If models (I) and (II) are compared, the influence of thegraded structure itself (without actual fibres) can be investigated.

Model type (B) and (C) do contain actual (circular) fibres. The fibres are represented bya material of higher stiffness. The design for the cross-section of model (B) is such that itcontains fibres of equal radius. The fibres are positioned in concentric layers (of equal width).The number of fibres in each layer is such that it fulfills the same fibre volume concentrationdistribution as in model (II), i.e. applying the rule of mixtures, an identical stiffness in thatlayer would be found. With comparison the mechanical behaviour of this model with model(I) and (II), more information should be gotten about the influence of the actual fibres.

Model (C) is created to investigate the possible advantages of using fibres of unequaldiameter. From Chapter 2 it could be seen that not only the fibre volume concentrationchanges over the thickness of the stem, but also the fibre size (and shape). The latter changes(in shape) are not considered here; only circular fibres are used. Considering the same fibrevolume concentration distribution along the thickness as in model (II), the influence of theunequal fibre size can be investigated with this model.

With use of these several model designs, a hypothesis can be formed about the structuralfeatures in the microstructure that help to resist bending moments. However, rapidprototyping may put further restrictions on the designs and may require small adaptationsto the design, as will be investigated in section 4.2. Therefore, the hypothesis will be formedafter the final test plan is formulated, which can be found in Chapter 5.

4.1.3 Biomimetic Design

The primary design parameter that is used, as stated before, is the volume fraction of fibresas function of the dimensionless thickness of the plant stem. In the next part of this chapter,the main mathematical expressions are given that are used for the design of the differentmodels, indicated with the Roman numbers (I to IV) before. These expressions are used ina series of MATLAB-files, used to assign the different parameter values in the different models.The structure of these files is explained in Appendix A.

At first, it should be noticed that for the calculation of the different model parameters,two coordinate systems are used. Because the design of the models is used to create the 2Dcross-section of the tubular structure, a 2D-coordinate system will be used. The origin of thefirst coordinate system is placed at the center of the cross-section, with radial coordinate R.Its dimensionless equivalent is taken as R/Rout and this radial coordinate is given by r. Theorigin of the second coordinate system used is placed at the inner radius Rin of the stem,and its radial coordinate represents the thickness of the stem T . This coordinate systemalso has a dimensionless equivalent, represented by t, which is calculated as T/dT in whichdT = Rout −Rin. These coordinate systems are given in figure 4.1

It was stated before that the fibre volume concentration distribution as function of the non-dimensional thickness can be approximated by a quadratic function. This general function isused for the volume fraction of fibres φf as function of the non-dimensional thickness t

φf (t) = at2 + bt+ c (4.1)in which the variables a, b and c are initially unknown. These can be solved using theboundary conditions, which are stated below.

φf (t = 0) = 0.20; φf (t = 1) = 0.60; φf,total = 0.40; (4.2)In the boundary conditions stated in expression 4.2, the total volume fraction of fibres φf,total

31

R, r T, t

x x

y y

Figure 4.1: The two coordinate systems used for the model design

can be calculated with the total volume of fibres Vf,total which will be divided by the totalvolume of the tubular structure Vtotal. The total volume of fibres can be calculated by usingan integral over the volume of function 4.1, with use of cylindrical coordinates. This is statedin equation 4.3.

Vf,total =∫ L

0

∫ 2π

0

∫ Rout

Rin

φf (R)R dR dθ dz (4.3)

It must be remarked that for the calculation of the total volume fraction, φf can not be takenas a function of t, but must be rewritten as function of R, since the integration is based on acylindrical coordinate system. The coordinate system used for the thickness (see figure 4.1,can not be transformed in a cylindrical coordinate system, since the axis of origin does notcoincide. To calculate the total volume concentration as function of R instead of on t, thefollowing correlation has to be used.

t =R−Rin

Rout −Rin=R−RindT

(4.4)

With these boundary conditions, the variables a, b and c defining equation 4.1 can be foundsolving a system of equations. This system is given below

φf (t = 0) = c = 0.20φf (t = 1) = a+ b+ c = 0.60

φf,total =Vf,totalVtotal

=

∫ L

0

∫ 2π

0

∫ Rout

Rin

[a(R−RindT

)2 + b(R−RindT

) + c

]R dR dθ dz

π(R2out −R2

in

)L

= 0.40

However, model III and model IV require the positioning of fibres in the cross-sectional areaof the stem. These fibres will set an upper constraint on the fibre volume, i.e. this gives amaximum reachable volume fraction of fibres. This maximum reachable fibre concentrationdepends on the model type, the minimum spacing Sp (the minimum feature size in a design,and depends on the manufacturing technique), the minimum fibre radius Rf and the numberof fibre layers. The latter is the number of virtual concentric layers in which the fibresare placed. A general situation for model III and IV is sketched in figure 4.2. From this,the angle α can be computed, which results in a maximum number of fibres that can bepositioned in that layer. Moreover, the maximum possible fibre concentration in that layer

32

R out, layer i R in, layer i

R fiber, i

Sp

α

Figure 4.2: General figure of fibre model III and IV with parameters needed to calculate αand φf,layer i

is calculated and compared with the needed fibre concentration at that radial position, asfollows from equation 4.1. If the maximum possible fibre concentration is not sufficient - i.e.if the maximum reachable fibre concentration in a virtual layer is below the fibre concentrationin that layer (as follows from the fibre volume concentration distribution in bamboo) - thefibre concentration function φf (t) must be scaled by a correction factor. This principle isshortly evaluated here, but for more details of the calculation is referred to Appendix A andcorresponding MATLAB-files. The angle α for a layer is found by

α = sin−1((Rf, i + Sp)

(Rin, i + Sp+Rf, i)) (4.5)

This angle is used to calculate the maximum number of fibres nf,max in that layer (for modeltype III and IV). This gives the maximum volume fraction of fibres that can be reached in thatparticular layer, and will be compared with the desired volume of fibres, given by equation4.1. If the volume of fibres is not sufficient, a scaling correction factor cf will be used to scalethe set of boundary conditions (by multiplication). This scaling factor is

cf =Vf, imaxVf, desired

(4.6)

This approach will of course influence the reproducibility of the volume concentration distri-bution as found in bamboo. However, this approach will guarantee the best approximationand same kind of distribution as found in bamboo, taking into account all boundary conditionsfor the design. The volume concentration distribution over the thickness will be adapted suchthat it can be reached by all different models, giving the same boundary conditions for thevolume concentration of fibres in all models.

The thickness of each layer depends on the fibre radius (and thus also on the numberof layers over the thickness). For model III all fibre radii are equal, but for model IV, afibre size distribution is used. This distribution is a linear relationship between fibre radius

33

Model III Model IV

φ f φ f

R f R f

t t

layer i layer i

0 1 0 1

Figure 4.3: Schematic distribution of volume fraction of fibres φf as function of dimensionlessthickness t and fibre radius Rf as function of layer number i for model III and IV

Rf and layer number i. The fibre radius at the most outer layer equals the minimum fibreradius, chosen to be 3 times the minimal spacing size Sp = 0.10mm. This is an arbitrarychosen value; the minimal fibre radius depends on the characteristics of the manufacturingtechnique. For more information ont the rapid prototyping process and accuracy (see section4.2). The possibilities need to be explored further, but with the given accuracy (42µm) thefibre size is expected to be applicable. The radius of the fibres is calculated such that theminimal spacing parameter Sp is used. The latter can also be seen in figure 4.2. The fibresize distribution for model III and IV as function of layer number is schematically representedin figure 4.3, together with the fibre volume concentration distribution along the thickness.

At the end of all calculations all models are compared with each other in a loop sequence,such that they all get exactly the same total volume fraction of fibres and have followedthe same fibre volume distribution function. The output of the MATLAB-files (see AppendixA) shows all the input parameters needed for the design of the different models I to IV inSolidWorks. As an example, for model IV the distribution of fibre volume fraction φf asfunction of the radius R is given in figure 4.4. In this figure, the width of the bars representthe width of the 4 virtual layers in which the fibres are placed. The blue line represents thefinal (scaled) volume concentration distribution as function of the radius. The accompanyingSolidWorks design is depicted in figure 4.5a. For comparison, a cross section of a bamboostem is depicted in figure 4.5b. The corresponding MATLAB-output is:

Model IV: Fibre model with size differenceFibre volume concentration = 0.30, phi_f_in = 0.15 and phi_f_out = 0.45

Concentric radius layer1 = 5.85 mm, nf = 7, R_f = 0.7343 mm and phi_f = 0.1897Concentric radius layer2 = 7.40 mm, nf = 16, R_f = 0.5924 mm and phi_f = 0.2710Concentric radius layer3 = 8.65 mm, nf = 33, R_f = 0.4495 mm and phi_f = 0.3505Concentric radius layer4 = 9.60 mm, nf = 72, R_f = 0.2990 mm and phi_f = 0.4192Vf_ModelIV = 70.6858 [mm^3] and phi_tot (check) = 0.30 [-]

34

Layer 1 Layer 4 Layer 3 Layer 2

Figure 4.4: Volume fraction of fibres φf as function of radial position R for model IV with 4layers

(a) Model design IV in SolidWorks (b) Transverse cross-section bamboo

Figure 4.5: Cross-sections of model IV (a) and bamboo (b)

35

4.2 Rapid Prototyping and Material Testing

In this exploratory work, the main research objective is to obtain a biomimetic designprocedure, as stated in Chapter 3. In the previous paragraphs, the modelling of differentbiomimetic designs (based on bamboo) was explained. One of the most important goals inthis survey, related to the main objective, is to manufacture these biomimetic designs. Asmentioned, the manufacturing technique that will be used is rapid prototyping. Possibilitiesand restrictions of a new rapid prototype machine have to be explored. Some features of thismachine will be listed in this chapter.

First, an introduction into the field of rapid prototyping will be given. Besides that,material properties given by the manufacturer are validated with tensile tests in order toobtain better predictions with FEA of the material behaviour and to select the matrix- andfibre-like materials. After that, two benchmark tests are performed and test results areanalysed. These will be used as a start for a new set-up of experiments. Aspects like sampleorientation during rapid prototyping are considered too.

4.2.1 Rapid Prototyping

The working principle of a rapid prototyping process can be seen as a 3D printer process.Rapid prototyping is an automatic manufacturing technique that is widely used to createcomplex shaped prototypes or products with high precision. Also production lines can makeuse of rapid prototyping, but this technique is generally only used on a relatively smallscale. Traditional techniques like injection molding are commonly less expensive for batchmanufacturing of polymer products. The main advantage of rapid prototyping is that itis flexible and relatively fast. With this technique (dependent on the printer) almost anycomplex shape can be made, unless the geometry contains any closed cavities. Furthermore,no expensive molds are required and switching between product geometries is easy. Theprototyping machine used here is the Connex350TM, which was recently purchased as partof the advanced manufacturing precinct (AMP) at RMIT University. For more informationabout this printer, see the manufacturers website from Objet [27].

From the geometry of the biomimetic design, as created in section 4.1, a model isconstructed in a modelling software package (CAD). In this software, a STL file formatcan be created, which can be read-in by the rapid prototyping software. Such a STL fileapproximates the shape of a part or assembly by the use of triangular facets. Smaller facetsproduce a higher quality surface. However, a larger file size is involved with STL files withhigher precision. However, in this project the STL files are relatively small, so the smallestfacets as possible are used (which resembles an angle tolerance of 0.5 degrees and an absolutedeviation tolerance of 0.01001935 mm, when an STL file is created in SolidWorks 2009 versionSP4.1).

When the STL file is read-in by the rapid prototyping software, the prototype designs canbe positioned and materials can be assessed. Furthermore, when a STL file is provided tothe rapid prototyping software, it also automatically creates a support material - surroundingthe part. This support material is a non-toxic gel-like photo-polymer and can easily bewashed away with water in a later stage. The Connex350TM has the function to use multiplepart materials. This gives the opportunity to create composite materials, required for thebiomimetic designs of bamboo. A gradient-like structure can be created with this machine,however some restrictions are currently given. Basically, two cartridges with pure materials

36

Figure 4.6: Example of slicing of a sample geometry in rapid prototyping

(so-called FullCure materials) are placed in the machine. From these materials (e.g. materialA and B), several mixtures can be created. These materials are called the Digital Materials(DMs) and they are actually not a real mixture. These materials contain a certain volumeconcentration of FullCure material A and a certain volume concentration of B, and these arecreated by making a fine, pattern-like structure.

The Connex350TM lays down successive layers of liquid polymer, using both the supportmaterial and the part materials (FullCure materials and/or DMs). After laying down onelayer, it immediately cures this layer with UV light, creating one solid part surrounded bythe support material. The successive layer built-up of the sample induces slicing of the part,represented by figure 4.6. The layer thickness (z-axis) is set to the minimal thickness of 16 µm.This multi-material 3D printing system has a build resolution in x- and y-direction of 42 µm(or 600dpi).

Material properties of the different polymeric materials that will be used with this rapidprototyping machine are stated in table 4.1 [27,28]. Only material properties that are relevantfor this thesis are stated. In this table, the test standards (ASTM) that were used to determinethese properties are given too [30]. Different materials are given: two FullCure materials,VeroWhite and TangoPlus. VeroWhite is an opaque material and TangoPlus is a more rubber-like flexible material. From these two materials, Digital Materials can be formed, in whichthe volume concentration of each FullCure material will differ. As can be obtained from table4.1, the materials do have a different modulus and tensile strength. However, the relativedifference between the modulus of any two materials is not as high as between the fibre andmatrix material in bamboo. All materials are very ductile, with a strain-to-failure of at least20%. Unfortunately, no plasticity data is given by the manufacturer, which is required for FEanalyses. To get an idea of the costs involved with the rapid prototyping process, an estimateof the costs is given in appendix D.

Prior to setting up an extensive test plan to test the biomimetic designs (created inparagraph 4.1) - which gives the opportunity to analyse structural design features that arefound in bamboo - and prior to manufacturing several test specimens with complex-shapedgeometries, first two simple 3-point-bending tests have been performed on simple tubularstructures. This benchmark test is done for several purposes, stated below:

37

Table 4.1: Material properties of FullCure and Digital Materials used for rapid prototypingwith the Connex350TM printer, as specified by manufacturer

PropertyTestStandard(ASTM)

UnitsVeroWhite(Full-Cure830)

DM8410 DM8420 DM8430 DM9795TangoPlus(Full-Cure930)

Tensilestrengthσ

D-638-03 MPa 50 49 44 39

D-412 MPa 20 1.5

Young’smodulusE

D-638-04 MPa 2495

D-638 MPa 2350 2150 1750

Elongationat Breakεfailure

D-638-05 % 20

D-638-03 % 35-45 50-60 60-70

D-412 % 30 218

• develop a material database with elasto-plastic properties

• investigate the influence of the sample orientation in rapid prototyping on themechanical behaviour, stability, manufacturing (time) and costs;

• validate if the roller fixture works properly (for the initial design of the rollers, seeAppendix C.1);

• check how the (simple) structure deforms and the material behaves under bending;compare this behaviour with material data that is provided by the manufacturer (Objet);

• analyse the failure behaviour of this material upon bending;

• validate if the test results can be predicted accurately by using FEA and/or analyticalexpressions;

4.2.2 Materials Testing

For future work it is required to create a material database of the materials used with rapidprototyping. The manufacturer provides information about the various digital and FullCurematerials in data sheets [27,28], but detailed elasto-plastic data are missing.

To obtain the material data, tensile tests are performed, following the ASTM standardD638-03 [30]. The tensile bar is modeled in SolidWorks and is depicted in Appendix C.2.The tensile bars are manufactured with rapid prototyping. The materials that are usedare all listed in table 4.1. For each material, two specimens are produced to validate thereproducibility of the tests. The tensile tests are performed on a Instron test machine, with a10 kN load cell. An extensometer is used which measures the strain in the tensile bar duringtesting. The gauge length (of extensometer) is 10 mm. The test set-up is schematicallyshown in figure 4.7. The velocity of the upper fixture clamp is set to 1 mm/min and samples

38

Figure 4.7: Schematic representation of test set-up for tensile testing with extensometer

are tested up to failure. Test results are post-processed with MATLAB, see also appendixB. With these MATLAB-files, several material properties can be determined, such as Young’smodulus, yield strength and elasto-plastic input data. The work principle of these files isfurther explained in appendix B.

Test results can be obtained from figure 4.10. Except for TangoPlus, test results arepost-processed, and true stress σtrue is plotted versus true strain εtrue in figure 4.10a -4.10e. Figure 4.10a provides test results for VeroWhite; from figure 4.10b - 4.10e the testresults of Digital Materials (mixtures of VeroWhite and TangoPlus) are given, in which theamount of VeroWhite decreases from figure 4.10b to 4.10e. In each of these subfigures, twographs can be found, corresponding with the two tests. As can be concluded from this,test results are reproducible. If the stress-strain responses of these materials are compared,the general behaviour is similar, only characteristic values change. An (intrinsically linear)elastic response is followed by a yield stress. In the area of elastic deformation, the straightlines represent the lines that were used to determine the Young’s modulus. After this yieldstress, the stress decreases. This part is called strain softening or yield stress relaxation and ischaracteristic for (amorphous) polymers. The blue triangle represents the lowest stress pointafter the yield point. After this point the stress increases again, due to strain hardening.Finally, the tensile bar breaks, indicated with the blue circle (failure point). In the legendof the subfigures, the test velocity (of the top fixture) is also given. It should be notedthat the deviation between the results of VeroWhite is due to different strain rates. Forone tensile bar made of VeroWhite a fixture velocity of 0.5 mm/min was used, and forthe other 1.0 mm/min. An increase in fixture velocity obviously results in a higher strainrate and this creates a higher yield strength, as can be seen in figure 4.10a. This gives alarger difference between the two tests, as can also be seen in the standard deviation ofthe VeroWhite material in table 4.2. In figure 4.10, it can be observed that the modulusremains unchanged. The test speed of 1 mm/min was therefore used for all subsequenttests. In table 4.2, information about both the Young’s Modulus E and yield stress σy aregiven of the tested materials. These parameters are determined in MATLAB, see AppendixB for more details. Also the standard deviation and relative error are listed in table 4.2.

39

Test Data

Material model; type simple

Test Data Material model; type normal

σ

σ

ε

ε

Figure 4.8: Schematic representation of thenormal material model (top) and simplematerial model (bottom)

As could be noticed before, no stress-straindiagrams for TangoPlus are given. Thematerial behaviour of TangoPlus is differentfrom the other materials that were tested.The material behaviour is rubbery (andtherefore too ductile) and the test procedureis therefore not applicable. Material dataof this material should be obtained usinga different test standard. However, forthe biomimetic designs this material is notregarded as a potential material. Therefore,no material data of TangoPlus is listed intable 4.2.From table 4.2, it can be concludedthat material properties as obtained fromtesting are not quite well in accordancewith the material data provided by Objet(manufacturer), see table 4.1 as reference.The material data from table 4.2 is used tocreate the material models.

Instead of copying the measured stress-strain curves directly to the FE program,two different models were developed. Thisis done because nonlinear stress analysisgenerally has problems with stress relaxation. For that reason, the softening behaviour isavoided in the new material models. The material models are showed schematically in figure4.8. With MATLAB, an average curve is created from the original set of curves (for eachmaterial). This curve is based on polynomial functions, and represents the average curveof the two stress-strain curves obtained from the tensile tests. This polynomial function isused for the creation of the material models. This procedure is explained in more detailin appendix B. The material model normal follows this polynomial exactly, but creates ahorizontal plateau from σy to the next point on the stress-strain curve where σ(ε) = σy, i.e.on the horizontal plateau dσ/dε = 0. Besides that, also a simplified version of this modelis created. In this material model, called simple, the stress increases linear elastically uponthe point of yielding, i.e. σ = Eε. From that point, the stress again remains the same untilthe next point on the curve where σ(ε) = σy. From there, the simple model is equal to the

Table 4.2: Material properties obtained from tensile tests

Property Units VeroWhite (FullCure830) DM8410 DM8420 DM8430 DM9795E MPa 1281.18 1501.59 1191.09 832.61 407.17std(E) MPa 96.78 128.09 10.08 47.45 19.96rel. error(E) [-] 0.076 0.085 0.008 0.057 0.049σy Mpa 25.73 26.96 24.67 18.21 14.01std(σy) MPa 1.882 0.017 0.384 0.308 0.035rel. error(σy) [-] 0.073 0.001 0.016 0.017 0.003

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

Figure 4.9: Material model normal for material DM8410

normal model. For the DM8410, the normal material model is given in figure 4.9.It should be noticed that sets of two tensile tests are not sufficient to capture all material

properties. Strain rate dependency, temperature effects and possible size effects can influencetests. However, the difference between material data given by the manufacturer and theobtained material data is significant. The new material models are expected to create abetter agreement between FEA and the benchmark test.

41

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

σσσσtrue

versus εεεεtrue

curve for VeroWhite

test #1, v = 0.5 [mm/min]

test #2, v = 1.0 [mm/min]

(a) VeroWhite

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

σσσσtrue

versus εεεεtrue

curve for DM8410

test #1, v = 1.0 [mm/min]

test #2, v = 1.0 [mm/min]

(b) DM8410

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

σσσσtrue

versus εεεεtrue

curve for DM8420

test #1, v = 1.0 [mm/min]

test #2, v = 1.0 [mm/min]

(c) DM8420

42

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

σσσσtrue

versus εεεεtrue

curve for DM8430

test #1, v = 1.0 [mm/min]

test #2, v = 1.0 [mm/min]

(d) DM8430

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

σσσσtrue

versus εεεεtrue

curve for DM9795

test #1, v = 1.0 [mm/min]

test #2, v = 1.0 [mm/min]

(e) DM9795

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Displacement [mm]

Fo

rce

[N

]

(f) TangoPlus

Figure 4.10: Test results from tensile tests on different materials

43

4.2.3 Sample Orientation in Rapid Prototyping

The main aspect that is considered first is the influence of the sample orientation in rapidprototyping. The sample orientation is the way the sample is positioned in the rapid prototypemachine. An infinite number of sample positions is possible, and they can easily be assessedby the rapid prototype machine software. Only the horizontal and vertical sample orientationswill be considered here, since this will use less support material and creates a higher accuracyfor fibre placement, i.e. the placement of the second material within a matrix material. Testresults will be given and compared with the results from FEA. Test results will give moreinsight in the preferable sample orientation, while the comparison with FEA might indicatethe material behaviour of the structure and, moreover, might suggest further changes in thetest set-up.

Test Set-up

3-Point-bending tests were performed on two tubular samples with different sample orienta-tion. These samples have the following dimensions:

Dout = 20 mm; Din = 14.399 mm; L = 100 mmThe specimens consist of only one material. The material used for these specimens is

VeroWhite. For material properties, see Table 4.1. One specimen is positioned with it is axialdirection aligned horizontally, the axial direction of the other specimen is positioned with therapid prototyping software in vertical direction. The 3-point-bending tests are performed ona INSTRON materials testing machine, with use of a 50 kN load cell. The specimen arebased on two (bottom) rollers, with a distance of 70 mm. The (upper) central roller willmove downwards, with a velocity of 0.5 mm/min. The maximum displacement of the upperroller is 10 mm.

As the rapid prototyping process is considered (see paragraph 4.2.1), it is expected thatlayer slicing, depicted in figure 4.6, might have (disadvantageous) effects on the materialbehaviour of the structure when the structure is loaded in bending and the slices are(depending on the specimen orientation) in or out of the loading direction. The way thespecimen is built-up might also influence the friction behaviour of the beam when the tubeis in contact with the fixture rollers. When the axial direction of the tube is in verticaldirection during rapid prototyping, the surface roughness might be larger, resulting in morefriction. Further on, accuracy, manufacturing time and usage of support material can alsobe influenced by orienting the sample in a different direction in the machine. Finally, it isexpected that the failure behaviour will not change upon sample positioning. It is expectedthat rapid prototyping creates a homogeneous material structure.

Test Results

The resulting force versus displacement graph is given in figure 4.11. In here, the solidlines represent the test results. As can be seen, the initial behaviour is similar for bothmanufacturing conditions, indicating that the bending stiffness of the material is equal.However, the vertical sample orientation (blue line) gives a slightly higher maximum forcethan the specimen that was placed horizontally in the rapid prototyping machine (red line).Although this might be a result of scattering in measurement results, it could also make a casefor the statement that the friction of the vertical oriented sample is higher. However, resultsseem to be quite well reproducible. Nevertheless, the failure behaviour is very ductile and is

44

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

1600

Displacement [mm]

Forc

e [N

]

RP orientation = VerticalRP orientation = Horizontal

Figure 4.11: Force-displacement response for 3-point-bending tests

similar for both samples. No delamination and/or cracking could be noticed during testing.Important to note that - using this test set-up - the upper roller indents in the specimens.

Influence of Sample Orientation

Some conclusions can be drawn out of this experiment, with respect to the sample orientationduring rapid prototyping. Table 4.3 gives an overview of the advantages and disadvantages ofthe two different sample orientations. Some points out of this table might need some furtherexplanation. The production time is favourable for the horizontal sample orientation. Forthis sample, the production time for the horizontally positioned sample is about 2 hours,while the production for the sample which axial direction is in in the build-up direction takes6 hours. It is noted that no machine or labour rates are paid (approx. $ 50 − 70), sincestudent work was non-profit. So, this aspect is less important here. The usage of supportmaterial is less for the vertically positioned sample. Costs induced with materials can befound in Appendix D. The third and fourth aspect in table 4.3 are closely related. Sincethe structure is relatively tall, the rapid prototyping of a vertically positioned sample canbe less stable, since there is contact during processing. This might decrease the accuracy.Placement of a second material in small amount to reproduce a fibre material, is assumed tobe more precise in the horizontally oriented sample. Production costs are, as stated before,only based on material costs. Since the support material is less expensive than the partmaterials, costs will be quite comparable. However, the horizontal positioning of the samplewould be preferable if manufacturing costs had to be taken into account. No differences werefound in failure behaviour and no delamination was found during testing. The friction ofthe vertically oriented sample is possibly slightly higher than for the horizontally positionedspecimen. However, the latter effect does not affect results as long as only one sample positionis used during rapid prototyping.

Conclusively, it is found that for this geometry the horizontal orientation (axial direction

45

of tube perpendicular to build-up orientation of rapid prototyping process) is the mostappropriate here. Therefore, biomimetic designs will be manufactured using this sampleorientation.

Table 4.3: Comparison between horizontal and vertical sample orientation in the rapidprototyping machine

Property Horizontal sample orientation Vertical sample orientationProduction time + −Usage support material − +Stability during processing + −Accuracy fibre placement + oProduction costs o oFailure behavior in bending o oDelamination during bending o oFriction o o

4.2.4 FEA

Finite Element Analyses - Work Plan

In the FE program (Abaqus, version 6.9.2), a certain model tree is used, to create an organisedCAE file. In this section, these consecutive steps are followed. It will give the general outlinethat is followed for the several numerical analyses in this project. However, later on morecomplex structures will require different meshing techniques. Generally, a CAE will be createdand from this, an input file (INP) will be provided to the Abaqus solver (nonlinear, static).Results can be obtained from ODB files.

Part

The model part, created with SolidWorks, can be imported as a single part using a STEPfile. It should be ensured that the part is positioned with the axial direction of the tube iny-direction. The latter is due to a restriction in Abaqus. In here, analytical surfaces (usedas contact surfaces) of the rollers have to be created, and these can only be extruded inz-direction. The part needs to be partitioned for boundary conditions, using a datum planrestricted to a xy-plane. These will be presented later. Besides this deformable body, threerollers are constructed. Three rigid contact surfaces are created that represent these rollers.These rollers are (for simplicity) modeled as circular rollers, with a diameter of 20 mm. Acurve, covering less than 180 degrees is extruded in z-direction.

Mesh

For the deformable tubular structure, a dependent mesh is created. To create the mesh,a bottom-up meshing technique is used. The element type used is C3D6, a 6-node lineartriangular solid prism (3D element). This element is capable of analysing bending. The seedsize used here is 1 mm and the number of elements over the length is 24. Using the bottom-upmeshing, the outer surface of the tube must be associated with the geometry in order to makecontact analyses possible. Besides that, nodes on the partitioned edges must be merged. Toprescribe the boundary conditions later, nodes have to be selected and placed in sets. Oneset contains all nodes in the xy-plane, splitting the part in two equal pieces. The other set

46

contains all nodes in the middle xz-plane, again splitting the tube structure in two equalparts. Seed size and the number of elements in axial direction can be refined of course toobtain more accurate results.

Materials

For the material properties, the material data that was obtained from material testing,see section 4.2.2, was used. Two different material models were used, indicated withsimple and normal, using simplified elastic-plastic behaviour (see figure 4.8 for a schematicrepresentation). Material data can be found in table 4.2.

Assembly

To create a 3-point-bending set-up, all parts have to be assembled. This is done by creatingan instance in the assembly section. This gives an overview of the whole set-up. Furthermore,3 surfaces (geometry) can be created from the contact surfaces of the roller parts. These willbe used to assign interactions.

Steps, Field Output Requests and History Output Requests

For the nonlinear (static) analysis, a (virtual) time step has to be created. Requirements ontime incrementation can influence the convergence behaviour of the analysis, due to contactand material nonlinearity. Furthermore, choice of solver, solution technique and load variationwith time can influence results. Initially, default values are used. The maximum numberof increments is 100, the initial and maximum time increment are 0.05 and the minimumtime increment is 1−5. In the menu Field Output Request, only relevant ouput variablesare selected. As History Output, the total forces and moments due to contact pressure arerequested. The latter values can be compared later with test data. For more details aboutthe output variables is referred to the CAE files.

Interactions and Interaction Properties

The tubular structure will slide over the rollers upon movement of the upper fixture clamp.This induces a certain amount of friction. There are many ways to prescribe the interactionproperties between fixture and sample. For simplicity, a penalty function is used in theInteraction Properties menu. The directionality of this friction coefficient is isotropic. Thecoefficient of friction for this polymer-steel interaction is initially unknown. Moreover, becausethe two different specimen orientations in rapid prototyping, different coefficients of friction forthe two specimen might be expected too. Therefore, different coefficients will be investigated,varying from 0 (no friction) to 0.3.

Boundary Conditions

The boundary conditions are listed below. For more convenience, in figure 4.12 the coordinatesystem is given together with boundary conditions itself, which are indicated with red dots.Note that the y-direction lies in the axial direction of the tube, as stated in the section Part.

47

z-symmetry z = 0

fixed (x,y,z=0)

y = 0

xmax = 10 mm

fixed (x,y,z=0)

Figure 4.12: Boundary conditions and coordinate system for 3-point-bending, as used inAbaqus

1. no displacements of the bottom 2 rollers (fixed)

2. a prescribed downwards x-displacement (ramped) of the central (upper) roller, with amaximum of 10 mm

3. no displacement in y-direction for the nodes on the central cross-sectional layer in thexz-plane

4. no displacement in z-direction for the nodes on the central cross-sectional layer in thexy-plane

5. no displacement in x-direction at the contact nodes with the bottom support rollers,due to contact.

As can be noticed from figure 4.12, boundary conditions on the rollers are prescribed usingReference Points (indicated with RP in figure 4.12). If a boundary condition is prescribedat this point, the whole fixture will behave as prescribed in that point. Note that all theseboundary conditions will be used for all future 3-point-bending analyses in this report. If4-point-bending tests are used (as will be discussed later), only the prescribed downwardsx-displacement will slightly differ from this set of BCs: this displacement will then be appliedat the upper two rollers. The y-symmetry condition remains for 4-point-bending.

Jobs

Finally, a job has to be created, covering a full analysis. Memory options and precision canbe changed here. From here, the input (INP) file can be written, which is provided to theAbaqus solver.

48

FEA Results

In this part, results of several finite element analyses are presented. These analyses are basedon the benchmark test (3-point-bending). The FE procedure is described in the previoussection. In here, not only different coefficients of friction, but also different material models(simple and normal), and the influence of mesh size are considered.

From the material data provided by Objet, see table 4.1, it can be noted that the materialproperties of VeroWhite and DM8410 are very similar. In the tensile tests, VeroWhite wastested at two different fixture speeds, resulting in larger standard deviations in the resultsfor VeroWhite compared to other materials (see also section 4.2.2). However, it can beassumed that the material behaviour of VeroWhite and DM8410 is comparable. Since thetest results for DM8410 are more reliable, the material models used for the following finiteelement analyses are based on material models determined for DM8410. It is expected thatthis will only slightly influence the results, since material data of these two materials aresimilar.

The coefficient of friction is varied between 0.3 and 0.4. In figure 4.13 the force-displacement curves for the benchmark tests are, again, represented by the solid lines. Thedashed lines represent the normal material model and dotted lines the simple material model.Black lines correspond with a value for the friction coefficient of 0.3 and green lines with avalue of 0.4. From this, it can easily be seen that all possible combinations of material modeland friction coefficient approximate the test results much better than before. Especially theinitial flexural stiffness and the maximum load is approximated well. However, still somedisagreements are found, like the point of displacement where the maximum force is found,probably due to shortcomings in the simplified material models (no softening).

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

1600

1800

Displacement [mm]

Fo

rce

[N

]

Material model = normal; Fr = 0.3

Material model = normal; Fr = 0.4

Material model = simple; Fr = 0.3

Material model = simple; Fr = 0.4

RP orientation = Vertical

RP orientation = Horizontal

Figure 4.13: Force-displacement response for 3-point-bending in both FEA and testing, wherenumerical results are now utilising the material data from tensile tests

49

Till now, the influence of the mesh density is not taken into account. Therefore, a meshrefinement study is done within Abaqus. First, the number of elements in axial direction ofthe tubular structure is varied to check whether this improves the accuracy of the results andthe agreement with test results. The number of elements over the length of the bar is variedbetween 12, 24, 36 and 48 elements. For this mesh refinement study the material data iskept constant. The material model normal is used together with a friction coefficient of 0.3.The seed size of the bottom-up mesh is still 1 mm. In figure 4.14, the results of the differentanalyses are shown, together with test results. Controversially, a smaller number of elementsseems to approximate the test results more. However, convergence to a solution is reachedupon mesh refinement in axial direction, indicating that 12 elements in axial direction arenot sufficient for this geometry. From this, it can be obtained that the original number ofelements in axial direction (24) was sufficient. Further refinement will not lead to significantimprovements while the number of elements (and computational effort) increases.

Besides changing the number of elements in axial direction, the number of elements inradial direction is varied too. In here, the number of elements in axial direction is 24. For aseed size of 0.50, 0.75 and 1.00 mm the results are given in figure 4.15. The result is quitesimilar to changing the friction factor, see also figure 4.13, so therefore the seed size is chosento be 1.00 mm. Further refinement of the mesh in radial direction does not change the overallbehaviour, while the file size (ODB file for analysis) and number of iterations will increasesignificantly (the file size increases from 64 Mb to 338 Mb for seed size 1.00 and 0.50 mmrespectively).

Concludingly, from this study on the material behaviour and mesh size dependency inFEA it appears that there are several factors in the FEA that can influence the results. Notonly the mesh size, the friction coefficient and material model, also factors as element type,solver, contact type and deformation rate contribute to the FE results. At the moment, fora simple tubular structure, the seed size of 1.00 mm, combined with 24 axial elements (trielements) seems to be sufficient to describe the material behaviour. If fibres are included inthe structure, meshing will become more complex, as will be seen later. Then the smallestfeature will give constraints to the minimal element size. Both material models lead to betteragreement between FEA and test results, although they are both simplifications of the realmaterial behaviour. Both material models will be used, and compared with test results. Thecoefficient of friction can also scale the FEA results to a certain amount. This value of thisfactor is estimated between 0.3 and 0.4.

50

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

1800

Displacement [mm]

Fo

rce

[N

]

Number of elements (axial) = 12

Number of elements (axial) = 24

Number of elements (axial) = 36

Number of elements (axial) = 48

RP orientation = Vertical

RP orientation = Horizontal

80 iterations; element size (axial) = 8.333 mm

109 iterations; element size (axial) = 4.167 mm

113 iterations; element size (axial) = 2.778 mm

114 iterations; element size (axial) = 2.083 mm

Figure 4.14: Force-displacement response for 3-point-bending in both FEA and testing, withmesh refinement in axial direction

0 2 4 6 8 10 120

200

400

600

800

1000

1200

1400

1600

Displacement [mm]

Fo

rce

[N]

Seed size = 0.50 mm

Seed size = 0.75 mm

Seed size = 1.00 mm

RP orientation = Vertical

RP orientation = Horizontal

109 iterations; approx. no. of elements in radial direction: 3

115 iterations; approx. no. of elements in radial direction: 4

166 iterations; approx. no. of elements in radial direction: 6

Figure 4.15: Force-displacement response for 3-point-bending in both FEA and testing, withmesh refinement in radial direction

51

4.2.5 Test Set-up Configurations

Figure 4.16: Indentation offixture roller into structure in 3-point-bending

As was obtained from the benchmark test, the current3-point-bending set-up induces indention of the rollersinto the material, rather than (pure) bending, see figure4.16. From chapter 2, it is clear the most natural loadingcondition of bamboo is bending. To be able to analysethe structural features in bamboo that can help to resistbending forces, biomimetic designs should be loaded inbending, and not indented. This states the need for a studyto an appropriate test set-up configuration. Because theindentation was found in both testing and FEA, severalnumerical analyses are performed to create an optimisedtest set-up configuration.

Different test set-up geometries are modelled in orderto obtain the best test set-up for the bending analysesof the biomimicked structures. The different set-ups thatare analysed are stated below. To create bending, both3- and 4-point-bending tests are considered. The bestway to compare different test set-up configurations is tomaintain a certain spacing between the support (bottom)rollers. For 3-point-bending, the moving (upper) roller isplaced in the center, in the 4-point-bending set-up, thespacing between all individual rollers is equal. For eachtest, the used material model, friction coefficient, boundaryconditions and so on are equal. The bottom-up meshingtechnique is used (with wedge elements, C3D6). Thenumber of axial elements is the same for all models and the cross-sectional area is seeded withan equal number of nodes. According to (material) costs involved with rapid prototyping,the maximum length of the tubular structure is limited to 200 mm.

• 3-point-bending tests, with

i. span width between support rollers is 70 mm; L = 100 mm (current configuration)

ii. span width between support rollers is 100 mm; L = 135 mm

iii. span width between support rollers is 150 mm; L = 200 mm

iv. rubber insert in the tubular structure, with a spacing between rollers of 35 mm;L = 100 mm

• 4-point-bending tests, with

v. span width between support rollers is 70 mm; L = 100 mm

vi. span width between support rollers is 100 mm; L = 135 mm

vii. span width between support rollers is 150 mm; L = 200 mm

As is stated in the enumeration above, also a model with a rubber insert is modeled.The hypothesis about the rubber insert model is that it creates opportunities to study thefailure behaviour of the biomimetic designs. The rubber insert could help to maintain the

52

inner volume in the tube without influencing the stresses in the structure, because a rubber is(almost) volume invariant and has a negligible stiffness compared to the structural material.This might prevent material indentation.

FEA results are stated in table 4.4. From this table, the most important parameter isthe maximum equivalent plastic strain, which represents the magnitude of a peak strain atthe point of indentation. From this, it is obtained that 4-point-bending, with a span widthof the support of 150 mm will give significantly less plastic strain. This implies that lessindentation of the roller is noticed in the structure. However, the area of maximum strainis spread over a much wider area, and on top of that, the strain at the outer surface is evenhigher as in the same set-up for 3-point-bending. This is visualised in figure 4.17, where theequivalent plastic strain is given in contour plots for both the 3-point-bending set-up (left)and the 4-point-bending set-up.

For the configuration with the rubber insert, some problems exist. Since the tubularstructure is not closed at the ends, the rubber insert can be pushed out on both sides. Howeverthe results in table 4.4 show that equivalent plastic strain decreases slightly (compared withthe 3-point-bending test with 70 mm roller spacing), the advantage of this set-up is expectedto be marginal. Besides that, a rubber insert will most probably cause other manufacturingproblems.

Conclusively, the 4-point-bending test with a span width of the bottom roller of 150 mmis more appropriate for bending tests of the biomimetic designs. With the use of theseknowledge, the test plan will be developed.

Table 4.4: Properties of different test set-up configurations (obtained with FEA)

Test set-up and number 3-point-bending set-up 4-point-bending set-up

Property Units i ii iii iv v vi vii

Span width (bot-tom rollers) mm 70 100 150 70 70 100 150

Logarithmic strainεln (max. princi-pal)

- 0.3753 0.2908 0.1354 0.4122 0.3194 0.2198 0.0743

Equivalent plasticstrain εp,eq (max.principal)

- 0.4270 0.3247 0.1314 0.2908 0.3249 0.2213 0.0663

Maximum VonMises stress σV M

MPa 39.02 34.73 34.66 35.38 41.86 33.90 32.42

53

Figure 4.17: Contour plots showing the equivalent plastic strain for 3-point-bending (left)and 4-point-bending (right) tests with a support span of 150 mm in FEA

54

Chapter 5

Test Plan

As seen in section 4.2, the influence of different sample orientations in rapid prototyping hasseveral effects on manufacturing. Material properties were determined with tensile tests andthis improved the numerical analysis predictions significantly. More realistic predictions of thestructural behaviour can be achieved based on these data. Furthermore, during benchmarktests, it was noticed that no pure bending occurred. This was also found in numerical analyses.The fixture indents into the tubular structure. Since indentation is not considered as a loadingcondition of interest, the bending test set-up was optimised numerically, to create a testconfiguration that creates bending rather than indentation. In this chapter, the test plan ispresented.

5.1 Test Set-up

In section 4.2.3 it was concluded that the preferred way of manufacturing of the tubularsamples is by use of a horizontal sample orientation. The specimens that will be created havethe following dimensions:

Din = 20 mm; Dout = 10 mm; L = 200 mmFor the materials, one material was selected as fibre material, and another material as matrixmaterial. Basically, the optimal (most realistic) way to create biomimetic structures wouldbe by using a very stiff, brittle fibre material and a much less stiff, more ductile matrixmaterial, as is found in bamboo (see also Chapter 2). This would make a complete analysisof the mechanical behaviour possible, including the failure behaviour. However, restrictionsare put on the choice of materials by the rapid prototyping process. Therefore, the mostrepresentative materials should be used that are available. Intuitively, the most suitable wayto create a gradient-like structure is by mixing the two FullCure materials (VeroWhite andTangoPlus) in certain specific volume ratios. First, from the whole range of mixture, onematerial (mixture) should be chosen as fibre material and one as matrix material. The choiceof these fibre and matrix materials is based on the material analysis, done in section 4.2.2.

For the fibre material, digital material DM8410 is picked, since this material hascompared to the other materials available the highest stiffness (together with VeroWhite).Besides that, for DM8410 the tensile test data are more reliable than for VeroWhite, showingless standard deviation, as explained before. The material data of DM8410 is more suitableto predict the material behaviour in numerical analyses.

As matrix material, DM8430 is chosen. Based on the tensile tests, this material has acomparable ductility to DM8410 (strain-to-failure is similar) while it has a lower stiffness and

55

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

(a) Material model normal for DM8410 (fibre)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

40

εtrue

[-]

σtr

ue [M

Pa

]

(b) Material model normal for DM8430 (matrix)

Figure 5.1: Material models (σ− ε -behaviour) for fibre-like (a) and matrix-like (b) materials

yield stress. Material DM9795 has an even lower stiffness and yield stress, as can be seen intable 4.2, but the strain-to-failure is much lower too. Because in bamboo the matrix materialis more ductile than the fibre material, material DM8430 is the best representative for thispart of the structure. In figure 5.1 the normal material models of the fibre material (DM8410)and the matrix material (DM8430) are given.

While the difference in material properties between the fibre and matrix material arenot as large as found in bamboo (see Chapter 2), it is still possible to create an anisotropicmaterial structure like bamboo. There is a significant difference in both Young’s modulus andyield stress. This makes it possible to create a functionally graded material. The stiffness ofthe biomimetic model will vary over the cross-section, however the anisotropy ratio is reducedcompared to bamboo. Based on the rule of mixtures (Equation 2.3), for the current designs(overall volume fraction of fibres is 0.3) the stiffness at the outer side will be 1134 MPa,while the stiffness at the most inner side will be 934 MPa. This still represents a certainanisotropy. However, the influence of a higher anisotropy ratio should be investigated morewith numerical analyses, by scaling material properties.

In the design of the biomimetic models, see paragraph 4.1, it can be seen that formodel type A.(I) (homogeneous mixture of fibre and matrix material) and model A.(II)(discrete layers with certain volume ratios of fibre and matrix material), the fibre material(DM8410) and matrix material (DM8430) have to be mixed in certain wanted specific volumeconcentrations. However, it is not possible to mix the two FullCure materials in specificvolume proportions to create materials with a stiffness in between the fibre material and thematrix material, since it is not possible to control the software code yet. Furthermore, nodata is given about volume fractions for the pre-defined Digital Materials. Besides that, itwas already noted that the mixture consists actually of a pattern-like structure rather than ahomogeneous mixture. However, the latter fact is assumed to have a negligible effect on thematerial behaviour.

Altogether, the rapid prototyping process not only puts restrictions on the choice of thematerials, it also puts restrictions on the mixture of these materials. This influence the abilityto manufacture certain biomimetic designs proposed in parapgraph 4.1. It is not possible tomanufacture models A.(I) and A.(II) at the moment - it might be possible to break in into

56

the software code in the future, but this has to be explored. This makes it more complex toanalyse the influence of the gradient of the fibre material over the cross-section experimentally.In chapter 2 the hypothesis was stated that the gradient in fibre volume concentration asfound in bamboo will create a more uniform stress distribution over the cross-section of thestem when loaded in bending. This creates a better resistance against bending loads. Withfinite element analyses, making use of the rule of mixtures for composite materials, materialproperties can be scaled such that model (A).I and model (A).II can still be created (basedon theoretical expressions). Nevertheless, it is not yet possible to validate these numericalanalyses with experiments. For now, the focus will lie on the analysis of structural featuresfound in bamboo that can be biomimicked using rapid prototyping.

Keeping the restrictions involved with rapid prototyping in mind, a simplified model ofmodel A.(II) can be created. This model consists of a cross-section with only two discreteconcentric layers, the outer made of the fibre material and the inner layer made of thematrix material. Besides that, the original models B.(III) and C.(IV) of section 4.1.2 canbe manufactured without restrictions. The design procedure described in that chapter ismaintained. The overall volume concentration of the fibre-like material (DM8410) is scaledwith reference to bamboo and will be 0.3. Other boundary conditions are scaled as describedin section 4.1.3. For models with the actual fibres inside ((B).III and (C).IV) the number oflayers in which the fibres are placed are chosen to be 2, 3 and 4. The models that will bemanufactured with rapid prototyping are stated in the overview below:

• Model IIb: two concentric layers with the fibre material in the outer ring and the matrixmaterial in the inner ring, see figure 5.3 for an image of the cross-section. This modelis a variation on the original model II, stated in section 4.1.2.

• Model III: Model with fibres of equal radius Rf , distributed along the wall thicknesswith the prescribed gradient of volume concentrations as found in bamboo. See figure5.4 for the three different variants of this model, with 2, 3 and 4 fibre layers.

• Model IV: Model with fibres of non-equal size, distributed along the wall thickness withthe prescribed gradient of volume concentrations as found in bamboo. For this model,the cross-sectional areas of the 3 variants are depicted in figure 5.5.

The model descriptions of model III and IV can also be found in section 4.1.2. Now it isclear which materials and model designs can be used for manufacturing, the test plan can befurther elaborated. This is done in the next section.

5.2 Test Plan

The biomimetic models stated in the overview of the last section are all manufactured induplex to validate the reproducibility of the material behaviour of the structure in bendingtests. In figure 5.6 an image is shown of model IV, with 3 fibre layers. This photograph istaken such that the different materials can be distinguished (this is problematic because thetwo materials have the same colour). In here one can clearly see the fibres embedded in thematrix material.

The bending tests themselves are − resulting from the bending set-up analysis in section4.2.5 − 4-point-bending tests, with a span width of the support rollers of 150 mm. Thespan width between the upper rollers is 50 mm. These tests are performed on a Instron

57

test machine, with a load cell of 10 kN . The test standards for 4-point-bending is D790-10 [30]. The bending tests are performed in displacement control, with a constant downwardsvelocity of the (upper) rollers of 1 mm/min. The maximum displacement that will be appliedis 50 mm. In figure 5.2 a photograph is given of the test set-up.

Figure 5.2: Image of the 4-point-bending test on one of the samples

NB. It should be noticed that during testing the roller fixture has changed to improve testingresults, this will be stated in Chapter 6

The direct output from the 4-point-bending tests on the Instron test machine will beforce versus displacement. Data files with this information will be created and will further beanalysed with MATLAB. Results will be compared with analyses done in the finite elementprogram (Abaqus). From here, the influence of the difference structural features of themicrostructure of bamboo will be distinguished, originating from the model designs stated insection 4.1. The results of the experiments and numerical analyses will be given in the nextchapter. Besides that, since the original goal was to analyse both bending and compressionloading conditions on the internodal structure of bamboo, in appendix E also compressionloading (buckling) is analysed with FEM. Moreover, analytical calculations are stated topredict buckling behaviour of tubular structures.

58

5.3 Hypothesis

Considering the restrictions put on the biomimicking process by the rapid prototyping process,mimicking of actual bamboo properties is not straightforward. However, rapid prototypingstill is a unique manufacturing method that enables us to create complex shaped compositestructures. We hope to capture structural features in the microstructure that help to resistbending moments.

With the analysis of the factor EI, a prediction can be made on the resistance againstbending. Model IIb will give the best resistance since this has all (stiff) fibre material on theoutside. As was stated in Chapter 2, the radius of gyration (see equation 2.1) can give anindication of the stiffness by the placement of fibres. In table 5.1 an overview is given forradius of gyration of the different fibre models. In this table, also the difference in effectivestiffness ∆EI with reference to the effective stiffness of Model IIb (which equals 16.458 Nm2)is given, together with the relative difference.

Table 5.1: Radius of gyration and difference in effective stiffness for different models

Model type Number of fibre layers Rg in mm ∆EI in Nm2 ∆EIEIModel IIb

in %

ModelIII

2 8.134 -1.068 -6.49

3 8.220 -1.002 -6.09

4 8.250 -0.978 -5.94

ModelIV

2 7.988 -1.180 -7.17

3 8.211 -1.009 -6.13

4 8.252 -0.977 -5.93

It will be interesting to see how model III and IV will behave compared to model IIb.Since in the biomimetic structures the ratio of the stiffness of the fibre and matrix materialis not as high as found in bamboo, the effect of the reinforcement by the fibre-like structureis significantly reduced. The stiffness ratio is now 1.8, whilst for bamboo this ratio is 23.Some advantage(s) of the highly anisotropic structure might show up during testing and/orFEA. With comparing the bending test results of model III and IV the influence of fibre sizewill possibly be captured. It is expected that a non-equal fibre size can represent a betterfibre distribution over the thickness of the tube wall, creating a more homogeneous stressdistribution and giving a better resistance against bending. However, the latter part is notfinished. No FE analyses were performed for the models with fibres (III and IV). For theseanalyses, a large number of elements and computational time is required. Numerical analysisof the models with fibres is therefore out-of-scope for the project.

From table 5.1 it can be seen that the radius of gyration becomes larger when morefibre layers are used and the difference between the stiffness (w.r.t. Model IIb) decreases.This indicates that the fibre placement for the models with more layers should give a stifferresponse in bending. However, from the comparison between the fibre placement in ModelIII and IV, it is expected that the stiffness should be comparable. The influence of the fibresize might therefore not show up during the experiments. However, it is interesting to see if

59

any effects can be seen in numerical analyses.Since the polymers that will be used are very ductile, analysis of the failure behaviour will

be quite complex, since the failure mode will change compared to bamboo. For the latter,FEA can give more insights, since material properties can be adapted easily. Nevertheless,failure analyses with FEA are beyond the scope of this project.

60

Figure 5.3: Cross-section of Model IIb in SolidWorks

Figure 5.4: Cross-section of Model III in SolidWorks, with from left to right respectively 2, 3and 4 fibre layers

Figure 5.5: Cross-section of Model IV in SolidWorks, with from left to right respectively 2, 3and 4 fibre layers

Figure 5.6: Cross-section of Model IV in SolidWorks, with 3 fibre layers (left) and photographof the cross-section of the same model, manufactured with rapid prototyping (right)

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62

Chapter 6

Bending Analysis

Following benchmark testing and material testing, the biomimicking procedure can becontinued that will be used to investigate the influence of the different microstructural featuresin bamboo on its mechanical behaviour. The original test set-up, biomimetic designs and testplan were adapted to former test results. With rapid prototyping, the selected biomimeticdesigns (see figure 5.3 to figure 5.5) are manufactured. In this chapter, first the test resultsfrom the 4-point-bending tests will be presented and analysed. Then, the FE procedurewill be explained in more detail, related to meshing techniques for models with fibres inside.Finally, a short discussion will relate the hypothesis (stated in section 5.3) to these testresults. Moreover, some points related to further exploitation of the bending analysis ofhighly anisotropic structures will be given.

6.1 4-Point-Bending Test Results

In this paragraph, test results of the performed 4-point-bending tests will be presented andanalysed. The force-displacement curves are given in figure 6.1. Before analysing these testresults, it must be stated that some problems with the roller fixture were initially present. Formodel IIb magnetic support rollers were used. However, at a displacement of about 17 mm thebottom rollers did not stay in position any more and lost contact with the tubular structure.At this point, the specimens had not failed yet. Since plastic deformation was already presentin the structure at that stage, it was impossible to test these models to a larger displacement(up to 50 mm) to continue the load-displacement curves to larger displacement. The latterhas been validated with repetitive tests that were performed because plastic deformation inthe structure was only slightly visible (and therefore seems to be insignificant). The force-displacement response in the repetitive test is much lower (difference with maximum force is> 50 N) indicating that the existing plastic deformation is still significant.

After the first two tests, the roller fixture was reviewed upon these experimental findings.Now, the bottom magnetic rollers were replaced by a more robust roller fixture, where therollers were fixed at both ends rather than maintained in position by a magnetic force. Thisroller fixture can be seen in figure 5.2. The upper roller fixture was still the original rollerconfiguration. Despite this change, upon testing of model III (with 2 virtual fibre layers)similar problems occurred, caused by loose contact between the sample and the top rollers.

Thus, also the top roller fixture had to be changed. Both the top en bottom rollers weremaintained in position by a solid fixture. This was working properly, and all subsequent testswere performed with a maximum displacement of 50 mm. This fixture is shown in figure

63

5.2. Altogether, the influence of this changing fixture might have influenced test results,making the comparison between model IIb and model III with 2 fibre layers, and the otherbiomimetic designs more complex based on the test data. Numerical analyses could give moreinsight. Nevertheless, the change in initial behaviour is assumed to be relatively small, sinceforce-displacement responses show similar curves for all tests.

In the top graph of figure 6.1 the force-displacement curves are given for model IIb.Model IIb is in fact a structure which is designed for optimal stiffness - having all the moststiff material (fibre) on the outside, see figure 5.3. From this graph, it can be obtained thattest results are quite well reproducible. The slight solid lines represent the test data on thetwo samples, the thick dashed line is an average curve for these two tests. As said, themaximum displacement is unfortunately about 17 mm, but the maximum force and initialflexural stiffness can still be obtained.

In the middle and bottom graph of this figure, the force-displacement curves for modelIII and IV are given respectively. The cross-sections of model III and IV can be seen infigure 5.4 and 5.5 respectively. Except for model III with 2 fibre layers, all samples have beentested upon a maximum displacement of 50 mm. The initial flexural stiffness of both modelIII and IV is comparable. Their values are all in the same range considering the standarddeviations involved. The only difference between the two biomimetic structures, that can beobtained from the curves, is a small increase in maximum force for model IV by comparisonto model III. The latter can also be obtained from table 6.1. This can give an indication thatthe structure with unequal fibres creates a more homogeneous stress distribution over thecross-section of the surface, giving an increased load carrying capability. An increased forceis needed to plastically deform the structure. However, it must be stated that the standarddeviation is large. The number of tests (in the same test conditions) has to be increased tovalidate this statement.Furthermore, the force-displacement responses for models III (middle) and IV (bottom)show that the failure behaviour is very ductile: no critical failure was obtained. Thisductile material behaviour was previously obtained during tensile tests in section 4.2.2.After bending, the structure deforms back slowly towards its original configuration. Plasticdeformation is only slightly visible. Therefore the influence of the different microstructuralfeatures on the failure behaviour of bamboo can not be analysed with these experiments, sincethe mechanical properties are too different compared to bamboo. The mechanical behaviourof a structure is the result of a subtle interplay between both the intrinsic constitutivebehaviour of the base materials and the complex microstructure of the material. The complexmicrostructure is controlled, but the choice of materials is restricted and material propertiesare different from the properties of bamboo. This of course has important effects on thematerial behaviour of the samples. Besides the difference in (relative) stiffness of the materials,

Table 6.1: Stiffness and maximum force for each model, obtained from 4-point-bending tests

PropertyModelIIb

Model III(2 layers)

Model III(3 layers)

Model III(4 layers)

Model IV(2 layers)

Model IV(3 layers)

Model IV(4 layers)

Stiffness[N/mm]

62.38 ±12.70

61.58 ±4.20

49.04 ±2.71

51.87 ±4.37

52.64 ±6.90

53.19 ±1.97

57.01 ±0.91

MaximumForce [N]

358.88 ±6.64

357.37 ±6.92

344.52 ±6.47

333.16 ±9.77

361.55 ±16.46

373.43 ±10.69

367.49 ±13.44

64

both rapid prototyping materials are also too ductile compared to bamboo.Referring to the latter statement, in the discussion of this chapter (see section 6.3), the

usefulness of rapid prototyping in the biomimetic design sequence will be discussed. It will beargued why biomimicking is a useful tool in the biomimetic design sequence, although materialproperties of bamboo can not be mimicked properly with this manufacturing technique.

Altogether, straightforward conclusions are hard to be made on the basis of these testresults alone. This where FEA comes into play. Numerical analyses can be used to createmore insight and optimise biomimetic designs. The first steps into numerical analyses ofcomplex shaped structures will be given in the next section.

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0 5 10 15 20 25 30 35 40 45 500

50

100

150

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Displacement [mm]

For

ce [N

]

Model IIb #1

Model IIb #2

Average response Model IIb

0 5 10 15 20 25 30 35 40 45 500

50

100

150

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250

300

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400

Displacement [mm]

Forc

e [N

]

Model III (2 layers) #1

Model III (2 layers) #2

Average response Model III (2 layers)

Model III (3 layers) #1

Model III (3 layers) #2

Average response Model III (3 layers)

Model III (4 layers) #1

Model III (4 layers) #2

Average response Model III (4 layers)

0 5 10 15 20 25 30 35 40 45 500

50

100

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Displacement [mm]

Forc

e [N

]

Model IV (2 layers) #1

Model IV (2 layers) #2

Average response Model IV (2 layers)

Model IV (3 layers) #1

Model IV (3 layers) #2

Average response Model IV (3 layers)

Model IV (4 layers) #1

Model IV (4 layers) #2

Average response Model IV (4 layers)

Figure 6.1: Test results for 4-point-bending; with results for model IIb (top), model III(middle) and model IV (bottom)

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6.2 Finite Element Procedure

In section 4.2.4, general parameter settings are given for the FEA of a bending test on ahomogeneous tubular structure with localised deformation under contact. Now, this tubularstructure has a more complex cross-section, which will drastically increase the number ofelements that is needed for convergence. Moreover, the method for meshing this cross-sectionhas to be reviewed. The latter is giving problems with connectivity of elements. This occursfrom the fact that the part consists of several cells (each fibre is at least one cell) whichhave to meshed separately. Afterwards, nodes have to be merged. In here, only the meshingtechnique itself is explained in detail. Due to a lack of time no analysis results were obtained.Therefore some opportunities and improvements for the use of FEA will be stated afterwards.

6.2.1 Meshing Technique

Although it was already mentioned that the application of finite element methods needs moreelaboration, some guidelines can be given about the parameter settings in the FE program,Abaqus. The example given here is based on model III, with 2 fibre layers. This model iscreated in SolidWorks and imported (as STEP file) into Abaqus as one single part. Thecross-section is given in figure 6.2. By importing this geometry, each fibre will become aseparate cell. The part has to be partitioned in the xy-plane in order to be able to prescribeboundary conditions properly, see therefore section 4.2.4. This means that the biomimeticmodels with the fibres inside consist of many cells which have all to be meshed separatelyusing the bottom-up meshing technique.

Figure 6.2: Cross-section of model III with 2 fibre layers in Abaqus

The strategy to create a consistent mesh is not straightforward. First, the circumferencesof the fibres (on top surface) have to be seeded. The seed size has to be chosen such thatthe circular shape can be approximated well by triangles. In here, the seed size of the fibres

67

is 0.75 mm. Then, bottom-up meshes have to be created for all fibres. After that, the twocells that represent the matrix material have to be meshed. The matrix material consists oftwo cells, since the whole part is partitioned along the xy-plane, to prescribe the z-symmetryboundary condition, see again figure 4.12. Therefore, the outer and inner ring of the tubularstructure have to be seeded. The value of the seed size has to be determined with trial anderror. Nodes of the mesh of the matrix material have to coincide with nodes of the fibremesh. In here, this seed size is determined to be 0.50 mm. However, at the partitioning line,which can be seen in figure 6.3, the nodes of the fibre mesh and matrix mesh will probablynot coincide yet. In that case, the matrix mesh has to be removed; and the partitioned fibrecells around the partition line have to be meshed again, using a different seed size. For thismodel, the seed size of the two partitioned fibres is 0.50 mm. Applying these values, a propermesh can be created for model III with 2 fibre layers. After these steps, nodes have to bemerged and the outer surface of the tubular structure has to be associated with the mesh.The resulting mesh is shown in figure 6.3, where a part of the mesh is visualised.

Figure 6.3: Top view of a part of the created mesh of model III with 2 fibre layers in Abaqus

As can be obtained from this description, mesh creation becomes complex and ispredominantly based on trial and error - even for relatively large fibre diameters. OtherFE programs might provide an easier way to create this mesh. In here, a bottom-up meshthat defines a 2D mesh is extruded in axial direction, but more options in meshing can beconsidered. The influence of mesh refinement should be investigated. The mesh in the contactzone should be fine enough to get rid of jumps in reaction forces caused by a instantaneous(large) deviation of contact nodes at a certain time increment. The latter is assumed to beone of the critical factors in the FEA of 4-point-bending. Altogether, test results should beused as reference for numerical analysis to validate the reliability.

6.2.2 Opportunities and Improvements

Although only a general description of meshing is given here rather than actual the resultsof FEA, it is assumed that the use of numerical analyses in combination with test results is

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useful. At the moment, numerical analyses in this project are not yet at the stage that themechanical behaviour can be predicted accurately, i.e. a relatively large discrepancy betweentest results and numerical results is found. Meshing techniques have to be explored in moredetail. Since the numerical analyses gave quite reliable predictions for simple structures in3-point-bending (see section 4.2.4), it is assumed that the method should be applicable forthe more complex biomimetic structures as well. Accurate predictions from FEA can offer avariety of opportunities in the field of biomimicking in Materials Science. One can think ofseveral options:

• Instead of using material properties of the materials used in rapid prototyping, useproperties of bamboo. The mimicked structure of bamboo, as found in the biomimeticdesigns, is optimised for material properties of bamboo in the FE program. The mi-crostructural optimisation will most probably become different if other materials (suchas DM8410 and DM8430) are used, especially when relative properties (proportions ofmaterial properties of the matrix material to the properties of the fibre material) change.The advantages of individual microstructural features that exist in bamboo might notbecome clear when the material properties of the rapid prototype sample are used, butthey could possibly give advantageous effects (e.g. homogeneous stress distribution) ifmaterial properties of bamboo are used. If the latter will be analysed carefully, thatknowledge can be used to create new biomimetic designs using specific base materials,in order to optimise a composite material structure. The functionality of that structure,of course, determines the way the structure should be optimised.

• Besides that, FEA in combination with actual testing of bamboo (in 4-point-bendingtests), results from finite element analyses can be validated. Can numerical analyses ofbiomimetic designs give similar results as test results for bamboo? In here, an otheroption that could be mentioned is to create a finite element model based on tomographydata, such that the actual fibre structure as found in bamboo is modelled. Is it possibleto predict the behaviour of bamboo in that way more accurately? Or are there somemicro-structural effects, or even effects originating from the hierarchical structure, thatlead to significant differences?

6.3 Discussion of Results

Before proceeding to the final conclusions, results that were given in this chapter are discussed.In this chapter it was already stated that the difference between model III (equal fibre size)and model IV (unequal fibre size) can not significantly be seen directly from test results. Alsothe statement (from section 5.3) that more fibre-layers give a higher flexural stiffness thanmodels with less fibre layers is not obtained from test results. The difference between differentmodels show too much standard deviation to give straightforward results based on the tests.Finite element analyses could give more insight in this, but this needs more elaboration.Besides that, from table 5.1, it was already predicted that - with these material properties -the difference in stiffness would be small, since the stiffness is similar (within a deviation ofa few per cent).

In section 6.1 the usefulness of rapid prototyping was mentioned, in relation to thetest results. With rapid prototyping, the microstructure can be manufactured followingthe biomimetic designs given in Chapter 5. However, material properties of bamboo can

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not be mimicked properly. Mechanical properties of the base materials are totally differentfrom the ones found in bamboo and furthermore, also the relative properties are different.The biomimetic designs of paragraph 4.1 mimic a certain fibre volume distribution in radialdirection of the tubular structure. Since not only the actual but also the relative materialproperties change by application of rapid prototyping, aspects as stiffness distribution overthe thickness become different too. The resulting mechanical behaviour of the biomimickedsamples is therefore totally different from bamboo. This is highlighted by the schematicfigure of the constitutive behaviour of a material, see figure 6.4. In here is visualised thatthe macroscopic constitutive behaviour of a material is the result of the interplay betweenthe complex microstructure and the intrinsic constitutive material behaviour of the two basematerials.

Figure 6.4: Schematic representation of the macroscopic constitutive behaviour of a material,by J.A.W. van Dommelen and J.G.F. Wismans, Faculty Mechanical Engineering, Universityof Technology Eindhoven

Thus, the microstructure that is mimicked following the characteristics found in bamboois most probably not the most favourable in combination with the materials used in rapidprototyping. Nevertheless, experimental test results of samples still contribute to a highextent to the reliability of numerical analyses. Together with the experimental results, optimalparameter settings for the FE program can be obtained. Rapid prototyping is a useful toolto create complex shaped structures and provides opportunities for validation of numericalanalyses. On the other hand, with the use of FE, it is easy to adapt material properties.Using the material properties of bamboo in FE, it can be validated if the biomimetic designindeed represents advantageous effects such as a more homogeneous stress distribution uponbending. With this strategy - in which experimental and numerical analyses are combined -more reliable insights can be achieved about the microstructural design as found in bamboo.

In addition to that, not only numerical analyses and experiments can be used inbiomimicking. Analytical expressions should be able to give more insight in the mechanicalbehaviour of highly anisotropic structure in bending as well. However, only a few studieshave been found on the effects of anisotropy [24,25]. Homogenisation techniques (such as ruleof mixture) can be applied to obtain the global behaviour of the structure, but local featurescan not be captured [7] with this. Despite analytical work being of the scope of this project,certainly more work has to be done in this area.

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Conclusions & Recommendations

Biomimicking is an exciting and relatively new research field in Materials Science andEngineering. In this survey, the biomimetic design procedure was based on bamboo. Usingrapid prototyping, the ability to manufacture biomimetic designs was explored, in order tocreate more insight in the beneficial microstructural features that are found in bamboo.Summarising, it can be stated that the application of this manufacturing technique incombination with biomimicking is not straightforward, putting several boundary conditionson the design. However, it can create more insights in the beneficial effects of certain aspectsof the microstructure on the mechanical behaviour. In the discussion of the results in section6.3 the advantages and disadvantages of the application of the rapid prototyping techniquein combination with the biomimetic design procedure were already mentioned. In here, somegeneral conclusions that were found in this project will be given.

First of all, it should be reconsidered that biomimicking of bamboo can be done for twodifferent purposes. The first goal of biomimicking is to understand the mechanical behaviourof bamboo in more detail. “Why is the microstructure of bamboo structured and optimisedthe way it is?” And the second goal is to design new materials, based on this knowledgeof the structural organisation of bamboo and its related mechanical properties. “Can theknowledge be used for the design of new lightweight composite structures with a high loadbearing efficiency?” The first goal of biomimicking is predominantly present in this thesis -in order to learn from nature, first more knowledge should be gotten from the structuring ofbamboo. Biomimetic designs were manufactured, consisting of microstructural features of thecross-section of the internodal part of a bamboo stem. Different designs gave the possibilityto individually analyse several features, like the gradient in fibre material, the use of fibresitself, and the use of fibres of unequal size. Manufacturing these designs was done by rapidprototyping. Understanding of the microstructural organisation in bamboo is needed to createnew insights in the development of new material structures and is critical to fulfill the secondgoal of biomimicking.

The statement in the beginning of this chapter, that the application of this manufacturingtechnique is not straightforward, can most well be explained using the design sequence used inthis internship project and shown in figure 6.5. This sequence is based on the scheme shown infigure 1.4 and on experience of this project. Starting with the analysis of the microstructureof bamboo, certain boundary conditions had to be taken into account before biomimeticdesigns could be created. In this thesis, microstructural properties of bamboo are simplifiedbased on a gradient of fibre volume concentration along the thickness, as found on averagein bamboo. Further on, the cross-sectional shape of the fibres was simplified to be circular-shaped, the composite consisted of only two base materials (a fibre material and a matrixmaterial), and no porosity was taken into account. Using these restrictions, several biomimeticdesigns were created. Before these biomimetic designs were manufactured, benchmark tests

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on simple tubular structures were performed. This can be seen as an initialisation of therapid prototyping process. These benchmark tests are 3-point-bending tests (described inparagraph 4.2) and form - together with the material analyses - the basis for the biomimeticdesign procedure using rapid prototyping. Conclusions that were obtained based on thebenchmark tests and the materials testing are stated below:

(i) Material properties given by manufacturer (Objet) are incorrect; material data obtainedby tensile tests are in better agreement with test results, when utilised in numericalanalyses. The materials that are available in the rapid prototyping process are all veryductile (compared to bamboo). Both actual and relative material properties of thematerials used with rapid prototyping are different from properties of bamboo, e.g. themodulus of the fibre-like material (DM8410) is 1.8 times higher than the modulus ofthe matrix-like material, whilst this factor for bamboo is 23.

(ii) To mimic the most natural loading condition of bamboo (bending due to wind forces),the span width of the support rollers must be evaluated with numerical analyses, anddepends on the geometry of the sample. 4-Point-bending is more favourable than 3-point-bending, creating a larger area on which maximum strain occurs and minimisingindentation of the fixture into the sample. Finite element analyses can give reliable andaccurate predictions of the structural behaviour of the simple tubular structure.

(iii) The influence of sample orientation during rapid prototyping is not significant;depending on the sample geometry the best orientation can be found using severalcriteria related to the rapid prototyping process. In here, the horizontal orientation ofthe sample is preferable, due to stability during manufacturing.

Figure 6.5: Biomimetic design sequence

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On the basis of the test results of both materials testing and benchmark tests, the originalbiomimetic designs and test plan had to be reviewed. Following the biomimetic designsequence from figure 6.5 it can be seen that rapid prototyping puts more boundary conditionsonto the biomimetic designs. Considering the accuracy of the rapid prototyping process, acritical minimal feature size had to be chosen such that the individual fibres are continuousin axial direction of the tubular structure. Since the in-plane accuracy is 42 µm, this criticalfeature size should be at least larger than that. In this thesis, no actual measurements weredone on the accuracy of rapid prototyping, but photographs of polished cross-sections ofsamples show that the chosen minimal feature size (100 µm for minimal spacing betweenfibres and 300 µm for minimal fibre radius) is sufficient.

Moreover, rapid prototyping creates some other restrictions for the biomimetic designsequence. Since this technique lays down discrete layers of material, it is not possible to createa continuous graded material, where the volume concentration of a material follows a smoothdistribution in radial direction of the tube. However, if the layer thickness is minimised, acontinuous graded material can be approximated. Nevertheless, it appeared that it is notyet possible to mix two materials in a certain user-specified volume concentration. And thevolume concentration proportions of pre-specified mixtures (Digital Materials) is unknown sofar. By contacting the manufacturer it could be possible to get more control over the machinesoftware. This might solve these problems and create more possibilities for the analyses ofthe microstructure of bamboo by the experimental procedure, since all original designs (asstated in parapgraph 4.1) can be fabricated.

However, it should be noted that these new boundary conditions in first instance onlyrelate to the ability to manufacture the designs. Finitie element analyses give more flexibility:material properties can be changed easily, new biomimetic designs can be evaluated andoptimised. Nevertheless, also with the use of FEA restrictions are put on the design. Again,minimal feature size and discontinuity of material properties are relevant. Both are relatedto the computational effort that is involved with more complex models that consist of a largeamount of elements.

The use of numerical analyses in combination with experiments on samples, manufacturedwith rapid prototyping, can give new insights in the development of materials with a high loadbearing efficiency. The combination of both experimental testing and FEA should be adapted,considering that the macroscopic constitutive behaviour of a structure is the result of a subtleinterplay of the intrinsic constitutive behaviour of the base materials and the complex micro-structure. Finite element analyses give great possibilities in biomimetic design, but withoutexperimental testing, numerical results will not make much sense. This was already stated insection 6.3.

From this work, it can finally be stated that a lot more work has to be done to actuallybe able to get more insight in the structural features of bamboo. Conclusively, a list ofrecommendations is given for future work on the biomimetic design based on bamboo:

(i) Perform tests on bamboo and compare this with FEA of biomimetic designs usingmaterial properties of bamboo. On top of that, tomography scans could also be used asa basis to create more realistic structures of bamboo, that can be analysed numerically.However, the latter will raise the need for further improvements of the FE procedure(related to meshing).

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(ii) Perform more bending tests, this will decrease the standard deviation and will probablygive more insight in the effect of certain microstructural features on the mechanicalbehaviour.

(iii) Explore the possibility of performing 4-point-bending tests at low temperatures; thismight create opportunities to actually test more brittle failure, since movement ofpolymer chains might be obstructed drastically. More research has to be done on thistopic. Starting point of this survey should be material testing at low temperatures.

(iv) More work has to be done related to analytical expressions for bending of highlyanisotropic structures and ovalisation of the cross-section upon bending.

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Discussion

In this project, one of the main objectives was to obtain the influence of different micro-structural features on the mechanical behaviour of bamboo. Why does bamboo have thisspecific microstructure (shown in Chapter 2) and can we learn from that and applicate theknowledge for the design of new materials? Bamboo-like structures are in the first instance notdesigned for high stiffness, as traditionally widely found in engineering structures. Bamboohas (as obtained from literature) uniform strength, the stress over the cross-section is uniformdue to a smart distribution of fibre material in the structure. This graded structure consistsbasically of a gradient in fibre volume concentration over the thickness, but also the shapeand size of the fibres change in radial direction.

This report gives advanced insight in the combination of rapid prototyping withbiomimicking. However the conclusions are currently not straightforward with reference tothe understanding of influence of the micro-structural features on the mechanical behaviour,this project shows that rapid prototyping is a useful tool in the biomimetic design sequence.Complex structures can be manufactured, as long as the structure does not have a closed-cell structure. The manufacturing process is relatively fast and cheap. However someimprovements in the manufacturing part are desirable, the technique is directly applicableand no further processing is required. At the moment, the choice in materials is a limitingfactor to actually study the mechanical behaviour of bamboo directly. Moreover, this impedesthe possibilities for the analysis of failure behaviour during experiments. However, one shouldalways notice that the mechanical behaviour of a structure originates from a subtle interplay ofthe intrinsic constitutive behaviour of the base material(s) and the complex micro-structure.This interplay is one of the reasons the biomimetic design procedure always should follow asequence as shown in figure 6.5: the microstructural optimisation is most possibly differentin case other base materials are used. FEA in combination with rapid prototyping is offeringgreat opportunities to create knowledge about several microstructural features in bamboo.The primary goal should be to understand the microstructural design of bamboo in moredetail, before new structural designs are created that are based on the microstructure ofbamboo. In biomimicking we should learn from nature, and the rapid prototyping techniqueis a helpful tool in this field. This work can be used as a start-up for future work; biomimickingin Materials Science and Engineering is a promising field to create new lightweight materialstructures, optimised for a high load bearing efficiency, and rapid prototyping can give -together with FEA - new insights in this field.

75

76

Bibliography

[1] Walter Liese The Anatomy of Bamboo Culms, 1998, International Network for Bambooand Rattan, Beijing Eindhoven New Delhi, ISBN 81-86247-26-2

[2] James M. Gere, Barry J. Goodno Mechanics of Materials, 7th Edition, 2009, CengageLearning, Stafmford USA, ISBN-13 978-0-495-43807-6

[3] Stefanie Feih Design of Composite Adhesive Joints, 2002, Cambridge University

[4] Jules J.A. Janssen Designing and Building with Bamboo , 2000, Technical Universityof Eindhoven, The Netherlands, International Network for Bamboo and Rattan, ISBN81-86247-46-7

[5] Michael F. Ashby Materials Selection in Mechanical Design, 1999, Boston, Butterworth-Heinemann, ISBN 07-506-4357-9

[6] Lorna J. Gibson and Michael F. Ashby Cellular solids - Structure and properties, 2001,2nd edition, Cambridge University Press, ISBN 0-521-49911-9

[7] Emılio Carlos Nelli Silva, Matthew C. Walters, Glaucio H. Paulino Modelling bambooas a functionally graded material: lessons for the analysis of affordable materials, 2006,Brazil and USA, J. Mater. Sci. (Vol. 41, p. 6991-7004)

[8] K. Ghavami, C.S. Rodrigues, S. Paciornik Bamboo: functionally graded compositematerial, 2003, Rio de Janeiro, Asian Journal of Civil Engineering (Vol. 4, No. 1, p.1-10)

[9] Hanns-Christof Spatz, Lothar Kohler, Thomas Speck Biomechanics and FunctionalAnatomy of Hollow-Stemmed Sphenopsids, 1998, Institute for Biology, University ofFreiburg Germany, American Journal of Botany (p. 305-314)

[10] S.H. Li, Q.Y. Zeng, Y.L. Xiao, S.Y. Fu, B.L Zhou Biomimicry of bamboo bast fiber withengineering composite materials, 1995, China, Materials Science and Engineering (C3, p.125-130)

[11] A.K. Ray, S. Mondal, S.K. Das, P. Ramachandrarao Microstructural characterization ofbamboo, 2004, India, Journal of Materials Science 39 (p. 1055-1060)

[12] A.K. Ray, S. Mondal, S.K. Das, P. Ramachandrarao Bamboo - A functionally gradedcomposite-correlation between microstructure and mechanical strength, 2005, India,Journal of Materials Science 40 (p. 5249-5253)

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[13] S. Amada et. al Fiber texture and mechanical graded structure of bamboo, 1996, GunmaUniversity Japan, Elsevier Composite Part B (Vol. 28B, p. 13-20)

[14] S. Amada, S. Untao Fracture Properties of Bamboo, 2001, Gunma University Japan,Elsevier Composite Part B (Vol. 32, p. 451-459)

[15] Linhao Zou, Helena Jin, Wei-Yang Lu, Xiadong Li, Nanoscale structural and mechanicalcharacterization of the cell wall of bamboo fibers, 2009, US, Materials Science andEngineering (Vol. C29, p. 1375-1379)

[16] Fumio Nogata, Hideaki Takahashi, Intelligent Functionally Graded Material: Bamboo,1995, Japan, Pergamon Composites Engineering (Vol. 5, No. 7, p. 743-751)

[17] Peter Fratzl Biomimetic materials research: what can we really learn from nature’sstructural materials?, 2007, Max Planck Institute of Colloids and Interfaces, PotsdamGermany, Journal of the Royal Society Interface (Vol. 4, p. 637-642)

[18] Marc Andre Meyers, Po-Yu Chen, Albert Yu-Min Lin, Yasuaki Seki Biological Materials:Structure and Mechanical properties, 2008, University of California, San Diego US,Elsevier, Progress in Materials Science (Vol. 53, p. 1-206)

[19] Jian-feng Ma, Wu-yi Chen, Ling Zhao, Da-hai Zhao Elastic Buckling of Bionic CylindricalShells Based on Bamboo, 2008, China, Journal of Bionic Engineering 5 (p. 231-238)

[20] Julian F. V. Vincent et. al Biomimetics: its practice and theory, 2006, Bath UK, J. R.Soc. Interface (Vol. 3, p. 471-482)

[21] Markus Antonietti, Peter Fratzl Biomimetic Principles in Polymer and Material Science,2010, Germany, Macromol. Chem. Phys. (Vol. 211, p. 166-170)

[22] Lorna J. Gibson Biomechanics of cellular solids, 2004, Cambridge, MassachusettsInstitute of Technology, Journal of Biomechanics (Vol. 38, p. 377-399)

[23] DuPont Technical Guide - Kevlar - Aramid Fibre, USA, www.kevlar.com

[24] Alpay Oral, Gunay Anlas Effects of radially varying moduli on stress distributionof nonhomogeneous anisotropic cylindrical bodies, 2005, Bogazici University IstanbulTurkey, International Journal of Solids and Structures (Vol. 42, p. 5568-5588)

[25] K. Schulgasser, A. Witzum On the Strength, Stiffness and Stability of Tubular PlantStems and Leaves, 1992, Israel, J. Theor. Biol. 155 (p. 497-515)

[26] M.J. Vaessen, J.J.A. Janssen Analysis of the critical length of culms of bamboo in four-point bending tests, 1997, Eindhoven University, Heron (Vol. 42, No.2) ISSN 0046-7316

[27] Objet Digital MaterialsTMData Sheets

http://www.objet.com/3D-Printer/Connex350/

[28] Objet FullCure R©Materials

http://www.objet.com/3D-Printer/Connex350/

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[29] EU Clean Sky Initiative

http://www.cleansky.eu/

[30] ASTM Book of Standards 2006, Section 8, ISBN 0-8031-4156

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Appendix A

M-files for Design of Models

This appendix gives some pointers about the routine used to design the different biomimeticdesigns. The MATLAB-files will not be stated literally here. In the subsequent item list, thesedifferent files are stated together with their working principles. This will give an idea of howthe models are designed. Finally, a block diagram will illustrate how these files work togetherin a loop sequence.

A1 Geometry_Computation_AllSamples_Final.m: This is the basic file that will be runned.In this file all model parameters are initialised, and boundary conditions are establishedfor the fibre volume concentrations across the cross-section of the tube. The differentmodels are treated in a loop sequence, to ensure the final designs have the same overallvolume concentration of fibre material.

A2 funphi.m: This function file contains the boundary conditions with reference to the fibrevolume concentration function, as stated in paragraph 4.1.

A3 phi_f_checkfile.m: In this file is verified if the model design is able to meet therequirements with respect to φf . Boundary conditions will be updated if needed,creating the opportunity to review the model design.

A4 modelIV.m: This file creates the geometry for model type IV, see paragraph 4.1 for anexplanation of this model type.

A5 modelIII.m: This file creates the geometry for model type III, see also paragraph 4.1.

A6 modelII_and_model_I.m: This file creates the geometry for model type I and II, see alsoparagraph 4.1.

A7 modelfigures.m: Creates figures for the volume fraction of fibres φf versus the radius,with bar plots indicating the width of the layers in which the fibres are placed.

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initialise modelparameters (A1)

solve function φf(A2) with BC’s

check ifmodel meets

requirements withrespect to φf

and adapt BC’sif needed (A3)

evaluate modelIV (A4)

evaluate modelIII (A5)

evaluate model IIand model I (A6)

postprocessing:figures (A7)

! Info: Model I to IV areevaluated in a loop sequence, sothe same fibre volume fractionsare guaranteed for all models

Figure A.1: Block scheme representing the matlab routine used to create the biomimeticdesigns

Appendix B

M-files for Testing of Materials

In this appendix, all MATLAB-files that are used for the post-processing of the test data ofthe tensile tests are given. The file structure is quite general and could be used for likewiseexperiments in future. Here is shortly explained how the program works and how differentfiles perform. Again, a block diagram is used to visualise the routine.

B1 Read_Data.m: This file contains all data about geometry of samples, as wel as data ofthe test set-up. It reads in all data that is available from testing: CSV files (Excel) with4 columns. The first column contains time (in s), the second is the load (in N), thethird is extension (in mm) and the fourth represents the strain. After the input datais read in, the data is filtered, such that the noise that exists in the measurements isreduced. It then calculates the yield stress, the lowest stress after yield stress relaxation(referred to here as ’softening stress’), the Young’s modulus and the strain-to-failure.Before plotting the results, this file continues in the next file.

B2 Output_Data.m: Now all experimental data is analysed, this file first calculates the meanvalues (for 2 identical experiments) and prints these values, together with standarddeviations and relative errors, on the screen. Now, material models has to be formedsuch that these are more general and easier to use in Abaqus. Therefore, the nextfunction file is used.

B3 Gs_Ge_polyfit.m: This m-file creates a polyfit for σ as function of ε and also providesthe input data for the material models used in Abaqus, i.e. stress σ as function of plasticstrain εp. In order to achieve this, the original stress-strain curve is divided into threeparts: (1) from ε = 0 to ε = εy, (2) from ε = εy to ε = εsof and part (3) from ε = εsofto ε = εfailure. In this three parts some boundary conditions are known now, since thematerial data from testing is analysed already. These data is used to create polyfits inthe 3 domains. For the creation of these polyfits, the three functions below are used,which also state the boundary conditions. Finally, the curve is more simplified becausethe yield stress relaxation behaviour gives problems in Abaqus. The material modelsused are stated in paragraph 4.2.2.

B4 polyn_Gs_1.m : Contains the function (3rd order polynom) with the boundary conditionsof part (1) of the stress-strain curve.

B5 polyn_Gs_2.m : Contains the function (3rd order polynom) with the boundary conditionsof part (2) of the stress-strain curve.

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B6 polyn_Gs_3.m : Contains the function (2nd order polynom) with the boundary conditionsof part (3) of the stress-strain curve.

Please note that throughout this files, the notation Gs refers to Greek s which representsthe stress σ. This is also valid for Ge and so on.

Read in testdata (CSV

files); calculateE, σy and

other materialproperties (B1)

Calculateaverage materialproperties (B2)

Function file withBC’s for part (2)of stress-strain

curve (B5)

Function file withBC’s for part (1)of stress-strain

curve (B4)

Function file withBC’s for part (3)of stress-strain

curve (B6)

Create materialmodels by solving

functions withBC’s and useof materialdata (B3)

Figure B.1: Block scheme representing the matlab routine used to create the material models

Appendix C

Additional SolidWorks Drawings

C.1 Testing Supply: Fixture for 3-Point-Bending

See figure C.1.

C.2 Tensile Bar

See figure C.2.

85

20 50

20

R25

4steel

BCD

12

A

32

14

B A

56

ALL DIM

ENSIO

NS IN

mm

UN

LESS OTH

ERW

ISE SPECIFIED

:

SUR

FACE FIN

ISH:

TOLER

ANC

ES LINEAR

: 0 PLAC

ES 1,0 1 PLAC

E 0,5 2 PLAC

ES 0,2

ANG

ULAR

: 0,5deg.

DR

AWN

:D

ATE:

DO

NO

T SCALE

IF IN D

OU

BT, ASK

TITLE:

DW

G N

o:

SCALE:

SHEET 1 O

F A4 C

SCH

OO

L OF AER

OSPAC

E, MEC

HAN

ICAL

& MAN

UFAC

TUR

ING

ENG

INEER

ING

T HIR

D AN

GLE

PRO

JECTIO

NPR

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T:

MATER

IAL:FIN

ISH:

HEAT

TREAT:

REM

OVE ALL BU

RR

S & SHAR

P EDG

ES

CH

ECKED

:

DATE:

SIGN

ED:

QTY:

ISSUE

CO

MM

ENTS

DATE

ITEMPAR

TS LIST CO

MPO

NEN

T / DESC

RIPTIO

NQ

TYPAR

T No.

Figure C.1: Drawing for the fixture rollers used as support in bending tests

19.05019.050

9.530

3.180

R12.700

2

7.935

9.530

7.935

BCD

12

A

32

14

B A

56

ALL DIM

ENSIO

NS IN

mm

UN

LESS OTH

ERW

ISE SPECIFIED

:

SUR

FACE FIN

ISH:

TOLER

ANC

ES LINEAR

: 0 PLAC

ES 1,0 1 PLAC

E 0,5 2 PLAC

ES 0,2

ANG

ULAR

: 0,5deg.

DR

AWN

:D

ATE:

DO

NO

T SCALE

IF IN D

OU

BT, ASK

TITLE:

DW

G N

o:

SCALE:

SHEET 1 O

F A4 C

SCH

OO

L OF AER

OSPAC

E, MEC

HAN

ICAL

& MAN

UFAC

TUR

ING

ENG

INEER

ING

T HIR

D AN

GLE

PRO

JECTIO

NPR

OJEC

T:

MATER

IAL:FIN

ISH:

HEAT

TREAT:

REM

OVE ALL BU

RR

S & SHAR

P EDG

ES

CH

ECKED

:

DATE:

SIGN

ED:

QTY:

ISSUE

CO

MM

ENTS

DATE

ITEMPAR

TS LIST CO

MPO

NEN

T / DESC

RIPTIO

NQ

TYPAR

T No.

Figure C.2: Drawing for the tensile bar [30]

Appendix D

Costs of Rapid Prototyping

In the next table, a small overview is given of the costs that are involved with rapidprototyping. This only includes material costs, since other costs (operation costs, machinecosts, and so on) are not of interest here. Note that costs are always subject to changes. Forthe final models that were manufactured, the mass is 57.4 grams. So this is the combinedmass of the VeroWhite and TangoPlus material. Cleaning of the machine is needed beforeprocessing, only when a material cartridge has to be replaced or changed to an other materialcartridge. The prototypes made in this project costs about 40 AUD each (only materialcosts). Machine and/or labour costs are not taken into account; this will be around 50 to 70AUD per hour. Manufacturing of a single (horizontally oriented) sample takes about 2 hours.

Table D.1: Material costs for rapid prototyping

Material Price per gram (AUD / gram)VeroWhite 0.40TangoPlus 0.52Support Material 0.19Model Cleaning Fluid 0.08Support Cleaning Fluid 0.08

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Appendix E

Buckling Analysis

Besides the bending loads (which are extensively analysed in the main body of this report), another natural loading condition that is of interest for the analysis of the bamboo strucuture iscompression. Due to the own mass of the bamboo stem, and natural conditions such as rain,the stem is loaded in compression. The microstructure of bamboo should also be organisedsuch that it can resist buckling. In this appendix, a first start is made on the analysis ofcompression loads. Some starting points for FEA are given and a baseline analysis is doneto compare the results to analytical solutions. For future work, the complex anisotropicbiomimicked models should be analysed too.

E.1 Analytical Results

As a starting point for the buckling analysis, a simple tubular structure is used. Thedimensions of the cross-sectional area are the same as for the first rapid prototype samplethat was used as a benchmark for the bending tests:

Dout = 20 mm; Din = 14.399 mm

With these values, the cross-sectional area, the second moment of area and the radius ofgyration (see equation 2.1) are calculated. For the material properties, a Young’s modulus Eof 2750 MPa is used, together with a Poissons ratio of ν = 0.3 [−]. The applied force P is1000 N . The critical buckling force is calculated for a column that is fixed at both ends [2].The first eigenvalue of Euler buckling is calculated with:

λ =π2EI

0.5LP(E.1)

in which L is the length of the column, which has been varied from 100 mm to 500 mm. Forthe more complex models with the fibres, the 2nd moment of area can be calculated with usethe parallel-axis theorem [2]. Furthermore, a slenderness ratio is defined as the ratio betweenthe effective length of the tubular structure and the radius of gyration:

Slenderness ratio =0.5LRg

(E.2)

E.2 FEA

The tubular structure is modeled in SolidWorks. A STEP file is created and imported inAbaqus. In this FE program the buckling analysis can be performed by use of a Bucklestep, which can be found under Linear Perturbation. The load has to be applied by using a

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Multiple Point Constraint (MPC). This MPC is a tie constraint. A the top side of the tubularstructure, all nodes are connected to this virtual node in the center of the beam. In here,Patran2010 has been used to create this extra node. This load can be seen in figure E.1.

Figure E.1: The load on the central node, applied on the structure by using a MPC

The boundary conditions in compression are a fixed displacement at the bottom and nodisplacement in radial direction (x- and y-direction) of the tube. Further details of the FEAcan be found in the CAE files. In figure E.2 the results of the buckling analyses are presented.In here, the y-axis gives the ratio between the 1st eigenvalue found with Abaqus and theeigenvalue found with equation E.1. On the x-axis, the number of elements in axial directionis given. The latter thus gives the influence of mesh refinement on the result. Graphs are givenfor beams with a different slenderness ratio, corresponding to a column length of 100, 200,300, 400 and 500 mm. From this figure, a couple of conclusions can be made. The slendernessratio should be at least about 30 to obtain reliable results. Otherwise, the buckling force willbe underestimated by using finite elements. As can be seen, for a slenderness ratio of 32.5and 40.6 the results converge to the analytical solution properly upon mesh refinement.

Note. this mesh refinement only includes mesh refinement by more elements in axialdirection. If the number of elements in radial direction is increased, only a slight improvementin accuracy was noticed. However, the effect might be larger for more complex structures.

E.3 Conclusion

Thus, from this small review it was noticed that buckling analyses in finite element methodscan be quite harsh. One should always consider the effect of the slenderness ratio to getreliable results. Also the effect of mesh refinement can improve result drastically. Similar tothe bending loading condition, the factor EI will determine the structural behaviour. A modelconform model IIb, with the material with the highest Young’s modulus on the outer side willtherefore be optimal. Further work should be done in this area to see if the cross-sectionalarea of bamboo is also in one or another way optimized for this loading condition. Assumedis, based on some papers [25], that the nodal structure of bamboo helps to prevent ovalizationand longitudinal crack growth of the stem, thereby increasing the resistance against bendingeven more.

25 30 35 40 45 50 55 60

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

number of axial mesh layers

λ / λ

an

aly

tica

l [-

]

Slenderness ratio = 8.1156

Slenderness ratio = 16.2312

Slenderness ratio = 24.3468

Slenderness ratio = 32.4624

Slenderness ratio = 40.5779

Analytical solution

Figure E.2: The ratio between the 1st mode eigenvalues from FEA and the analytical solutionof the eigenvalue versus the number of axial elements for columns with a different slendernessratio